Improved Integral Form of the Compressible Flowfield in Thin Channels with Injection

Abstract

This work seeks to obtain an improved integral formulation for the rotational, inviscid, compressible motion in a solid rocket motor. Assuming a slender porous chamber, the method in this study reduces the problem to a single integral equation that can be solved numerically. Alternatively, closed-form analytical approximations are shown to exist for particular values of the specific heats ratio. These are obtained using anAbel transformation of the pressure equation. For the case of uniform surface mass flux, a recursion is derived for the pressure as a function of space and specific heats ratios. Here, the dependence of sidewall injection on chamber pressure is modeled according to SaintRobert’s power law. After overcoming some deficiencies encountered in previous work, results are presented and compared with two closed-form analytical solutions developed under oneand two-dimensional isentropic flow conditions for either uniformly distributed mass flux or wall injection velocity. Furthermore, agreement with an existing one-dimensional solution is established for the case of uniform mass flux. For constant sidewall injection velocity, the formulation is shown to compare favorably with a two-dimensional solution obtained by Maicke and Majdalani (“On the Rotational Compressible Taylor Flow in Injection-Driven Porous Chambers,” Journal of Fluid Mechanics, Vol. 603, 2008, pp. 391–411). A collection of experimental data acquired by Traineau et al. (Cold-Flow Simulation of a Two-Dimensional Nozzleless Solid-RocketMotor,”AIAAPaper 1986-1447, July 1986) is also used in the validation process, including a computational fluid dynamics verification carried out using the Reynolds stress model.