Introduction
Lagrangian hydrodynamic methods applied to two-dimensional calculations have had striking success in problems involving small fluid distortions. Although the interfluid boundary motions can be calculated well, the methods break down when the fluids (and thus the coordinate system) become strongly distorted. Eulerian methods can survive strong distortions but suffer from false diffusion of boundaries. It is the purpose of this note to describe a method of calculation that, in preliminary tests, has achieved some success in solving large distortion, two-dimensional problems involving several fluids.

Tests of the Method
Numerous calculations have been performed to test various aspects of the method. These have been carried out on IBM Electronic Data Processing Machines, types 701 and 704.

One-dimensional experiments were very successful in reproducing the theoretical characteristics of interactions of shock waves with density discontinuities, and subsequent propagation of shocks and rarefactions. Shock waves were generally smeared out to a width of two cells. The lack of velocity fluctuations behind the shock shows that the correct amount of entropy is created by the dissipation, in all examples, the approximate solution converged to the exact solution (where known) at a rate which increased with the ratio of the significant dimensions of the system to the cell size. Calculations of energy transport and partition for more complicated problems were compared with similar calculations using conventional Lagrangian hydrodynamic methods. The results agreed well, and it was not possible to decide which more closely approximated the true solution.

For two-dimensional calculations, there have been no comparison solutions except in the simplest cases. The maintenance of sphericity of an expanding gas sphere in cylindrical coordinates was tested. Results show an average deviation from sphericity of less than 3 % after a three and one-half fold expansion. In two-dimensional problems with two fluids, no interparticle diffusions are observed, and the boundaries are well preserved. One calculation of a shock hitting an irregular interface between two fluids showed stability in the density-decrease case, and a very interesting pattern of instability in the density-increase case both as expected qualitatively.

The experiments made with this method of calculation showed that it is nearly as good as conventional methods in one-dimensional problems, and that in its extension to two-dimensional, large distortion problems, there is very little, if any, loss of accuracy.

Article from: Francis H. Harlow. Hydrodynamic problems involving large fluid distortions. Journal of the ACM, 4(2):137-142, April 1957.