Dynamics of thin vortex rings in a low-viscosity fluid

In order to describe the unsteady flow of a viscous fluid induced by a toroidal vorticity distribution we use the two-scale expansion method [6], By this means we obtain a vorticity distribution in the core of the thin vortex ring that is consistent with the external potential flow. The time dependence of the flow characteristics obtained confirms the experimental results for the inertial regime. The interaction of coaxial vortex rings is investigated as an example.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 52–59, September–October, 1992.     

Long-wave instability of a vortex ring

The instability of barrel-shaped vibrations of a vortex ring in an ideal fluid is investigated. These vibrations, stable for a vortex ring with a piecewise-uniform vorticity profile, appear to be unstable for a vortex ring with a smooth vorticity profile. The instability growth rate is found on the basis of the energy balance equation determining the energy transport from perturbations with negative energy in the critical layer to perturbations with positive energy in the rest of the flow. The curvature of the vortex ring, by virtue of which the perturbations with energies of different signs appear to be connected, plays a prominent role in the mechanism under consideration.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 72–78, November–December, 1995.     

Turbulent heat—Concentration plume in the case of high viscosity in a nonstratified medium

In this paper, an approximate self-similar solution for the structure of the heat—concentration plume produced by an instantaneous point source of heat in the presence of multicomponent admixture and when the coefficient of turbulence is large is found on the basis of the turbulent transfer equations of vorticity, energy, and matter. Analytical expressions are obtained for the propagation velocity of the buoyancy core and the toroidal vortex formed. The influence of the source parameters and of the coefficient of turbulence on the structure and dynamics of thermal lift is investigated. A comparison is made with the laminar regime of motion of similar formations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 153–163, November–December, 1984.     

Basic Displacements in the Problem of Core Perturbations of a Thin Isochronous Vortex Ring

Periodic perturbations in the core of a thin isochronous vortex ring in an inviscid incompressible fluid are investigated in the linear approximation. The aim of the study is to construct the system of basic displacements, namely, the complete system of solutions of the Helmholtz equation for vorticity perturbations inside the core of a vortex ring with a given frequency in the form of expansion in the ring thinness parameter μ. The structure of basic displacements depends substantially on the fact to what extent the frequency of the forcing action is close to the resonance frequencies of the system. If the difference between these frequencies is small, then, in addition to the ring thinness μ, the second small parameters arises in the problem. This leads to significant complication of the procedure of obtaining the solution and appearance of considerable corrections in the subsequent approximations of the expansion procedure. The case of isochronous vortex ring in which the periods of revolution of liquid particles are identical is considered. Obtaining the threedimensional oscillations in such flows turns out to be the simplest since there are no perturbations of the continuous spectrum for the isochronous ring. The system of basic displacements is the necessary element in deriving the dispersion relation for the eigen-oscillations of the vortex ring. The solutions obtained can also serve as an instrument to analyze the reaction of flows with curvilinear vortex lines or flows localized in toroidal regions to the external excitation.     

Model of two-dimensional vortex motion with an entrainment mechanism

A model for describing the vertically averaged vortex motions of an incompressible viscous fluid with an arbitrary vertical structure of the bottom Ekman boundary layer is proposed. An approach similar to that adopted in [1] is used: the second moments of the deviations from the average velocities required in order to close the vorticity equation are calculated by means of the Ekman solution for gradient flows, which makes it possible to take the integral bottom boundary layer effect into account. As a result, these terms lead to a specific form of nonlinear friction with a coefficient that depends on the vorticity of the average flow. In the particular case of a constant vertical turbulent transfer coefficient the inaccuracies of the model described in [1] can be eliminated. The generalized vorticity equation obtained has solutions of the vorticity spot type with a uniform internal vorticity distribution, which can be effectively investigated by means of appropriate algorithms [2]. The mechanism of entrainment at the vorticity front is illustrated with reference to the example of the evolution of vorticity spots. An exact solution of the problem of the evolution of an elliptic vortex (generalized Kirchhoff vortex), which in the case of fairly strong anticyclonic vorticity degenerates first into a line segment (vortex sheet) and then into a point vortex, is constructed. Equations describing the dynamics of an elliptic vorticity spot in an external field with a linear dependence of the velocity on the horizontal coordinates and generalizing the classical Chaplygin-Kida model [3, 4] are constructed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 49–56, November–December, 1992.     

