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In this work, we present two simple mean flow solutions that mimic the bulk gas motion inside a full-length, cylindrical hybrid rocket engine. Two distinct methods are used. The first is based on steady, axisymmetric, rotational, and incompressible flow conditions. It leads to an Eulerian solution that observes the normal sidewall mass injection condition while assuming a sinusoidal injection profile at the head end wall. The second approach constitutes a slight improvement over the first in its inclusion of viscous effects. At the outset, a first order viscous approximation is constructed using regular perturbations in the reciprocal of the wall injection Reynolds number. The asymptotic approximation is derived from a general similarity reduced Navier–Stokes equation for a viscous tube with regressing porous walls. It is then compared and shown to agree remarkably well with two existing solutions. The resulting formulations enable us to model the streamtubes observed in conventional hybrid engines in which the parallel motion of gaseous oxidizer is coupled with the cross-streamwise (i.e., sidewall) addition of solid fuel. Furthermore, estimates for pressure, velocity, and vorticity distributions in the simulated engine are provided in closed form. Our idealized hybrid engine is modeled as a porous circular-port chamber with head end injection. The mathematical treatment is based on a standard similarity approach that is tailored to permit sinusoidal injection at the head end.

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This work seeks to obtain an improved integral formulation for the rotational, inviscid, compressible motion in a solid rocket motor. Assuming a slender porous chamber, the method in this study reduces the problem to a single integral equation that can be solved numerically. Alternatively, closed-form analytical approximations are shown to exist for particular values of the specific heats ratio. These are obtained using an Abel transformation of the pressure equation. For the case of uniform surface mass flux, a recursion is derived for the pressure as a function of space and specific heats ratios. Here, the dependence of sidewall injection on chamber pressure is modeled according to Saint-Robert's power law. After overcoming some deficiencies encountered in previous work, results are presented and compared with two closed-form analytical solutions developed under one-and two-dimensional isentropic flow conditions for either uniformly distributed mass flux or wall injection velocity. Furthermore, agreement with an existing one-dimensional solution is established for the case of uniform mass flux. For constant sidewall injection velocity, the formulation is shown to compare favorably with a two-dimensional solution obtained by Maicke and Majdalani ("On the Rotational Compressible Taylor Flow in Injection-Driven Porous Chambers,"

This work focuses on the development of a semi-analytical model that is appropriate for the rotational, steady, inviscid, and compressible motion of an ideal gas, which is accelerated uniformly along the length of a right-cylindrical rocket chamber. By overcoming some of the difficulties encountered in previous work on the subject, the present analysis leads to an improved mathematical formulation, which enables us to retrieve an exact solution for the pressure field. Considering a slender porous chamber of circular cross-section, the method that we follow reduces the problem's mass, momentum, energy, ideal gas, and isentropic relations to a single integral equation that is amenable to a direct numerical evaluation. Then, using an Abel transformation, exact closed-form representations of the pressure distribution are obtained for particular values of the specific heat ratio. Throughout this effort, Saint-Robert's power law is used to link the pressure to the mass injection rate at the wall. This allows us to compare the results associated with the axisymmetric chamber configuration to two closed-form analytical solutions developed under either one-or two-dimensional, isentropic flow conditions. The comparison is carried out assuming, first, a uniformly distributed mass flux and, second, a constant radial injection speed along the simulated propellant grain. Our amended formulation is consequently shown to agree with a one-dimensional solution obtained for the case of uniform wall mass flux, as well as numerical simulations and asymptotic approximations for a constant wall injection speed. The numerical simulations include three particular models: a strictly inviscid solver, which closely agrees with the present formulation, and both k-ω and Spalart-Allmaras computations.