T b l M d li iTurbulence Modeling in Computational Fluid Dynamics (CFD)p y ( )

Jun Shao, Shanti Bhushan, Tao Xing and Fred Stern

OutlineOutline1. Characteristics of turbulence 2. Approaches to predicting turbulent flows2. Approaches to predicting turbulent flows 3. Reynolds averaging4. RANS equations and unknowns5. The Reynolds-Stress Equations6. The closure problem of turbulence7 Ch t i ti f ll b d t b l t fl7. Characteristics of wall-bound turbulent flows8. Turbulence models and ranges of applicability

7 1 RANS7.1. RANS7.2. LES/DES7.3. DNS

9 E l diff

2

9. Example: diffuser

Characteristics of turbulence • Randomness and fluctuation: u U u′= +

• Nonlinearity: Reynolds stresses from the nonlinear convective terms

• Diffusion: enhanced diffusion of momentum, energy etc.

• Vorticity/eddies/energy cascade: vortex

3

• Vorticity/eddies/energy cascade: vortex stretching

Characteristics of turbulence

•Dissipation: occurs at smallest scales •Three-dimensional: fluctuations are always 3DThree dimensional: fluctuations are always 3D •Coherent structures: responsible for a large part of the mixing

•A broad range of length and time scales: making

4

•A broad range of length and time scales: makingDNS very difficult

Approaches to predicting turbulent flows• AFD, EFD and CFD:

– AFD: No analytical solutions exist EFD E i ti i d ti– EFD: Expensive, time-consuming, and sometimes impossible (e.g. fluctuating pressure within a flow)

– CFD: Promising, the need for turbulence modeling g, g• Another classification scheme for the approaches

– The use of correlations: ( )ReDC f=

– Integral equations: reduce PDE to ODE for simple cases – One-point closure: RANS equations + turbulent models – Two-point closure: rarely used FFT of Two-pointTwo point closure: rarely used, FFT of Two point

equations – LES: solve for large eddies while model small eddies

5

– DNS: solve NS equations directly without any model

Deeper insights on RANS/URANS/LES

RANS

URANS

LES

6

Reynolds Averaging

• Time averaging: for stationary turbulenceturbulence

• Spatial averaging: for homogenous t b lturbulence

• Ensemble averaging: for any turbulence • Phase averaging: for turbulence with

periodic motion p

7

RANS equations and unknowns• RANS equation

( )2i iU U PU Sρ ρ μ ρ∂ ∂ ∂ ∂+ + ′ ′

• RANS equation in conservative form

( )2i ij ji

jj i

i j

U St x

u ux x

ρ ρ μ ρ+ = − +∂ ∂ ∂

−∂

q

( ) ( )2ij i ji

j i jj i

U PU U St x

ux x

uρ ρ μ′ ′∂ ∂ ∂ ∂+ + = − +∂ ∂ ∂ ∂

• Numbers of unknowns and equations–Unknowns: 10 = P (1) + U (3) + (6) ( )i ju u′ ′−–Equations: 4 = Continuity (1) + Momentum (3)

( )

8

The Reynolds-Stress EquationD i i T ki f h NS i•Derivation: Taking moments of the NS equation.

Multiply the NS equation by a fluctuating property and time average the product. Using this procedure, one can derive a g p g p ,differential equation for the Reynolds-stress tensor.

2ij ij j i ji uuU UUτ τ

τ τ ν∂ ∂ ∂ ∂+

′+

∂′∂2j j j ik ik jk

k k k

ji

k kx xU

t x x xτ τ ν+ = − − +

∂ ∂ ∂ ∂ ∂ ∂

ji ijuu p p τ⎛ ⎞ ∂⎛ ⎞∂⎜ ⎟ ⎜ ⎟

′′ ′ ′∂ ∂ ′ ′ ′jii j k

j

ij

ki k

u p p u u ux x x x

νρ ρ

⎛ ⎞∂+ + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ′ ′ ′

∂ ∂⎝+

∂ ⎠⎝ ⎠∂

•NEW Equations: 6 = 6 equation for the Reynolds stress tensorNEW Equations: 6 6 equation for the Reynolds stress tensor •NEW Unknowns: 22 = 6 + 6 + 10

6ji uu p p unkownsx xρ ρ

⎛ ⎞′′ ′ ′∂ ∂+ →⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

92 6ji

k k

uu unkownsx x

ν′∂′∂ →

∂ ∂

j ix xρ ρ⎜ ⎟∂ ∂⎝ ⎠

10i j ku u u unkowns′ ′ ′ →

The closure problem of turbulence

• Because of the non-linearity of the Navier-Stokes equation as we take higher and higher momentsequation, as we take higher and higher moments, we generate additional unknowns at each level.

