4.5.2 Shear-Stress Transport (SST) $k$- $\omega$ Model

Overview

The shear-stress transport (SST) $k$- $\omega$ model was developed by Menter [ 224] to effectively blend the robust and accurate formulation of the $k$- $\omega$ model in the near-wall region with the free-stream independence of the $k$- $\epsilon$ model in the far field. To achieve this, the $k$- $\epsilon$ model is converted into a $k$- $\omega$ formulation. The SST $k$- $\omega$ model is similar to the standard $k$- $\omega$ model, but includes the following refinements:

• The standard $k$- $\omega$ model and the transformed $k$- $\epsilon$ model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the near-wall region, which activates the standard $k$- $\omega$ model, and zero away from the surface, which activates the transformed $k$- $\epsilon$ model.

• The SST model incorporates a damped cross-diffusion derivative term in the $\omega$ equation.

• The definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress.

• The modeling constants are different.

These features make the SST $k$- $\omega$ model more accurate and reliable for a wider class of flows (e.g., adverse pressure gradient flows, airfoils, transonic shock waves) than the standard $k$- $\omega$ model. Other modifications include the addition of a cross-diffusion term in the $\omega$ equation and a blending function to ensure that the model equations behave appropriately in both the near-wall and far-field zones.

Transport Equations for the SST $k$- $\omega$ Model

The SST $k$- $\omega$ model has a similar form to the standard $k$- $\omega$ model:

and

In these equations, $\widetilde{G}_k$ represents the generation of turbulence kinetic energy due to mean velocity gradients, calculated as described in Section  4.5.1. $G_\omega$ represents the generation of $\omega$, calculated as described in Section  4.5.1. ${\Gamma}_k$ and ${\Gamma}_\omega$ represent the effective diffusivity of $k$ and $\omega$, respectively, which are calculated as described below. $Y_k$ and $Y_\omega$ represent the dissipation of $k$ and $\omega$ due to turbulence, calculated as described in Section  4.5.1. $D_{\omega}$ represents the cross-diffusion term, calculated as described below. $S_k$ and $S_{\omega}$ are user-defined source terms.

Modeling the Effective Diffusivity

The effective diffusivities for the SST $k$- $\omega$ model are given by

$${\Gamma}_k = \mu + \frac{\mu_t}{\sigma_k}$$ (4.5-34)

$${\Gamma}_\omega = \mu + \frac{\mu_t}{\sigma_\omega}$$ (4.5-35)

where $\sigma_k$ and $\sigma_\omega$ are the turbulent Prandtl numbers for $k$ and $\omega$, respectively. The turbulent viscosity, $\mu_t$, is computed as follows:

where $S$ is the strain rate magnitude and

$$\sigma_{k} = \frac{1}{F_1/\sigma_{k,1} + (1 - F_1)/\sigma_{k,2}}$$ (4.5-37)

$$\sigma_{\omega} = \frac{1}{F_1/\sigma_{\omega,1} + (1 - F_1)/\sigma_{\omega,2}}$$ (4.5-38)

$\alpha^*$ is defined in Equation  4.5-6. The blending functions, $F_1$ and $F_2$, are given by

$$F_1 = \tanh \left(\Phi_1^4 \right)$$ (4.5-39)

$${\Phi}_1 = \min \left[\max \left (\frac{\sqrt{k}}{0.09 \omega y}, \frac{500 ... ... \omega} \right ), \frac{4 \rho k} {\sigma_{\omega,2} D_{\omega}^+ y^2} \right]$$ (4.5-40)

$$D_{\omega}^+ = \max \left[2 \rho \frac{1}{\sigma_{\omega,2}} \frac{1}{\omega} \f... ...\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}, 10^{-10} \right]$$ (4.5-41)

$$F_2 = \tanh \left(\Phi_2^2 \right)$$ (4.5-42)

$$\Phi_2 = \max \left [ 2 \frac{\sqrt{k}}{0.09 \omega y}, \frac{500 \mu}{\rho y^2 \omega} \right]$$ (4.5-43)

where $y$ is the distance to the next surface and $D_{\omega}^+$ is the positive portion of the cross-diffusion term (see Equation  4.5-52).

Modeling the Turbulence Production

Production of $k$

The term $\widetilde{G}_k$ represents the production of turbulence kinetic energy, and is defined as:

where $G_k$ is defined in the same manner as in the standard $k$- $\omega$ model. See Section  4.5.1 for details.

Production of $\omega$

The term $G_\omega$ represents the production of $\omega$ and is given by

Note that this formulation differs from the standard $k$- $\omega$ model. The difference between the two models also exists in the way the term $\alpha_{\infty}$ is evaluated. In the standard $k$- $\omega$ model, $\alpha_{\infty}$ is defined as a constant (0.52). For the SST $k$- $\omega$ model, $\alpha_{\infty}$ is given by

where

$$\alpha_{\infty,1} = \frac{\beta_{i,1}}{\beta^*_\infty} - \frac{\kappa^2}{\sigma_{w,1} \sqrt{\beta^*_\infty}}$$ (4.5-47)

$$\alpha_{\infty,2} = \frac{\beta_{i,2}}{\beta^*_\infty} - \frac{\kappa^2}{\sigma_{w,2} \sqrt{\beta^*_\infty}}$$ (4.5-48)

where $\kappa$ is 0.41.

Modeling the Turbulence Dissipation

Dissipation of $k$

The term $Y_k$ represents the dissipation of turbulence kinetic energy, and is defined in a similar manner as in the standard $k$- $\omega$ model (see Section  4.5.1). The difference is in the way the term $f_{\beta^*}$ is evaluated. In the standard $k$- $\omega$ model, $f_{\beta^*}$ is defined as a piecewise function. For the SST $k$- $\omega$ model, $f_{\beta^*}$ is a constant equal to 1. Thus,

Dissipation of $\omega$

The term $Y_\omega$ represents the dissipation of $\omega$, and is defined in a similar manner as in the standard $k$- $\omega$ model (see Section  4.5.1). The difference is in the way the terms $\beta_i$ and $f_\beta$ are evaluated. In the standard $k$- $\omega$ model, $\beta_i$ is defined as a constant (0.072) and $f_\beta$ is defined in Equation  4.5-24. For the SST $k$- $\omega$ model, $f_{\beta}$ is a constant equal to 1. Thus,

Instead of having a constant value, $\beta_i$ is given by

and $F_1$ is obtained from Equation  4.5-39.

Cross-Diffusion Modification

The SST $k$- $\omega$ model is based on both the standard $k$- $\omega$ model and the standard $k$- $\epsilon$ model. To blend these two models together, the standard $k$- $\epsilon$ model has been transformed into equations based on $k$ and $\omega$, which leads to the introduction of a cross-diffusion term ( $D_\omega$ in Equation  4.5-33). $D_\omega$ is defined as

For details about the various $k$- $\epsilon$ models, see Section  4.4.

Model Constants

All additional model constants ( $\alpha_{\infty}^*$, $\alpha_{\infty}$, $\alpha_0$, $\beta_{\infty}^*$, $R_\beta$, $R_k$, $R_\omega$, $\zeta^*$, and M $_{t0}$) have the same values as for the standard $k$- $\omega$ model (see Section  4.5.1).