Influence of second-order boundary layer effects in hypersonic flow over slender blunt cones

The aim of this study is to determine the influence of second-order effects in the aggregate on supersonic axisymmetric flow over slender blunt cones and also to determine the region of applicability of approximate methods of taking into account the strongest of these second-order effects — entropy layer absorption. A system of complete viscous shock layer equations containing all the terms of the gas dynamic Euler equations and all the second-order terms of asymptotic boundary layer theory is chosen as the gas-dynamic model. Within the calculation domain the problem is solved in a unified manner.Translated from Izvestiya Rossiiskoi Akademii Nauk, Meknanika Zhidkosti i Gaza, No.4, pp. 129–134, July–August, 1992.     

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.     

Viscous shock layer near the stagnation point on a rotating body with unsteady injection and surface cooling

Unsteady supersonic flow regimes in the neighborhood of a stagnation point are investigated on the basis of a system of viscous shock layer equations [10] containing all the terms of the Euler equations and the boundary layer equations. An analytic solution of the unsteady equations valid near the surface of the body is found in the case of strong injection. The unsteady equations of the viscous shock layer are solved numerically on the basis of a divergent implicit scheme of the second order of approximation across the shock layer, using Newtonian linearization and vector sweep methods with allowance for the boundary relations on the surface of the body and at the isolated bow shock. Certain calculation results illustrating the effect of injection, surface cooling, the swirl of the external flow and the angular velocity of the body on the structure of the steady and unsteady viscous shock layer are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 114–122, September–October, 1987.     

Investigation of the hypersonic viscous shock layer around blunt bodies in a nonuniform flow

Unseparated viscous gas flow past a body is numerically investigated within the framework of the theory of a thin viscous shock layer [13–15]. The equations of the hypersonic viscous shock layer with generalized Rankine-Hugoniot conditions at the shock wave are solved by a finite-difference method [16] over a broad interval of Reynolds numbers and values of the temperature factor and nonuniformity parameters. Calculation results characterizing the effect of free-stream nonuniformity on the velocity and temperature profiles across the shock layer, the friction and heat transfer coefficients and the shock wave standoff distance are presented. The unseparated flow conditions are investigated and the critical values of the nonuniformity parameter a_k [10] at which reverse-circulatory zones develop on the front of the body are obtained as a function of the Reynolds number. The calculations are compared with the asymptotic solutions [10, 12].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–159, May–June, 1987.     

Unsteady viscous shock layer near the leading edge of a wing of infinite span

The effect of unsteady injection and wall temperature variation on the parameters of the viscous shock layer near the stagnation line of a wing of infinite span at an angle of slip is investigated on the basis of the viscous shock layer model. An analytic solution of the nonstationary problem, valid near the surface of the wing for strong injection, is obtained. A numerical investigation is carried out and some results of calculating the unsteady viscous shock layer equations for various forms of the time dependence of the injection velocity and wing surface temperature are presented. The calculations are based on a finite-difference method of the second order of approximation in the space variable and the first order of approximation in time, which makes use of expression of the equations in divergence form, Newtonian linearization and vector sweeps across the shock layer. In the steady-state case the results of the calculations are in good agreement with the data of [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 90–95, January–February, 1988.     

Influence of vibration-dissociation coupling on heat transfer and hypersonic drag

A numerical investigation is carried out within the framework of the multicomponent total viscous shock layer model [6, 7], according to which when Re 100 the flow near a blunt body can be divided into a shock wave zone and a viscous shock layer. At the inner edge of the shock wave the generalized Rankine-Hugoniot relations are imposed, and in the shock layer the complete system of viscous shock layer equations is solved with allowance for vibrational relaxation and nonequilibrium dissociation and ionization reactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 141–151, May–June, 1990.     

Investigation of hypersonic flow of rarefied gas past wings of infinite span

In the framework of the locally self-similar approximation of the Navier-Stokes equations an investigation is made of the flow of homogeneous gas in a hypersonic viscous shock layer, including the transition region through the shock wave, on wings of infinite span with rounded leading edge. The neighborhood of the stagnation line is considered. The boundary conditions, which take into account blowing or suction of gas, are specified on the surface of the body and in the undisturbed flow. A method of numerical solution of the problem proposed by Gershbein and Kolesnikov [1] and generalized to the case of flow past wings at different angles of slip is used. A solution to the problem is found in a wide range of variation of the Reynolds numbers, the blowing (suction) parameter, and the angle of slip. Flow past wings with rounded leading edge at different angles of slip has been investigated earlier only in the framework of the boundary layer equations (see, for example, [2], which gives a brief review of early studies) or a hypersonic viscous shock layer [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 150–154, May–June, 1984.     

