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© 2006 ANSYS, Inc. All rights reserved. 1 ANSYS, Inc. Proprietary

Near-Wall Modelling: FLUENT Capabilities

Dr Aleksey GerasimovSenior CFD EngineerFluent Europe LtdOctober 2006


Modelling the Turbulent Transport

Mean Flow:

TwoEquation Turbulence Models ReynoldsStress Turbulence Closures LargeEddy Simulation

NearWall Flow:

LowReynoldsNumber Approach Wall Functions

Modelling of the mean flow & nearwall behaviour are equally important in complex industrial flows

The thin viscous sublayer is very influential in determining wall friction and wall heat transfer in turbulent flows


Near-Wall Flow Phenomena

The treatment of wall boundaries requires particular attention in turbulence modelling due to the presence of the viscositydominated region which is responsible for extremely sharp gradients of mean and turbulent flow variables

DNS data (Kim et al) DNS data (Kim et al)


Importance of the Importance of the Near-Wall TurbulenceNear-Wall Turbulence

• Walls are main source of the shear & vorticity and are responsible for the turbulence generation and dissipation mechanisms

• Accurate nearwall modelling is important for most engineering applications:

– Successful prediction of frictional drag for external flows, or pressure drop for internal flows, depends on fidelity of local wall shear predictions

– Pressure drag for bluff bodies is dependent upon extent of separation

– Thermal performance of heat exchangers, etc., is determined by wall heat transfer whose prediction depends upon nearwall effects


Importance of the Importance of the Near-Wall Turbulence (2)Near-Wall Turbulence (2)

dP/dx=0 dP/dx<0 dP/dx>0

• Flows under pressure gradients, flow separation


• Buoyancyassisted and buoyancyopposed flows

Importance of the Importance of the Near-Wall Turbulence (3)Near-Wall Turbulence (3)

q wal

l

(a)

q wal

l

(b)

Examples of the velocity profiles distortions


Structure of Turbulent Boundary Layer

Near-Wall Modelling Experimental Evidence


Near-Wall Modelling IssuesNear-Wall Modelling Issues

• HighReynolds Number (HRN) kε and RSM models have been derived for free shear flows and are valid in the mean part of the turbulent flow away from walls– Some of the modelled terms in these equations are based on

isotropic behavior• Isotropic diffusion ( µt /σ )• Isotropic dissipation• Pressurestrain redistribution• Some model parameters based on experiments of isotropic

turbulence– Nearwall flows are anisotropic due to the presence of walls

• Special nearwall treatments are necessary since equations cannot be resolved down to the walls in their original form due to the strong gradients in mean and turbulent quantities


Flow Behaviour in the Near-Wall Region Flow Behaviour in the Near-Wall Region Equilibrium Flows (1)Equilibrium Flows (1)

Concept of local equilibrium:

Turbulence Production = Turbulence Dissipation

kU/uτ

Examples: Flow over a Flat Plate & Fully Developed Pipe Flow


Flow Behaviour in the Near-Wall Region Flow Behaviour in the Near-Wall Region Equilibrium Flows (2)Equilibrium Flows (2)

• Velocity profile exhibits layer structure identified from dimensional analysis– Inner layer: viscous forces rule, U = f(ρ, τw, µ, y)– Outer layer: dependent upon mean flow– Overlap region: Loglaw applies

• k production and dissipation are nearly equal in overlap layer– ‘turbulent equilibrium’

• dissipation >> production in sublayer region


• In many circumstances, even far enough from the wall for viscous stresses to be negligible, one still finds a region where the local mean velocity U depends only on

U = f(ρ, τw, μ, y)

• Note, velocity is not dependent on the channel width D or boundary layer thickness δ; nor on “upstream history” (i.e. convection effects)

• Since U = f(ρ, τw, μ, y) the Buckingham theorem tells us there are:

5 (variables) – 3 (basic dimensions: M, L, T) = 2 dimensionless groups

usually presented as:

• The quantity evidently has the dimension of velocity and it is usually written as Uτ the friction velocity

