**4. Conclusions**

For four decades and counting, the motion of incompressible fluids through porous tubes with wall-normal injection (or suction) has been extensively used in the propulsion and flow separation industries. 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. The procedure that we follow starts with Euler's steady equations, and ends with an approximation that is exact only at the sidewall, the centerline, or when using similarity-conforming inlet velocities. For similarity-nonconforming profiles, our approach becomes increasingly more accurate as the distance from the headwall is increased; this property makes our model well suited to describe the bulk motion in simulated solid and hybrid rockets where the blowing speed is assumed to be uniformly distributed along the grain surface. The justification for using a linear summation of eigensolutions and the reason for its increased accuracy in elongated chambers may be connected, in part, to the quasi-linear behavior of the vorticity transport equation for large *z*. Such behavior is corroborated by the residual error analysis that we carried out in Section 2.6. Furthermore, as carefully shown by Kurdyumov (2008), the vorticity-streamfunction relation appears to be strongly nonlinear in the direct vicinity of the headwall, yet becomes increasingly more linear with successive increases in *z*.

Another advantage of the present formulation may be ascribed to its quasi-viscous character, being observant of the no-slip requirement at the sidewall. Based on numerical simulations conducted under both inviscid and turbulent flow conditions, the closed-form expressions that we obtain appear to provide reasonable approximations for several headwall injection patterns associated with conventional laminar and turbulent flow profiles. Everywhere, our comparisons are performed for the dual cases of small and large headwall injection in an 34 Will-be-set-by-IN-TECH

by branching out to the Type II region, it may be argued that such development is not possible for two reasons. Firstly, the character of the two types of solutions is sharply dissimilar,

maximizes the entropy for the Type I branch, it can be viewed as a local equilibrium state. As such, there is no necessity for the system to switch branches once it reaches the Taylor–Culick

If the system is initialized on the Type II branch, it will approach the solution with most vorticity (i.e. Type II, *q* = 2). Although this may be a mathematically viable outcome, it may not be physically realizable because it would be practically impossible to initialize a system with such a high level of vorticity without the aid of external work. The most natural flow evolution corresponds to an irrotational system originally at rest in which vorticity generation is initiated at the sidewall during the injection process. The ensuing motion will subsequently progress until it reaches the stable Taylor–Culick equilibrium state wherein it can settle with

For four decades and counting, the motion of incompressible fluids through porous tubes with wall-normal injection (or suction) has been extensively used in the propulsion and flow separation industries. 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

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. The procedure that we follow starts with Euler's steady equations, and ends with an approximation that is exact only at the sidewall, the centerline, or when using similarity-conforming inlet velocities. For similarity-nonconforming profiles, our approach becomes increasingly more accurate as the distance from the headwall is increased; this property makes our model well suited to describe the bulk motion in simulated solid and hybrid rockets where the blowing speed is assumed to be uniformly distributed along the grain surface. The justification for using a linear summation of eigensolutions and the reason for its increased accuracy in elongated chambers may be connected, in part, to the quasi-linear behavior of the vorticity transport equation for large *z*. Such behavior is corroborated by the residual error analysis that we carried out in Section 2.6. Furthermore, as carefully shown by Kurdyumov (2008), the vorticity-streamfunction relation appears to be strongly nonlinear in the direct vicinity of the

headwall, yet becomes increasingly more linear with successive increases in *z*.

Another advantage of the present formulation may be ascribed to its quasi-viscous character, being observant of the no-slip requirement at the sidewall. Based on numerical simulations conducted under both inviscid and turbulent flow conditions, the closed-form expressions that we obtain appear to provide reasonable approximations for several headwall injection patterns associated with conventional laminar and turbulent flow profiles. Everywhere, our comparisons are performed for the dual cases of small and large headwall injection in an

*<sup>n</sup>* . Secondly, given that the Taylor–Culick solution

*<sup>n</sup>* and *α*<sup>+</sup>

especially in the expressions for *α*−

no further tendency to branch out.

that may be attributed to Majdalani & Saad (2007b).