Hypersonic viscous shock layer on sweptback wings of infinite span at different angles of attack

A study is made of the flow of a viscous compressible gas in a hypersonic shock layer on sweptback wings of infinite span with blunt leading edge at different angles of attack. The equations of the hypersonic viscous shock layer with modified Rankine-Hugoniot relations across the shock wave and boundary conditions on the surface of the body that take into account slip and discontinuity of the temperature are solved by a method of successive approximation which yields not only an analytic solution for the first approximations but also an exact numerical solution when the method is implemented on a computer. The analytic solution of the problem is found in the first approximation. Expressions are obtained for the coefficients of friction and heat transfer on the surface of the body, and also for the profiles of the velocities and the temperature across the shock layer. Comparison of the analytic solution with the numerical solution reveals a satisfactory accuracy of the analytic solution for not too large Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 91–102, March–April, 1979.We thank G. A. Tirskii for his interest in the work and valuable discussions.     

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.     

Analytic investigation of friction and heat transfer in the neighborhood of a three-dimensional stagnation point at low and moderate Reynolds numbers

A study is made of hypersonic three-dimensional flow of a viscous gas past blunt bodies at low and moderate Reynolds numbers with allowance for the effects of slip and a jump of the temperature across the surface. The equations of the three-dimensional viscous shock layer are solved by an integral method of successive approximation and a finite-difference method in the neighborhood of the stagnation point. In the first approximation of the method an analytic solution to the problem is found. Analysis of the obtained solution leads to the proposal of a simple formula by means of which the calculation of the heat flux to a three-dimensional stagnation point is reduced to the calculation of the heat flux to an axisymmetric stagnation point. A formula for the relative heat flux obtained by generalizing Cheng's well-known formula [1] is given. The accuracy and range of applicability of the obtained expressions are estimated by comparing the analytic and numerical solutions. Three-dimensional problems of the theory of a supersonic viscous shock layer at small Reynolds numbers were considered earlier in [2–5] in a similar formulation but without allowance for the effects of slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–150, March–April, 1988.     

Three-dimensional chemically nonequilibrium viscous shock layer on a catalytic surface with allowance for associated heat transfer

A three-dimensional flow of dissociating air past blunt bodies is investigated in the framework of the thin viscous shock layer theory. Multicomponent diffusion and homogeneous chemical reactions, including dissociation, recombination, and exchange reactions, are taken into account. The generalized Rankine-Kugoniot conditions are specified on the shock wave and the conditions which take into account the heterogeneous catalytic reactions, on the surface of the body. The viscous shock layer equations are solved together with the heat equations inside the coating, which is carbon with a deposited thin film of SiO₂, or quartz. The case of a thermally insulated surface is also considered. The problem for the case of the motion of a body along the re-entry trajectory into Earth's atmosphere is investigated numerically. The temperature of the surface and the heat flux toward it are given as a dependence on the height (tine) of the flight for different cases of the specification of the catalytic reactions. It is shown that the difference between the heat fluxes towards the thermally insulated surface and the fluxes toward the heat-conducting surface in the neighborhood of the stagnation point is of the order of 6–12% for all the cases considered. This makes it possible to decouple the solution of the problem of heat conduction in the body.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–146, November–December, 1985.deceased     

Three-dimensional thin viscous shock layer in the absence of planes of symmetry in the flow

The basic laws of viscous homogeneous gas flow at high supersonic speeds past smooth blunt bodies with a permeable surface are investigated within the framework of the thin viscous shock layer model. An efficient numerical method of solving these equations, which makes it possible to consider cases of flow past bodies at angles of attack and slip, when there are no planes of symmetry in the flow, is proposed. Some results of calculating the flow past a triaxial ellipsoid with an axial ratio of 103n73 at angles of attack =0–45° and slip angles =0–45° over a broad interval of Reynolds numbers are presented as an example. The effect of the principal determining parameters of the problem on the flow structure in the shock layer and the surface friction and heat transfer coefficients is analyzed. An expression for calculating the heat fluxes to the impermeable surface of smooth blunt bodies in a supersonic homogeneous viscous gas flow over a broad interval of Reynolds numbers is proposed on the basis of the solutions obtained and the results of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 150–158, March–April, 1989.     

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.     

Calculation of supersonic viscid flow near stagnation line of a blunt body

A numerical study is made of supersonic flow of a viscous gas in the vicinity of the stagnation line of plane and axisymmetric blunt bodies (cylinder, sphere). As in [1–5], which consider the compressed layer of a viscous gas in the vicinity of the stagnation point, use is made of the locally self-similar approximation, which is used to transform the Navier-Stokes equations into a system of ordinary differential equations. In the present paper the solution is sought with the simplifications of [5] and with more general conditions, which makes it possible to study a broad class of flows. The proposed numerical algorithm permits obtaining the structure of the compressed layer near the stagnation line, including the shock wave and the boundary layer. The calculations made on a computer for different flow conditions are illustrated by graphs.The author wishes to thank G. I. Petrov, G. F. Telenin, and L. A., Chudov for their interest in the study and for their helpful discussions. discussions.     

