Biglobal Instability of the Bidirectional Vortex. Part 2: Complex Lamellar and Beltramian Motions

Batterson, Joshua W.; Majdalani, Joseph

doi:10.2514/6.2011-5649

Cited by 15 publications

(9 citation statements)

References 15 publications

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“…His approach was further extended to accommodate arbitrary endwall injection by Akiki and Majdalani [73,74]. As for compressibility effects and stability of cyclonic motions, these were addressed in a series of studies by Maicke and Majdalani [75][76][77] and, for the stability assessment, by Batterson and Majdalani [78,79]. The last set supported the notion that flow stability could be continuously enhanced with successive increases in the swirl velocity.…”

Section: Introductionmentioning

confidence: 93%

Unified Framework for Modeling Swirl Dominated Helical Motions

Majdalani¹

2014

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

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In this article, the fundamental characteristics of swirl dominated motions are examined in the context of both external and internal flows, single and two-cell vortices, as well as unidirectional and multidirectional configurations. Our coverage begins with the single cell Rankine, Lamb-Oseen, and Burgers-Rott solutions, evolves into the two-cell vortices by Sullivan and Kuo, and then transitions into the family of multidirectional motions in both conically and cylindrically confined cyclonic chambers. In the process, a judicious renormalization of variables enables us to recast the modest selection of models in question into a selfsimilar, universal form that is mainly dependent on an Ekman-like, off-swirl parameter. This is achieved by rescaling external vortices by their peak tangential velocity and core radius, which is the distance from the axis of rotation to the point of maximum swirl. For internal vortices, the renormalization is based on the average tangential injection speed and characteristic chamber radius. By providing a basic correction to Kuo's two-cell motion, straightforward reconciliation with traditional models is attained, and this includes the ability to bring into perspective Kuo's strong similarity with Sullivan's vortex, a property that stands in fulfilment of the Prandtl-Bradshaw analogy between temperature-induced buoyancy and curvature-based rotation. Furthermore, by providing a subtle correction to the Bloor-Ingham vortex, an exact solution for conical cyclones is made possible. This is followed by a novel extension to the matched-asymptotic, viscous layer analysis of wall-bounded cyclonic flowfields. Our work thus culminates in the presentation of a generic, uniformly valid viscous approximation, which is applicable to the special class of bidirectional vortices that arise in the context of a right-cylindrical Vortex Combustion Cold-Wall Chamber (VCCWC), a prototypical thrust chamber that is well known in the propulsion community. The ensuing discussion encompasses mutually complementary sets of helical profiles of the complex lamellar, linear Beltramian, and nonlinear Beltramian types. Our viscous analysis also extends to a generalized Beltramian formulation that can assimilate a variable endwall injection pattern. At the outset, several dimensionless parameters are derived from first principles and compared to traditional analogues that are conjectured via guesswork, scaling, or rationalization. These include a modified form of the swirl number that remains configuration-independent, and unique forms of the vortex Reynolds number that incorporate the effects of swirl, finite chamber length, and tangential injection patterns at entry. After identifying commonalities in vortex patterns, such as the composite structure of the swirl velocity and its containment of forced, free, and transitional vortex layers, the fundamental features of the aforementioned models are expressed as a function of dimensionless parameters and then briefly described. These include the core and wall boundary ...

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Section: Introductionmentioning

confidence: 93%

Unified Framework for Modeling Swirl Dominated Helical Motions

Majdalani¹

2014

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

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“…The method has been proven to be unconditionally stable and, hence, effective at solving stiff, oscillatory ODEs and PDEs. Batterson and Majdalani 45,46 and Batterson 16 have successfully employed this technique to study biglobal instability and vortical motion by solving systems of PDEs that arise in the context of hydrodynamic and vorticoacoustic waves in simulated solid and liquid rockets. In these and many such studies, Chebyshev polynomial collocation has shown distinct advantages in the…”

Section: Third Gst Correction For the Type II Solutionmentioning

confidence: 99%

Generalized Scaling Technique for the Solution of the Vortical Wave Eigenfunction Equation

Batterson¹,

Majdalani²

2014

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

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The description of the unsteady wave motion in an acoustic chamber is relevant to a multitude of aerospace applications such as rockets, gas turbines, scramjets, and preburners. Several related problems arise in the development and propagation of vorticoacoustic boundary layers when considering the effects of arbitrary injection profiles on acoustic wave motions. The reverse problems correspond to arbitrary suction profiles associated with chambers that are intended for filtration or particle separation. Two wave solutions can be developed for such flow regimes: The first being a relatively straightforward potential flow motion for the compressible, irrotational acoustic wave, while the other leads to a complex eigenfunction solution for the rotational, incompressible vortical wave. Unlike the acoustic wave equation, the eigenfunction equation does not readily lend itself to straightforward analysis. Its exact solutions are tractable only for simple base flow profiles and geometries. Moreover, the inherently stiff nature of the eigenfunction equation makes a numeric solution to the resulting problem a non-trivial task. Previous work has tackled the solution of this class of problems by solving equations that are derived for specific geometric configurations and injection conditions. The resulting approximations have relied heavily on multiple-scales and WKB approximations, as these have the ability to accommodate both linear and nonlinear scales. The present work provides a generalized formulation to the rotational wave eigenfunction equation assuming arbitrary coordinate systems and mean flow profiles. The ensuing solutions are obtained using an extension of multiple-scales theory, namely, the Generalized Scaling Technique (GST). The GST method extends the applicability of multiple scales by determining the problem's underlying scales through analysis rather than conjecture. The problem is also approximated using the WKB expansion method. In doing so, we not only confirm our GST results, but also supply sufficient evidence to re-establish the WKB method as a special case of the GST framework. Finally, our solution is validated numerically using a spectral Chebyshev collocation approach. Nomenclature a = chamber radius or half-height s a = speed of sound Re = injection Reynolds number,

