The Menter [1992] k-ω SST (shear stress transport) model is an extension of Menter's k-ω model that accounts for the transport of the turbulent shear stress and so offers improved predictions of flow separation under adverse pressure gradients. The model differs from Menter's k-ω model in that a limiter is applied to the eddy-viscosity relationship, but the complete set of mode equations are given below for completeness.
Menter's k-ω SST model may be summarised as follows:
∂(ρ*k)/∂t + ∇.(ρ*U*k) = ∇.(ρ*{ν_{l}+ν_{t}*σ_{k}} ∇k)+ ρ*(P_{k} - ε) | (2.1) |
∂(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 | (2.2) |
ν_{t} = ( a_{1}*k )/ max [a_{1}*ω, S*F_{2}] | (2.3) |
P_{k} = ν_{t}*(∇U + ∇U)^{t}):∇U | (2.4) |
ε = C_{D}*ω*k | (2.5) |
S = (∇U + ∇U)^{t})/2 | (2.6) |
wherein: ρ is the density; ν_{l} and ν_{t} are the laminar and turbulent kinematic viscosities; P_{k} is the volumetric production rate of k; and F_{1} and F_{2} are blending functions defined by:
F_{1} = tanh(A_{1}^{4}) | (2.7) |
A_{1} = min [ max { √k/(C_{D}ωδ), 500ν/(ωδ^{2}) }, 4*k*σ_{ω2}/(D_{ω}δ^{2}) ] | (2.8) |
D_{ω}= max [ 2{σ_{ω2}/ω}*∇k∇ω , 10^{-10} ] | (2.9) |
F_{2} = tanh(A_{2}^{4}) | (2.10) |
A_{2} = max [ 2√k/(C_{D}ωδ), 500ν/(ωδ^{2})] | (2.11) |
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_{1} | (2.12) |
where φ_{1} represents constant 1 and φ_{2} represents constant 2. The constants are:
σ_{k1}=0.5, σ_{ω1}=0.5, β_{1}=0.075,
σ_{k2}=1.0, σ_{ω2}=0.856, β_{2}=0.0828,
C_{D}=0.09, C_{1ω}=5/9, C_{2ω}=3/40.
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} | (3.1) |
ω=U_{τ}/(√C_{D}*κ*δ) | (3.2) |
where U_{τ} is the resultant friction velocity ( = √(τ_{w}/ρ) ), τ_{w} 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.*ν_{l}/(C_{2ω}*δ^{2}) | (3.3) |
k = (I*U)^{2} | (3.5) |
ω = ε/(C_{D}*k) | (3.6) |
ε=C_{D}^{3/4}*k^{3/2}/L_{m} | (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_{m} ~ 0.1H, where H is a characteristic inlet dimension, say the hydraulic radius of the inlet pipe.
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).
The high-Re form of k-ω model is activated by inserting the PIL command TURMOD(KWSST) 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 =19 ; 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,BF2,GEN1)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(KWSST-LOWRE), which is equivalent to TURMOD(KWSST) but with IENUTA=20.
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.
A number of Q1 files may be found in the advanced-turbulence-models library which demonstrate the use of the model.
See also the Instruction Course lectures on Turbulence Modelling.