Discrete Continuity Equation Research Articles

Parallel multigrid solution of the Navier-Stokes equations on general 2D domains

Multigrid methods are distinguished by their optimal (sequential) efficiency and by the fact that all their algorithmical components are fully parallelizable. For this reason, this class of numerical methods is especially attractive for use on parallel (MIMD, local memory) computers. In this paper, we describe a parallel multigrid solver for steady-state incompressible Navier-Stokes equations on general domains which is currently being developed at the GMD. Due to the geometrical generality of the problem, our approach is based on a non-staggered (nodal-point) finite volume scheme on multi-block boundary fitted grids. The typical instability of non-staggered schemes is overcome by suitably modifying the discrete continuity equation without affecting the overall order of consistency. Starting from the most simple Cartesian case, we discuss several possible multigrid approaches to the general 2D-problem. This motivates the basic design decisions of our multigrid solver in regard to both the discretization and the choice of multigrid components (smoothing schemes). Furthermore, the principal technique of parallelization (grid partitioning) is described as well as some fundamental aspects of the implementation (communication library).

A segregated formulation of Navier-Stokes equations with finite elements

A finite element procedure is presented for the solution of the Navier-Stokes equations, adopting a segregated velocity-pressure formulation. The procedure is based on the derivation of an approximate pressure equation, which allows the uncoupling of pressure and velocity solutions. This equation is derived from the discretized continuity equation by considering the relationships established between velocity and pressure in the discretized momentum equations. The numerical performance of the proposed method is demonstrated by investigating several examples. The results are compared with finite difference predictions.

The general dynamic equation for aerosols. Theory and application to aerosol formation and growth

Conservation equations for aerosol size distribution dynamics are derived and compared. A new discrete-continuous conservation equation is derived that overcomes the limitations of the purely discrete or continuous equations in simulating aerosol dynamics over a broad particle size spectrum. Several issues related to aerosol formation and growth not previously amenable to exact analysis are studied in detail using the discrete-continuous equation: (1) the establishment of a steady state concentration profile of molecular clusters in the presence of a preexisting aerosol; (2) the relative importance of nucleation, condensation, and scavenging in gas-to-particle conversion; and (3) the importance of cluster-cluster agglomeration relative to other processes. The formation and growth of a sulfuric acid/water aerosol in a smog chamber is simulated using the discrete-continuous equation, and the results are compared with recent experimental data. The above issues related to aerosol formation and growth are investigated for the simulated experiment.