turbulence theory and modelingcfdntlab/myongpdf/tbtheory/%b... · 2011-05-17 · numerical...

CFD amp Turbulence Lab httpcfdkookminackr

School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG

Turbulence Theory and Modeling

Hyon Kook MYONG

School of Mechanical Engineering

httpcfdkookminackr



Contents

서론

기초방정식

난류의 발생

난류의 생성 및 소산

와동(Vortex) Dynamics난류 스케일

상관

난류 스펙트럼

난류모델

난류 현상의 예측 예



Turbulence Modeling

단순화

亂流

어지러울 난

模型化

흐를 류

TurbulenceTurbulence ModelingModeling++



Historical Development of Turbulence Modeling-1

15~16 15~16 CenturyCentury

L da Vinci (1452-1519) Visual and descriptive (available)

Mechanistic theory of nature + () Math modelNote no mathematical model available to describe flow motions

17~18 Century17~18 Century

I Newton (1643-1727) L Euler (1707-1783)D Bernoulli (1700-1782) J DrsquoAlembert (1717-1783)

Continuum inviscid Newtonrsquos law (available)

Viscous fluid model

Note no mathematical model available to describe viscous flows

( )+minusminus=i

ii

xPg

DtDU

partpartρρ




1919thth CenturyCentury

L M H Navier (1785-1836) J B Fourier (1768-1830)B de Saint Venant (1797-1886) G G Stokes (1819-1903)

Continuum viscous Stokes postulations on and Fourier postulations on (available)

Turb flow model

Turb heat flux model

Note no mathematical model available to describe turbulent flow and heat transfer

τ ijqi

( )++minusminus=j

ij

ii

i

xxPg

DtDU

partτpart

partpartρρ

( )ρ τpartpart

partpart

CDTDt

Ux

DPDt

qxp ij

i

j

i

i= + + +





O Reynolds (1842-1912) T von Karman (1881-1963)L Prandtl (1875-1953) G I Taylor (1880-1975)

Continuum averaging viscous turbulent postulations

Note Development of turbulence model

( )

( )ρ ρpartpart

partτpart

partτpart

DUDt g

Px x x

ii

i

ij

j

tij

j= minus + + + +

( )

( )ρ τpartpart

partpart

partpart

CDTDt

Ux

DPDt

qx

qxp ij

i

j

i

i

ti

i= + + + +



Turbulence Model Postulations

Before we take on the postulation let us ask ifMomentum diffusion

(N-S)Heat diffusion

Mass diffusion

then shouldTurb M diffusion

Turb H diffusion

( )ρUi iij UGrad~τ

( )ρC Tp q GradTi ~

( )ρC M GradCi ~

( )ρu ui j

( )ρ θC up i

( )u u up

Grad u ui j i j+ρ

~ ~

u up

Grad ui iθθρ

θ+ ~ ~



Turbulence Scale

Small Scale

UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena

Large Scale

Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy

Effective in transport phenomena

u t η υ τ



Turbulence Model

Definition of an Ideal Turbulence Model

ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo

How Complex does a Turbulence Model have to be

ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo



Inviscid Estimate for Dissipation Rate

Rate of energy supply (=Production rate)

Production rate = Dissipation rate

Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity

32 υυυ =asymp

3~ υε



Fundamental Equations

Navier Stokes equation

Continuity equation Time averaging

Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

ρu u u u uu u u u uu u u u u

12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0



Transport Equation forReynolds Stresses

( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus

minusPressure strain

i j

k kViscous diffusion

i

k

j

kViscous dissipation

u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2



Transport Equation of Turbulent Kinetic Energy

U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction

i iMolecular diffusion

j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32



Transport Equation of Turbulent Dissipation Rate

ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model

Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model

k-kl model k- modelAlgebraic stress modelReynolds stress model

Structural model

Large eddy simulation

Turbulence Models

ω



Time Averaged Models

Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---

Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde

pde partial differential equation ale algebraic equation asm assumption



Basic Concepts in Turbulence Models-1

Characterization of local state of turbulence by only few parameters

V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)

(or alternatively LV = time scale)

Task of turbulence model

1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow

uui j uiϕ




Boussinesq eddy viscositydiffusivity conceptFor general flows

- These quantities are not fluid properties but depend strongly on the state of turbulence

- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and

momentum transfer

ydiffusiviteddyorturbulentityviseddyorturbulent

t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t

t turbulent prandtl or Schmidt number

=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ




From dimensional analysis

Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative

-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering

flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence

( )τν 2ˆˆ VorLVt prop

υρ uminus part partU yνt

νt Γtεminusk



Mixing Length Model-1

In 1925 Prandtlrsquos proposal

This yields

V L Uy

=partpart

ν partpartt mUy

= 2

κ λκ λ= = rarr= = rarr

0435 009041 0085

Patankar and SpaldingCrawford and Kays

λ κy δ

m

m = λδm y= κ




Mixing length has to be prescribed empiricallyVery close to the wall

Cebeci-Smith model Uses in outer layer

For general shear layers

( )[ ]m y y A= minus minus + +κ 1 exp

( )ν α αt U U dy= minus =infininfinint0 0 0168

( )ν α δ δt U U U dy= prop = minus infininfinint 10

m bprop

bκ =

=+

+

von Karman constantvon Driests damping factor

function ofA

A dp dx~




Baldwin-Lomax modelInner layerOuter layer

Here Fmax is the maximum value of

and ymax is the y value at that time

μ ρ ω ωti l vorticity= 2

( )μ ρκto cp wake klebC F F y=

( ) ( )[ ]F y y y A= minus minus + +ω 1 exp

[ ]F y F C y U Fwake wake dif= min max max max max2

( ) ( )[ ]F y C y ykleb kleb= +minus

1 55 6 1 max

Klebanoffrsquos intermittency function

κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb

Ddif = difference between the max and min values



Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)

Buoyancy effect is characterized by the gradient Richardson number

For (Monin-Oboukhov relation)

For (KEYPS formula)

Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence

(=ratio of centrifugal to inertial forces)

( )R g P y

U yi = minus

ρpart partpart part 2

Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4

2= minus congminusβ β

R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0



Mixing Length Model-5 (Discussion on Mixing Length Model)

Lack of universality of the empirical input

Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)

Examples1) grid turbulence

model yields

2) Channel flow model yields

at symmetry plane where

-- The model is not very suitable when convective and diffusive transport and history effects are important

In complex flows is difficult to prescribe empirically

νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U



Mixing Length Model-6 (Mixing Length Model Assessment)

AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established

DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress



One-Equation Model -1(Energy Equation Model-1)

Transport and history effects are accounted for by transport equation for velocity scale V

Physically most meaningful scale is kinetic energy of the turbulent motion

k- equation at high Reynolds numbers

k where k uui i =12

ndissipatioviscous

j

i

j

i

ndestructioproductionbouyantG

ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2



One-Equation Model -2 (Energy Equation Model-2)

Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms

With eddy viscositydiffusivity relations for and the k- equation reads

Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming

or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity

Length scale L needs to be prescribed empirically

minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2

ν μt c kL= primeuv

uv kprop( )uv k= 03

k prop



One-Equation Model -3(S-A (Spalart-Allmaras) model)

2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus

νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤

⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla

σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1

The dependent variable of the length-scale-determining equation must not be the length scale L itself

1 Diffusion2 Source interaction with mean motion3 Sink self interaction

Additional diffusion usually Additional source or sink

for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ

yDtDZ prime+prime+minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ

primeSZprime =SZ 0 prime =DZ 0

y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr



Second Equation-2(ε-Equation)

- equation

-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다

--확산항 구배 확산 가정

minus minus⎛

⎝⎜⎜

⎞

⎠⎟⎟ = minus minus2 2

22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k

stretchingvortextodueproduction

l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε



Two-Equation Models-1(Standard k-ε Model)

( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

Cμ C1ε C2ε C3ε kσ

σε σt

009 144 192 0-02 1when Glt0 when Ggt0

1 13 05-07 09free shear near-wall

layers flows



Two-Equation Models-2(RNG k-ε Model)

( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv



Two-Equation Models-3(Realizable k-ε Model)

( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr

ijijijij WWSSU minusequiv()

As = 6cosφ

φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3

A0 = 40



Two-Equation Models-4(Wilcoxrsquos Model)

ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=



Two-Equation Models-5(Shear-Stress Transport Model)

)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart

minus+minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε

εω φφφ minusminus minus+= kk FFconModel )1( 11

]4)500090

min[max(arg)tanh(arg 221411 yCD

kyy

kwhereFkw ωωσ

ρω

νω

==

)5000902max(arg)tanh(arg 22

222 yy

kwhereFω

νω

==

ωminusk

Hybrid k-ε Model Model+ ωminusk



Wall functionWall function

Bridging of viscous sublayer by

Assume Local equilibriumUniversal logarithmic laws

Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=

Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)



ProcedureProcedure⑴ Solve k - equation up to yP

neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws

Merits Easy to use Small mesh number

Weakness Questionable for complex flow must begt12 for all region


k U yP Pminus minus +

( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+



TwoTwo--layer methodlayer method

Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription

-- Iacovides and Launder (1987) Mixing length model

-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein

-- Rodi (1988) Norris-Reynoldsrsquo energy equation model

Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)



