Normal interaction of shock waves in the presence of vibrational relaxation

The effect of nonequilibrium physicochemical processes on the flow resulting from the normal collision and reflection of shock waves is studied by the example of nonequilibrium excitation of molecular oscillations in nitrogen. It is shown that the thermal effect of vibrational relaxation is small and the problem can be linearized around a known solution [1]. A similar approach to the solution of the problem of flow around a wedge and certain one-dimensional non-steady-state problems was used earlier in [2–4]. The solution of these problems was constructed in an angular domain, bounded by the shock wave and a solid wall (or the contact surface) and was reduced to a well-known functional equation [6]. The solution of this problem, because of the presence of two angular domains divided by a tangential discontinuity, reduces to a functional equation of more general form than in [6]. The results are obtained in finite form. In the special case of shocks of equal intensity, the normal reflection parameters are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 90–96, July–August, 1976.     

Theory of detachment of a laminar boundary layer

In this paper, we investigate the structure of the flow that arises in the case of uniformly distributed injection from a plane permeable surface, under conditions that ensure a transition from a flow in a boundary layer to a detached flow including a mixing layer and a region of inviscid wall flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 16–20, November–December, 1987.The author thanks V. Ya. Neiland for his discussion of the results and valuable advice.     

Effect of accuracy of solving integral equations for the thermal and viscous Chapman functions on the value of gas-kinetic coefficients

The dependence of the gas-kinetic coefficients on the accuracy of calculating the thermal and viscous Chapman functions for the case of a simple gas in the neighborhood of a plane rigid surface is studied. Expressions for the gas-kinetic coefficients are obtained by solving the Boltzmann equation using the Loyalka method. In order to find the temperature jump we use boundary conditions which take into account the accommodation both on the energy and the momentum. The effect of the accuracy of solving the integral equations for the thermal and viscous functions on the value of the temperature jump and the thermal and isothermal slip coefficients was studied by taking into account one, two or three terms in expansions of these functions in Sonine polynomials. The dependence of the results on the choice of the molecule interaction potential model is analyzed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 190–198, March–April, 1995.     

Averaged equations and wave jumps describing the propagation of stokes waves with slowly varying parameters

The equations describing the stationary envelope of periodic waves on the surface of a liquid of constant or variable depth are investigated. Methods previously used for investigating the propagation of solitons [1–5] are extended to the case of periodic waves. The equations considered are derived from the cubic Schrödinger equation assuming slow variation of the wave parameters. In using these equations it is sometimes necessary to introduce wave jumps. By analogy with the soliton case a wave jump theory in accordance with which the jumps are interpreted as three-wave resonant interactions is considered. The problems of Mach reflection from a vertical wall and the decay of an arbitrary wave jump are solved. In order to provide a basis for the theory solutions describing the interaction of two waves over a horizontal bottom are investigated. The averaging method [6] is used to derive systems of equations describing the propagation of one or two interacting wave's on the surface of a liquid of constant or variable depth. These systems have steady-state solutions and can be written in divergence form.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for useful discussions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1989.     

Vortex flows of a viscous heat-conducting gas in a slightly divergent tube with volume energy supply

The results of a numerical investigation of viscous vortex flow in a slightly divergent tube with thermal energy supplied to the flow are presented. The initial stage of vortex flow development is considered for two different longitudinal velocity distributions simulating the velocity profiles in jet-like and wake-like vortex flows in the vicinity of the vortex axis. The first type of flow can be considered as a model for the near-axis region of the vortex formed in the flow around a delta wing at incidence. The second type can serve as a model for the near-axis region of the trailing vortex downstream of a high-aspect-ratio wing. The development of the two flows is studied for a constant area tube, a slightly divergent tube, and in the case of thermal energy supply from a volume energy source at a constant wall temperature.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 90–97, September–October, 1996.     

