- Introduction
- Description of the model
- Boundary conditions
- Activation of the model
- Sources of further information

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.

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

∂(ρ*k)/∂t + **∇.**(ρ***U***k) =
**∇.**(ρ*{ν_{l}+ν_{t}/σ_{k}}
**∇**k)+ ρ*(P_{k} - ε)

∂(ρ*ω)/∂t + **∇.**(ρ***U***ω) =
**∇.**(ρ*{ν_{l}+ν_{t}/σ_{ω}}**∇**ω)
+ ρ*ω*(γ*P_{k}/k - β*ω)
+ 2*(1-F_{1})*ρ/[σ_{ω2}*ω]***∇**k**∇**ω

ν_{t} = k/ω

P_{k} = ν_{t}*(∇**U** + ∇**U**)^{t})**:**∇**U**

ε = C_{D}*ω*k

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

wherein: ρ is the density; ν

F_{1} = tanh(Γ^{4})

Γ = min [ max { √k/(C_{D}ωδ), 500ν/(ωδ^{2}) },
4*k/(σ_{ω2}D_{ω}δ^{2}) ]

D_{ω}= max [ 2/{σ_{ω2}*ω}***∇**k**∇**ω , 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 φ_{1}
and φ_{2}, as follows:

φ=F_{1}*φ_{1}+(1-F_{1})*φ_{2}

where φ_{1} represents constant 1 and φ_{2} represents
constant 2. The constants are:

σ_{k1}=2.0, σ_{ω1}=2.0, β_{1}=0.075,

σ_{k2}=1.0, σ_{ω2}=1.168, β_{2}=0.0828,

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=U_{τ}^{2}/√C_{D}

ω=U_{τ}/(√C_{D}*κ*δ)

(3.1)

(3.2)

where U

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.*ν_{l}/(C_{2ω}*δ^{2})

(3.3)

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/(C_{D}*k)

ε=C_{D}^{3/4}*k^{3/2}/L_{m}

(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 L

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).

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:

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.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 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.

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

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

See also the Instruction Course lectures on Turbulence Modelling.