7.3 Calculation of the Pressure
177
Another similar method of this kind was proposed by Patankar (1980) and
is called SIMPLER. In it, the pressure-correction equation (7.39) is solved
first with the last term neglected as in SIMPLE. The pressure correction so
obtained is used only to correct the velocity field so that it satisfies continuity
i.e. to obtain u y . The new pressure field is calculated from the pressure
equation (7.35) using G T instead of Oy*. This is possible because u y is now
available.
As already noted, due to the neglect of O!, in Eq. (7.39) (which is equivalent
to neglecting it in Eq. (7.37)), the SIMPLE algorithm does not converge
rapidly. Its performance depends greatly on the size of time step, or - for
steady flows - on the value of the under-relaxation parameter used in the
momentum equations. It has been found by trial and error that convergence
can be improved if one adds only a portion of p' to pm-l, i.e. if one takes
pm = pm-l + appl
(7.45)
after the pressure-correction equation is solved, where 0 5 a, 5 1. SIMPLEC,
SIMPLER and PISO do not need under-relaxation of the pressure correction.
One can derive an optimum relation between the under-relaxation factors
for velocities and pressure by the following argument1.
The velocities in the SIMPLE method are corrected by
i.e., O!,,, is neglected. To make up for this crudeness, we may now go back
to the momentum equations (7.31) and look for pressure which will satisfy
these equations when u r * is replaced by corrected velocities u y , which now
satisfy the continuity equation (this is the path that leads to the pressure
equation in SIMPLER). By assuming that the final pressure correction is
aPp1, we arrive a t the following equation:
By making use of Eq. (7.46), we arrive at the following expression for a,:
We can calculate O:,p using Eq. (7.38) but, in multi-dimensional problems,
we would have more than one equation from which a, can be calculated.
However, if instead of calculating GIlp, we use the approximation (7.41) used
in SIMPLEC, then the above equation reduces to:
Raithby and Schneider (1979) found this relation following a different route.
177
Another similar method of this kind was proposed by Patankar (1980) and
is called SIMPLER. In it, the pressure-correction equation (7.39) is solved
first with the last term neglected as in SIMPLE. The pressure correction so
obtained is used only to correct the velocity field so that it satisfies continuity
i.e. to obtain u y . The new pressure field is calculated from the pressure
equation (7.35) using G T instead of Oy*. This is possible because u y is now
available.
As already noted, due to the neglect of O!, in Eq. (7.39) (which is equivalent
to neglecting it in Eq. (7.37)), the SIMPLE algorithm does not converge
rapidly. Its performance depends greatly on the size of time step, or - for
steady flows - on the value of the under-relaxation parameter used in the
momentum equations. It has been found by trial and error that convergence
can be improved if one adds only a portion of p' to pm-l, i.e. if one takes
pm = pm-l + appl
(7.45)
after the pressure-correction equation is solved, where 0 5 a, 5 1. SIMPLEC,
SIMPLER and PISO do not need under-relaxation of the pressure correction.
One can derive an optimum relation between the under-relaxation factors
for velocities and pressure by the following argument1.
The velocities in the SIMPLE method are corrected by
i.e., O!,,, is neglected. To make up for this crudeness, we may now go back
to the momentum equations (7.31) and look for pressure which will satisfy
these equations when u r * is replaced by corrected velocities u y , which now
satisfy the continuity equation (this is the path that leads to the pressure
equation in SIMPLER). By assuming that the final pressure correction is
aPp1, we arrive a t the following equation:
By making use of Eq. (7.46), we arrive at the following expression for a,:
We can calculate O:,p using Eq. (7.38) but, in multi-dimensional problems,
we would have more than one equation from which a, can be calculated.
However, if instead of calculating GIlp, we use the approximation (7.41) used
in SIMPLEC, then the above equation reduces to:
Raithby and Schneider (1979) found this relation following a different route.