u u i i j j i x x j j

10. DNS, LES, and RANS

~ attempts to predict turbulence

0=∂

∂=⋅∇

j

j

xu

u

( )i

jj

i

ij

jii fxx

uxp

xuu

tu

+∂∂

∂ν+

∂∂

ρ−=

∂

∂+

∂∂ 1

~ four equations for four unknowns uj and p

DNS (Direct Numerical Simulation):

~ solve Navier-Stokes equations directly without modeling

~ finest grid Δx ~ Kolmogorov’s dissipation length scale

~ time increment Δt ~ Kolmogorov’s dissipation time scale

~ simulation time period = several L/q’s

Resolution 1:


Resolution 2:

Resolution 3:

Resolution 4:

10. DNS, LES, and RANS 10. DNS, LES, and RANS

Estimation of the number of grids required for a DNS with ReL = 104:

Kolmogorov’s dissipation time scale ~ ( ) 2121 Re−η =εν=τ LqL

43Re −=η LLKolmogorov’s dissipation length scale ~

( ) 9493 10~Re~~# LL η

Estimation of the number of time steps required for a DNS with ReL = 104:

100~Re~~# 21L

qL

ητ

10. DNS

This animation demonstrates the evolution of near-wall coherent structures as they convect downstream in a fully-developed turbulent channel flow. The gray structures indicate where the fluid is "spinning" fastest: such regions are identified by solids (also known as "isosurfaces") which enclose all regions of the flow in which the discriminant of the velocity gradient tensor is greater than some positive threshold, indicating fluid motion which is focus in nature. Such regions also roughly (but not exactly) indicate where the pressure of the fluid is lowest, as do the visible cores of tornados; thus, the shaded regions above roughly correspond to tiny unstable "tornado-like" structures in the turbulent flow. Only one eighth of the computational domain used for this simulation, which used approximately 8.5 million grid points, is shown. The animation is shown in a reference frame which convects at approximately 2/3 of the bulk velocity to slow the apparent movement of these structures so that their evolution may be more easily observed. Simulation by Thomas Bewley (UCSD). Visualization by Ned Hammond (Stanford).

Color contours of the vorticity magnitude in Direct Numerical Simulations of supersonic turbulent jet with near acoustic field. The simulations were performed by Jonathan Freund, ParvizMoin and Sanjiva K. Lele.

10. DNS

Color surfaces (red is positive, blue is negative) of the instantaneous streamwise vorticity in Direct Numerical Simulations of flow over a cylinder at Re=300. The simulations were performed by Arthur Kravchenko.

10. DNS

a spatially evolving free shear layer Bert Debusschere

10. DNS


Resolution 3: LES

Resolution 4: DNS

10. LES

LES (Large Eddy Simulation) ~ spatial average

1. select a space filter );( σ− xyG

( ) )2exp(21; 22 σ−−σπ

=σ− xyxyG

( ) ( )xyxy

xyG−π

σ−π=σ−

sin2;

( )⎩⎨⎧ σ≤−σ

=σ−otherwise. 0

;2 if 1;

xyxyG

x2σ

y

ydxyGtyutxu ii ∫ σ−≡ );(),(),(

2. define large-eddy quantity:

),(),(),( txutxutxu iii ′+=

jjjj xtxuydxyGtyu

xydxyG

ytyu

xtxu

∂∂

=σ−∂∂

=σ−∂

∂=

∂∂

∫∫),();(),();(),(),(

ttxu

ttxu

∂∂

=∂

∂ ),(),(

0

i

ii

u

uu

′ ≠≠

jijijiijjijiji uuuuuuuuuuuuuu −+′′+′+′+=

subgrid stresses, ηij

Leonard stresses, Lij

10. LES

3. solve large-eddy motion :iu

0=∂

∂=⋅∇

j

j

xu

u

( )i

jj

i

ij

jii fxx

uxp

xuu

tu

+∂∂

∂ν+

∂∂

ρ−=

∂

∂+

∂∂ 21

~ four equations for four unknowns uj and p

0=∂

∂

j

j

xu

( ) ( )ijsubij

ji

jj

ikk

ij

jii Lx

fxx

upxx

uut

u−τ

∂∂

++∂∂

∂ν+⎟⎟

⎠

⎞⎜⎜⎝

⎛η+

ρ∂∂

−=∂

∂+

∂∂ 2

31

( )ijkkijsubij δη−η−≡τ 3

1 traceless

10. LES

4. Subgrid-stress models ~ modeling small-scale turbulence

( )ijkkijsubij δη−η−≡τ 3

1

jijiij uuuuL −≡

jiijjiij uuuuuu ′′++′+′≡η

~ effects of small eddies on large eddies through interactions in between

~ effects of small eddies generated by large eddies on large eddies

~ theoretically computable (no need in modeling)

