LECTURE ON

AN INTRODUCTION TO TURBULENCE

1

An Introduction to Turbulence

similarity universality intermittency intermittency models statistical modelsfractal models Karman-Howarth equation (approximation theory) DIA theory (direct-interaction theory)averaged motion LES DNS

2

1.

2. (scales) (energy cascade)

3. small-scale universality Kolmogorov

4. (intermittency)

5. (isotropic homogeneous turbulence)

6.

7. DNS LES

8. Reynolds equations closure problem

9.

10.

3

1.

(laminar flow) (turbulent flow) Osborne Reynolds (1883, 1884) (fluctuations) (mean motion)

u u

u u u

(t) = T

(t' )dt'

(t) = (t) + ' (t)

t-T/2

t+T/21

ensemble average< >u ergodic hypothesis u u= < > u f(t) = A sin(2 t) u f

ff (t) = A

sin( T)T

sin(2 t)

M sin( T)T

ff

2

( f > 1/T ) () T (1/T )

(unstable) (stable solution) (disturbance) (solutions) x = x x3 0x=0 (steady solution)(stable)

4

>0 x = 0 x = x = x=0

(3D unsteady) (vorticity structures) (mixing) (adverse pressure gradients) (separate) Navier-Stokes (Newtonian fluids)

2. (scales) (energy cascade)

(time and length scales) (internal flows) (external flows) (characteristic dimension) (viscosity) Kolmogorov length scale Kolmogorov length scale (large eddies) (smaller eddies) () (cascade) (source) (stationary)()

(multi-scale)

5

(L) L ~ Re Taylor microscale

5 2q

( J kg sec ) Re q q q ' 'i i2 u u u'i ith kinematic viscosity Kolmogorov length scale ()

/

31 4

~ Re 1/2

L ~ Re ~ ReL3/4 /3 2 ( Re ~ ReL1/2 Re qLL ) (L )()

3. small-scale universality Kolmogorov

(boundary flows) (channel flows) (jets) (mixing layers)

(isotropic) (small-scale eddies) (universality)

6

Kolmogorov 1941 (universal theory) () L ~ ReL3/4 (large eddies) (small eddies) ReL3/4 >> 1

Kolmogorov theory ( universal eddies) (inertial subrange) L L L L LL

7

(similarity) Cantor Set

Kolmogorov theory Kolmogorov Obukov 1962 E(k) k

E( ) = C K r2/3k k 5 3/

r r ~ 1/k (volume average)1941 (x,t) = constant r2/3 = = 2 3 2 3/ / Kolmogorov Obukov (1962) r logarithmic normal distribution (LN theory)Gurvich Yaglom (1967) Central Limit Theorem

f ( ) = 1

2

1 explog(

2r

r r

r r

r

[ ) ] +

2 2

2

2

r2 r [log( = A( , t) + log(L r) )]2 x

r 2/3 ~ ( L)2 3 9/ /k LN theory

E( ) = C L)Kk k k 2 3 5 3 9/ / /( 0.5

Novikov Stewart 1964 (statistical model)Frisch et al (1978) (fractal) (dynamical model) -model o C (>>1) 1 1 1 3 ~ Co / C M (

8

box-counting (fractal dimension)D D limit log( )

log( )limit log(M

log( C 3log(M)

log(C)n n

n

on /3

=

=

))

r ~ ~ Cp o p/3 ~ MC rn

rn

p

n=1

r rp

~ MC

~ r

n np(n-1)

~ MC

= 3D rn n o (1-n) ~ ( / r)

E( ) ~ ( ok k k 2 3 5 3 3/ / /) multi-fractal Meneveau & Sreenivasan

(1987) p-model Benzi et al (1984) random- model ODE She(1991) two-fluid model phenomenological models Tennekes (1968) vortex tube modelCorrsin (1962) vortex sheet modelPullin et al (1994) Townsend-Lundgren vortex model 5. (isotropic homogeneous turbulence)

() (translation) () (rotation) () (spatial gradients) < > = 0u u u=

(two-time-two-point velocity correlation) R , , t , t ) u , t )u ( , t ) ij 2 1 2 i 1 1 j 2 2( (x x x x1 ui ith ergodic ensemble (stationary) R ij R , , t , t ) R , , | t t | )ij 2 1 2 ij 2 1 2( (x x x x1 1 R ij R , , t , t ) ij 2 1 2(x x1 = R , | t t | ) ij 1 2(r r x x 1 2(one-time-two-point) R , , t) u , t)u ( , t) ij 2 i 1 j 2( (x x x x1 R , , t)ij 2(x x1 = R ( )ij r Navier-Stokes

2i i

i m im i m m

u u1 p(u u ) = ft x x x x

+ + + (5.1)

9

fi ith (external force) ui(x) uj( x ) uj( x ) ui(x)

