On the Viscous Bidirectional Vortex. Part 2: Nonlinear Beltramian Motion

Batterson, Joshua W.; Majdalani, Joseph

doi:10.2514/6.2010-6764

Cited by 11 publications

(10 citation statements)

References 16 publications

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“…It may be easily shown that this composite approximation will be uniformly valid for 0 < z < l. Although its behavior near the core matches the results obtained for (i) the complex lamellar model by Majdalani [60], and (ii) the Beltramian models by Batterson and Majdalani [61,62], the boundary layer treatment near the sidewall is different here. Contrary to the previous formulation, which made use of ad hoc assumptions near r = 1, the matched-asymptotic expansion unravelled here proceeds directly from first principles.…”

Section: Composite Uniformly Valid Swirl Approximationsupporting

confidence: 66%

“…Most of these will be connected with the author's multiple flowfield solutions introduced in the context of cylindrical cyclones and VCCWC thrust chambers. These will refer to successively emerging work by Vyas and Majdalani [59], Majdalani and Rienstra [66], Maicke and Majdalani [75], Barber and Majdalani [69], Majdalani and Chiaverini [60], Batterson and Majdalani [61,62], Akiki and Majdalani [73,74], and Majdalani [72].…”

Section: Emerging Bidirectional Vortex-dominated Flowfieldsmentioning

confidence: 96%

“…For the nonlinear case, we follow Batterson and Majdalani [62] in approximating the harmonically varying function over the length of the chamber. This is done by taking the spatial average, …”

Section: American Institute Of Aeronautics and Astronauticsmentioning

confidence: 99%

“…Despite its inherent simplicity and cold-flow basis, this so-called complex lamellar model helped to dispel some of the skepticism that lingered at that point against the physicality of cyclonic motions within a right-cylindrical VCCWC. The ensuing profile 4 American Institute of Aeronautics and Astronautics quickly became the launchpad for a number of complementary sequels in which the treatment of boundary layers could be partially addressed [60][61][62][63][64]. In separate work, Majdalani [65] subdivided the hybrid rocket flowfield within VIHRE into a leading-order approximation, for which the sidewall could be kept inert, and a first-order correction, which proved to be commensurate with the wall blowing speed relative to the tangential injection velocity.…”

Section: Introductionmentioning

confidence: 99%

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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: Composite Uniformly Valid Swirl Approximationsupporting

confidence: 66%

Section: Emerging Bidirectional Vortex-dominated Flowfieldsmentioning

confidence: 96%

Section: American Institute Of Aeronautics and Astronauticsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unified Framework for Modeling Swirl Dominated Helical Motions

Majdalani¹

2014

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

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“…For the sake of brevity, we refer the reader to the papers by Vyas and Majdalani, 15 Majdalani and Chiaverini, 16 and those by Batterson and Majdalani 17,[19][20][21] for the comprehensive characterization of the various bidirectional vortex models. In short, we find the uniformly valid, ad hoc velocity and pressure solutions for each flowfield to be:…”

Section: Bidirectional Vortex Modelsmentioning

confidence: 99%

Biglobal Instability of the Bidirectional Vortex. Part 1: Formulation

Batterson¹,

Majdalani²

2011

47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference &Amp;amp; Exhibit

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Hydrodynamic instability research has begun to shift from the one-dimensional analysis of the Orr-Sommerfeld, local nonparallel, and parabolic type to the two-dimensional, biglobal analysis. Classic methods, such as the local nonparallel approach, specify a one-dimensional amplitude function a priori so that streamwise and azimuthal variations become characterized by a complex wave and integer mode number, respectively. This formulation is heavily dependent on the so-called parallel flow assumption. In brief, this property requires the streamlines of the baseflow to be parallel or nearly parallel, a condition that may be quite restrictive at times. It can therefore be seen that the use of biglobal analysis becomes necessary to encompass a broader range of baseflows. The relatively novel, two-dimensional characterization of the hydrodynamic waveform gives rise to a more complete analysis that intrinsically captures spatial instability behavior while remaining unrestricted to special geometries, such as elongated rocket chambers. The biglobal approach only requires periodicity in the tangential direction and seems to be tailor-made for cylindrical axisymmetric flows, such as those arising in the bidirectional vortex engine. Moreover, for short rocket chambers such as those associated with bidirectional vortex engines, the parallel flow assumption cannot be justified, and one finds the biglobal analysis as a more suitable alternative to the local nonparallel (LNP) approach used extensively in solid rocket motor stability investigations. However, being multi-dimensional, the biglobal technique requires more effort than its predecessor, specifically in the analytical derivation, numerical discretization, and spectral collocation tools necessitated by the treatment of the resulting partial differential equations. The present work serves to offer a detailed description of the theoretical framework and numerical tools needed to tackle this problem in the context of a confined vortex chamber such as that investigated by NASA/ORBITEC.

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On the Viscous Bidirectional Vortex. Part 3: Multiple Mantles

Batterson¹,

Majdalani²

2010

46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference &Amp;amp; Exhibit

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This study seeks to characterize the multiple flow reversals that may arise in the context of bidirectional vortex motion. It constitutes an essential sequel to the analysis of the linear and nonlinear Beltramian motions wherein the formation of only one mantle is considered. The mantle in cyclonic flows refers to the non-translating interface separating the updraft from the downdraft. In this work, we show that an odd number of mantles may develop and these are accompanied by endwall flow reversals that may be predicted by sequentially higher eigenvalues derived directly from first principles. The multiplicity of mantle alignments is corroborated by experimental measurements obtained in swirl-driven vortex chambers. The resulting flowfield is characterized by multilayered coaxial vortex shells with sign-alternating axial velocity. In this article, the multiple solutions associated with different modes of flow reversal are ascertained using as a baseline, each of the linear and nonlinear Beltramian motions obtained by Majdalani (Majdalani, J., "Exact Eulerian Solutions of the Cylindrical Bidirectional Vortex," AIAA Paper 2009-5307, Denver, Colorado, Aug. 2009). Our analytical approximations are shown to predict the positioning of internal mantles with substantial accuracy when compared to computational fluid dynamics (CFD) and experimental particle-image velocimetry (PIV) data. In addition, viscous layers that appear in the core and sidewall regions are quantified and appended to the inviscid models as required.

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On the Viscous Bidirectional Vortex. Part 2: Nonlinear Beltramian Motion

Cited by 11 publications

References 16 publications

Unified Framework for Modeling Swirl Dominated Helical Motions

Unified Framework for Modeling Swirl Dominated Helical Motions

Biglobal Instability of the Bidirectional Vortex. Part 1: Formulation

On the Viscous Bidirectional Vortex. Part 3: Multiple Mantles

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