~ usually modeled together with subgrid stresses because its magnitude is usually on the order of truncation errors. In that case,

( ) ( )ijkkijijkkijsubij LL δ−−δη−η−≡τ 3

131

10. LES

0=∂

∂

j

j

xu

( )j

subij

ijj

i

ij

jiix

fxx

upxx

uut

u∂

τ∂++

∂∂∂

ν+⎟⎟⎠

⎞⎜⎜⎝

⎛ρ∂

∂−=

∂

∂+

∂∂ 2

mod

( ) ( ) ijkkkkijijsubij LL δ+η−+η−≡τ 3

1

( )kkkk Lpp+η+

ρ≡

ρ 31mod

~ solve for the large eddy averaged velocity and the modified pressure

~ subgrid stresses need modeling

10. LES

( )j

subij

ijj

i

ij

jiix

fxx

upxx

uut

u∂

τ∂++

∂∂∂

ν+⎟⎟⎠

⎞⎜⎜⎝

⎛ρ∂

∂−=

∂

∂+

∂∂ 2

mod

10. LES

iu

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

( )212122

mod

iij

j

sub subi i i i ii i ij i i ij

i j j j j j

uu ut x

u p u u u uu u u fx x x x x x

∂∂ +∂ ∂

⎛ ⎞⎛ ⎞ ∂ ∂ ∂ ∂∂ ∂ ⎟⎜⎟⎜ ⎟=− + ν + τ −ν + −τ⎜⎟⎜ ⎟⎟ ⎜⎟⎜ ⎟∂ ρ ∂ ∂ ∂ ∂ ∂⎜⎝ ⎠ ⎝ ⎠

Energy exchange rate between resolved and subgrid eddies

sub iij

j

ux

∂τ∂

(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99)~ simplest, commonly used

( ) ( ) 212 2 /ijijSt SSσCν =

• eddy viscosity:

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=

−=

i

j

j

iij

ijtsubij

xu

xu

S

Sντ

21

2

• mixing length assumption:

( ) 212 , ~~ ijijt SSSSuxν ≡σ⋅σΔ⋅Δ

~ always positive (no backward cascade)

~ isotropic

~ incorrect near walls

10. LES

(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99)

•Determination of the proportional constant:

Energy exchange rate between resolved and subgrid eddies:

( ) ><σ>=∂∂

τ>=<ε<32 SC

xu

Sj

isubijg

Recall for homogeneous isotropic turbulence:

∫σπ

≈>ωω=<0

2 )( 2 dkkEkS ii

At large Reynolds number:

( )∫∫σπ

−σπ

>ε<≈≈>ωω>=<<0

35322

0

22 2)( 2 dkkCkdkkEkS Kii

232223

32

><⎟⎠⎞

⎜⎝⎛πσ

⎟⎟⎠

⎞⎜⎜⎝

⎛>=ε< S

CK

10. LES

(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99)

• Determination of the proportional constant:

At large Reynolds number: >ε>≈<ε< g

213

43243

321

><

><⎟⎟⎠

⎞⎜⎜⎝

⎛π

=S

S

CC

KS

( ) 212 ijij SSS ≡

( ) ( ) 212 2 /ijijSt SSσCν =

ijtsubij Sντ 2−=

43

321

⎟⎟⎠

⎞⎜⎜⎝

⎛π

≈KC

10. LES

( )3 2 2

3 22 3 223S

K

C S SC

⎛ ⎞ ⎛ ⎞σ⎟⎜ ⎟⎜σ < >≈ < >⎟⎜ ⎟⎜⎟ ⎟⎜⎜ ⎟ ⎝ ⎠π⎝ ⎠

(b) structure-function model (1992, JFM 239,157-194)

structure function: ( ) >+−≡< 22 ),(,),( trxutxurxF

For homogeneous isotropic turbulence:

>+<−><== )()(2)()(2)(),( 22 rxuxuxuxurFrxF iiii

∫∞

⎟⎠⎞

⎜⎝⎛ −=

0

sin1)(4 dkkr

krkE

xkdk

krkrkErF c

kc

Δπ

=⎟⎠⎞

⎜⎝⎛ −≈ ∫ , sin1)(4)(

02

( )( ) >+−+−≡< )()()()()(2 rxuxurxuxurF iiii

structure function of resolved eddies:

∫ ⎟⎠⎞

⎜⎝⎛

ΔΔ

−≈Δck

dkxk

xkkExF0

2sin1)(4)(

10. LES 10. LES

( ) ( )( ) ( ) expi i iiu x u x r k ik r dk∫∫∫< + >= Φ ⋅

( ) ( )max 2 2

0 0 0exp cos sin

k

ii k ikr k d d dkπ π

∫ ∫ ∫= Φ θ θ φ θ

For isotropic homogeneous turbulence:

( ) ( )( )max 2

0 02 exp cos sin

k

iik k ikr d dkπ

∫ ∫= π Φ θ θ θ

( ) ( )0

0exp cos s

exp cosinikr d

ikrikr

ππ

∫θ

=−θ θ θ( ) ( )exp expikr ikr

ikr ikr−

= −

( ) ( )2 si inn 2si krikr

krkr

==

max1 12 2

0( , ) ( )

k

i i iiu u k t dk E k dk∫∫∫ ∫= Φ =

∫ ∫π π

φθθΦ=2

0 0

221 sin),()( ddktkkE ii

22 ( , )iik k t= π Φ

( ) ( )max 2

0

sin( ) ( ) 4

k

i i ii

kru x u x r k k dk

kr∫< + >= π Φ

10. LES For isotropic homogeneous turbulence:

Recall

( ) ( )max

0

sin( ) ( ) 2

k

i i

kru x u x r E k dk

kr∫< + >=


At large Reynolds number:

∫ ⎟⎠⎞

⎜⎝⎛

ΔΔ

−⎟⎟⎠

⎞⎜⎜⎝

⎛≈Δ

−ck

cc dk

xkxk

kkkExF

0

35

2sin1)( 4)(

c

ckkE

xA )(4

2

38

Δ

π=

476738.0sin1 0

35 ≈ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ξξ

−ξ≡ ∫π

− dA

On the other hand, expect from mixing length assumption:

( )21

211 )()(~~ ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⋅Δ⋅Δν −

c

cccct k

kEkEkkux

21

238

221

)(4

)(⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Δπ

Δ=⎟⎟

⎠

⎞⎜⎜⎝

⎛≡ν xF

AxC

kkE

C Fc

cFt

10. LES

(b) structure-function model (1992, JFM 239,157-194)• Determination of proportional constant (CF):

At large Reynolds number: ∫ν>=ε>≈<ε<ck

tg dkkEk0

2 )(2

23

23 −= KF CC

10. LES

2

02 ( )

ck

t k E k dk∫< ε>= ν

2 2/3 5 3

02

ck

t Kk C k dk−∫= ν ⋅ < ε>

2/3 4/3324t K cC k= ν < ε> ⋅

1/3 4/331 24t K cC k−= ν < ε> ⋅

( ) 3/ 21 22/3 1/2

51

/3 231 (2

)K K

cc

cF cC kEC kk

kC

−−⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⋅ ⋅ < ε> ⋅ ⋅⎝ ⎠

3/ 2312 F KC C= ⋅


23

23 −= KF CC

)(105.0)(43

2 212

2321

238

223 xFxCxF

AxC KKt Δ⋅Δ⋅⋅≈

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Δπ

Δ=ν −−

• computation of structure function

( )( ) >Δ+−Δ+−≡<Δ )()()()(),(2 xxuxuxxuxuxxF iiii

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ⎪

⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

Δ−−+Δ+−+

Δ−−+Δ+−+

Δ−−+Δ+−+

≈2

3332

333

2222

2222

2111

2111

61

xxuxuxxuxu

xxuxuxxuxu

xxuxuxxuxu

iiii

iiii

iiii

10. LES

21

238

221

)(4

)(⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Δπ

Δ=⎟⎟

⎠

⎞⎜⎜⎝

⎛≡ν xF

AxC

kkE

C Fc

cFt

•Comparison between Smagorinsky’s model and structure function model:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

