Flow past a sphere of two co-axial supersonic streams

We investigate the flow past a sphere of a parallel supersonic stream which is nonuniform in magnitude; such a flow can be considered as two co-axial streams of an ideal gas. The problem is solved numerically by the method of establishment [1]. Supersonic flow of nonuniform magnitude and direction past blunt bodies and a plane wall was investigated in [2–5],Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 89–94, September–October, 1970.The author wishes to thank G. F. Telenin for his attention to the paper.     

Impact of a figure of revolution in a stream of incompressible fluid with separation of a jet

The impact interaction of bodies with a fluid in a flow with jet separation has been considered in [1–3], for example. This investigation was in the two-dimensional formulation. The present paper considers the three-dimensional problem of impact of a figure of revolution in a stream of an ideal incompressible fluid with separation of a jet in accordance with Kirchhoff's scheme. A boundary-value problem is formulated for the impact flow potential and solved by the Green's function method. A method for constructing the Green's function is described. Expressions are given for the coefficients of the apparent masses. The results are given of computer calculations of these coefficients in the case of a cone using the flow geometry of the corresponding two-dimensional problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 176–180, November–December, 1980.     

Blowing into a supersonic stream through a permeable surface

Distributed blowing of gas into a supersonic stream from flat surfaces using an inviscid flow model was studied in [1–9]. A characteristic feature of flows of this type is the influence of the conditions specified on the trailing edge of the body on the complete upstream flow field [3–5]. This occurs because the pressure gradient that arises on the flat surface is induced by a blowing layer whose thickness in turn depends on the pressure distribution on the surface. The assumption of a thin blowing layer makes it possible to ignore the transverse pressure gradient in the layer and describe the flow of the blown gas by the approximate thin-layer equations [1–5]. In addition, at moderate Mach numbers of the exterior stream the flow in the blowing layer can be assumed to be incompressible [3]. In [7, 8] a solution was found to the problem of strong blowing of gas into a supersonic stream from the surface of a flat plate when the blowing velocity is constant along the length of the plate. In the present paper, a different blowing law is considered, in accordance with which the flow rate of the blown gas depends on the difference between the pressures on the surface over which the flow occurs and in the reservoir from which the gas is supplied. As in [8, 9], the solution is obtained analytically in the form of universal formulas applicable for any pressure specified on the trailing edge of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 108–114, September–October, 1980.I thank V. A. Levin for suggesting the problem and assistance in the work.     

Wave motions in viscoelastic tubes. Forced oscillations

A characteristic of small blood and lymphatic vessels is the capacity of the wall to change its rheological properties and lumen by active contraction of the annular muscle cells contained in it [1–3]. A model of flow in the vessels taking this feature into account has been proposed in [4, 5], where a linear stability analysis is also given. A consequence of wall activity is the existence of auto-oscillatory flow conditions [6–8], which have also been discovered in the numerical solutions of the corresponding problems [9, 10]. Up to the present time flows have only been studied under steady conditions at the ends of the vessel and in the environment. The wall of an actual blood vessel is subject to various actions, frequently of a periodic nature: pressure pulsations at entry and rhythmically changing external forces applied from the surrounding tissues. Data exist on the sensitivity of vessels to transient actions [11–13], in particular on the relationship of their hydraulic resistance to frequency and amplitude of the action. There has been frequent discussion of the hypothesis that bv contraction of muscles in its walls or by external compression the vessel can act as a valveless pump [14, 15]. Within the framework of the quasione-dimensional approximation given below [4] the movement of liquid along a viscoelastic tube in the presence of small amplitude periodic external actions has been studied. A general solution of the problem has been constructed and concrete examples are given illustrating the features of forced wave motions in a tube having passive and active properties.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 4, pp. 94–99, July–August, 1984.     

The boundary layers of a fully ionized two-temperature plasma with given component temperatures at an electrode

The flow of a plasma with different component temperatures in the boundary layers at the electrodes of an MHD channel is investigated without any assumptions as to self-similarity. For the calculation of the electron temperature, the full energy equation for an electron gas [1] is solved with allowance for the estimates given in [2]. In contrast to [3, 4], the calculation includes the change in temperature of electrons and ions along the channel caused by the collective transport of energy, the work done by the partial pressure forces, and the Joule heating and the energy exchange between the components. The problem of the boundary layers in the flow of a two-temperature, partially ionized plasma past an electrode is solved in simplified form by the local similarity method in [5–7]. In these papers, either the Kerrebrock equation is used [5, 6] or the collective terms are omitted from the electron energy equation [7].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1972.The author thanks V. V. Gogosov and A. E. Yakubenko for interest in this work.     

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).     

