Turbulence equations (compressible flow)

Here, the two-parameter turbulence models BSL (baseline) and SST (shear stress transport) are treated [51]. The $ k-\epsilon$ [37] and the $ k-\omega $-model [90] are special cases of the BSL-model. The two parameters are the turbulent kinetic energy $ k$ and the turbulence frequency $ \omega$. The equation for $ k$ reads [51]:

$\displaystyle \frac{\partial \rho k}{\partial t} + \nabla \cdot (\rho k \boldsy...
... \right ] + t_{ij}^t \frac{\partial v_i}{\partial x_j} - \beta^* \rho \omega k.$ (713)

Since

$\displaystyle t_{ij}^t = 2 \mu_t (d_{ij}-\frac{1}{3} d_{kk} \delta_{ij}) - \frac{2}{3} \rho k \delta_{ij}$ (714)

and

$\displaystyle t_{ij}^l = 2 \mu(d_{ij}-\frac{1}{3} d_{kk} \delta_{ij}),$ (715)

one can write

$\displaystyle t_{ij}^t = \frac{\mu_t}{\mu } t_{ij}^l - \frac{2}{3} \rho k \delta_{ij}.$ (716)

This leads to:

$\displaystyle \frac{\partial \rho k}{\partial t} + \nabla \cdot (\rho k \boldsy...
...u_t}{\mu } t_{ij}^l d_{ij} - \frac{2}{3} \rho k d_{kk} - \beta^* \rho \omega k.$ (717)

Notice that $ \mu_t t_{ij}^l d_{ij}/\mu$ can be written as:

$\displaystyle \frac{\mu_t}{\mu} t_{ij}^l d_{ij}$ $\displaystyle = \mu_t \left ( 2 \left [ \left ( \frac{\partial v_1}{\partial x_...
... ) ^2 + \left ( \frac{\partial v_3}{\partial x_3} \right ) ^2 \right ] \right .$    
  $\displaystyle + \left [ \left ( \frac{\partial v_1}{\partial x_2} + \frac{\part...
...ial v_2}{\partial x_3} + \frac{\partial v_3}{\partial x_2} \right ) ^2 \right ]$    
  $\displaystyle - \left . \frac{2}{3} (\nabla \cdot \boldsymbol{v})^2 \right )$ (718)

In the above conservation equation $ \sigma_k$ is a function of the flow characteristics through the blending factor $ F1$ [51] and the dynamic turbulent viscosity satisfies:

$\displaystyle \mu_t=\rho \frac{k}{\omega }.$ (719)

In fact, the BSL model is a linear combination of the $ k-\omega $ model and the $ k-\epsilon$-model with coefficients $ F_1$ and $ 1-F_1$, respectively. Near a wall $ F_1$ tends to 1, thus favoring the $ k-\omega $-model which is particularly good in the near-wall region, far away from a wall $ F_1$ tends to zero, leading to a pure $ k-\epsilon$-model.

The blending factor used in iteration $ (m)$ is $ F_1 ^{(m-1)}$, i.e. its calculation is mainly based on the results from the previous iteration and satisfies the following equations:

$\displaystyle (F_1)_P ^{(m-1)}$ $\displaystyle = \tanh \left [ \left ( {{\text{arg}}_1}_P ^{(m-1)} \right ) ^4 \right ]$ (720)
$\displaystyle {{\text{arg}}_1}_P ^{(m-1)}$ $\displaystyle = \min \left [ \max \left ( \frac{\sqrt{k_P ^{(m-1)} }}{0.09 \ome...
...a_\omega }_2 k_P ^{(m-1)} }{{(\text{CD}_{k \omega}})_P ^{(m-1)} y_P^2} \right ]$ (721)
$\displaystyle {(\text{CD}_{k \omega}})_P ^{(m-1)}$ $\displaystyle = \max \left ( 2 \rho_P ^{(m)} {\sigma_\omega }_2 \frac{1}{\omega...
...1)} } \nabla k_P ^{(m-1)} \cdot \nabla \omega _P ^{(m-1)} , 10^{-20} \right ) ,$ (722)

where $ y_P$ is the distance from the element center P to the next solid surface.

Using the blending factor one obtains the correct parameter values, e.g.:

$\displaystyle \sigma_k = F_1 {\sigma_k}_1 + (1-F_1) {\sigma_k}_2.$ (723)

For completeness the parameters are listed here:

$\displaystyle \gamma_1$ $\displaystyle = \frac{\beta_1}{\beta^*} - \frac{{\sigma_\omega}_1 \kappa^2}{\sq...
...2 = \frac{\beta_2}{\beta^*} - \frac{{\sigma_\omega}_2 \kappa^2}{\sqrt{\beta^*}}$ (724)
$\displaystyle {\sigma_k}_1$ $\displaystyle = 0.5 \;\;\;\;\; {\sigma_\omega }_1 = 0.5 \;\;\;\;\; \beta_1 = 0.075$ (725)
$\displaystyle {\sigma_k}_2$ $\displaystyle = 1.0 \;\;\;\;\; {\sigma_\omega }_2 = 0.856 \;\;\;\;\; \beta_2 = 0.0828$ (726)
$\displaystyle \beta^*$ $\displaystyle = 0.09 \;\;\;\;\; \kappa=0.41$ (727)

In the conservation equation for k one very easily identifies the time-dependent, convective, diffusive and body terms. They are treated in a completely analogous way to the energy equation. The convective boundary conditions amount to:

For the convective interpolation of $ k$ the modified smart algorithm has not shown any advantages, therefore, the upwind difference scheme is always used.

The diffusion boundary conditions are:

The source terms are treated in the following way:

The equation for the turbulence frequency $ \omega$ runs [51]:

$\displaystyle \frac{\partial \rho \omega }{\partial t} + \nabla \cdot (\rho \om...
...rho (1-F_1) {\sigma_\omega} _2 \frac{1}{\omega } \nabla k \cdot \nabla \omega ,$ (738)

which can be rewritten as:

$\displaystyle \frac{\partial \rho \omega }{\partial t} + \nabla \cdot (\rho \omega \boldsymbol{v})$ $\displaystyle = \nabla \cdot \left [ ( \mu + \sigma_\omega \mu_t) \nabla \omega...
...t ] + \frac{\gamma }{\nu_t} \left [ \frac{\mu_t}{\mu } t_{ij}^l d_{ij} \right ]$    
  $\displaystyle - \frac{2}{3} \gamma \rho \omega d_{kk} - \beta \rho \omega ^2 + 2 \rho (1-F_1) {\sigma_\omega} _2 \frac{1}{\omega } \nabla k \cdot \nabla \omega .$ (739)

One easily recognizes the time-dependent, convective, diffusive and source terms. The convective boundary conditions amount to:

For the convective interpolation of $ \omega$ the modified smart algorithm has not shown any advantages, therefore, the upwind difference scheme is always used.

The diffusion boundary conditions are:

The source terms are treated as follows: