Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of the Guderley Problem

Abstract

This work is concerned with the use of Guderley's converging shock wave solution of the inviscid compressible flow equations as a verification test problem for compressible flow simulation software. In practice, this effort is complicated by both the semi-analytical nature and infinite spatial/temporal extent of this solution. Methods can be devised with the intention of ameliorating this inconsistency with the finite nature of computational simulation; the exact strategy will depend on the code and problem archetypes under investigation. For example, scale-invariant shock wave propagation can be represented in Lagrangian compressible flow simulations as rigid boundary-driven flow, even if no such “piston” is present in the counterpart mathematical similarity solution. The purpose of this work is to investigate in detail the methodology of representing scale-invariant shock wave propagation as a piston-driven flow in the context of the Guderley problem, which features a semi-analytical solution of infinite spatial/temporal extent. The semi-analytical solution allows for the derivation of a similarly semi-analytical piston boundary condition (BC) for use in Lagrangian compressible flow solvers. The consequences of utilizing this BC (as opposed to directly initializing the Guderley solution in a computational spatial grid at a fixed time) are investigated in terms of common code verification analysis metrics (e.g., shock strength/position errors, global convergence rates). For the examples considered in this work, the piston-driven initialization approach is demonstrated to be a viable alternative to the more traditional, direct initialization approach.