Simulate with complex geometries and complex physics
GeneralizedUpwind
Generalized way of computing upwind terms
This section here generalizes the way of computing upwind pressure and upwind velocity, stated in SimplifiedFastUpwindTerms and ClassicalUpwindTerms .
Method 1: original method, see ClassicalUpwindTerms
\begin{align} \begin{array}{l}
\bar p_{orig} = p - \frac{{\rho c}}{2}\left( {{\mathbf{v}}^ + - {\mathbf{v}}^ - } \right)^T{\mathbf{n}}{\text{ with }}{\mathbf{v}}^ + \left( {\mathbf{x}} \right) = {\mathbf{\tilde v}}\left( {{\mathbf{x}} + \alpha h{\mathbf{n}}} \right){\text{ and }}{\mathbf{v}}^ - \left( {\mathbf{x}} \right) = {\mathbf{\tilde v}}\left( {{\mathbf{x}} - \alpha h{\mathbf{n}}} \right)\\
{\mathbf{\bar v}}_{orig} = {\mathbf{v}} - \frac{1}{{2\rho c}}\left( {p^ + - p^ - } \right){\mathbf{n}}{\text{ with }}p^ + \left( {\mathbf{x}} \right) = \tilde p\left( {{\mathbf{x}} + \alpha h{\mathbf{n}}} \right){\text{ and }}p^ - \left( {\mathbf{x}} \right) = \tilde p\left( {{\mathbf{x}} - \alpha h{\mathbf{n}}} \right)
\end{array}\end{align}
With
\( \alpha h\) the upwind step length
\( {\mathbf{n}} = \frac{1}{{\left\| {\nabla p} \right\|}}\nabla p\) the upwind direction
Method 2: simplified and fast upwind quantities, see SimplifiedFastUpwindTerms
\begin{align} \begin{array}{l}
\bar p_{simp} = p - \rho cL\left( {\tilde \nabla ^T{\mathbf{v}}} \right)\\
{\mathbf{\bar v}}_{simp} = {\mathbf{v}} - \frac{L}{{\rho c}}\tilde \nabla p
\end{array}\end{align}
With
\( L = \left\{ {\begin{array}{*{20}{c}}
{{\text{either }}\beta \cdot \Delta t \cdot c}\\
{{\text{or }}\gamma \cdot h{\text{ }}}
\end{array}} \right.\) the upwind step length (needed in the Taylor series expansion)
Hint: choosing \( \beta = 0.5\) will lead to second order in time integration.
Combined method
We can bring both methods together into one
\begin{align} \begin{array}{l}
\bar p_{general}{\rm{ }} = {\rm{ }}p{\rm{ }} - {\rm{ }}\frac{{\rho c}}{2}\left( {{\mathbf{v}}^ + - {\mathbf{v}}^ - } \right)^T{\mathbf{n}}{\rm{ }} - {\rm{ }}\rho cL\left( {\tilde \nabla ^T{\mathbf{v}}} \right)\\
{\mathbf{\bar v}}_{general}{\rm{ }} = {\rm{ }}{\mathbf{v}}{\rm{ }} - {\rm{ }}\frac{1}{{2\rho c}}\left( {p^ + - p^ - } \right){\mathbf{n}}{\rm{ }} - {\rm{ }}\frac{L}{{\rho c}}\tilde \nabla p
\end{array}\end{align}
- Define the parameter of \( \alpha\) in GASDYN_UpwindOffset,
- define the parameter of \( \beta\) in GASDYN_Upwind_Lbeta,
- define the parameter of \( \gamma\) in GASDYN_Upwind_Lgamma.
- FPM1 (original) \begin{align} \begin{array}{l} \alpha = 0.2\\ \beta = 0\\ \gamma = 0 \end{array}\end{align}
- FPM1 (simplified, currently implemented in VPS) \begin{align} \begin{array}{l} \alpha = 0\\ \beta = 0\\ \gamma = 0.2 \end{array}\end{align}
- FPM3 \begin{align} \begin{array}{l} \alpha = 0\\ \beta = \left\{ {\begin{array}{*{20}{c}} {0.5{\text{ }}if{\text{ }}\nabla ^T{\mathbf{v}} > 0{\text{ (rarefaction)}}}\\ {0{\text{ elsewise }}} \end{array}} \right.\\ \gamma = \left\{ {\begin{array}{*{20}{c}} {0.0{\text{ }}if{\text{ }}\nabla ^T{\mathbf{v}} > 0{\text{ (rarefaction)}}}\\ {0.2{\text{ elsewise }}} \end{array}} \right. \end{array}\end{align}