5.12 Steady-state solution
The equations in Sec. 5.10 are combined using algorithms that couple the solutions for and . 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 () terms are omitted due to the steady-state assumption.
The algorithm involves an iterative sequence with steps, . It begins by constructing a matrix equation for energy which is under-relaxed by a factor . The equation is solved for , which is used to update according to an equation of state. A matrix equation is then constructed using all the terms from the momentum equation excluding , i.e.
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(5.21) |
and are then evaluated from (the momentum matrix), as described on page 351. They are used to form the pressure equation, which is solved for .
The new pressure is used to correct the flux so that it obeys mass conservation better (the flux corrector). It is then under-relaxed by a factor before correcting before the next solution step begins (the momentum corrector).