Two-Equation Models-9 (Low-Reynolds k-ε Model-1)

tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=minus=jxkD νεεε

( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=

where

etcRRyRoffunctionsf yt εμ+

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412

equivequivequivequiv +

2 etcRRyRoffunctionsf yt ε+

011 etcRRoffunctionsorf yt



Two-Equation Models-10(Low-Reynolds k-ε Model-2)

Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer

After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng

Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer

( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν

( )kk εω equivminus

( )( )ννεε 21 yRy equivrarr+

( )ετ kk equivminus




After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM

Sakar (1995) MS Thesis Arizona State Univ

Use


3εP εkP

( ) ⎟⎠⎞

⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf

εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε



Two-Equation Models-12(k-ε Model Assessment)

AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model

DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct



Modification for Curvature -1

Launder et al(1977)

-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets

Hanjalic and Launder (1980)

-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers

Leschziner-Rodi (1981)

( ) ( )C C R where R k Ur

Urr

Cc it it cε εpartpart2

2

2 21 0 25rarr minus = cong

( )P C uv Uy

C u v Ux k

C CrArr minus + minus⎡

⎣⎢

⎤

⎦⎥ ltε ε ε ε

partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572




Leschziner and Rodi (1981)

-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)

--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)

( )PkC P C Sk t nsε ε ε

ε ν= prime minus primeprime1 1

P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2

( ) ( )C C M where M k S Cc ft ft ij ij cε ε2

2

22 21 0 20rarr minus = minus congΩ

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21




Use of Invariant TermsUse of Invariant Terms

Rodi (1972)

Cotton et al (1992)

where are constants or functions of Yakhot et al (1992) additional term in eq

Kato amp Launder (1993)

( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=

jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε

εε μμμμ2



Additive term in -equation

Original Launder amp Sharma model

Myong model(1989)

(Example) In 2-D riblet flow

(Ref) Additive term in k-equation in 2-D riblet flow

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2

ν part part νpartpart

partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε



Non-linear k-ε Model-1

SpezialeSpeziale et al (1987)et al (1987)

where

Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow

εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞

⎜⎝⎛ minusminus

⎟⎠⎞

⎜⎝⎛ minusminusminus=

ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21




NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)

where

Test-- Low-Re channel flow Couette flow secondary flow in a square duct

⎟⎠⎞

⎜⎝⎛ minus+minus= sum

=ijijijtijji SSCkSkuu δ

ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3




Myong amp Myong amp KasagiKasagi (1990)(1990)

where

Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel

( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞

⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+

⎟⎠⎞

⎜⎝⎛ minus+minus=

εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21

jmimjninijijW δδδδδ 4+minusminus=




Craft et al (1996 1997)Craft et al (1996 1997)

where

Test-- pipe flow impinging jet flow around turbine blade transitional flow

( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞

⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+

Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ



Reynolds Stress Equation Model-1

Reynolds stress equations

( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

대류항 확산항 확산항

응력 생산항 압력

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항

점성확산항 점성소산항

νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i



Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)

AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows

DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling

εminuskεminusk



Algebraic Stress Model-1

Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg

Incorporation into model - equation

P and G are stress and buoyant production of k

u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus+⎟⎟

⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji



Algebraic Stress Model-2(Algebraic Stress Model Assessment)

AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)

DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits

εminusk

εminusk

εminusk



Explicit Algebraic Stress Model

Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem

Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope

Test-- homogeneous flows high Re flows

Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability

Test--- homogeneous flows channel flow backward facing step flow

εminusk



LES(Large Eddy Simulation)-1(Concept of LES)

Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly

Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models



LES(Large Eddy Simulation)-2(LES and DNS)

Schematic representation of turbulent motion

Time dependence of a velocity component



LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)



LES(Large Eddy Simulation)-4(Filtering Operation)

Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part

A filtered (or resolved or large-scale) variable denoted by an overbar is defined as

If G is a function of x-xrsquo only differentiation and the filtering operation commute

int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=



LES(Large Eddy Simulation)-5 (Typical Filter Functions)



LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)

0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(

jijiij UUUU minus=τ

Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies



LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)

ijijtij kS δντ322 +minus=

Boussinesqrsquos concept

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv



Smagorinsky model (1963)

ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K

23K2S )( asympasymp= minus

π

S23

k CC2C minus=ε




LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)

sC reduces from 004 to 0004 near a wall

)( ++minusminus= Ay0ss e1CC

The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard

Near a wall the flow structure is very anisotropic( ) 3

1zyx ΔΔΔ=Δ rarr ( ) 2

1222 zyx Δ+Δ+Δ=Δ



LES(Large Eddy Simulation)-10(Grid Parameter R)

de

eRminusΔminus

=

)( ε23e kscaleenergeticthe=

413d scalendissipatiothe )( εν=

1R0 ltlt



LES(Large Eddy Simulation)-11(Dynamic SGS Model)

