K-Omega Model

Model Variants

Two variants of the K-Omega model are implemented in Simcenter STAR-CCM+:

Standard K-Omega

Wilcox revised his original model in 1998, and then in 2006, to account for several perceived deficiencies in the original version (1988). These revisions include:

• A revised set of model coefficients
• Two corrections to account for sensitivity to free-stream/inlet conditions, both based on products of $\nabla k$ and $\nabla \omega$. The first model, introduced in 1998, is a modification of the turbulent kinetic energy equation. The more recent revision [329] is the introduction of a cross-diffusion term that is similar to that used in the SST K-Omega model.
• A correction to improve the free-shear-flow spreading rates of the model
• A compressibility correction
• Low-Reynolds number corrections that allow the K-Omega model to be better applied in the prediction of low-Reynolds number and transitional flows

However, the validation results published in Wilcox’s book are typically for two-dimensional, primarily parabolic, flows. Until further validations for complex flows are widely published, the corrections should be used with caution. Therefore, each correction is included as an option in Simcenter STAR-CCM+.

SST K-Omega

The problem of sensitivity to free-stream/inlet conditions was addressed by Menter [323], who recognized that the $\epsilon$ transport equation from the Standard K-Epsilon model could be transformed into an $\omega$ transport equation by variable substitution.

The transformed equation looks similar to the one in the Standard K-Omega model, but adds an additional non-conservative cross-diffusion term containing the dot product $\nabla k\cdot \nabla \omega$. Inclusion of this term in the $\omega$ transport equation potentially makes the K-Omega model give identical results to the K-Epsilon model. Menter suggested using a blending function (which includes functions of wall distance) that would include the cross-diffusion term far from walls, but not near the wall. This approach effectively blends a K-Epsilon model in the far-field with a K-Omega model near the wall. Purists may object that the blending function crossover location is arbitrary, and could obscure some critical feature of the turbulence. Nevertheless, the fact remains that this approach cures the biggest drawback to applying the K-Omega model to practical flow simulations.

Menter also introduced a modification to the linear constitutive equation and named the model containing this modification the SST (shear-stress transport) K-Omega model. However, the linear relation between the Reynolds stresses and the mean strain rate tends to strongly under predict the anisotropy of turbulence. Turbulence is anisotropic in most complex flows—for example in strong swirl, streamline curvature, shear layer, or boundary layer flows. The anisotropy of the Reynolds stresses not only affects the flow field but also the turbulent transport of scalars (temperature, concentration, passive scalar).

To account for anisotropy of turbulence, Simcenter STAR-CCM+ offers the following non-linear constitutive relations:

• Quadratic—as suggested by Spalart [325].
• Cubic—as proposed by Wallin and Johansson [327] and further improved by Hellsten [322]

The cubic constitutive relation is derived from a Reynolds stress transport model and, thus, represents an Explicit Algebraic Reynolds Stress Model (EARSM). Reynolds stress transport models perform much better than eddy viscosity models in anisotropic turbulence, but are often unstable when used in complex flows. Therefore, the cubic model is a good compromise between the two.

The SST model has seen fairly wide application in the aerospace industry, where viscous flows are typically resolved, and turbulence models are applied throughout the boundary layer.

Relation for Turbulent Viscosity

The turbulent eddy viscosity ${\mu }_t$ is calculated as:

where:

• $\rho$ is the density.
• $T$ is the turbulent time scale.

The turbulent time scale is calculated using Durbin's realizability constraint [321] or the Vorticity Limiter realizability option [310] which is only applicable to Volume of Fluid (VOF) multiphase wave flow.

• ${\alpha }^{*}$ , $a_1$ , and $C_T$ are Model Coefficients.
• ${\lambda }_2\frac{\beta }{{\beta }^{\text{*}}\alpha }$ is the vorticity coefficient. See Model Coefficients.
• $S$ is given by Eqn. (1129).

$F_2$ is a blending function calculated as:

where:

• ${\beta }^{*}$ is a Model Coefficient.
• $d$ is the distance to the wall.