Sound absorption near the boundary dividing two liquids

It is well known that sound absorption in finite media is caused mainly by fluid viscosity and thermal conductivity. Kirchhoff [1] developed a general theory describing the mechanism of such absorption and applied it to the particular case of sound propagating in tubes. Rayleigh [2] used Kirchhoff's theory to study sound absorption by a porous wall with normal incidence of the sound wave. Konstantinov [3] also used Kirchhoff's theory to solve the problem of sound absorption by a rigid, isothermal (with infinite thermal conductivity) and a thermally insulating plane wall with arbitrary angle of sound-wave incidence. A natural extension of these efforts is a study of sound absorption on the boundary dividing two liquids. Aside from its scientific interest, such a problem is of practical significance, for example, in hydroacoustics or in creating methods for visualization of sound in gases and liquids [4]. The present study will attempt to solve this problem. The results can be applied to both liquid and solid (resinlike) materials.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 6–9, January–February, 1984.The author thanks T. P. Zhizhina for much assistance in the study.     

Laminar boundary layer on the moving surface of a rankine oval

A numerical investigation has been made of the laminar boundary layer that arises on the moving surface of a cylindrical body (Rankine oval with relative elongation 4) that moves with constant velocity in an incompressible fluid. The distributions of the frictional stress on the surface of the cylinder for different velocities of the wall motion are found. Numerical integration was employed to determine the work needed to overcome the frictional drag, the pressure, and also the work expended on the motion of the moving surface of the body in the case of constant velocity. In the presence of a separation region the drag forces are calculated under the assumption that in the separation region the pressure and the frictional stress on the wall are constant and equal to the corresponding values at the singular point of the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 3, pp. 171–174, May–June, 1984.I thank G. G. Chernyi for constant interest in the work and discussing the results.     

Upper bound limit analysis of active earth pressure with different fracture surface and nonlinear yield criterion

Conventional calculations of static and seismic active earth pressures of soils on a retaining wall are formulated assuming the soils obeying a linear Mohr–Coulomb yield criterion. However, experimental evidences show that the strength envelopes of almost all geomaterials are nonlinear in nature over a wide range of normal stresses. In this paper, the strength envelope of the backfill behind a retaining wall is considered to follow a nonlinear yield criterion. A simple method is proposed for calculating the static and seismic active earth pressures acting against a retaining wall using a nonlinear yield criterion. This method is based on the upper bound theorem of limit analysis. Both translational and rotational fracture surfaces are employed in the formulation for calculating active earth pressures. Quasi-static representation of earthquake effects using a seismic coefficient concept is adopted for seismic active earth pressure calculations. Instead of using directly the actual nonlinear yield criterion, a linear Mohr–Coulomb yield criterion, which is tangential to the nonlinear yield criterion, is used to formulate the active earth pressure problem as a classical nonlinear programming problem. A nonlinear sequential quadratic programming algorithm is used to search for the maximum solution. In order to assess the validity of the proposed method, values of active earth pressures for different values of seismic coefficients and nonlinear parameters in the yield criterion are calculated and compared with solutions obtained using an extended Rankine’s active earth pressure theory. For the case of static active earth pressure, the upper bound solutions using the present method with a translational fracture surface are equal to the extended Rankine’s theoretical solutions and are slightly smaller than those obtained using the present method with a rotational fracture surface. For the case of seismic active earth pressure, numerical results obtained using the present method with a rotational fracture surface is very close to the extended Rankine’s theoretical solutions. A study is conducted to investigate the effects of the parameters in the nonlinear yield criterion on the active earth pressures.     

