7.4 Other Methods
185
To connect this method with the ones described above, we note that the
intermediate velocity field ( ~ f ) " + ~ ,
obtained from the momentum equations
using old pressure, does not satisfy the incompressible continuity equation,
i.e.,
at the time step n + l . The derivatives on the left hand side of this equation are
evaluated by some finite difference approximation; the choice is not important
here, which is why symbolic difference notation is used.
For solving these equations, many methods are available. In fact, because
each equation now contains a time derivative, methods employed to solve
them can be modeled after ones used to solve ordinary differential equations
presented in Chap. 6. Because the artificial compressibility method is principally intended for steady flows, implicit methods should be favored. Another
important point is that the principal difficulty faced in compressible flow,
namely the possibility of transition from subsonic to supersonic flow and, especially, the possible existence of shock waves can be avoided. The best choice
for a solution method for two or three dimensional problems is an implicit
method that does not require the solution of a full two or three dimensional
problem at each time step, which means that an alternating direction implicit or approximate factorization method is the best choice. An example of
a scheme for deriving a pressure equation using artificial compressibility is
presented below.
The simplest scheme uses a first order explicit discretization in time; it
enables pointwise calculation of pressure, but places a severe restriction on the
size of time step. Since the time development of pressure is not important
and we are interested in obtaining the steady state solution as quickly as
possible, the fully implicit Euler scheme is a better choice:
The problem is that the velocity field at the new time level is not known.
However, one can linearize about the known old state and transform the
above equation into a Poisson equation for pressure or pressure correction!
Let us see how this can be done.
The unknown quantity
may be approximated as:
By inserting this expression into the continuity equation (7.69) we obtain an
equation for the new pressure pn+l.
185
To connect this method with the ones described above, we note that the
intermediate velocity field ( ~ f ) " + ~ ,
obtained from the momentum equations
using old pressure, does not satisfy the incompressible continuity equation,
i.e.,
at the time step n + l . The derivatives on the left hand side of this equation are
evaluated by some finite difference approximation; the choice is not important
here, which is why symbolic difference notation is used.
For solving these equations, many methods are available. In fact, because
each equation now contains a time derivative, methods employed to solve
them can be modeled after ones used to solve ordinary differential equations
presented in Chap. 6. Because the artificial compressibility method is principally intended for steady flows, implicit methods should be favored. Another
important point is that the principal difficulty faced in compressible flow,
namely the possibility of transition from subsonic to supersonic flow and, especially, the possible existence of shock waves can be avoided. The best choice
for a solution method for two or three dimensional problems is an implicit
method that does not require the solution of a full two or three dimensional
problem at each time step, which means that an alternating direction implicit or approximate factorization method is the best choice. An example of
a scheme for deriving a pressure equation using artificial compressibility is
presented below.
The simplest scheme uses a first order explicit discretization in time; it
enables pointwise calculation of pressure, but places a severe restriction on the
size of time step. Since the time development of pressure is not important
and we are interested in obtaining the steady state solution as quickly as
possible, the fully implicit Euler scheme is a better choice:
The problem is that the velocity field at the new time level is not known.
However, one can linearize about the known old state and transform the
above equation into a Poisson equation for pressure or pressure correction!
Let us see how this can be done.
The unknown quantity
may be approximated as:
By inserting this expression into the continuity equation (7.69) we obtain an
equation for the new pressure pn+l.