Higher Mean-Flow Approximation for Solid Rocket Motors with Radially Regressing Walls

Abstract

The bulk gas motion in a circular-port rocket motor is described using a rotational, incompressible, and viscous flow model that incorporates the effect of wall regression. The mathematical idealization developed is also applicable to semi-open porous tubes with expanding walls. Based on mass conservation, a linear variation in the mean axial velocity is ascertained. This relationship suggests investigating a spatial transformation of the Proudman-Johnson form. With the use of similar arguments, a temporal transformation is also introduced. When these transformations are applied in both space and time, the Navier-Stokes equations are reduced to a single, nonlinear, fourth-order differential equation. Following this exact Navier-Stokes reduction, the resulting problem is solved using variation of parameters and small-parameter perturbations. The asymptotic solutions for the velocity, pressure, vorticity, and shear are obtained as function of the injection Reynolds number Re and the dimensionless regression ratio a. By way of verification, it is shown that, as α/Re → 0, Yuan and Finkelstein's solutions can be restored from ours. Similarly, as α/Re → 0, Culick's inviscid profile is recovered. It is demonstrated that, for a range of small α/Re, inviscid solutions are practical. However, for fast burning propellants under development, the inviscid assumption deteriorates. Because it is applicable over a broader range of operating parameters, the current analysis leads to a closed-form mean-flow solution that can be used, instead of the inviscid profile, to 1) prescribe an adjusted aeroacoustic field, 2) describe the so-called acoustic boundary layer, 3) evaluate the viscous and rotational contributions to the acoustic stability growth rate factor, 4) track the evolution of hydrodynamic instability, and 5) accurately simulate the internal gasdynamics in rapidly regressing motors and cold-flow experiments with medium-to-high levels of injection.