7.5 Solution Methods for the Navier-Stokes Equations
195
Finally, substitution of the above expressions (7.107) and (7.109) for u'
and v' into the continuity equation leads to the pressure-correction equation:
where the coefficients are:
The term Amf, is analogous to Am;, with ii' and 6' replacing u* and v*,
see Eqs. (7.101) and (7.110). Since the velocity corrections are not known
prior t o the solution of the pressure-correction equation, this term is usually
neglected, as mentioned in the previous section. We then have the SIMPLE
algorithm.
After the pressure-correction equation has been solved, the velocities and
pressure are corrected. As noted in Sect. 7.3.4, if one tries to calculate steady
flows using very large time steps, the momentum equations must be underrelaxed as described in Sect. 5.4.2, so only part of the pressure correction
p' is added to pm-'. Under-relaxation may also be required in unsteady
calculations with large time steps.
The corrected velocities satisfy the continuity equation to the accuracy
with which the pressure-correction equation is solved. However, they do not
satisfy the non-linear momentum equation, so we have to begin another outer
iteration. When both the continuity and momentum equations are satisfied
to the desired tolerance, we can proceed to the next time level. To begin
the iterations at the new time step, the solution at the previous time step
provides the initial guess. This may be improved by use of extrapolation. For
small time steps extrapolation is fairly accurate and saves a few iterations.
A computer code employing this algorithm is available via Internet; see
Appendix for details. Some examples of its application and performance are
presented below.
The above algorithm is easily modified t o give the SIMPLEC method described in Sect. 7.3.4. The pressure-correction equation has the form (7.111),
but A; and A; are replaced by A; + El A; and A; + El A:, respectively.
The extension to the P I S 0 algorithm is also straightforward. The second
pressure-correction equation has the same coefficient matrix as the first one,
but the source term is now -Am;. This term was neglected in the first
pressure-correction equation, but it can now be calculated using the first
velocity correction ui.
195
Finally, substitution of the above expressions (7.107) and (7.109) for u'
and v' into the continuity equation leads to the pressure-correction equation:
where the coefficients are:
The term Amf, is analogous to Am;, with ii' and 6' replacing u* and v*,
see Eqs. (7.101) and (7.110). Since the velocity corrections are not known
prior t o the solution of the pressure-correction equation, this term is usually
neglected, as mentioned in the previous section. We then have the SIMPLE
algorithm.
After the pressure-correction equation has been solved, the velocities and
pressure are corrected. As noted in Sect. 7.3.4, if one tries to calculate steady
flows using very large time steps, the momentum equations must be underrelaxed as described in Sect. 5.4.2, so only part of the pressure correction
p' is added to pm-'. Under-relaxation may also be required in unsteady
calculations with large time steps.
The corrected velocities satisfy the continuity equation to the accuracy
with which the pressure-correction equation is solved. However, they do not
satisfy the non-linear momentum equation, so we have to begin another outer
iteration. When both the continuity and momentum equations are satisfied
to the desired tolerance, we can proceed to the next time level. To begin
the iterations at the new time step, the solution at the previous time step
provides the initial guess. This may be improved by use of extrapolation. For
small time steps extrapolation is fairly accurate and saves a few iterations.
A computer code employing this algorithm is available via Internet; see
Appendix for details. Some examples of its application and performance are
presented below.
The above algorithm is easily modified t o give the SIMPLEC method described in Sect. 7.3.4. The pressure-correction equation has the form (7.111),
but A; and A; are replaced by A; + El A; and A; + El A:, respectively.
The extension to the P I S 0 algorithm is also straightforward. The second
pressure-correction equation has the same coefficient matrix as the first one,
but the source term is now -Am;. This term was neglected in the first
pressure-correction equation, but it can now be calculated using the first
velocity correction ui.