This is to include variable density in the the 1-2eq turbulence models.
Explanation, using the K-omega model as an example:
Previously implemented (K):
\frac{\partial \left( k \right)}{\partial t}
+ \nabla \cdot \left( \mathbf{v} k \right)
- \nabla \cdot \left[ \left( \nu + \sigma_\textrm{k} \frac{k}{\omega}\right) \nabla k \right]
- (2 \nu_t \mathbf{S} \cdot \mathbf{S})
+ \beta_k k \omega = 0
Introduced in this MR (K):
\frac{\partial \left( \rho k \right)}{\partial t}
+ \nabla \cdot \left( \rho k \mathbf{v} \right)
- \nabla \cdot \left[ \left( \mu + \rho \sigma_\textrm{k} \frac{k}{\omega}\right) \nabla k \right]
- (2 \mu_t \mathbf{S} \cdot \mathbf{S})
+ \rho \beta_k k \omega = 0
Previously implemented (\omega
):
\frac{\partial \left( \omega \right)}{\partial t}
+\nabla \cdot \left( \mathbf{v} \omega \right)
-\nabla \cdot \left[ \left( \nu + \sigma_{\omega} \frac{k}{\omega}\right) \nabla \omega \right]
-\alpha \frac{\omega}{k} (2 \nu_t \mathbf{S} \cdot \mathbf{S})
+\beta_{\omega} \omega^2
-\frac{\sigma_d}{\omega} \nabla k \nabla \omega = 0
Introduced in this MR (\omega
):
\frac{\partial \left( \rho \omega \right)}{\partial t}
+\nabla \cdot \left( \rho \mathbf{v} \omega \right)
-\nabla \cdot \left[ \left( \mu + \rho \sigma_{\omega} \frac{k}{\omega}\right) \nabla \omega \right]
-\alpha \frac{\omega}{k} (2 \mu_t \mathbf{S} \cdot \mathbf{S})
+\rho \beta_{\omega} \omega^2
-\rho \frac{\sigma_d}{\omega} \nabla k \nabla \omega = 0
20 \beta_K k \omega \text{ to } 20 \rho \beta_K k \omega
Introduce this additionally for the:
Fix failing kepsilon wall function test
relevant documentation: https://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node99.htm#eq6.3.8bc
- 2/3 \rho k \delta_{ij}
in the production term in the 2-eq models (see: Computational Fluid Dynamics, Versteeg& Malalasekera page 90)
\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}