• In essence, Reynolds averaging is a brutal simplification that loses much of the informationsimplification that loses much of the informationcontained in the Navier-Stokes equation.

• The function of turbulence modeling is to devise i i f h k l i iapproximations for the unknown correlations in

terms of flow properties that are known so that a sufficient number of equations exist.

• In making such approximations, we close the system.

10

Characteristics of Wall-Bound Turbulent Flows• The turbulent boundary layer (zero-

pressure gradient) has universal velocity distribution near the wallvelocity distribution near the wall (inner-layer) (Clauser 1951)

5: <=== +++ yyyuuu τ 5: <=== yyuu ντ

3010:1.5)ln(4.0

1 4 ≥>+= +++ yyuk 4.0

3010:3.0 42 ≥>= +y

uk

τ

• Th l it d f t h l tt d• The velocity defect when plotted vs. y/δ collapse on a single curve (outer-layer)

11

2)1(6.9δτ

yu

uu −=−∞

Turbulent models (RANS) ( )• Boussinesq eddy-viscosity approximation• Algebraic (zero-equation) modelsAlgebraic (zero equation) models

– Mixing length – Cebeci-Smith Model – Baldwin Lomax Model– Baldwin-Lomax Model

• One-equation models– Baldwin-Barth model – Spalart-Allmaras model

• Two-equation models – k ε model– k-ε model – k-ω model

• Four-equation (v2f) models

12

• Reynolds-stress (seven-equation) models

Turbulent models (RANS) ( )• Boussinesq eddy-viscosity approximation

23

jT

ii j ij

j i

UUu ux x

kδν⎛ ⎞∂∂′ ′− = + −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠j⎝ ⎠

• Dimensional analysis shows: , where q is a b l l l d b l l h

T C qLμν =turbulence velocity scale and L is a turbulence length scale. Usually where is the turbulent kinetic energy. Models that do not provide a length

2q k=12 i ik u u′ ′=

gy p gscale are called incomplete.

13

Turbulent models (0-eqn RANS) ( q )• Mixing length model: 2

T mixdUldy

ν =

• Assume for free shear flow, then( )mixl xαδ=

─ α=0.180 for far wake

─ α=0.071 for mixing layer

─ α=0.098 for plane jet

─ α=0.080 for round jet • Comments:

─Reliable only for free shear flows with different α values

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─Not applicable to wall-bounded flows

Turbulent models (0-eqn RANS) ( q )Cebeci-Smith Model (Two-layer model)

,iT my yν ≤⎧⎪

⎨ h i h ll l f fy,i

o

T mT

T m

y y

y yν

ν⎪= ⎨ >⎪⎩

Where is the smallest value of for my yi oT Tν ν=

Inner layer:1 22 2

2iT mix

U Vly x

ν⎡ ⎤⎛ ⎞∂ ∂⎛ ⎞= +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

01 y Amixl y eκ

+ +−⎡ ⎤= −⎣ ⎦

Cl ffi i t

Outer layer:y x∂ ∂⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦

( )0

* ;T e v KlebU F yν α δ δ=

0 40 0 0168α1 2

26 1 dP dxA y−

+ ⎡ ⎤= +⎢ ⎥Closure coefficients: 0.40κ = 0.0168α = 226 1A y

Uτρ= +⎢ ⎥

⎣ ⎦

( )16

; 1 5.5KlebyF y δδ

−⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦( )*

01v eU U dy

δδ = −∫Comments:

Th k difi ti t i i l th d lThree key modifications to mixing length model; Applicable to wall-bounded flows;Applicable to 2D flows only;Not reliable for separated flows;

