Fluent-Adv Turbulence 15.0 L03 Wall Modeling

1 © 2014 ANSYS, Inc. April 23, 2014 ANSYS Confidential

15.0 Release

Lecture 3:

Near-Wall Modeling and Transition Modeling

Turbulence Modeling Using ANSYS Fluent


Topics to be Discussed

• Near wall modeling options

• Low Reynolds number turbulence models

• V2F model

• Laminar to turbulent transition in boundary layers


Importance of Near-Wall Modeling • Walls are main source of vorticity and turbulence

• The presence of walls usually gives rise to turbulent momentum and thermal boundary layers: the steepest variations are in the very near-wall regions

– 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 zones

– Thermal performance of heat exchangers, etc., is determined by turbulent wall heat transfer whose prediction depends upon near-wall modeling

• Use of very fine mesh to resolve the steep profiles is still too expensive for many industrial CFD simulations

• Hence accurate near-wall modeling is important for most industrial CFD applications


Near-Wall Modeling Issues

• k-e and RSM models are valid in the turbulent core region and through the log layer

– Some of the modeled terms in these equations are based on isotropic behavior

• Isotropic diffusion ( m t / s )

• Isotropic dissipation

• Pressure-strain redistribution

• Some model parameters based on experiments of isotropic turbulence

– Near-wall flows are anisotropic due to presence of walls

• Special near-wall treatments are necessary for these models since equations cannot be integrated down to wall


• Velocity profile exhibits layer structure identified from dimensional analysis

– Inner layer

• viscous forces rule, U = f ( r, tw , m , y )

– Outer layer

• dependent upon mean flow

– Overlap layer

• log-law applies

Flow Behavior in Near-Wall Region

U/ut


• Turbulent kinetic energy production and dissipation are nearly equal in the log-law region

– ‘Turbulent equilibrium’

• dissipation >> production in the viscous sublayer region

TKE Budget in Near-Wall Region k


Near-Wall Modeling Options • In general, ‘wall functions’ are a collection or set of laws that serve as boundary

conditions for momentum, energy, and species as well as for turbulence quantities

• Wall Function Options – The Standard and Non-equilibrium Wall Function options

refer to specific ‘sets’ designed for high Re flows • The viscosity affected, near-wall region is not resolved

• Near-wall mesh is relatively coarse

• Cell center information bridged by empirically-based wall functions

• Enhanced Wall Treatment or Low-Re Option – This near-wall model combines the use of enhanced wall

functions and a two-layer model • Used for low-Re flows or flows with complex near-wall

phenomena

• Generally requires a very fine near-wall mesh capable of resolving the near-wall region

• Turbulence models are modified for ‘inner’ layer

inner layer

outer layer


Wall Functions

• Wall functions consist of ‘wall laws’ for mean velocity and temperature and formulas for turbulent quantities

• The Universal Wall Laws

– Viscous sublayer

u+ = y+

– Log layer

• Formulas for k and e

– Local equilibrium

• k = ut2/Cm

1/2

• e = ut3/ky

– Precludes transport of turbulence in log layer

• k and e are functions of ut only tk u

UuCyu ;)ln(

1


• Fluent uses Launder-Spalding Wall Functions

– U = U ( r , t , m , y , k )

• Introduces additional velocity scale, U*, for ‘general’ application

• In an equilibrium boundary layer, where production is equal to dissipation, u* and u+ are identical (see right hand side of Slide 8)

• Similar ‘wall laws’ apply for energy and species

Standard Wall Functions: Velocity

rt

m

/

2/14/1

*

w

PP kCUU

m

r m PP ykCy

2/14/1

where * * ln1 EyU

k

* * yU for

** vyy ** vyy


• “Wall Functions” does not just mean the velocity profile

– Turbulence quantities also require special consideration

• 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

• e calculated at wall-adjacent cells using local equilibrium assumption • e = Cm

3/4 k3/2 / k y

• Wall functions less reliable when cell is down to the viscous sublayer

– Due to the fact that production << dissipation in the viscous sublayer while production is roughly equal to dissipation in the log-law layer

Standard Wall Functions: Turbulence


Limitations of Standard Wall Functions • Wall functions become less reliable when flow departs from the

conditions assumed in their derivation: – Local equilibrium assumption is not valid in the following:

• Strong pressure gradient p (strong acceleration or deceleration of the flow)

• Transpiration through wall

• Strong body forces

• Large curvature

• Stagnation and stream-line re-attachment

• Highly 3D flow (e.g., cross flows)

• Rapidly changing fluid properties near wall

– Low-Re flows are pervasive throughout model

– Small gaps are present


• Log-law is sensitized to pressure gradient for better prediction of adverse pressure gradient flows and separation

• Relaxed local equilibrium assumptions for TKE in the turbulent region of the wall-neighboring cells

m

r

krt

mm ykCE

kCU

w

2/14/12/14/1

ln1

/

~

mrkrk

y

k

yy

y

y

k

y

dx

dpUU vv

v

v

2

2/12/1ln

21~

where

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

Non-Equilibrium Wall Functions


Scalable Wall Functions • For k-e models, the scalable wall functions introduces a limiter in the y* calculations

such that y* = max ( y*, 11.225 )

– Essentially like saying if the first grid point is too close, the wall surface is shifted so that the first cell centroid is just outside the viscous sublayer

– In an equilibrium boundary layer where u* = u+, 11.225 is the value of y* where the viscous sublayer velocity profile and the logarithmic velocity profile intersect

• Can help to reduce y+ dependency when wall functions are employed

– It’s not magic: adequate resolution of boundary layer is necessary

~

Example: Grid sensitivity for simulation of heat transfer in a sudden pipe expansion

Standard WF’s Scalable WF’s


Enhanced Wall Treatment • Enhanced wall functions

– Momentum boundary condition based on blended law-of-the-wall (Kader)

– Similar blended ‘wall laws’ apply for energy, species, and w – Kader’s form for blending allows for incorporation of additional physics

• Pressure gradient effects • Thermal (including compressibility) effects

• Two-layer model – A blended two-layer model is used to determine near-wall e field

• Domain is divided into viscosity-affected (near-wall) region and turbulent core region – Based on ‘wall-distance’ turbulent Reynolds number: – Zoning is dynamic and solution adaptive

• High Re turbulence model used in outer layer • ‘Simple’ turbulence model used in inner layer

– Solutions for e and m t in each region are blended, e.g.,

The Enhanced Wall Treatment is an option for the k-e and RSM turbulence models

turblam ueueu1

mr /ykRey

innerouter

1 tt mm ee


• Enhanced Wall Treatment (EWT) and Scalable Wall Functions are both y+ insensitive near wall modeling approaches – This means that as the grid is refined, the solution will not show excessive sensitivity to the y+

value of the first cell, as on the left hand side of Slide 13

– Scalable wall functions are not grid sensitive like standard or non-equilibrium wall functions but that does not mean they are more accurate

• In addition to being a y+ insensitive approach, EWT is also a viscous sublayer resolving approach – This means that when it is used with a mesh that is fine enough to capture the flow behavior

in the viscous sublayer, more accurate results would be expected with EWT than with scalable wall functions

• EWT must be used to get accurate results in such a case (details in a later slide)

• In a case where the first grid point is always in the log layer, there should not be much difference between EWT and scalable wall functions

Comparison of EWT and Scalable Wall Functions


Two-Layer Zonal Concept

• Approach is to divide flow domain into two regions – Viscosity affected near-wall region

– Fully turbulent core region

• Use different turbulence models for each region

– One-equation ( k ) near-wall model for the viscosity affected near-wall region

– High-Re k-e or RSM models for turbulent core region

• Wall functions, and their limitations, are avoided


Two-Layer Zones

• The two regions are demarcated on a cell-by-cell basis:

– Based on Rey = r k1/2y / m

– Rey > 200, it is called the

turbulent core region

– Rey < 200, it is called the

viscosity affected region

– y is the shortest distance to the

nearest wall

– zoning is dynamic and solution

adaptive


Models Used in the Two-Layer Zones

• In the turbulent region, the chosen high-Re turbulence model is used

• In the viscosity-affected region, a one-equation model is used

– k equation is same as high-Re model

– Length scale used in evaluation of m t is not from e • m t = r Cm k

1/2 lm

• lm = cl y ( 1 – exp (- Rey /Am ))

• cl = k Cm3/4

– Dissipation rate, e , is calculated algebraically (not from transport equation) • e = k3/2/le

