Elsevier

Applied Mathematics Letters

Volume 25, Issue 11, November 2012, Pages 1681-1688
Applied Mathematics Letters

A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier–Stokes problems

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Abstract

We present a new fast vector penalty-projection method (V PPε) to efficiently compute the solution of unsteady Navier–Stokes problems governing incompressible multiphase viscous flows with variable density and/or viscosity. The key idea of the method is to compute at each time step an accurate and curl-free approximation of the pressure gradient increment in time. This method performs a two-step approximate divergence-free vector projection yielding a velocity divergence vanishing as O(εδt), δt being the time step, with a penalty parameter ε as small as desired until the machine precision, e.g. ε=1014, whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed vector correction step typically requires only a few iterations of a suitable preconditioned Krylov solver whatever the spatial mesh step. The method is numerically validated on three benchmark problems for non-homogeneous or multiphase flows where we compare it to the Uzawa augmented Lagrangian (UAL) and scalar incremental projection (SIP) methods. Moreover, a new test case for fluid–structure interaction problems is also investigated. That results in a very robust method running faster than usual methods and being able to efficiently and accurately compute sharp test cases whatever the density, viscosity or anisotropic permeability jumps, whereas other methods crash.

Keywords

Vector penalty-projection method
Divergence-free penalty-projection
Penalty method
Splitting prediction–correction scheme
Navier–Stokes equations
Incompressible non-homogeneous or multiphase flows

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