15

Not reliable for separated flows;difficult to determine in some cases;*, ,v eUδ δ

Turbulent models (0-eqn RANS) ( q )Baldwin-Lomax Model (Two-layer model)

,iT my yν ≤⎧⎪

⎨ h i h ll l f fy,i

o

T mT

T m

y y

y yν

ν⎪= ⎨ >⎪⎩

Where is the smallest value of for my yi oT Tν ν=

Inner layer: 2iT mixlν ω= 01 y A

mixl y eκ+ +−⎡ ⎤= −⎣ ⎦

Closure coefficients:Outer layer:

⎣ ⎦( )

0 max;T cp wake Kleb KlebC F F y y Cν α=

0.40κ = 0.0168α = 26A+ = 1.6cpC = 0.3KlebC = 1wkC =

Comments:A li bl t 3D fl

( )16

maxmax

; 1 5.5Kleb KlebKleb

yF y y Cy C

−⎡ ⎤⎛ ⎞⎢ ⎥= + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

1 22 2 2V U W V U Wx y y z z x

ω⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= − + − + −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Applicable to 3D flows;Not reliable for separated flows; No need to determine *, ,v eUδ δ

16

v e

Turbulent models (1-eqn RANS) ( q )Baldwin-Barth model

Kinematic eddy viscosity : 1 2T TC R D Dμν ν=y y

Turbulence Reynolds number :1 2T Tμ

( ) ( ) ( ) ( ) ( ) ( )21T T T TT

R R RU C f C R P

Rνν ν νν ν ν σ ν∂ ∂ ∂

+ + +∂∂

Closure coefficients and auxiliary relations:

( ) ( ) ( ) ( ) ( ) ( )2 2 1j T

k

T

kT

k kj

U C f C R Pt x x x x xε

ε ε εν ν ν σσ

+ = − + + −∂ ∂ ∂∂ ∂∂

1 2C 2 0C 0 09C 26A+ 10A+ = ( )1 CC C μ 0 41κ1 1.2Cε = 2 2.0Cε = 0.09Cμ = 0 26A+ = 2 10A = ( )1 2 2C C μ

ε εεσ κ

= − 0.41κ =

23

ji i k kT

UU U U UP ν⎡ ⎤⎛ ⎞∂∂ ∂ ∂ ∂= + −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎢ ⎥

01 1 y AD e

+ +−= − 22 1 y AD e

+ +−= −3T

j i j k kx x x x x⎜ ⎟∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦1

0 21 1 2 12 1 2 1 2

2 2 0 2

1 11 y A y AC C D Df D D D D e eC C y A AD D

ε ε

κ+ + + +− −

+ + +

⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞= + − + ⋅ + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

17

2 2 0 21 2C C y A AD Dε ε κ⎝ ⎠ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Turbulent models (1-eqn RANS) ( q )Spalart-Allmaras model

Kinematic eddy viscosity : fν ν=Kinematic eddy viscosity :

Eddy viscosity equation:

1T vfν ν=

2 1 cνν ν ν ν ν⎡ ⎤∂ ∂∂ ∂ ∂ ∂⎛ ⎞

Closure coefficients and auxiliary relations:

( ) 11 1

1 bj b w w

j kk kk

cU c S c ft x d xx x x

νν νσ

ν ν ν ν ννσ

⎡ ⎤∂ ∂+⎢ ⎥∂ ∂ ∂ ∂⎛ ⎞+ = − + +⎜ ⎟∂ ∂ ∂⎣ ∂⎦∂ ∂⎝ ⎠

1 0.1355bc = 2 0.622bc = 1 7.1vc = 2 3σ = ( )211 2

1 bbw

cccκ σ

+= +

2 0.3wc = 2 2.0wc = 0.41κ = νχ = ( )62wg r c r r= + − 2 2r

S dν

κ=2w 2w

3

1 3 31

vv

fc

χχ

=+ 2

1

11v

v

ff

χχ

= −+

1 663

6 63

1 ww

w

cf gg c

⎡ ⎤+= ⎢ ⎥+⎣ ⎦

χν ( )2wg 2 2S dκ

22 2 vS S fd

νκ

= + 2 ij ijS = Ω Ω

18

3w⎣ ⎦

Turbulent models (1-eqn RANS) ( q )

Comments on one-equation models:1.One-equation models based on turbulence kinetic energy are incomplete as q gy p

they relate the turbulence length scales to some typical flow dimension. They are rarely used.

