5.12 Steady-state solution

The equations in Sec. 5.10 are combined using algorithms that couple the solutions for eqn and eqn. One algorithm is SIMPLE (Semi-Implicit Method for Pressure-Linked Equations)3, which is presented here with a modern interpretation.

The SIMPLE algorithm is generally used to produce steady flow solutions in CFD. These solutions are directly applicable for flows that reach a steady state, i.e. when flow variables stop changing in time. They can also provide approximate solutions to flows that are moderately unsteady, usually at a lower cost than a more exact transient solution.

An example of the algorithm is shown for the system of equations presented in Sec. 5.9 . The time derivative (eqn) terms are omitted due to the steady-state assumption.

The algorithm involves an iterative sequence with steps, eqn. It begins by constructing a matrix equation for energy which is under-relaxed by a factor eqn. The equation is solved for eqn, which is used to update eqn according to an equation of state. A matrix equation is then constructed using all the terms from the momentum equation excluding eqn, i.e.

 --(T;-p) A u b r (uu) r ( ru) 0 g \relax \special {t4ht=
The matrix equation is under-relaxed by a factor eqn before equating with eqn and solving for eqn (the momentum predictor).

eqn and eqn are then evaluated from eqn (the momentum matrix), as described on page 351. They are used to form the pressure equation, which is solved for eqn.

The new pressure eqn is used to correct the flux eqn so that it obeys mass conservation better (the flux corrector). It is then under-relaxed by a factor eqn before correcting eqn before the next solution step begins (the momentum corrector).

PICT\relax \special {t4ht=

3Suhas Patankar and Brian Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, 1972.

Notes on CFD: General Principles - 5.12 Steady-state solution