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.     

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.     

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.     

Asymptotic behavior of the far region of turbulent wake vortices

An approximate mathematical model, formulation of the problem, and its approximate solution are proposed for the far region of a turbulent vortex wake past a moving body, where the departure of the horizontal velocity component from the uniform flow is slight. It is assumed that the single important parameter that defines the main flow characteristics in this region is the vortex momentum per unit length produced in the fluid by the lift equal to the weigth of the moving body uncompensated by the buoyancy force. Thus, the flow is self-similar, and the self-similarity law determines the intensity, shape, and location of vortex lines as functions of the downstream distance with accuracy up to a constant factor, which cannot be determined theoretically and should be obtained by comparison of theory with experiment. A boundary-value problem is formulated to determine the flow structure of vortex lines (vorticity distribution). A solution of the problem is obtained numerically in the limit of “vanishing turbulent viscosity.” The variation in the maximum velocity of a vortex line with distance, determined by self-similarity, is in agreement with available experimental data. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 13–23, March–April, 1999.     

Supersonic flow over a slender wing with subsonic edges

The method of integral equations is generalized to calculate steady flow past wings with an arbitrary shape in plan with subsonic leading and trailing edges. The determination of the velocity potential in the leading part of the wing, where there is no influence of the vortex sheet, is reduced to the solution of a two-dimensional integral equation of the second kind. The trailing part, which is subject to the influence of the vortex sheet, is divided into a number of subregions, in which the calculation of the acceleration potential reduces to the solution of one-dimensional equations of the type of Fredholm equations of the second kind and to quadrature. The unique solvability of the obtained integral equations is investigated; it is shown that they can be solved by successive approximation. As an example, the solution to the problem of flow past a flat delta-shaped wing is found and compared with the exact solution to the problem found by the method of conic flows [4, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 119–127, September–October, 1981.I thank G. Yu. Stepanov for discussing the paper.     

Viscous flow past an airfoil

Incompressible viscous flow past an airfoil at low Reynolds numbers is investigated on the basis of a numerical solution of the complete Navier-Stokes equations. Steady flow regimes, with and without separation, are obtained and, moreover, periodic regimes with the formation of a vortex trail in the wake. The frequency of vortex formation is determined by the linear dimension of the projection of the airfoil on the normal to the freestream velocity. The relation between the Strouhal and Reynolds numbers, determined from this linear dimension, depends only slightly on the angle of attack and shape of the airfoil and is similar to the experimental dependence for circular cylinders.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 29–36, July–August, 1986.In conclusion the authors wish to express their warm thanks to G. I. Petrov for his interest in their work and valuable discussion of their results, and to V. P. Shkadova for discussing the formulation of the problem and the method of solution and for her constant readiness to advise on the organization of the calculations.     

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.     

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.     

Visual study of the three-dimensional flow diagram around a delta wing in a subsonic stream

Experimental methods, particularly visualization methods, permit a sufficiently detailed representation of the flow around bodies of complex shape, whose analysis meets with a considerable number of difficulties. The flow around a delta wing in the 1–90-m/sec free-stream velocity range is studied in this paper by using three-dimensional visual methods. Since stream separation and vortex-system formation play the main role in the flow formation over a wing surface, the main purpose of the experiment was to trace the physical process of dynamic development of the flow resulting in separation and vortex formation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 190–194, March–April, 1976.     

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.     

Effect of buoyancy on the wakes of circular and square cylinders: a schlieren-interferometric study