2.One-equation models based on an equation for the eddy viscosity are complete such as Baldwin-Barth model and Spalart-Allmaras model.

3.They circumvent the need to specify a dissipation length by expressing the decay, or dissipation, of the eddy viscosity in terms of spatial gradients.

4 S l All d l di b l h B ld i B h4 Spalart-Allmaras model can predicts better results than Baldwin-Barth model, and much better results for separated flow than Baldwin-Barth model and algebraic models.

5 Also most of DES simulations are based on the Spalart Allmaras model5 Also most of DES simulations are based on the Spalart-Allmaras model.

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Turbulent models (2-eqn RANS) ( q )k-ε model: 2

T C kμν ε=

( )ij ij T k

j j j j

Uk k kUt x x x x

τ ε ν ν σ⎡ ⎤∂∂ ∂ ∂ ∂+ = − + +⎢ ⎥∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦j j j jt x x x x∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦

( )2

1 2i

j ij Tj j j j

UU C Ct x k x k x xε ε εε ε ε ε ετ ν ν σ

⎡ ⎤∂∂ ∂ ∂ ∂+ = − + +⎢ ⎥∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦

1 1 44C = 1 92C = 0 09C = 1 0σ = 1 3σ =,

( )C kω ε= 3 2l C k ε=1 1.44Cε 2 1.92Cε = 0.09Cμ 1.0kσ = 1.3εσ,

,

k-ω model:( )C kμω ε= l C kμ ε=

T kν ω=

( )* *ij ij T

Uk k kU kτ β ω ν σ ν⎡ ⎤∂∂ ∂ ∂ ∂+ = − + +⎢ ⎥∂ ∂ ∂ ∂ ∂⎢ ⎥

,

( )j ij Tj j j jt x x x x

β ⎢ ⎥∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦

( )2ij ij T

j j j j

UUt x k x x xω ω ω ωα τ βω ν σν

⎡ ⎤∂∂ ∂ ∂ ∂+ = − + +⎢ ⎥∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦131325

α =0 fββ β= *

* *0 f

ββ β= 1

2σ = * 1

2σ = 0

9125

β = 1 701 80

f ωβ

ω

χχ

+=+

3ij jk kiS

ωχΩ Ω

= * 9β = *2

1, 0

1 680k

fχ

χ≤⎧

⎪= +⎨ 3

1k

k ωχ ∂ ∂=

20

( )3*0

ωχβ ω 0 100

β = *

2

1 680 01 400

kk

k

fβ χ χχ

+⎨ >⎪ +⎩3k

j jx xχ

ω ∂ ∂

* kε β ω= 1 2l k ω=

Turbulent models (2-eqn RANS) ( q )

Comments on two-equation models:1 Two equation models are complete;1. Two-equation models are complete;2. k-ε and k-ω models are the most widely used two-

equation models and a lot of versions exist. For example, a , popular variant of k-ω model introduced by Menter has

been used in our research code CFDSHIP-IOWA. There are also a lot of low-Reynolds-number versions with different ydamping functions.

3. k-ω model shows better results than k-ε model for flows with adverse pressure gradient and separated flowsflows with adverse pressure gradient and separated flows as well as better numerical stability.

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Turbulent models (4-eqn RANS) ( q )v2f-kε model: 2

T C v Tμν =

( )2 2 2

tj j

Dv v vkfDt k x x

ε ν ν⎡ ⎤∂ ∂= − + +⎢ ⎥

∂ ∂⎢ ⎥⎣ ⎦

22 2 1

223

kPC vL f f CT k k

⎛ ⎞∇ − = − −⎜ ⎟⎜ ⎟

⎝ ⎠

,

j j⎢ ⎥⎣ ⎦ ⎝ ⎠

t

j k j

Dk kPDt x x

νε νσ

⎡ ⎤⎛ ⎞∂ ∂= − + +⎢ ⎥⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

2

1 2t

j j

D C P CDt k k x xε ε

ε

νε ε ε ενσ

⎡ ⎤⎛ ⎞∂ ∂= − + +⎢ ⎥⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

1 0,

, v2f-kω model:

1 0.4C = 2 0.3C = 0.3LC = 70Cη = 2 1.9Cε = 1.0εσ =( )

421 1.3 0.25 1 2LC C d Lε

⎡ ⎤= + +⎣ ⎦ max ,6kT νε ε

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

1 43 2 3

max ,LkL C Cη

νε ε

⎧ ⎫⎛ ⎞⎪ ⎪= ⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

12

nn

T C k vμν ω−

=,

v2f kω model: T C k vμν ω

( )2 2

26 tj j

Dv vkf vDt k x x

ε ν ν⎡ ⎤∂ ∂= − + +⎢ ⎥

∂ ∂⎢ ⎥⎣ ⎦( )

2 22 2

1 21 2 11 5

3kPv vL f C C

T k T k k

⎛ ⎞ ′∇ − = − − − −⎜ ⎟⎜ ⎟

⎝ ⎠

*i tij

j j k j

UDk kkDt x x x

ντ β ω νσ

⎡ ⎤⎛ ⎞∂ ∂ ∂= − + +⎢ ⎥⎜ ⎟∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

12

2

n

i tij

j j j

UD vDt k x k x xω

νω ω ωα τ βω νσ

−⎛ ⎞ ⎡ ⎤⎛ ⎞∂ ∂ ∂= − + +⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠

0.45977α = * 0.09β = 3 40β = 1.0kσ = 1.5ωσ =

22

β β k ω

max ,6kT νε ε

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

1 43 2 3

max ,LkL C Cη

νε ε

⎧ ⎫⎛ ⎞⎪ ⎪= ⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

Turbulent models (4-eqn RANS) ( q )

Comments on four-equation models:1 For two-equation models a major problem is that it is hard to1. For two-equation models a major problem is that it is hard to

specify the proper conditions to be applied near walls. 2. Durbin suggested that the problem is that the Reynolds number is

low near a wall and that the impermeability condition (zero normal ,

p y (velocity) is far more important. That is the motivation for the equation for the normal velocity fluctuation.

3. It was found that the model also required a damping function f, hence the name v2f model.

4. They appear to give improved results at essentially the same cost as the k-ε and k-ω models especially for separated flows. Hopefully the v2f-kω model can have better numerical stability than v2f-kε model as their counterparts behave in two-equation models.

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Turbulent models (7-eqn RANS) ( q )Some of the most noteworthy types of applications for which models based on the Boussinesq approximation fail are:

1. Flows with sudden changes in mean strain rate2. Flow over curved surfaces3. Flow in ducts with secondary motions4 Fl i t ti fl id4. Flow in rotating fluids5. Three-dimensional flows6. Flows with boundary-layer separation

In Reynolds-stress models, the equations for the Reynolds stress tensor are modeled and solved along with the ε-equation:

2ij ij j ijik ik jk

k

j ji ii j k

k k jk kik k

u uu u p p uU UU

t x xu

xu

x x x x x xτ τ τ

τ νρ ρ

τ ν′ ′∂′ ′ ′ ′∂ ∂ ∂ ′ ′ ′+

∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞∂ ∂+ = − − + + + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

24

Turbulent models (7-eqn RANS) ( q )These are some versions of Reynolds stress models:

1. LRR rapid pressure-strain model 2. Lumley pressure-strain model3. SSG pressure-strain model 4. Wilcox stress- model

Comments on Reynolds stress models:Comments on Reynolds stress models:1.Reynolds stress models require the solution of seven additional

PDEs and those equation are even harder to solve than the two-equation models.models.

2. Although Reynolds stress models have greater potential to represent turbulent flow more correctly, their success so far has been moderate.

3. There is a lot of current research in this field, and new models are often proposed. Which model is best for which kind of flow is not clear due to the fact that in many attempts to answer this question numerical

25

errors were too large to allow clear conclusions to be reached.

Turbulent models (LES)

• Large scale motions are generally much more energetic than the small scale ones.

• The size and strength of large scale motions make them to be the most effective transporters of the conserved properties.the most effective transporters of the conserved properties.

