Two-Dimensional Lagrangian Fluid Dynamics Using Triangular Grids

Abstract

Section 5.1 stressed the deleterious effects of mesh distortion on Lagrangian methods. The loss in accuracy due to mesh distortion applies, of course, to higher dimensions as well. In practice, Lagrangian techniques have been limited to one-dimensional calculations or flows in higher dimensions which are very “well behaved,” since shear, fluid separation, and large-amplitude motions produce severe grid distortions. The mesh points commonly used to evaluate gradients and Laplacians are shown in Fig. 6-1a for a regular two-dimensional grid. Figure 6-1b illustrates a simple grid distortion produced by shear flow. A well-formulated Lagrangian finite-difference algorithm will properly account for the angle between grid lines and the variable mesh spacing produced by this distortion. Nevertheless, numerical approximations based on this mesh can still be grossly in error because differences no longer involve neighboring vertices. Mesh points which are now closer to the central vertex do not enter into the approximation, while those farther away do.