Rotating Flows in Acoustically Unstable Rocket Motors

Citation

Flandro, Gary Arnold (1967) Rotating Flows in Acoustically Unstable Rocket Motors. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BYWJ-5Q96. https://resolver.caltech.edu/CaltechETD:etd-02272004-154250

Abstract

One of the most interesting manifestations of acoustic combustion instability in solid propellant rocket motors is the formation of strong vortices in the combustion chamber. A single vortex filament stretching along the motor axis from the head-end has been observed in several experiments in association with gas oscillations in the frequently occurring traveling tangential mode of instability. These flows are sometimes accompanied by a quite noticeable axial torque on the motor itself, and this effect has given rise to flight performance difficulties in a number of instances. Previous theoretical studies of the vortex generation effect have been inadequate in several respects. The present work is an attempt to place the theory on a more firm base and to clarify the connection between traveling wave motions and the generation of vortices and torques.

It is readily shown that traveling waves transport momentum, and in the case of traveling tangential waves in a cylindrical combustion chamber this represents a steady axial component of angular momentum in the gas. This observation gives rise to a simple conceptual model of the vortex generation effect. Thus the presence of a steady mass flux about the axis implies the existence of a layer of vorticity at the chamber walls which may be represented by a vortex sheet composed of axially oriented bound vortex filaments. In the three-dimensional case these vortices are shed either at the end of propellant grain or at the periphery of the nozzle; the other ends of the filaments traverse the fore-end closure to the center and are combined and shed in an intense vortex filament along the symmetry axis of the motor.

Due to the production of gas at the chamber wall, tangential forces at the wall are produced by the interaction of this mean flow with the bound vortices. Angular momentum arguments must be used in this conceptual mechanism to estimate the strength of the axial vortex filament, and it is readily shown that the sense of the vortex must be opposite to the direction of travel of the waves. The direction and magnitude of the torque on the motor depend on the mean flow Mach number at the wall and must be established by calculation of the wall shear stresses.

The detailed calculations are guided by the mechanism just outlined. All physical features of the problem which appear to be significant are simulated mathematically. In particular, the effects of the three dimensional mean flow pattern in the chamber and the pressure-sensitive combustion region at the burning surface are represented. Also considered are the effects of freedom of motion of the rocket motor in the plane normal to the symmetry axis. Both inviscid and viscous theories are developed using multi-parameter asymptotic perturbation expansion techniques. It is proved that traveling tangential waves are subject to amplification under conditions existing in typical solid propellant rockets, and that a steady transport of gas about the chamber axis accompanies this motion as a second-order perturbation. The equations of motion admit of only a vortex-like steady second-order azimuthal solution. This must be superimposed on the acoustic wave motions in such a way that angular momentum is conserved (due consideration being given to body forces on the gas and tangential forces at the wall). Thus the net pattern of steady circumfer vential mass flux at a given motor cross-section consists of a drift of fluid in the direction of the wave adjacent to the wall with a rapid transition to an oppositely spinning vortex flow as the longitudinal axis is approached. Introduction of the viscous corrections gives rise to a boundary. condition which sets the vortex strength, and a formal connection with the classical acoustic streaming effect is established. Since momentum is dissipated in the shear region at the wall, a torque appears on the chamber itself. This roll moment is opposite in sense to the wave travel during amplification of the acoustic waves, and numerical calculations give torque magnitudes which are in agreement with experimental data from several sources.

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