Rising vortex ring in a viscous fluid

Most theoretical results for thermals, whose motion is determined by the complex interaction between dynamics and buoyancy, have been obtained numerically [1–4]. The analytic solutions for a convection element have been limited to consideration of the self-similar regime [5]. At the same time, the preself-similar stage of development of a vortex ring of dynamic origin has been described analytically [6]. This approach is now extended to a rising vortex ring. In this case a modification of the traditional formulation of the problem makes it possible to obtain an analytic solution of the problem of a weak thermal in the form of unsteady temperature, vorticity and stream function fields that tend in the limit to the self-similar regime. The rate of ascent of the convective vortex ring is found. A solution is obtained for the two-dimensional analog of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–48, May–June, 1989.     

Theory of nonstationary vortices in an ideal fluid

Equations that describe the evolution of a region with nonzero vorticity are formulated. These equations are solved on a bounded time interval for regions having the shape of a sphere or a circular cylinder at the initial time. It is shown that a spherical vortex formed in a medium at rest begins to move, and is stretched in the direction of the motion; a cylindrical vortex, under the influence of the nonuniform intensity of the vorticity on its boundary, changes both the magnitude and direction of its velocity, and describes a curvilinear trajectory. Expressions are obtained which describe the initial evolution of a fluid sphere of one density in a fluid medium of another density.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–11, November–December, 1980.     

Steady Vortex Ring with Isochronous Flow in the Vortex Core

Steady-state solutions to the problem of a thin vortex ring in an inviscid incompressible fluid in infinite space are investigated. The Fraenkel procedure is used to construct the steady-state solutions. In this procedure a given vorticity distribution in plane flow with circular streamlines is transformed into a steady vortex ring using an expansion in the ring thinness parameter. For example, a two-dimensional vortex of constant vorticity is transformed into a steady vortex ring with the uniform distribution in which the absolute value of vorticity is proportional to the distance from the axis of symmetry. The principal aim of our study is to construct the algorithm of finding the flow for an isochronous vortex ring in which the periods of revolution are the same for all the liquid particles in the vortex core. The problem is that the two-dimensional distribution which goes over in the isochronous ring in accordance with the Fraenkel procedure is unknown in advance. In particular, the ring with the uniform distribution is not isochronous despite the isochronism of the initial two-dimensional flow. In this connection the Fraenkel procedure is significantly modified so that the initial two-dimensional vorticity distribution is determined in each of the steps of the iteration procedure. The solution for the vortex ring with the uniform distribution obtained in the present study is significantly used to construct the isochronous solution. The necessary corrections to the former solution are calculated in each step. Obtaining of the isochronous flow is the key step for the investigation of stability of three-dimensional oscillations of the vortex ring since the oscillation spectrum of this flow is discrete.     

Nonlinear interaction of a large vortex with a boundary layer

The nonlinear problem of boundary layer instability under the influence of a plane vortex is investigated for high Reynolds numbers. The vortex occupies the entire thickness of the boundary layer and has a longitudinal dimension of the order of the Tollmien-Schlichting wavelength. The initial vortex is rapidly swept away by the flow, inducing a Stokes layer near the surface of the plate. Expanding, this layer reaches the dimensions of the viscous sublayer of free interaction theory, where wave packet generation takes place. In the case in question a feature of the nonlinear stage of development of the disturbances is the formation of a concentrated vortex, which arises in the Stokes layer and grows rapidly, whereas the wave packet propagated ahead of it remains linear. From the calculations there emerges a tendency for the new vortex to be formed above the wail, whereas the maximum vorticity of the vortex generated in the Stokes layer corresponds to the wall itself.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 70–77, January–February, 1993.The authors are grateful to V. V. Kozlov for his interest in their work.     

Numerical simulation and PIV study of compressible vortex ring evolution

Formation and evolution of a compressible vortex ring generated at the open end of a short driver section shock tube has been simulated numerically for pressure ratios (PR) of 3 and 7 in the present study. Numerical study of compressible vortex rings is essential to understand the complicated flow structure and acoustic characteristics of many high Mach number impulsive jets where simultaneously velocity, density and pressure fields are needed. The flow development, incident shock formation, shock diffraction, vortex ring formation and its evolution are simulated using the AUSM+ scheme. The main focus of the present study is to evaluate the time resolved vorticity field of the vortex ring and the shock/expansion waves in the starting jet for short driver section shock tubes—a scenario where little data are available in existing literature. An embedded shock and a vortex induced shock are observed for PR =  7. However the vortex ring remains shock free, compact and unaffected by the trailing jet for PR =  3. Numerical shadowgraph shows the evolution of embedded shock and shock/expansion waves along with their interactions. The velocity and vorticity fields obtained from simulation are validated with the particle image velocimetry results and these data match closely. The translational velocity of the vortex ring, velocity across the vortex and the centre line velocity of the jet obtained from simulation also agree well with the experimental results.     

