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3.4.8 The Wilcox (1988) k-ω turbulence model

Contents

  1. Introduction
  2. Description of the model
  3. Boundary conditions
  4. Activation of the model
  5. Sources of further information

1. Introduction

The first two-equation turbulence model was the k-ω model of Kolmogorov [1942], which involves the solution of transport equations for the turbulent kinetic energy k and the turbulence frequency ω. It should be mentioned that other workers define ω as the specific dissipation rate ε/k where ε is the dissipation rate of k.

Several different and improved versions of Kolmogrov's model have been proposed, including those of: Saiy [1974], Spalding [1979], Wilcox [1988, 2008], Speziale et al [1990] and Menter [1992].

The section describes standard k-ω model of Wilcox [1988] which includes a low-Reynolds-number extension for near-wall turbulence.

Although the k-ω model is not as popular as the k-ε model, it does have several advantages, namely that:

  1. the model is reported to perform better in transitional flows and in flows with adverse pressure gradients;
  2. the model is numerically very stable, especially the low-Re version, as it tends to produce converged solutions more rapidly than the k-ε models; and
  3. the low-Re version is more economical and elegant than the low-Re k-ε models, in that it does not require the calculation of wall distances, additional source terms and/or damping functions based on the friction velocity.

The main weakness of the standard k-ω model is that unlike the k-ε model, it is sensitive to the free-stream boundary condition for ω in free-shear flows. Modified variants exist which claim to remove this sensitivity by including cross-diffusion source terms in the ω transport equation. Two of these variants are provided in PHOENICS, namely the Menter k-ω model and its extension to include the effects of shear-stress transport - the k-ω SST model.

2. Description of the model

The k-ω model may be summarised as follows:

∂/∂t (ρ*k) + ∇.(ρ*U*k) = ∇.(ρ*{νl+νt/σk} ∇ k)+ ρ*(Pk - f2*ε)

∂/∂t (ρ*ω) + ∇.(ρ*U*ω) = ∇.(ρ*{νl+νt/σω}∇ ω) + ρ*ω*(f1*C1ω*Pk/k - C2ω*ω)

νt = fμk/ω

Pk = νt*(∂Ui/∂xj + ∂Uj/∂xi) ∂Ui/∂xj

ε = CD*ω*k

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

 

wherein: ρ is the density; νl and νt are the laminar and turbulent kinematic viscosities; Pk is the volumetric production rate of k; and fμ, f1 and f2 are low-Re damping functions.

The following constants are normally used:

σk=2.0, σω=2.0, CD=0.09, C1ω=5/9, C2ω=3/40.

The damping functions, which are set to unity in the high-Re model, are defined by:

fμ = (1/40 + RT/RK)/(1+RT/RK)

f1 = {1/fμ}*(0.1+RT/RW)/(1+RT/RW)

f2 = (5/18 + (RT/RB)4)/(1.+(RT/RB)4)

RT = k/(ω*νl)

(2.6)

(2.7)

(2.8)

(2.9)

 

where RB=8, RK=6.0, RW=2.7 and RT is the turbulent Reynolds number. In regions where RT is high, fμ, f1 and f2 tend to unity.

3. Boundary Conditions

3.1 Wall boundary conditions

The high-Re k-ω model can be used with equilibrium (GRND2), non-equilibrium (GRND3) and scalable wall functions, as well as with fully-rough wall functions (GRND5). For GRND2 wall functions, the following boundary conditions are applied for the turbulence variables:

k=Uτ2/√CD

ω=Uτ/(√CD*κ*δ)

(3.1)

(3.2)

 

where Uτ is the resultant friction velocity ( = √(τw/ρ) ), τw is the wall shear stress, δ is the normal distance of the first grid point from the wall, and k is von Karman's constant.

If the low-Re version is selected, then k=0 at the wall and the following condition is applied for ω at the near-wall grid point:

ω=2.*νl/(C2ω*δ2)

(3.3)

 

The alternative condition of

ω=2.*νl/(CD*f2*δ2),

(3.4)

 

also proposed by Wilcox [1988], produces nearly identical results, and so it has not been coded in PHOENICS.

