Conservation of the resultant force vector in the plane of the angle of attack for slender star-shaped bodies in supersonic flows

At high supersonic flight speeds bodies with a star-shaped transverse and power-law longitudinal contour are optimal from the standpoint of wave drag [1–3]. In most of the subsequent experimental [4–6] and theoretical [6–9] studies only conical star-shaped bodies have been considered. For these bodies in certain flow regimes ascent of the Ferri point has been noted [10]. In [11] the boundary-value problem for elongated star-shaped bodies with a power-law longitudinal contour was solved for the case of supersonic flow. The present paper deals with the flow past these bodies at an angle of attack. It is found that for arbitrary star-shaped bodies with any longitudinal (in particular, conical) profile the aerodynamic forces can be reduced to a wave drag and a lift force, the lateral force on these bodies being equal to zero for any position of the transverse contour.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 135–141, November–December, 1989.     

Calculation of aerodynamic characteristics of arrays of profiles of arbitrary shape

It is well known that the problem on nonseparating potential flow of an incompressible fluid about an array of profiles reduces to an integral equation for a certain real function, determined on the contours of the profiles of the array. As such a function one can take, as was done, for instance, in [1–5], the relative velocity of the fluid on the profiles of the array. For arrays of profiles of arbitrary shape it is necessary to solve the corresponding integral equation numerically. In the particular examples of the calculation of aerodynamic arrays that are available [1–3] the numerical methods used were based on the approximate evaluation of contour integrals by rectangle formulas. As investigations showed, sizeable errors arose thereby in the approximate solution obtained, these being especially significant in the case of curved profiles of relatively small bulk. In the present paper a method for the numerical solution of the integral equation obtained in [5] is proposed. The method is based on the replacement of a profile of the array with an inscribed N polygon, the length of whose sides is of the order N^–1 and whose internal angles are close to . Convergence with increasing N of the numerical solution to an exact solution of the integral equations at the reference points is demonstrated. Examples of the calculation are given.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 105–112, March–April, 1972.     

Shock wave in a gas-liquid medium

The propagation of weak shock waves and the conditions for their existence in a gas-liquid medium are studied in [1]. The article [2] is devoted to an examination of powerful shock waves in liquids containing gas bubbles. The possibility of the existence in such a medium of a shock wave having an oscillatory pressure profile at the front is demonstrated in [3] based on the general results of nonlinear wave dynamics. It is shown in [4, 5] that a shock wave in a gas-liquid mixture actually has a profile having an oscillating pressure. The drawback of [3–5] is the necessity of postulating the existence of the shock waves. This is connected with the absence of a direct calculation of the dissipative effects in the fundamental equations. The present article is devoted to the theoretical and experimental study of the structure of a shock wave in a gas-liquid medium. It is shown, within the framework of a homogeneous biphasic model, that the structure of the shock wave can be studied on the basis of the Burgers-Korteweg-de Vries equation. The results of piezoelectric measurements of the pressure profile along the shock wave front agree qualitatively with the theoretical representations of the structure of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 65–69, May–June, 1973.     

Interaction between the boundary layer and a supersonic flow when part of the wall is rapidly heated

There have been many theoretical studies of aspects of the unsteady interaction of an exterior inviscid flow with a boundary layer [1–9]. The mathematical flow models obtained in these studies by the method of matched asymptotic expansions describe a wide range of phenomena observed experimentally. These include boundary layer separation near the hinge of a flap, the flow in the neighborhood of the trailing edge of an oscillating airfoil [1–2], and the development and propagation of perturbations in a boundary layer excited by an oscillating wall or some other way [3–5]. The present paper studies the interaction of an unsteady boundary layer with a supersonic flow when a small part of the surface of a body in the flow is rapidly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–70, January–February, 1984.     

Plane incompressible fluid flow past an array of arbitrary profiles vibrating with arbitrary phase shift

We consider the problem of the vibration of an array of arbitrary profiles with arbitrary phase shift. Account is taken of the influence of the vortex wakes. The vibration amplitude is assumed to be small. The problem reduces to a system of two integral Fredholm equations of the second kind, which are solved on a digital computer. An example calculation is made for an array of arbitrary form.A large number of studies have considered unsteady flow past an array of profiles. Most authors either solve the problem for thin and slightly curved profiles or they consider the flow past arrays of thin curvilinear profiles [1].In [2] a study is made of the flow past an array of profiles of arbitrary form oscillating with arbitrary phase shift in the quasi-stationary formulation. The results are reduced to numerical values. Other approaches to the solution of the problem of unsteady flow past an array of profiles of finite thickness are presented in [3–5] (the absence of numerical calculations in [3, 4] makes it impossible to evaluate the effectiveness of these methods, while in [5] the calculation is made for a symmetric profile in the quasi-stationary formulation).     

Radiation of waves by a sphere harmonically oscillating to-and-fro in a viscous medium

The theory of the radiation of sound by a sphere in an ideal medium is presented in detail in [1–3]. The emission of waves by a sphere oscillating to-and-fro in a viscous incompressible liquid is analyzed in [4, 5]. The present paper gives a precise solution to the problem of the radiation of sound by a sphere oscillating in a viscous medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 101–106, September–October, 1970.     

Stability of nonplane-parallel three-dimensional jet flows

The problem of the stability of nonplane-parallel flows is one of the most difficult and least studied problems in the theory of hydrodynamic stability [1]. In contrast to the Heisenberg approximation [1], the basic state whose stability is investigated depends on several variables, and the stability problem reduces to the solution of an eigenvalue problem for partial differential equations in which the coefficients depend on several variables [2–7]. In the case of a periodic dependence of these coefficients on the time [2] or the spatial coordinates [3, 4], the analog of Floquet theory for the partial differential equations is constructed. With rare exceptions, the case of a nonperiodic dependence has usually been considered under the assumption of weak nonplane-parallelism, i.e., a fairly small deviation from the plane-parallel case has been assumed and the corresponding asymptotic expansions in the linear [6] and nonlinear [7] stability analyses considered. The present paper considers the case of an arbitrary dependence of the velocity profile of the basic flow on two spatial variables. The deviation from the plane-parallel case is not assumed to be small, and the corresponding eigenvalue problem for the partial differential equations is solved by means of the direct methods of [5], which were introduced for the first time and justified in the theory of hydrodynamic stability by Petrov [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 21–28, May–June, 1987.     

Hydrodynamic Load Associated with the Oscillations of a Cylinder at the Interface in a Two-Layer Fluid of Finite Depth

The results of numerically solving the linear problem of the small steady-state oscillations of a horizontal cylinder located at the interface between two fluids of different densities are presented. The hybrid element method is used. In this method the velocity potentials are represented by means of the finite element method in a narrow zone in the neighborhood of the body and by means of the boundary integral equations in the outer domain. The Green’s functions for an oscillating source in a two-layer fluid bounded from above by a free boundary and from below by an even horizontal bottom are derived. Numerical calculations of the apparent mass and damping coefficients are carried out for an elliptic cylinder beneath a free surface and for a cylinder with the cross-section in the form of a Lewis rib contour which floats on the free surface.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 122–131Original Russian Text Copyright © 2005 by Sturova and Syui.     

The zonal linearization method for multidimensional mass transfer problems in porous media

A number of methods have been proposed in recent years for calculating the combined flows of immiscible and miscible liquids in strata to systems of boreholes. We propose a method which can naturally be called the zonal linearization method [1]. It is more compact than the usual finite-difference method and has high accuracy, in particular, in the neighborhood of a borehole, since it is closely similar to the method of characteristics. The method can be applied to both continuous and discontinuous flows and in principle makes it possible to investigate the formation and breakdown of discontinuities. As distinct from the method of characteristics, it is well suited to programming and implementation on a computer, and it also makes it possible to obtain an approximate analytic solution of the problem in many cases and to estimate the accuracy of the solution. The method is based on the zonal linearization of the equation for mass conservation in the total flow between chosen surfaces or contour lines (lines of equal saturation or concentration). Determination of the dynamics of the contour surfaces leads to a Cauchy problem for a system of integrodifferential equations involving partial derivatives. The zonal linearization method is a development of the scheme described in [2–4], and the method of solving the Cauchy problem is a generalization of the methods described in [4–13]. The essence of the method and its convergence are illustrated by means of two-dimensional problems in two-phase filtration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 66–80, July–August, 1973.     

