Shocks generated in a confined gas due to rapid heat addition at the boundary. I. Weak shock waves

Abstract

An inert compressible gas, confined between infinite parallel planar walls, is in an equilibrium state initially. Subsequently energy is added at the boundary during a period that is short compared to the acoustic time of the slot t' a (the wall spacing divided by the equilibrium sound speed), but larger than the mean time between molecular collisions. Conductive heating of a thin layer of gas adjacent to the wall induces a gas motion arising from thermal expansion. The small local Mach number at the layer edge has the effect of a piston on the gas beyond. A linear acoustic wave field is then generated in a thicker layer adjacent to the walls. Eventually nonlinear accumulation effects occur on a timescale that is longer than the initial heating time but short compared with t' a . A weak shock then appears at some well defined distance from the boundary. If the heating rate at the wall is maintained over the longer timescale, then a high temperature zone of conductively heated expanding gas develops. The low Mach number edge speed of this layer acts like a contact surface in a shock tube and supports the evolution of the weak shock propagating further from the boundary. One-dimensional, unsteady solutions to the complete Navier-Stokes equations for an inert gas are obtained by using perturbation methods based on the asymptotic limit t' a / t' c → 0, where t' c , the conduction time of the region, is the ratio of the square of the wall spacing to the thermal diffusivity in the initial state. The shock strength is shown to be related directly to the duration of the initial boundary heating.