Near-Wall Distribution of Mean Velocity

U

τ w/ρ= f yτw /ρ

ν τw/ρ

‘Law of the wall’


Near-Wall Distribution of Mean Velocity (2)

• Now define U + = U / Uτ , y + = y Uτ / ν and obtain a universal relationship for the nearwall flows

U+ = f ( y+ )• While written especially for the turbulent region, it applies also to the

viscous sublayer if

f ( y+ ) = ρyUτ /μ = y+

or

• In the fully turbulent region, while U certainly depends on μ, the difference in velocity between adjacent layers should not. Thus

should be independent of μ

UUτ

=y Uρ τ

Uy= Uρ τ

2=τw

∂U∂ y

=U τdf

dydy

dy=U τ

U τ ρ df

dy


Near-Wall Distribution of Mean Velocity (3)

• This can only happen if:

• After integration wrt y +

• The edge of the viscous sublayer is where the laminar and the turbulent regions meet

• With κ = 0.42 and C = 5.45

• y + < 5 – laminar region

• y + > 35 – fully turbulent logarithmic region

df

dy=Const⋅ y Uρ τ

−1

f =U=1κ

ln yC

yv Uτ

ν=

1κ

ln yvU τ

ν C

yv=

yv Uτ

ν≈11


Near-Wall Distribution of Temperature

• Reynolds' analogy between momentum and energy transport gives a similar logarithmic law for the mean temperature

• The usual thermal logarithmic law is

• It is possible to replace the thermal loglaw with the help of the hydrodynamic one

T=σT U

Π Π

yv

yP

yn

P

Velocity loglaw

yθ

yP

yn

P

Temperature loglaw

T=1χ

ln yCθ Cθ= f Pr = f σ

Pee function by Jayatilleke

Pr ~δδ T

~yv

yθ


Flow Behaviour in the Near-Wall Region Flow Behaviour in the Near-Wall Region Non-Equilibrium Flows (1)Non-Equilibrium Flows (1)

• Equilibrium flows are very rare• Great majority of flows are nonequilibrium

Examples of NonEquilibrium Flows

– Flows developing under pressure gradients– Buoyant flows

dP/dx=0 dP/dx<0 dP/dx>0


• Two distinct modelling strategies exist for the nearwall region

– LowReynoldsnumber approach (fine mesh)– Wallfunction approach (coarse mesh)

• The choice of modelling strategy depends on the application, the engineering question, available resources…

Boundary Layer Wallfunction MeshLRN Mesh

Near-Wall Modelling OptionsNear-Wall Modelling Options


Low-Reynolds-Number Modelling

• Fine nearwall grid

• Includes nearwall damping terms

+ Accurate (using appropriate turbulence model)

− Slow convergence due to elongated cells− High storage requirements

Too expensive for most industrial applications


Conventional Wall Functions

• Coarse nearwall grid: first node in fullyturbulent region (y+≥30)• Based on prescribed U and T profiles in nearwall cell

+ Fast solution (at least 10x faster than LRN)

+ Low storage requirements

− Poor results in nonequilibrium flows− Nearwall cell size dependency− Does not take into account such effects as:

Insufficient breadth of applicability

pressure gradients, convective transport andgravitational forces


Near-Wall Modeling Options in FLUENT Near-Wall Modeling Options in FLUENT Wall-Function ApproachWall-Function Approach

τw=∂U∂ y

• Wall functions provide boundary conditions for momentum, energy, species and turbulent quantities

• The Standard and Nonequilibrium Wall Functions(StWF and NEWF) use the law of the wall

• Original flow equations are modified in accord with the chosen lawofthewall

yv

yP

yn

P

yv

yP

yn

P

Velocity profile Temperature profile

qw=−k∂T∂ y

Wall Shear Stress Fick’s Law


Near-Wall Modeling Options in FLUENT Near-Wall Modeling Options in FLUENT Wall-Function Approach (2)Wall-Function Approach (2)

