Energy Steepened States of the Taylor-Culick Profile

Majdalani, Joseph; Saad, Tony

doi:10.2514/6.2007-5797

Cited by 8 publications

(13 citation statements)

References 34 publications

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“…While the incompressible motion is relatively well understood (Taylor 1956;Culick 1966;Majdalani & Akiki 2010), recent advances have enabled us to account for the presence of arbitrary headwall injection (Majdalani & Saad 2007b;Saad & Majdalani 2009), wall regression (Majdalani, Vyas & Flandro 2002;Zhou & Majdalani 2002), grain taper (Saad, Sams & Majdalani 2006;Sams, Majdalani & Saad 2007), variable cross-section (Kurdyumov 2006), headwall singularity (Chedevergne, Casalis & Féraille 2006), viscous effects (Majdalani & Akiki 2010), and stability (Chedevergne, Casalis & Majdalani 2012). Furthermore, flow approximations exhibiting smoother or steeper profiles than the cold flow equilibrium state have been studied in connection with their energy content by Majdalani & Saad (2007a) and Saad & Majdalani (2008). In what concerns compressible flow effects, these have been first investigated by Traineau, Hervat & Kuentzmann (1986) in the context of two-dimensional porous tubes and channels with sidewall injection.…”

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confidence: 99%

Compressible integral representation of rotational and axisymmetric rocket flow

Akiki

Majdalani

2016

J. Fluid Mech.

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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.

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confidence: 99%

Compressible integral representation of rotational and axisymmetric rocket flow

Akiki

Majdalani

2016

J. Fluid Mech.

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“…Pursuant to the original discussion by Majdalani & Saad (2007a), it may be hypothesized that two complementary families of solutions exist with the unique characteristics of exhibiting continually varying energy levels from which the Taylor-Culick model may be recovered. To this end, it may be useful to seek mean-flow solutions with either increasing or decreasing energies.…”

Section: Generalization (A) Type I Solutions With Increasing Energy Lmentioning

confidence: 99%

“…In the process, we unravel a family of approximate solutions that satisfy the problem's fundamental constraints (see also Majdalani & Saad 2007a). Among those, we apply the Lagrangian optimization principle to identify the particular forms that require the most or the least kinetic energy to appear.…”

Section: Introductionmentioning

confidence: 99%

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

Saad

Majdalani

2009

Proc. R. Soc. A.

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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.

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“…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.…”

Section: Generalized Taylor-culick Formulationmentioning

confidence: 99%

“…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.…”

Section: Generalizedmentioning

confidence: 99%