170
7. Solution of the Navier-Stokes Equations
to which this pressure belongs is arbitrary. If the pressure gradient term had
been treated implicitly, we would have pn+' in place of pn but everything
else would remain unchanged.
This provides the following algorithm for time-advancing the NavierStokes equations:
Start with a velocity field 2 1 7 at time t n which is assumed divergence free.
(As noted, if it is not divergence free this can be corrected.)
Compute the combination, HI", of the advective and viscous terms and its
divergence (both need to be retained for later use).
Solve the Poisson equation for the pressure pn.
Compute the velocity field at the new time step. It will be divergence free.
The stage is now set for the next time step.
Methods similar to this are commonly used t o solve the Navier-Stokes
equations when an accurate time history of the flow is required. The principal
differences in practice are that time advancement methods more accurate
than the first order Euler method are usually used and that some of the
terms may be treated implicitly. Some of these methods will be described
later.
We have shown how solving the Poisson equation for the pressure can
assure that the velocity field satisfies the continuity equation i.e. that it is
divergence free. This idea runs through many of the methods used to solve
both the steady and unsteady Navier-Stokes equations. We shall now study
some of the more commonly used methods for solving the steady NavierStokes equations.
7.3.3 A Simple Implicit Time Advance Method
To see what additional difficulties arise when an implicit method is used to
solve the Navier-Stokes equations, let us construct such a method. Since we
are interested in illuminating certain issues, let us use a scheme based on the
the simplest implicit method, the backward or implicit Euler method. If we
apply this method to Eq. (7.17), we have:
We see immediately that there are difficulties that were not present in the
explicit method described in the preceding section. Let us consider these one
at a time.
First, there is a problem with the pressure. The divergence of the velocity
field a t the new time step must be zero. This can be accomplished in much
the same way as in the explicit method. We take the divergence of Eq. (7.22),
assume that the velocity field a t time step n is divergence free (this can be
7. Solution of the Navier-Stokes Equations
to which this pressure belongs is arbitrary. If the pressure gradient term had
been treated implicitly, we would have pn+' in place of pn but everything
else would remain unchanged.
This provides the following algorithm for time-advancing the NavierStokes equations:
Start with a velocity field 2 1 7 at time t n which is assumed divergence free.
(As noted, if it is not divergence free this can be corrected.)
Compute the combination, HI", of the advective and viscous terms and its
divergence (both need to be retained for later use).
Solve the Poisson equation for the pressure pn.
Compute the velocity field at the new time step. It will be divergence free.
The stage is now set for the next time step.
Methods similar to this are commonly used t o solve the Navier-Stokes
equations when an accurate time history of the flow is required. The principal
differences in practice are that time advancement methods more accurate
than the first order Euler method are usually used and that some of the
terms may be treated implicitly. Some of these methods will be described
later.
We have shown how solving the Poisson equation for the pressure can
assure that the velocity field satisfies the continuity equation i.e. that it is
divergence free. This idea runs through many of the methods used to solve
both the steady and unsteady Navier-Stokes equations. We shall now study
some of the more commonly used methods for solving the steady NavierStokes equations.
7.3.3 A Simple Implicit Time Advance Method
To see what additional difficulties arise when an implicit method is used to
solve the Navier-Stokes equations, let us construct such a method. Since we
are interested in illuminating certain issues, let us use a scheme based on the
the simplest implicit method, the backward or implicit Euler method. If we
apply this method to Eq. (7.17), we have:
We see immediately that there are difficulties that were not present in the
explicit method described in the preceding section. Let us consider these one
at a time.
First, there is a problem with the pressure. The divergence of the velocity
field a t the new time step must be zero. This can be accomplished in much
the same way as in the explicit method. We take the divergence of Eq. (7.22),
assume that the velocity field a t time step n is divergence free (this can be