Analysis of Wide-Driver Shock Tube Flow with Sharp Area Transition

Abstract

Consider a shock tube configuration with a sharp transition between a wide driver segment and a much narrower driven segment. The flow commencing upon the rupture of the diaphragm that separates the two segments is studied both numerically and by analytic modeling. This analysis is aimed in particular at understanding the converging flow through the sharp area contraction, noting the vena-contracta formed by the separation bubble that evolves as the flow negotiates the sharp corner at the entrance to the driven segment. For a high driver-to-driven pressure ratio (typically 100) the large-time flow at the entrance to the driven segment is found to be nearly steady. The separation bubble downstream of the corner seems to produce a choked vena-contracta flow with a smooth transition from subsonic to supersonic velocity (at high driver-to-driven pressure ratio). It is this flow that drives the shock wave into the long driven segment. The numerical study was performed on an axisymmetric configuration with an area ratio 36 and initial pressure ratio 100, first by solving the Euler equations (using the GRP method), then by a commercial code (FLUENT) based on the compressible Navier-stokes equations. On the Mach number flow map, a sonic surface (M = 1) is clearly visible at the entry to the driven segment, just ahead of a separation bubble. While CFD solutions do provide a detailed view of the flow field, it is generally difficult to get an understanding of the observed phenomena in terms of well-known concepts of fluid dynamics. Here we seek such insight by resorting to analytic modeling where the large-time flow model is based on simple one-dimensional fluid-dynamical “building blocks”, such as steady quasi-one-dimensional isentropic flow, or simple waves: centered rarefaction wave, shock discontinuity, and contact discontinuity. The resulting flow-field consists of a fixed-length isentropic flow producing a steady sonic or supersonic flow. A self-similar set of flow segments (CRW, constant flow segments, contact discontinuity) is “matched” to the steady flow and produces the (constant intensity) driven shock wave.