• LES treats the large eddies more exactly than the small ones may make sense

• LES is 3D, time dependent and expensive but much less costly than DNS

26

than DNS.

Turbulent models (LES, filtering)• LES needs a velocity field that contains only the large scale

components of the total field, which is achieved by filtering the velocity field (Leonard 1974)the velocity field (Leonard, 1974)

( ) ( ) ( ) ξξξ duxGxu ii ∫ −=• G(x-ξ) is the filter kernel, is a localized function, which

includes a Gaussian, a box filter (a simple local average) and a cutoff (a filter which eliminates all Fourier coefficients belonging to wavenumbers above a cutoff)belonging to wavenumbers above a cutoff)

• Each filter has a length scale associated with it, Δ.• Eddies of size large than Δ are large eddies while those

smaller than Δ are small eddies and need to be modeled. ⎧

( )1

0G x ξ

⎧⎪− = Δ⎨⎪⎩

if2Δ≤−ξx

otherwise( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ−−

⎟⎠⎞

⎜⎝⎛

Δ=− 2

221

2 expξγ

πγξ

xxG ( ) ( )( )

( )ξξξ

−−=−

xkxkxG

c

csin

Δ= π

ck

27

Box or top-hat filter Gaussian filterΔc

Cutoff filter

Turbulent models (LES, Governing Equations)g q )

• Filtered Navier-Stokes equations (constant density, incompressible):

( )∂ ρ

( ) ( )⎥⎤

⎢⎡

⎟⎞

⎜⎛ ∂∂∂∂∂∂ jji uupuuu ρρ

( ) 0=∂

∂

i

i

xuρ

( ) ( )⎥⎥⎦⎢

⎢⎣

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

∂∂+

∂∂−=

∂∂

+∂

∂

i

j

j

i

jij

jii

xu

xu

xxp

xuu

tu μ

ρρ

• Note that: jiji uuuu ≠ jiji uuuu ≠

( )jijiSij uuuu −−= ρτIntroducing Subgrid-scale

Reynolds Stress

• The filter width Δ> grid size h• The models used to approximate the SGS Reynolds

stress are called subgrid-scale (SGS) or subfilter-scale d l

28

models.

Turbulent models (LES, Smagorinsky model)• The earliest and most commonly used subgrid scale

model is one proposed by Smagorinsky (1963), which is an eddy viscosity model.

• As the increased transport and dissipation are due to the viscosity in laminar flow, it seems reasonable to assume that

jiSS Suu μμδττ 21 ⎟

⎞⎜⎛ ∂

+∂ijt

i

j

j

itji

Skk

Sji S

xxμμδττ 2

3 ,, =⎟⎟⎠

⎜⎜⎝ ∂

+∂

=−

Strain rate of the large scale or resolved fieldijS22

Eddy viscosity tμ SCSt22 Δ= ρμ

2.0≈SC Model constant• D b k 1 C i t t t• Drawbacks: 1. Cs is not constant

2. Changes of Cs are required in all shear flows3. Need to be reduced near the wall.4 Not accurate for complex and/or higher

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4. Not accurate for complex and/or higherReynolds number flows.

Turbulent models (LES, Scale-similarity model and D namic model)and Dynamic model)

• Dynamic model: 1. filtered LES solutions can be filtered again using a filter g gbroader than the previous filter to obtain a very large scale field.2. An effective subgrid-scale field can be obtained by subtraction of the two fields.subtraction of the two fields.3. model parameter can then be computed.Advantages: 1. model parameter computed at every spatial grid point and every time step from the results of LES

lf b d l d l2. Self-consistent subgrid-scale model3. Automatically change the parameter near the wall and in shear flowsDisadvantages: backscatter (eddy viscosity<0) may causeDisadvantages: backscatter (eddy viscosity<0) may cause instability.

