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 $ω = ε / k$ $ω = ε / k$ . 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)

\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}

$\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}$

Eddy Frequency (Specific Dissipation Rate) ω

(2)

\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}

$\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}$

Production Modeling

Turbulent Kinetic Energy k

(3)

P_{k} = μ_{t} S^{2}

$P_{k} = μ_{t} S^{2}$

Eddy Frequency ω

(4)

P_{ω} = \frac{γ ω}{k} μ_{t} S^{2}

$P_{ω} = \frac{γ ω}{k} μ_{t} S^{2}$

where $γ = \frac{β_{0}}{β^{*}} - \frac{σ_{ω} κ^{2}}{\sqrt{β^{*}}}$ $γ = \frac{β_{0}}{β^{*}} - \frac{σ_{ω} κ^{2}}{\sqrt{β^{*}}}$ , $β = β_{0} f_{β}$ $β = β_{0} f_{β}$ , $f_{β} = \frac{1 + 85 χ_{ω}}{1 + 100 χ_{ω}}$ $f_{β} = \frac{1 + 85 χ_{ω}}{1 + 100 χ_{ω}}$ , $χ_{ω} = ∣ ∣ ∣ \frac{Ω_{i j} Ω_{j k} {ˆ S}_{k i}}{{(β^{*} ω)}^{3}} ∣ ∣ ∣$ $χ_{ω} = | \frac{Ω_{i j} Ω_{j k} {\hat{S}}_{k i}}{{(β^{*} ω)}^{3}} |$ , ${ˆ S}_{k i} = S_{k i} - \frac{1}{2} \frac{\partial ¯ ¯¯¯ ¯ u_{m}}{\partial x_{m}} δ_{k i}$ ${\hat{S}}_{k i} = S_{k i} - \frac{1}{2} \frac{\partial \bar{u_{m}}}{\partial x_{m}} δ_{k i}$ , $S_{i j} = \frac{1}{2} (\frac{\partial ¯ ¯ ¯ u_{i}}{\partial x_{j}} + \frac{\partial ¯ ¯¯ ¯ u_{j}}{\partial x_{i}})$ $S_{i j} = \frac{1}{2} (\frac{\partial \bar{u_{i}}}{\partial x_{j}} + \frac{\partial \bar{u_{j}}}{\partial x_{i}})$ , $Ω_{i j} = \frac{1}{2} (\frac{\partial ¯ ¯ ¯ u_{i}}{\partial x_{j}} - \frac{\partial ¯ ¯¯ ¯ u_{j}}{\partial x_{i}})$ $Ω_{i j} = \frac{1}{2} (\frac{\partial \bar{u_{i}}}{\partial x_{j}} - \frac{\partial \bar{u_{j}}}{\partial x_{i}})$

Dissipation Modeling

Turbulent Kinetic Energy (k) (5)

D_{k} = - ρ β^{*} k ω

$D_{k} = - ρ β^{*} k ω$

Eddy Frequency (ω) (6)

D_{ω} = - ρ β ω^{2}

$D_{ω} = - ρ β ω^{2}$

Modeling of Turbulent Viscosity $μ_{t}$ $μ_{t}$

(7)

μ_{t} = \frac{k}{´ ω}

$μ_{t} = \frac{k}{\overset{´}{ω}}$

where $´ ω = m a x [ω, C_{l i m} \sqrt{\frac{2 {¯ ¯ ¯ S}_{i j} {¯ ¯ ¯ S}_{i j}}{β^{*}}}]$ $\overset{´}{ω} = m a x [ω, C_{l i m} \sqrt{\frac{2 {\bar{S}}_{i j} {\bar{S}}_{i j}}{β^{*}}}]$ , ${¯ ¯ ¯ S}_{i j} = S_{i j} - \frac{1}{3} \frac{\partial ¯ ¯¯ ¯ u_{k}}{\partial x_{k}} δ_{i j}$ ${\bar{S}}_{i j} = S_{i j} - \frac{1}{3} \frac{\partial \bar{u_{k}}}{\partial x_{k}} δ_{i j}$ , $C_{l i m} = \frac{7}{8}$ $C_{l i m} = \frac{7}{8}$ ,

Model Coefficients

$σ_{k}$ $σ_{k}$ = 0.6, $σ_{ω}$ $σ_{ω}$ = 0.5, $β^{*}$ $β^{*}$ = 0.09, $β_{0}$ $β_{0}$ = 0.0708, $κ$ $κ$ = 0.4, $σ_{d} = ⎧ ⎨ ⎩ \begin{matrix} 0.0 f o r \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}} \leq 0 \frac{1}{8} f o r \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}} > 0 \end{matrix}$ $σ_{d} = {\begin{matrix} 0.0 f o r \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}} \leq 0 \\ \frac{1}{8} f o r \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}} > 0 \end{matrix}$ .