Grid filter with Test filter with

Δ

Δ

GG


jijiij UUUUT minus=



Principal Challenges of Turbulence Predictions

(I) Growth and separation of the boundary layer

(II) Momentum transfer after separation

Challenge (I)

- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)

- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage

Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models

LES and the new methods- time-dependent three-dimensional simulations



Numerical Challenges

Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)

- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)



DES (Detached Eddy Simulation)-1

The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls

- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness

- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal

δgtgtΔ||

LltltΔ




A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary

layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady

eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information

- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies



S-A (Spalart-Allmaras) model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Hybrid RANSLES (or DES) Method

- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall

- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and

- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing

- Thus the S-A model with replaced by a length proportional to can be a SGS model

- If we now replace in the S-A destruction term with defined by

we have a single model that acts as S-A when and a SGS model when

ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~

)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv

ΔltltdΔgtgtd



DES model

2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus

νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤

⎢⎣⎡minusnabla+nabla+sdotnabla+=d

fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc




Application of DES to Channel Flow-1

The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives

- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed

- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid

Ref) Myong(2002)

δltltΔ||

||Δδ



Application of DES to Channel Flow-2(Numerical Method)

Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -

Calculation Code (extended version of Dr Choi)

Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ

)52000(Re2000Re asympν==τ HUmm

π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +

wwclwcl yyyyy



Application of DES to Channel Flow-3(Skin friction coefficient)

Calculated error in skin friction coefficient ref

)2000Re( =Δ τatCf22)Re(log123750 minus= mfC

- 60 - 57 - 20 - 3

No modelLESDES (hybrid RANSLES)RANS

method fCΔ

Ref) Myong(2002)



Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)

Ref) Myong(2002)



Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)

y+

100 101 102 103 104

U+

0

10

20

30

40

50no modelLESRANSDES

U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)



Application of DES to Channel Flow-6(Shear Stresses)

RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10

viscousmodeledresolved

DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



Constraint to the Spread of CFD

In addition to a substantial investment outlay an organization needs qualified people

to run the codes and communicate their results and briefly consider the modeling skills required by CFD users

The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software



Problem Solving with CFD

In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator







Contents

서론

기초방정식

난류의 발생

난류의 생성 및 소산

와동(Vortex) Dynamics난류 스케일

상관

난류 스펙트럼

난류모델

난류 현상의 예측 예



Turbulence Modeling

단순화

亂流

어지러울 난

模型化

흐를 류











Viscous fluid model


( )+minusminus=i

ii

xPg

DtDU

partpartρρ







Turb flow model



τ ijqi

( )++minusminus=j

ij

ii

i

xxPg

DtDU

partτpart

partpartρρ

( )ρ τpartpart

partpart

CDTDt

Ux

DPDt

qxp ij

i

j

i

i= + + +








( )

( )ρ ρpartpart

partτpart

partτpart

DUDt g

Px x x

ii

i

ij

j

tij

j= minus + + + +

( )

( )ρ τpartpart

partpart

partpart

CDTDt

Ux

DPDt

qx

qxp ij

i

j

i

i

ti

i= + + + +





(N-S)Heat diffusion

Mass diffusion


Turb H diffusion



( )ρC M GradCi ~

( )ρu ui j

( )ρ θC up i

( )u u up

Grad u ui j i j+ρ

~ ~

u up

Grad ui iθθρ

θ+ ~ ~



Turbulence Scale

Small Scale


Large Scale



u t η υ τ



Turbulence Model











32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)

















Turbulence Modeling

단순화

亂流

어지러울 난

模型化

흐를 류











Viscous fluid model


( )+minusminus=i

ii

xPg

DtDU

partpartρρ







Turb flow model



τ ijqi

( )++minusminus=j

ij

ii

i

xxPg

DtDU

partτpart

partpartρρ

( )ρ τpartpart

partpart

CDTDt

Ux

DPDt

qxp ij

i

j

i

i= + + +








( )

( )ρ ρpartpart

partτpart

partτpart

DUDt g

Px x x

ii

i

ij

j

tij

j= minus + + + +

( )

( )ρ τpartpart

partpart

partpart

CDTDt

Ux

DPDt

qx

qxp ij

i

j

i

i

ti

i= + + + +





(N-S)Heat diffusion

Mass diffusion


Turb H diffusion



( )ρC M GradCi ~

( )ρu ui j

( )ρ θC up i

( )u u up

Grad u ui j i j+ρ

~ ~

u up

Grad ui iθθρ

θ+ ~ ~



Turbulence Scale

Small Scale


Large Scale



u t η υ τ



Turbulence Model











32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)
























