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.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 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.     

Plane problem of wave motions occurring in a continuously stratified fluid during the flow around a submerged body

The plane stationary problem of wave motions occuring in the flow of a uniform inviscid incompressible gravity fluid with an arbitrary continuous (stable) change in density with depth around submerged sources and sinks of equal intensity is investigated in a linear formulation. An analysis of the structure of the wave motion in a flow with an arbitrary density change is performed in [1].Paper presented at the First Soviet-American Symposium on Internal Waves in the Ocean, Novosibirsk, December 3–8, 1976.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 148–152, July–August, 1978.     

Steady two-phase flow in a vertical tube subject to combined free and forced convection

Using the two-velocity, two-temperature model of a continuous medium, the viscousgravitational flow of a mixture of incompressible liquid and solid particles in a vertical round tube is considered. The free-convection equations are written down on the basis of the general equation of motion and the energy equation of a two-phase medium [1, 2]. Using a finite Hankel integral transformation, a solution is constructed for the case of a linear wall-temperature distribution along the tube. The results of some practical calculations of the velocity and temperature fields over the cross section of the tube are presented, together with the dimensionless heat-transfer coefficient expressed as a function of the Rayleigh number and phase concentration. Here it is assumed that the dynamic and thermal-interaction coefficients between the phases correspond to the Stokes mode of flow for each particle, as a result of which the velocity and thermal phase lag is very small [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 132–136, July–August, 1975.     

Arbitrary body,close to a wedge,in a supersonic flow

The problem of supersonic flow around bodies close to a wedge was first discussed in the two-dimensional case in [1]. The shock wave was assumed to be attached, and the flow behind it to be supersonic; taking this into account, the angle of the wedge was assumed to be arbitrary. The surface of the body was also arbitrary, provided that it was close to the surface of the wedge. In solution of the three-dimensional problem, there was first considered flow around two supporting surfaces with only slightly different angles of attack [2], and then around a delta wing [3, 4]. In all these articles, the Lighthill method was used to solve the Hilbert boundary-value problem [5, 6]. A whole class of surfaces of bodies with arbitrary edges, under the assumption that the surface of the body was cylindrical, with generatrices directed along the flow lines of the unperturbed flow behind an oblique shock wave, was discussed in [7]. In the present work, the problem is regarded for a broad class of surfaces of bodies, using a new method which generalizes the results of [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1974.The author thanks G. G. Chernyi for his direction of the work.     

Similarity solutions of the equations of a boundary layer in a nonuniform external flow

Similarity solutions of the equations of a laminar incompressible boundary layer, formed in a rotational external flow, are investigated. Such problems arise in the analysis of the flow in a boundary layer when there is an abrupt change in the boundary conditions (for example, in the case of a discrete inflation of the boundary layer, in hypersonic flow about blunt bodies, etc.). Various approaches to their solution have been proposed earlier in [1–4]. Solved below is the so-called inverse problem of boundary layer theory (see [3], p. 200), where the contour of the body that causes a given flow outside the boundary layer is unknown beforehand and is found during the course of solution of the problem in connection with the coupling of the longitudinal and transverse velocity components. The cases of a parabolic (u_e ~ y²) and a linear (u_e=a(x)+b(x)y) variation in the velocity of the external flow with distance along the transverse direction are considered in detail. The latter includes an investigation of the flow in the neighborhood of the critical point of a blunt body, taking account of the vorticity of the flow in the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 78–83, March–April, 1971.     

On the optimal distribution of the flow rate of gas blown through the side walls of the channel of a de plasmotron (plasma generator)

It is well known that the blowing of cold gas through the side walls of the channel of a dc plasmotron (plasma generator) with longitudinal blowing over the arc leads to an increase in the useful power of the plasmotron [1]. The increase is due to the increase in the combustion voltage of the arc and also the decrease in the heat fluxes to the wall of the channel. The present paper solves the problem of the optimal distribution of the flow rate of gas blown through the side walls into the channel of a dc plasmotron of arbitrary shape F(x). The flow in the main channel and in the ducts in the side walls is described by the quasi-one-dimensional gas-dynamic equations investigated qualitatively in [2] and verified experimentally in [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 120–124, May–June, 1981.     

Supersonic unsteady-state flow around bodies of variable form

This paper discusses supersonic flow around a spherical blunt body, whose surface is deformed according to a known law over a certain period of time. An investigation is made of unsteady-state flows during the change in the form of the body as a function of the deflection and the rate of change in the form of the body. The calculations were made for Mach number M =2, 10. The solution is obtained by the grid-characteristic method [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 187–191, November–December, 1978.     

