Extension of the Taylor-Culick Profile to Rockets with Noncircular Grain Perforations

摘要

The Taylor-Culick mean flow profile is universally employed to describe the bulk gaseous motion in solid rocket motors with circular cross sections. In this study, an extended form of the solution is obtained for motors with noncircular grain perforations. This is achieved through the use of a judicious set of mathematical assumptions and asymptotic approximations that take into account the periodic deviations of the grain surface from the fixed radius associated with a circular port. Our analysis leverages the periodicity of the geometric irregularities that typically accompany the burn-back behavior of several practical grain configurations. These include the wagon-wheel, dendrite, and dog-bone perforations, specifically those which comprise three or more axisymmetric lobes. Then using a sinusoidal function to capture the grain's inner circumference, an extended form of the Taylor-Culick profile is derived as function of both the number of axisymmetric lobes that orbit the axis of the motor, and the periodic deviation amplitude relative to a strictly circular grain. The analysis also accounts for the radial velocity fluctuations that are induced by the undulating grain surface, and the deficiencies in the inward radial injection velocity that accompany the continual and periodic reorientations of the normal unit vector along the inner circumference. Finally, a small parameter perturbation in the periodic deviation amplitude is pursued to extract an approximate, potential solution for the mean flow profile, stream function, and corresponding flow characteristics.