Self-Similar Solutions to the Compressible Euler Equations and their Instabilities

This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. One of these endpoints is a shock formation that happens before the singularity at the origin, showing that the implosion process is unstable.Comment: Main text: 24 pages, 13 figures. Appendices: 12 pages, 3 figures. v2: extended materia

Exploration of a superposition and reconciliation based approach to cell-centered Lagrangian hydrodynamic methods

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Nuclear Science and Engineering, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 84-90).Applications and experiments involving the hypervelocity deformation of solids are difficult to devise, implement, and occur on microsecond time scales. As a result, simulations play a large role in the study of hypervelocity deformation. This study explored a superposition and reconciliation based approach using cell-centered Lagrangian hydro methods. The reconciliation forces that are not explicitly calculated for mesh movement were analyzed on an existing hydrocode by Pierre-Henri Maire (PHM) and a truncated form of the Runnels-Gilman method (implemented without using the reconciliation forces as additional forces to form a new hydro method called the Runnels-Gilman method). Results from both the 1D Piston and Saltzman test problems illustrate that the unaccounted reconciliation forces are acting on the mesh both at the shock front and behind the shock wave in PHM's method, while in the truncated Runnels- Gilman method, reconciliation forces are acting only on the vertices at the shock front. In test problems using PHM's method, reconciliation forces may be capturing the additional forces that account for more stable density and internal energy solution during shock wave propagation as compared to the truncated Runnels-Gilman method.by Lindsey Anne Gilman.S.M