Experimental investigation of vortex rings impinging on inclined surfaces

Vortex–ring interactions with oblique boundaries were studied experimentally to determine the effects of plate angle on the generation of secondary vorticity, the evolution of the primary vorticity and secondary vorticity as they interact near the boundary, and the associated energy dissipation. Vortex rings were generated using a mechanical piston-cylinder vortex ring generator at jet Reynolds numbers 2,000–4,000 and stroke length to piston diameter ratios (L/D) in the range 0.75–2.0. The plate angle relative to the initial axis of the vortex ring ranged from 3 to 60°. Flow analysis was performed using planar laser-induced fluorescence (PLIF), digital particle image velocimetry (DPIV), and defocusing digital particle tracking velocimetry (DDPTV). Results showed the generation of secondary vorticity at the plate and its subsequent ejection into the fluid. The trajectories of the centers of circulation showed a maximum ejection angle of the secondary vorticity occurring for an angle of incidence of 10°. At lower incidence angles (<20°), the lower portion of the ring, which interacted with the plate first, played an important role in generation of the secondary vorticity and is a key reason for the maximum ejection angle for the secondary vorticity occurring at an incidence angle of 10°. Higher Reynolds number vortex rings resulted in more rapid destabilization of the flow. The three-dimensional DDPTV results showed an arc of secondary vorticity and secondary flow along the sides of the primary vortex ring as it collided with the boundary. Computation of the moments and products of kinetic energy and vorticity magnitude about the centroid of each vortex ring showed increasing asymmetry in the flow as the vortex interaction with the boundary evolved and more rapid dissipation of kinetic energy for higher incidence angles.     

Reynolds number dependence of two-dimensional laminar co-rotating vortex merging

The paper reports unsteady Navier–Stokes calculations of laminar two-dimensional co-rotating vortex merging for various Reynolds numbers. The unsteady, incompressible two-dimensional Navier–Stokes equations were solved with fourth-order Runge–Kutta temporal discretization and fourth-order symmetric compact schemes for spatial discretization. Calculations of the unsteady Taylor vortex benchmark showed that fourth-order accurate solutions for all primitive variables were indeed achieved. Calculations for a pair of equal-strength co-rotating vortices show good agreement with reported direct numerical simulation and experiments for the evolution of the separation distance and core radius. It is found that the time required for merging is inversely proportional to the square root of the Reynolds number. According to previous experimental research, it was also found that complete merging in laminar regime undergoes four stages with physical meaning. The physical mechanism responsible for the merging process is investigated and it is found that the antisymmetric vorticity dynamics plays an important role until full merging.     

The pinch-off process in a starting buoyant plume

The vortex ring formation process of a starting buoyant plume was studied experimentally using digital particle image thermometry and velocimetry (DPITV). The vortex ring was observed to pinch-off, or become disconnected, from the trailing plume. Pinch-off occurred at non-dimensional times, or formation numbers, between 4.4 and 4.9. The observed pinch-off process is consistent with an explanation based upon the Kelvin–Benjamin variational principle. This is analogous to the pinch-off of a vortex ring generated using a piston–cylinder apparatus, suggesting that pinch-off is a general component of the vortex ring formation process for various generation mechanisms.     

Method of contour dynamics for plane flows of an ideal fluid

The method of contour dynamics is generalized for plane flows of a general form when, apart from vortices, distributed mass sources (or sinks) are present in the fluid. The laws of variation of the vorticity and divergence of the fluid particles with time are obtained for this case which makes it possible to use the method of contour dynamics for piecewise-constant vortex sink distributions.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 16–19, September–October, 1993.     

Plane vortex flow induced by a mass source (sink) in a rotating viscous fluid

A solution of the self-similar type, describing the development with time of a plane vortex flow excited by an axisymmetric mass source (sink) in a rotating viscous fluid, is obtained. Sources of two kinds — impulsive and of constant strength — are considered. The solutions for the velocity and vorticity fields are expressed in the form of functions similar to incomplete gamma functions and are presented in the form of graphs for various flow Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 172–175, July–August, 1991.     

