Wilcox k-ω Model

Since all three k-ε turbulence models cannot be integrated all the way to walls, wall damping wall functions must be employed to provide correct near wall behavior. It is also known that the standard k-ε turbulence model fails to predict the flow separation under adverse pressure gradients.

Wilcox proposed a turbulence model similar to the standard k-ε turbulence model but replaced the dissipation rate (ε) equation with the eddy frequency (ω) equation (Wilcox, 2006; Wilcox, 2008). The eddy frequency (ω) is often referred to the specific dissipation rate and is defined as ω=ε/kω=ε/k . The Wilcox k-ω turbulence model has an advantage over the k-ε turbulence model as the k-ω model does not require any wall functions for the calculation of the velocity distribution near walls. As a result, the k-ω turbulence model has better performance for flows with adverse pressure gradient when compared to the k-ε turbulence models. However, the k-ω model exhibits a strong sensitivity to the freestream boundary condition (Wilcox, 2006) for external flow applications.

Transport Equations

Turbulent Kinetic Energy k (1)
(ρk)t+(ρ¯ujk)xj =xj[(μ+σkμt)kxj]+Pk+Dk(ρk)t+(ρ¯¯¯¯ujk)xj =xj[(μ+σkμt)kxj]+Pk+Dk
Eddy Frequency (Specific Dissipation Rate) ω (2)
(ρk)t+(ρ¯ujk)xj =xj[(μ+σkμt)kxj]+Pk+Dk(ρk)t+(ρ¯¯¯¯ujk)xj =xj[(μ+σkμt)kxj]+Pk+Dk

Production Modeling

Turbulent Kinetic Energy k (3)
Pk=μtS2Pk=μtS2
Eddy Frequency ω (4)
Pω=γωkμtS2Pω=γωkμtS2

where γ=β0β*σωκ2β*γ=β0βσωκ2β , β=β0fββ=β0fβ , fβ=1+85χω1+100χωfβ=1+85χω1+100χω , χω=|ΩijΩjkˆSki(β*ω)3|χω=ΩijΩjkˆSki(βω)3 , ˆSki=Ski12¯umxmδkiˆSki=Ski12¯¯¯¯¯umxmδki , Sij=12(¯uixj+¯ujxi)Sij=12(¯¯¯uixj+¯¯¯¯ujxi) , Ωij=12(¯uixj¯ujxi)Ωij=12(¯¯¯uixj¯¯¯¯ujxi)

Dissipation Modeling

Turbulent Kinetic Energy (k) (5)
Dk=ρβ*kωDk=ρβkω
Eddy Frequency (ω) (6)
Dω=ρβω2Dω=ρβω2

Modeling of Turbulent Viscosity μtμt

(7)
μt=kˊωμt=k´ω

where ˊω=max[ω,Clim2ˉSijˉSijβ*]´ω=max[ω,Clim2¯¯¯Sij¯¯¯Sijβ] , ˉSij=Sij13¯ukxkδij¯¯¯Sij=Sij13¯¯¯¯ukxkδij , Clim=78Clim=78 ,

Model Coefficients

σkσk = 0.6, σωσω = 0.5, β*β = 0.09, β0β0 = 0.0708, κκ = 0.4, σd={0.0 for kxjωxj018 for kxjωxj>0 .