introduction to turbulence and turbulent flows: … · ¥ non-universality and problems in complex...

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Introduction to Turbulence and its Numerical Simulation Charles Meneveau Department of Mechanical Engineering Center for Environmental and Applied Fluid Mechanics Johns Hopkins University Mechanical Engineering SCAT 1st Latin-American Summer School January 9th, 2007 Objectives of course (I & II): • Give an introduction to turbulent flow characteristics and physics, • Provide a first, basic understanding of standard turbulence models used in Computational Fluid Dynamics (CFD) • Provide basic understanding of Direct Numerical (DNS) and Large Eddy Simulation (LES) • Illustrate state-of-the-art subgrid model for LES and applications. Prerequisites: • Basic Fluid Mechanics, • Tensors and Index Notation Outline (I): • Overview of turbulent flow characteristics, • Reynolds decomposition, • Turbulence physics and energy cascade, •Turbulence modeling for CFD: Eddy-viscosity and k-! model • Filtering, Large Eddy Simulation (LES) • Direct Numerical Simulation (DNS) Outline (II): • Smagorinsky model and coefficient calibration, • Non-universality and problems in complex flows, • Dynamic model and applications Turbulent flows: multiscale, mixing, dissipative, chaotic, vortical well-defined statistics, important in practice From: Multimedia Fluid Mechanics, Cambridge Univ. Press

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Introduction to Turbulence and

its Numerical Simulation

Charles Meneveau

Department of Mechanical Engineering

Center for Environmental and Applied Fluid Mechanics

Johns Hopkins University

Mechanical Engineering

SCAT 1st Latin-American Summer School

January 9th, 2007Objectives of course (I & II):

• Give an introduction to turbulent flow

characteristics and physics,

• Provide a first, basic understanding of standard

turbulence models used in Computational

Fluid Dynamics (CFD)

• Provide basic understanding of Direct Numerical (DNS)

and Large Eddy Simulation (LES)

• Illustrate state-of-the-art subgrid model for LES

and applications.

Prerequisites:

• Basic Fluid Mechanics,

• Tensors and Index Notation

Outline (I):

• Overview of turbulent flow characteristics,

• Reynolds decomposition,

• Turbulence physics and energy cascade,

•Turbulence modeling for CFD:

Eddy-viscosity and k-! model

• Filtering, Large Eddy Simulation (LES)

• Direct Numerical Simulation (DNS)

Outline (II):

• Smagorinsky model and coefficient calibration,

• Non-universality and problems in complex flows,

• Dynamic model and applications

Turbulent flows:

•multiscale,

•mixing,

•dissipative,

•chaotic,

•vortical

•well-defined statistics,

•important in practice

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Turbulent flows:

•multiscale,

•mixing,

•dissipative,

•chaotic,

•vortical

•well-defined statistics,

•important in practice

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

e.g. 50 m

H

Turbulence in atmospheric boundary layer

Turbulence in reacting flows:

Premixed flame in I.C.

engine, combustion

Numerical simulation of flame

propagation in decaying

isotropic turbulence

Turbulence in aerospace systems:

LES of flow in thrust-reversers

Blin, Hadjadi & Vervisch (2002)

J. of Turbulence.

Turbulence in thermofluid equipment:

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Turbulence in turbomachines:

Lattice of wakes:

Superimposed Image of 10 Different Rotor Phases

at 500 rpm and at the Mid-span location. x=0 is at

rotor leading edge. The stage length is Ls = 203

mm.

Optical velocity field measurements in index-matched axial pumpPhase-averaged turbulent kinetic energy distribution

(Uzol, Katz & Meneveau, J. Turbomach. 2003)

Simplest turbulence: Isotropic decaying turbulence

Contractions

Active Grid

M = 6 "

1x

2x

Test Section

1u

2u

Flow

Active Grid

M = 6 "

JHU Corrsin wind tunnel

Physical laws governing fluid flow

•Conservation of mass

•Newton’s second law

•First law of thermodynamics

•Equation of state

•Some constraints in closure relations from second law of TD

•Density field

•Velocity vector field

•Pressure field

•Temperature field (or internal energy, or enthalpy etc..)

