Hydrodynamic moment acting on a moving deformable body in a linear potential flow of an ideal incompressible liquid

The solution of the problem of a solid body motion in a parallel flow is discussed in [1]. To investigate the motion of a body with a deformable surface in a potential flow, one can use the following approximate method. By the external flow outside the body we shall understand the difference between the resulting flow, and the flow with a potential regular outside the body due to the presence of the body. For an infinite problem the notions of external and unperturbed flow (the flow in the absence of the body) coincide. We investigate the class of external flows for which the Taylor series of the velocity field converges in some sphere containing the body. The flows described by the partial sums S₁, S₂,... of this series approximate with increasing accuracy the external flow in the sphere. The exact solution S_n of the body motion in a flow without boundaries is an approximate solution of the original problem (not necessarily without boundaries). The flow S₁ is parallel. We shall call the flow S₂ linear because it is a linear function of the radius vector. Evidently, S₂ is the simplest nonparallel flow S_n for n 2. Apparently, [2] is the first work that investigated the linear flow in three dimensions. Yakimov [3] obtained an expression for the force acting on a deformable body in the linear flow. Votsnov et al. [4] also solved the problem of hydrodynamic action on the solid body, in the linear-flow approximation, but in the expression for the moment he did not obtain all terms that appear in the exact solution. In the present work we shall obtain an expression for the moment acting on an arbitrary (deformable) body in a linear potential flow. The result is expressed in terms of the local characteristics of the external flow, and in terms of the shape of the body. We investigate bodies whose surface has planes of symmetry, and revolution bodies. We compare our results with the results for parallel flow and with the results of [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 88–93, September–October, 1977.The author thanks Yu. L. Yakimov for supervision, and G. Yu. Stepanov for a number of comments.     

Vibrations of a body in a bounded volume of viscous fluid

The problem of the vibrations of a body in a bounded volume of viscous fluid has been studied on a number of occasions [1–4]. The main attention has been devoted to determining the hydrodynamic characteristics of elements in the form of rods. Analytic solution of the problem is possible only in the simplest cases [2]. In the present paper, in which large Reynolds numbers are considered, the asymptotic method of Vishik and Lyusternik [5] and Chernous' ko [6] is used to consider the general problem of translational vibrations of an axisymmetric body in an axisymmetric volume of fluid. Equations of motion of the body and expressions for the coefficients due to the viscosity of the fluid are obtained. It is shown that in the first approximation these coefficients differ only by a constant factor and are completely determined if the solution to the problem for an ideal fluid is known. Examples are given of the determination of the “viscous” added mass and the damping coefficient for some bodies and cavities. In the case of an ideal fluid, general estimates are obtained for the added mass and also for the influence of nonlinearity. Ritz's method is used to solve the problem of longitudinal vibrations of an ellipsoid of revolution in a circular cylinder. The hydrodynamic coefficients have been determined numerically on a computer. The theoretical results agree well with the results of experimental investigations.     

Asymptotic laws of degeneracy of the shape of thin cavities

Differential equations are obtained in the approximation of a slender body for the shape of thin nonstationary axisymmetric cavities. In contrast to the equations obtained earlier by Grigoryan [1] and the author [2], the equations derived in the present paper contain the well-known asymptotic behavior for a flow of Kirchhoff type in the stationary case. For the case of compressible fluid, the dependence found for the asymptotic behavior of the cavity on the Mach number differs from the result obtained by Gurevich [3]. When compressibility is taken into account, the cavity area grows less rapidly. It is shown that the solutions of the obtained differential equations satisfy the well-known separation conditions [3, 4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–10, May–June, 1981.I thank L. I. Sedov, S. S. Grigoryan, V. P. Karlikov, V. A. Eroshin and others who participated in a discussion of the work, and also L. A. Barmina for assisting in the preparation of the paper.     

Linear theory of unsteady gas flow under the influence of nonpotential external forces

We determine in the linear formulation the velocity and pressure fields excited in a compressible medium by a lifting filament that displaces and deforms arbitrarily. For general unsteady motion of such a filament we give explicit formulas that express the velocity at a given point in terms of the intensity of the free vortices entering the audio signal audibility zone constructed for this point. We examine gas flow caused by an arbitrary external body force field.Studies devoted to the determination of gas velocity fields for flow past slender bodies relate primarily to translational motion of a body with a dominant constant velocity [1–3]. Gas velocities for helical motion of a rectilinear lifting filament within the gas have been examined in [4].     

Explosion of a body in flight

Little attention has been paid up to now in the theory of explosions to such an important and interesting problem as the explosion of a body in flight. A formulation of the problem presented by such an explosion in connection with the problem of simulating the explosion of a meteorite body flying at cosmic velocity is given in [1]. In this case the kinetic energy of the translational motion may be comparable to or even in excess of the internal energy of the explosive transformation, which will lead to a significant distortion of the flow pattern compared with the usual explosion process. An analysis of the effect of the initial velocity of particles on the course of an explosion in idealized formulations in the framework of one-dimensional flows with plane, cylindrical, and spherical waves was first made in [2–4]. The asymptotic flow properties were found in these papers. It is shown that if the internal energy E_o and the kinetic energy K_o are separated (the specification of the latter for a fixed mass is equivalent to specification of the initial momentum), some intermediate self-similar regimes corresponding to a short pulse [2] or to flows with a sink [3, 4] are observed, these becoming the solution for a strong explosion at long times [1]. The time of transition from one qualitative regime to another depends on the ratio K_o/E_o. In the present paper the next step in the investigation of the question is taken. An axisymmetric, basically realistic flow model is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 126–129, September–October, 1984.     

Calculating the flows of a viscoplastic medium in tubes using the local variation method

This article considers the problem of the motion of a visco-plastic medium in tubes and channels. The results of [1, 2] are used, which present the variational formulation and the qualitative analysis of this problem. The method of local variations suggested in [3] is used for the numerical solution of the variational problem. A more detailed presentation of the algorithm of this method in application to boundary-value and variational problems is given in [4]. Results of calculations of certain concrete problems on an electronic computer are presented.The author wishes to express his sincere gratitude to F. L. Chernous'ko for the problem formulation and helpful counsel, and G. I. Barenblatt and S. S. Grigoryan for useful discussions.     

Cavitation flow around a plate in a channel by a stream of heavy liquid

The nonlinear problem of cavitation flow around a plate by a stream of heavy liquid is investigated in precise formulation; the plate is located on the horizontal floor of a channel when the gravity vector is directed perpendicular to the wall of the channel. Two flow systems are considered-Ryabushinskii's and Kuznetsov's system [1]. This problem was investigated in linear formulation in [2], Similar problems were considered earlier in [3–7] for unrestricted flow. Below, on the basis of a method proposed by Birkhoff [8, 9], all the principal hydrodynamic and geometric characteristics are calculated for the problem being considered.Translated from Ivestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1973.     

Computations of hydrodynamic coefficients, displacement-amplitude ratios and forces for a floating cylinder due to wave diffraction and radiation

Computations of the hydrodynamic coefficients, displacement-amplitude ratios and loadings on floating vertical circular cylinder due to diffraction and radiation are presented here. The boundary value problem (BVP) is solved in terms of diffraction potential and three potentials due to radiation, two translational motions about x-axis (surge) and about z-axis (heave), one rotational motion about y-axis (pitch). The analytical expressions for the hydrodynamic coefficients, displacement-amplitude ratios and loadings for this case were obtained previously by Bhatta and Rahman [1]. In this paper, we present the computational aspects of those analytical results for different depth to radius and draft to radius ratios. JMSL (Java Mathematical and Statistical Library) is used to compute special functions and solve complex matrix equations.     

Horizontal impact of a sphere half immersed in a liquid of finite depth

We consider the problem of bottom influence during horizontal hydrodynamic impact of a spherical solid body of diameter 2a which is half submerged in a liquid layer of finite depth. The sphere is subjected to the action of a shock pulse, as a result of which it acquires an initial translational velocity u directed along the x-axis.The influence of the spherical bottom on the sphere impact phenomenon was first studied by Zhukovskii [1]. Vertical impact in a layer of finite thickness was considered in [2]; horizontal impact in a halfspace was examined in [3].     

Asymptotic law of expansion of a cavity in the presence of hydrodynamic singularities in the flow

The asymptotic law of expansion of an axisymmetric half-body of finite resistance exposed to a flow of an ideal incompressible fluid and its relation with the force acting on the half-body were obtained in [1]. It is obvious that the deformation of the nose of the body will not change the form of the expressions found. It will be shown below how the asymptotic law and magnitude of hydrodynamic reactions change for half-bodies of finite resistance in the presence of sources (sinks) in the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 153–155, May–June, 1977.The author thanks G. Yu. Stepanov for attention to the work.     

Angular vibrations of an ellipsoid of revolution in a confined viscous fluid

Vibrations of bodies in confined viscous fluids have been studied on a number of occasions, transverse vibrations of rods being the main subject of investigation [1–3]. The present authors [4] have considered the general problem of translational vibrations of an axisymmetric body in an axisymmetric region containing a low-viscosity fluid. The present paper follows the same approach and considers the problem of small angular vibrations of an ellipsoid of revolution in a circular cylinder with flat ends. In the general case, the hydrodynamic coefficients in the equation of motion of the ellipsoid are determined numerically for different values of the dimensionless geometrical parameters using Ritz's method. In the case of an unconfined fluid, analytic dependences in terms of elementary functions are obtained for the hydrodynamic coefficients. The theoretical results agree well with experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 34–39, March–April, 1984.     

