Improved flowfield models in rocket motors and the Stokes layer with sidewall injection

The boundary-layer structure is investigated in rectangular and cylindrical tubes where sidewall injection is imposed upon an oscillatory flow. A rotational and inviscid solution based on Culick's approach is assumed for the mean flow. The oscillatory velocity is obtained by superimposing the acoustic (compressible, irrotational) and the vortical (incompressible, rotational) velocity vectors. Two-dimensional time-dependent multiple scales perturbations that employ the proper scaling coordinates are applied to the vortical momentum equations. A uniformly valid, closed form expression for the oscillatory velocity is derived to the order of the injection Mach number, which is practically small. This analytic solution is indistinguishable from the numeric solution and agrees well with cold flow experimental data, and with the recent two-dimensional solution by Flandro which solves the vorticity transport equation using regular perturbations. Analytic formulations of various existing models are furnished in details. A new similarity parameter that controls the boundary-layer thickness is derived. A closed form expression for the boundary-layer thickness is furnished. The role of the Strouhal number in controlling the wavelength of rotational waves is established. An accurate assessment of the amplitude and phase relations between oscillatory velocity and pressure is obtained. The boundary-layer thickness is found to decrease with increasing frequency and kinematic viscosity, depending on the penetration number, axial location, and mode of oscillations. At very low injections, the multiple scales expression approaches the exact Stokes' solution without injection. Overall, this analysis closely follows the current modeling procedures of gas dynamics in the internal combustion of solid rocket motors. The new axisymmetric solution of the time-dependent rotational flow leads to marked improvements over existing models in the combustion stability of solid rocket motors, resulting in enhanced predictive power and design ameliorations. The results are also of interest in bioengineering, in the study of the respiratory and blood circulation systems. By itself, the multiple scales approach, which employs one composite scale to match several different length scales, is a novel technique in perturbation theory that can be used in similar situations arising in fluid mechanics, solid mechanics, mass or heat transfer, and other applications characterized by decaying vibrations or oscillations.