Flow Characteristics in a Vortex Chamber

Vatistas, Georgios H.; Jawarneh, Ali M.; Hong, Henry

doi:10.1002/cjce.5450830305

Cited by 12 publications

(8 citation statements)

References 23 publications

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“…At the sidewall, a companion boundary layer ␦ w develops parallel to the core. As shown by Vatistas et al, 24 the presence of ␦ c and ␦ w is characteristic of most vortex chambers including those driven by either unidirectional or bidirectional flow motion. Today, interest in vortex-driven combustors continues to be evidenced in developmental programs such as the vortex combustion cold-wall chamber by Chiaverini et al, 25 the vortex fired hybrid engine by Gloyer et al, 26 the vortex injection hybrid rocket engine by Knuth et al, 27 and the reverse vortex combustor, also known as the tornado combustor by Matveev et al 28 Despite the abundance of numerical and experimental investigations of cylindrical cyclones, limited attention has been given to the advancement of viscous analytical models.…”

Section: Introductionmentioning

confidence: 75%

On steady rotational cyclonic flows: The viscous bidirectional vortex

Majdalani

Chiaverini

2009

Physics of Fluids

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This study is focused on the tangential boundary layers of a bidirectional vortex, specifically those forming at the core and the sidewall of a swirl-driven cyclonic chamber. Our analysis is based on the regularized, tangential momentum equation which is rescaled in a manner to capture the forced vortex near the chamber axis and the no slip requirement at the sidewall. After identifying the coordinate transformations needed to resolve the rapid changes in the regions of nonuniformity, two inner expansions are constructed. These expansions are then matched with the outer, free vortex solution that is sandwiched between the core and the hard wall. By combining inner and outer expansions, uniformly valid approximations are subsequently obtained for the swirl velocity, vorticity, and pressure. These are shown to be strongly influenced by a dimensionless grouping that we refer to as the vortex Reynolds number, V. This keystone parameter appears as a ratio of the mean flow Reynolds number and the product of the swirl number and the chamber aspect ratio. Based on V, several fundamental features of the bidirectional vortex are quantified. Among them are the thicknesses of the viscous core and sidewall boundary layers; these decrease with V 1/2 and V, respectively. The converse may be said of the peak velocity which increases with V 1/2. In the same vein, the angular speed of the rigid-body rotation of the forced vortex is found to be linearly proportional to V. Our laminar swirl velocity is reminiscent of Sullivan's two-cell vortex except for its additional dependence on the aspect ratio of the chamber. For the purpose of verification, theoretical predictions are compared to particle image velocimetry measurements and Navier-Stokes simulations at high vortex Reynolds numbers. By properly accounting for the turbulent eddy viscosity in the analytical model, local agreement is obtained with both laboratory measurements and computer simulations.

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Section: Introductionmentioning

confidence: 75%

On steady rotational cyclonic flows: The viscous bidirectional vortex

Majdalani

Chiaverini

2009

Physics of Fluids

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“…In these models, the forced vortex diameter increases while the maximum swirl velocity diminishes with successive increases in viscosity (see Vatistas et al. [17][18][19]). …”

