5.19 Transient solution

A modern interpretation of the SIMPLE algorithm was described in Sec. 5.12 to couple steady solutions for eqn and eqn. An equivalent transient solution follows an iterative sequence in which equations for eqn and eqn are solved over successive time intervals eqn, between a start time eqn and end time eqn.

In a transient simulation, eqn needs to be relatively small to maintain sufficient accuracy as the solution evolves over time eqn. The equations do not then generally require under-relaxation to converge due to the increase of eqn to the diagonal coefficients of the matrix from discretisation of the time derivative eqn.

A transient simulation could follow the same sequence as the SIMPLE algorithm on page 359 , but iterating over that sequence within each time step is costly. A more efficient algorithm solves the sequence only once per time step but adds an iterative loop which substitutes eqn from the momentum corrector into the eqn term of the pressure equation.

The pressure equation is then solved a second time, followed by a second momentum corrector, in the style of the PISO algorithm.7 The addition of this “PISO loop” improves the accuracy of eqn, eqn and eqn within each time step. The improved eqn becomes eqn in the next time iteration, which critically increases the accuracy of the time derivative eqn in the momentum equation.

Without any iteration over the entire system of equations, the advection terms are discretised using eqn from the previous time step. This “lagging” of eqn does not compromise accuracy significantly, if eqn is sufficiently small, e.g. corresponding to eqn.

A further iteration of the PISO loop can be introduced to solve a third pressure equation and momentum corrector, in particular as part of an update to non-orthogonality described in Sec. 5.20 . Increasing the iterations still further is generally not beneficial.

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7Pressure Implicit with Splitting of Operators, 1986

Notes on CFD: General Principles - 5.19 Transient solution