GASDYN_Upwind2ndOrder
DEPRECATED!!! (GASDYN parameter for FPM1)
This is a deprecated parameter. Do not use anymore. Instead, use the parameters GASDYN_Upwind_Lbeta and GASDYN_Upwind_Lgamma .
Default:
GASDYN_Upwind2ndOrder = 0.5
Represents gamma in the improved (practically second order} upwind velocity \( \mathbf{v}_{uw}^{*} = \mathbf{v}_{uw} - \gamma \frac{\Delta t}{\rho} \nabla p\)
Second order is reached, if
GASDYN_Upwind2ndOrder = 0.5 .
Please remember that the classical upwind velocity is given by \( \mathbf{v}_{uw} = \mathbf{v} - \frac{1}{2\rho c} \left( p^+ - p^- \right)\)
The distance between the upwind locations at the plus(+) and minus(-)-points is ruled by the parameter
GASDYN_UpwindOffset.
This second order idea comes from the following consideration: First order (for example for the density) is given by
\begin{align}
\frac{\rho^{n+1}-\rho^{n}}{\Delta t} = - \bar{\rho} \cdot \nabla^T \left( \mathbf{v} \right)
\end{align}
Higher order (second order) improvement is given by
\begin{align}
\frac{\rho^{n+1}-\rho^{n}}{\Delta t} = - \bar{\rho} \cdot \nabla^T \left( \mathbf{v} + \gamma \Delta t \frac{d \mathbf{v}}{dt} \right) = - \bar{\rho} \cdot \nabla^T \left( \mathbf{v} - \gamma \frac{\Delta t}{\rho} \nabla p \right)
\end{align}
OPTION:
Choose this parameter negative, i.e.
This will lead to the improved upwind velocity
\begin{align} \mathbf{v}_{uw}^{*} = \mathbf{v} - \left| \gamma \right| \frac{h}{c} \frac{1}{\rho} \nabla p\end{align}
This improved upwind idea comes from the consideration that
\begin{align}
\mathbf{v}_{uw} = \mathbf{v} - \frac{1}{2\rho c} \left( p^+ - p^- \right) \approx \mathbf{v} - \gamma \frac{h}{c} \frac{1}{\rho} \nabla p \right)
\end{align}
So, the classical upwind velocity can be approximated in this way. The nice side effect is, that the divergence of the upwind velocity leads to laplace-lke term (damping!!!) in the numerical scheme, i.e.
\begin{align}
\frac{\rho^{n+1}-\rho^{n}}{\Delta t} = - \bar{\rho} \cdot \left[ \nabla^T \left( \mathbf{v} \right) + \nabla^T \left( \left| \gamma \right| \frac{h}{c} \frac{1}{\rho} \nabla p \right) \right]
\end{align}
Note: This parameter can also be set chamberwise for multiphase simulations (see also
KindOfProblem,
CHAMBER).
If it is not set for specific chambers, it is automatically set according to the non-chamberwise definition for all chambers.