Viscous fluid model


( )+minusminus=i

ii

xPg

DtDU

partpartρρ







Turb flow model



τ ijqi

( )++minusminus=j

ij

ii

i

xxPg

DtDU

partτpart

partpartρρ

( )ρ τpartpart

partpart

CDTDt

Ux

DPDt

qxp ij

i

j

i

i= + + +








( )

( )ρ ρpartpart

partτpart

partτpart

DUDt g

Px x x

ii

i

ij

j

tij

j= minus + + + +

( )

( )ρ τpartpart

partpart

partpart

CDTDt

Ux

DPDt

qx

qxp ij

i

j

i

i

ti

i= + + + +





(N-S)Heat diffusion

Mass diffusion


Turb H diffusion



( )ρC M GradCi ~

( )ρu ui j

( )ρ θC up i

( )u u up

Grad u ui j i j+ρ

~ ~

u up

Grad ui iθθρ

θ+ ~ ~



Turbulence Scale

Small Scale


Large Scale



u t η υ τ



Turbulence Model











32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)





















Turb flow model



τ ijqi

( )++minusminus=j

ij

ii

i

xxPg

DtDU

partτpart

partpartρρ

( )ρ τpartpart

partpart

CDTDt

Ux

DPDt

qxp ij

i

j

i

i= + + +








( )

( )ρ ρpartpart

partτpart

partτpart

DUDt g

Px x x

ii

i

ij

j

tij

j= minus + + + +

( )

( )ρ τpartpart

partpart

partpart

CDTDt

Ux

DPDt

qx

qxp ij

i

j

i

i

ti

i= + + + +





(N-S)Heat diffusion

Mass diffusion


Turb H diffusion



( )ρC M GradCi ~

( )ρu ui j

( )ρ θC up i

( )u u up

Grad u ui j i j+ρ

~ ~

u up

Grad ui iθθρ

θ+ ~ ~



Turbulence Scale

Small Scale


Large Scale



u t η υ τ



Turbulence Model











32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)






















( )

( )ρ ρpartpart

partτpart

partτpart

DUDt g

Px x x

ii

i

ij

j

tij

j= minus + + + +

( )

( )ρ τpartpart

partpart

partpart

CDTDt

Ux

DPDt

qx

qxp ij

i

j

i

i

ti

i= + + + +





(N-S)Heat diffusion

Mass diffusion


Turb H diffusion



( )ρC M GradCi ~

( )ρu ui j

( )ρ θC up i

( )u u up

Grad u ui j i j+ρ

~ ~

u up

Grad ui iθθρ

θ+ ~ ~



Turbulence Scale

Small Scale


Large Scale



u t η υ τ



Turbulence Model











32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















(N-S)Heat diffusion

Mass diffusion


Turb H diffusion



( )ρC M GradCi ~

( )ρu ui j

( )ρ θC up i

( )u u up

Grad u ui j i j+ρ

~ ~

u up

Grad ui iθθρ

θ+ ~ ~



Turbulence Scale

Small Scale


Large Scale



u t η υ τ



Turbulence Model











32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)

















Turbulence Scale

Small Scale


Large Scale



u t η υ τ



Turbulence Model











32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)

















Turbulence Model











32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)





















32 υυυ =asymp

3~ υε






Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















Reynolds stresses

jj

i

ij

ij

i

xxU

xP

xUU

tU

partpartpart

ρμ

partpart

ρpartpart

partpart 21

+minus=+

0=i

i

xU

partpart

( ) iii

t

t ii uUUdttUtt

U +equivminus

equiv int 1 1

001

u ui j

minus

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


12

1 2 1 3

2 1 22

2 3

3 1 3 2 32

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus=+

=

jij

i

jij

ij

i

i

i

uuxU

xxP

xUU

tU

xU

ρpartpartμ

partpart

ρpartpart

ρpartpart

partpart

partpart

11

0




( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















( )Uu ux x

u u uu px

u px

u uUx

u u Ux

p ux

ux

ki j

kConvection

kk i j

Diffusion

j

i

i

j

Diffusion

i kj

kj k

i

kGeneration

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart

= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

1 2

1

⎠⎟

+ minus


i j


i

k

j


u ux x

ux

ux

νpartpart part

ν partpart

partpart

2

2




U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















U kx x

uu u p u u U

x

kx x

ux

ux

ii

Convectioni

ii j

Turbulent diffusion

i ji

j

oduction


j

i

j

iDissipation

partpart

partpart ρ

partpart

ν partpart part

νpartpart

partpart

= minus +⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪minus

+ minus

2

2

Pr

( )k u u u= + +12 1

222

32




ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















ndissipatioViscous

lk

i

Diffusion

l

i

lij

ik

k

Generation

l

k

l

i

k

i

Generation

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

Convection

ii

xxu

xu

xp

xxuu

xxu

xu

xu

xuu

xxU

xu

xu

xu

xu

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε ν partpart

partpart

=ux

ux

i

l

i

l



Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)

















Time averaged model



Structural model


Turbulence Models

ω















uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)





























uui j uiϕ







momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)





















momentum transfer


t

t

)(cos)(

=Γ

=ν

νt Γ t

Γtt

t


=

=

νσ

σ

minus = +⎛

⎝⎜

⎞

⎠⎟ minus

minus =

u u Ux

Ux

k

ux

i j ti

j

j

ii j

i ti

ν partpart

partpart

δ

ϕ partφpart

23

Γ










νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)
























νt Γtεminusk





This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















This yields

V L Uy

=partpart

ν partpartt mUy

= 2


0435 009041 0085


λ κy δ

m

m = λδm y= κ










m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)
























m bprop

bκ =

=+

+


function ofA

A dp dx~












1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


























1 55 6 1 max


κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb







For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















For (KEYPS formula)



( )R g P y

U yi = minus


Ri gt 0m

miR

0

1 5 101 1= minus =β β ~

( )m

miR

0

1 1421 4


R U RU nis c

s=part part

β1 6 14= ~

Ri lt 0







model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)





















model yields





νt t= =Γ 0

part partU y = 0

m

k

νt t= =Γ 0tμ

from MLM

U












k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


























k where k uui i =12

ndissipatioviscous

j

i

j

i


ii

shearbyproductionP

j

iji

transportdiffusive

jji

i

transportconvective

ii

changeofrate

xu

xu

ug

xU

uupuuu

xxkU

tk

==

=

minusminus

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=+

ε

partpart

partpart

νϕβ

partpart

ρpartpart

partpart

partpart

partpart

2









minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)























minus +⎛⎝⎜

⎞⎠⎟ = =u

u u p kx

C kLi

j j t

k iD2

3 2

ρνσ

partpart

ε

u ui j uiϕpartpart

partpart

partpart

νσ

partpart

ν partpart

partpart

β νσ

partφpart

ε

ktU k

x

xkx

Ux

Ux

gx

C kL

ii

i

t

k it

i

j

j

i

P

it

k iG

D

+

=⎛⎝⎜

⎞⎠⎟ + +

⎛

⎝⎜

⎞

⎠⎟ + minus

3 2


uv kprop( )uv k= 03

k prop





νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb



Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)

















Second Equation-1




for k-ε model

eg)

model

1011-105-115

Z k La b=

ZZttZ

t SDZk

CyU

kZC

yZ


⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

3

2

2

2

1

εpartpartν

partpart

σν

partpart

primeDZ


y

x

u

a bεminusk

kk minusωminusk

minusk

kC

yU

kC

yyDtD

ttt

2

2

2ε

partpartνε

partεpart

σν

partpartε

ε

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

2

ωpartpartνω

partωpart

σν

partpartω

ω

CyU

kC

yyDtD

ttt minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= )( ωε k=lArr




- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















- equation



minus minus⎛

⎝⎜⎜

⎞


22

1 2

2

νpartpart

partpart

partpart

νpartpart part

ε partpart

ελε ε

Ux

Ux

Ux

Ux x

Ckuu U

xC

kl

k l

k

l

i

k li j

i

j

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛minus

i

t

il

i

lij

ik

k xxxU

xP

xxU

Ux part

εpartσν

partpart

partpart

partpart

partpart

ρν

partpart

partpartν

ε

2

ndestructioviscous

lk

i

transportdiffusive

l

i

lij

ik

k


l

k

l

i

k

i

production

l

ik

lk

i

k

l

i

l

l

k

l

i

k

i

transportconvective

ii

xxU

xU

xP

xxUU

xxU

xU

xU

xUU

xxU

xU

xU

xU

xU

xU

xU

22

2

2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛minusminus

minus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=

partpartpartν

partpart

partpart

partpart

ρν

partpart

partpartν

partpart

partpart

partpartν

partpart

partpartpartν

partpart

partpart

partpart

partpart

partpartν

partεpart

ε




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε


σε σt

009 144 192 0-02 1when Glt0 when Ggt0


layers flows




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

30

3

22 1)1(

βηηηη

μεε +minus

+rArr CCC

εη kSequiv 21)

21( ijij SSS equiv




( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















( )

εν

εεpartεpart

σν

partpart

partεpart

partεpart

εpartφpart

σνβ

partpart

partpart

partpartν

partpart

σν

partpart

partpart

partpart

μ

εεεε

2

2

231

kC

kCGCP

kC

xxxU

t

xg

xU

xU

xU

xk

xxkU

tk

t

Pi

t

iii

G

it

ti

P

j

i

i

j

j

it

ik

t

iii

=

minus++⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

minusε

ε

μ kUAAC

s()