Investigation of the influence of blowing on the flow in a hypersonic viscous shock layer near the stagnation streamline on a blunt body

In the framework of the approximation of local similarity to the Navier-Stokes equations, an investigation is made of the axisymmetric flow of homogeneous gas in a hypersonic shock layer, this including the region of transition through the shock wave. Boundary conditions, which take into account blowing of gas, are specified on the surface of the body and in the undisturbed flow. A numerical solution to the problem is obtained in a wide range of variation of the Reynolds number and the blowing parameter. Expressions are found for the dependences on the blowing parameter usually employed in boundary layer theory of the coefficients of friction and heat transfer on the surface of the body, which are divided by their values obtained for blowing parameter equal to zero. It is shown that these dependences are universal and the same as the dependences obtained from the solution of the equations of a hypersonic viscous shock layer with modified Rankin-Hugoniot relations across the shock wave and from the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 199–202, January–February, 1980.     

Body shape with minimum total radiant energy flux toward its surface

During a space vehicle's entry into a planet's atmosphere at hypersonic speed one of the important problems is the aerodynamical surface heating due to convective and radiant heat fluxes from the gas after passing through a strong shock wave. Due to the high destructive action of this heating, an important problem is the selection of the aerodynamic shape allowing the minimum heat influx to its surface. The problem of determining the shapes of an axisymmetric body from the condition of minimum total convective heat flux along the lateral face of the body was considered under various assumptions in [1–7]. There are a number of entry conditions (for example, into the earth's atmosphere with a speed of 11 km/ sec at an altitude of about 60 km [12]) during which the radiative component becomes dominant in the total heat flux toward the body. A numerical solution of the problem of hypersonic flow of a nonviscous, non-heat-conducting radiating gas around a body is obtained at this time only for a limited class of bodies and primarily for certain entry conditions (for example, [8–12]). On the basis of these calculations it is impossible to make general conclusions concerning arbitrary body shapes. Therefore, approximate methods were proposed which permit the distribution of radiant heat flux to be obtained for an arbitrary axisymmetric body in a wide range of flight conditions [13–15]. In the present work an expression is derived for the total radiant heat flux over the entire body surface and similarity criteria are found. A variational problem is formulated to determine the shape of an axisymmetric body from the condition of minimum total radiant-heat flux over the entire body surface. It is solved analytically for the class of thin bodies and in the case of a strongly radiating gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 84–89, July–August, 1976.     

Flow past a plate lattice with developed cavitation

Problems of streamline cavitation flow past a lattice were studied in [1–8] using the Kirchhoff scheme. In this scheme the magnitude of the velocity at the free surface is equal to the stream velocity behind the lattice, and the cavitation number is zero (for a lattice the relative velocity and the cavitation number are defined from the stream velocity behind the lattice). In [4, 7] a solution is given of the problem of flow past a lattice using a scheme with an Efros-Gilbargreturn streamline, which permits considering arbitrary cavitation numbers; however, a unique solution is not given. Some other streamline schemes are mentioned in [8].In the following we consider the cavitational flow of an ideal incompressible inviscid and weightless fluid past an infinite lattice of flat plates, using the streamline wake model previously utilized by Wu [9] in studying cavitational flow past an isolated obstacle. In accordance with this model, the streamlines which separate from the body and bound the cavity behind it pass into two curvilinear infinitely long walls, along which the pressure increases and approaches the pressure in the undisturbed stream.It is further assumed that in the hodograph plane there corresponds to the curvilinear walls a cut along some line and that the complex potential takes the same values at points lying on opposite sides of the cut. In particular, at the points of contact of the streamlines with the curvilinear walls the complex potential is the same. In the Wu scheme the latter condition leads to vanishing of the velocity circulation along the contour CABC₁ (Fig. 1).In conclusion the author wishes to thank N. V. Yurtaeva for the accurately performed numerical work.     

The spherical problem of the filtration of a liquid and a gas with a change in the permeability of rocks in accordance with an exponential law

The article considers the problem of the filtration of liquids (or gases), pumped through a borehole at a constant rate with elastic filtration conditions. The permeability of the stratum is assumed to be an exponential function of the coordinates. The viscosities of the injected and displaced liquids are assumed to be different. To increase the capacity of strata, i.e., of collectors used for the burial of industrial waste flows and gases, various methods are employed to increase the fracturing and the permeability of the rocks (hydro-pulse techniques, explosions, and other methods). As a result of this, a spherical region is formed in the rocks, in which the permeability varies along the radius. The character of this change is well described by an exponential function. The pumping of waste flows or industrial gases into such a cavity leads to the displacement of the stratum liquid (or gas). The problem of the displacement of one liquid by another liquid not miscible with it under rigid filtration conditions was first discussed in [1–5]. Here a study was made of a region of finite dimensions, bounded by two boundaries, with given pressures or mass flow rates (the linear and axisymmetric flow problems). The permeability of the stratum was assumed to be independent of the coordinates. A special characteristic of these problems is the fact that it is impossible to consider unbounded or semi-bounded filtration conditions in them since, under rigid filtration conditions, the condition of bounded character of the pressure (the head) is not satisfied at infinity. Elastic filtration conditions for two immiscible liquids were first discussed in [6], and later in [7, 8] and other reports. Here an investigation was made of the linear and axisymmetric problems for an unbounded region. In [9, 10] solutions are given to some problems with spherical symmetry for an unbounded region, with rigid filtration conditions and a jumpwise change of the permeability along the radius. In the problems of [6–10] the condition of the bounded character of the pressure is satisfied. In [11] the case of a hyperbolic change in the permeability of the rocks is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 42–51, November–December, 1974.     

