Secondary thermocapillary motions of soliton type

An exact analytic solution is obtained for the problem of the stability of the axisymmetric thermocapillary motion due to a point heat source of constant power located on the horizontal free surface of a viscous fluid. Analytic expressions are found for monotonic neutral disturbances of hydrodynamic and thermal type. The critical values of the dimensionless source power for disturbances with arbitrary quantum numbersl andm are determined, together with the secondary motions near the stability threshold. An exact solution of the problem of the axisymmetric thermocapillary motion due to a spherical heat source is presented and its stability is investigated. It is shown that it is always possible to select physical heater properties such that for arbitrarily small source power, the axisymmetric motion is unstable relative to the vortex motion. A comparison is made with experiment.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 20–27, July–August, 1992.     

On neutrally buoyant laminar spherical vortices

It is shown that the similarity solution for an unbounded viscous vortex ring, first given by Phillips and considered in detail by Morton, may be modified to represent a finite spherical volume of fluid moving relative to the fluid at infinity; that is, a laminar spherical vortex is obtained. The gross behaviour of the vortex is discussed and compared with the observations of Keedy. Good agreement between the predicted and experimental results is found.     

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.     

Projective Submodel of the Ovsyannikov Vortex

A submodel of the Ovsyannikov vortex with projective symmetry is studied. Integration of the factor system of the submodel reduces to solving a first-order differential equation which is not solved with respect to the derivative. The properties of the solutions of this equation are studied. It is shown that the submodel describes gas ow with a nonstationary source and a nonstationary sink. The problem of the motion of a gas volume between pistons of cylindrical shapes is studied, and its solution with an invariant shock wave is obtained.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 3–16, July–August, 2005.     

Spherical Vortex Structures with a Core and a Shell

In this paper we construct and investigate the vortex structure consisting of a spherical vortex (vortex core) inside a spherical vortex layer (shell). A partial case of this structure is a spherical vortex with uniformly helical motion of the fluid within the core and the shell. The strengths of the helical flows in the core and the shell are generally different. The case of identical strengths is analyzed in detail. The streamline pattern is presented. The vortex velocity limit at which the vortex does not collapse is found. This proves to be less by a factor 1.7 than the analogous quantity for a vortex without a shell and 4 times lower than the maximum velocity of the Hill vortex.     

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.     

THE VORTICITY STRUCTURE OF HOMOGENEOUS ISOTROPIC TURBULENCE IN IT5 FINAL PERIOD OF DECAY

流体力学工作者很早就认识到,湍流运动是由许许多多微小的涡旋廷动所组成.本论文的目的,在于从粘性不可压缩流体的线性性化以后的运动方程的漪旋运动解,来说明均匀各向同性湍流在后期衰变时期的运动现象.确定在这时期作为湍流元随机运动着的涡旋的物理条件是:首先,涡旋的涡量分布必须集中在涡旋的几何中心附近,其次,每一个涡旋的总角动量是一常数.我们运用辛和林家翘的方法来计算湍流中的二元速度关联函数,从而导致米里洪西科夫解.这个解的速度关联和湍能耗拍定律都和白却勒尔与爆生德的实验相符合.最后,我仍指出了湍流的湍性结构是和作为湍流元的涡旋的涡性结构有密切联系以及依据现有涡旋运动解来计算出高元速度关联和其他种类关联函数的可能性.     

“Bunched” model of corona discharge in a gas-dynamical flow

In the present paper, a theoretical model of corona discharge is proposed for the case when electric charge transport is implemented by means of the motion of discrete charged bunches of finite dimensions. A system of equations and boundary conditions is formulated for the study of unsteady cyclic processes in a corona discharge. The electric field induced by the space charge of bunches and the presence of an external electric circuit are taken into account. A solution of the formulated system of equations for corona discharge with spherical geometry is obtained. The integrated (current-voltage) characteristics and the amplitude-frequency characteristics of the corona discharge are found. The proposed theory is generalized to the case of a corona discharge in a moving gas. The unsteady characteristics of corona discharge with spherical geometry for gas motion in a radial direction are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 153–160, January–February, 1986.The authors wish to express their gratitude to V. A. Likhter and V. I. Shul'gin for their useful discussions and valuable observations.     

Vortex motion on a sphere: barrier with two gaps

Techniques based on the conformal mapping and the numerical method of contour dynamics are presented for computing the motion of a finite area patch of constant vorticity on a sphere in the presence of a thin barrier with two gaps. Finite area patch motion is compared with exact point vortex trajectories and good agreement is found between the point vortex trajectories and the centroid motion of finite area patches when the patch remains close to circular. Patch centroids are, in general, closely constrained to follow point vortex trajectories. However, Kelvin’s theorem constrains the circulation about the barrier to be a constant of the motion, thus, forcing a time-dependent volume flux through the gaps. More exotic motion is observed when the through-gap flow forces the vortex patch close to an edge of a barrier, resulting in the vortex splitting with only part of the patch passing through the gap. As the gap width is decreased this effect becomes more dramatic.     