Wakes behind heated cylinders, circular, and square have been experimentally investigated at low-Reynolds numbers. The electrically heated cylinder is mounted in a vertical airflow facility such that buoyancy aids the inertia of main flow. The operating parameters, i.e., Reynolds number and Richardson number are varied to examine flow behavior over a range of experimental conditions from forced to mixed convection regime. Laser schlieren-interferometry has been used for visualization and analysis of flow structures. Complete vortex shedding sequence has been recorded using a high-speed camera. The results on detailed dynamical characteristics of vortical structures, i.e., their size, shape and phase, Strouhal number, power spectra, convection velocity, phase shift, vortex inception length, and fluctuations are reported. On heating, alteration of organized (coherent) structures with respect to shape, size and their movement is readily perceived from instantaneous Schlieren images before they reduce to a steady plume. For both cylinders, Strouhal number shows a slow increase with an increase in Richardson number. At a critical value, there is complete disappearance of vortex shedding and a drop in Strouhal number to zero. The corresponding spectra evolve from being highly peaked at the vortex shedding frequency to a broadband appearance when vortex shedding is suppressed. The geometry of vortex structures transforms to a slender shape before shedding is suppressed. At this heating level, absence of multiple peaks in power spectra at cylinder centerline indicates absence of interaction between opposite shear layers. The convection velocity of vortices increases in stream wise direction to an asymptotic value and its variation is a function of Richardson number. The convection speed abruptly falls to zero at critical Richardson number. The phase difference of shed vortices between upstream and downstream location increases with an increase in Richardson number. Velocity profiles show an increase in fluid speed and beyond the critical point, buoyancy forces add enough momentum to cancel momentum deficit due to the cylinder. Overall, the combined effect of temperature gradient on the separating shear layer velocity profile in near field and vortical structures interaction in far field influences wake instability of a heated cylinder. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Calculation of flow around a lifting surface in the case of large deformations

Many studies have been made of the nonstationary flow of an ideal incompressible fluid around a lifting surface. The present state of the numerical methods of solution of this problem is reviewed in [1]. The present paper studies three-dimensional nonstationary flow around a lifting surface which undergoes deformation and behind which a wake vortex surface is formed. The lifting and wake vortex surfaces are represented in parametric form. The metrics of these surfaces are used, and the introduced vortex function is approximated by bicubic splines. For the convenient application of the theory developed here to the flapping flight of insects, for which it is sometimes difficult to distinguish the lateral and trailing edges of the wings, the following terminology is introduced. The part of the edge of the lifting surface from which the wake vortex surface is shed is called the trailing edge. The remaining part is called the leading edge. On the leading edge, the velocity has a singularity. Test calculations have demonstrated the effectiveness of the method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 72–79, July–August, 1980.     

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.     

Convective flow with uniform vorticity in a vertical layer

The free convective circulation of liquid in plane vertical slits of circular and square cross section with a longitudinal horizontal temperature gradient at the boundaries was investigated experimentally. It was found that under such heating conditions there is a uniform-vorticity flow with a region of quasirigid rotation, which has the shape of a disk in a circular slit and the shape of a cross in a square slit; in each longitudinal section of this zone the liquid moves along concentric trajectories with constant angular velocity. Dimensionless numbers for the problem were established by tests with various liquids and cavities of different dimensions. In dimensionless form, the angular velocity of the vortex and the temperature gradient in it depend linearly on the temperature difference at the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 160–165, May–June, 1984.     

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.     

Axisymmetric cavitation flow around a body in a tube

The problem of the axisymmetric flow around a body in a circular tube with arbitrary shape of the meridian section is reduced to the numerical solution of a system of two integral equations to determine the shape of the cavern and the intensity of the vortex rings arranged on the solid boundaries and the cavern boundary. Results of computations of the cavitation flow around a sphere, ellipsoid of revolution, and cone in a cylindrical tube, and also for a cone in converging and expanding tubes and in a hydrodynamic tunnel with the actual shape of the converging and working sections, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 50–55, July–August, 1976.     

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.     

Formation of a vortex ring during the rise of a heated air mass in the stratified atmosphere

The motion of a strongly heated air mass in the earth's atmosphere is studied numerically. The turbulence effects are modeled by the assignment of constant coefficients of viscosity and thermal conductivity. It is shown that a vortex torus is formed during the rise of a hot mass initially having a spherical shape. The velocity, density, and temperature distributions over the significant time interval are obtained for a series of variants differing in the values of the Reynolds and Prandtl numbers. A satisfactory correspondence between the integral characteristics (altitude of rise, dimensions of disturbed region) and the data obtained by the approximate method of A. T. Onufriev [1] is established for several variants.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 186–189, March–April, 1978.The author thanks L. A. Chudov for guidance of the 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.     

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.