∂∂

≈Δ2

3

2

2

2

12 222

61),( x

xux

xux

xuxxF iii

xxi Δ=Δ

2

1

1211111 )()( x

xu

xxuxu Δ∂∂

≈Δ+−

2

1

2211212 )()( x

xuxxuxu Δ∂∂

≈Δ+−

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ⎪

⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

Δ−−+Δ+−+

Δ−−+Δ+−+

Δ−−+Δ+−+

≈2

3332

333

2222

2222

2111

2111

2 61

xxuxuxxuxu

xxuxuxxuxu

xxuxuxxuxu

F

iiii

iiii

iiii

10. LES

iiKKFt SxCxFxC ωω+⋅Δ⋅⋅=Δ⋅Δ⋅⋅=ν −− 222321

223 043.0)(105.0

( ) 21243

23

21 /ijij

K

St SSσ

Cν ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛π

= SxCK ⋅Δ⋅⋅= − 223055.0

( )j

i

j

i

xu

xuxxxF

∂∂

∂∂Δ

=Δ3

,2

2 ( )ijijijij SSxΩΩ+

Δ=

3

2

( )iiijij SSxωω+

Δ= 2

6

2

•Comparison between Smagorinsky’s model and structure function model:

10. LES

(c) dynamic model (1992, JFM 238,352-336; 1991, Phys. Fluids A 3, 1760-1765)

idea: use two filters, one with width Δx and the other with αΔx (α>1)

ydxxyGtyutxu ii ∫ Δ−≡ );(),(),(

ydxxyGtyutxu ii ∫ Δα−≡ );(),(),(~

ydxxyGtyutxu ii ∫ Δα−≡ );(),(),(~

0=∂

∂

j

j

xu

( )j

iji

jj

i

ij

jiixT

fxx

upxx

uut

u∂

∂++

∂∂∂

ν+⎟⎟⎠

⎞⎜⎜⎝

⎛ρ∂

∂−=

∂

∂+

∂∂

jijiij uuuuT −≡

10. LES

( )j

iji

jj

i

ij

jiix

fxx

upxx

uut

u∂

∂ℑ++

∂∂∂

ν+⎟⎟⎠

⎞⎜⎜⎝

⎛ρ∂

∂−=

∂

∂+

∂∂ ~~~~~~

jijiijij uuuuT −+≡ℑ ~~~ ~ jijiij uuuuL ~~~ −≡ ~~

jijiij uuuuT −≡~ ~

η Δx αΔx L

ijij LT ~~ −=

( )j

iji

jj

i

ij

jiixT

fxx

upxx

uut

u∂

∂++

∂∂∂

ν+⎟⎟⎠

⎞⎜⎜⎝

⎛ρ∂

∂−=

∂

∂+

∂∂

ijT

ijℑ

10. LES

ijijij TL ℑ−= ~~Germano identity:

If Smagorinsky’s model is applied, then

ijijkkij SSxCTT 231 2 Δ=δ−

= effect of eddies of size < Δx on those of size > Δx

( ) ijijkkij SSxC~~

2 231 Δα=δℑ−ℑ

= effect of eddies of size < αΔx on those of size > αΔx

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

iij x

uxu

S~~~

21

⎟⎠⎞⎜

⎝⎛ Δα−Δ=δ− ijijijkkij SSxCSSxCLL

~~ 2~~ 222

31

10. LES

Assume C = constant:

• One possibility:

( ) ijijijkkijij SCMLLS 2~~31 =δ−

mnmn

ijij

SMSL

C~

21

=

ijijijijkkij CMSSSSxCLL 2~~

2~~ 2231 ≡⎟

⎠⎞⎜

⎝⎛ α−Δ=δ−

• Another possibility: choose C to be the one that minimizes E(C)

( )( )ijijkkijijijkkij CMLLCMLLCE 2~~2~~)( 31

31 −δ−−δ−≡

( )13

( ) 0 2 2ij ij kk ij ijE C M L L CM

C∂ = =− − δ −∂

10. LES

mnmn

ijij

MMML

C~

21

=

10. LES( )1

30 2 2ij ij kk ij ijM L L CM=− − δ −

130 2ij ij ij kk ij ij ijM L M L CM M= − δ −

13 2ij ij jj kk ij ijM L M L CM M− =

( )2 2 ij ij ijM x S S S S≡Δ −α

10. LES

mnmn

ijij

MMML

C~

21

=

( )2 2 ij ij ijM x S S S S≡Δ −α

jijiij uuuuL ~~~ −≡

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

iij x

uxu

S~~~

21

( )j

iji

jj

i

ij

jiixT

fxx

upxx

uut

u∂

∂++

∂∂∂

ν+⎟⎟⎠

⎞⎜⎜⎝

⎛ρ∂

∂−=

∂

∂+

∂∂

0i

i

ux

∂ =∂

ijijkkij SSxCTT 231 2 Δ=δ− (subgrid stresses)

Other subgrid models:

• variations of dynamic model

(1992) Phys. Fluids A 4, 633-635

(1994) Center for Turbulence, Proc. Of the Summer Program 271-299

(1995) JFM 286, 229-255

• k-equation dynamic model

dynamic model:

• C = 0 in laminar flows and at walls.

