DerivePoissonEquationForPressure

The Poisson equation for the pressure can be derived by application of the divergence-operator to the equation of momentum: \begin{align} \nabla ^{T}\mathbf{\dot{v}}+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p \right) \right)=\nabla ^{T}\left( \frac{1}{\rho }\nabla \mathbf{S}_{s} \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla \mathbf{S}_{v}\left( \mathbf{v} \right) \right)+\nabla ^{T}\mathbf{g}-\nabla ^{T}\left( \beta \cdot (\mathbf{v} - \mathbf{v}_{\beta}) \right)\end{align} That gives \begin{align} \frac{d}{dt}\left( \nabla ^{T}\mathbf{v} \right)+\Phi \left( \mathbf{v} \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p \right) \right)=\nabla ^{T}\left( \frac{1}{\rho }\nabla \mathbf{S}_{s} \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla \mathbf{S}_{v}\left( \mathbf{v} \right) \right)+\nabla ^{T}\mathbf{g}-\nabla ^{T}\left( \beta \cdot (\mathbf{v} - \mathbf{v}_{\beta}) \right)\end{align} using the definitions \( \mathbf{v}=\left( u , v , w \right)^{T}\) \( \Phi \left( \mathbf{v} \right)\equiv \left( \nabla u \right)^{T}\cdot \frac{\partial \mathbf{v}}{\partial x}+\left( \nabla v \right)^{T}\cdot \frac{\partial \mathbf{v}}{\partial y}+\left( \nabla w \right)^{T}\cdot \frac{\partial \mathbf{v}}{\partial z}\) More simplifications can achieved by defining \( \Psi \left( \mathbf{v} \right)\equiv \nabla ^{T}\left( \frac{1}{\rho }\nabla \mathbf{S}_{v}\left( \mathbf{v} \right) \right)\) \( \Pi \left( \mathbf{v} \right)\equiv \nabla ^{T}\left( \frac{1}{\rho }\nabla \mathbf{S}_{s} \right)\) \( \Theta \left( \mathbf{v} \right)\equiv \nabla ^{T}\left( \beta \cdot (\mathbf{v} - \mathbf{v}_{\beta}) \right)\) That means \begin{align} \frac{d}{dt}\left( \nabla ^{T}\mathbf{v} \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p \right) \right)=\Pi +\nabla ^{T}\mathbf{g}+\Psi \left( \mathbf{v} \right)-\Theta \left( \mathbf{v} \right)-\Phi \left( \mathbf{v} \right)\end{align} In a numerical sense, this is \begin{align} \frac{\left( \nabla ^{T}\mathbf{v} \right)^{n+1}-\left( \nabla ^{T}\mathbf{v} \right)^{n}}{\Delta t}+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p^{n+1} \right) \right)=\Pi ^{n+1}+\nabla ^{T}\mathbf{g}+\Psi \left( \mathbf{v}^{n+1} \right)-\Theta \left( \mathbf{v}^{n+1} \right)-\Phi \left( \mathbf{v}^{n+1} \right)\end{align} Splitting this equation into a hydrostatic and a dynamic part yields \begin{align} \frac{\left( \nabla ^{T}\mathbf{v} \right)_{hyd}^{n+1}+\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_{hyd}^{n+1}+p_{dyn}^{n+1} \right) \right)=\Pi ^{n+1}+\nabla ^{T}\mathbf{g}+\Psi \left( \mathbf{v}^{n+1} \right)-\Theta \left( \mathbf{v}^{n+1} \right)-\Phi \left( \mathbf{v}^{n+1} \right)\end{align} The different parts of pressure are more precisely described HydrostaticPressure and DynamicPressure . The splitting of \( \Theta \left( \mathbf{v}^{n+1} \right)\) into hydrostatic and dynamic parts is explained in ComputationOfTHETA .