0

1

+rArr


As = 6cosφ


A0 = 40




ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















ων

βωωαpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ω

k

Pkxxx

Ut

kxU

xU

xU

xk

xxkU

tk

t

i

t

iii

j

i

i

j

j

it

ik

t

iii

=

minus+⎟⎟⎠

⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

2

kωβε =

ωminusk

ωβ

k=




)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















)max(

11)1(2

21

1

21

2

Faka

xxkFP

kxxxU

t

kxU

xU

xU

xk

xxkU

tk

t

iii

t

iii

j

i

i

j

j

it

ik

t

iii

Ω=

partpart

partpart


⎞⎜⎜⎝

⎛=+

minus⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

ων

ωωσ

βωωγpartωpart

σν

partpart

partωpart

partωpart

ωβpartpart

partpart

partpart

νpartpart

σν

partpart

partpart

partpart

ωε


]4)500090


kyy

kwhereFkw ωωσ

ρω

νω

==


222 yy

kwhereFω

νω

==

ωminusk







Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















Resulting

PPP

PP

yu

CukEuy

uU

κε

νκτ

μ

ττ

τ

32

ln1==⎟

⎠⎞

⎜⎝⎛=

( )PP

P

yu

Cuk

κε τ

μτ

3

2

=

=










( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)























( )( )Pr+++

+++

=

=

PP

PP

yTT

yUU

yP+












tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


























tt kTfC μμν =

DxUuu

xk

xDtDk

j

iji

jk

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

= εσνν ~

ET

fCxUuu

TfC

xxDtD

tj

iji

tj

t

j

+minuspartpart

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛+

partpart

=εε

σννε

εεε

~12211

2

2~⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart



where


+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε

( )ν

νενννε ε

τ yRykRyuykR yt

412



















Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)






























Use


3εP εkP

( ) ⎟⎠⎞


εμ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

partpartpart

=+=2

2

33jk

it xx

UCEP ννεε









Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)























Launder et al(1977)






Urr


2


( )P C uv Uy

C u v Ux k


⎣⎢

⎤


partpart

partpart

ε1 3

2 21 3

Ck U

nUR

UR

s s

c

s

c

μ

εpartpart

=+ +

⎛⎝⎜

⎞⎠⎟

0 09

1 0 572









P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)























P Ck

Uxti

jε ε

ε ν partpart

=⎛

⎝⎜

⎞

⎠⎟1

2


2


⎟⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

=i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

21

21





Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















Rodi (1972)

Cotton et al (1992)



( )εμ kPfuncC =

jiij SSkSwithS

Cεβ

αμ equiv

+= 21

βα εkPε

( )kS

SSSCR

2

30

3

11 ε

βμ

+

minusminus=


εε μμμμ2





Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















Myong model(1989)



+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

it xx

UPpartpart

partννε

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

22

3 2nm

imnt xx

UPpartpart

partδννε

222

2

22

2

222

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

WyW

xW

xxU

nm

i

partpartpart

partpart

partpart

partpartpart

δpart

part partpartpart

partpartmn

i

m n

Ux x

Wx

Wy

2 2 2

2

2 2

2

2⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

( )2 21 2 22 2


partpart

k x kx

kyj

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ε





where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















where


εδ

δδ23

2

2

31

31

32

kCwithDDC

DDDDCDkkuu

ijmmijE

ijmnmnmjimDijijji

=⎟⎠⎞


⎟⎠⎞


ki

k

jkj

k

iijij D

xU

DxU

DtDDD

part

partminus

partpart

minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

==i

j

j

iijij

xU

xUSD

21





where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















where


⎟⎠⎞



ενδ βααβ

ββ 3

1232 3

12

3

21 21 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart

part

part+

part

part

part

part=

part

part

partpart

=γ

γ

γ

γ

γγ xU

xU

xU

xU

SxU

xUS i

j

j

iij

jiij

jiij x

Ux

US

part

part

part

part= γγ

3





where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















where


( )

( )mnWx

kk

kCSSkC

SSSSkCSkuu

ijn

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

32

31

312

32

2

32

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

+

⎟⎠⎞


⎟⎠⎞


εν

δεν

εν

δεννδ

kijki

j

j

iij

i

j

j

iij x

UxU

xU

xUS Ωminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

part

partminus

partpart

=Ω⎟⎟⎠

⎞⎜⎜⎝

⎛

part

part+

partpart

= ε21

21






where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















where


( )

