Philip M. Gresho

A MODIFIED FINITE ELEMENT METHOD FOR SOLVING THE TIME-DEPENDENT, INCOMPRESSIBLE NAVIER-STOKES EQUATIONS. PART 1: THEORY*

Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier-Stokes equations, the spatial approximation is modified in two ways in the interest of cost-effectiveness: the mass matrix is ‘lumped’ and all coefficient matrices are generated via 1-point quadrature. After appending an hour-glass correction term to the diffusion matrices, the modified semi-discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection-dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid-driven cavity at Re ≤ 10,000, flow past a circular cylinder at Re ≤ 400, and the simulation of a heavy gas release over complex topography.

Don't suppress the wiggles—They're telling you something!☆

The subject of oscillatory solutions (wiggles), which sometimes result when the conventional Galerkin finite element method is employed to approximate the solution of certain partial differential equations, is addressed. It is argued that there is an important message behind these wiggles and that the appropriate response to it usually involves a combination of: re-examination of the imposed boundary conditions, judicious mesh refinement (via isoparametric elements) in critical areas, and sometimes even admitting that the problem, as posed, is just too difficult to solve adequately on an “affordable” mesh. It is further argued that it is usually an inappropriate response to develop methods which a priori suppress these wiggles and thereby make claims that these unconventional FEM techniques are actually improvements and can be used to solve difficult problems on coarse meshes.

A comparison of various mixed-interpolation finite elements in the velocity-pressure formulation of the Navier-Stokes equations☆

Abstract It is generally recognized that mixed interpolation should be used in the velocity-pressure finite element formulation of incompressible viscous flow problems. In this paper, four types of mixed interpolation elements are considered and compared. These are namely: six-node triangular elements, eight-node serendipity elements, nine-node Lagrangian elements and four-node quadrilateral elements. The comparison is made via two numerical examples concerning steady flow through a sudden expansion and steady free thermal convection in a square cavity. Results indicate that for the same number of pressure unknowns, serendipity elements can give considerably less accurate pressure fields than most other types of elements. Lagrangian elements give the most accurate pressure and velocity distributions. The numerical performance of triangular elements is intermediate in accuracy and is dependent on the triangular pattern used. Finally, the four-node element may generate spurious pressure modes depending on the boundary condition specifications.

Finite element simulations of steady, two-dimensional, viscous incompressible flow over a step

The Galerkin finite element method is utilized to obtain quite detailed results for flow through a channel containing a step at Reynolds numbers of 0 and 200. This technique, however, like its centered finite difference counterpart, is prone to generating wiggles or oscillations when streamwise gradients become too large to be resolved by the mesh. These wiggles are carefully analyzed and are used as a guide in obtaining accurate solutions via successive mesh refinement. It is argued that the appropriate solution to the wiggle problem is the utilization of selective grid refinement (easily available via isoparametric finite elements) rather than taking recourse to upwind methods which effectively reduce the local Reynolds number and thereby generate deceptively smooth and often inaccurate results.

FEM solution of the Navier-Stokes equations for vortex shedding behind a cylinder: experiments with the four-node element

Results from three variations on a simple 4-node element (bilinear velocity and piecewise constant pressure) are compared with those from a higher order element (9-node biquadratic velocity and 4-node bilinear pressure) on the same problem and on the same grid. (GHT)

Simulation of LNG vapour spread and dispersion by finite element methods1

Abstract Two finite element models, one based on solving the time-dependent, two-dimensional conservation equations of mass, momentum, and energy, with buoyancy effects included via the Boussinesq approximation, the other based on solving the otherwise identical set of equations except using the hydrostatic assumption, are described and used to predict some aspects of the vapour dispersion phenomena associated with LNG spills. A number of controlled numerical experiments, representing a reasonable expected range of LNG spill scenarios and atmospheric conditions, have been carried out. Based on a comparison of the results obtained with these finite element models, some data regarding the applicability and limitations of the hydrostatic assumption for predicting LNG vapour spread and dispersion are established.

On the Finite Element Modeling of the Scalar Transport Equation

ABSTRACT This material was presented (by RL.S.) as part of a three hour lecture on finite element modeling of incompressible and Boussinesq flows at the summer school organized by IUSTI, Universite de provence.