• le = cl y ( 1 – exp (-Rey /Ae ))

– The two e - fields can be quite different along the interface in highly non-equilibrium turbulence


Blended e − equations • The transition (of e − field) from one zone to another can be made

smoother by blending the two sets of e − equations (Jongen, 1998)

e

e

23

k

Wall

Inner layer

Outer layer

A

ReRe

kSaa

*

yy

PPP

nb

nbnbPP

tanh12

1

123

e

e

eee

eee

with

P

nb

nbnbPP SaaDt

Deee

er

e

e

23

PP

k


m

mmmm

ee

rme

rm

mm

AyckC

kC

y

tt

tt

Reexp1, ,

1

inner

2

outer

inner outer

Blended Turbulent Viscosity

• Turbulent viscosity ( m t ) is also blended using the individual formulations

e

e

23

k

Wall

Inner layer

Outer layer e

rm m

2

outer k

Ct

mmrm kCt inner


Blended Wall Laws

• Mean velocity

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

turb

1

lam

ueueu

yu lam

yEu ln1

turbk

1 exp,5

,01.0,1

4

E

Ec

cb

cayb

ya

yEu

yu

ln1

turb

lam

k

where


‘Wall Law’ Sub-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 • Non-adiabatic wall heat transfer effects

• Compressibility effects - takes effect when ideal-gas option is chosen


Sub-models and Options • The base laws-of-the-wall (mean velocity and temperature)

are modified using (White and Christoph, 1972) :

– Pressure-gradient contribution comes from:

– Thermal contribution comes from:

21

2

111

effects thermal

effectsgradient pressure

uuyydy

du

k

xd

pd

uy

xd

pd

w

ww

tt

tt ,

parameterility compressib

2

parameterfer heat trans

2,,1

2

wp

t

wwp

wtw

TC

u

TC

uquu tt s

t

s

r

r


Wall boundary conditions for w − equation

• In SST and standard k-w, the wall value of omega can be given as

– In the viscous sublayer we have

– And in the log layer we have

• A blending function is used to switch automatically from the sublayer form to the log layer form depending on the grid – The use of the blending function means that the SST and k-w models can be used in a robust

manner on meshes suitable for wall functions

– However, keep in mind that many of the applications these models are intended to address require the viscous sublayer to be resolved for optimum accuracy

wm

rw

2 * uw

2

6

yi

w

dy

du turb

*

1

w


Grid Considerations (1)

• Cases where the highly resolved Low-Re meshes can be afforded or are absolutely necessary

1st cell centroid at y+ ~ 1

The viscous sublayer is resolved Can use • k-e family or RSM with EWT • S-A, k-w, SST • transition models


• Cases where the resolving the viscous sublayer is unaffordable – Majority of industrial 3D flows.

1st cell centroid at 30 < y+ < 300

Grid Considerations (2)

WF’s bridge the gap between the wall and the fully turbulent region where the first cell centroid is placed

Can use • k-e family or RSM with WF’s


Example in Predicting Near-wall Cell Size • During the pre-processing stage, you will need to know a suitable size for the first layer of grid

cells (inflation layer) so that Y+ is in the desired range.

• The actual flow-field will not be known until you have computed the solution (and indeed it is sometimes unavoidable to have to go back and remesh your model on account of the computed Y+ values).

• To reduce the risk of needing to remesh, you may want to try and predict the cell size by performing a hand calculation at the start. For example:

• For a flat plate, Reynolds number ( ) gives Rel = 1.4x106

Recall from earlier slide, flow over a surface is turbulent when ReL > 5x105

Flat plate, 1m long

Air at 20 m/s r = 1.225 kg/m3

m = 1.8x10-5 kg/ms

y

The question is what height (y) should the first row of grid cells be. We will use SWF, and are aiming for Y+ 50

m

rVLl Re


• Re is known, so use the definitions to calculate the first cell height

• We know we are aiming for y+ of 50, hence:

our first cell height y should be approximately 1 mm.