( )

kkijt

kkijt

ijnmnmmjmimjmit

kikjjkit

ijkkkjikt

kijkkjikt

ijkkkjikt

ijtijji

SkCSSSkC

SSSkC

SSSkC

kCSSkC

SSSSkCSkuu

ΩΩ++

⎟⎠⎞


Ω+Ω+

⎟⎠⎞


⎟⎠⎞


2

2

72

2

6

2

2

5

2

2

4

32

1

32

31

312

32

εν

εν

δενεν

δεν

εν

δεννδ

( )Ω= SfuncCμ





( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















( )Uu ux x

u u uu Px

u Px

u uUx

u u Ux

P ux

ux

ki j

k kk i j

j

i

i

j

i kj

kj k

i

k

i

j

j

i

partpart

partpart ρ

partpart

partpart

partpart

partpart ρ

partpart

partpart



= minus minus +⎛

⎝⎜⎜

⎞

⎠⎟⎟

minus minus + +⎛

⎝⎜

⎞

⎠⎟

1 2

1

minus

+ minus

변형률 상관항


νpartpart part

ν partpart

partpart

2

2u ux x

ux

ux

i j

k k

i

j

j

i






εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















εminuskεminusk







u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)





















u ui j

( ) ( )

( )ϕϕβ

partpart

partpart

ε

εδ

εεδ

εδ

ijjiji

l

jlj

l

jliji

jiji

jiji

jiji

ugugG

xU

uuxU

uuPwhere

GPC

GCCPP

Ckuu

+minus=

minusminus=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minus+

+

⎟⎟⎠

⎞⎜⎜⎝


⎠

⎞⎜⎜⎝

⎛minusminus

+=1

321

321

32

1

32

u ui j

( ) ( ) ( )εminus+=⎟⎟⎠

⎞⎜⎜⎝

⎛minus=minus GP

kuu

kDifftDkD

kuu

uuDifftDuuD jiji

jiji






εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















εminusk

εminusk

εminusk









εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)























εminusk




















int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


































int Δ=D

xdxxGxfxf )()()(

)()()( txftxftxf +=







0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)





















0xU

i

i =partpart

j

ij

jj

i2

iji

j

i

xxxU

xP1UU

xtU

partτpart

partpartpart

ρμ

partpart

ρpartpart

partpart

minus+minus=+ )(








⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

part

part+

partpart

=i

j

j

iij x

UxU

21S

iik τ21equiv




ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















ij2

St SC Δ=ν

ijijij SSS = ( ) 31

zyx ΔΔΔ=Δ

)( ijij2

k SSCk Δ=

23ijij

223k

23

SSCCkC

)(Δ=Δ

= εεε

20202C 2k ==π

501Cwith0420C232C K


π

S23

k CC2C minus=ε














de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



























de

eRminusΔminus

=



1R0 ltlt





Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)



















Δ

Δ

GG


jijiij UUUUT minus=






Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















Challenge (I)
















δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


























δgtgtΔ||

LltltΔ












νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


























νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21

~~~~1~~~⎥⎦⎤


σ+ν=

νd

fccScDtD

wwbb

12222 1

1~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

17230)1(

41062203213550

13222

11

21

===σ++κ=

=κ==σ=

υcccccc

cc

wwbbw

bb










ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)
























ν~2)~( dν d

dS 2~ SdpropνS

Δ 2Δpropν SSGS

d Δ

d d~


ΔltltdΔgtgtd



DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)

















DES model


νν

equivχ+χχ

=ν=νυ

υυ

~~

31

3

3

11 cfft

( )( ) ( )[ ]2

12

21 ~~~~~1~~~⎥⎦⎤


fccScDtD

wwbbννννν

σνν

12222 1

1~~~

υυυ χ+

χminus=

κν

+equivf

ffd

SS

226

2

61

63

6

63 ~~

~)(1

dSrrrcrg

cgcgf w

w

ww κ

νequivminus+=⎥

⎦

⎤⎢⎣

⎡++

=

65017230)1(

41062203213550

13222

11

21

====σ++κ=

=κ==σ=

υ DESwwbbw

bb

Ccccccc

cc








Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)





















Ref) Myong(2002)

δltltΔ||

||Δδ






Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















Ref) Myong(2002)

j

ij

j

i

ji

i

xxu

xxP

DtDu

part

partminus

partpart

partpart

+partpart

minus=τ

Re1

0=partpart

i

i

xu

RANSforS

LESforS

ijtij

ijSGSij

νminus=τ

νminus=τ



wwclwcl yyyyy






- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)




















- 60 - 57 - 20 - 3


method fCΔ

Ref) Myong(2002)




Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















Ref) Myong(2002)




y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















y+

100 101 102 103 104

U+

0

10

20

30

40


U+ = lny+041 + 52

yH5 1500 10 20

ν t+

0

50

100

150

200

250

LESRANSDES

Ref) Myong(2002)




RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)


















RANS

yH25 50 75000 100

τ+

2

4

6

8

00

10


DES

yH25 50 75000 100

τ+

2

4

6

8

00

10


LES

yH25 50 75000 100

τ+

2

4

6

8

00

10


Ref) Myong(2002)















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