Theory of turbulent mixing at the interface of fluids in a gravity field

The theory of turbulent mixing at the interface of two media in accelerated motion was constructed in [1], and an approximate solution was given for incompressible fluids. The time variation of kinetic energy was neglected in the equation of balance for the kinetic energy of the turbulent motion. In [2] the characteristic turbulent velocity is averaged over the mixing region. This allows the initial equations to be solved allowing for the time variation of kinetic energy. It turns out that the resulting density profile roughly coincides with the profile of [1] within a wide range of variation of the initial density differential. In the present paper the equations for the mixing of incompressible fluids are studied in their complete form. It is established that the solutions of [1, 2] are applicable within a limited region, valid for small density ratios. The resulting solution is analyzed qualitatively, and it is shown that the density gradient at the mixing front is discontinuous. The dependence of the solution on two empirical constants is investigated. An approximate choice of the values of these constants is made on the basis of the theoretical considerations of [2, 3], and by comparison with the solution of [1]. The mixing asymmetry is found numerically as a function of the initial density differential. Quantitative characteristics of the solution are illustrated in graphs.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 74–81, July–August, 1976.     

Region of applicability and the main features of the generalized Chapman-Enskog method

The present paper is made necessary by the publication of the foregoing paper in this issue by Kolesnichenko [1]. It considers the basic propositions of the generalized Chapman-Enskog method and analyzes the arguments put forward by Kolesnichenko [1] and the validity of the method. The position of the results obtained by Kolesnichenko [14–17] is indicated. Nonequilibrium flows of multiatomic gases in which there occur processes of exchange of internal energy of the molecules in collisions between them and chemical reactions (such processes are called inelastic) are encountered frequently in nature and technology. It is therefore naturally of interest to derive gas-dynamic equations for such flows. The methods of the kinetic theory of gases were first used to obtain equations describing the limiting cases of very fast inelastic processes that take place in times of the order of the molecule-molecule collision times (equilibrium case) and very slow inelastic processes that take place over times of the order of the characteristic flow time (relaxation case). In [2–5], an algorithm was proposed for deriving gas-dynamic equations valid for arbitrary ratios of the rates of the elastic and inelastic processes and reducing to the well-known equations for the limiting cases already mentioned. The algorithm is called the generalized Chapman-Enskog method (abbreviated to the generalized method). The development, modification, and analysis of its properties can be found in [4, 6–13]. In [1], Kolesnichenko has questioned the validity of this algorithm.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 126–136, May–June, 1984.We thank V. A. Rykov for helpful and constructive discussions of the work.     

Hydrodynamic methods in the inverse problem of gravitational prospecting

In [1, 2], a dynamical method is proposed for solving stationary inverse problems of potential theory, including the inverse problem of gravitational prospecting. It is based on analogy with the problem of establishing the interface of two immiscible fluids flowing in a porous medium. In the present paper, a system of two functional equations is derived from which one can obtain, as special cases, an equation corresponding to the method of [1, 2], and also a system of equations that enables one to propose a new and different method for solving the inverse problem of gravitational prospecting. Equations are derived in polar coordinates for plane Cauchy problems corresponding to both methods, and the results are also given of the solution of some model problems by these methods. Finally, ways of generating new methods of solution of the inverse problem of gravitational prospecting are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 63–71, July–August, 1980.     

Convection in an oscillating force field and microgravity

Convective flows in a plane layer of viscous fluid in the presence of an oscillating external force are investigated numerically [1 – 8]. The layer is assumed to be placed in a gravitational field. The cases in which the external field oscillations are generated by rotation about the horizontal axis or by vibration in the longitudinal direction are considered. The Navier-Stokes equations and the Boussinesq approximation are used for describing the fluid motion. The flows developing in the layer in the presence of a transverse temperature gradient are determined, the stability boundaries of these flows are found, and the supercritical motion regimes are studied. These investigations are carried out using the averaging method (in order to find the stability limits for high rotation velocities and vibration frequencies) and the Galerkin method.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–106, September–October, 1994.     

Design of a wing airfoil in a flow with a separated turbulent boundary layer

The problem of designing the contour of an airfoil in a viscous (incompressible and compressible) flow with a separated turbulent boundary layer from a pressure distribution given on the separationless part of the contour is solved using the boundary layer theory together with the separated flow model proposed in [1]. Numerical calculations are carried out to demonstrate the possibilities of the method.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 83–91, May–June, 1994.     