**3.12.2 Type II branching**

**4. Conclusions**

state.

effort to mimic the internal flow character in either SRM or hybrid rocket motors. Overall, we find that the flow field evolves to the self-similar Taylor–Culick sinusoid far downstream irrespective of the headwall injection pattern. Nonetheless, the details of headwall injection remain important in hybrid motors, short chambers, and T-burners where the foregoing approximations may be applied. In hybrid rockets, our models seem to capture the streamtube motion quite effectively.

The other chief contribution of this chapter is the discussion of variational solutions that may be connected with the Taylor-Culick problem. Based on the Lagrangian optimization of the total volumetric energy in the chamber, we are able to identify two families of solutions with dissimilar energy signatures. These are accompanied by lower or higher kinetic energies that vary, from one end of the spectrum to the other, by up to 66 percent of their mean value. After identifying that *α*− *<sup>n</sup>* <sup>∼</sup> (−1)*n*(2*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)−<sup>2</sup> yields the profile with least kinetic energy, a sequence of Type I solutions is unraveled in ascending order, *α*− *<sup>n</sup>* <sup>∼</sup> (−1)*n*(2*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)−*q*; *<sup>q</sup>* <sup>&</sup>gt; 2, up to Taylor-Culick's. The latter is asymptotically recovered in the limit of *q* → ∞, a case that corresponds to an equilibrium state with maximum entropy. In practice, most solutions become indiscernible from Taylor-Culick's for *q* ≥ 5. Indeed, those obtained with *q* = 2, 3, and 4 exhibit energies that are 18.9, 8.28, and 2.73 percent lower than their remaining counterparts. The least kinetic energy solution with *q* = 2 returns the classic, irrotational Hart-McClure profile. It can thus be seen that the application of the Lagrangian optimization principle to this problem leads to the potential form that historically preceded the Taylor-Culick motion. It can also be inferred that the Type I solutions not only bridge the gap between a plain potential representation of this problem and a rotational formulation, but also recover a continuous spectrum of approximations that stand in between. When the same analysis is repeated using *α*+ *<sup>n</sup>* <sup>∼</sup> (2*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)−*q*; *<sup>q</sup>* <sup>≥</sup> 2, a complementary family of Type II solutions is identified with descending energy levels. These are shown to be purely academic, although they represent a class of exact solutions to the modified Helmholtz equation. Their most notable profiles correspond to *q* = 2, 3, and 4 with energies that are 47.0, 8.08, and 2.40 percent higher than Taylor-Culick's. Their entropies are also higher than that associated with the equilibrium state. Despite their dissimilar forms, both Type I and II solutions converge to the Taylor-Culick representation when their energies are incremented or reduced. Yet before using the new variational solutions to approximate the mean flow profile in porous tubes or the bulk gaseous motion in simulated rocket motors, it should be borne in mind that no direct connection exists between the energy steepened states and turbulence. For this reason, it is hoped that additional numerical and experimental investigations are pursued to test their physicality and the particular configurations in which they are prone to appear. As for the uniqueness of the Taylor-Culick equilibrium state, it may be confirmed from the entropy maximization principle and the Lagrangian-based solutions where, for a given set of boundary conditions, the equilibrium state may be asymptotically restored as *q* → ∞ irrespective of the form of *<sup>α</sup><sup>n</sup>* <sup>∼</sup> (−1)*n*(*p n* <sup>+</sup> *<sup>m</sup>*)−*q*, provided that the Lagrangian constraint <sup>∑</sup> (−1)*nα<sup>n</sup>* <sup>=</sup> 1 is faithfully secured.

Lastly, we note that the collection of variational solutions that admit variable headwall injection increase our repertoire of Euler-based approximations that may be used to model the incompressible motion in porous tubes. For the porous channel flow analogue, the planar solutions are presented by Saad & Majdalani (2008a; 2009b). As for tapered grain configuration, the reader may consult with Saad et al. (2006) or Sams et al. (2007).

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