Flow behind the boundary layer separation point in a supersonic stream

The flow structure behind the separation point of a laminar boundary layer in a supersonic stream has been investigated. Analytic and numerical solutions are obtained for simple semiinfinite separation zones starting from the leading edge or a point on the smooth surface. The question of the pressure plateau in a separation zone of finite length is discussed and its value is calculated on the basis of asymptotic theory. The asymptotic theory of flow [1, 2] in the neighborhood of the separation point of the laminar boundary layer in a supersonic gas stream (region of free interaction) is employed. The local solution obtained is subsequently used to construct the flow pattern in the separation zone [3]. An analysis is made of the behavior of the solution for the free-interaction region on transition to the region of reverse flows. The results make it possible actually to compute (in the first approximation) the pressure in the plateau region after establishing the mathematical significance of this concept, previously introduced on the basis of the experimental results. At the same time relatively simple solutions are obtained for semiinfinite separation zones.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 19–25, May–June, 1971.     

Flow properties of a dusty-gas point source in a supersonic free stream

By using Lagrangian method, the flow properties of a dusty-gas point source in a supersonic free stream were studied and the particle parameters in the near-symmetry-axis region were obtained . It is demonstrated that fairly inertial particles travel along oscillating and intersecting trajectories between the bow and termination shock waves . In this region, formation of " mufti- layer structure" in panicle distribution with alternating low- and high density layers is revealed. Moreover, sharp accumulation of particles occurs near the envelopes of particle trajectories .     

Flow in the neighborhood of the stagnation point of an axisymmetric wake

The solutions of the equations of parabolic type describing the development of the flow in an axisymmetric wake under the Influence of viscosity and an adverse pressure gradient are considered. It is then shown that in the general case in the neighborhood of the stagnation point on the axis of the wake the solution is a singular one, the possibility of its continuation beyond the stagnation point being excluded. The following solutions are also obtained: a regular solution in the neighborhood of the stagnation point and a singular solution continuable downstream. This singular solution is the limit for the class of regular solutions having a miniumum in the velocity distribution on the axis as the minimum velocity tends to zero.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–59, March–April, 1986.The author is grateful to V. Ya. Neiland and Vik. V. Sychev for discussing the results and offering useful advice.     

Investigation of heat flux at the forward critical point of bodies placed in a high-temperature subsonic stream

We present the results of measurements of the total heat flux at the forward critical point of axially symmetric models placed in a high-temperature subsonic stream of air and nitrogen in an apparatus which uses a high-frequency inductive discharge [1] to heat the gas. The heat fluxes were measured for cylindrical models whose forward part had one of three possible shapes: a hemisphere, a hemisphere with a blunt nose, or a flat circular end-face. A water-cooled calorimeter sensor was set up at the forward critical point of the model; the calorimeter sensor was made of different materials, so that it was possible to estimate the radiant and convective components of the total heat flux and determine the effect of the sensor material on the heat flux measured. The convective component of the heat flux was compared to a calculated value obtained by Fay and Riddell's formula. The heat-flux values found for two shapes of models were used in determining the effective radius of streamline flow for a model with a plane end-face.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 133–141, September–October, 1973.The authors are grateful to N. I. Nesterov for his help with the work.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Flow of gas with a cellular vortical structure through a nozzle

The operation of rocket motors is often accompanied by the development of powerful secondary vortices in the combustion chamber [1–3], The superposition of the secondary vortices on the main flow leads to the formation of a cellular flow structure. Each of the cells represents a three-dimensional vortical circulation of the gas, and this causes a change in the working conditions of the nozzle. The model of helical motion [4] is used in this paper in considering the influence of the three-dimensional behavior of an adiabatic flow on the flow and traction characteristics of the nozzle.     

Fluid motion in the stagnation point neighborhood

An analytical solution of the inviscid incompressible fluid equations (outer flow), which serves as the asymptote for the numerical solution of the viscous-fluid equations (inner flow), is presented. Using this solution, the effect of the spreading rate and the vorticity of the outer flow on the heat transfer (heat flux "peaks") can be studied. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 97–101, July–August, 2000.     

Heat transfer in the neighborhood of the critical point or line in the presence of hydrogen blowing and combustion

The results of numerical calculations of the thin viscous shock layer equations are presented. The dependence of the heat exchange on the generalized blowing parameter, which depends on the blown gas flow rate and the Reynolds number, is obtained.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 144–149, May–June, 1995.     