)(Frr

)( R 2)(P)(T =

t)( R

ijmm

ij2

ijijij r

rrr

r +++

(5.2)

xxr ij i m j i m j

m

ij j ii j

ij i j j i

T ( , t) u ( ,t)u ( ,t)u ( ,t) u ( ,t)u ( ,t)u ( ,t) r

1P ( , t) u ( ,t)p( ,t) u ( ,t)p( ,t) r r

F ( , t) u ( ,t)f ( ,t)+u ( ,t)f ( ,t)

r x x x x x x

r x x x x

r x x x x

(5.3)

Fourier-Stieltjes transform ( k wave vector k = |k| )

( )

t= ( ) ( ) ( ) ( )ij ij ij ij ij

k k k k k+ +2 2k (5.4) R ( ) = exp(i ) dij ijr k k r k ( )

T ( ) = exp(i ) d

P ( ) = exp(i ) d

F ( ) = exp(i ) d

ij ij

ij ij

ij ij

r k k r k

r k k r k

r k k r k

( )

( )

( )

(5.5)

i = j r = 0 R (0) = u ( )u ( ) = ( )d = qii i i ii 2x x k k 3 2u iiii ii F

iii x=

x=

r

u u )i i (x

0= }pux

pux

{ 1= pu r

pu r

1 )(P ii

ii

ii

ii

ii +

r

k ii d = 0( )k k (directional distribution) (conserved)

( )t

= ( ) ( ) ( )ii ii ii ii k k k k +2 2k (5.6)

ii ( )k k k E( ) = {41

2 iik k ( )}2 k E( )d q2k k

0

12

=

E( , t)

t= T( , t) E( , t) F( )k k k k k +2 2 (5.7)

T( , t) = 2 iik k ( )2 k k

10

T( , t)dk k0

0 = (5.8)

q

t= E( ,t)d F( )d +

12

2 +

2 20 0

k k k k k

12

q2 = =

(5.2)(5.3) u pi i- 180 u ui i= u p u pi i = = 0 =u pi 0Pij = 0 u = u = u q12 22 32 13

2= u 2 i j u ui j = 0 u ui j

P PP

p

Q QQ

q p q pq

PQP P' PQ, PQQ", QQ'Q" Q Q PQP p1 , p2 , p3 q1 , q2 , q3 P Q PQ PQ Q Q Q Q

p = p cos + p sin1 2 . q q cos sin q sin sin + q cos1 2 3= +

pq p q cos cos sin + p q cos sin sin p q cos cos

p q sin cos sin p q sin sin sin p q sin cos1 1 1 2 1 3

2 1 2 2 2 3

= ++ + +

p q , i ji j = 0 f(r) g(r) f(r) = p q

g(r) = p q

1 1

2 2

u

u

2

2 (5.9)

1 r f(r) longitudinal correlation coefficient2 r

11

g(r) transverse correlation coefficient

( )pq f(r, t) cos cos + g(r, t)sin sin sin =u 2 (5.10)

R r, t) = u , t)u ( , t) = f(r, t) g(r, t)

r r r g(r)ij i j

22 i j ij

( ( x x r+ +

u (5.11)

9 Rij(independent)

R r

f(r) + r2

f r

= g(r)iji

= 0 (5.12)

S u u uij,m i j m= 18 S 13 u up

2p u u up n1 n1 u u up n2 n2 u un12 p u un22 p p r

n1n2 r ppp,3 S=(r)ku pnn,3 S=(r)hu nnp,3 S=(r)qu

+++= )

rr

rr

(r

rr

rrr)2(S ijm

jim

mij3

mji3mij, qhqhku (5.13)

kh

kq

21

2

m

mij,

=

)r(r 4r

1 = 0r

S

=

(5.14)

6.

Karman-Howarth similarity Kraichnan DIA Hinze Turbulence Batchelor Homogeneous turbulence BatchelorProudmanSaffmanTaylorTownsend Yakhot & Orszag (1986) RNG (renormalization group theory) Orszag (1970) EDQNM (eddy-damped quasi-normal Markovian theory)

(i) Karman-Howarth similarity

(5.11)(5.12)(5.13)(5.14)(5.2) T r, t) = r

S Sijm

im,j mj,i( [ ]

+ Pij = 0

()

+

+

r f r

r r12

r4

r =f)(

t4

4232 uuu kk (6.1)

12

(6.1) 1938 Karman & Howarth Karman-Howarth equation

(6.1) similarity solution Karman & Howarth

)(~

)L(t)r(~

=t)(r,

)(f~ )L(t)r( f~=t)f(r,

kkk (6.2)

(6.1)

) f~()L2()~

4~

(L

= f~ dtdL

L1f~

dtd 44221322 ++

uuuu kk

similarity L d

dtconstant

LL

dLdt

1 dLdt

constant

L L L ) constant

3

32

=

= =

= =

u

u

u

uu

uu u

2

3

2

2 12 2 (

L(t) t , t1/2 1/2 u t n u 2 ()

Taylor microscale = =

5

2 2

2

q uu

10

d dt t1/2 (6.3)

L(t) f(r, t) = f (r / ) )(r/~=t)(r, kk u 2 t n Karman-Howarth ()Huang & Leonard (1994)

