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

Abstract

SUMMARYThis work describes a projection method for approximating incompressible viscous ows of nonuniform density. It is shown that unconditional stability in time is possible provided two projections are performed per time step. A ÿnite element implementation and a ÿnite 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 conÿrming a conjecture by Birkho stating that the inviscid problem is not well-posed. Furthermore, we show that at even moderate Reynolds numbers, this problem is extremely sensitive to mesh reÿnement and to the numerical method adopted.

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

“…The data are exactly the same as in the case referred to by 'Re = 1000' in [36]. The chosen mesh is the same as in [36] for the finite volume computation, namely a 256×512 uniform grid. A MUSCL technique with a Van Leer limiter is implemented for the solution of the balance equation for y.…”

“…We now address a case studied in [35,36], consisting in a Rayleigh-Taylor instability flow. The data are exactly the same as in the case referred to by 'Re = 1000' in [36].…”

An hybrid finite volume–finite element method for variable density incompressible flows

“…At each step of this integration method, we have to evaluate the right hand side of (20). Having at hand two interpolation values e q þ A and e q þ Ai (respectively e q À A and e q À Ai ) for q on C þ i (respectively…”

“…(2), the constraint (3) being treated through a projection method, see [21]. This is the methodology followed in [23,20]. These contributions use the formulation (7) since a specific care is given to the kinetic energy relation.…”

Computational Strategies for Micro‐ and Nanofluid Dynamics

“…18. Pressure projection methods have also been used in conjunction with finite element schemes, 38,39 where an unconditionally stable method was developed based on two projections per time step and its performance was investigated both in finite volume and in finite-element implementations.…”