The particular difficulty of incompressible flow is this: the momentum equations give velocity, but there is no separate transport equation for pressure. Pressure is derived indirectly from the continuity condition, and the method of that derivation sets the character of the solver.
Segregated solvers
The momentum and pressure-correction equations are solved in sequence; the SIMPLE family and its derivatives are the classic examples of this approach [1]. For transient problems, the operator-splitting based PISO method reduces the number of iterations needed per step [2].
Advantage: low memory requirement, economical on large meshes. Limit: because the coupling between pressure and velocity is weak, convergence is slow and sensitive to relaxation factors.
Coupled solvers
Momentum and continuity are solved together in a single system.
Advantage: converges in far fewer iterations; markedly superior in problems where buoyancy dominates and the pressure-velocity coupling is strong. Natural convection is the typical example. See the natural convection note. Price: higher memory and work per iteration.
Practical choice
| Problem | Preference |
|---|---|
| Large mesh, simple physics, ample iteration budget | Segregated |
| Natural convection, strong buoyancy | Coupled |
| Transient with small time steps | Segregated (PISO type) |
| Steady state that will not converge | Try coupled first |
One caution
Solver type changes the rate of convergence, not the solution. Two converged results, one segregated and one coupled, must agree. If they do not, at least one has not converged. See the convergence note.
References
- Patankar, S. V. (1980). Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing. ISBN 978-0-89116-522-4.
- Issa, R. I. (1986). Solution of the implicitly discretised fluid flow equations by operator-splitting. Journal of Computational Physics, 62(1), 40–65. DOI: 10.1016/0021-9991(86)90099-9
- Ferziger, J. H., Perić, M. & Street, R. L. (2020). Computational Methods for Fluid Dynamics, 4th ed. Springer. DOI: 10.1007/978-3-319-99693-6