Interaction of coaxial vortex rings in an ideal fluid

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 78–84, March–April, 1988.     

Acoustics and dynamics of coaxial interacting vortex rings

Using a contour dynamics method for inviscid axisymmetric flow we examine the effects of core deformation on the dynamics and acoustic signatures of coaxial interacting vortex rings. Both “passage” and “collision” (head-on) interactions are studied for initially identical vortices. Good correspondence with experiments is obtained. A simple model which retains only the elliptic degree of freedom in the core shape is used to explain some of the calculated features.     

Head-on collision of two coaxial vortex rings in an ideal fluid

The opposing motion of two coaxial vortex rings is classified in relation to the initial parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 60–64, July–August, 1989.The authors are grateful to V. V. Meleshko for discussing their results.     

Cooling of liquid in the presence of natural convection

At present a large number of theoretical and experimental studies have been devoted to the problems of initiation and development of natural convection [1–4]. However, so far investigators have not paid much attention to the study of the damping characteristics of natural convection. In the present work the problem of cooling of a liquid in the presence of natural convection is solved by numerical methods.     

Numerical study of interaction of two vortex rings

Three-dimensional interaction of vortices with finite core is investigated numerically using the Rosenhead-Moore approximation. As the computational scheme in this approximation cannot deal with the structure of the vortex core, a bundle of vortex elements is employed to represent a single physical vortex tube. After the validity of this method was confirmed by comparing the numerical result for a single vortex ring composed of various number of the elements with the analytical solution, two cases of interaction of two vortex rings were studied. The first case is two vortex rings traveling along a common axis, and the second is two vortex rings moving side by side along parallel axes. Comparison with the experiments showed good agreement.     

Vortex dynamics simulation of interacting vortex rings and filaments

Interaction between a vortex ring and a vortex filament of the same strength has been simulated by a three-dimensional vortex method which employs vortex arrows with the Rosenhead-Moore core structure. The power spectrum of the velocity field and the enstrophy spectrum are obtained in a closed form whose limit of the core radius tending to zero is equivalent to that given be Aksman et al. (1985) Phys. Rev. Lett. 54. 2410. Four basic modes of the interaction are shown to exist. Temporal evolution of the spectrum during the interaction is also obtained for the basic modes: the energy of the velocity field is generally transferred to the high-wavenumber range as the interaction proceeds.     

Adaptive panel representation for oblique collision of two vortex rings

In this article, we use a hierarchical panel method for representing vortex sheet surface motion in 3D flow to investigate the oblique collision of two vortex rings. The particles representing the sheet are advected by a regularized Biot-Savart integral with smoothed Rosenhead-Moore kernel. The particle velocities are evaluated by an adaptive treecode algorithm based on Taylor expansions in Cartesian coordinates. The method allowed us to consider late stages of a vortex rings collision, producing a funnel region. Vorticity iso-surfaces evolution is also investigated.     

On the evolution of laminar vortex rings

Using Laser Doppler Anemometry (LDA) and Digital Particle Image Velocimetry (DPIV), the physical properties of laminar vortex rings are investigated in the Reynolds-number range 830 ≤ Re ≤ 1650. The measured initial circulations of the vortex rings are found to agree well with corrected versions of the vorticity-flux (slug-flow) model proposed by Didden and Pullin. The DPIV and LDA data show excellent agreement regarding local velocities and vortex-ring circulations. The DPIV data depict the distribution of the vorticity and circulation in the core regions, where the resulting vorticity distributions are found to be self-similar Gaussian profiles. The propagation velocity of the vortex rings is well approximated by an analytical model of Saffman for large core sizes. In the asymptotic limit t → ∞, the trajectories are in excellent agreement with the exact Stokes-dipole solution of Cantwell and Rott.     

Perfect gas convection in a porous medium between two coaxial horizontal cylinders of large length

Natural convection of a perfect gas in a porous medium between two coaxial horizontal cylinders of large length located in heat-conducting space is considered. The two-dimensional problem (thin porous ring) is investigated in a plane orthogonal of the axis of the cylinders. The dependence of the criterion of the onset of convection on the non-Boussinesq parameters is studied. In the steady-state case an analytic solution of the nonlinear problem is obtained and its asymptotic behavior is considered for large Rayleigh numbers and when the compressibility criterion tends to zero. The gas flow rate in the ring and other characteristics of convection are studied as functions of the gas compressibility criterion and a constant temperature gradient given far away from the contour.     