Development of magnetohydrodynamic boundary layers

Many authors have studied the problem of the development of a hydrodynamic boundary layer when a body is suddenly set in motion. The results obtained are presented most fully in the monographs of H. Schlichting [1] and L. G. Loitsyanskii [2]. In magnetohydrodynamics the development of the boundary layer over the surface of an infinite flat plate for uniform oncoming flow has been closely studied (for example [3, 4]). Below, the problem of the development of a plane magnetohydrodynamic boundary layer is considered in a different formulation. We shall suppose that the distributions of velocity U(x) and enthalpy h(x) are given along the body contour for t=0. At that moment the viscosity and thermal conductivity mechanisms are instantaneously switched on. Viscous and thermal boundary layers begin to grow in a direction normal to the body. The medium in the boundary layer interacts with the magnetic field. This formulation corresponds to the development of a magnetohydrodynamic boundary layer on a body which is set in motion with a jerk, in the case where the rate of establishment of magnetohydrodynamic flow of the inviscid, thermally nonconducting fluid around the body is much less than the rate of development of the boundary layer. Then U(x) and h(x) are found by solving the problem of stationary magnetohydrodynamic flow of an inviscid thermally nonconducting fluid around a body, or simply the hydrodynamic flow if the medium interacts with the field only in the boundary layer.     

Wave flow of a liquid film together with a flow of gas

The wave flow of a thin layer of viscous liquid in conjunction with a flow of gas was considered in a linear formulation earlier [1, 2]. In this paper the problem of the wave flow of a liquid film together with a gas flow is solved in a nonlinear setting. On this basis relationships are derived for calculating the parameters of the film and the hydrodynamic quantities.Ivanovo. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 12–18, January–February, 1972.     

Stability of quasistatic equilibrium state of an inhomogeneous viscous layer on an inclined plane

A study is made of the stability against small perturbations [1] of a slow flow of an incompressible inhomogeneous linearly viscous liquid under the influence of a force of gravity on an unbounded inclined plane. Problems of such kind arise in glaciology when one estimates the stability of snow on mountain slopes or determines the catastrophic movement of a glacier; the results can also be applied to solifluction phenomena [2, 3]. Equations for perturbations of parallel flows of linearly viscous fluids in the case of a continuous variation of the viscosity and density across the flow were derived in [4]. An attempt to solve the hydrodynamic problem with allowance for a perturbation of the viscosity was made in [5]; however, in the equations for the perturbations, simplifications resulted in the neglect of terms that take into account perturbations of the viscosity. In the quasistatic formulation considered here in the case when allowance is made for perturbation of the density and viscosity, the equation for the perturbation amplitudes is an ordinary differential equation with variable coefficients; analytic solution of the equation is very difficult, even for long-wave perturbations. In this connection a study is made of the stability of a laminar model; the viscosity and density are constant within each layer. A similar hydrodynamic problem in the long-wave approximation was considered in [6]. In the present paper an exact solution is constructed in the quasistatic formulation for a two-layer model; the solution shows that the viscosity of the lower layer has an important influence on the stability. Essentially, instability is observed when the lower layer acts as a lubricant.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 20–24, November–December, 1973.     

Determining the form of a body resulting in minimum heat flux,with laminar flow in the boundary layer

The exact solution of the problem of determining the optimal body shape for which the total thermal flux will be minimal for high supersonic flow about the body involves both computational and theoretical difficulties. Therefore, at the present time wide use is made of the inverse method, based on comparing the thermal fluxes for bodies of various specified form [1, 2]. The results of such calculations cannot always replace the solution of the direct variational problem. Therefore it is advisable to consider the direct variational problem of determining the form of a body with minimal thermal flux by using the approximate Newton formula for finding the gasdynamic parameters at the edge of the boundary layer. This approach has been used in finding the form of the body of minimal drag in an ideal fluid [3–5] arid with account for friction [6], and also for determining the form of a thin two-dimensional profile with minimal thermal flux for given aerodynamic characteristics [7].     

Mass transfer between particles and a flow in the presence of a volume chemical reaction

The exterior problem of the mass transfer between a spherical drop and a linear shear flow in the presence of a first-order volume reaction is solved in the diffusion boundary layer approximation. A simple approximate expression for calculating the average Sherwood number for a drop or solid particle of arbitrary shape is proposed. At large Péclet numbers this expression is applicable to any type of flow over the entire range of variation of the reaction rate constant. The problem of diffusion to a spherical drop in a translational Stokesian flow in the presence of a first-order volume reaction was investigated in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 109–113, November–December, 1987.     

Shear flow over a porous particle

Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytic solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flow as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered, for example, in [1,2]. A solution of the problem of translational Stokes flow over a porous spherical particle was given in [3]. Linear shear-strain Stokes flow over a spherical drop was investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1995.     

Unsteady weakly perturbed motion of a body of revolution in a liquid with jet separation

The unsteady weakly perturbed motion of a body in a liquid with jet separation has been investigated on various occasions in the twodimensional formulation [1–3]. The present paper gives a generalization of the formulation of this two-dimensional problem to the threedimensional case of flow past a body of revolution in accordance with Kirchhoff's scheme. A method is proposed for solving the obtained boundary-value problem using a Green's function. This function is constructed in a special system of curvilinear coordinates. To obtain an effective solution, a Laplace transformation is used. Expressions are given for the Laplace transforms of the vectors of the force and torque acting on the body in the unsteady motion.