• A negative C is possible (backward cascade).

• mathematically inconsistent (dynamic localization methods)

• Too strong negativeness causes instability.

• The real C is found to be intermittent.

10. LES ( )1 ,2

ij ij

mn mn

L MC C x t

M M= =

1/ 213 2ij kk ij ijT T C x k S− δ = Δ ⋅ ⋅

10. LES

(d) spectral models (models in the wave space)

)(2)()( 2 kEkkTtkE

ν−=∂

∂

22 ( ) ( )k E k T kt

⎛ ⎞∂ ⎟⎜ + ν =⎟⎜ ⎟⎜⎝ ⎠∂ 0( ; , )

p q kk k

S k p q dpdq dk+ + ==

∫ ∫∫⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

0( ; , )

p q kT k p q dpdq

+ + =∫∫≡

and or 0 0

( ; , ) ( ; , )c cp q

p q k p q kk p q k

T k p q dpdq T k p q dpdq+ + = + +

<=>

∫∫ ∫∫= +

: resolved eddies: subgrid eddies

c

c

k kk k

⎧ <⎪⎪⎨ >⎪⎪⎩

cascade due to subgrid stresses, to be modeled

10. LES

2

0 0 and or

2 ( ) ( ; , ) ( ; , )c cp q k p

p q k p q kq k

k E k T k p q dpdq T k p q dpdqt + + = + + =

< >

∫∫ ∫∫⎛ ⎞∂ ⎟⎜ + ν = +⎟⎜ ⎟⎜⎝ ⎠∂

and0

2( ; , ) 2 ( )c

tpp q k

q kT k p q dpdq k E k

+ + =<

∫∫= + ν

( )

2

nd0

a

2 ( ) ( ; , )c

tp q kp q k

k E k T k p q dpdqt

<+ + =∫∫

⎛ ⎞∂ ⎟⎜ + ν+ν =⎟⎜ ⎟⎜⎝ ⎠∂

0or

2

2 ( ) ( ; , )c

tp q

kk

p q

k E k T k p q dpdq>

+ + =∫∫ν =

10. LES0

or

2

2 ( ) ( ; , )c

tp q

kk

p q

k E k T k p q dpdq>

+ + =∫∫ν =

Isotropic homogeneous turbulence theory:

EDQNM (Eddy-damped quasi-normal Markovian by Orszag)

DIA (direct interaction approximation by Kraichnan)

( ) ( ) ( ) ( )( )3

2 2, , kpqxy zT k p q E q k E p p E k

q+= θ ⋅ ⋅ ⋅ −

xy

zp

qk

x,y,z : cosine value of the corresponding angles

( )

( )

2 / 3 5 / 3

1

1/ 3 2 / 3

K

kpq k p q

k K

E k C k

C k

−

−

= ε

θ = η +η +η

η = μ ε

10. LES

( ) ( )2

0ˆ ˆ ˆ i ij m j m

p q kk u i k u p u q dpdq

t + − =∫∫

⎛ ⎞∂ ⎟⎜ +ν =−Δ⎟⎜ ⎟⎜⎝ ⎠∂

for ck k<

( ) ( ) ( )2

,

ˆ ˆ ˆ c

i ij m j mt

p qp q k

k

k u i k u p u q dpdqt + =

<

∫∫⎛ ⎞∂ ⎟⎜ + =−Δ⎟⎜ ⎟⎜⎝ ⎠∂

ν+ν

Kraichnan,1976, Eddy viscosity in two and three dimensions, J. Atmos. Sci. 33 1521–36.

10. LES

( )( )( )

0

c

c

k

ck

c k

c

T k k dkk k

T k k dk

∫

∫

′ ′Π ≡−

′ ′= the fraction of the total energy

transfer across kc which comes from wavenumbers between k and kc

Color shades of the instantaneous passive scalar in Large Eddy Simulations of turbulent flow in coaxial jet combustor with swirl. The simulations were performed by Charles Pierce and Parviz Moin.

10. LES

Color shades of the instantaneous streamwise velocity in Large Eddy Simulations of turbulent flow in plane diffuser. The simulations were performed by Massimiliano Fatica, RajatMittal, Hans Kaltenbach and Parviz Moin.

10. LES

Color contours of the instantaneous vorticity magnitude in Large Eddy Simulations of flow over a cylinder at Re=3900. The simulations were performed by Arthur Kravchenko.

