7.4 Other Methods
187
This equation is very similar to the pressure-correction equation of the SIMPLE method, Eq. (7.39). It thus appears that all the pressure calculation
methods presented so far, although arrived a t via different routes, reduce
to the same basic method. Again, it is important that the pressure derivatives inside brackets are approximated in the same way as in the momentum
equations, while the outer derivative is the one from the continuity equation.
The crucial factor for convergence of a method based on artificial compressibility is the choice of parameter p. The optimum value is problem dependent, although some authors have suggested an automatic procedure for
choosing it. A very large value would require the corrected velocity field to
satisfy the incompressible continuity equation. In the above version of the
method, this corresponds to the SIMPLE scheme without under-relaxation
of pressure correction; the procedure would then converge only for small At.
However, if only a portion of p' is added to the pressure as in SIMPLE,
infinitely large p could be used.
On the other hand, the lowest value of p allowed can be determined by
looking a t the propagation speed of pressure waves. The pseudo-sound speed
is:
By requiring pressure waves to propagate much faster than the vorticity
spreads, the following criterion can be derived for a simple channel flow (see
Kwak et al., 1986):
where XL is the distance between inlet and outlet, x6 is half the distance
between two walls and x,,f is the reference length. Typical values of p used in
various methods based on artificial compressibility were in the range between
0.1 and 10.
Obviously, 1/(pAt) should be small compared to the coefficients arising
from the second term in Eq. (7.78) if the corrected velocity field is to closely
satisfy the continuity equation. This is a necessity if rapid convergence is to
be obtained. For some iterative solution methods (e.g. using domain decomposition technique in parallel processing or block-structured grids in complex geometries) it has been found useful to divide the Ap coefficient of
the pressure-correction equation in SIMPLE by a factor smaller than unity
(0.95 to 0.99). This is equivalent to the artificial compressibility method with
l / ( p A t ) M (0.01 to 0.05)Ap.
187
This equation is very similar to the pressure-correction equation of the SIMPLE method, Eq. (7.39). It thus appears that all the pressure calculation
methods presented so far, although arrived a t via different routes, reduce
to the same basic method. Again, it is important that the pressure derivatives inside brackets are approximated in the same way as in the momentum
equations, while the outer derivative is the one from the continuity equation.
The crucial factor for convergence of a method based on artificial compressibility is the choice of parameter p. The optimum value is problem dependent, although some authors have suggested an automatic procedure for
choosing it. A very large value would require the corrected velocity field to
satisfy the incompressible continuity equation. In the above version of the
method, this corresponds to the SIMPLE scheme without under-relaxation
of pressure correction; the procedure would then converge only for small At.
However, if only a portion of p' is added to the pressure as in SIMPLE,
infinitely large p could be used.
On the other hand, the lowest value of p allowed can be determined by
looking a t the propagation speed of pressure waves. The pseudo-sound speed
is:
By requiring pressure waves to propagate much faster than the vorticity
spreads, the following criterion can be derived for a simple channel flow (see
Kwak et al., 1986):
where XL is the distance between inlet and outlet, x6 is half the distance
between two walls and x,,f is the reference length. Typical values of p used in
various methods based on artificial compressibility were in the range between
0.1 and 10.
Obviously, 1/(pAt) should be small compared to the coefficients arising
from the second term in Eq. (7.78) if the corrected velocity field is to closely
satisfy the continuity equation. This is a necessity if rapid convergence is to
be obtained. For some iterative solution methods (e.g. using domain decomposition technique in parallel processing or block-structured grids in complex geometries) it has been found useful to divide the Ap coefficient of
the pressure-correction equation in SIMPLE by a factor smaller than unity
(0.95 to 0.99). This is equivalent to the artificial compressibility method with
l / ( p A t ) M (0.01 to 0.05)Ap.