8.8 Pressure-Correction Equation
249
(8.55)
The second term on the right hand side disappears when the line connecting
nodes P and E is orthogonal to the cell face i.e. when P and PI and E and
El coincide. If the objective is to prevent pressure oscillations on colocated
grids, it is sufficient to use just the first term on the right-hand side of Eq.
(8.56), i.e. one can approximate Eq. (8.51) as:
v :
,
'
= (v,F*), -
(8.56)
The correction term in square brackets thus represents the difference between
the pressure difference p~ - pp and the approximation to it calculated using
interpolated pressure gradient, (gradp), - (TE - r p ) . For a smooth pressure
distribution, this correction term is small and it tends to zero as the grid
is refined. The pressure gradient at the CV centers is available as it was
calculated for use in the momentum equations.
These mass fluxes calculated using the interpolated velocity,
do not satisfy the continuity requirement, so their sum results in a mass
source:
which must be made to be zero. The velocities have to be corrected so that
mass conservation is satisfied in each CV. In an implicit method it is not
necessary to satisfy mass conservation exactly at the end of each outer iteration. Following the method described earlier, we correct the mass fluxes
by expressing the velocity correction through the gradient of the pressure
correction, thus:
249
(8.55)
The second term on the right hand side disappears when the line connecting
nodes P and E is orthogonal to the cell face i.e. when P and PI and E and
El coincide. If the objective is to prevent pressure oscillations on colocated
grids, it is sufficient to use just the first term on the right-hand side of Eq.
(8.56), i.e. one can approximate Eq. (8.51) as:
v :
,
'
= (v,F*), -
(8.56)
The correction term in square brackets thus represents the difference between
the pressure difference p~ - pp and the approximation to it calculated using
interpolated pressure gradient, (gradp), - (TE - r p ) . For a smooth pressure
distribution, this correction term is small and it tends to zero as the grid
is refined. The pressure gradient at the CV centers is available as it was
calculated for use in the momentum equations.
These mass fluxes calculated using the interpolated velocity,
do not satisfy the continuity requirement, so their sum results in a mass
source:
which must be made to be zero. The velocities have to be corrected so that
mass conservation is satisfied in each CV. In an implicit method it is not
necessary to satisfy mass conservation exactly at the end of each outer iteration. Following the method described earlier, we correct the mass fluxes
by expressing the velocity correction through the gradient of the pressure
correction, thus: