Asymptotic theory of vortex breakdown

Axisymmetric breakdown of a vortex in a steady flow of incompressible fluid is investigated by means of an asymptotic analys is of the Navier-Stokes equations for large Reynolds number (R). A criterion for vortex breakdownis formulated. The region of slow motion resulting from vortex breakdown is shown to be bounded by an ellipsoid with length of order R^–1/4 and thickness of the order R^–3/8.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 78–90, May–June, 1993.     

Vortex motion of a disperse mixture

A study is made of the motion of fine spherical particles in a given steady vortex flow of an incompressible fluid. The results are given of an investigation into the Lyapunov stability of a particle trajectory coincident with the vortex axis, and of trajectories from which the distance to the vortex axis is determined by the condition of equality of the radial components of the force of the phase interaction and of the centrifugal force which acts on the particle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–169, November–December, 1984.     

Calculation of the flow of an ideal fluid past a wing of finite thickness

The problem of irrotational flow past a wing of finite thickness and finite span can be reduced by Green's formula to the solution of a system of Fredholm equations of the second kind on the surface of the wing [1]. The wake vortex sheet is represented by a free vortex surface. Besides panel methods (see, for example, [2]) there are also methods of approximate solution of this problem based on a preliminary discretization of the solution along the span of the wing in which the two-dimensional integral equations are reduced to a system of one-dimensional integral equations [1], for which numerical methods of solution have already been developed [3–6]. At the same time, a discretization is also realized for the wake vortex sheet along the span of the wing. In the present paper, this idea of numerical solution of the problem of irrotational flow past a wing of finite span is realized on the basis of an approximation of the unknown functions which is piecewise linear along the span. The wake vortex sheet is represented by vortex filaments [7] in the nonlinear problem. In the linear problem, the sheet is represented both by vortex filaments and by a vortex surface. Examples are given of an aerodynamic calculation for sweptback wings of finite thickness with a constriction, and the results of the calculation are also compared with experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 124–131, October–December, 1981.     

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.     

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.     

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.     

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.     

Design of airfoil with flap simulated by a point vortex

The method of quasisolutions of inverse boundary-value problems (see [4]) is used to solve the problem of designing an airfoil with a flap, replaced by a fixed vortex, from given velocity distribution along the contour of the wing main part. Profiles are constructed and the effect of the flap (vortex) on the shape and aerodynamic properties of the mechanized wing is examined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–9, January–February, 1992.     

Formation of a vortex street behind an oscillating cylinder

The paper presents the results of an experimental study of the vortex street behind a circular cylinder oscillating across a water stream at Reynolds numbers less than the critical value at which a vortex street is formed behind a stationary cylinder. The asymptotic dependence of the nondimensional vortex separation frequency (the Strouhal number) on the main relevant parameters is obtained. The value of a constant coefficient appearing in this relationship has been found experimentally.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 51–56, January–February, 1995.     

Numerical study of eccentric Couette–Taylor flows and effect of eccentricity on flow patterns

In this study, the differential quadrature (DQ) method was used to simulate the eccentric Couette–Taylor vortex flow in an annulus between two eccentric cylinders with rotating inner cylinder and stationary outer cylinder. An approach combining the SIMPLE (semi-implicit method for pressure-linked equations) and DQ discretization on a non-staggered mesh was proposed to solve the time-dependent, three-dimensional incompressible Navier–Stokes equations in the primitive variable form. The eccentric steady Couette–Taylor flow patterns were obtained from the solution of three-dimensional Navier–Stokes equations. The reported numerical results for steady Couette flow were compared with those from Chou [1], and San and Szeri [2]. Very good agreement was achieved. For steady eccentric Taylor vortex flow, detailed flow patterns were obtained and analyzed. The effect of eccentricity on the eccentric Taylor vortex flow pattern was also studied.     

Solution of linear problems of the uniform motion of a vortex source in a multilayer fluid

A method of solving linear problems of the uniform motion of a vortex source in a multilayer fluid having an arbitrary finite number of layers is proposed. As an example, the problem of the motion of a vortex source of given intensity in a three-layer fluid is solved. Formulas for the complex velocities and hydrodynamic reactions are obtained.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 127–132, May–June, 1995.     

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.     