Surface activity and bubble motion

This paper reviews recent progress in the theories of the surface boundary conditions of adsorbed solutes in liquids, and of the effects of those solutes on the steady motion of a bubble or drop in the liquid. Both singular perturbation theory and numerical solutions have useful roles in this problem, and their relationship is explored. In addition, analytical solutions are given to two problems concerning a spherical bubble rising steadily at low Reynolds number in a viscous fluid. One of these is displacement of the internal vortex centre from its position in the absence of surface activity when there is a small stagnant cap of surfactant at the rear. The results agree with experimental data in the direction of that displacement but give only about half its amount. The other problem is the velocity perturbation all round the surface caused by a very dilute solution of a weak surfactant at high Péclet number. This compares quite well with the numerical solution for a Péclet number of 60, having relative errors of order (60)^–1/2 as would be expected.     

Problem of gas motion in tubes of variable cross section and its application to solar wind perturbations

A qualitative study of self-similar flows with consideration of possible initial gas motion is performed for the problem of motion of a variable density gas in shock tubes with varying cross section. Asymptotic formulas are obtained near the contact discontinuity, and a numerical solution is obtained for the case of finite density on the contact surface. The equations are applied to the problem of gas motion in solar flares, and the parameters of the disturbed magnetic field are calculated. The results of the computation are compared with data from shock-wave observation in outer space.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 84–89, May–June, 1976.The author thanks V. P. Korobeinikov for his valuable advice and interest in the study.     

Asymptotic model of the development of separations inside a boundary layer under the action of a traveling pressure wave

A model of the nonlinear interaction between a pressure perturbation traveling at a constant velocity and an incompressible boundary layer is constructed when its near-wall part is described by the “inviscid boundary layer” equations. A steady-state solution is managed to obtain in the finite form under the assumption that it exists in a moving coordinate system. It is shown that the boundary layer can easily overcome pressure perturbations whose amplitude is not higher than the dynamic pressure calculated from the velocity of the pressure perturbation. At the higher pressure perturbation amplitudes a vortex sheet sheds from the body surface to the boundary layer. The vortex sheet represents an unstable surface of the tangential discontinuity which separates the regions of the direct and reverse separation flows. In the case of an arbitrary shape of the pressure perturbation the surface of the tangential discontinuity sheds from the body surface at a finite angle with the formation of a stagnation point. An example of the pressure perturbation in which the vortex sheet sheds from the body surface along the tangent is constructed.     

Three-dimensional analog of the Sokhotsky-Plemelj formulas and its application in the wing theory

A solution of a hydrodynamic problem of motion of an ideal incompressible fluid in a finite-thickness vortex layer is obtained. In the limiting case (infinitely thin layer), this layer transforms to a vortex surface. Formulas are derived for limiting values of the velocity vector of the fluid approaching this surface; these formulas extend the Sokhotsky-Plemelj formulas for a singular integral of the Cauchy type to a three-dimensional space. Three integral equations are derived on the basis of these formulas and the proposed method of modeling a finite-thickness wing by a closed vortex surface. It is shown that only one equation is left in the case of an infinitely thin wing, which corresponds to the condition of fluid non-penetration through the wing surface.     

Evolution of Two Vortices Near a Solid Surface

A numerical solution is found for the problem of evolution of two oppositely rotating vortices in a viscous incompressible fluid near a solid surface. The mechanism of vortex dissipation is determined. The trajectory of vortex motion is constructed.     

Free-surface deformation due to spiral flow owing to a source/sink and a vortex in Stokes flow

The free-surface shape and cusp formation are analyzed by considering a viscous flow arising from the superposition of a source/sink and vortex below the free surface where the strength of the source and vortex are arbitrary. In the analysis, Stokes’ approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex function theory. From the solution, shapes of the free surface are obtained, and the formation of a cusp on the free surface is discussed. Above some critical capillary number with a sink, the free-surface shape becomes singular and an apparent cusp should form on the free surface below a real fluid. On the other hand, no cusp would occur for sources of zero or positive strength. Typical streamline patterns are also shown for some capillary numbers. As the capillary number vanishes, the solution is reduced to a linearized potential flow solution.     

Simulation of free shear flow by the continuous vortex sheet method

A method is developed for simulating the motion and deformation of a tangential discontinuity with stability characteristics similar to the real ones. As distinct from the discrete vortex method, which forms the basis of the method proposed, the motion of a continuous vortex sheet of finite thickness is considered. The equations of motion are derived on the basis of an analysis of the physical reasons for the stability of this sheet with respect to small-scale perturbations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 42–50, January–February, 1999.     

Motion of a spherical particle in a fluid forced vortex by DQM and DTM

In this study,coupled equations of the motion of a particle in a fluid forced vortex were investigated using the differential transformation method(DTM) with the Pade approximation and the differential quadrature method(DQM).The significant contribution of the work is the introduction of two new,fast and efficient solutions for a spherical particle in a forced vortex that are improvements over the previous numerical results in the literature.These methods represent approximations with a high degree of accuracy and minimal computational effort for studying the particle motion in a fluid forced vortex.In addition,the velocity profiles(angular and radial) and the position trajectory of a particle in a fluid forced vortex are described in the current study.