10. LES 10. DNS, LES, and RANS

RANS (Reynolds Averaged Navier-Stokes Equations)

0=∂

∂

j

j

xu

( )tijij

jii

j

ij

iix

Kpx

gxu

ut

uDtuD

τ+τ∂∂

ρ+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

ρ∂∂

−=∂∂

+∂∂

≡1

32

(Here overbar represents an ensemble average.)

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

μ=τi

j

j

iij x

uxu

mean viscous

stress tensor

tensorstress )(turbulent Reynolds 32 =δρ+′′ρ−=τ ijji

tij Kuu

Zero equation One equation Model : model K-equationTwo equation Model : equation equation −ε+−KReynolds stress models: model t

ijτ -equations

10. RANS

§ Zero equation

Slt2≈ν• Mixing length models:

2yl ≈Sublayer:

yl κ≈Overlap layer:

constant≈lOuterlayer:

¶ Prandtl and Karman: ¶ van Driest Model

flow plate-flatfor 26 ; exp1 ≈⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−κ≈

+A

Ayyl

damping factor

A varies with flow conditions (pressure gradient, wall roughness, blowing/suction etc)

• Eddy Viscosity Concept: free)-e(divergenc 2 ijt

tij Sν=ρ

τ

: eddy viscosity (isotropic herein)tν

§ One-Equation Model

• dimensional analysis: ),( ε=ν Kft

sm2 22 sm 32 sm

ε= μ2 KC

[ ]L

KL

2/3

3

2~

velocityareavelocity~

massvelocitydrag

ρ

×××ρ×=ε

LKC

2/3

ε=ε∝L

2/1 ∝K eddy velocity

effective eddy size

• turbulent kinetic energy dissipation rate:

• Eddy Viscosity Concept: free)-e(divergenc 2 ijt

tij Sν=ρ

τ

10. RANS

turbulent diffusion term:

1 2

21

jKjiij x

Kt

Cuuuup∂∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡≅′′′−′′

ρ− :

2

jK x

KKC∂∂

ε≡

turbulent production term:

( )j

itij

j

iij

tij

j

iji x

uτ

xu

K τxu

uu∂∂

=∂∂

δρ−=∂∂

′′ρ− 32

• turbulent kinetic energy per unit mass K:

ijijj

iji

jjiij

j

jj

SSxu

uuxKuuuup

x

xKu

tK

DtDK

′′μ−∂∂′′ρ−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂∂

μ+′′′ρ−′′−∂∂

=

∂∂

ρ+∂∂

ρ=ρ

2 21

ε=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

ε=ν=

ρ

τμ

i

j

j

iijt

tij

xu

xuKCS

222

10. RANS

ε−∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

ε+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂∂

ν+∂∂

ε∂∂

=∂∂

+∂∂

μ 22

j

i

i

j

j

i

jjK

jjj x

uxu

xuKC

xK

xKKC

xxKu

tK

0=∂

∂

j

j

xu

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

ε+

∂∂

ν∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

ρ∂∂

−=∂∂

+∂∂

μi

j

j

i

j

i

jii

j

ij

ixu

xuKC

xu

xKp

xg

xu

ut

u 2

32 2

ρτtij

LKC

2/3

ε=ε

6 equations for 6 unknowns ⎟⎟⎠

⎞⎜⎜⎝

⎛ε+

ρ , , , 3

2 KKpui

with 3 empirical constants ( )LCCCK εμ , ,

§ One-Equation Model

10. RANS

§ Two-Equation Model

• turbulent kinetic energy dissipation rate (per unit mass):j

i

j

ixu

xu

∂′∂

∂′∂

ν≡ε

mm

i

jj

i

m

j

m

i

j

i

j

m

j

i

m

j

i

j

m

i

mj

i

m

ij

jmm

j

m

i

m

ij

j

xxu

xxu

xu

xu

xu

xu

xu

xu

xu

xu

xxu

xu

u

xxp

xu

xu

xu

uxDt

D

∂∂′∂

∂∂′∂

ν−∂

′∂

∂′∂

∂′∂

ν−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂′∂

∂′∂

+∂

′∂

∂

′∂

∂∂

ν−∂∂

∂∂

′∂′ν−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂ε∂

ν+∂

′∂∂

′∂

ρν

−∂

′∂∂

′∂′ν−

∂∂

=ε

22

2

22

22

2

exact equation:turbulent diffusion molecular diffusion

production

destruction

10. RANS

ε⋅ε

−≡ ε KC 2:

mm

j

m

i

m

ij x

pxu

xu

xu

u∂

′∂∂

′∂

ρν

−∂

′∂∂

′∂′ν−2

turbulent diffusion terms:

jxKC

∂ε∂

ε≡ ε

2:

production terms:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂′∂

∂′∂

+∂

′∂

∂

′∂

∂∂

ν−∂∂

∂∂

′∂′ν−j

m

j

i

m

j

i

j

m

i

mj

i

m

ij x

uxu

xu

xu

xu

xxu

xu

u 22 2

j

iji x

uuu

KC

∂∂′′ε

−≡ ε1:j

itij x

uskg

m∂∂

τ⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡

⋅≅

3

jxsm

∂ε∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡≅

2

mm

i

jj

i

m

j

m

i

j

ixx

uxx

uxu

xu

xu

∂∂′∂

∂∂′∂

ν−∂

′∂

∂′∂

∂′∂

ν−22

22

destruction terms:

ε⎥⎦⎤

⎢⎣⎡≅

sec1

j

i

i

j

j

ixu

xu

xu

KCC∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

≡ εμ 12:

10. RANS

6 equations for 6 unknowns ⎟⎟⎠

⎞⎜⎜⎝

⎛ε+

ρ , , , 3

2 KKpui

with 5 empirical constants ( )21,, , , εεεμ CCCCCK

ε−∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

ε+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂∂

ν+∂∂

ε∂∂

=∂∂

+∂∂

μ 22

j

i

i

j

j

i

jjK

jjj x

uxu

xuKC

xK

xKKC

xxKu

tK

0=∂

∂

j

j

xu

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

ε+

∂∂

ν∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

ρ∂∂

−=∂∂

+∂∂

μi

j

j

i

j

i

jii

j

ij

ixu

xuKC

xu

xKp

xg

xu

ut

u 2

32 2

§ Two-Equation Model

KC

xu

xu

xu

KCCxx

KCxDt

D

j

i

i

j

j

i

lll

2

21

22 ε

−∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂ε∂

ν+∂ε∂

ε∂∂

=ε

εεμε

10. RANS

§ Reynolds-stress Model

RANS (Reynolds Averaged Navier-Stokes Equations)

0=∂

∂

j

j

xu

( )tijij

jii

j

ij

iixx

pgxu

ut

uDtuD

τ+τ∂∂

ρ+

∂∂

ρ−=

∂∂

+∂∂

≡11

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

μ=τi

j

j

iij x

uxu

mean viscous

stress tensor

tensorstress )(turbulent Reynolds =′′ρ−=τ jitij uu

Model equations directly.tijτ

10. RANS

m

mji

m

imj

m

jmi

m

j

m

i

mm

ji

ji

ij

m

jim

jiji

xuuu

xu

uuxu

uu

xu

xu

xxuu

xpu

xpu

xuu

utuu

DtuuD

∂

′′′∂−

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂′′+

∂

∂′′−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

′∂

∂′∂

−∂∂

′′∂ν+

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂′∂′+

∂′∂′

ρ−=

∂

′′∂+

∂

′′∂=

′′

2

1

2

Reynolds stress tensor equations

~ mean motion Lagrangian

~ pressure effects (nonlocal, linear, and nonlinear)

~ viscous diffusion/dissipation effect

~ production and reorientation by the mean motion

~ turbulent advection

nonlocal

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂′∂

+∂

′∂

ρ′

+⎭⎬⎫

⎩⎨⎧

ρ′

δ′+δ′∂∂

−=j

i

i

jjmiimj

m xu

xuppuu

x

10. RANS

turbulent diffusion terms:

( ) mjijmiimj uuupuu ′′′−ρ′

δ′+δ′−m

ji

xuum

∂

′′∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡≅

sec

2

m

jiK x

uuKC∂

′′∂⋅

ε≡ :

2

pressure-strain terms:

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂′∂

+∂

′∂

ρ′

j

i

i

j

xu

xup

⎭⎬⎫

⎩⎨⎧

∂∂′′δ−

∂∂′′+

∂

∂′′≡

m

nmnij

m

imj

m

jmi x

uuu

xu

uuxu

uuC32: 2

traceless, expected to be able to

be expressed in terms of jij

i uuxu ′′−∂∂

and

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

′′−≅j

iji x

uuu

dissipation terms:

m

j

m

ixu

xu

∂

′∂

∂′∂

ν− 2 ⎟⎠⎞

⎜⎝⎛ δ−′′

ε−εδ−≡ Kuu

KC ijjiij 3

232 : 1

isotropic part

non-isotropic part

10. RANS

⎟⎠⎞

⎜⎝⎛ δ−′′ε

−εδ−

⎭⎬⎫

⎩⎨⎧

∂∂

′′δ−∂∂

′′+∂

∂′′+

⎭⎬⎫

⎩⎨⎧

∂∂′′+

∂

∂′′−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂

′′∂⎟⎟⎠

⎞⎜⎜⎝

⎛ν+

ε∂∂

=′′

KuuK

C

xu

uuxu

uuxu

uuC

xu

uuxu

uux

uuKCxDt

uuD

ijjiij

m

nmnij

m

imj

m

jmi

m

imj

m

jmi

m

jiK

m

ji

32

32

32

1

2

2

modeled Reynolds stress tensor equations

i=j :

ε−⎭⎬⎫

⎩⎨⎧

∂∂′′+

∂∂′′−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ν+

ε∂∂

= 212

m

imi

m

imi

mK

m xu

uuxu

uuxKKC

xDtDK

6 equations for 6 new unknowns ⎟⎠⎞⎜

⎝⎛ ′′′′′′′′′ , , , , , 133221

23

22

21 uuuuuuuuu

10. RANS

KC

xu

uuK

Cxx

KCxDt

D

j

iji

lll

2

21

2 ε−

∂∂

′′ε−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂ε∂

ν+∂ε∂

ε∂∂

=ε

εεε

0=∂

∂

j

j

xu

⎟⎟⎠

⎞⎜⎜⎝

⎛′′−

∂∂

ν∂∂

+∂∂

ρ−=

∂∂

+∂∂

jij

i

jii

j

ij

i uuxu

xxpg

xu

ut

u 1

⎟⎠⎞

⎜⎝⎛ δ−′′ε

−εδ−

⎭⎬⎫

⎩⎨⎧

∂∂

′′δ−∂∂

′′+∂

∂′′+

⎭⎬⎫

⎩⎨⎧

∂∂′′+

∂

∂′′−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂

′′∂⎟⎟⎠

⎞⎜⎜⎝

⎛ν+

ε∂∂

=′′

KuuK

C

xu

uuxu

uuxu

uuC

xu

uuxu

uux

uuKCxDt

uuD

ijjiij

m

nmnij

m

imj

m

jmi

m

imj

m

jmi

m

jiK

m

ji

32

32

32

1

2

2

11 equations for 11 unknowns

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′′′′′′ερ

, , , , , ,, , , , 1332212

32

22

1321 uuuuuuuuupuuu

§ Reynolds-stress Model

10. RANS

§ Algebraic Stress Model

Assume negligible turbulent convection and diffusion

⎟⎠⎞

⎜⎝⎛ δ−′′ε

−εδ−

⎭⎬⎫

⎩⎨⎧

∂∂

′′δ−∂∂

′′+∂

∂′′+

⎭⎬⎫

⎩⎨⎧

∂∂′′+

∂

∂′′−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂

′′∂⎟⎟⎠

⎞⎜⎜⎝

⎛ν+

ε∂∂

=′′

KuuK

C

xu

uuxu

uuxu

uuC

xu

uuxu

uux

uuKCxDt

uuD

ijjiij

m

nmnij

m

imj

m

jmi

m

imj

m

jmi

m

jiK

m

ji

32

32

32

1

2

2

⎭⎬⎫

⎩⎨⎧

∂∂

′′δ−∂∂

′′+∂

∂′′+

⎟⎠⎞

⎜⎝⎛ δ−′′ε

−εδ−⎭⎬⎫

⎩⎨⎧

∂∂′′+

∂

∂′′−=

m

nmnij

m

imj

m

jmi

ijjiijm

imj

m

jmi

xu

uuxu

uuxu

uuC

KuuK

Cxu

uuxu

uu

32

32

320

2

1

~ algebraic equations for the Reynolds stresses ~

10. RANS

Masud Behnia, Sacha Parneix& Paul Durbin

10. RANS

Masud Behnia, Sacha Parneix& Paul Durbin

10. RANS 10. DNS, LES, and RANS

Comparisons among DNS, LES and RANS:

DNS

• simulate motion of all scales (limited only by numerical errors)

• able to capture all details of the turbulent flow fields

• tremendously large amount of computations and huge data size

LES

• simulate motion of large eddies only (less computations)

• need modeling effects of subgrid eddies on large eddies

• able to see fluctuations over length scale >= Δx

RANS

• compute only ensemble averaged quantities (least computations)

• need strong assumptions to model effects of fluctuations on the mean motion

u u i i j j i x x j j

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