172
7. Solution of the Navier-Stokes Equations
needs to be found iteratively. An interesting possibility is t o use the alternating direction implicit (ADI) method to split the equations into a series of
one dimensional problems, each of which is block tridiagonal. The solution a t
the new time step can then be found with sufficient accuracy with just one
iteration (one set of block tridiagonal solutions in each direction).
So a reasonable strategy is to use the local linearization based on Eq.
(7.24) and update the equations by the AD1 method using the old pressure
gradient. We can then correct the velocity field using the following scheme.
0 Call the velocity field computed by updating the momentum equations with
the old pressure gradient uf . It does not satisfy the continuity equation.
0 Solve a Poisson equation for the pressure correction:
0 Update the velocity:
which does satisfy continuity.
With these tricks, the method suggested here is about twice as expensive as
the explicit method per time step.
The method described above is designed to produce an accurate solution
of an unsteady problem. In problems of that kind, the required accuracy in
time usually sets the time step, which will be rather small. Because they
allow large time steps to be used without instability, implicit methods are
often used to solve steady state problems. The idea is to compute in time
until a steady solution is obtained. In this type of calculation, the error made
in linearizing the problem is no longer negligible and the type of method
described here may not be the best choice. Methods designed for solving
steady state problems are given in the next section. They introduce other
means of getting around the problems we encountered here.
7.3.4 Implicit Pressure-Correction Methods
As noted in Chap. 6, many methods for steady problems can be regarded as
solving an unsteady problem until a steady state is reached. The principal
difference is that, when solving an unsteady problem, the time step is chosen
so that an accurate history is obtained while, when a steady solution is sought,
large time steps are used to try to reach the steady state quickly. Implicit
methods are preferred for steady and slow-transient flows, because they have
less stringent time step restrictions than explicit schemes (they may not have
any).
7. Solution of the Navier-Stokes Equations
needs to be found iteratively. An interesting possibility is t o use the alternating direction implicit (ADI) method to split the equations into a series of
one dimensional problems, each of which is block tridiagonal. The solution a t
the new time step can then be found with sufficient accuracy with just one
iteration (one set of block tridiagonal solutions in each direction).
So a reasonable strategy is to use the local linearization based on Eq.
(7.24) and update the equations by the AD1 method using the old pressure
gradient. We can then correct the velocity field using the following scheme.
0 Call the velocity field computed by updating the momentum equations with
the old pressure gradient uf . It does not satisfy the continuity equation.
0 Solve a Poisson equation for the pressure correction:
0 Update the velocity:
which does satisfy continuity.
With these tricks, the method suggested here is about twice as expensive as
the explicit method per time step.
The method described above is designed to produce an accurate solution
of an unsteady problem. In problems of that kind, the required accuracy in
time usually sets the time step, which will be rather small. Because they
allow large time steps to be used without instability, implicit methods are
often used to solve steady state problems. The idea is to compute in time
until a steady solution is obtained. In this type of calculation, the error made
in linearizing the problem is no longer negligible and the type of method
described here may not be the best choice. Methods designed for solving
steady state problems are given in the next section. They introduce other
means of getting around the problems we encountered here.
7.3.4 Implicit Pressure-Correction Methods
As noted in Chap. 6, many methods for steady problems can be regarded as
solving an unsteady problem until a steady state is reached. The principal
difference is that, when solving an unsteady problem, the time step is chosen
so that an accurate history is obtained while, when a steady solution is sought,
large time steps are used to try to reach the steady state quickly. Implicit
methods are preferred for steady and slow-transient flows, because they have
less stringent time step restrictions than explicit schemes (they may not have
any).