Improved Time-Dependent Flowfield Solution for Solid Rocket Motors

Abstract

A model of the time-dependent velocity field in solid rocket motors is derived analytically for an oscillatory field that is subject to steady sidewall injection. The oscillatory pressure amplitude is assumed to be small by comparison to the mean pressure. The mathematical approach includes solving the momentum equation governing the rotational flow using separation of variables and multiple scales. This requires identifying scales at which unsteady inertia, convection, and diffusion are significant. A composite scale is obtained that combines three disparate scales. The time-dependent axisymmetric solution obtained incorporates the effects of unsteady inertia, viscous diffusion, and the radial and axial convection of unsteady vorticity by Culick's mean flow components (Culick, F. E. C., Rotational Axisymmetric Mean Flow and Damping of Acoustic Waves in a Solid Propellant Rocket, AIAA Journal, Vol. 4, No. 8, 1966, pp. 1462-1464). The resulting agreement with tbe numerical solution to the momentum equation is remarkable. The uncertainty in a short analytical expression is found to be smaller than the injection Mach number, which represents the error associated with the mathematical model itself. The multiple-scales solution agrees extremely well with Flandro's recent flowfield solution (Flandro, G. A., On Flow Turning, AIAA Paper 95-2730, July 1995). The present solution has the advantage of being shorter, more manageable in extracting quantities of interest, and capable of showing the significance of physical parameters on the solution.