On thermal convection between two coaxial square containers inclined by forty five degrees

This paper is concerned with flow visualization, velocity, streakline, and heat transfer measurements on thermal convection between two coaxial square containers inclined by forty five degrees. It is found that there is no critical Rayleigh number for the onset of thermal convection in the test section. There appear two great circulations, which are symmetrical with respect to the diagonal vertical plane of the two coaxial square containers. Received on 18 June 1997     

Dynamics of vortex rings in viscous fluids

As part of a long range study of vortex rings, their dynamics, interactions with boundaries and with each other, we present the results of experiments on thin core rings generated by a piston gun in water. We characterize the dynamics of these rings by means of the traditional equations for such rings in an inviscid fluid suitably modifying them to be applicable to a viscous fluid. We develop expressions for the radius, core size, circulation, and bubble dimensions of these rings.     

Dynamics of vortex line in presence of stationary vortex

The motion of a thin vortex with infinitesimally small vorticity in the velocity field created by a steady straight vortex is studied. The motion is governed by non-integrable PDE generalizing the Nonlinear Schrodinger equation (NLSE). Situation is essentially different in a co-rotating case, which is analog of the defocusing NLSE and a counter-rotating case, which can be compared with the focusing NLSE. The governing equation has special solutions shaped as rotating helixes. In the counter-rotating case all helixes are unstable, while in the co-rotating case they could be both stable and unstable. Growth of instability of counter-rotating helix ends up with formation of singularity and merging of vortices. The process of merging goes in a self-similar regime. The basic equation has a rich family of solitonic solutions. Analytic calculations are supported by numerical experiment.     

Mechanism of formation of vortex rings

It is demonstrated experimentally that a vortex ring is created by the slowing down of the central part of a stream and that during formation of this ring the amount of the mass of liquid participating in its formation can change. The possibility is also established of obtaining liquid vortex blobs in the form of hemispheres or spheroids.Deceased.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 19–26, March–April, 1973.In conclusion, the author thanks G. A. Lyubimov and A. T. Onufriev for useful discussions of the data from the investigations, and O. M. Belotserkovskii for interest in the work.     

Coaxial interactions of two vortex rings or of a ring with a body

Inviscid coaxial interactions of two vortex rings, including head-on collisions and leapfrogging motions, are considered using a contour dynamics technique. Interactions of vortex rings with solid bodies are also investigated by combining the contour dynamics technique with a boundary integral equation method. Numerical results show that a clean, successful passage motion is possible for two vortex rings with not too thick cores. In both cases of head-on collisions and leapfrogging motions, very large core deformations are observed when two vortex rings get close to each other. A head-tail structure is formed in the later stage of a head-on collision of two fat vortices. Numerical results also show that a vortex ring will stretch and slow down when it moves toward a solid boundary, will shrink and speed up when it moves away from a solid boundary, and will either translate steadily or approach an oscillating asymptotic state when it is far away from any boundaries. The project supported by The National Education Commission of China and NASA under cooperative grant agreement #NCC5-34.     

Evolution of single elliptic vortex rings

The evolution of single elliptic vortex rings for initial aspect ratio (AR)=2,4,6 has been studied. The incompressible Navier-Stokes equations are solved by a dealiased pseudo-spectral method with 64³ grid points in a periodic cube. We find that there are three kinds of vortex motion asAR increases and bifurcation occurs at certainAR. The processes of advection, interaction and decay of vortex ring are discussed. Numerical results coincide with experiments and other authors' numerical simulation. The project is supported by National Natural Science Foundation of China and Doctoral Program of Institution of Higher Education     

The decay of confined vortex rings

Vortex rings are produced during the ejection of fluid through a nozzle or orifice, which occurs in a wide range of biological conditions such as blood flow through the valves of the heart or through arterial constrictions. Confined vortex ring dynamics, such as these, have not been previously studied despite their occurrence within the biological flow conditions mentioned. In this work, we investigate laminar vortex rings using particle image velocimetry and develop a new semi-empirical model for the evolution of vortex ring circulation subject to confinement. Here we introduce a decay parameter ?? which exponentially grows with increasing vortex ring confinement ratio, the ratio of the vortex ring diameter (D _VR) to the confinement diameter (D), with the relationship $\beta=4.38 \exp(9.5D_{\rm VR}/D),$ resulting in a corresponding increase in the rate of vortex ring circulation decay. This work enables the prediction of circulation decay rate based on confinement, which is important to understanding naturally occurring confined vortex ring dynamics.     

Dynamics of vortex rings moving counter the lift

The motion of a buoyant vortex ring counter the direction of the lift of given value is experimentally investigated on a wide range of the initial velocity of the ring. The dynamics of its parameters are determined. The experimental results are compared with the calculations according to the earlier developed theoretical model. It is established in which cases the theoretical model describes the dynamics of buoyant vortex ring in motion counter the lift.