ComputationOfPHI

From DerivePoissonEquationForPressure we have a formulation for \( \Phi(\mathbf{v})\), which is \begin{align} \Phi(\mathbf{v}) = \nabla^T \left( \frac{d \mathbf{v}}{dt} \right) - \frac{d}{dt}\left( \nabla ^{T}\mathbf{v} \right) \end{align} Numerically, we have several choices to compute this term, they all might differ depending on the approximation quality of the differential operators.

• Variant 1: The formal way from the equation above yields \begin{align} \Phi \left( \mathbf{v} \right) = ( u_x u_x + v_x u_y + w_x u_z ) + ( u_y v_x + v_y v_y + w_y v_z) + ( u_z w_x + v_z w_y + w_z w_z )\end{align}
• Variant 2: We try to isolate the divergence of velocity by \begin{align} \Phi \left( \mathbf{v} \right) = 2 (v_x u_y - u_x v_y) + 2( w_x u_z - u_x w_z ) + 2( w_y v_z - v_y w_z) + \left( \nabla^T \mathbf{v} \right)^2\end{align}
• Variant 3: take the divergence of the stationary part of the substantial velocity \begin{align} \Phi \left( \mathbf{v} \right) = \nabla^T \left( u \mathbf{v}_x + v \mathbf{v}_y + w \mathbf{v}_z \right) - u ( \nabla^T \mathbf{v} )_x - v ( \nabla^T \mathbf{v} )_y - w ( \nabla^T \mathbf{v} )_z\end{align}
• Variant 4: take the divergence of the true substantial derivative of the velocity (use \( \frac{d\mathbf{v}}{dt}\) as computed due to reconstructLagrangianAcceleration, triggered by the second digit of FLIQUID_ConsistentPressure_Version ) \begin{align} \Phi \left( \mathbf{v} \right) \approx \nabla^T \left( \frac{ d \mathbf{v} }{ dt} \right) - \frac{ \nabla^T \mathbf{v}^{n+1} - \nabla^T \mathbf{v}^{n} }{\Delta t}\end{align}
• Variant 5: same as version 1, using derived gradient operator (see DIFFOP_gradient)
• Variant 6: same as version 2, using derived gradient operator (see DIFFOP_gradient)
• Variant 7: same as version 3, using derived gradient operator (see DIFFOP_gradient)
• Variant 8: same as version 4, using derived gradient operator (see DIFFOP_gradient)
• Variant 9: \begin{align} \Phi \left( \mathbf{v} \right) = \frac{1}{\Delta t} \nabla^T \left( \mathbf{v}_0 - (\mathbf{v}_{Euler}^T \nabla) \mathbf{v}_0 \right)\end{align}

Especially variants 2 and 3 explicitly contain terms with the divergence of velocity \( \nabla^T \mathbf{v}\) . Even for incompressible flows, the numerical evaluation of this term will not entirely be zero. In order to neglect the divergence anyways, use the option FLIQUID_ConsistentPressure_UseDivV. The hope is to gain smoothness of the pressure solution.