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### 1. Introduction

The Wilcox (2008) revised k-ω model is an improved version of Wilcox's (1988) model that incorporates the addition of a cross-diffusion term, a stress-limiter modification to the eddy viscosity; and a vortex-stretching modification to the ω equation.

### 2. Description of the model

The revised k-ω model may be summarised as follows:

∂/∂t (ρ*k) + ∇.(ρ*U*k) = ∇.(ρ*{νltk} k)+ ρ*(Pk - ε)

∂/∂t (ρ*ω) + ∇.(ρ*U*ω) = ∇.(ρ*{νltω} ω) + σd*(ρ/ω)*∇k*∇ω + ρ*ω*(C*Pk/k - C*ω)

Pk = νt*∇U:(∇U+∇UT)

ε = CD*ω*k

νt = k/W ,    W = max(ω, Clim√(2.*SijSij*),  Clim=7/8

Sij= 0.5*(∂ui/∂xj  + ∂uj/∂xi )

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

wherein: ρ is the density; νl and νt are the laminar and turbulent kinematic viscosities; and Pk is the volumetric production rate of k.

The following coefficients are normally used:

σk=5/3, σω=2.0, CD*=0.09, C=13/25,

and

C = β0fβ

fβ = (1 + 85χω)/(1+100χω)

χω=|(ΩijΩjkSki)/(β*ω)3|

ωij= 0.5*(∂ui/∂xj  - ∂uj/∂xi )

σd = 0 ,  if  k.∇ω≤ 0

σd = 1/8 ,  if  k.∇ω> 0

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

where β0=0.0708 and χω is the vortex stretching parameter.

### 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/(C2)

(3.3)

The alternative condition of

ω=2.*νl/(CD*f22),

(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-ω' SST models.

### 4. Activation of the model

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

```
TURMOD(KWMODL);IENUTA=15;PRT(KE)=5./3.;PRT(OMEG)=2.0
STORE(XWP,FBP);STORE(GEN1)
PATCH(KWSOGD,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
COVAL(KWSOGD,OMEG, FIXFLU,GRND4)
STORE(DUDX,DUDY,DUDZ,DVDX,DVDY,DVDZ,DWDX,DWDY,DWDZ)
```

The revised 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.

### 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. D.C.Wilcox, 'Reassessment of the scale determining equation for advanced turbulence models', AIAA J., Vol.26, No.11, p1299, (1988).
2. D.C.Wilcox, 'Turbulence modelling for CFD' DCW Industries, La Canada, California, USA, (1993).
3. D.C.Wilcox, 'Formulation of the k-ω turbulence model revisited', AIAA J., Vol.46, No.11, p2823, (2008).