Reverse flow radius in vortex chambers

Vatistas, Georgios H.; Song, Lin; Kwok, C. K.

doi:10.2514/3.9539

Cited by 50 publications

(21 citation statements)

References 5 publications

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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%

“…This is done in the interest of consistency with the axial and tangential solutions. The radial analysis begins with " @ 2 u r @r 2 1 r @u r @r u r r 2 u r @u r @r u 2 r u z @u r @z @p @r (17) where ( u r 1 0 no slip lim r!0 u r r u o r merging with outer solution (18) Here u o r r 1 sinr 2 . The next step is to ignore axial derivatives by insisting that radial effects dominate.…”

Section: B Radial Correctionsmentioning

confidence: 99%

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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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“…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%

Section: B Radial Correctionsmentioning

confidence: 99%

Sidewall Boundary Layers of the Bidirectional Vortex

Batterson

Majdalani

2010

Journal of Propulsion and Power

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“…They will be used as the main propulsion system of the cargo for a potential manned mission to Mars [1], [2], as well as for the construction of heavy space structures like a space solar power system [3], [4]. There are many candidates for high power electric propulsion; Hall thrusters [5]- [7] offer an attractive combination of high thrust efficiency, exceeding 50%, with a specific impulse range of 1000-3000 s, a higher ion beam density than ion thrusters and a larger expected total impulse. In addition, in the near future, Hall thrusters will be a main propulsion system for super low orbit satellites.…”

Section: Introductionmentioning

confidence: 99%

Development of a Novel Power Processing Unit for Hall Thrusters

Yamamoto

Takegahara

Aoyagi

et al. 2015

IEEE Trans. Plasma Sci.

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A newly developed power processing unit (PPU) offers the advantages of smaller size and lighter weight than conventional PPUs. The thrust performance of a magnetic layer type Hall thruster developed at Kyushu University with this new PPU was investigated; it showed a good performance as compared with conventional power supplies. The thrust to power ratio was improved to 58 mN/kW at discharge voltage of 150 V and anode xenon mass flow rate of 1.0 mg/s.

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“…Theoretical analyses have also been carried out by Fontein and Dijksman [5], Smith [3,4], and Bloor and Ingham [6][7][8]. Semi-empirical models are, in turn, available from Reydon and Gauvin [9], Vatistas, Lin and Kwok [10,11], Vatistas [12], and others. In view of continual progress in computational mechanics, numerical simulations have been recently undertaken by Hsieh and Rajamani [13], Hoekstra, Derksen and Van den Akker [14], Derksen and Van den Akker [15], and Fang, Majdalani and Chiaverini [16].…”

Section: Introductionmentioning

confidence: 99%

On the bidirectional vortex and other similarity solutions in spherical coordinates

Majdalani

Rienstra

2006

Z. angew. Math. Phys.

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The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid, rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill's vortex. This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach to other complex fluid motions that are easier to capture using spherical coordinates.

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Reverse flow radius in vortex chambers

Cited by 50 publications

References 5 publications

Sidewall Boundary Layers of the Bidirectional Vortex

Sidewall Boundary Layers of the Bidirectional Vortex

Development of a Novel Power Processing Unit for Hall Thrusters

On the bidirectional vortex and other similarity solutions in spherical coordinates

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