(r/L)~

Re=t)(r,

(r/L)f~Re+(r/L) f~=t)f(r,1b

l

2bl1

kk

(6.3)

f1 regular confluent hypergeometric function ( , ( )f r / ) = M n, n

r1

52

54

2 f r as r 1 2n

u 2 t nf k~ f2k~ (closure problem)

(ii) Direct Interaction Approximation

13

Kraichnan 1959 DIA LES DIA DIA Kraichnan 1959 6070 (a)

Navier-Stokes equations

( ) u

tu i k p i k u u + fi i i m i m i

~+ = k 2 1 (6.4)

g( t) = g( , t)exp(i )dx k k x x, i k u = 0i i solenoidal ()0 = p k k u u )i m i m

~1 2

(k +

1 p = k k u u )i m i m

~

( k

2 (6.5)

(6.4)

( )

ut

u i k P ( ) u u + f

P ( ) i k k

ii m ij j m i

ij ij i j

~+ =

k

k

2

2

k

k

(6.6)

L (L )

g( , t) g( , t)exp(i )ki=1,2,3

i

x k k x= =

(6.7)

(g g g )g ( ) 1 2

1 2

~)( ) (

,k p q

p q kp q

=+ =

(6.6)

Lu t

u ( ) u ( )u ( ) + f

( ) k P k P

i ii2 ijm j m i

ijm m ij j im

( ) ( )

( ) ( )

+ =

++ = k2 k k p q k

k k k

p q k

(6.8)

u u ( , t) , u u ( , t )i i i i( ) ( ) k k k k

14

t u ( )u ( ) u ( )u ( )u f ( ) u

( ) S f ( ) u (6.9)

i ni2 ijm j m n i n

i2 ijm njm i n

+ = +

= + + =

+ =

( ) ( ) ( )

( , , ) ( )

k2 k k k p q k k k

k k p q k k

p q k

p q k

S u u u , = 0njm n j m( , , ) ( ) ( ) ( )k p q k p q k p q + + (6.10) (b) DIA

DIA u ( )u i np q ( ) Snjm ( , , )k p q (closed)

1) kpqSnjm ( , , )k p q (6.8) u ( )uj m p q( ) u( )k (disturbance)(operator) ( , )ij t, tk Green function u( )k

( ) ( (

( ) ( )

u , t, t )

t

t b , t ) dt

b , t i u ( )u ( )

i ij

o

j

i ijm j m

k k k

k k p q

=

(6.11)

2) u( )k 3) u( )k (Weak Dependence

Principle, WDP)

S u u u + u u u + u u u njm n j m n j m n j m( , , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )k p q k p q k p q k p q u( )k u( )p u( )q kpq Fourier

components DIA

L2

u ( , t + )u , t) P ) U( , ) = P )(4 E( r(

L2

f ( , t + )f , t) P ) F( , ) = P )(4 F( (

, t + , t) P g( , )

i j*

ij ij

i j*

ij12 ij

ij ij

=

=

=

312

2 1

312

2 1

k k k k

k k k k

k k

( ( ( ) ) , )

( ( ( ) ) , )

( ( )

k k k k

k k k k

k

(6.12)

DIAKraichnan

15

2 2

0

k k k k

k k p q p q

k k k k

E( T

T T( d d

F g s s ds

i

i

, ) ( , ) ( , )( , ) | , )

( , ) ( ) ( , ) ( , )

= +=

=

(6.13)

k = |k| , p = |p| , q = |q|

T( E E E Ek p q xy zq

k p q k p q p q k| , ) ( , , ) ( ) ( ) ( ) ( )= +

32 2 (6.14)

( , , )k p q k p q= g( , s) r( , s) r( , s)ds0

(6.15)

T(k | p, q) k, p, q k i(k,) (k, p, q) x, y, z kpq cosine g(k,) r(k,) dg( , )

dg( d d b( E g( , s)g( ,s)r( ,s)ds

0

k k k k p q pq

k p q q k p q

+ =

2 2, ) , , ) ( ) (6.16)

dr( , )d

r( a( E( ) E( )E

g( ,s )r( ,s)r( ,s)ds

b( E g( ,s)r( ,s )r( ,s)ds

k k k k dpdq k p q p qk

k p q

k p q q p k q

kpq

k pq

+ =

2

0

22

2

, ) , , ) ( )

, , ) ( )

(6.17)

a

b

( , , )

( , , )

k p q xyz y z

k p q p k xy z

=

= +

12

1 2 2 2

3 (6.18)

(5.8)T(k | p, q) + T(p | k, q) + T(q | k, q) = 0Fourier modes km km

( ) |k k k k k k k kk

k k

m0

m0

T( , )d T( , )d T( )d

m

m m

= = (6.19)

16

T( ) T( | , )d dmk k k p q p q|

(6.20) kpq k k p q k >m m , or

t m mT( E( | ) | ) ( )k k k k k k 2 2 (6.21)

(6.13)T( , )k k p q k m, , k k p q k >m m , or ()

( ) E( , t)

tT t E( t t

T t T( d d

t m i m

, m

k k k k k k k k k

k k p q p qk p q k

= + +