178
7. Solution of the Navier-Stokes Equations
We may now recall that all conservative schemes, in the absence of any contribution to Ap from source terms, lead to Ap = - El A1 + A;, where A :
is the contribution from the unsteady term. If a steady solution is sought
through iterating for an infinite time step, A; = 0, but we have to use
under-relaxation, as explained in Sect. 5.4.2. In that case, Ap = - El A l / a u ,
where a, is the under-relaxation factor for velocities (usually the same for
all components, but it need not be so). We then obtain:
which has been found to be nearly optimum and yields almost the same
convergence rate for outer iterations as the SIMPLEC method.
The solution algorithm for this class of methods can be summarized as
follows:
1. Start calculation of the fields a t the new time t,+' using the latest solution UT and pn as starting estimates for ul+' and pn+'.
2. Assemble and solve the linearized algebraic equation systems for the velocity components (momentum equations) to obtain uy'.
3. Assemble and solve the pressure-correction equation to obtain p'.
4. Correct the velocities and pressure to obtain the velocity field u y , which
satisfies the continuity equation, and the new pressure pm.
For the PIS0 algorithm, solve the second pressure-correction equation
and correct both velocities and pressure again.
For SIMPLER, solve the pressure equation for pm after u y is obtained
above.
5. Return to step 2 and repeat, using u y and pm as improved estimates for
ul+' and pnf ', until all corrections are negligibly small.
6. Advance to the next time step.
Methods of this kind are fairly efficient for solving steady state problems; their convergence can be improved by the multigrid strategy, as will be
demonstrated in Chap. 11. There are many derivatives of the above methods
which are named differently, but they all have roots in the ideas described
above and will not be listed here. We shall show below that the artificial
compressibility method can also be interpreted in a similar way.
7.4 Other Methods
7.4.1 Fractional Step Methods
In the methods of the preceding section, the pressure is used to enforce continuity. It is also used in computing the velocity field in the first step of the
7. Solution of the Navier-Stokes Equations
We may now recall that all conservative schemes, in the absence of any contribution to Ap from source terms, lead to Ap = - El A1 + A;, where A :
is the contribution from the unsteady term. If a steady solution is sought
through iterating for an infinite time step, A; = 0, but we have to use
under-relaxation, as explained in Sect. 5.4.2. In that case, Ap = - El A l / a u ,
where a, is the under-relaxation factor for velocities (usually the same for
all components, but it need not be so). We then obtain:
which has been found to be nearly optimum and yields almost the same
convergence rate for outer iterations as the SIMPLEC method.
The solution algorithm for this class of methods can be summarized as
follows:
1. Start calculation of the fields a t the new time t,+' using the latest solution UT and pn as starting estimates for ul+' and pn+'.
2. Assemble and solve the linearized algebraic equation systems for the velocity components (momentum equations) to obtain uy'.
3. Assemble and solve the pressure-correction equation to obtain p'.
4. Correct the velocities and pressure to obtain the velocity field u y , which
satisfies the continuity equation, and the new pressure pm.
For the PIS0 algorithm, solve the second pressure-correction equation
and correct both velocities and pressure again.
For SIMPLER, solve the pressure equation for pm after u y is obtained
above.
5. Return to step 2 and repeat, using u y and pm as improved estimates for
ul+' and pnf ', until all corrections are negligibly small.
6. Advance to the next time step.
Methods of this kind are fairly efficient for solving steady state problems; their convergence can be improved by the multigrid strategy, as will be
demonstrated in Chap. 11. There are many derivatives of the above methods
which are named differently, but they all have roots in the ideas described
above and will not be listed here. We shall show below that the artificial
compressibility method can also be interpreted in a similar way.
7.4 Other Methods
7.4.1 Fractional Step Methods
In the methods of the preceding section, the pressure is used to enforce continuity. It is also used in computing the velocity field in the first step of the