Section: Nomenclaturementioning

confidence: 95%

Sidewall Boundary Layers of the Bidirectional Vortex

Batterson

Majdalani

2010

Journal of Propulsion and Power

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This study seeks to resolve the sidewall boundary layers forming in the axial and radial directions of a bidirectional vortex chamber. Our analysis is initiated by the formulation of the laminar boundary-layer equations via an order of magnitude reduction of the incompressible Navier-Stokes equations at the wall. Asymptotic theory is then applied to linearize and systematically truncate the governing equations, thus converting them from partial differential equations to more manageable ordinary differential equations. Scaling transformations are additionally applied to resolve the rapid changes arising near the sidewall. Because of the spatial character of the outer solutions, further transformations of the dependent variables are undertaken to secure the axially changing outer conditions. Through the use of matched-asymptotic expansions, we recover similar boundary-layer structures in all three orthogonal directions: the axial and radial components presented here, and the wall-tangential boundary layer obtained previously. This behavior is consistent with the resultant velocity being dominated by its tangential component and with the tangential boundary layer being axially invariant. These factors cause the axial layer to remain uniform in the streamwise direction. Based on the ensuing asymptotic results, viscous corrections at the wall are seen to be mainly dependent on the vortex Reynolds number, V. The latter combines the swirl number, the Reynolds number, and the chamber aspect ratio. Having obtained the three components of the velocity, essential flow characteristics, such as pressure, vorticity, swirling intensity, and wall shear stresses, are evaluated and discussed. Nomenclature A i = inlet area a = chamber radius b = chamber outlet radius, b < a l = chamber aspect ratio, L=a p = normalized pressure, p=U 2 Q i = normalized volumetric flow rate, Q i =Ua 2 1 Q i = inlet volumetric flow rate Re = injection Reynolds number, Ua= 1=" r, z = normalized radial or axial coordinates, r=a, z=a S = swirl number, ab=A i s = scaled transformation variable, = U = average inflow velocity in the tangential direction, u a; L u = normalized velocity u r ; u z ; u =U u = normalized swirl/spin/tangential velocity, u =U V = vortex Reynolds number, Q i Rea=L "l 1 = constant, 1 6 2 1 ' 0:644934 = normalized outlet radius, b=a = minus rescaled layer w = wall boundary-layer thickness, w =a " = perturbation parameter, 1=Re =Ua = transformed variable, r 2 = inflow parameter, Q i =2l 2l 1 = kinematic viscosity, = = density = modified swirl number, Q 1 i S= Subscripts i = inlet property r = radial component z = axial component = azimuthal component Superscripts o = outer (inviscid) solution = dimensional variables

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“…Given the shortage of purely analytical models of axisymmetric cyclonic flows, 27 an Eulerian based solution was developed by Vyas, Majdalani and Chiaverini 28 for a right-cylindrical VCCWC chamber model. Although their effort only produced a simple solution for the problem at hand, 29 it set the pace for a laminar boundary layer treatment of the viscous core 30 along with a cursory characterization of the multiple-mantle paradigm.…”

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confidence: 99%

Exact Eulerian Solutions of the Cylindrical Bidirectional Vortex

Majdalani¹

2009

45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference &Amp;amp; Exhibit

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In this work, new types of Eulerian motions are investigated as potential flow candidates for describing the bidirectional vortex field in a right-cylindrical chamber. These basic solutions apply to industrial cyclone separators and to idealized representations of either liquid or hybrid rocket engines. The latter correspond to a bidirectional vortex chamber with weak sidewall permeability. As usual, we take the bulk motion to be isentropic along streamlines, with no concern for reactions, heat transfer, viscosity, compressibility, or unsteadiness. Our mathematical approach is anchored on the cylindrical Bragg-Hawthorne equation which is concurrently applied in its spherical form to the treatment of the conical cyclone (Barber, T. A., and Majdalani, J., "Exact Eulerian Solution of the Conical Bidirectional Vortex," AIAA Paper 2009-5306, August 2009). Among the unique characteristics of the new solutions, we cite the axial dependence of the swirl velocity, the Trkalian and Beltramian characters of the helical motions, the appreciable sensitivity to the outlet radius, the alternate location of the mantle, and the increased axial and radial velocity magnitudes, including the rate of mass transfer across the mantle, for which explicit approximations are obtained. Our results are compared to one another and to an existing, complex lamellar solution in which the swirl velocity collapses into a free vortex. In this vein, we find the strictly Beltramian flows to share virtually identical pressure variations and radial pressure gradients with those associated with the complex lamellar motion. By the same token, both families necessitate an asymptotic treatment to overcome their endpoint deficiencies caused by their dismissal of viscous stresses. From a broader perspective, the work delineates a logical framework through which self-similar, axisymmetric solutions to bidirectional and multidirectional vortex motions may be rigorously pursued. Furthermore, it illustrates the manner through which multiple configurations may be arrived at depending on the types of boundary conditions imposed. For example, both the slip condition at the sidewall and the inlet flow pattern at the headwall may be enforced or relaxed. Our analysis is carried out in the context of a right-cylindrical chamber first without and then with allowance for sidewall injection. The latter enables us to model the basic flow in the socalled Vortex Injection Hybrid Rocket Engine (VIHRE). Finally, the alternate mantle location and swirl velocity are verified in the light of existing measurements and numerical simulations performed with the Reynolds Stress-transport Model (RSM).

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Flow Characteristics in a Vortex Chamber

Cited by 12 publications

References 23 publications

On steady rotational cyclonic flows: The viscous bidirectional vortex

On steady rotational cyclonic flows: The viscous bidirectional vortex

Sidewall Boundary Layers of the Bidirectional Vortex

Exact Eulerian Solutions of the Cylindrical Bidirectional Vortex

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