Flow and heat exchange in the neighborhood of a subterranean pipeline

The present study considers flow and heat exchange in the neighborhood of a subterranean pipeline on irrigated sector of a route. The flow is caused by thermal convection of the liquid in the field of gravitational forces, this convection being connected with the cooling or heating of the liquid near the surface of the pipeline.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 51–56, November–December, 1987.     

Dependence of the aerodynamic forces on the Reynolds number with the three-dimensional flow of a vortical stream around bodies

The article discusses the dependence of the viscous stresses on the Reynolds number Re in three-dimensional flows around bodies of arbitrary form. It is shown that, with an infinite growth of a vortex with the approach to a body, singular terms appear in an asymptotic expansion in terms of ε=Re^?1/2. The infinite values of the derivatives of the velocity in flows of an incompressible liquid are due to the initial vorticity; in a supersonic flow, they can be connected with the absence of a maximum of the entropy at the critical flow line behind a leading shock wave. The singularity in the tangential stresses brings about the appearance of additional terms in the total aerodynamic forces and moments acting on the body.     

Viscous sublayer in the neighborhood of a corner point in supersonic flow past a cooled body

The behavior of the flow functions at the upper edge of the viscous sublayer is subjected to an asymptotic analysis. The results obtained permit a more accurate formulation of the boundary-value problem and an improvement in the quality of numerical calculation of the flow in the viscous sublayer. The heat flux and friction distributions on the surface of the body are obtained for precritical and postcritical interaction regimes; for expansion flows the calculated results are presented in the form of the dependence of the maximum values of the friction and heat flux on the temperature factor and the angle of deflection of the flap; approximate expressions suitable for practical use are proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 142–147, November–December, 1989.     

Flow in the neighborhood of the cusp of an axisymmetric cavity

The steady-state axisymmetric potential flow of an ideal incompressible fluid is considered. The structure of the singularity at the point where the free stream surface leaves the axis of symmetry is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–176, September–October, 1989.     

Breaking of gravity waves in a neighborhood of their second critical propagation speed

Experimental data on gravity shallow-water waves generated by a vertical plate moving in a predetermined manner are given. The plate completely covers the cross section of the channel. It is found that with when the wave speed exceeds the first critical value known in hydraulics, the wave retains smoothness. Breaking of the waves begins at the second critical speed (which is about 1.3 times as high), whose value coincides with the limiting propagation speed of a solitary wave. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 52–58, March–April, 1998.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

Hypersonic three-dimensional viscous shock layer in a nonuniform gas flow in the neighborhood of the stagnation point

Three-dimensional hypersonic viscous gas flow past smooth blunt bodies in the presence of injection or suction is considered. The effect of the nonuniformity of the approach stream on the shock-wave standoff, the flow structure and the friction and heat transfer coefficients is investigated with reference to the examples of flow from a supersonic spherical source and flow of the far wake type. It is shown that this effect depends importantly on the Reynolds number, the nature of the nonuniformity and the shape of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 136–145, November–December, 1987.     

Flow of a sonic free gas stream around a profile

The stream function far from a profile in a two-dimensional sonic free gas jet is constructed. The stream function satisfies the Tricomi equation and is constructed on the , plane by the method of singular integral equations. With unlimited increase of the jet width and by satisfying a certain condition, the stream function transforms to the self-similar solution of Frank [1, 2] and Guderley, which describes an unbounded sonic stream far from the profile. In conclusion, flow of a sonic jet issuing from a duct about a profile is considered.The author wishes to thank S. V. Fal'kovich for valuable suggestions in discussing this article.     

Flow in the neighborhood of the trailing edge of a plate in a transonic flow of viscous gas

An investigation is made into the influence of the Mach number and the viscosity on the flow in the neighborhood of the trailing edge of a plate. The Mach number is assumed to satisfy m_∞ ² ? 1 = 0(R^?l/5), which corresponds to the regime of transonic interaction. It is shown that if the Mach number is such that ¦M_∞ ² ? 1¦ > O(R^?1/5) the problem in the region of free interaction can be reduced by an appropriate transformation to the already known solutions for an incompressible fluid [5] and supersonic flow [7].     

Heat transfer in the neighborhood of a point of inflection in the leading edge of a plate in hypersonic flight

A study is made of the three-dimensional flow of a viscous gas around a flat plate with an inflection in the generator of the leading edge in the case of strong interaction between the exterior hypersonic flow and the boundary layer. Numerical solutions to the problem are obtained. It is shown that near points of inflection of the profile of the leading edge of a flat wing strong self-induced secondary flows can be formed together with associated local peaks of the heat fluxes and the friction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1980.