Internal waves arising in the presence of the unsteady motion of a source in a continuously stratified fluid

A general approach to the investigation of linear internal and surface waves in a stably stratified fluid, arising from different types of perturbations, is presented in [1]. The methods of calculation of the internal waves generated by the motion of a mass source are developed for particular cases of steady horizontal motion in a continuously stratified fluid in [2] and arbitrary motion in an unbounded exponentially stratified fluid in [3]. Internal waves generated by other types of perturbations have also been investigated but only for particular cases of motion (see, for example, [4]). This paper presents a method for the calculation of unsteady, linear, internal gravity waves arising in an inviscid incompressible fluid with continuous stable stratification.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–130, July–August, 1985.     

Internal waves generated in an exponentially stratified fluid by an arbitrarily moving source

A theoretical investigation is made into the development of linear internal waves in an exponentially stratified flow of an ideal incompressible fluid in the Boussinesq approximation. The waves are generated by an arbitrarily moving point mass source. The obtained solution is used to investigate three special cases of motion: uniform motion at an angle to the horizontal, nonstationary motion during a finite interval of time, and uniform motion in a circular path. The method of solution of this problem is similar to that used by Wolfe and Lewis [1], who studied the generation of acoustic waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–74, May–June, 1980.I thank V. V. Sazonov for assistance in the calculations.     

Structure of internal waves in a channel

One method of describing wave motion in a fluid with continuous stratification is to use normal waves (modes). The propagation of internal gravity waves in closed rectangular regions whose boundaries coincide with planes through which there is no normal motion is essentially different from wave motion in an unbounded medium [1, 2]. This paper describes a theoretical and experimental investigation of the propagation of internal waves in an exponentially stratified fluid in a horizontal channel of finite height.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–110, January–February, 1935.In conclusion the authors wish to express their gratitude to A. T. Onufriev for his interest in their work.     

Waves in an inhomogeneous fluid due to periodic pressures in the presence of a thin plate

The problem of motion of a two-layer fluid of finite depth subjected to pressure applied to its surface in the presence of a semiinfinite plate is considered in a linear formulation. Expressions governing the form of the originating surface and internal waves are obtained. Results are presented of a numerical computation of the elevation of the free surface and the interface caused by the application of the pressure.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–96, September–October, 1976.     

Internal wave structure in a three-layer fluid with a stratified middle layer

The behavior of internal waves in a vertically bounded channel differs considerably from the wave motion in an infinite stratified fluid. In [1] the phase structure of the internal waves in an exponentially stratified layer of fluid between rigid horizontal planes was experimentally and theoretically investigated. A characteristic feature of such a channel is the boundedness of the phase and group velocities of each mode. Below, the case of an exponentially stratified channel between layers of homogeneous unbounded fluid is considered.In conclusion, the authors wish to thank A. T. Onufriev for his interest in their work.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–132, May–June, 1987.     

Long waves in two superposed layers of magnetic fluid

The equations and boundary conditions describing plane-parallel potential motions of two superposed layers of stably stratified magnetic fluid are formulated. The fluid is assumed to fill entirely a horizontal plane channel in the presence of a uniform longitudinal magnetic field induced by external sources. With reference to the case of long waves propagating over the interface between the upper and lower layers, it is shown that the action of the field may be interpreted as the result of an increase in the nondimensional surface tension by an amount proportional to the square of the undisturbed field. In the linear formulation the effect of the field on the evolution of a long-wave perturbation of the initially plane interface is investigated. Korteweg-de Vries equations with quadratic and cubic nonlinearities are derived and the action of the field on the internal solitary waves is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 126–133, May–June, 1993.     