3.2 Inlet conditions


At mass-inflow boundaries, the inlet values of k and ω are usually unknown, and one needs to take guidance from experimental data for similar flows. The simplest practice is to assume uniform values of k and ω computed from:

k = (I*U)2

ω = ε/(CD*k)

ε=CD3/4*k3/2/Lm

(3.5)

(3.6)

(3.7)

 

where U is the bulk inlet velocity, I is the turbulent intensity (typically in the range 0.01<.I< 0.05) and the mixing length Lm ~ 0.1H, where H is a characteristic inlet dimension, say the hydraulic radius of the inlet pipe.

3.3 Free-stream conditions


At free (entrainment) boundaries, where a fixed-pressure condition is employed, it is necessary to prescribe free stream values for k and ω. If the ambient stream is assumed to be free of turbulence, then k and ε can be set to negligibly small values and ω can then be calculated from eqn(2.5).

It should be mentioned that when using ω=0 in the free stream, the k-ω model consistently predicts spreading rates of free-shear layers that exceed measured values by more than 20%. As was noted earlier and discussed by Wilcox [1993], these solutions are in fact quite sensitive to the free-stream value of ω.

Speziale et al [1990] and Menter [1992] have proposed the inclusion of cross-diffusion source terms in the ω equation which remove the sensitivity to free-stream conditions. These terms are included in the Menter k-ω and k-&omega' SST models.

4. Activation of the model

The high-Re form of k-ω model is activated by inserting the PIL command TURMOD(KWMODL) in the Q1 file, which is equivalent to the following PIL commands: 

  
  SOLVE(KE,OMEG);ENUT=GRND7;EL1=GRND5;KELIN=0;EL1A=0.6
  PATCH(KWSOURCE,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
  COVAL(KWSOURCE,KE,GRND4,GRND4)
  COVAL(KWSOURCE,OMEG,GRND4,GRND4);GENK=T;IENUTA=10
  PRT(KE)=2.0;PRT(OMEG)=2.0
  TERMS(KE,N,Y,Y,Y,Y,N);TERMS(OMEG,N,Y,Y,Y,Y,N)
	STORE(EP)
The Wilcox k-ω is coded in Subroutine GXKW_WILCOX, apart from the eddy viscosity, which is calculated in Subroutine GXKNVST.FOR. The generation rate used in the source terms can be stored by the command STORE(GENK), and likewise the mean rate of strain by STORE(GEN1). If required the dissipation rate of k can be stored by setting STORE(EP) in the Q1 input file.

The low-Re form is activated by the setting TURMOD(KWMODL-LOWRE), which is equivalent to TURMOD(KWMODL) but with IENUTA=11.

The WALL and CONPOR commands automatically create the required COVALs for wall boundaries, i.e.

COVAL(WALLN,KE,GRND2,GRND2); COVAL(WALLN,OMEG,GRND2,GRND2)

for the high-Re version, and

COVAL(WALLN,KE,1.0,0.0); COVAL(WALLN,OMEG,GRND2,GRND2)

for the low-Re version.

Further information on wall functions can be found at here.

Information and advice on the use of low-Re models in general can be found in Section 3.4.4 on the Lam-Bremhorst k-ε model.

5. Sources of further information

A number of Q1 files may be found in the advanced-turbulence-models library which demonstrate the use of the model.

  1. A.N.Kolmogorov, 'Equations of turbulent motion of an incompressible fluid', Izv Akad Nauk SSR Ser Phys, 6, Vol 1/2, 56, (1942).
  2. F.R.Menter, 'Improved two-equation k-ω turbulence model for aerodynamic flows', NASA TM-103975, (1992).
  3. M.Saiy, 'Turbulent mixing of gas streams', PhD Thesis, Imperial College, University of London, (1974).
  4. D.B.Spalding, 'Mathematical models of turbulent transport processes' HTS/79/2, Imperial College, Mech.Eng.Dept., (1979).
  5. C.G.Speziale, R.Abid and E.C.Anderson, 'A critical evaluation of two-equation turbulence models for near-wall turbulence', AIAA Paper 90-1481, (1990).
  6. D.C.Wilcox, 'Reassessment of the scale determining equation for advanced turbulence models', AIAA J., Vol.26, No.11, p1299, (1988).
  7. D.C.Wilcox, 'Turbulence modelling for CFD' DCW Industries, La Canada, California, USA, (1993).
  8. D.C.Wilcox, 'Formulation of the k-ω turbulence model revisted', AIAA J., Vol.46, No.11, p2823, (2008).

See also the Instruction Course lectures on Turbulence Modelling.