Exact Solution of the Bidirectional Vortex

Vyas, Anand B.; Majdalani, Joseph

doi:10.2514/1.14872

Cited by 75 publications

(106 citation statements)

References 22 publications

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“…Applying (24) and the vorticity transport equation (for additional detail see Vyas & Majdalani, 2006) leads to the following relation in ψ, namely,…”

Section: Laminar Core Modelmentioning

confidence: 99%

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry

Maicke¹,

Majdalani²

2012

The Particle Image Velocimetry - Characteristics, Limits and Possible Applications

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“…Applying (24) and the vorticity transport equation (for additional detail see Vyas & Majdalani, 2006) leads to the following relation in ψ, namely,…”

Section: Laminar Core Modelmentioning

confidence: 99%

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry

Maicke¹,

Majdalani²

2012

The Particle Image Velocimetry - Characteristics, Limits and Possible Applications

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“…The (r, z) versus q segregation allows us to introduce a swirl velocity model after Vyas & Majdalani (2006a). Their exact inviscid solution is simply u ZKk sinðpr 2 Þ r e r C 1 r e q C 2pkz cosðpr 2 Þe z ;…”

Section: (B ) Basic Formulationmentioning

confidence: 99%

“…Given that the inner core velocity is bounded at the centreline, a companion pressure may be obtained that does not exhibit the inviscid singularity of its predecessor (see Vyas & Majdalani 2006a Integration and combination of these equations provides the pressure distribution…”

Section: ð3:6þmentioning

confidence: 99%

A constant shear stress core flow model of the bidirectional vortex

Maicke

Majdalani

2008

Proc. R. Soc. A.

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In this paper, we discuss the merits of two models for the swirl velocity in the core of a confined bidirectional vortex. The first is piecewise, Rankine-like, based on a combinedvortex representation. It stems from the notion that a uniform shear stress distribution may be assumed in the inner vortex region of a cyclone, especially at high Reynolds numbers. Thereafter, direct integration of the shear stress enables us to retrieve an expression for the swirl velocity that overcomes the inviscid singularity at the centreline. The second model consists of a modified asymptotic solution to the problem obtained directly from the Navier-Stokes equations. Both solutions we present transition smoothly to the outer, free-vortex approximation at some intermediate position in the chamber. This position is deduced from available experimental data to the extent of providing an accurate swirl velocity distribution throughout the chamber. By scaling the constant shear radius to the core layer thickness, the constant of proportionality is readily calculated using the method of least squares. Interestingly, the constant of proportionality is found to be invariant at several vortex Reynolds numbers, thus helping to achieve closure. The combined-vortex representation is validated against a large body of experimental measurements and through comparisons to a laminar core model that is enhanced through the use of an eddy viscosity. Other heuristic schemes are discussed and the two most suitable models to capture realistic flow behaviour at high vortex Reynolds numbers are identified. Our two models are first derived analytically and then anchored on the available experimental measurements.

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“…These configurations have been thoroughly explored by Saad & Majdalani (2007b, 2009), Majdalani & Rienstra (2007) and Vyas & Majdalani (2006) using physical conditions that are furnished in figure 1b,c. Being a member of the same family of injection driven flows, these profiles exhibit physical characteristics that are similar to those ascribed to the axisymmetric problem at hand.…”

Section: (G) Other Geometric and Flow Configurationsmentioning

confidence: 99%

“…(7.20) Vyas & Majdalani (2006) derived an inviscid rotational model given by (7.21) where, for simplicity, we have set κ = 1 in the complex lamellar Vyas-Majdalani profile. By evaluating equation (7.10a) along the open boundary at z = L, we recover…”

Section: (I) Poiseuille Flow In Ducts Of Arbitrary Cross Sectionsmentioning

confidence: 99%

On the Lagrangian optimization of wall-injected flows: from the Hart–McClure potential to the Taylor–Culick rotational motion

Saad

Majdalani

2009

Proc. R. Soc. A.

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The Taylor-Culick solution for a porous cylinder is often used to describe the bulk gas motion in idealized representations of solid rocket motors. However, other approximate solutions may be found that satisfy the same fundamental constraints. In this vein, steeper or smoother profiles may be observed in either experimental or numerical tests, particularly in the presence of intense levels of acoustic energy. In this study, we use the Lagrangian optimization principle to arrive at multiple solutions that can help to explain the practically observed motions. We then search for the extreme states that display either the least or the most kinetic energy. These are derived and found to be dependent on the chamber's aspect ratio. By assuming a slender case, simple expressions are retrieved from which both the rotational Taylor-Culick and the irrotational HartMcClure solutions are recovered as special cases. At the outset, a new family of flow approximations is derived extending from purely potential to highly rotational fields. These are constructed, verified and catalogued based on their kinetic energies. To help understand the tendency of a motion to shift energy states, a second law analysis is performed and used to rank these solutions based on their entropy content. Interestingly, the Taylor-Culick profile is found to represent an equilibrium state entailing the most energy and entropy among those starting from the rest. A formal extension of Kelvin's minimum energy theorem to an open region is then pursued, followed by a direct implementation to the problem at hand. Three other flow motions with open boundaries are examined to further exemplify the application of Kelvin's extended criteria. Our efforts illustrate the overarching harmony that exists between Lagrange's minimization principle and Kelvin's minimum energy theorem; both converge on the potential solution as the least kinetic energy bearer even when the velocity at the open barrier is finite.

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Exact Solution of the Bidirectional Vortex

Cited by 75 publications

References 22 publications

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry

A constant shear stress core flow model of the bidirectional vortex

On the Lagrangian optimization of wall-injected flows: from the Hart–McClure potential to the Taylor–Culick rotational motion

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