Secondary flows in a layer with a free surface

The loss of stability of a plane-parallel incompressible viscous heat-conducting fluid flow in a horizontal layer subject to a longitudinal temperature gradient is considered. The lower surface of the layer is assumed to be rigid, while the upper one is free with a surface tension coefficient depending linearly on temperature. Both boundaries are assumed to be thermally-insulated. The critical value of the temperature gradient as a function of other relevant parameters is determined by analyzing the spectrum of the linearized problem. Secondary flows arising after the onset of instability are determined from an analysis of the full nonlinear problem using the expansion of the solution in a power series in terms of a supercritical state parameter in the vicinity of the bifurcation point. Three types of secondary flows are studied: plane two-dimensional waves propagating along the temperature gradient; plane waves travelling at a certain angle to the gradient; and three-dimensional waves propagating along the gradient. A numerical method of problem solution, based on the polynomial approximation, is described.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 85–98, September–October, 1994.     

Effect of internal linear waves on the hydrodynamic characteristics of a vortex source

The results of an investigation to estimate the effect of surface and internal waves on the hydrodynamic characteristics are presented for the problem of the uniform motion of a vortex source in a three-layer fluid. The behavior of the lift force and wave drag is studied in the neighborhood of the critical Froude number. Some results of the numerical experiments are presented. An analogous investigation is also carried out for the motion in a two-layer fluid beneath a rigid top and in the presence of a bottom.Omsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 146–153, September–October, 1996.     

Propagation of a soliton along a fluid-filled pipe

In [1, 2] a mathematical model of the motion of a fluid in a pipe whose axis is a curve in space was discussed and certain simplifications of the problem were studied. The propagation of linear and nonlinear waves in the framework of the model was studied. In the present paper we consider a simple wave flow in a pipe with elastic walls suing one of the models introduced in [1], which, unlike [2], takes into account axial displacements of the pipe. The basic equations describing the propagation of waves in the pipe are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 3, pp. 58–63, May–June, 1986.     

Stokes waves on a cavity surface in a rotating fluid

The axisymmetric flow of an inviscid incompressible fluid rotating about a cavity with constant pressure is considered. Due to the centrifugal force, on the cavity surface waves may exist, in particular, waves with a break in the wave base where the cavity meridional sections form the angle 2/3, i.e. Stokes waves. A method of finding these waves from the boundary-value problem for the fluid velocity potential is described. For an infinite cavity, the dependence of the wave parameters on the cavitation number, calculated using the pressure in the cavity, is given.St. Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 105–110, November–December, 1996.     

Phase structure of three-dimensional internal waves in a channel

The results of a theoretical and experimental investigation of the phase structure of three-dimensional internal waves in a horizontal channel occupied by an exponentially stratified fluid are presented. The stationary internal waves excited by a source moving uniformly along the bottom of a channel with rigid covers are considered. A dispersion relation, on the basis of which the phase structure of the internal waves in the far zone is investigated, is obtained within the context of the linear theory. The dependence of the boundary of the region of wave disturbances on the internal Froude number F is found. The equation of the lines of constant phase is obtained in the first approximation for large F. By means of an IAB-451 shadowgraph instrument, using the dark field method with vertical illumination, the phase patterns of the internal waves are obtained. The angles of the wave disturbance zone are measured in the region of relatively large F. Similarity of the phase patterns for the first three-dimensional mode is obtained experimentally for large F.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–135, January–February, 1989.The authors wish to thank A. T. Onufriev for his interest in their work.     

Viscous flow in a partially-filled horizontal cylinder rotating at constant speed

The problem under consideration is that of the stationary shape of the free surface of a viscous fluid in a steadily rotating horizontal cylinder. In the majority of investigations of this problem the thickness of the fluid layer coating the inner surface of the cylinder is assumed to be small [1–3]. The case of a near-horizontal free surface, with the bulk of the fluid at the cylinder bottom, was considered in [4], where, after considerable simplification, the governing equations were reduced to ordinary differential equations. In the present study the behavior of the free surface is investigated using a creeping flow approximation. The controlling parameters vary over a wide range. In the numerical computations a boundary element method was used. The numerical results have been confirmed experimentally.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–30, May–June, 1993.     

