ClassicalDPA

This algorithm is invoked if the first digit of the variable FLIQUID_ConsistentPressure_Version is put to 1. According to DynamicPressure, the precise model for the dynamic pressure is \begin{align}\frac{\left( \nabla ^{T}\mathbf{v} \right)_{dyn}^{n+1}-\left( \nabla ^{T}\mathbf{v} \right)^{n}}{\Delta t}+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p_{dyn}^{n+1} \right) \right)=\Psi \left( \mathbf{v}^{n+1} \right)-\Theta_{dyn} \left( \mathbf{v}^{n+1} \right)-\Phi \left( \mathbf{v}^{n+1} \right)\end{align} Which, in another way, is \begin{align}\frac{\left( \overline{\nabla ^{T}\mathbf{v}} \right)_{dyn}^{n+1}-\frac{1}{\Delta t}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}\left( p_{dyn}^{n+1}-p_{dyn}^{n} \right) \right)-\left( \nabla ^{T}\mathbf{v} \right)^{n}}{\Delta t}+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p_{dyn}^{n+1} \right) \right)=\Psi \left( \mathbf{v}^{n+1} \right)-\Theta_{dyn} \left( \mathbf{v}^{n+1} \right)-\Phi \left( \mathbf{v}^{n+1} \right)\end{align} Reorganization (step by step) yields \begin{align}\frac{1}{\Delta t}\left( \left( \overline{\nabla ^{T}\mathbf{v}} \right)_{dyn}^{n+1}-\left( \nabla ^{T}\mathbf{v} \right)^{n} \right)-\frac{1}{\Delta t^{2}}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}\left( p_{dyn}^{n+1}-p_{dyn}^{n} \right) \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p_{dyn}^{n+1} \right) \right)=\Psi \left( \mathbf{v}^{n+1} \right)-\Theta_{dyn} \left( \mathbf{v}^{n+1} \right)-\Phi \left( \mathbf{v}^{n+1} \right)\end{align} And finally \begin{align}\begin{array}{*{35}{l}} -\frac{1}{\Delta t^{2}}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}p_{dyn}^{n+1} \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p_{dyn}^{n+1} \right) \right)= & -\frac{1}{\Delta t}\left( \left( \overline{\nabla ^{T}\mathbf{v}} \right)_{dyn}^{n+1}-\left( \nabla ^{T}\mathbf{v} \right)^{n} \right) \\ {} & -\frac{1}{\Delta t^{2}}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}p_{dyn}^{n} \right) \\ {} & \text{+}\Psi \left( \mathbf{v}^{n+1} \right)-\Theta_{dyn} \left( \mathbf{v}^{n+1} \right)-\Phi \left( \mathbf{v}^{n+1} \right) \\ \end{array}\end{align} The numerical discretization of this PDE is \begin{align} \sum\limits_{j=1}^{N(i)} \left( c^{\nabla^T \frac{1}{\rho} \nabla}_{ij} - \delta_{ij} \frac{1}{\Delta t^2} \frac{1}{\rho} \frac{\partial \rho }{\partial p} \right) p_{j} = \sum\limits_{j=1}^{N(i)} W_{ij} p_{j} = Q_i\end{align} with \( i\) running over all MESHFREE point indices, \( W_{ij}\) being the matrix indices and \( Q_i\) the right hand side vector of the global, sparse linear system.

• the derivation of the divergence of the velocity terms is found in DeriveDivergenceOfVelocity
• The value of \( \overline{\left( \nabla ^{T}\mathbf{v} \right)}_{dyn}^{n+1}\) is temporarily stored in the varible %ind_div_bar% . Moreover, for postprocessing reasons, it can be retrieved from the variable %ind_div_bar_pDyn% .
• the value of \( \rho_{dyn}^n\), needed for the term \( \overline{\left( \nabla ^{T}\mathbf{v} \right)}_{dyn}^{n+1}\), is found in the variable %ind_r_pDyn% .