MASS_correction_DivergenceVelocity
Mass Correction for weakly compressible flow problems
Default:
MASS_correction_DivergenceVelocity = 'NON'
It only works for the LIQUID solver and for problems with pressure dependent densities!
Furthermore at the moment an inflow and/or outflow boundary condition is required to determine the target mass!
The idea is to add a source term \( \tilde{q}\) to the continuity equation
\begin{align}
\frac{D \varrho}{D t} + \varrho \, \nabla \cdot \mathbf{v} = \tilde{q}
\end{align}
in order to compensate the mass loss resp. the gain in mass. Hence the desired divergence of velocity for the
CorrectionPressureAlgorithm
(see
DesiredAndNominalDivergenceOfVelocity) is computed by
\begin{align}
\nabla \cdot \mathbf{v} = -\frac{1}{\varrho} \left(\frac{D \varrho}{D t} - \tilde{q}\right)
\end{align}
This is also used for the computation of the dynamic pressure (see
FLIQUID_ConsistentPressure_Version,
ClassicalDPA,
RegularizeDPA,
AlternativeDPA).
Therefore it can be interpreted as a correction method of the dynamic pressure as well.
For the computation of the source term \( \tilde{q}\) the relative error \( \mathrm{err}\) of target mass
\begin{align}
M_{t} = \int\limits_{t^0}^{t^n} \int\limits_{\partial\Omega_{\mathrm{in/out}}} \varrho(t) \big({\bf v}(t) \cdot {\bf n}(t)\big) \, dA \, dt \approx \sum_{k=0}^{n} \sum_{i \in P} \varrho_{i}(t^k) \big({\bf v}_i(t^k) \cdot {\bf n}_i(t^k)\big) A_i
\end{align}
and current mass
\begin{align}
M_{c} = \int\limits_{\Omega} \varrho \, dV - \int\limits_{\Omega} \varrho^{(\mathrm{start})} \, dV \approx \sum_{i \in P} \varrho_{i} \cdot V_i - \sum_{i \in P} \varrho_{i}^{(\mathrm{start})} \cdot V_i
\end{align}
is computed by
\begin{align}
\mathrm{err} = \frac{M_t - M_c}{M_t}.
\end{align}
Moreover the relative error \( \mathrm{err}\) is weighted with a coefficient \( d \in [1,1200]\), which depends on the absolute error, the smaller the mass difference the higher \( d\).
But overall the product \( \mathrm{coeff} = d \cdot \mathrm{err}\) is limited by
\begin{align}
-12 \leq \mathrm{coeff} \leq 12.
\end{align}
This results in the source term
\( \tilde{q} = \mathrm{coeff} \cdot \tilde{\varrho} = d \, \dfrac{M_t - M_c}{M_t} \, \tilde{\varrho},\)
where
\begin{align}
\tilde{\varrho} = \frac{1}{N} \sum_{i = 1}^{N} \varrho_{i}
\end{align}
is the average density.