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.     

Evolution of solitary vortices in a rotating fluid

The evolution of cyclones and anticyclones with a characteristic scale substantially exceeding the radius of deformation is investigated numerically. An equation obtained by an asymptotic method for small values of the Kibel'-Rossby number is employed. The model ensures conservation of the potential vorticity in the fluid particles for motions of finite amplitude (when the thickness of the layer H deviates considerably from the undisturbed value H₀). It is shown that anticyclones of a certain type adapt themselves to a steady shape and travel in a direction opposite to the direction of rotation of the sphere (westwards). It is found that the existence of a steadily migrating anticyclone requires the presence of a region of closed isolines of the potential vorticity (trapping region), in which the fluid is transported together with the anticyclone. Cyclones drift westwards more slowly than anticyclones and migrate towards the poles, losing energy by emitting Rossby waves. It is found that in cyclones the trapping region retains an almost circular shape, the decay of the eddy, associated with wave emission, showing as its initial intensity increases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–59, July–August, 1986.     

Analytic investigation of a nonlinear problem of the effect of buoyancy on the evolution of an annular vortex

The evolution of a heavy axisymmetric vortex whose density differs from that of the surrounding irrotational ideal fluid is investigated analytically. If the vortex had no buoyancy (i.e., if the densities were identical), it would preserve its shape and velocity. An approximate analytic solution of the problem is obtained. This solution describes two types of evolution of the vortex shape corresponding to different values of the initial velocity and small buoyancy. The spectrum of a nonlinear wave developing on the vortex boundary is estimated.Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 56–66, January–February, 1996.     

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.     

The N-vortex problem on a sphere: geophysical mechanisms that break integrability

We describe the dynamical system governing the evolution of a system of point vortices on a rotating spherical shell, highlighting features which break what would otherwise be an integrable problem. The importance of the misalignment of the center-of-vorticity vector associated with a cluster of point vortices with the axis of rotation is emphasized as a crucial factor in the interpretation of dynamical features for many flow configurations. We then describe two important physical mechanisms which break what would otherwise be an integrable problem—the interactions between the local center-of-vorticity vectors of more than one region of concentrated vorticity, and the coupling between the center-of-vorticity vector and the background vorticity field which supports Rossby waves. Focusing on the Polar vortex splitting event of September 2002, we describe simple (i.e., low dimensional) mechanisms that can trigger instabilities whose subsequent development cause the onset of chaotic advection and global particle transport. At the linear level, eigenvalues that oscillate between elliptic and hyperbolic configurations initiate the pinch-off process of a passive patch representing the Polar vortex. At the nonlinear level, the evolution and topological bifurcations of the streamline patterns are responsible for its further splitting, stretching, and subsequent transport over the sphere. We finish by briefly describing how to incorporate conservation of potential vorticity and the development of a model governing the probability density function associated with the point vortex system.     

Flow past a sphere in density-stratified fluid

Stratified flow past a three-dimensional obstacle such as a sphere has been a long-lasting subject of geophysical, environmental and engineering fluid dynamics. In order to investigate the effect of the stratification on the near wake, in particular, the unsteady vortex formation behind a sphere, numerical simulations of stratified flows past a sphere are conducted. The time-dependent Navier–Stokes equations are solved using a three-dimensional finite element method and a modified explicit time integration scheme. Laminar flow regime is considered, and linear stratification of density is assumed under Boussinesq approximation. The effects of stratification is implemented by density transport without diffusion. The computed results include the characteristics of the near wake as well as the effects of stratification on the separation angle. Under increased stratification, the separation on the sphere is suppressed and the wake structure behind the sphere becomes planar, resembling that behind a vertical cylinder. With further increase in stratification, the wake becomes unsteady, and consists of planar vortex shedding similar to von Karman vortex streets.     

Dynamics of vorticity at a sphere

The article discusses the two-dimensional flow of an incompressible liquid between two infinitely close concentric spheres, due to an initial distribution of the vorticity differing from zero. The concept of point singularities (vortices, sources, and sinks) at a sphere is introduced. Equations of motion are obtained for point vortices, as well as invariants of the motion, known for the plane case [1]. The simplest case of the mutual motion of a pair of vortices is considered. Equations are obtained for the motion of point vortices at a rotating sphere. Integral invariants for the continuous distribution of the vorticity are obtained, having the dynamic sense of the total kinetic energy and the momentum of the liquid at the sphere. The effect of the topology of the sphere on the dynamics of the vorticity is noted, and a comparison is made with the plane case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 57–65, November–December, 1977.     

