DynamicPressure

This pressure only occurs (different from zero) if the fluid is in motion. Therefore it represents the dynamic forces or compression forces. Its basic equation stems from the considerations in DerivePoissonEquationForPressure and is given by \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} We take into account equation (1.10), hence we obtain \begin{align} \begin{array}{*{35}{l}} \frac{-\frac{1}{\Delta t}\frac{1}{\rho }\frac{\partial \rho }{\partial p}\left( p_{dyn}^{n+1}-p_{dyn}^{n} \right)+\overline{\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{array}\end{align} and after sorting terms, the final representation of the dynamic pressure is \begin{align} \begin{matrix} -\frac{1}{\Delta t^{2}}\frac{1}{\rho }\frac{\partial \rho }{\partial p}\left( p_{dyn}^{n+1}-p_{dyn}^{n} \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p_{dyn}^{n+1} \right) \right)= & \frac{1}{\Delta t}\left( \nabla ^{T}\mathbf{v} \right)^{n}-\frac{1}{\Delta t}\overline{\left( \nabla ^{T}\mathbf{v} \right)}_{dyn}^{n+1} \\ {} & +\Psi \left( \mathbf{v}^{n+1} \right)-\Theta_{dyn} \left( \mathbf{v}^{n+1} \right)-\Phi \left( \mathbf{v}^{n+1} \right) \\ \end{matrix}\end{align} Remarks :

• The numerical way of solving this (extremely non-trivial) equation is found in DynamicPressureAlgorithm .
• 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% .