Stability of laminar boundary layer on an elastic surface in an incompressible fluid

Interest in the present problem arose after the publication of the results of the experiments of Kramer [1–3]. In addition to the studies indicated in [4], the articles [5–8] are devoted to the question of the interaction of a flexible elastic surface with the boundary layer. In the present paper the problem of the interaction of an elastic surface with disturbances arising in the boundary layer is posed as in [4]. The approximate nature of the methods of solving the problem of the hydrodynamic stability of the laminar boundary layer leads to a difference in the final computational formulas even in the case when authors use the same Heisenberg-Tollmien-Schlichting-Lin scheme. Therefore, in what follows we present a comparison of the data on the stability of the boundary layer on a solid wall obtained by several authors with the calculations using the formulas, which are then generalized to the case of the elastic surface.The author wishes to thank G. I. Petrov and V. A. Medvedev for discussions of the present study.     

Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.     

Turbulent boundary layers on porous surfaces in the case of blowing at different angles to the wall

The results are given of an experimental investigation of turbulent boundary layers on porous plates in the case of homogeneous blowing at various angles to the wall. It is shown that the intensity and angle of the blowing influence the profiles of the averaged and pulsating components of the velocities, the characteristic thicknesses, the surface friction, and the turbulent tangential stresses in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 59–64, May–June, 1982.We thank V. I. Kondrat'ev and V. S. Senterov for participating in the work.     

Calculation of the boundary layer of a transparent gas on a radiating surface

Heat transfer in the laminar boundary layer of a transparent gas flowing aroud a plane radiating surface is studied. Radiative heat-transfer processes in gases may be divided into two main groups. The first involves heat transfer in absorbing and radiating media. In this case, the effect of radiation lies in the introduction of new terms into the energy equation, representing internal heat sources and sinks. The second group embraces heat-transfer processes in a transparent gas when the effect of radiation on convection expresses itself solely by way of the boundary conditions. Here we study a case of practical importance belonging to the second group: heat transfer in the laminary boundary layer of a transparent gas flowing around a flat plate with the thermal flux q_w specified on its surface.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 107–110, January–February, 1972.     

Existence and nonuniqueness of local separation zones in viscous jets

The plane steady problem of the flow of a viscous wall jet past a smoothed break in the contour of a body is considered. For convenience, the flow in the neighborhood of the junction between two flat plates inclined at an angle to each other is chosen for study. As a result of the small extent of the region investigated the flow field is divided into two layers: the main part of the jet, which undergoes inviscid rotation, and a thin sublayer at the wall, which ensures the satisfaction of the no-slip condition. Particular interest attaches to the flow regime in which the solution in the sublayer satisfies the Prandtl boundary layer equations with a given pressure gradient. A similar problem was studied in [1–4]. The present case is distinguished by the structure of the free interaction region in a small neighborhood of the point of zero surface friction stress. By means of the method of matched asymptotic expansions, applied to the analysis of the Navier-Stokes equations, it is established that the interaction mechanism is that described in [5–7]. As a result, an integrodifferential equation describing the behavior of the surface friction stress function is obtained. A numerical solution of this equation is presented. The range of plate angles on which solutions of the equation obtained exist and, therefore, flows of this general type are realized is determined. The essential nonuniqueness of the possible solutions is established, and in particular attention is drawn to the possible existence of six permissible friction distributions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 38–45, January–February, 1986.The author wishes to thank V. V. Sychev and A. I. Ruban for their useful advice and discussion of the results.     

Construction of liquid flow past a solid body with an air layer

It is known that the friction drag for a body moving in a liquid may be reduced by varying the physical properties of the liquid at the body surface and, in particular, by creating an artificial air cavity on the body surface.The question of the magnitude of the drag gain in the case of the presence of a wall fluid layer with different physical constants was considered in [1–3].In [4, 5] results are presented of theoretical and experimental studies on the question of determining the parameters of artificial air cavities created on the lower surface of a horizontal wall in order to reduce the drag.In the following an attempt is made to construct the three-dimensional flow of an ideal liquid past a body which is enveloped entirely in an air cavity.     