Diffusion of a ring vortex

The diffusion of a ring vortex is investigated in the present paper with allowance for the influence of the initial radius of the toroidal vorticity distribution on the flow structure. The statement of the problem in such a formulation makes it possible to classify and reinterpret results obtained previously. A vortex pair is studied together with a vortex ring. The toroidal vorticity and stream function distributions are obtained analytically. The self-induced lift velocity of the ring vortex is found. The influence of inertial terms is investigated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 10–15, November–December, 1987.     

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.     

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.     

Numerical study of an inviscid incompressible flow through a channel of finite length

A two‐dimensional inviscid incompressible flow in a rectilinear channel of finite length is studied numerically. Both the normal velocity and the vorticity are given at the inlet, and only the normal velocity is specified at the outlet. The flow is described in terms of the stream function and vorticity. To solve the unsteady problem numerically, we propose a version of the vortex particle method. The vorticity field is approximated using its values at a set of fluid particles. A pseudo‐symplectic integrator is employed to solve the system of ordinary differential equations governing the motion of fluid particles. The stream function is computed using the Galerkin method. Unsteady flows developing from an initial perturbation in the form of an elliptical patch of vorticity are calculated for various values of the volume flux of fluid through the channel. It is shown that if the flux of fluid is large, the initial vortex patch is washed out of the channel, and when the flux is reduced, the initial perturbation evolves to a steady flow with stagnation regions. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Direct Numerical Simulation of Transitional Noncircular Buoyant Reactive Jets

The near field dynamics of transitional buoyant reactive jets established on noncircular geometries, including a rectangular nozzle with an aspect ratio of 2:1 and a square nozzle with the same cross-sectional area, are investigated by three-dimensional spatial direct numerical simulations. Without applying external perturbations at the inflow boundary, large vortical structures develop naturally in the flow field due to buoyancy effects. Simulation results and analysis describe the details and clarify mechanisms of vortex dynamics of the noncircular buoyant reactive jets. The interaction between density gradients and gravity initiates the flow vorticity. Among the major vorticity transport terms, the gravitational term mainly promotes flow vorticity in the cross-streamwise direction. For the baroclinic torque, it can either create or destroy flow vorticity depending on the local flow structure. The vortex stretching term has different effects on the streamwise and cross-streamwise vorticity. Streamwise vorticity is mainly created by vortex stretching, while this term can either create or destroy cross-streamwise vorticity. Under the coupling effects of buoyancy and noncircular nozzle geometry, three-dimensional vortex interactions lead to the transitional behavior of the reactive jets. Simulations also show that the rectangular jet is more vortical than the square jet. The rectangular jet has a stronger tendency of transition to turbulence at the downstream due to the aspect ratio effect. Mean flow property calculations show that the rectangular buoyant reactive jet has a higher entrainment rate than its square counterpart. Received 13 December 2000 and accepted 24 July 2001     

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.     

Unsteady flow around impacting bluff bodies

The flow resulting from the collision without rebound of generic bluff bodies with a wall in a still viscous fluid is investigated both computationally and experimentally. Emphasis is on the case of a circular cylinder impact (two-dimensional geometry), but comparisons with the flow generated by the impact of a sphere (axisymmetric geometry) are included. For normal cylinder impacts, the two counter-rotating vortices forming behind the body during its motion continue their trajectory towards the wall after the collision, leading to the generation of opposite-signed secondary vorticity at the cylinder and wall surfaces. Secondary vortices forming from this vorticity at higher Reynolds numbers exhibit a short-wavelength three-dimensional instability. Comparison with the sphere impact reveals significant differences in the scales of the vortices after the collision, due to the additional vortex stretching acting in the axisymmetric geometry. This leads to a delay in the onset of three-dimensionality and to a different instability mechanism. Oblique cylinder impacts are also considered. For increasing impact angles, the wall effect is gradually reduced on one side of the cylinder, which favours the roll-up of the secondary vorticity and increases the rebound height of the vortex system.     

From generation to chaotic motion of a ring configuration of vortex structures on a sphere

This is a review article of recent research developments on the motion of a polygonal ring configuration of vortex structures with singular vorticity distributions in incompressible and inviscid flows on a non-rotating sphere. Numerical computation of a single vortex sheet reveals that the Kelvin-Helmholtz instability gives rise to the formation of a polygonal ring arrangement of rolling-up spirals. An application of methods of Hamiltonian dynamics to the N-vortex problem on the sphere shows that the motion of the ring configuration of homogeneous point vortices, which is a simple model for the rolling-up spirals, becomes chaotic after a long time evolution. Some remarks on an extension of the present research and a generic non-self-similar collapse are also provided.     

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.     

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.     

平行涡管三维合并的直接数值模拟

采用拟谱方法求解Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程，对两个同向平行涡管的三维合并过程进行直接数值模拟，分析了它的不同于二维合并的“缠绕式”特性，特别地，对在合并过程中产生的垂直于初始涡管方向的非主体涡量的意义和物理机制了探讨。