Notes on Computational Fluid Dynamics: General Principles

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5.12 Steady-state solution

The equations in Sec. 5.10 are combined using algorithms that couple the solutions for $\text{[math]}$ and $\text{[math]}$ . 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 ( $\text{[math]}$ ) terms are omitted due to the steady-state assumption.

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

--(T;-p) A u b r (uu) r ( ru) 0 g \relax \special {t4ht=

(5.21)

The matrix equation is under-relaxed by a factor $\text{[math]}$ before equating with $\text{[math]}$ and solving for $\text{[math]}$ (the momentum predictor).

$\text{[math]}$ and $\text{[math]}$ are then evaluated from $\text{[math]}$ (the momentum matrix), as described on page 351. They are used to form the pressure equation, which is solved for $\text{[math]}$ .

The new pressure $\text{[math]}$ is used to correct the ﬂux $\text{[math]}$ so that it obeys mass conservation better (the ﬂux corrector). It is then under-relaxed by a factor $\text{[math]}$ before correcting $\text{[math]}$ 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