K-omega turbulence model

In computational fluid dynamics, the k–omega (k–ω) turbulence model is a common two-equation turbulence model, that is used as a closure for the Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the turbulence kinetic energy (k) while the second (ω) is the specific rate of dissipation (of the turbulence kinetic energy k into internal thermal energy). The eddy viscosity νT, as needed in the RANS equations, is given by: νT = k/ω, while the evolution of k and ω is modelled as: ∂ ( ρ k ) ∂ t + ∂ ( ρ u j k ) ∂ x j = ρ P − β ∗ ρ ω k + ∂ ∂ x j [ ( μ + σ k ρ k ω ) ∂ k ∂ x j ] , with  P = τ i j ∂ u i ∂ x j , ∂ ( ρ ω ) ∂ t + ∂ ( ρ u j ω ) ∂ x j = γ ω k P − β ρ ω 2 + ∂ ∂ x j [ ( μ + σ ω ρ k ω ) ∂ ω ∂ x j ] + ρ σ d ω ∂ k ∂ x j ∂ ω ∂ x j . {displaystyle {egin{aligned}&{frac {partial ( ho k)}{partial t}}+{frac {partial ( ho u_{j}k)}{partial x_{j}}}= ho P-eta ^{*} ho omega k+{frac {partial }{partial x_{j}}}left,qquad { ext{with }}P= au _{ij}{frac {partial u_{i}}{partial x_{j}}},\&displaystyle {frac {partial ( ho omega )}{partial t}}+{frac {partial ( ho u_{j}omega )}{partial x_{j}}}={frac {gamma omega }{k}}P-eta ho omega ^{2}+{frac {partial }{partial x_{j}}}left+{frac { ho sigma _{d}}{omega }}{frac {partial k}{partial x_{j}}}{frac {partial omega }{partial x_{j}}}.end{aligned}}} For recommendations for the values of the different parameters, see Wilcox (2008).