Taylor's incompressible and rotational profile is extended to a porous cylinder with arbitrary headwall injection. This profile, often referred to as Culick's mean flow, is now generalized to permit the imposition of reactive headwall conditions. Starting with Euler's steady equations, the solution that we derive is approximate, being exact only at the sidewall, the centerline, or for similarity-conforming inlet profiles. Furthermore, the approximation is quasiviscous, being observant of the no slip requirement at the sidewall. Based on numerical experiments under inviscid flow conditions, the closed-form approximation that we obtain appears to be well suited to describe the bulk flow field in basic models of solid and hybrid rockets where uniform sidewall injection is imposed at the propellant surface. For similarity-nonconforming profiles, the approximation becomes more accurate as we move away from the headwall. Results are verified using computational fluid dynamics for several headwall injection patterns.
The Taylor-Culick solution for a porous cylinder is often used to describe the bulk gas motion in idealized representations of solid rocket motors. However, other approximate solutions may be found that satisfy the same fundamental constraints. In this vein, steeper or smoother profiles may be observed in either experimental or numerical tests, particularly in the presence of intense levels of acoustic energy. In this study, we use the Lagrangian optimization principle to arrive at multiple solutions that can help to explain the practically observed motions. We then search for the extreme states that display either the least or the most kinetic energy. These are derived and found to be dependent on the chamber's aspect ratio. By assuming a slender case, simple expressions are retrieved from which both the rotational Taylor-Culick and the irrotational HartMcClure solutions are recovered as special cases. At the outset, a new family of flow approximations is derived extending from purely potential to highly rotational fields. These are constructed, verified and catalogued based on their kinetic energies. To help understand the tendency of a motion to shift energy states, a second law analysis is performed and used to rank these solutions based on their entropy content. Interestingly, the Taylor-Culick profile is found to represent an equilibrium state entailing the most energy and entropy among those starting from the rest. A formal extension of Kelvin's minimum energy theorem to an open region is then pursued, followed by a direct implementation to the problem at hand. Three other flow motions with open boundaries are examined to further exemplify the application of Kelvin's extended criteria. Our efforts illustrate the overarching harmony that exists between Lagrange's minimization principle and Kelvin's minimum energy theorem; both converge on the potential solution as the least kinetic energy bearer even when the velocity at the open barrier is finite.
In this paper, Taylor's incompressible and rotational flow in a porous channel with surface mass addition is extended to account for arbitrary headwall injection. Our analysis considers Euler's steady-state equations from which an approximate solution is derived. The resulting mean flow representation satisfies the vanishing axial velocity condition at the blowing walls and is confirmed through comparisons with inviscid finite volume numerical simulations. For a given class of symmetric headwall injection profiles, our solutions are exact at the sidewall and along the midsection plane, where the error in the approximations vanishes identically. The case of uniform injection stands as an exception due to its discontinuity at the sidewall. All solutions become increasingly more accurate in the downstream direction. They hence enable us to approximate the bulk flowfield in both hybrid and solid rockets with variable headwall injection.
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