A stabilized edge-based implicit incompressible flow formulation

Abstract

An edge-based implementation of an implicit, monolithic, finite element (FE) scheme for the solution of the incompressible Navier–Stokes (NS) equations is presented. The original element formulation is based on the pressure stability properties of an implicit second-order in time fractional step (FS) method, which is conditionally stable. The final monolithic scheme preserves the second-order accuracy of the FS method, and is unconditionally stable. Furthermore, it can be demonstrated that the final pressure stabilizing term is practically the same fourth-order pressure term added by some authors (but following different arguments) to obtain high order accurate results, and that the final discretized convective terms are formally a second-order discretization of the respective continuous one.

The development of the edge implementation is supported by two criteria: the properties of the element based one, which has already been extensively tested and for which convergence and stability analysis has already been presented, and on the enforcement of global conservation and symmetry at the discrete level. A monotonicity preserving term which decreases the discretization order in sharp gradient regions to avoid localized oscillations (overshoots and undershoots), is formulated and tested. Some numerical examples and experimental comparisons are presented.

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