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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
© 2006 ANSYS, Inc. All rights reserved. 2 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 3 ANSYS, Inc. Proprietary
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)
© 2006 ANSYS, Inc. All rights reserved. 4 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 5 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 6 ANSYS, Inc. Proprietary
• 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
© 2006 ANSYS, Inc. All rights reserved. 7 ANSYS, Inc. Proprietary
Structure of Turbulent Boundary Layer
Near-Wall Modelling Experimental Evidence
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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
© 2006 ANSYS, Inc. All rights reserved. 9 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 10 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 11 ANSYS, Inc. Proprietary
• 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’
© 2006 ANSYS, Inc. All rights reserved. 12 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 13 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 14 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 15 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 16 ANSYS, Inc. Proprietary
• 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
© 2006 ANSYS, Inc. All rights reserved. 17 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 18 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 19 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 20 ANSYS, Inc. Proprietary
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:
© 2006 ANSYS, Inc. All rights reserved. 21 ANSYS, Inc. Proprietary
• 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
© 2006 ANSYS, Inc. All rights reserved. 22 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 23 ANSYS, Inc. Proprietary
• 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
© 2006 ANSYS, Inc. All rights reserved. 24 ANSYS, Inc. Proprietary
• 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)
© 2006 ANSYS, Inc. All rights reserved. 25 ANSYS, Inc. Proprietary
• 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 ]
© 2006 ANSYS, Inc. All rights reserved. 26 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 27 ANSYS, Inc. Proprietary
Thermal Wall Functions - Limitations (3)
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:
Thermal Wall Functions - Limitations (4)
© 2006 ANSYS, Inc. All rights reserved. 29 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 30 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 31 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 32 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 33 ANSYS, Inc. Proprietary
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 ]
© 2006 ANSYS, Inc. All rights reserved. 34 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 35 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 36 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 37 ANSYS, Inc. Proprietary
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 ]
© 2006 ANSYS, Inc. All rights reserved. 38 ANSYS, Inc. Proprietary
‘‘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
© 2006 ANSYS, Inc. All rights reserved. 39 ANSYS, Inc. Proprietary
‘‘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 ¿
¿
© 2006 ANSYS, Inc. All rights reserved. 40 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 41 ANSYS, Inc. Proprietary
• 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
© 2006 ANSYS, Inc. All rights reserved. 42 ANSYS, Inc. Proprietary
• 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
© 2006 ANSYS, Inc. All rights reserved. 43 ANSYS, Inc. Proprietary
• 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
© 2006 ANSYS, Inc. All rights reserved. 44 ANSYS, Inc. Proprietary
• 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
© 2006 ANSYS, Inc. All rights reserved. 45 ANSYS, Inc. Proprietary
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 ]
© 2006 ANSYS, Inc. All rights reserved. 46 ANSYS, Inc. Proprietary
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
© 2006 ANSYS, Inc. All rights reserved. 47 ANSYS, Inc. Proprietary
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
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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
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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:
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• 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
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• 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
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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
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• 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
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