Some effects of strong blowing of gas into a supersonic flow

The intense evaporation of bodies moving in the atmospheres of planets at high supersonic velocities has been partly simulated both theoretically [1–5] (numerical calculations of strong blowing in the framework of the Navier-Stokes equations were also made at the Scientific-Research Institute of Mechanics at the Moscow State University by É. A. Gershbein and A. F. Kolesnikov [6]) as well as experimentally [7–9]. Below, the results are given of investigations of strong blowing of gas from the flat end of a cylinder into a supersonic flow at Reynolds numbers such that the mixing layer separating the blown and the oncoming gas is fairly thin. In this case, the mixing layer can be regarded as a contact surface, so that the problem of blowing can be solved in the framework of Euler's equations. The results of a numerical solution are compared with the results of experiments on the separation and profile of the shock wave, the thickness of the blowing layer on the axis, and also on the pressure distribution on the end of the cylinder. It was established experimentally, and then confirmed numerically that there is a downwash of the blown gas on the periphery of a porous end. It is shown that for the same blowing parameter K, which is equal to the ratio of the dynamic head of the blown gas to the dynamic head of the oncoming gas, and for a given distribution of K over the surface of the body the contact surface tends to a certain limiting position with increasing Mach number of the oncoming flow, i.e., the profile of the contact surface is stabilized. The influence of the adiabatic exponent on the thickness of the blowing layer is estimated. The present investigations continue earlier experimental studies, the main results of which have been presented in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–98, January–February, 1980.     

Optimal shapes of bodies with attached shock waves

We consider the problem of finding the shape of two-dimensional and axisymmetric bodies having minimal wave drag in a supersonic perfect gas flow. The solution is sought among bodies having attached shock waves. The limitations on the body contour are arbitrary: these constraints may be body dimensions, volume, area, etc. Such problems with arbitrary isoperimetric conditions may be solved by the method suggested in [1, 2]. This method involves the use of the exact equations of gasdynamics which describe the flow as additional constraints. This method was developed further in [3–6] in the solution of several problems.The author wishes to thank V. M. Borisov, A. N. Kraiko and Yu. D. Shmyglevskii for their interest in this study.     

Nonisothermal Couette flow of a non-Newtonian fluid under the action of a pressure gradient

Nonisothermal Couette flow has been studied in a number of papers [1–11] for various laws of the temperature dependence of viscosity. In [1] the viscosity of the medium was assumed constant; in [2–5] a hyperbolic law of variation of viscosity with temperature was used; in [6–8] the Reynolds relation was assumed; in [9] the investigation was performed for an arbitrary temperature dependence of viscosity. Flows of media with an exponential temperature dependence of viscosity are characterized by large temperature gradients in the flow. This permits the treatment of the temperature variation in the flow of the fluid as a hydrodynamic thermal explosion [8, 10, 11]. The conditions of the formulation of the problem of the articles mentioned were limited by the possibility of obtaining an analytic solution. In the present article we consider nonisothermal Couette flows of a non-Newtonian fluid under the action of a pressure gradient along the plates. The equations for this case do not have an analytic solution. Methods developed in [12–14] for the qualitative study of differential equations in three-dimensional phase spaces were used in the analysis. The calculations were performed by computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 26–30, May–June, 1981.     

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.     

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.     

Heat exchange of sphere with blowing for slow flows

An analytic solution is given for the problem of convective heat- and mass-exchange of a sphere with transverse flow of matter along the surface for values of Peclet numbers smaller than one and for blowing velocity smaller than that of the incoming gas flow. The solution for velocity field obtained by the authors in a previously published publication is employed on flow past a sphere with blowing; the method of asymptotic expansions of Acrivos and Taylor is also used. Expressions to the second approximation are determined for temperature field and for the values of local and averaged Nusselt numbers. It is shown that blowing reduces the temperature gradient or the concentrations at the surface.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 89–94, July–August, 1972.     

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.