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.
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GASDYN_Upwind2ndOrder DEPRECATED!!! (GASDYN parameter for FPM1)