An 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 timedependent 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.The resulting agreement with the 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. 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. Nomenclature a Q = mean stagnation sound speed, ^/p 0 / p 0 A p = dimensional oscillatory pressure amplitude k m = wave number, mnR IL = 0) 0 R I a 0 L = internal chamber length M h -wall injection Mach number, V b I a 0 Po = mean chamber pressure, p 0

Cold-flow analysis of the gas dynamics in a slab rocket motor including curvature effects

“…In general, the time-dependent field is found to be controlled by the injection Strouhal number, Unsurprisingly, this parameter is identical to the one encountered in the circular-port geometry. 8 By comparison with the mean flow, the character of the unsteady field is remarkably analogous. Accordingly, unsteady vorticity is more pervasive in the slab configuration.…”

“…Equations (7)- (8) can be reduced to Berman's equation 24 and then solved asymptotically following Yuan. 25 Defining the streamfunction to be of the Berman type, (i.e., that varies linearly with the distance from the head end due to geometric similarity) we let…”

“…Most analytical work in the past has addressed axisymmetric flow structures in circular-port geometries. Such configurations mimic commonly produced rocket motors (see for example Culick, 1 Flandro, [2][3][4][5][6] Majdalani and Van Moorhem, 7,8 and Kirkkopru et al 9 ). Much can be gained from these traditional idealizations that produce closed-form solutions.…”

“…The procedure employs a multiple-scale perturbation solution presented recently by the authors. 8 Furthermore, a formal WKB solution is constructed to provide a separate asymptotic formulation that is valid to any desired order. In Sec.…”

Improved mean-flow solution for slab rocket motors with regressing walls

“…ATHEMATICAL flow models of the internal gas dynamics in solid rocket motors have relied on subdividing the field into a mean and a time-dependent component. Recent analyses by Flandro, [1][2][3] Majdalani, 4,5 and Majdalani and Van Moorhem 6,7 have been successful in producing time-dependent solutions that could incorporate both viscous and rotational effects. At the outset, both viscous and rotational effects were shown to play important roles in prescribing the temporal flow behavior.…”