Reconnections of vortex filaments

The process of break-down and reconnection of vortex filaments is considered by the method of three-dimensional vortex singularities (vortons) in various situations, including oblique interaction of a vortex ring with a boundary in shear flow, shedding of a vortex ring from a horseshoe vortex, instability of elliptic vortex ring, Crow instability of two perturbed antiparallel vortex filaments, merging and subsequent splitting of vortex rings. Special attention is paid to the global integrals (vorticity, momentum, angular momentum) and to the inviscid dissipation of energy. The visualization of the effective vortex core, created by the interference of the vorticity fields of vortons, is presented. The comparisons with other methods of simulation of three-dimensional vortex interactions and with the observations have been made.     

On the flow field generated by a gradually varying flow through an orifice

The motion of a vortex ring generated by gradually varied flows through a thin-edged orifice has been investigated experimentally using particle image velocimetry. This flow reproduces the primary characteristics of many biological flows, such as cardiac flows through valves or jellyfish and squid propulsion. Even though vortex ring formation has been extensively studied, there is still interest in gradually varying inflows, i.e. the ones that are mostly found in previous conditions. The main purpose of this paper is to extend the time scaling already proposed in the literature to the entire cycle of vortex ring formation, pinch-off and free motion. To this end, eight inflow time laws have been tested, with different acceleration and deceleration phases. They have been selected in relation to practical applications by their resemblance to the main characteristics of cardiovascular and pulsed locomotion flows. Analysis of measured velocity and vorticity fields suggested a general criterion to establish the instant of vortex pinch-off directly from the imposed velocity program. This allows the proper scaling of the entire time evolution of the vortex ring for all tested inflows. Since it is quite easy to identify this instant experimentally, these results give a simple, practical rule for the computation of scales in vortex ring formation and development in the case of gradual inflows. The “slug model” has been used to test the proposed scaling and to obtain predictions for the vortex position, circulation and vorticity which are in agreement with experimental data.     

Radiation and scattering of sound from a vortex ring

The mechanisms of generation and scattering of sound by a vortex ring are investigated on the basis of fluid dynamics. The vortex ring can serve as a simple dynamic model of the large-scale structures observed in shear flows. Moreover, it is probably the most easily studied vortex element that can be created experimentally. The sound scattering investigation also served to determine the extent to which the vortex is affected by sound, its selectivity with respect to such parameters as the acoustic frequency, the angle of incidence of the wave, etc. The perturbed motion is considered against the background of the steady-state motion of the ring. The perturbed motion in the vortex core is determined on the basis of linear incompressible fluid dynamics. Two terms of the expansion in the M number of the far acoustic field generated by the perturbations in the core are found in accordance with Lighthill's theory. The acoustic power and directivity of the radiation and the acoustic instability growth rate are calculated. It is shown that the scattering of sound by the vortex ring is a resonance effect, and the scattering amplitude near resonance is determined. The acoustic action on the hydrodynamic structure of the flow in the core of the ring is especially intense near the resonances and extends over a period short as compared with the characteristic time of the acoustic instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 83–95, May–June, 1987.     

Particle Simulation of Swirling Flows

The vortex particle method has been applied to the axisymmetric swirling flow of a viscous fluid. The formulation used yields two transport equations which have been solved within the lagrangian framework of particle method. The diffusion operator for both equations has been approximated by means of a Particle Strength Exchange scheme. Applications to the cases of one isolated vortex ring and two co-rotating vortex rings illustrate the interest of this new method. Special attention has been devoted to the vorticity production resulting from the interaction between the azimuthal components of vorticity and velocity. The generation of small eddies at the boundary of the vortex ring cross-section has been particularly investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of homogeneous stratification of the fluid on the dynamics of a kármán vortex street behind a circular cylinder

The experimental results of studying the effect of homogeneous stratification of the fluid on the conditions of generation of a Kárman vortex street [1] developing in the wake of a cylinder in steady horizontal motion are described. In a homogeneous medium at Reynolds numbers Re >5 two symmetrical regions of vorticity of opposite sign are formed behind the cylinder and move together with the latter. As the speed of the cylinder increases, the link between the vortices and the cylinder grows weaker, the vortices are stretched out along the flow and at Re > 40 begin to separate alternately, forming a vortex street in the wake. At first, the frequency of vortex separation increases sharply with increase in Re, but then levels off. It is found that in a uniformly stratified fluid the onset of vortex separation from the moving cylinder is delayed. The dependence of the critical Reynolds number (onset of vortex separation) on the internal Froude number is obtained. The effect of stratification of the fluid on the frequency of separation of the vortices in the Kármán street is investigated. The effect of the Froude number on the dependence of the Strouhal number on the Reynolds number is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 83–86, January–February, 1986.In conclusion the authors wish to thank A. T. Onufriev for his interest in their work and useful discussions of the results.