Finite element solution of the incompressible Navier‐Stokes equations by a Helmholtz velocity decomposition

Abstract

SUMMARYFinite 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.

Simplified level set method coupled to stabilised finite element flow solver for moving boundaries

“…is the Reynolds number in which ρ is the density, and μ is dynamic viscosity coefficient. Equation ( 2) is the standard momentum equation for Newtonian fluids while equation ( 3) is the modified continuity equation through which the connection between the continuity equation and the momentum equation is recovered using a simplified pressure stabilisation technique which was devised to treat the well-known incompressibility constraint problem (Peeters et al, 1991).…”

“…The bi-linear quadrilateral element is C 1 continuous on the element interior and C 0 continuous at element interfaces. Making use of the modified continuity equation introduced inPeeters et al (1991) allows for using equal-order shape functions namely for the velocity and pressure variables.…”

Numerical solutions of incompressible Navier–Stokes equations using modified Bernoulli's law

“…Similar decompositions have been used in the literature for ow simulations in many ways. See for example, the recent work of Nikfetrat and Hafez [23,24] for a review and in particular, see References [15,[25][26][27][28][29][30][31][32][33].…”

A generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces

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