Efficient Algorithm on a Non-staggered Mesh for Simulating Rayleigh-Benard Convection in a Box

An efficient semi-implicit second-order-accurate finite-difference method is described for studying incompressible Rayleigh-Benard convection in a box, with sidewalls that are periodic, thermally insulated, or thermally conducting. Operator-splitting and a projection method reduce the algorithm at each time step to the solution of four Helmholtz equations and one Poisson equation, and these are are solved by fast direct methods. The method is numerically stable even though all field values are placed on a single non-staggered mesh commensurate with the boundaries. The efficiency and accuracy of the method are characterized for several representative convection problems.Comment: REVTeX, 30 pages, 5 figure

Finite element simulation of three-dimensional free-surface flow problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet

Modification of the FEM3 model to ensure mass conservation

The problem of global mass conservation (lack thereof) in the current anelastic equations solved by FEM3 is described and its cause explained. The additional equations necessary to solve the problem are presented and methods for their incorporation into the current code are suggested. 14 refs

An update on projection methods for transient incompressible viscous flow

Introduced in 1990 was the biharmonic equation (for the pressure) and the concomitant biharmonic miracle when transient incompressible viscous flow is solved approximately by a projection method. Herein is introduced the biharmonic catastrophe that sometimes occurs with these same projection methods

Development of a three-dimensional model of the atmospheric boundary layer using the finite element method

This report summarizes our current effort and ideas toward the development of a model for the planetary boundary layer using the finite element technique. As an initial step, the finite element methodology is applied to simpler version of the boundary layer equations given by the two-dimensional, constant-property, incompressible conservation equations (Navier-Stokes equations). Solution procedures for both the steady-state and transient equations are discussed. For the transient problem, a variable time-step, trapezoid-rule algorithm with dynamic time-truncation error control is presented. The resulting system of nonlinear algebraic equations is solved by a Newton iteration procedure with a frontal solution scheme used for the linear set of equations. The need to develop a suitable linear equation solver, with respect to minimization of computer storage and execution costs, particularly for large (three-dimensional) finite element problems, is also discussed

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 involves a combination of reexamination 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 lead to claims that these unconventional FEM techniques are actually improvements and can be used to solve difficult problems on coarse meshes. 9 figures

On the spurious pressures generated by certain GFEM solutions of the incompressible Navier-Stokes equations

The spurious pressures and acceptable velocities generated when using certain combinations of velocity and pressure approximations in a Galerkin finite element discretization of the primitive variable form of the incompressible Navier-Stokes equations are analyzed both theoretically and numerically for grids composed of quadrilateral finite elements. Schemes for obtaining usable pressure fields from the spurious numerical results are presented for certain cases

Finite element simulations of thermally induced convection in an enclosed cavity

The simulations reported establish benchmark comparisons of different techniques of solving the Navier-Stokes and thermal energy equations for a particular thermally driven flow. Included also is a brief analysis of the results and some comparisons with previous solutions. The second section of this paper contains a result at a higher Rayleigh number and some tilted cavity results, which were performed, in part, as potential candidates for more difficult follow-up benchmark problems