Nonequilibrium viscous flow of a multicomponent gas in the vicinity of the stagnation point of a blunt body

This paper gives the results of numerical calculations characterizing the effect of variation of the shock layer parameters on the heat transfer in the case of a multicomponent nonequilibrium-dissociating air on a wall with finite catalycity in the vicinity of the stagnation point of a spherical blunt body. Similar results for the case of a binary mixture can be found in [1–3]. It is shown that a consideration of the variation of the parameters in the nonequilibrium shock layer leads to a significant increase in heat flux to the noncatalytic wall in comparison with the theory of an asymptotically thin nonequilibrium boundary layer with equilibrium parameters on its outer boundary.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–147, March–April, 1971.The author thanks V. V. Lunev for useful comments in the discussion of this work.     

Nonstationary laminar heat convection in a closed domain for a given heat flux

Characteristic modes of the time development of nonstationary heat convection in a closed planar domain upon a sudden supply of heat from the lateral surface are considered for Rayleigh numbers 10³–10⁷. Estimates of the boundaries of the beginning of the influence of convection on the temperature field and the buildup of a quasistationary convection mode in the range of Rayleigh and Fourier numbers are given. Characteristics of the circulation flow, the singularities of the temperature-field configuration and of the heat transfer from the wall to the fluid, are investigated. The mechanism for the origination and disappearance of vertical temperature differences, caused by convection, and the dependence of the vertical temperature differences on the Rayleigh and Fourier numbers, on the thermal mode of the boundary, and the domain geometry, are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1970.The author is grateful to T. D. Pirumov and T. V. Volokitin for assistance in performing the computations.     

Viscous incompressible flow in a narrow channel with one free wall and fluid supply through a porous insert

The problem investigated relates the plane unsteady flow of a viscous incompressible fluid in a narrow channel one of whose walls is free and acted upon by a given load, while the other is rigidly fixed. The fluid enters the channel through a porous insert in the stationary wall. A model of the flow of a thin film of viscous incompressible fluid and Darcy's law for flow in a porous medium are used to find the distribution of fluid pressure and velocity in the channel and the porous insert in the two-dimensional formulation for fairly general boundary conditions in the case where the length of the porous insert exceeds the length of the free wall. In the particular case where the length of the porous insert is equal to the length of the free wall an exact stationary solution of the problem is obtained for a given value of the channel height. The stability of the equilibrium position of the free wall supported on a hydrodynamic fluid film is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–24, January–February, 1986.     

Investigation of the propagation of a two-dimensional wall jet over a step

The propagation of an underexpanded sonic jet over a flat end face has been experimentally investigated. As distinct from previous studies, the object of investigation is not a free jet, but a jet flowing from a nozzle along a horizontal surface. The total separation of the jet from the surface and its attachment to the end wall are related to the propagation characteristics of underexpanded wall jets. The effect of the total pressure in the jet and the height of the step on the separation of the jet and its attachment to the wall and, moreover, on the principal characteristics of the flow — the pressure in the base region, the extent of the circulation zone, the jet trajectory — is examined. The associated hysteresis effects are studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 61–66, July–August, 1991.     

Ablation under the combined erosive and thermal influence of a two-phase flow

A study is made of the combined effect of a high-velocity two-phase (gas—solid particles) flow to the surface of a body. Different ablation models are proposed for simultaneous thermal (or thermochemical) and impact influence of the flow on the material. A numerical investigation is made of the unsteady thermal state of a body on which solid particles impinge, and the ranges of the determining parameters in which the thermal effect is negligibly small compared with the erosive effect are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–126, January–February, 1984.     

Dependence of Poiseuille slip and thermal creep on the law of interaction of gas molecules with the boundary surface

The problem of gas slip at a plane wall is solved on the basis of a linearized kinetic equation with a model collisional term (the S-model, which provides the correct value of the Prandtl number). Expressions are obtained for the Poiseuille slip and thermal creep which contain integrals of the scattering kernel over the velocity half-space. These expressions are made concrete for three particular cases corresponding to well-known models of the scattering kernel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 107–114, November–December. 1977.