DDt

Uρ =∂

∂ y [ t ∂U∂ y ]SU SU ~τw= f UP , k p , yP

DDt

Tρ =∂

∂ y [σ t

σ t∂T∂ y ]ST ST ~ qw= f ' τw , T w , k p , UP

DDt

kρ =∂

∂ y [ t

σk ∂ k∂ y ]G k−ρε G k⇒Gk= f ' ' τw , k p , y n , yv ε ⇒ ε

DDt

ρε =∂

∂ y [ t

σε ∂ε∂ y ]C 1ε

εk

G k− Cρ 2ε

ε2

k εP=

k P3/ 2

l=

kP3/2

c l yP

C =0 . 09 , C 1ε =1. 44 , C 2ε =1. 92 , σ k=1. 0, σε=1. 3

• Streamwise momentum equation:

• Energy equation:

• Turbulent kinetic energy equation:

• Turbulent energy dissipation rate equation:

NOT SOLVED

In the NearWall Row of Cells:


• FLUENT uses LaunderSpalding Wall Functions– U = U(ρ, τ, µ, y, k )

• Introduces additional velocity scale for ‘general’ application

• Similar ‘wall laws’ apply for energy and species.– Generally, k is obtained from solution of k transport equation

• Cell center is immersed in log layer• Local equilibrium (production = dissipation) prevails∀ ∇k∙n = 0 at surface

– ε calculated at walladjacent cells using local equilibrium assumption

• ε = Cµ3/4k3/2/κy

– Wall functions less reliable when cell intrudes viscous sublayer• Forcing (Production = Dissipation) over (Production << Dissipation)

U ¿≡UP C 1/4 k P

1/ 2

τw /ρy¿≡

ρ C 1/ 4 k P1/2 y Pwhere

U ¿=

1κκ

ln Ey ¿U ¿

=y¿

fory¿yv

¿

y¿yv¿

Standard Wall FunctionsStandard Wall Functions


Wall functions become less reliable when flow departs from the conditions assumed in their derivation:

– Local equilibrium assumption fails• Severe ∇p• Transpiration through wall• strong body forces• highly 3D flow• rapidly changing fluid properties near wall

– LowRe flows are pervasive throughout model– Small gaps are present

Limitations of Standard Wall FunctionsLimitations of Standard Wall Functions


• Loglaw is sensitized to pressure gradient for better prediction of adverse pressure gradient flows and separation

• Relaxed local equilibrium assumptions for TKE in wallneighboring cells

U C1/4 k1/2

τ w/ρ=

1κ

ln E ρC1/4 k1/2 y U=U−

12

dpdx [ y

v

ρκ k1/2ln y

yv y−y

v

ρκ k 1/2

yv

2

]

Rij, k, ε are estimated in each region and used to determine average ε and production of k

where

Non-Equilibrium Wall FunctionsNon-Equilibrium Wall Functions


• within thermal viscous sublayer

• in the fully turbulent region

Thermal Wall Functions

T ¿≡T w−T P ρ c p C

14 k P

12

q

T ¿=Pr y¿12ρ Pr

C1/4 k P1/2

qw

U P2

Temperature loglaw

yθ

yP

yn

P

T=σT U

Π

Similar to the hydrodynamic wall functions thermal wall functions in Fluent are based on y* , U* , T* rather than y+ , U+ , T+

T ¿=Prt [ 1κ

ln Ey¿ Π ] 12ρ

C1

4 kP

12

q [ Prt UP2 Pr−Pr t U c

2 ]

terms account for heating due to viscous dissipation (supersonic flows)


• Hydrodynamic loglaw is not valid when the flow departs from the state of local equilibrium

• Thermal loglaw is expressed in terms of hydrodynamic one and inevitably suffers from the same limitations

• Plus there is a dependence on molecular and turbulent Prandtl numbers

• Example: Prandtl number for water ranges from 10 to 1.5 when the temperature changes from 0°C to 100°C

