18. SIMPLE and related methods for the solution of Navier-Stokes equations

This lecture is about SIMPLE and related methods for the solution of Navier-Stokes equations. This class of methods is most frequently used in both commercial and public CFD codes. After an introduction to principles behind these methods, four algorithms are described in detail: SIMPLE, SIMPLEC, SIMPLER, and PISO. There are more versions of such algorithms in literature, but they differ only in small details. It will be shown that even the fractional-step methods are not much different from SIMPLE-type methods - they are all based on the same idea: first predict the new velocity using a suitable estimate of the pressure field, then correct velocity and pressure to enforce the continuity constraint.
While fractional-step methods always march in time toward a steady-state solution, SIMPLE-type methods can drop the rate-of-change term altogether and use an under-relaxation parameter to promote convergence of outer iterations in a single, infinite time step. It will be shown that this practice is related to time marching with a time step that varies within the solution domain.
Finally, a complete step-by-step discretization and solution process using collocated grids and SIMPLE algorithm is presented, including the derivation of pressure-correction equation. A computer code in which all these steps can be identified will be provided as a supplement to lectures 19 and 20.
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