On the Lagrangian optimization of wall-injected flows: from the Hart–McClure potential to the Taylor–Culick rotational motion

Abstract

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.

Internal Flows Driven by Wall-Normal Injection

“…This principle states that a system will tend to maximize entropy at equilibrium and may hence be applied to our problem by considering the different energy solutions as different states of the same system. As shown by Saad & Majdalani (2010), the second law analysis reveals that the volumetric entropy of the Type I family grows with successive increases in q but depreciates in the Type II case. So given an initial profile, the system may evolve according to one of two scenarios that are described below.…”

“…While β n was prescribed by the headwall injection pattern, the choice of α n appeared to be flexible provided that the constraint given by (22) remained satisfied. In this section, we follow Majdalani & Saad (2007a) by applying the Lagrangian optimization technique to the total kinetic energy of the generalized Taylor-Culick solution to the extent of producing a variational constraint on α n (see also Saad & Majdalani, 2010). After some effort, two types of solutions will be identified with increasing or decreasing kinetic energies; of the two families, the Taylor-Culick model will be recovered as a special case.…”

“…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. In subsequent work, Majdalani & Saad (2007a) and Saad & Majdalani (2010) manage to introduce a variational procedure based on Lagrangian multipliers to identify solutions of the Taylor-Culick type with varying kinetic energies. As it will be seen in Section 3, these will help to uncover a wide array of motions ranging from purely irrotational to highly rotational fields.…”

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

“…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]. This potential mean flow preceded the use of Culick's model in several fundamental investigations of combustion instability [52][53][54].…”

On the swirling Trkalian mean flow field in solid rocket motors

“…Among relevant applications, one may bring into perspective those concerned with combustion instability (Flandro & Majdalani 2003;Majdalani, Fischbach & Flandro 2006;Flandro, Fischbach & Majdalani 2007) and vortico-acoustic wave propagation Majdalani 2001bMajdalani , 2009, as well as those devoted to hydrodynamic stability analyses in porous chambers both with and without particle interactions (Féraille & Casalis 2003;Chedevergne et al 2006;Abu-Irshaid et al 2007;Chedevergne et al 2012). Using variational calculus, the Lagrangian optimization principle is further extended to this problem where two types of continual spectra of quantum-like energy states of the TC solution are identified, and these have been essentially shown to evolve from a purely irrotational Hart-McClure mean flow motion (with least kinetic energy) to a fully rotational profile requiring the most energy to excite (Saad & Majdalani 2010). In all cases considered, simple sinusoidal approximations are obtained assuming sufficiently long chambers.…”