Cascaded thin lightly loaded airfoils vibrating in a subsonic separated flow

The problem of calculating the nonstationary aerodynamic characteristics of a cascade of thin lightly loaded airfoils in a subsonic flow with the formation of thin separation zones of finite extent is solved approximately. As in [1–5], an approach based on a linear small-perturbation analysis, within which the flow is assumed to be inviscid, is employed and the boundaries of the unsteady separation zones are simulated by oscillating lines of contact discontinuity. However, instead of the requirement of a given fixed pressure at the boundary of the separation zone, used in [1–5], this study proposes a more general condition according to which on each element of length of the thin separation layer the pressure oscillates with an amplitude proportional to the local value of the amplitude of its thickness oscillations. The problem is reduced to a system of two singular integral equations which can be solved numerically.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–191, January–February, 1995.     

Extremal nozzle contours for gas flows with particle lag

The determination of the extremal nozzle contour for gas flow without foreign particles has been carried out in several studies [1–6], based on the calculation of the flow field using the method of characteristics.In [7, 8] the equations are derived for the characteristics and the relations along the streamlines which are required for calculating two-dimensional gas flow with foreign particles. The variational problem for two-phase flow in the two-dimensional formulation may be solved by the method of Guderley and Armitage [9] with the use of equations given in [7] or [8]; however this method is very tedious, even with the use of high-speed computers.In [10, 11] studies are made of two-phase one-dimensional flows by expanding the unknown functions in series in a small parameter, defined by the particle dimensions. In [12] a solution is given for the variational problem (in the one-dimensional formulation) of designing the contour of a nozzle with maximal impulse. However that study does not take account of the static term appearing in the impulse and the solution is obtained in relative cumbersome form. Moreover, the question of account for the losses due to nonparallelism and nonuniformity of the discharge was not considered.The present paper considers in the one-dimensional formulation the flow of a two-phase medium in a Laval nozzle with small particle lags (in velocity and temperature). The variational problem of determining the maximal nozzle impulse is formulated along the nozzle contour for fixed geometric expansion ratio. The impulse losses due to nonparallelism of the discharge are simulated by a function which depends on the ordinates which are variable along the contour and on the slope of the tangent to the contour.The author wishes to thank Yu. D. Shmyglevskii and A. N. Kraiko for helpful discussions and V. K. Starkov for carrying out the calculations on the computer.     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.     

System of equations for describing dynamical problems associated with the mechanics of lung parenchyma

In [1], which is an expanded version of the paper [2], the equations of conservation of mass and momentum are shown to be valid for dynamical problems of lung parenchyma. This system of equations is now closed by means of the heat flux equation. As in [1, 2], the heat flux equation is obtained on the basis of the methods of the mechanics of heterogeneous media [3] and anatomical data on the structure of lung parenchyma.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 21–29, May–June, 1988.     

Water hammer and propagation of perturbations in elastic fluid-filled pipes

A Korteweg—de Vries (KV) approximation is constructed in this paper for the perturbations being propagated in elastic pipes filled with fluid. On the basis of the approximation constructed and the equation obtained for the perturbations of a finite-amplitude velocity, the water-hammer phenomenon is analyzed in the Zhukovskii formulation, and the water hammer in systems with preliminary longitudinal tension is considered separately. Special attention in the study of the perturbations is paid to the signal structure and evolution in the hydraulic line. Taking account of the inertial properties of the pipe in the approximation mentioned permitted the indication of new effects, in principle, which are essential for applied problems of the propagation of perturbations in elastic hydraulic lines. In particular, it is shown that the initial signal can be doubled in such lines by redistributing its intensity over the frequencies. It is established that the origination of an oscillating forerunner is possible in hydraulic lines with preliminary tension. Starting with [1], the water-hammer phenomenon was investigated in many papers, in [2], for example. The main attention in these papers was paid to the propagation velocity of the water hammer and its intensity. After simplification, the initial system of Zhukovskii equations contains no mechanism hindering the twisting of the wave profile, and, therefore, there is no possibility of stationary shock formation within the framework of this theory. Moreover, the Zhukovskii theory of the water hammer and of propagation of perturbations in elastic pipes results in the conclusion that the wave structure, velocity, and amplitude depend essentially on the characteristics of the initial perturbation and can differ significantly from the water hammer predicted by theories for powerful signals in sufficiently long pipes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–8, July–August, 1976.