Robert L. Lee

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.

A mixed finite element formulation for Maxwell's equations in the time domain

Abstract A Galerkin finite element solution technique for the Maxwells equations is discussed. This new formulation can be viewed as a generalization of certain staggered-grid finite difference schemes to arbitrary meshes. It is shown that this technique is simple to implement and is more accurate as well as more cost effective than the standard equal-order finite element approach. Numerical are presented to evaluate the performance of this new element relative to the standard element.

Flow around a complex building: Experimental and large-eddy simulation comparisons

A field program to study atmospheric releases around a complex building was performed in the summers of 1999 and 2000. The focus of this paper is to compare field data with a large-eddy simulation (LES) code to assess the ability of the LES approach to yield additional insight into atmospheric release scenarios. In particular, transient aspects of the velocity and concentration signals are studied. The simulation utilized the finite-element method with a high-fidelity representation of the complex building. Trees were represented with a canopy term in the momentum equation. Inflow and outflow conditions were used. The upwind velocity was constructed from a logarithmic law fitted to velocities obtained on two levels from a tower equipped with a 2D sonic anemometer. A number of different kinds of comparisons of the transient velocity and concentration signals are presented—direct signal versus time, spectral, Reynolds stresses, turbulent kinetic energy signals, and autocorrelations. It is concluded that the LES approach does provide additional insight, but the authors argue that the proper use of LES should include consideration of cost and may require an increased connection to field sensors; that is, higher-resolution boundary and initial conditions need to be provided to realize the full potential of LES.

Lagrangian stochastic particle model simulations of turbulent dispersion around buildings

Abstract This paper describes a numerical modeling approach that can be used to provide estimates of air concentrations due to emissions at industrial sites or other sites where buildings may have an important impact on the dispersion patterns. The procedure consists of two sequential steps: (i) Prediction of mean flow and turbulence fields via a turbulent flow model; and, (ii) Employment of the calculated flow and turbulence fields to drive a Lagrangian stochastic particle model. Two flow scenarios are used as input to a Lagrangian model; the scenarios assume neutral atmospheric conditions and incoming winds that are from 270 and 240°. The problem configuration for the first calculation is based on an earlier transport and diffusion simulation that employed an existing particle-in-cell flux-gradient dispersion model. The second simulation is used to demonstrate the strong spatial variations that the concentration field exhibit within the highly complex separation zones of building wakes. The relationship between concentration levels and toxic load are discussed for the case of a chemical spill.

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)

Numerical simulation of drainage flow in Brush Creek, Colorado

Abstract One of the objectives of the Atmospheric Studies in Complex Terrain (ASCOT) program is to develop numerical models that can be used to aid in the understanding and prediction of flow patterns observed over complex terrain. As part of this program, we have developed a three-dimensional dynamic model that uses a simplified finite element scheme combined with a variation of the forward Euler time integration scheme to solve the Boussinesq equations of motion over irregular terrain. We have used the model to simulate the development of nocturnal down-valley circulation in the area where the ASCOT field observations were made at Brush Creek, Colorado. The model and the simulation results are described in this paper.

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.

Numerical modeling of three-dimensional flow and pollutant dispersion around structures

Abstract This paper describes a numerical modeling approach that can be used to provide estimates of emissions at industrial sites. In particular the models presented are capable of simulating the wind flow and dispersion of airborne pollutants around surface-mounted structures such as buildings or building complexes. The calculational procedure in this approach consists of two sequential steps, namely: (i) prediction of the mean flow via a turbulent flow model; and (ii) employment of the calculated flow field to drive a particle-in-cell transport and diffusion model. A benchmark simulation is performed in which numerical results from the flow model are compared with other numerical models and with experimental data for flow over a backward-facing step. Results from three-dimensional simulations of flow and dispersion over a two-building complex are also presented.