Nonstationary waves in the continuously stratified flow of a fluid of finite depth

A theoretical investigation is made of the development of linear two-dimensional waves in a continuously stratified flow of an ideal incompressible fluid. The waves are generated by pressures that are independent of time and that are applied at time t=0 to a bounded region on the free surface of an initially undisturbed flow. The stationary internal waves generated by such a disturbance have been investigated in [1–3]. The nonstationary waves in a continuously stratified fluid that are generated by initial disturbances or periodic surface pressures applied to the entire free surface have been studied in [4–7] in the absence of a slow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 87–93, November–December, 1976.     

Effect of continuous variation of the density of a liquid on the waves generated by moving surface pressures

This study investigates the plane linear problem of steady-state internal waves in an ideal incompressible liquid with nonuniform density. The waves are generated by surface pressures applied in a bounded region which moves at constant velocity. It is assumed that the density in the unperturbed state varies continuously with depth, remaining constant in the upper and lower layers and varying according to an exponential law in the middle layer. The problem may be regarded, in particular, as a hydrodynamic model for the study of internal waves produced by a cyclone moving over the surface of the ocean. Analogous investigations for a homogeneous liquid were carried out in [1–3]; internal waves for a liquid with the above-mentioned law of density variation but with stationary pressure changes which are periodic with respect to time were studied in [4]. Problems analogous to the one considered here, both for exponential variation of density in the entire layer and for the case of a nonuniform layer near the surface, were investigated in [5, 6]. An analysis of non-linear waves of the steady-state type with arbitrary distribution of vorticity and density with respect to depth was carried out in [7, 8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 55–62, November–December, 1973.     

Wave flow of a liquid film in a horizontal separator

The calculation of the motion of separated moisture in a linear horizontal separator is made on the basis of the analysis of the development of the waves in a flow of a thin layer of liquid along a vertical surface without allowance for the transverse flow of mass [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–176, March–April, 1985.     

Asymptotic analysis of the unsteady problem of waves in a two-layer flow

The problem investigated is the unsteady problem of the internal waves generated in a two-layer flow by a certain periodic perturbation which leads to small deviations from the basic flow. A method of constructing an approximate solution uniformly valid throughout the region of variation of the variables and the parameters of the problem is indicated. It is confirmed that for large times and near-resonance parameters the motion of the fluid is described by the mixed problem for a cubic Schrödinger equation. Certain qualitative properties of the solution of this nonlinear problem are noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 82–90, November–December, 1987.The author is grateful to V. I. Bukreev and to I. V. Sturova for their interest in his work.     

Motion of two viscous fluids in a precessing vessel

An approximate solution of the problem of the motion of two viscous fluids in a cylindrical vessel executing slow regular precession with an arbitrary angle of nutation is constructed. The axial component of the moment of the forces exerted by the fluid on the lateral surface of the vessel is determined.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 27–34, September–October, 1995.     

Free vibrations of two capillary liquids rotating in a cylindrical vessel

The problem of the small natural vibrations of two coaxially disposed ideal liquids rotating in a cylindrical vessel under conditions of complete weightlessness is considered. The set of normal vibrations of the system consists of internal wave motions and surface waves. Asymptotic formulas are derived for the vibration frequencies of the surface waves. The results of computer calculations are presented in the form of graphs and tables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 97–104, September–October, 1976.     

Impact of a circular disk and washer on a fluid in a cylindrical vessel

The solution of the problem of the axisymmetric motion of an ideal incompressible fluid in a cylindrical vessel of finite depth is obtained for small vibrations of a flexible circular disk and washer (disk with centered hole) on the surface of the fluid. On the basis of this solution the virtual mass is determined as a function of the dimensions of the vessel, disk and washer for the special case of a rigid nondeformable disk and washer.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 103–111, January–February, 1995.     

Unsteady motion of viscous fluid in an expanding tube

The problem of unsteady motion of a viscous compressible fluid in a semiinfinite tube with horizontal axis is solved by successive approximation. The circular cross section of the tube depends exponentially on the coordinate measured along the tube axis. At the end of the tube there is a unit such as a sliding valve, compressor, reciprocating pump, or turbine that changes the flow rate. The process is assumed to be barotropic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 79–82, May–June, 1983.