Calculating Wall Distance for a Given y+ • Begin with the definition of y+ and rearrange:

• The target y+ value and fluid properties are known, so we need Ut, which is defined as:

• The wall shear stress ,tw ,can be found from the skin friction coefficient, Cf:

• A literature search suggests a formula for the skin friction on a plate1 thus:

1 An equivalent formula for internal flows, with Reynolds number based on the pipe diameter is Cf = 0.079 Red-0.25

2.0Re058.0 lfC

221

UC fw rt

r

tt

wU

r

m

tU

yy

m

r t yUy

mU

yy 4-9x10

r

m

t

m/s 0.82 r

tt

wU

22

21 smkg/ 0.83 UC fw rt

.0034 Re058.0 2.0

lfC


erm mm

2

k

Cft

erme

ee 22

2

11 CfSCfk

t

Damping-Function Low-Re Models

• Standard k-e model modified by damping functions:

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

e − transport equation

turbulent viscosity

21, , fffm

jj xxDt

D e

s

mm

er

e

t


12

Re exp1

36

Re exp

9

21

1

Re41 Re008.0tanh

2

2

1

4/3

yt

ty

f

f

fm

Typical Damping Functions • Damping functions written in terms of Reynolds numbers:

• e.g., Abid’s model:

m

remr

m

r

em

r yykkεyt

4/1 2 / Re;Re;

Re


Low-Re k-e Models • Several full low-Re k-e models now available

– Lam Bremhorst

– Launder-Sharma

– Abid

– Chang et al.

– Abe-Kondo-Nagano

– Yang-Shih

• Enables modeling of low-Re effects including transitional flows – Implementations are problem specific

• Features are not visible in GUI – Access from TUI


V2F low-Re k-e Model

• Durbin (1990) suggests that wall normal fluctuations, , are responsible for the near-wall transport

• behaves quite differently than and

– attenuation of is a kinematic effect

– whereas the damping of is a dynamic effect

• Model instead of

• Requires two additional equations:

– transport equation for wall-normal fluctuations,

– equation for an elliptic relaxation function, f

2 v

2 v2 u

2 w2 v

2 u

Tv ~ t

2m kTt ~ m

2 v


V2F k-e Model Equations e − transport equation

v2 − transport equation

Relaxation equation

Scales

j

t

j xxtD

D

e

s

mm

er

e

1

2

2

1 erm ee CSCT

t

kvfk

x

v

xtD

vD

jk

t

j

err

s

mmr 2

2

2

Tk

vN

k

SC

k

v

T

C

xx

fLf t

jj

1

3

2

22

2

2

1

22

r

m

e

ee

e

3

2

322 , max;6, max C

kCL

kT L


V2F k-e Model Pros and Cons

• 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 eddy-viscosity model

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


Modeling Laminar to Turbulent Transition • Until recently, most engineering CFD simulations did not include laminar to

turbulent transition

– Low Re turbulence models cannot reliably predict transition

• The transition process strongly affects heat transfer, skin friction and flow separation that are central to the efficiency of many technical devices

• Fluent includes three turbulence models developed specifically for prediction flows with laminar to turbulent transition

– Models can predict laminar to turbulent transition in boundary layers

– Models can predict numerous transition regimes

• Natural transition, bypass transition, separation-induced transition, crossflow transition


Transition Models in Fluent • Three transition models are available

– Transition SST (Menter and Langtry, 2004) • Correlation based model using transport equations and local formulation • Uses SST k and w equations plus two additional transport equations

(intermittency and Req ) – Intermittency Transition Model (ANSYS, R15) • Correlation based model using transport equations and local formulation • Uses SST k and w equations plus one additional transport equation for

intermittency • Only model that can predict crossflow transition

– k-kl-w (Walters and Cokljat, 2007) • Based on laminar kinetic energy concept • Uses k and w equations plus one additional transport equation (laminar

kinetic energy)


Appendix


Guidelines for Transition Model Usage • Mesh

– First cell centroid at y+ ~ 1

– Stretching (expansion ratio) of grid in wall normal direction should not exceed ~ 1.1

• Transition SST – Converge 2 equation SST

– Switch to Transition SST

– Leave intermittency as it is

– In initialize panel see what value you will get for Req when you initialize from the inlet (4th transport equation) and use that value to patch Req into the field

– Ensure that all schemes and all URF are the same for all 4 turbulence equations

– Use Coupled PBNS solver

• k-kl-w – Converge any k-w model (optional)

– Switch to 3 equation k-kl-w model

– Inflow value of kl should be 10-6

Fluent-Adv Turbulence 15.0 L03 Wall Modeling

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