Analytic solutions to the problem of jets with time-dependent injection velocities

Abstract

It has frequently been stated (in particular by us) that the problem of a jet with a periodically varying injection velocity does not allow full analytic solutions. A description of such jets in terms of a sequence of free-streaming, continuous jet beam segments, separated by narrow 'internal working surfaces' results in a set of simple equations, which have then been solved numerically. Also, an asymptotic analytic solution (for large distances from the source) has been found. In this paper we present a new method for solving the equations for a hypersonic jet with a time-dependent injection velocity, based on a consideration of the momentum conservation for the internal working surfaces. With this formulation, it is possible to derive parametric analytic solutions for the full flow. We derive such solutions for specific time dependencies of the injection velocity and density: a sinusoidal velocity variability with constant mass injection rate, and with constant injection density. These analytic solutions have clear applications for the interpretation of observations of HH jets.