Physical quantities describing fluid flow

Navier Stokes equations for a Newtonian, incompressible fluid

Navier-Stokes equations, incompressible, Newtonian

!uj

!t+

!uku

j

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j

!uj

!x j= 0

aj=Fj

m

Turbulence: Reynolds decomposition

!uj

!t+

!uku

j

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j

!uj

!x j= 0

t

Turbulence: Reynolds decomposition

!uj

!t+

!uku

j

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j

!uj

!x j= 0

!uj

!t+!u

ju

k

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j"

!

!xk

uju

k" u

ju

k( )

!uj

!xj

= 0

Reynolds’ equations:

t

• Reynolds stress

• Energy cascade

• Spectral energy tensor

• Isotropic turbulence

• Kolmogorov spectrum (1941)

!uj

!t+!u

ju

k

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j"

!

!xk

uju

k" u

ju

k( )

!uj

!xj

= 0

! jk

R= ujuk " ujuk

Written as velocity co-variance tensor:

! jk

R= (uj + "uj )(uk + "uk ) # ujuk = ujuk + uj "uk + uk "uj + "uj "uk # ujuk = "uj "uk

0 0

0

• Kinematic Reynolds stress (minus): • Kinematic Reynolds stress (minus):

!uj

!t+

!uju

k

!xk

= "1

#

!p

!xj

+ $%2uj+ g

j"!&

jk

R

!xk

!uj

!xj

= 0

! jk

R" #uj #uk

• Deviatoric (anisotropic part):

!uj

!t+

!uju

k

!xk

= "1

#

!p *

!xj

+ $%2uj+ g

j"!&

jk

R

!xk

! jk

R" # jk

R$1

3#mm

R% jk

• Kinematic Reynolds stress (minus):

!uj

!t+

!uju

k

!xk

= "1

#

!p *

!xj

+ $%2uj+ g

j"!&

jk

R

!xk

!uj

!xj

= 0

u1(x1, x

2, x

3,t),

u2(x1, x

2, x

3,t),

u3(x1, x

2, x

3,t),

p(x1, x

2, x

3,t)

Unknowns: mean velocity and

pressure field

Closure required for Reynolds

stress tensor: express stress

in terms of mean velocity field…

!

jk

R= func(u)

• Spectral representation of co-variance tensor:

!uj !uk =1

(2" )3

# jk (k1,k

2,k

3) d

3k$$$

! jk (k1,k2 ,k3) : Spectral tensor of turbulence

(how much energy there is in each wave vector k)

In homogeneous isotropic turbulence (simplest case, with no preferred

directions) the spectral tensor function of a vector can be expressed based on

a single scalar function of magnitude of wavenumber, E(k):

! jk (k) =1

4"k2# jk $

kjkk

k2

%&'

()*

E(k)

• What is the dependence of energy density E(k)

with wavenumber?

We now discuss the energy cascade

and Kolmogorov theory of turbulence

• Turbulence Physics: the energy cascade

(Richardson 1922, Kolmogorov 1941)

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

L

!K

• Turbulence Physics: the energy cascade

(Richardson 1922, Kolmogorov 1941)

Big whorls have little whorls,

which feed on their velocity,

and little whorls have lesser whorls,

and so on to viscosity (in the molecular sense)

• Turbulence Physics: the energy cascade

(Richardson 1922, Kolmogorov 1941)

Injection of kinetic energy into

turbulence (from mean flow)

!u2

Time!

!u2

L / !u!

!u3

L

Dissipation of kinetic energy into

heat (due to molecular friction)

! !"u3

L

! !"u (r1)

3

r1

! = 2"# $ui#x

j

# $ui#x

j

+# $uj

#xi

%

&'

(

)*

! !"u (r2 )

3

r2

L

!K

Co

nsta

nt

Flu

x o

f e

ne

rgy

acro

ss s

ca

les: !

!u "1

3ui!u

i!

• Turbulence Physics: the energy cascade

(Richardson 1922, Kolmogorov 1941)

Injection of kinetic energy into

turbulence (from mean flow)

Dissipation of kinetic energy into

heat (due to molecular friction)

!

Co

nsta

nt

Flu

x o

f e

ne

rgy

acro

ss s

ca

les: !

k

kL=1

L

k! =1

!K

k

E = f (k,!)

Dimensional Analysis (Pi-theorem: 3-2=1):

E k5 /3

!2 /3

= const

E(k) = cK!2 /3k"5 /3

Solid experimental

support for K-41:

Supports (approximately)the notion that ! is the only

relevant physical scale in

the inertial range,

+ L at large scales+ " at small scales

!

