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
Production Modeling
where γ=β0β*−σωκ2√β*γ=β0β∗−σωκ2√β∗ , β=β0fββ=β0fβ , fβ=1+85χω1+100χωfβ=1+85χω1+100χω , χω=|ΩijΩjkˆSki(β*ω)3|χω=∣∣∣ΩijΩjkˆSki(β∗ω)3∣∣∣ , ˆSki=Ski−12∂¯um∂xmδkiˆSki=Ski−12∂¯¯¯¯¯um∂xmδki , Sij=12(∂¯ui∂xj+∂¯uj∂xi)Sij=12(∂¯¯¯ui∂xj+∂¯¯¯¯uj∂xi) , Ωij=12(∂¯ui∂xj−∂¯uj∂xi)Ωij=12(∂¯¯¯ui∂xj−∂¯¯¯¯uj∂xi)
Dissipation Modeling
Modeling of Turbulent Viscosity μtμt
where ˊω=max[ω,Clim√2ˉSijˉSijβ*]´ω=max[ω,Clim√2¯¯¯Sij¯¯¯Sijβ∗] , ˉSij=Sij−13∂¯uk∂xkδij¯¯¯Sij=Sij−13∂¯¯¯¯uk∂xkδij , 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 ∂k∂xj∂ω∂xj≤018 for ∂k∂xj∂ω∂xj>0 .