Thermal Wall Functions - Limitations

T=PrT U

Π

Π=9 . 24 {PrPr T

3/4

−1}[10 . 28 exp −0 . 007PrPrT ]


Thermal Wall Functions - Limitations (2)

100 101 102

Y+

0

2

4

6

8

10

12

14

16

18

20

U+

x = 6.25 m, Re=17561 , Grq=3.89e+9 , Bo=0.205x = 7.50 m, Re=17682 , Grq=4.00e+9 , Bo=0.208x = 8.75 m, Re=18031 , Grq=4.27e+9 , Bo=0.211x = 10.0 m, Re=18339 , Grq=4.52e+9 , Bo=0.214log law

LRN: q = 9 kW/m2, Re = 16000

100 101 102

Y+

0

25

50

75

100

125

150

175

200

225

250

T+

x = 6.25 m, Re=17561 , Grq=3.89e+9 , Bo=0.205x = 7.50 m, Re=17682 , Grq=4.00e+9 , Bo=0.208x = 8.75 m, Re=18031 , Grq=4.27e+9 , Bo=0.211x = 10.0 m, Re=18339 , Grq=4.52e+9 , Bo=0.214T+=(0.42/0.46)*(π+U+)

LRN: q = 9 kW/m2, Re = 16000q wal

l

(a)

q wal

l

(b)

• Buoyancyassisted flow in a pipe – distortion of the velocity profile

– distortion of the temperature profile



BuoyancyAided Upflow of Air in Heated Pipes

Re=15000

(Experimental data of J. Li, 1994)

50 100 150x/d

0

20

40

60

80

100

Nu

Exp.data of Li (1994)y*

n=50yn

*=75y*

n=100yn

*=150Nu=0.023 Pr0.333 Re0.8

LRN Calculation

Inlet: Re=15023, Gr=2.163*108, Bo=0.1124

(a)

LRN terms included

50 100 150x/d

0

20

40

60

80

100

120

140

160

180

200

220

240

Nu

y*n=48

yn*=100

y*n=250

yn*=500

Nu=0.023 Pr0.333 Re0.8

LRN Calculation

Inlet: Re=49635, Gr=1.745*109, Bo=0.0156

(b)

LRN terms included

q wal

l

(a)

q wal

l

(b)

© 2006 ANSYS, Inc. All rights reserved. 28 ANSYS, Inc. Proprietary0.2 0.4 0.6 0.8 1.0

(ry)/r

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

U/U

bulk

neutralmoderate heatingsevere heating

(a)

0.2 0.4 0.6 0.8 1.0(ry)/r

0.005

0.0025

0

0.0025

0.005

ρuv/

(ρU

2 bulk)

neutralmoderate heatingsevere heating

(b)

Buoyant effects cause significant changes in Heat Transfer

q wal

l

(a)

q wal

l

(b)

Ability of buoyant forces to modify turbulent shear stress

Consequential changes in the turbulence production

Altered turbulent diffusivity

Impaired or enhanced heat transfer

Dominant Factor:



Near-Wall Modelling Options in FLUENT Near-Wall Modelling Options in FLUENT EEnhanced nhanced WWall all TTreatment (EWT)reatment (EWT)

• Enhanced Wall Treatment is a ‘QuasiLRN’ approach which, ideally, requires fine grid

– Combines the use of a twolayer zonal model and blended lawofthe wall. Thus coarser mesh is less of a problem than in LRN models

– Suitable for lowRe flows or flows with complex nearwall phenomena as it employs fine mesh and directly resolves turbulence variables all the way to the wall

– Turbulence models are modified for the inner layer

– Generally requires a fine nearwall mesh capable of resolving the viscous sublayer (more than 10 cells within the inner layer)

– The EWT is an option for the kε and RSM turbulence models


EEnhanced nhanced WWall all TTreatment (EWT)reatment (EWT)Two-Layer ModelTwo-Layer Model

• A blended twolayer model is used to determine nearwall ε field:– Domain is divided into viscosityaffected (nearwall) region and

turbulent core region– High Re turbulence model used in outer layer– ‘Simple’ turbulence model used in inner layer

• Solutions for ε and μt in each region are blended

The two regions are demarcated on a cellbycell basis in a dynamic, solutionadaptive way:

– Turbulent core region y* > 200– Viscosity affected region y* < 200

Where y* = ρk1/2y/μ and y is the shortest distance to a nearest wall


Models used in two-layer zonesModels used in two-layer zones

• In the turbulent region, the selected highRe turbulence model is used

• In the viscosityaffected region, a oneequation model is used

– k – equation is same as highRe model– Length scale used in evaluation of µt is not from ε

• μt = ρ Cμ k1/2 lμ

• lµ = cl y (1exp(y* /Aµ ))• cl = κ Cµ

3/4

– Dissipation rate, ε, is calculated algebraically and not from the transport equation• ε = k3/2 / lε • lε = cl y (1exp(y* /Aε ))

– The two ε fields can be quite different along the interface in highly nonequilibrium turbulence


Two-Layer Zonal Model

• The k – equation is solved throughout the whole domain, but ε – field is determined by two different formulations

• The two ε – fields can be quite different along the interface in highly nonequilibrium conditions

Wall

Inner layer

Outer layer

ε=k

32

ε

ε=k

32

ε

DDt

ρε =¿⋅¿

y¿200

y¿≥200


Blended Blended εε – – equationsequations

• The transition (of ε – field) from one zone to another can be made smoother by blending the two sets of ε – equations (Jongen, 1998)

ε=k

32

ε

Wall

Inner layer

Outer layer ρDεDt

=¿⋅¿ aPεP∑nb

anb εnb=S ε P

εP=

k

P

32

ε

λε × [aPεP∑nb

anb εnb=Sε P ] 1−λ ε × [εP=k P

3/ 2

ε ]with λ ε=

12 [1tanh Rey−Rey

¿

A ]


Blended Turbulent ViscosityBlended Turbulent Viscosity

• Turbulent viscosity, μt , is also blended using the individual

formulations λ ε t outer

1−λ ε t inner

t outer=ρ Ck2

ε, t inner=ρ C k , =c y [1−exp − Rey

A ]

ε=k

32

ε

Wall

Inner layer

Outer layer t outer=ρ Ck2

ε

t inner=ρ C k


EEnhanced nhanced WWall all TTreatment (EWT)reatment (EWT)Enhanced Wall FunctionsEnhanced Wall Functions

• Momentum boundary condition based on blended lawofthewall (Kader):

• Similar blended ‘wall laws’ apply for energy, species, and ω

• Kader’s form for blending allows for incorporation of additional physics

– Pressure gradient effects – Thermal (including compressibility) effects

inner layer

outer layer

u=eΓ u lam e

1Γu turb


Blended HydrodynamicLaw-of-the-Wall

• Mean velocity

• Blended ‘wall laws’ for temperature and species as well

u=eΓ u lam e

1Γu turb

where u lam

= y

u turb

=1κ

ln E y

u lam =y

u turb =

1κ

ln E y

Γ=−a y

4

1by, a=0 . 01 c ,

b=5c

, c=exp E' '

E−1


Blended ThermalLaw-of-the-Wall

• Mean temperature

t=eΓ t lam e

1Γ t turb

Γ=−a σ y

4

1b σ3 y, a=0 . 01 c , b=

5c

, c=exp E ' '

E−1

where

t≡T−T w cρ pu¿

qw' '

t lam

=σ y

t turb =σ t u turb

P

P=9 . 24[σσ t 3/4

−1] [10 . 28 exp −0 . 007σσ t ]


‘‘Wall-Law’ Wall-Law’ Sub-models and OptionsSub-models and Options

• “Pressure Gradient Effects” option– Always available

deactivated by default

• “Thermal Effects” option

– Available only when energy equation is turned on deactivated by default

– Accounts for• Nonadiabatic wall heat

transfer effects• Compressibility effects

takes effect when idealgas option is chosen


‘‘Wall-Law’ Wall-Law’ Sub-models and Options (2)Sub-models and Options (2)

