Steady vortex rings with swirl in an ideal fluid: asymptotics for some solutions in exterior domains

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder (rd) where (r, , z) denotes the cylindrical co-ordinates in ³ is considered. The motion is with swirl (i.e. the -component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. [9] that for the problem without swirl (f _q = 0 in (f)) in the whole space, as the flux constant k tends to 1) dist(0z, A) = O(k ^1/2); diam A = O(exp(–c ₀ k ^3/2));2) k^1/2)_k converges to a vortex cylinder U _m (see (1.2)).We show that for the problem with swirl, as k , 1) holds; if m q + 2 then 2) holds and if m > q + 2 it holds with U _q+2 instead of U _m. Moreover, these results are independent of f ₀, f _q and d > 0.     

Nested axi-symmetric vortex rings

In an inviscid and incompressible fluid, we prove the existence of nested co-axial vortex rings moving at the same speed.     

Existence and numerical modelling of vortex rings with elliptic boundaries

We revisit in this paper the theory of axisymmetric vortex rings in an ideal fluid. The boundary separating the vortex ring from the external (potential) flow is assumed of elliptic shape. For a given distribution of vorticity in the vortex core, we theoretically put into evidence the critical parameter for the existence of non-trivial solutions, thus confirming the numerical observation of Durst et al. [ZAMP 32 (1981) 156]. A sharp estimation of the critical threshold is analytically derived. Theoretical predictions are confirmed by numerical simulations using finite elements. A new numerical algorithm is presented and shown to display better performances compared to previous published algorithms using finite differences. The convergence of the iterative algorithm is proved using the theory of elliptic partial differential equations with discontinuous nonlinearities.     

On the existence of global vortex rings

Sans résumé Supported in part by a grant from the National Science Foundation. The results of this paper were obtained while the author was partially supported by U. S. Army Research Office grant No. DAA-29-78-6-0127 at Courant Institute, New York University.     

Short-wave instabilities on a vortex pair of unequal strength circulation ratio

The creation of vortex pairs occurs in a range of industries, including mixing, transport, and plastic moulding. In particular, vortex pairs are observed in the wake of aircraft, and are the cause of a significant hazard in the aviation industry. Instabilities, which grow on vortex pairs, have the potential to lead to enhanced dissipation, thus limiting this safety concern, in addition to enhancing mixing in chemical engineering industries. To date research has mostly considered instabilities growing on a vortex pair where each vortex has the same magnitude of circulation. However, in practice it is unusual to have an equal-strength vortex pair. This investigation is the first to consider the instability modes that may develop on a Lamb–Oseen vortex pair of arbitrary circulation ratio. We find a significant change in the growth rates of all instability modes reported previously for an equal-strength vortex pair. All simulations employ an accurate spectral-element method to discretise the domain coupled with a three-step time splitting scheme. A wide range of instability wavelengths is considered to ensure that all instability modes are captured. By identifying and enhancing the leading instability modes, we are able to enhance the dissipation of the vortex pair.     

On the formation of vortex rings: Rolling-up and production of circulation

The temporal development of the rolling-up process of vortex rings produced in water at a circular nozzle is investigated by flow visualization. The results are compared with similarity laws for the rolling-up of vortex sheets in plane flow. The unsteady flow field in the nozzle-exit plane is measured by laser anemometry. The vorticity distribution in the boundary layers at the inner and outer nozzle wall and the vorticity flux through the exit plane are derived from the measurements. The flow conditions measured at the nozzle are used to explain the production of the ring-vortex circulation (measured three nozzle-diameters downstream) in relation to the generation conditions.

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Zusammenfassung Durch Strömungs-Sichtbarmachung wird für Ringwirbel, die an einer kreisförmigen Rohrmündung in Wasser erzeugt werden, der zeitliche Verlauf des Aufrollvorgangs untersucht und mit Ähnlichkeitsgesetzen für die Aufrollung von ebenen Wirbelschichten verglichen. Aus Messungen des instationären Strömungsfeldes im Mündungsquerschnitt mit einem Laser-Anemometer wird die Wirbelstärke-Verteilung in den Grenzschichten an der inneren und äußeren Rohrwand sowie der Wirbelfluß durch den Mündungsquerschnitt bestimmt. Dadurch wird die Entstehung der Zirkulation, die im fertig ausgebildeten Ringwirbel drei Rohrdurchmesser stromabwärts gemessen wird, in Abhängigkeit von den Erzeugungsbedingungen verständlich.