Motion of an intense vortex on a rotating globe

The evolution of an initially circular vortex is considered in terms of the relation between its dimensions and the screening scale — the radius of deformation of the quasigeostrophic single-layer model in the betaplane. It is shown that the beta-effect causes the displacement of the center of the vortex as a result of wave drift and secondary flows of dipole structure, whose development is analyzed asymptotically. It is found that with increase in the radius of deformation relative to the dimensions of the vortex the velocity of its center with respect to latitude becomes greater than the velocity with respect to longitude. The change in the intensity of the vortex due to the motion of its center with respect to latitude is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 68–77, March–April, 1988.     

Relaxation-Induced Suppression of Vortex Disturbances in a Molecular Gas

The influence of thermal excitation on a finite-amplitude vortex disturbance in a shear flow of a molecular gas is studied in a model problem. The evolution of such vortex structures is typical of both the nonlinear stage of the laminar–turbulent transition and for developed turbulence. Since the excitation level was assumed to be comparatively low, full Navier–Stokes equations for a compressible heat-conducting gas were used in calculations; nonequilibrium was taken into account by the coefficient of bulk viscosity. As the bulk viscosity increases in the range of realistic values, the disturbance-energy damping rate in a weakly compressible flow increases approximately by 10%. The increase in the Mach number enhances the effect of disturbance suppression.     

Vortex flows of a viscous heat-conducting gas in a slightly divergent tube with volume energy supply

The results of a numerical investigation of viscous vortex flow in a slightly divergent tube with thermal energy supplied to the flow are presented. The initial stage of vortex flow development is considered for two different longitudinal velocity distributions simulating the velocity profiles in jet-like and wake-like vortex flows in the vicinity of the vortex axis. The first type of flow can be considered as a model for the near-axis region of the vortex formed in the flow around a delta wing at incidence. The second type can serve as a model for the near-axis region of the trailing vortex downstream of a high-aspect-ratio wing. The development of the two flows is studied for a constant area tube, a slightly divergent tube, and in the case of thermal energy supply from a volume energy source at a constant wall temperature.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 90–97, September–October, 1996.     

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.     

Motion of two vortex rings

It is well-known [1] that two coaxial rings which are moving in the same direction pass through each other alternately. In the case of thin vortex rings this phenomenon was first considered qualitatively in [2]. The assumption that the vortex rings are thin means that when their interaction is considered they can be assumed to be annular vortex filaments. In the present paper, on the basis of the approach suggested in [2], certain new properties are determined for a system of two coaxial vortex rings of the same intensity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 176–177, September–October, 1985.The authors express their sincere thanks to A. A. Aleksandrov for his interest in their work.     

Thermocapillary Vortices Induced by a Light Beam near a Bubble Surface in a Hele-Shaw Cell

This paper studies thermocapillary vortices induced by local heating of a bubble surface in a Hele-Shaw cell by a light beam. It is found that the vortex rotation frequency and its depth depend on the distance from the light-beam projection onto the layer to the bubble boundary. The surface velocity of the thermocapillary flow is calculated using the balance of the near-surface and return flows of the thermocapillary vortex and the equality of capillary and dynamic pressures. It is shown that a decrease in the surface velocity and the vortex rotation frequency with increase in the distance from the light beam to the bubble surface is due to a decrease in the temperature gradient between the illuminated and cold poles of the bubble.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 93–99, September–October, 2005.     

Determining the radius of the air vortex during the laminar flow of a liquid in a centrifugal atomizer

In existing theories of centrifugal atomizers, such as that of Abramovich [1], in order to determine the radius r₀ of the air vortex the conditions of the maximum rate of flow or some other extremal principle are conventionally employed. In this paper the radius of the air vortex will be determined from the equations of motion of a viscous incompressible liquid.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 54–57, May–June, 1976.     

Asymptotic model of the axisymmetric breakdown of a vortex filament in an incompressible fluid

The axisymmetric flow of an incompressible fluid is considered. An exact solution of the Euler equations corresponding to the breakdown of a straight vortex filament of intensity ₀ into a vortex filament of lesser intensity and a conical vortex surface is obtained. It is shown that beyond the breakdown point in the region bounded by the conical vortex surface reverse flows occur. An investigation of the problem with allowance for viscous effects at large Reynolds numbers makes it possible to establish a relation between the free parameters entering into the solution of the Euler equations. The results obtained are useful for investigating the problem of the breakdown of a swirled jet, whose solution has recently been receiving much attention [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 47–52, November–December, 1989.