Approximation of variable density incompressible flows by means of finite elements and finite volumes

This work describes a projection method for approximating incompressible viscous flows of non‐uniform density. It is shown that unconditional stability in time is possible provided two projections areperformed per time step. A finite element implementation and a finite volume one are described and compared. The performance of the two methods are tested on a Rayleigh–Taylor instability. We show that the considered problem has no inviscid smooth limit; hence confirming a conjecture by Birkhoffstating that the inviscid problem is not well‐posed. Furthermore, we show that at even moderate Reynolds numbers, this problem is extremely sensitive to mesh refinement and to the numerical method adopted. Copyright © 2001 John Wiley & Sons, Ltd.