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Finite element solution of the incompressible Navier‐Stokes equations by a Helmholtz velocity decomposition
Author(s) -
Peeters M. F.,
Habashi W. G.,
Nguyen B. Q.
Publication year - 1991
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650130202
Subject(s) - mathematics , linearization , finite element method , mathematical analysis , discretization , pressure correction method , compressibility , solver , galerkin method , helmholtz equation , navier–stokes equations , boundary value problem , nonlinear system , physics , mathematical optimization , mechanics , quantum mechanics , thermodynamics
Abstract Finite element solution methods for the incompressible Navier‐Stokes equations in primitive variables form are presented. To provide the necessary coupling and enhance stability, a dissipation in the form of a pressure Laplacian is introduced into the continuity equation. The recasting of the problem in terms of pressure and an auxiliary velocity demonstrates how the error introduced by the pressure dissipation can be totally eliminated while retaining its stabilizing properties. The method can also be formally interpreted as a Helmholtz decomposition of the velocity vector. The governing equations are discretized by a Galerkin weighted residual method and, because of the modification to the continuity equation, equal interpolations for all the unknowns are permitted. Newton linearization is used and at each iteration the linear algebraic system is solved by a direct solver. Convergence of the algorithm is shown to be very rapid. Results are presented for two‐dimensional flows in various geometries.