On the Compressible Bidirectional Vortex. Part 1: A Bragg-Hawthorne Stream Function Formulation

Maicke, Brian A.; Majdalani, Joseph

doi:10.2514/6.2012-1103

Cited by 11 publications

(9 citation statements)

References 6 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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“…In Part 1 of this series, 29 the compressible Bragg-Hawthorne framework was established and perturbed using the Rayleigh-Janzen expansion technique. For the reader's convenience, the essential stream function equations entailed in this model are reproduced below:…”

Section: A Compressible Bragg-hawthorne Frameworkmentioning

confidence: 99%

“…As alluded to earlier, 29 the motivation for the present study centers on the bidirectional vortex flowfield that is relevant to a number of cyclonic-flow applications, including the VCCWC engine prototype. In its basic form, the VCCWC may be modeled as a closed-open cylinder of radius a and height L. When all spatial coordinates are normalized by the radius, an idealized chamber of unit radius emerges, as depicted in coordinate system may be conveniently placed a the center of the headwall with the dimensionless radial and axial coordinates being r and z, respectively.…”

Section: B the Bidirectional Vortexmentioning

confidence: 99%

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On the Compressible Bidirectional Vortex. Part 2: A Beltramian Flowfield Approximation

Maicke

Majdalani

2012

50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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“…In the pursuit of a compressible solution for the bidirectional vortex, Maicke and Majdalani 34 have developed a well-posed compressible Bragg-Hawthorne framework in which a Rayleigh-Janzen expansion is systematically unraveled. The corresponding procedure consists of sequentially solving the problem's conservation equations to the extent of producing an analytical approximation for the compressible motion under axisymmetric, inviscid, and isentropic flow conditions.…”

Section: Introductionmentioning

confidence: 99%

On Steady Trkalian High Speed Flows: Swirling Compressible Motions in Rockets with Headwall Injection

Cecil¹,

Majdalani²,

Batterson³

2015

51st AIAA/SAE/ASEE Joint Propulsion Conference

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This work considers the wall-injected swirling motions evolving inside a right-cylindrical solid rocket motor (SRM) with and without headwall injection and, alternatively, a hybrid rocket engine (HRE) with an axisymmetric oxidizer showerhead at its forward closure. The bulk gaseous motion is modeled as a non-reactive, inviscid flow with a swirl velocity component that increases linearly along the axis of the chamber. Our approach is initiated from the compressible Bragg-Hawthorne equation, which is systematically solved using Rayleigh-Janzen perturbation expansions, to the extent of producing closed-form semi-analytical approximations for the stream function of a Trkalian motion from which all other flow attributes may be readily inferred. In the process, the case of a similarity-conforming headwall injection profile is considered, where the axial flow entering the chamber at the forward end enables us to simulate conditions associated with an idealized solid rocket motor with a reactive fore-end closure, or a simplified hybrid rocket engine with an oxidizer flux along its chamber's head-end section. Results are then compared to their counterparts obtained using a strictly incompressible Trkalian profile (Majdalani, J., and Fist, A., Improved Mean Flow Solution for Solid Rocket Motors with a Naturally Developing Swirling Motion, AIAA Paper 2014-4016, July 2014). They are also benchmarked against available compressible solutions in an effort to characterize the dilatational effects that are precipitated by flow acceleration in long rocket chambers or chambers with sufficiently large sidewall injection. Besides the stream function, the velocity, pressure, temperature, and density are evaluated over a range of physical parameters corresponding to both SRM and HRE flows. Finally, the distortions affecting the velocity profiles are characterized and shown to result in a blunter motion near the center and a steeper curvature near the sidewall as a consequence of high speed flow acceleration. In comparison to nonswirling complex-lamellar solutions, we find the Trkalian solution to be generally faster and therefore able to reach sonic conditions in a shorter distance from the headwall.

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On the Compressible Bidirectional Vortex. Part 1: A Bragg-Hawthorne Stream Function Formulation

Cited by 11 publications

References 6 publications

Unified Framework for Modeling Swirl Dominated Helical Motions

Unified Framework for Modeling Swirl Dominated Helical Motions

On the Compressible Bidirectional Vortex. Part 2: A Beltramian Flowfield Approximation

On Steady Trkalian High Speed Flows: Swirling Compressible Motions in Rockets with Headwall Injection

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