Self-similar problems of convective diffusion in the presence of heterogeneous chemical reactions with allowance for thermal diffusion effects

A number of studies have been made of the problem of the effect of a temperature gradient on mass transfer in a forced viscous fluid flow. The question of allowing for thermal diffusion effects has been examined in connection with flow around bodies [1–4], duct flow [5], and jet flows [6,7]. Recently, in addition to the problem of thermal diffusion separation, the attention of investigators has been claimed by the problem of taking into account the effect of thermal diffusion on mass transfer in a convective flow in the presence of chemical reactions on the flow surfaces [4].     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

Flow past a plate under the free surface of a weightless liquid

A study is made of the nonlinear problem of the flow without separation of a perfect weightless liquid past a plate near the free surface. This problem was first posed by Gurevich [1]. At present, there are only a general solution to the problem [2–4] and some numerical calculations [5], which have been made under definite restrictions and are inadequate for detailed information about the interaction between the free surface and the plate. In the present paper, a complete investigation of the problem is given. Convenient computational formulas are obtained together with asymptotic expansions of them, and detailed calculations are made for all depths of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–162, January–February, 1980.     

Computation of base flow characteristics with uniform blowing

Blowing at bluff body base was considered under different conditions and for small amount of blowing this problem was solved using dividing streamline model [1]. The effect of supersonic blowing on the flow characteristics of the external supersonic stream was studied in [2–4]. The procedure and results of the solution to the problem of subsonic blowing of a homogeneous fluid at the base of a body in supersonic flow are discussed in this paper. Analysis of experimental results (see, e.g., [5]) shows that within a certain range of blowing rate the pressure distribution along the viscous region differs very little from the pressure in the free stream ahead of the base section. In this range the flow in the blown subsonic jet and in the mixing zones can be described approximately by slender channel flow. This approximation is used in the computation of nozzle flows with smooth wall inclination [6, 7]. On the other hand, boundary layer equations are used to compute separated stationary flows with developed recirculation regions [8] in order to describe the flow at the throat of the wake. The presence of blowing has significant effect on the flow structure in the base region. An increasing blowing rate reduces the size of the recirculation region [9] and increases base pressure. This leads to a widening of the flow region at the throat, usually described by boundary-layer approximations. At a certain blowing rate the recirculation region completely disappears which makes it possible to use boundary-layer equations to describe the flow in the entire viscous region in the immediate neighborhood of the base section.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 76–81, January–February, 1984.     

Filtration of a three-phase liquid (Buckley-Leverett model)

Buckley and Leverett [1] formulated the problem of the displacement of immiscible liquids in a porous medium and obtained a very simple one-dimensional solution for a two-phase flow. Different generalizations of it are known [2]. In [3, 4], a method of characteristics is proposed for numerical solution of the problem of three-phase flow. Articles [5, 6] consider the problem of the injection (at a given pressure) of two incompressible liquids into a porous stratum previously saturated with a third, elastic liquid. The authors started from the assumption of the existence, for this problem, of zones of three-, two-, and single-phase flow, separated by unknown mobility gradients. The present work investigates the solution for a three-phase flow, analogous to the Buckley-Leverett solution for two phases. It is shown that the character of the degrees of saturation depends essentially on the initial saturation of the porous stratum and on the phase composition of the mixture being injected.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 39–44, January–February, 1972.     

Drag on a body placed between two parallel plates in a hele-shaw flow

In [1], the drag was found that acts on a circular gas bubble between two parallel plates in a slow viscous flow. In the present paper, the problem considered in [1] is solved for a body of arbitrary shape under the assumption that the conditions of a Hele-Shaw flow are satisfied. An expression is obtained for the drag containing only one coefficient in the expansion of the complex potential in a Laurent series.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–162, September–October, 1979.     

Viscoplastic flow between two approaching or parallel circular plates

The problem of the flow of a viscoplastic medium between two parallel circular plates in translatory coaxial relative motion is solved. The Bingham model [1] of a viscoplastic medium is assumed. The problem is solved in the inertialess thin layer approximation [2] for arbitrary values of the viscosity coefficient and yield stress.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 9–17, January–February, 1996.     

Stability of the motion of a rigid body in an unsteady flow

The problem of rigid-body motion in an unsteady gas flow is considered using a flow model [1] in which the motion of the body is described by a system of integrodifferential equations. The case in which among the characteristic exponents of the fundamental system of solutions of the linearized equations there are not only negative but also one zero exponent is analyzed. The instability conditions established with respect to the second-order terms on the right sides of the equations are noted. The problem may be regarded as a generalization of the problem of the lateral instability of an airplane in the critical case solved by Chetaev [2], pp. 407–408.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 18–22, May–June, 1989.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Unsteady aerodynamic interaction of two annular blade rows of thin lightly loaded blades rotating one relative to the other in a subsonic flow

The problem of subsonic unsteady ideal-gas flow over two annular blade rows of thin lightly loaded blades rotating one relative to the other is solved within the framework of linear small perturbation theory. As in the case of the interaction of two-dimensional cascades [1], the problem reduces to an infinite system of singular integral equations for the harmonic components of the oscillations in the distribution of the unknown aerodynamic load on one blade of each row. The system of integral equations for a finite number of harmonics is solved numerically by the collocation method. The kernels of the integral equations are regularized on the basis of the method proposed in [2].Translated from Izvestiya Akademii Nauk.SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 165–174, May–June, 1991.The author is grateful to A. A. Osipov and K. K. Butenko for their considerable assistance in the preparation of this paper.     

