PERFORMANCE OF PROJECTION METHODS FOR LOW-REYNOLDS-NUMBER FLOWS

Abstract

There exists growing interest in modelling flows at millimetric and micrometric scales, characterised by low Reynolds numbers (Re << 1). In this work, we investigate the performance of projection methods (of the algebraic-splitting kind) for the computation of steady-state simple benchmark problems. The most popular approximate factorization methods are assessed, together with two so-called exact factorization methods. The results show that: (a) The error introduced by non-incremental schemes on the steady state solution is unacceptably large even in the simplest of flows. This is well-known for the basic first order scheme and motivated variants aiming at increased accuracy. Unfortunately, the variants studied either become unstable for the time steps of interest, or yield steady states with larger error than the basic first order scheme. (b) Incremental schemes have an optimal time step delta t* so as to reach the steady state with minimum computational effort. Taking delta t = delta t* the code reaches the steady state in not less than a few hundred time steps. Such a cost is significantly higher than that of solving the velocity-pressure coupled system, which can compute the steady state in one shot. (c) The main difficulty, however, is that if delta t is chosen too large (in general delta t* is not known), then thousands or tens of thousands of time steps are required to reach the numerical steady state with incremental projection methods. The numerical solutions of these methods follow a time-step-dependent spurious transient which makes the computation of steady states prohibitively expensive.