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. (5.21)
The matrix equation is under-relaxed by a factor before equating with and solving for (the momentum predictor). 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). 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 