• The base lawsofthewall (mean velocity and temperature) are modified using (White and Christoph, 1972) :

― “Pressure Gradient Effects” option

― “Thermal Effects” option

τ≃τwdpdx

y , α≡νw

τw uτ

dpdx

ρw

ρ≃1− uβ

− uγ

2¿ , β≡

σ t qw' ' uτ

Cpτw T wheat transfer parameter

, γ≡σt uτ

2

2Cp T wcompressibility parameter

¿

pressure gradient

effects

1 yα underbracealignl ¿¿

yβ − yγ 2

¿

thermal effects

1−¿

¿ ¿ ]1

2

¿

du

dy=

1

yκ ¿

¿


Enhanced Wall TreatmentWall Boundary Conditions

• Zero normalgradient is applied to TKE at walls

• Other turbulence quantities at walladjacent cells are computed, whenever possible, using the blended twolayer formulation– Production of TKE at walladjacent cells is computed using the velocity

gradient given by the blended lawofthewall for mean velocity

• Dissipation rate, ε, at wall cells is computed using the inner layer formula

• EWT is the only nearwall option available for the kω models

∂ k∂ n

=0


• FullyDeveloped Channel Flow (Ret = 590)

– For fixed pressure drop cross periodic boundaries, different nearwall mesh resolutions yielded different volume flux as follows

– The enhanced nearwall treatment gives a much smaller variation for different nearwall mesh resolutions compared to the variations found using standard wall functions

Enhanced Wall TreatmentEnhanced Wall TreatmentSome ResultsSome Results


• For standard or nonequilibrium wall functions, each walladjacent cell’s centroid should be located within:

• For the enhanced wall treatment (EWT), each walladjacent cell’s centroid should be located: – Within the viscous sublayer, , for the twolayer zonal

model:– Preferably within for the blended wall

function

• How to estimate the size of walladjacent cells before creating the grid:– ,

– The skin friction coefficient can be estimated from empirical correlations:

uτ≡τw /ρ=U ec f / 2

y p≈30−300

y p≈1

y p≡ y p uτ/ν ⇒ y p≡y p

ν /uτ

Placement of the Near-Wall Grid PointPlacement of the Near-Wall Grid Point

y p≈30−300


• Use StWF or NEWF in highRe applications (Re > 106) where you cannot afford to resolve the viscous sublayer

– Use NEWF for mildly separating, reattaching, or impinging flows

• You may consider using EWT if:

– Nearwall characteristics are important.– The physics and nearwall mesh of the case is such that y+ is

likely to vary significantly over a wide portion of the wall region.

• Try to make the mesh either coarse or fine enough to avoid placing the walladjacent cells in the buffer layer (y+ = 5 ~ 30)

Near-Wall Modelling: Near-Wall Modelling: Recommended StrategyRecommended Strategy


• Original ε – equation for HRN flows

• Std. kε model modified by damping functions:

• k and ε equations solved on fine mesh (required) right to the wall

DDt

ρε =∂

∂ x j [ t

σ ε ∂ε∂ x j ] f 1⋅C 1ε

εk

G k− f 2⋅ Cρ 2εε2

k

ε ε −− transport equationtransport equation

turbulent viscosityturbulent viscosityt=ρ f C

k 2

ε

f , f 1 , f 2

Near-Wall Modelling: Near-Wall Modelling: Low-Reynolds-Number ModellingLow-Reynolds-Number Modelling

Damping Function LRN Models

DDt

ρε =∂

∂ x j [ t

σ ε ∂ε∂ x j ] C 1ε

εk

G k− Cρ 2εε2

k


Typical damping functionsTypical damping functions

• Damping functions written in terms of Reynolds numbers:

• e.g., Abid’s model:

ℜt =kρ 2

με; ℜy=

ρ k yμ

; ℜε=ρ με /ρ

1/ 4 yμ

f = tanh0 . 008 Rey 14 Ret−3 /4

f 1=1

f 2=[1−29

exp −Ret2

36 ][1−expRey

12 ]