     

Ordered and random movement in the dynamics of three coaxial vortex rings

The article presents the results of the theoretical investigation of the movement of a system of three coaxial vortex rings in an ideal liquid. It is shown that when the rings interact with each other, the process may become randomized in time. The conditions of ordered and random movement of three vortex rings are determined. The article presents the paths and Poincaré mappings for a number of characteristic situations.Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 100–104, 1990.     

Axisymmetric magnetic vortices with swirl

New solutions for compact vortex rings and vortices in the presence of swirl and toroidal magnetic field are presented. Steady solutions are found for thin-core vortex rings, generalizing the classical results of Kelvin and Hicks. An exact family of steady spherical solutions is found in the case when the underlying governing equation, originally due to Hicks, is linear. Proper matching of the pressure is critical in obtaining these solutions, for which another 19th century equation proves key. Contour dynamics simulations show the existence of counter-propagating dipolar structures when both swirl and magnetic fields are present.     

The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem

We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.     

Prime rings with PI rings of constants

It is shown that if the ring of constants of a restricted differential lie algebra with a quasi-Frobenius inner part satisfies a polynomial identity (PI) then the original prime ring has a generalized polynomial identity (GPI). If additionally the ring of constants is semiprime then the original ring is PI. The case of a non-quasi-Frobenius inner part is also considered.     

Dynamics and self-propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid

We describe a model for the dynamic interaction of a sphere with uniform density and a system of coaxial circular vortex rings in an ideal fluid of equal density. At regular intervals in time, a constraint is imposed that requires the velocity of the fluid relative to the sphere to have no component transverse to a particular circular contour on the sphere. In order to enforce this constraint, new vortex rings are introduced in a manner that conserves the total momentum in the system. This models the shedding of rings from a sharp physical ridge on the sphere coincident with the circular contour. If the position of the contour is fixed on the sphere, vortex shedding is a source of drag. If the position of the contour varies periodically, propulsive rings may be shed in a manner that mimics the locomotion of certain jellyfish. We present simulations representing both cases.     

Radicals of Jordan rings connected with alternative rings

Subject to a certain restriction on the additive group of an alternative ring A, we prove that R(A)=R(A⁽⁺⁾), where A⁽⁺⁾ is a Jordan ring and R is one of the following radicals: the Jacobson radical, the upper nil-radical, the locally nilpotent radical, or the lower nil-radical. For the proof of these relationships Herstein's well-known construction for associative rings is generalized to alternative rings.     

Semigroup rings and group rings with large center

A ring R is called a ring with a large center or an IIC-ring if any nonzero ideal of R has a nonzero intersection with the center of R. We consider conditions which guarantee that a semigroup ring over an IIC-ring is an IIC-ring.     

On exchange rings with primitive factor rings artinian

Abstract

In this paper we introduce generalized ideal-stable regular rings. It is shown that if a regular ring R is a generalized I-stable ring, then every square matrix over I is the product of an idempotent matrix and an generalized invertible matrix and admits a diagonal reduction by some generalized invertible matrices.     

Stability criteria for swirl flows with free surfaces

A swirl flow with a free surface is susceptible to centrifugal, capillary, and Rayleigh-Taylor instabilities. The present work extends the stability criteria of Rayleigh and Synge for inviscid and viscous fluids, respectively, to free-surface configurations.

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Résumé Un écoulement tourbillonnaire admettant une surface libre est susceptible de présenter des instabilités du type centrifuge, capillaire et Rayleigh-Taylor. Notre travail étend les critères de stabilité obtenus par Rayleigh et Synge respectivement pour des fluides parfaits et visqueux aux cas des configurations à surface libre.

     

Similarity solution of axisymmetric non-Newtonian wall jets with swirl

The similarity solution of axisymmetric wall jets with swirl on bodies of revolution for non-Newtonian power-law fluids is presented. The physical and geometrical meaning of all parameters appearing in the course of similarity procedure is treated in detail. Especially, the functional dependence of length, velocity, and pressure similarity scales on the shape and swirl parameters and the power-law flow behaviour index is determined. The already published results related to the similarity solution obtained are discussed.