E(k) = c

K!

2/3k"5/3, c

K~ 1.6

Solid line is equivalent of a 3D radial spectrum equal to

Back to Closure problem for Reynolds equations:

!T ! (velocity scale) " (length scale)

!uj

!t+

!uju

k

!xk

= "1

#

!p *

!xj

+ $%2uj+ g

j"!&

jk

R

!xk

!

jk

R= func(u)

In analogy with molecular friction:

!jk

R= "#

T

$uj

$xk

+$u

k

$xj

%

&'

(

)*

“Eddy-viscosity”

Kolmogorov:

Characterization of turbulence: minimum 2 variables

! !"u3

L

Turbulent kinetic energy: K =1

2!ui!ui=1

2!u2

! !K3/2

L

Need (K,L) or (!,L) or (K,!)

Eddy-viscosity scaling:

!T ! (velocity scale) " (length scale)

~ !u

~ K1/2

~ L

~K

3/2

!

L

!K

!u

!T= Cµ

K2

"

!T(x,t) = Cµ

K2(x,t)

"(x,t)

Transport equations for K(x,t) and !(x,t)

Empirical calibrations:

5 adjustable coefficients

!K

!t+ u

k

!K

!xk

= "#ij

R!u

i

!xj

+!!x

j

$T

%k

+ $&

'()

*+!K

!xj

&

'(

)

*+ " ,

!,!t

+ uk

!,!x

k

= C,1"#

ij

R!u

i

!xj

&

'(

)

*+,K

+!!x

j

$T

%,

+ $&

'()

*+!,!x

j

&

'(

)

*+ " C, 2

,,K

Launder & Spalding (1972) - but earlier (1940s) Kolmogorov K-# model

Cµ = 0.09

C!1= 1.44

C! 2= 1.92

"K= 1.0

"!= 1.3

!uj

!t+!u

ju

k

!xk

= "1

#!p

!xj

+ gj+

!!x

k

$T+ $( )

!uj

!xk

+!u

k

!xj

%

&'

(

)*

+

,--

.

/00

!uj

!xj

= 0

K-! model for turbulence mean flow predictions:

+ boundary (& initial) conditions

!K

!t+ u

k

!K

!xk

= "#ij

R!u

i

!xj

+!!x

j

$T

%k

+ $&

'()

*+!K

!xj

&

'(

)

*+ " ,

!,!t

+ uk

!,!x

k

= C,1"#

ij

R!u

i

!xj

&

'(

)

*+,K

+!!x

j

$T

%,

+ $&

'()

*+!,!x

j

&

'(

)

*+ " C, 2

,,K

Direct Numerical Simulation:

N-S equations:

!uj

!t+!uku j

!xk= "

!p

!x j+ #$

2uj + gj

!uj

!x j= 0

Moderate Re

(~ 103),

DNS possible

High Re

(~ 107),

DNS impossible

From: Multimedia Fluid Mechanics, Cambridge Univ. Press

Regions of large vorticity in isotropic turbulence

World-record DNS (Nagoya

group in Japan):

Source: Kaneda & Ishihara,

J. of Turbulence, 2006 © Taylor & Francis

On Earth Simulator (2003)

4,0963 grid points

~ 2 Terabytes at each time-step

Thousands of time-steps

$

G(x): Filter

Large-eddy-simulation (LES) and filtering:

!˜ u j

!t+ ˜ u k

!˜ u j

!xk

= "!˜ p

!x j

+ #$2˜ u j "

!

!xk

% jk

$

$

G(x): Filter

Large-eddy-simulation (LES) and filtering:

N-S equations:

!˜ u j

!t+ ˜ u k

!˜ u j

!xk

= "!˜ p

!x j

+ #$2˜ u j "

!

!xk

% jk

Filtered N-S equations:

!uj

!t+!ukuj

!xk= "

!p

!x j

+ #$2

u j

!˜ u j

!t+!ukuj

!xk

= "!˜ p

!x j

+ #$2

˜ u j

where SGS stress tensor is:

! ij = uiu j " ˜ u i ˜ u j

!uj

!x j= 0

Useful references:

• S. Pope: Turbulent Flow

(Cambridge Univ. Press, 2000)

• J. Ferziger & M. Peric: Computational Methods for

Fluid Dynamics (Springer, 1996)