Mass conservation equation

For incompressible flow the mass flow as expressed by Equation () can be written as:

\dot{m} _f ^{(m)} = \overrightarrow{\rho }_f ^{(m-1)} A_f \boldsymbol{v}_f ^{(m)} \cdot \boldsymbol{n}_f , (762)

where \overrightarrow{\rho }_f^{(m-1)} is a function of \overrightarrow{T }_f ^{(m-1)} only. Therefore, the density is constant while treating the mass conservation equation. Using Equation () this can be further rewritten as:

\dot{m} _f ^{(m)} = \overrightarrow{\rho }_f ^{(m-1)} A_f \left ( \overline{\boldsymbol{v} }_f ^{(m)} + \overline{D}_f \left [ \overline{\nabla p}_f ^{(m)} - \nabla p_f ^{(m)} \right ] \right ) \cdot \boldsymbol{n}_f . (763)

In the next step Equation () is approximated by:

\boldsymbol{v}_P ^{(m)} + \boldsymbol{H}_P (\boldsymbol{v}^* ) = B_P ^{(m-1)} - D_P \nabla p ^{(m)} _P (764)

Notice that \boldsymbol{v} ^{(m)} in the \boldsymbol{H}-term was replaced by \boldsymbol{v}^*. This does not correspond to the SIMPLE algorithm, since the term is not neglected, but also not quite corresponds to the SIMPLEC algorithm, since \boldsymbol{v}^* is used. Writing this equation for the neighboring element E and taking the mean leads to:

\overline{\boldsymbol{v}}_f ^{(m)} + \overline{\boldsymbol{H}}_f (\boldsymbol{v}^* ) = \overline{B}_f ^{(m-1)} - \overline{D}_f \overline{\nabla p} ^{(m)} _f, (765)

in which the second order correction \overline{D_f \nabla p ^{(m)} _f} \approx \overline{D}_f \overline{\nabla p} ^{(m)} _f was used. Therefore Equation () now reduces to:

\dot{m} _f ^{(m)} = \overrightarrow{\rho }_f ^{(m-1)} A_f \left ( \overline{B}_f ^{(m-1)}- \overline{\boldsymbol{H}}_f (\boldsymbol{v}^* ) - \overline{D}_f \nabla p_f ^{(m)} \right ) \cdot \boldsymbol{n}_f , (766)

or, with the abbreviation \boldsymbol{v}_f^{\Box}:= \overline{B}_f ^{(m-1)}- \overline{\boldsymbol{H}}_f (\boldsymbol{v}^* ):

\dot{m} _f ^{(m)} = \overrightarrow{\rho }_f ^{(m-1)} A_f \left ( \boldsymbol{v}_f^{\Box} - \overline{D}_f \nabla p_f ^{(m)} \right ) \cdot \boldsymbol{n}_f . (767)

If the velocity on the face is known the mass flow can be calculated directly. Therefore, Equation () now reads:

\sum_{f\setminus BC}^{} \overrightarrow{\rho }_f ^{(m-1)} A_f \overline{D}_f \nabla p_f ^{(m)} \cdot \boldsymbol{n}_f = \sum_{f\setminus BC}^{} \overrightarrow{\rho }_f ^{(m-1)} A_f \boldsymbol{v}_f^{\Box} \cdot \boldsymbol{n}_f +\sum_{BC}^{} \dot{m} _f ^{(m)}, (768)

where BC stands for the faces on which the velocity is defined by the user. Notice that the unknown here is the pressure and not the correction to the pressure as in compressible flow.

The pressure gradient can be treated as in Equation (). Alternatively, one can also write (Figure ):

Figure: Alternative derivation of the pressure gradient

\nabla p_f = \frac{p_{F'}-p_{P'}}{(\boldsymbol{r}_F - \boldsymbol{r}_P)\cdot \boldsymbol{n} }. (769)

Since

p_{F'} \approx p_F + \nabla p_F \cdot (\boldsymbol{r}_{F'…} - \boldsymbol{r}_F) (770)

and

p_{P'} \approx p_P + \nabla p_P \cdot (\boldsymbol{r}_{P'…} - \boldsymbol{r}_P) (771)

one obtains:

(\nabla p_f \cdot \boldsymbol{n}_f)^s \approx \frac{p_{F}^s-p_{P}^s}{(\boldsymbol{r}_F - \boldsymbol{r}_P)\cdot \boldsymbol{n} } + \left ( \frac{\nabla p_F \cdot (\boldsymbol{r}_{F'} - \boldsymbol{r}_F) - \nabla p_P \cdot (\boldsymbol{r}_{P'} - \boldsymbol{r}_P) }{(\boldsymbol{r}_F - \boldsymbol{r}_P)\cdot \boldsymbol{n}} \right ) ^{s-1}. (772)

If the pressure is known on the face, the expression reduces to:

(\nabla p_f \cdot \boldsymbol{n}_f)^s \approx \frac{p_{f}-p_{P}^s}{(\boldsymbol{r}_f - \boldsymbol{r}_P)\cdot \boldsymbol{n} } + \left ( \frac{- \nabla p_P \cdot (\boldsymbol{r}_{P'} - \boldsymbol{r}_P) }{(\boldsymbol{r}_f - \boldsymbol{r}_P)\cdot \boldsymbol{n}} \right ) ^{s-1}. (773)