Boundary-layer separation on a cooled body and interaction with a hypersonic flow

In previous papers, e.g., [1, 2], boundary-layer separation was investigated by analyzing solutions of the boundary-layer equations with a given external pressure distribution. In general, this kind of solution cannot be continued after the separation point. Study of the asymptotic behavior of solutions of the Navier-Stokes equations [3–5] shows that, in boundarylayer separation in supersonic flow over a smooth surface, the main effect on the flow in the immediate vicinity of the separation point is a large local pressure gradient induced by interaction with the external flow. The solution can be continued beyond the separation point and linked to the solutions in the other regions, located downstream [5]. Similar results for incompressible flow were recently obtained in [6]. We can assume that in general there is always a small region near the separation point in which separation is self-induced, and where the limiting solution of the Navier-Stokes equations does not contain unattainable singular points. However, this limiting slope picture can be more complex and can contain more regions where the behavior of the functions differed from that found in [3–6]. The present paper investigates separation on a body moving at hypersonic speed, where the ratio of the stagnation temperature to the body temperature is large. It is shown that both. for hypersonic and supersonic speeds the flow near the separation point is appreciably affected by the distribution of parameters over the entire unperturbed boundary layer, and not only in a narrow layer near the body, as was true in the flows studied earlier [3–6]. Regions may appear with appreciable transverse pressure drops within the zone occupied by layers of the unperturbed boundary layer. Similarity parameters are given, the boundary problems are formulated, and the results of computer calculation are presented. The concept of subcritical and supercritical boundary layers is refined, and the dependence of pressure coefficients responsible for separation on the temperature factor is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–109, November–December 1973.     

Supersonic nonuniform viscous gas flow past a blunt body with supply of gas from the surface

The problem of axisymmetric nonuniform gas flow past smooth blunt bodies at high Mach numbers is investigated. The approach stream is a parallel axisymmetric flow in which the velocity and temperature depend on the radial distance from the axis of symmetry and the pressure is constant. On the axis of symmetry the velocity has a minimum and the temperature a maximum. A characteristic feature of this flow is the existence of two qualitatively different flow regimes: separated [1-4], when in the shock layer on the front of the body there is a closed region of reverse-circulating flow, and unseparated [5, 6], when there is no such zone. In this study the case of unseparated flow is investigated. The equations of a thin viscous shock layer with generalized Rankine-Hugoniot conditions at the shock and boundary conditions on the body that take into account the supply of gas from the surface are solved numerically. The effect of the gas supply on the conditions of unseparated flow is analyzed in relation to the Reynolds number, and the critical values of the nonuniformity parameter a = a_k [5] are obtained. It is shown that at high Reynolds numbers the supply of gas from the surface has practically no effect on a_k, while at low and intermediate Reynolds numbers it reduces the region of unseparated flow. For high Reynolds numbers and an intense supply of gas from the surface an asymptotic solution of the problem is obtained for the neighborhood of the stagnation point. This is compared with the numerical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–129, July–August, 1988.The authors wish to thank G. A. Tirskii for useful discussions of the results.     

Rayleigh stability of shear flow in relaxing media

The stability of steady-state flow is considered in a medium with a nonlocal coupling between pressure and density. The equations for perturbations in such a medium are derived in the linear approximation. The results of numerical integration are given for shear motion. The stability of parallel layered flow in an inviscid homogeneous fluid has been studied for a hundred years. The mathematics for investigating an inviscid instability has been developed, and it has been given a physical interpretation. The first important results in flow stability of an incompressible fluid were obtained in the papers of Helmholtz, Rayleigh, and Kelvin [1] in the last century. Heisenberg [2] worked on this problem in the 1920's, and a series of interesting papers by Tollmien [3] appeared subsequently. Apparently one of the first problems in the stability of a compressible fluid was solved by Landau [4]. The first investigations on the boundary-layer stability of an ideal gas were carried out by Lees and Lin [5], and Dunn and Lin [6]. Mention should be made of a series of papers which have appeared quite recently [7–9]. In all the papers mentioned flow stability is investigated in the framework of classical single-phase hydrodynamics. Meanwhile, in recent years, the processes by which perturbations propagate in media with relaxation have been intensively studied [10–12].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 87–93, May–June, 1976.     

Laminar mixing of homogeneous currents in the presence of a pressure gradient

The problem of the mixture of a current with a quiescent gas was solved by Chapman [1]. In this study, the results of certain calculations on the laminar mixing zone of homogeneous gas currents with a pressure gradient will be presented. On the basis of the data calculated and evaluations, it is shown that the concept of similarity, formulated by Less [2], is applicable to the problem of mixture with a pressure gradient. In variable similarities, the velocity profiles for gradient flow practically coincide with the profiles of nongradient flow for the same parameter values at the interior and exterior boundaries of the mixture zone. Moreover, it proves to be the case that the excess velocity profile depends weakly on the specific parameters of the problem.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 67–71, March–April, 1972.