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Turbulent models (DES)Turbulent models (DES)

• Massively separated flows at high Re usually involveboth large and small scale vortical structures and veryboth large and small scale vortical structures and verythin turbulent boundary layer near the wall

• RANS approaches are efficient inside the boundarylayer but predict very excessive diffusion in the separatedlayer but predict very excessive diffusion in the separatedregions

• LES is accurate in the separated regions but is unaffordablef l i thi ll t b l t b d l tfor resolving thin near-wall turbulent boundary layers atindustrial Reynolds numbers

• Motivation for DES: combination of LES and RANS. RANSi id tt h d b d l d LES i th t dinside attached boundary layer and LES in the separatedregions

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DES Formulation • Modification to RANS models was straightforward by substituting the length scale , which is the distance to the closest wall with the new DES length scale defined as:l

wd

closest wall, with the new DES length scale, defined as:

• h i th DES t t i th id i d i

l

min( , )w DESl d C= Δ max( , , )x y zδ δ δΔ =

C Δ• where is the DES constant, is the grid spacing and is based on the largest dimensions of the local grid cell, and are the grid spacing in x, y and z coordinates respectively

DESC Δ, ,x y zδ δ δ

• Applying the above modification will result in S-A Based, standard k-ε (or k-ω) based and Menter’s SST based DES models etc

* 3 2

3 2

/kRANS k

k

D k k l

kωρβ ω ρ −= =

models, etc.

1 2 *( )kl kω β ω− =

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3 2kDES

kDl

ρ= min( , )k DESl l Cω−= Δ

Resolved/Modeled/Total Reynolds stress (DES)stress (DES)

TKE

ModeledModeled

ResolvedTotal

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Modeled Reynolds stress: ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂+

∂∂−=

xv

yuvu tν''

Resolved/Modeled/Total Reynolds stress (URANS)stress (URANS)

Modeled TotalModeled Total

Resolved

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Deep insight into RANS/DES (Point oscillation on free surface with power spectral analysis)

EFD DES URANS

-5/3

352HZ

Direct Numerical Simulation (DNS)• DNS is to solve the Navier-Stokes equation directly

without averaging or approximation other than numerical discretizations whose errors can be estimated (V&V) and

t ll dcontrolled.• The domain of DNS must be at least as large as the

physical domain or the largest turbulent eddy (scale L)• The size of the grid must be no larger than a viscouslyThe size of the grid must be no larger than a viscously

determined scale, Kolmogoroff scale, η• The number of grid points in each direction must be at

least L/ η• Th t ti l t i ti l t• The computational cost is proportional to

• Provide detailed information on flow field

( ) 434/3 Re01.0Re ≈L

• Provide detailed information on flow field• Due to the computational cost, DNS is more likely to be a

research tool, not a design tool.

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Examples (Diffuser)• Asymmetric diffuser with

separation is a good test case for turbulence models.

• A inlet channel was added at• A inlet channel was added at the diffuser inlet to generate fully developed velocity profile

• Boundary layer in the lower diffuser wall will separate duediffuser wall will separate due to the adverse pressure gradient.

• Results shown next include i b t V2f dcomparisons between V2f and

k-ε• LES simulation of this geometry

can be found in: M. Fatica, H. J. Kaltenbach, and R. Mittal, “Validation of LES in a Plain Asymmetric Diffuser”, center for turbulence research

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center for turbulence research, annual research briefs, 1997

Examples (Diffuser)

• Mean velocity predicted by V2f agreed very well

ith EFD d twith EFD data, particular the separation region isregion is captured.

• K-ε model fails to predict theto predict the separation caused by adverse pressure gradient.

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Examples (Diffuser)v2f

k-ε

• TKE predicted by V2f agreed better with EFD data than k- ε model, particular the asymmetric distribution

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asymmetric distribution.• Right column is for the skin friction coefficient

on the lower wall, from which the separation and reattachment point can be found.

2x/H

References

1. J. H. Ferziger, M. Peric, “Computational Methods for FluidDynamics,” 3rd edition, Springer, 2002.

2 D C Wilcox “Turbulence Modeling for CFD” 19982. D. C. Wilcox, Turbulence Modeling for CFD , 1998.3. S.B.Pope, “Turbulent Flows,” Cambridge, 20004. P. A. Durbin and B. A. Pettersson Reif, “Statistical Theory and

Modeling for Turbulent Flows,” John Wiley & Sons, LTD, 2001.5. P.A. Davidson, “Turbulence: An Introduction for Scientists

and Engineers,” Oxford, 2004.g , ,

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