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### 1. Introduction

The Menter [1992] k-ω model combines the k-ω and k-ε models so as to take advantage of the former's robustness near wall surfaces due to its simple low Reynolds number formulation and its ability to compute flows with weak adverse pressure gradients accurately, and the latter's better performance near the boundary layer edge and away from walls, due to its insensitivity to the free stream values. The basis of this technique is the transformation of the k-ω model to a k-ε formulation. This is an exact conversion, except for small contributions from the diffusion term due to the difference in the diffusion coefficients of the k and e equations.

### 2. Description of the model

Menter's k-ω model may be summarised as follows:

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

∂(ρ*ω)/∂t + ∇.(ρ*U*ω) = ∇.(ρ*{νltω}ω) + ρ*ω*(γ*Pk/k - β*ω) + 2*(1-F1)*ρ/[σω2*ω]*kω

νt = k/ω

Pk = νt*(∇U + ∇U)t):U

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 F1 is a blending function defined by:

F1 = tanh(Γ4)

Γ = min [ max { √k/(CDωδ), 500ν/(ωδ2) },  4*k/(σω2Dωδ2) ]

Dω= max [ 2/{σω2*ω}*kω , 10-10 ]

(2.6)

(2.7)

(2.8)

where δ is the distance to the nearest wall.

The constants φ of the model are calculated from the constants φ1 and φ2, as follows:

φ=F11+(1-F1)*φ2

where φ1 represents constant 1 and φ2 represents constant 2. The constants are:

σk1=2.0, σω1=2.0, β1=0.075,

σk2=1.0, σω2=1.168, β2=0.0828,

. CD=0.09, C=5/9, C=3/40.

### 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 κ=0.41 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)

### 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(3.6).

### 4. Activation of the model

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

```
PATCH(KWSOURCE,PHASEM, 1, NX, 1, NY, 1, NZ, 1, 1)
COVAL(KWSOURCE,KE , GRND4 , GRND4 )
COVAL(KWSOURCE,OMEG, GRND4 , GRND4 )
IENUTA =17 ; DISWAL
PRT(KE)=1.E10;PRT(OMEG)=1.E10
PRNDTL(KE)=GRND6;PRNDTL(OMEG)=GRND6
PATCH(KWSOGD,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
COVAL(KWSOGD,OMEG,GRND4,GRND4)
STORE(EP,BF1)
```
The coding for the Menter k-ω model is mainly in the file GXKW_MENTER.for, but the eddy-viscosity relationship is coded in the file GXKNVST.FOR, and the variable turbulent Prandtl numbers for k and ω are computed in the file GXPRNDTL.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), which is equal to √GEN1.

The low-Re form is activated by the setting TURMOD(KWMENTER-LOWRE), which is equivalent to TURMOD(KWMENTER) but with IENUTA=18.

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. F.R.Menter, 'Improved two-equation k-ω turbulence model for aerodynamic flows', NASA TM-103975, (1992).
2. 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).
3. D.C.Wilcox, 'Reassessment of the scale determining equation for advanced turbulence models', AIAA J., Vol.26, No.11, p1299, (1988).
4. D.C.Wilcox, 'Turbulence modelling for CFD' DCW Industries, La Canada, California, USA, (1993).