Low-Re Model Fundamentals

• Most models try to achieve asymptotic consistency

• The models usually end up looking like

U∂ k∂ x

V∂ k∂ y

=ν t ∂U∂ y

2

−ε∂

∂ y [ν ν t

σk ∂ k∂ y ]

u≈A x , z ,t yO y2

v≈B x , z , t y2O y3

w≈C x , z , t yO y2

as y 0k≈

1

2 A2C2 y2O y3

ε≈ν A2C 2O y

τxy≈AB y3O y4

U∂ ε∂ x

V∂ ε∂ y

=C 1ε f 1

εkν t ∂U

∂ y 2

−C 2ε f 2

ε2

kE

∂

∂ y [ν ν t

σ ε ∂ ε∂ y ]

where ε=ε0ε and ν t=C f k2

/ ε


LRN kε Models

• Several full lowRe kε models now available (not documented or officially supported)– Lam Bremhorst– LaunderSharma– Abid– Chang et al.– AbeKondoNagano

• Enables modeling of lowRe effects including transitional flows

• Features not visible in GUI, but must be accessed via TUI


Two-Layer Near-Wall RSM Model

• Twolayer zonal modeling approach combining

– Launder & Sharma’s lowRe model

– Wolfstein’s oneequation model

• RSM can be used with fine nearwall meshes

• RSM can be used for lowRe turbulent flows

DkDt

=∂

∂ x i t

σ k

∂ k∂ x i t

∂U i

∂ x j ∂U i

∂ x j

∂U j

∂ x i −C Dρk 3/ 2


Near-Wall Adjustments in RSM models

In secondmomentclosures explicit transport equations for the Reynolds stresses are solved:

Du i u j

Dt=PijGijφijd ij−εij

Two different models of this type are implemented in Fluent

• ‘Basic’ secondmomentclosure model

• Quadratic pressure strain model

φ ij=φ ij1φij 2φ ijw

Pressure Strain term:


• Wall reflection term

• If EWT option is chosen and grid is fine then LRN modifications are applied to all three pressure redistribution terms

• Quadratic pressure strain model does not require wallreflection terms

• Quadratic pressure strain model cannot be employed when EWT is used

Near-Wall Adjustments in RSM models (2)

φij1 φ ijw

φ ij1 , φij 2 , φ ijw= f ReT , a ij


• Durbin (1990) suggests that wall normal fluctuations, , are responsible for nearwall transport

• behaves quite differently than and – attenuation of is a kinematic effect– damping of is a dynamic effect

• Model instead of

• Requires two additional transport equations:– equation for wallnormal fluctuations, – equation for an elliptic relaxation function, f

v 2

u2v 2 w2

t ~ v 2 T t ~ kT

v 2

u2

v 2

VV22F Low-Re F Low-Re k-k-εε ModelModel


ρDεDt

=∂

∂ x j [ t

σε ∂ ε∂ x j ] 1

T C1ε t S 2−ρC 2εε

εε−−transporttransport equationequation

ρD v2

Dt=

∂

∂ x j [ t

σk ∂ v2

∂ x j ] kfρ −ρ v2 εk

vv22 transporttransport equationequation

f −L2 ∂2 f

∂ x j∂ x j

=C1

T 23−

v2

k C2t S2

kρ

N−1 v2

kT

relaxation equationrelaxation equation

T=max [ kε

, 6 νε ] ; L2=C L

2 max [ k3

ε2 ,Cη ν3

ε ]scalesscales

VV22F F kkεε model equationsmodel equations


• Very promising results for a wide range of flow and heat transfer test cases– at least as good as the best of the damping

function approaches in most test cases

• Still an isotropic eddyviscosity model– Can be extended for RSM

• Needs 2 additional equations, so requires more memory and CPU than damping functions

VV22F F kkεε Model Pro’s and Con’sModel Pro’s and Con’s


Thank You

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