Calculation of nonstationary viscous flows

The present study will consider the problem of formation of a nonstationary flow of a viscous, thermally conductive gas around a semiinfinite plate, with the gas being set in motion instantaneously from a state of rest (problem I) and behind a shock wave incident upon the gas (problem H). A numerical method will be used, based on integration of the complete system of Navier-Stokes equations. Special attention is given in the solution to these times when the simplifying assumptions of boundary layer theory are invalid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–171, November–December, 1978.The authors thank Yu. A. Dem'yanov for evaluating the results of the study.     

Influence of rotation of the body and exterior vorticity on the heat transfer near the stagnation point of a blunt body in a supersonic stream

A study is made of the influence of longitudinal vorticity on the hypersonic viscous shock layer near an axisymmetric cooled surface that rotates about the longitudinal axis with angular velocity ₁ [1, 2]. The equations of the viscous shock layer for the neighborhood of the stagnation point are simplified on the basis of the theory of a thin three-dimensional shock layer [1, 2]. The results are given of some calculations of the influence of the parameters and ₁ on the heat transfer and the structure of the shock layer in the case of steady flow. An iterative numerical method proposed by the author [1] is used, and a modification to accelerate the convergence of the iterations is proposed. It is noted that the parameters and ₁ have characteristic ranges of variation, which depend on the Mach and Reynolds numbers, for which the distributions of the streamlines in the shock layer are qualitatively different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 179–182, May–June, 1984.I thank V. Ya. Neiland and Yu. D. Shevelev for helpful discussions of this paper.     

Numerical investigation of the hypersonic viscous shock layer on wings of infinite span at angles of attack and slip

In the framework of the theory of a hypersonic viscous shock layer [1] with modified Rankine-Hugoniot relations [2] at the shock wave a study is made of flow past wings of infinite span with a rounded leading edge. A numerical solution to the problem has been obtained in a wide range of variation of the Reynolds number (5–10⁶), the blowing-suction parameter, the angle of attack (0–45 °), and the angle of slip (0–70 °). Data are given on the influence of the angle of slip on the profiles of the temperature and the velocity across the shock layer. A study is made of the dependence of the distributions of the pressure, the heat flux, and the friction coefficients along the surface of the body on the blowing-suction parameter and the angles of attack and slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–108, March–April, 1984.     

Nonequilibrium viscous shock layer on blunt cones

Some results are given of the numerical investigation into the parameters of the nonequilibrium flow of air in a viscous shock layer in the case of blunt circular cones at zero angle of attack; they are also compared with experimental data obtained during re-entry of ballistic objects into the Earth's atmosphere. The calculations were made with allowance for the nonequilibrium processes of dissociation and ionization, and also vibrational relaxation. The influence of viscosity, heat conduction, and diffusion is taken into account in the complete shock layer. The conditions on the shock wave are posed with allowance for its finite thickness. The characteristic profiles of the velocity, temperature, and electron concentration in the shock layer are given. Good agreement is obtained between the calculated and experimental data on the level and the profiles of the electron concentration. The parameters of the shock layer were determined by a method that is a natural extension of the numerical method of [1] to the case of nonequilibrium flow in a viscous shock layer. Because of this, only the main differences of the method when applied to the calculation of nonequilibrium flows of a multicomponent mixture such as dissociated and ionized air are described.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 15–20, November–December, 1979.     

Laminar mixing of planar supersonic chemically reacting jets of unequal pressure

A study is made of the influence of a transverse pressure gradient on the mixing of reacting jets of fluorine and hydrogen. The real picture is simulated by a system of simplified Navier-Stokes equations. A comparison is made with calculations based on the complete Navier-Stokes equations and also boundary layer equations. Features of the shock waves are analyzed under conditions of strong heat release in the mixing layer. The influence of these features on the gain in laser situations is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 18–23, May–June, 1982.We thank G. N. Volchkova, Yu. P. Golovachev, V. A. Pospelov, M. Kh. Strel'ts, and M. L. Shur for helpful discussions.     

Viscous hypersonic flows for various aerophysical models

The results of mathematical simulation of hypersonic air flow past a sphere with allowance for chemical nonequilibrium and ionization are presented. The calculated data are obtained on the basis of a numerical solution of the Navier-Stokes equations and the hypersonic viscous shock layer equations. The equilibrium radiation temperature of the body surface, the pressure, and the shock wave detachment are compared for these models. The combined effect of molecular transport and the catalytic properties of the body surface on the aerodynamic heating is analyzed. The results of the effect of certain models of the thermochemical disequilibrium of the gas on the temperature of the body surface are also analyzed along a heat-stressed interval of the reentry trajectory of an aerospace vehicle.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 151–161, July–August, 1996.