The Taylor-Culick profile with arbitrary headwall injection

Abstract

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.

Internal Flows Driven by Wall-Normal Injection

“…In this chapter, the focus has been on the inviscid form of the Taylor-Culick family of incompressible solutions. The originality of the analysis stands, perhaps, in the incorporation of variable headwall injection using a linear series expansion that may be attributed to Majdalani & Saad (2007b). The extended Taylor-Culick framework has profound implications as it permits the imposition of realistic conditions that may be associated with solid or hybrid propellant rockets with reactive fore ends or injecting faceplates.…”

“…As we move closer to the central topic of this chapter, we consider recent work in which the Taylor-Culick solution is reconstructed for the case of solid rocket motors with headwall injection or hybrid motors with a large headwall-to-sidewall velocity ratio (Majdalani, 2007a). The corresponding problem is analyzed in both axisymmetric and planar configurations by Majdalani & Saad (2007b) and Saad & Majdalani (2009b), respectively. This will be the topic of Section 2 where the solutions for the Taylor-Culick flow with arbitrary headwall injection are derived and compared to steady state, second order accurate inviscid computations.…”

“…Our approach is based on a technique introduced by Majdalani (2007a) and Majdalani & Saad (2007b). The ability to account for arbitrary headwall injection will extend the Taylor-Culick approximation to a wider range of problems.…”

“…As shown by Majdalani & Saad (2007b) and detailed in Section 2.1, orthogonality may be applied to obtain β n for an axisymmetric headwall injection pattern. Application of Lagrangian optimization in conjunction with the large L approximation yield identical results for α n as those obtained in (92) and (98).…”

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

“…The Taylor-Culick solution was originally verified to be an adequate representation of the expected flowfield in SRMs both numerically by Sabnis et al [44] and experimentally by Dunlap et al [45,46], thereby confirming its character in a nonreactive chamber environment. It was extended by Majdalani and Akiki [4] to include effects of viscosity and headwall injection, by Saad et al [42] and Sams et al [43] to account for wall taper, by Kurdyumov [47] to capture effects of irregular cross sections, and by Majdalani and Saad [48] to allow for arbitrary headwall injection. Then, using variational calculus and the Lagrangian optimization principle, Saad and Majdalani [49] uncovered a continuous spectrum of Taylor-like solutions exhibiting increasing kinetic energy signatures, while ranging from the traditional Culick profile down to its predecessor, the irrotational mean flow known as the Hart-McClure profile [50,51].…”