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ComputationOfTHETA
how to numerically compute the source term due to the Darcy forces
We see in DerivePoissonEquationForPressure , HydrostaticPressure , and DynamicPressure that the source term due to the Darcy-forces is subdivided into its hydrostatic and dynamic parts.
For this we provide 4 different variants, which are meant for experimenting. The Darcy-term has the property, that it might produce huge acceleration forces at very
low velocities, as \( \beta\), the Darcy constant \( \tilde{\beta}\) divided by \( \rho\), becomes big (which is naturally possible).
So, the numerics is very sensitive in these cases, and a final general stability condition could not yet be determined.
- Variant 1: the original and numerically most natural version \begin{align} \Theta^{n+1}_{hyd} = \nabla^T \left( \beta \cdot \mathbf{v}_{\beta} \right)\end{align} \begin{align} \Theta^{n+1}_{dyn} = \nabla^T \left( \beta \cdot \mathbf{v}^{n+1} \right)\end{align}
- Variant 2: bring the Darcy-contributions mainly to the hydrostatic part of the pressure \begin{align} \Theta^{n+1}_{hyd} = \nabla^T \left( \beta \cdot ( \mathbf{v}_{\beta} - \mathbf{v}^{n}) \right)\end{align} \begin{align} \Theta^{n+1}_{dyn} = \nabla^T \left( \beta \cdot ( \mathbf{v}^{n+1} - \mathbf{v}^{n} ) \right) \approx 0\end{align}
- Variant 3: bring the Darcy-contributions mainly to the dynamic part of the pressure \begin{align} \Theta^{n+1}_{hyd} = 0\end{align} \begin{align} \Theta^{n+1}_{dyn} = \nabla^T \left( \beta \cdot (\mathbf{v}^{n+1} - \mathbf{v}_{\beta}) \right)\end{align}
- Variant 4: assume perfect adaption of the fluid velocity to the Darcy basis velocity, i.e. assume \( \left(\mathbf{v}^{n+1} - \mathbf{v}_{\beta}) \right) = 0\): \begin{align} \Theta^{n+1}_{hyd} = 0\end{align} \begin{align} \Theta^{n+1}_{dyn} = 0\end{align}