The mean gaseous motion in solid rocket motors has been traditionally described using an inviscid solution in a porous tube of fixed radius and uniform wall injection. This model, usually referred to as the Taylor–Culick profile, consists of a rotational solution that captures the bulk gaseous motion in a frictionless rocket chamber. In practice, however, the port radius increases as the propellant burns, thus leading to time-dependent effects on the mean flow. This work considers the related problem in the context of viscous motion in a porous tube and allows the radius to be time dependent. By implementing a similarity transformation in space and time, the incompressible Navier–Stokes equations are first reduced to a nonlinear fourth-order ordinary differential equation with four boundary conditions. This equation is then solved both numerically and asymptotically, using the injection Reynolds number Re and the dimensionless wall regression ratio α as primary and secondary perturbation parameters. In this manner, closed-form analytical solutions are obtained for both large and small Reynolds number with small-to-moderate α. The resulting approximations are then compared with the numerical solution obtained for an equivalent third-order ordinary differential equation in which both shooting and the irregular limit that affects the fourth-order formulation are circumvented. This code is found to be capable of producing the stable solutions for this problem over a wide range of Reynolds numbers and wall regression ratios.
Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls
2017
Article (Journal)
English