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“…Following previous works, 26,27,35 Chebyshev collocation is selected over other discretization methods due to its simplicity and straightforward implementation. We define a discrete Chebyshev polynomial of order N as …”

Section: Chebyshev Discretizationmentioning

confidence: 99%

“…24 In the latter, direct numerical simulations of cylindrically-perforated solid rocket motor grains are shown to spectacularly agree with biglobal stability predictions of the chamber's unsteady motion when properly augmented by the vorticoacoustic wave contribution, formulated by Majdalani and Van Moorhem. 25 In liquid rocket flowfield stability, the first noteworthy investigation may be attributed to Batterson and Majdalani 26,27 and their biglobal stability analysis of the bidirectional vortex motion; the corresponding problem arises in the context of the so-called Vortex Combustion Cold Wall Chamber (VCCWC) by Chiaverini et al [28][29][30][31] This two-part series provides the detailed analytical derivation, numerical discretization steps, and spectral collocation tools needed to handle the complete set of partial differential equations that emerge in a problem where the parallel flow assumption proves to be unsuitable. The Batterson-Majdalani series also illustrates the practical applications of biglobal stability to incompressible helical profiles.…”

Section: Introductionmentioning

confidence: 98%

Two-Phase Flow Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Particle Entrainment

Elliott¹,

Majdalani²

2014

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

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Particle mass addition is often used to increase specific impulse and suppress high frequency instability. This study investigates the addition of particles to the biglobal stability framework, which has been recently introduced in the context of rocket flow modeling. In the process of framework integration, the metallized particles are considered to be spherical, chemically inert, and invariant in size. Pursuant to several previous investigations of biglobal instability in the context of rocket chambers, the present analysis bases itself on the incompressible Navier-Stokes equations and the extended Taylor-Culick mean flow profile, which incorporates Berman's self-similar, half-cosine pattern at the chamber headwall. To this end, particle velocities and spatial concentrations throughout the chamber are realized by invoking flow similarity. Steady-state particle mass concentrations are subsequently obtained using a judicious execution of the method of characteristics on the particle species equation. The biglobal stability framework developed here is not limited to a streamfunction formulation; instead, it employs the complete Navier-Stokes equations to the extent of producing a linearized eigenproblem, namely, one that can be readily solved using pseudo-spectral methods. At the outset, the stability of the motors with particle entrainment is evaluated and compared to results previously acquired through the use of the biglobal stability framework with no particle integration. Our findings indicate that the most amplified eigenmodes with particle injection are the same as those found without. The spectra, however, show that the introduction of particles leads to more stable modes. Finally, a comparison to previous work based on the streamfunction approach is provided and discussed. Nomenclatureij A = operator matrix a = chamber radius ij B = the right-hand-side coefficient matrix of a matrix pencil N D = Chebyshev pseudo-spectral derivative matrix of size N d = weight coefficients for pseudo-spectral derivative matrices N I = identity matrix of size N L = chamber length M = base flow component M  = instantaneous flow component m = general amplitude function m  = acoustic fluctuation m  = unsteady hydrodynamic fluctuation m  = vortical fluctuation  = Landau order symbol P = base flow pressure 2 p = pressure amplitude function p  = hydrodynamic pressure fluctuation q = azimuthal integer wave number r = normalized radial coordinate Re = Reynolds number N T = Chebyshev polynomial of the first kind U = base flow velocity, r U ,U θ , z U u = velocity amplitude function u  = hydrodynamic velocity fluctuation z = normalized axial coordinate Greek ∇ = gradient operator ε = 1/ Re η = streamwise Chebyshev variables mapped between [-1, 1] λ = eigenvalue ω = complex frequency of oscillations, r i i ω ω + i ω = temporal stability growth rate r ω = dimensionless circular frequency θ = tangential coordinate ξ = radial Chebyshev variables mapped between [-1, 1]Subscripts and Superscripts i = denotes an imaginary component i...

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Biglobal Instability of the Bidirectional Vortex. Part 2: Complex Lamellar and Beltramian Motions

Cited by 15 publications

References 15 publications

Unified Framework for Modeling Swirl Dominated Helical Motions

Unified Framework for Modeling Swirl Dominated Helical Motions

Generalized Scaling Technique for the Solution of the Vortical Wave Eigenfunction Equation

Two-Phase Flow Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Particle Entrainment

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