Transport Equations

The transport equations for the kinetic energy $k$ and the specific dissipation rate $\omega$ are:

where:

• $\overline{v}$ is the mean velocity.
• $\mu$ is the dynamic viscosity.
• ${\sigma }_k$, and ${\sigma }_{\omega }$ , $C_{\epsilon 1}$, and $C_{\epsilon 2}$ are Model Coefficients.
• $P_k$ and $P_{\omega }$ are Production Terms.
• $f_{{\beta }^{*}}$ is the free-shear modification factor.
• $f_{\beta }$ is the vortex-stretching modification factor.
• $S_k$ and $S_{\omega }$ are the user-specified source terms.
• $k_0$ and ${\omega }_0$ are the ambient turbulence values that counteract turbulence decay [326].

Production Terms

The formulation of the production terms $P_k$ and $P_{\omega }$ depends on the K-Omega model variant:

where:

• the contributions to the production terms are:

  	Description	Formulation	where:
  $G_k$	Turbulent production	$${\mu }_t{f}_c{S}^2-\frac{2}{3}\rho k\nabla \cdot \overline{\mathbf{\text{v}}}{-\frac{2}{3}\mu }_t{(\nabla \cdot \overline{\mathbf{\text{v}}})}^2$$ (1217)	$f_c$ is the curvature correction factor given by Eqn. (1287).
  $G_b$	Buoyancy production	$$\beta \frac{{\mu }_t}{{\mathrm{Pr}}_t}(\nabla \overline{T}\cdot \mathbf{\text{g}})$$ (1218)	$\beta$ is the coefficient of thermal expansion. For constant density flows using the Boussinesq approximation, $\beta$ is user-specified. For ideal gases, $\beta$ is given by $\beta =-\frac{1}{\rho }\frac{\partial \rho }{\partial \overline{T}}$ ${\mathrm{Pr}}_t$ is the turbulent Prandtl number. $\overline{T}$ is the mean temperature. $\mathbf{\text{g}}$ is the gravitational vector.
  $G_{nl}$	"Non-linear" production	$$\begin{array}{c}\left({\mathbf{\text{T}}}_{\mathrm{RANS},NL}\right):\nabla \overline{\mathbf{\text{v}}}\end{array}$$ (1219)	${\mathbf{\text{T}}}_{\mathrm{RANS},NL}$ is a non-linear Constitutive Relation.
  $G_{\omega }$	Specific dissipation production	SKO: $$\rho \alpha \left[\left({{\alpha }^{\text{*}}S}^2-\frac{2}{3}{(\nabla \cdot \overline{\mathbf{\text{v}}})}^2\right)-\frac{2}{3}\omega \nabla \cdot \overline{\mathbf{\text{v}}}\right]$$ (1220) SSTKO: $$\rho \gamma \left[\left(S^2-\frac{2}{3}{(\nabla \cdot \overline{\mathbf{\text{v}}})}^2\right)-\frac{2}{3}\omega \nabla \cdot \overline{\mathbf{\text{v}}}\right]$$ (1221)	$\alpha$, ${\alpha }^{*}$, and $\gamma$ are Model Coefficients.
  $D_{\omega }$	Cross-diffusion term	$$2\rho \left(1-F_1\right){\sigma }_{{\omega }_2}\frac{1}{\omega }\nabla k\cdot \nabla \omega$$ (1222)	$F_1$ is given by Eqn. (1230). ${\sigma }_{{\omega }_2}$ is a Model Coefficient.

Model Coefficients

The blending function $F_1$ combines the near-wall contribution of a coefficient with its value far away from the wall and is defined as:

where:

• $d$ is the distance to the wall.
• $\nu$ is the kinematic viscosity.
• ${\text{CD}}_{k\omega }=\text{max}\left(\frac{1}{\omega }\nabla k\cdot \nabla \omega ,{10}^{-20}\right)$ (cross-diffusion coefficient)