Improved energy normalization function in rocket motor stability calculations

Majdalani, Joseph; Fischbach, Sean R.; Flandro, Gary A.

doi:10.1016/j.ast.2006.06.002

Cited by 24 publications

(13 citation statements)

References 17 publications

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“…(20), subject to the fundamental constraints given by Eq. (25). By substituting the Taylor-Culick profile for the leading-order solution into Eq.…”

Section: A First-order Correctionmentioning

confidence: 98%

Rotational Approximation of the Taylor-Culick Profile for Rockets with Noncircular Grains

Majdalani¹,

Horn²

2014

50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference

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The Taylor-Culick mean flow profile is universally employed to describe the bulk gaseous motion in solid rocket motors with circular cross sections. In this study, an extended form of the solution is obtained for motors with noncircular grain perforations. This is achieved through the use of a judicious set of mathematical assumptions and asymptotic approximations that take into account the periodic deviations of the grain surface from the fixed radius associated with a circular port. Our analysis leverages the periodicity of the geometric irregularities that typically accompany the burn-back behavior of several practical grain configurations. These include the wagon-wheel, dendrite, and dog-bone perforations, specifically those which comprise three or more axisymmetric lobes. Then using a sinusoidal function to capture the grain's inner circumference, an extended form of the Taylor-Culick profile is derived as function of both the number of axisymmetric lobes that orbit the axis of the motor, and the periodic deviation amplitude relative to a strictly circular grain. The analysis also accounts for the radial velocity fluctuations that are induced by the undulating grain surface, and the deficiencies in the inward radial injection velocity that accompany the continual and periodic reorientations of the normal unit vector along the inner circumference. Previous work in this area focused on a small parameter perturbation in the periodic deviation amplitude to extract an approximate, potential solution for the mean flow profile, streamfunction, and corresponding flow characteristics. The present article updates and supersedes the previous effort identically by replacing the one-term potential solution with a three-term, judiciously constructed rotational approximation for the mean flowfield.

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“…(20), subject to the fundamental constraints given by Eq. (25). By substituting the Taylor-Culick profile for the leading-order solution into Eq.…”

Section: A First-order Correctionmentioning

confidence: 98%

Rotational Approximation of the Taylor-Culick Profile for Rockets with Noncircular Grains

Majdalani¹,

Horn²

2014

50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference

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“…The so-called blown-off layer has been the subject of several dedicated studies including a seminal paper by Cole and Aroesty. 69 (sidewall) (42) where (o) refers to the exact, outer, farfield solution; (i), to the inner, viscous correction in the core region; and (w), to the wall correction. At both endpoint regions of non-uniformity, the implementation of the farfield matching requirement enables us to recover the exact outer solution (43) which was previously depicted in Fig.…”

Section: Viscous Solutionmentioning

confidence: 99%

Improved Mean Flow Solution for Solid Rocket Motors with a Naturally Developing Swirling Motion

Majdalani¹,

Fist²

2014

50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference

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In this work, a new exact Euler solution is derived under the same fundamental contingencies of axisymmetric, steady, rotational, incompressible, single phase, non-reactive, and inviscid fluid, which stand behind the ubiquitously used mean flow profile named "Taylor-Culick." In comparison to the latter, which proves to be complex lamellar, the present model is shown to be of the Trkalian type, hence capable of generating a nonzero swirl component that increases linearly in the streamwise direction. This enables us to provide an essential mathematical representation that is appropriate of flow configurations where the bulk gaseous motion is permitted to swirl. Examples abound and one may cite the classic experiments of Dunlap and co-workers, which focus on porous chambers with circular cross-sections, and where the inevitable presence of swirl as a natural flow companion is clearly demonstrated (). From a procedural standpoint, the new Trkalian solution is deduced directly from the Bragg-Hawthorne equation, which has been repeatedly shown to possess sufficient platitude to reproduce several existing profiles such as Taylor-Culick's as special cases. Throughout this study, the fundamental properties and benefits of the present model are highlighted and discussed in the light of existing flow approximations and experiments. Consistent with the original Taylor-Culick mean flow motion, the Trkalian velocity is seen to exhibit both axial and tangential components that increase linearly with the distance from the headwall, and a radial component that remains axially invariant. However, unlike its axial and radial counterparts, which identically satisfy the no slip requirement at the sidewall (even in their purely inviscid form), the tangential velocity must be treated asymptotically so as to capture the evolving sidewall shear layer while resolving the core layer near the centerline. At the outset, a matched-asymptotic expression for the swirl velocity is obtained and discussed. Finally, our Trkalian model is shown to form a subset of the Beltramian class of solutions for which the velocity and vorticity vectors are not only parallel but also directly proportional. This characteristic feature is interesting as it stands in sharp contrast to the complex lamellar nature of the Taylor-Culick flow, where velocity and vorticity vectors remain orthogonal. Nomenclature a = chamber radius B = tangential angular momentum, ru θ B = axially normalized angular momentum, / ( ) B z rU r θ = C = swirl momentum constant 1 c = radial velocity constant multiplier, 1 1 1.9 1/ ( 26 34 ) 2 8 J λ ≈ D = tangential surface parameter H = stagnation pressure head 0 L = chamber length L = chamber aspect ratio, 0 / L a 2 p = normalized pressure, 2 / ( ) w p U ρ Q = normalized volumetric flow rate, 2 / ( ) w Q U a q = auxiliary constant, 1 1 1 1 1 4 ( ) 1.1580647 J λ λ − ≈ Re = crossflow or wall injection Reynolds number, 1 / w U a ν ε − = , , r z θ = radial, tangential, and axial coordinates U = mean tangential (inflow) velocity c U = headwall inj...

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“…1,2 Owing to this fact, analytical methodologies put forward to describe flow oscillations lean heavily on the assumption of small acoustic disturbances. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] Contrary to this assumption, however, a vast body of experimental evidence conveys a dissimilar picture, specifically, one involving large amplitude oscillations with steep gradients in flow variables. For example, in the extensive experimental studies of Clayton, Sotter, and co-workers, [18][19][20][21] a heavily instrumented, laboratory scale, 20 000 lbf thrust engine was used to investigate high amplitude tangential oscillations.…”

Section: Introductionmentioning

confidence: 99%

Acoustic streaming in simplified liquid rocket engines with transverse mode oscillations

2010

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This study considers a simplified model of a liquid rocket engine in which uniform injection is imposed at the faceplate. The corresponding cylindrical chamber has a small length-to-diameter ratio with respect to solid and hybrid rockets. Given their low chamber aspect ratios, liquid thrust engines are known to experience severe tangential and radial oscillation modes more often than longitudinal ones. In order to model this behavior, tangential and radial waves are superimposed onto a basic mean-flow model that consists of a steady, uniform axial velocity throughout the chamber. Using perturbation tools, both potential and viscous flow equations are then linearized in the pressure wave amplitude and solved to the second order. The effects of the headwall Mach number are leveraged as well. While the potential flow analysis does not predict any acoustic streaming effects, the viscous solution carried out to the second order gives rise to steady secondary flow patterns near the headwall. These axisymmetric, steady contributions to the tangential and radial traveling waves are induced by the convective flow motion through interactions with inertial and viscous forces. We find that suppressing either the convective terms or viscosity at the headwall leads to spurious solutions that are free from streaming. In our problem, streaming is initiated at the headwall, within the boundary layer, and then extends throughout the chamber. We find that nonlinear streaming effects of tangential and radial waves act to alter the outer solution inside a cylinder with headwall injection. As a result of streaming, the radial wave velocities are intensified in one-half of the domain and reduced in the opposite half at any instant of time. Similarly, the tangential waves are either enhanced or weakened in two opposing sectors that are at 90°angle to the radial velocity counterparts. The second-order viscous solution that we obtain clearly displays both an oscillating and a steady flow component. The steady part can be an important contributor to wave steepening, a mechanism that is often observed during the onset of acoustic instability.

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Improved energy normalization function in rocket motor stability calculations

Cited by 24 publications

References 17 publications

Rotational Approximation of the Taylor-Culick Profile for Rockets with Noncircular Grains

Rotational Approximation of the Taylor-Culick Profile for Rockets with Noncircular Grains

Improved Mean Flow Solution for Solid Rocket Motors with a Naturally Developing Swirling Motion

Acoustic streaming in simplified liquid rocket engines with transverse mode oscillations

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