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.     

Investigation of a hypersonic viscous shock layer on rotating axisymmetric bodies in the presence of blowing

The hypersonic flow of a laminar stream of viscous compressible gas past blunt axisyrametric bodies rotating about the longitudinal axis is considered. It is assumed that gas blows from the surface of the body. The solution of the problem is obtained by a finite-difference method in a wide range of Reynolds numbers and blowing and rotation parameters. Some results of the calculations characterizing the effect of the rotation on the velocity and temperature profiles across the shock layer, on the friction and heat transfer coefficients, and the shock wave separation are given for the neighborhood of the stagnation point. For large Reynolds numbers and strong blowing an analytic solution of the problem is found in an approximation of two inviscid layers separated by a contact surface. The calculations are made for the flow past a sphere and a paraboloid and it is shown that in the presence of rotation the maximum of the heat flux is shifted from the stagnation point onto the side surface of the body. The dependence of the pressure distribution, the heat flux, and the friction coefficient is investigated for cases of constant and variable blowing over the contour of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–114, January–February, 1986.     

Asymptotic solution of the euler equations in the shock layer on a blunt body in nonuniform flow with gas injection from the body surface

Supersonic nonuniform gas flow over blunt bodies without surface injection has previously been investigated by both numerical [1–3] and experimental [3] methods. The processes of surface vaporization under the influence of an intense heat flux, artificial gas injection and surface combustion [4] are all worthy of study. The problem of the interaction between a nonuniform supersonic flow and a body in the presence of intense gas injection from the surface is examined and an analytical solution is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–134, November–December, 1989.     

Axisymmetrical bodies with minimal resistance and with minimal flow toward the surface of the body,with different characters of the flow in the boundary layer

Approximate formulas are obtained which permit calculating the distribution of the friction stress and of the local heat flux over the surface of a body of arbitrary form, with a given pressure distribution, both with laminar and with turbulent flow conditions in the boundary layer. These approximate equations were used to solve the variational problems of determination of the form of axisymmetrical bodies with a minimal resistance or with a minimal total flow of heat to the surface (in the class of bodies consisting of a flat leading edge and a lateral surface) in a hypersonic flow of viscous gas. In the solution of variational problems of determination of the optimal form of a body from the conditions of minimal resistance or of a minimal total heat flux toward the surface, we must be able to determine the distribution of the pressure, the friction stress, and the local heat flux along the surface of a body of arbitrary form. At large Reynolds numbers, the problem of determining the pressure distribution comes down to solving the Euler equation with corresponding boundary conditions. However, at the present time there are no effective methods for solving this problem (at least from the point of view of using the methods for solution of variational problems); in the solution of variational problems, to determine the pressure distribution this forces us to use various approximate methods (for example, the method of tangential wedges or cones, the Newton method, etc.). The use of such approximate formulas renders unfeasible on exact solution of the equations of the boundary layer, for which the distribution of the gas-dynamic parameters at the outer limit of the boundary layer (including also the pressure distribution) must be known previously. This makes it necessary to construct approximate formulas to determine the friction stress, _w, and the local heat flux, q_w, whose accuracy in an arbitrary body will be determined by the accuracy of the assignment of the gas-dynamic parameters at the outer limit of the boundary layer. We give below the derivation of such dependences for laminar and turbulent flow in a boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–102, March–April, 1971.     

Heat transfer at and near the stagnation point in turbulent flow over bodies

Results are presented of experimental investigations of heat transfer in the neighborhood of the stagnation point in flow of a turbulent gas over bodies. It is assumed that the outer flow is capable of rendering the boundary layer turbulent over the whole body surface, i.e., the hypothesis is invoked that there is a turbulent stagnation point. Using the method of integral relations [1] and the flat plate heat-transfer law, transformed in such a way as to satisfy the heat-transfer conditions at the stagnation point, simple formulas have been obtained for calculating the heat flux.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–181, July–August, 1975.     

Two-phase mass injection from the stagnation zone of a blunt body in a hypersonic flow

A mathematical model of the hypersonic steady gas flow over the stagnation zone of an axisymmetric blunt body with given two-phase injection from the surface is proposed. The two-continuum model of a dusty gas [3] is used for describing the flow in the region of the wall. The problem is solved in the boundary layer and thin viscous shock layer approximations. On the basis of the numerical calculations the distribution of the parameters of the carrier and dispersed phases near the axis of symmetry is obtained. The similarity parameters determining the convective heat transfer are found. The stagnation point heat fluxes with and without particles are compared. The range of parameters on which particles can significantly reduce the heat transfer is determined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 60–66, July–August, 1992.     

Determination of the body shape for minimum radiative heating along a flight trajectory

Space vehicles are subject to intense aerodynamic heating in planetary entry. According to estimates in [1], the heat shield mass for entry of a probe into the atmospheres of the outer planets can make up 20–50% of its total mass; here the radiative component predominates in the aerodynamic heating. It is therefore interesting to investigate methods of reducing the heat flux to the nose region of a vehicle. Analysis shows [2–6] that, for a given atmospheric composition, the heat-shield weight is determined by the trajectory, the body shape, the heat-protection method, and the chemical composition and the thermophysical and optical properties of the heat shield material. In such a general statement of the problem, optimization of the heat-shield mass depends on many parameters, and has not been solved hitherto. A number of papers have examined simpler problems, associated with reducing spacevehicle heating: optimization of the trajectory from the condition that the total heat flux to the body stagnation point should be a minimum for given probe parameters [2, 3], optimization of the characteristic probe size for a given trajectory [2–4], and optimization of the probe shape in a class of conical bodies at a given trajectory point [3, 5, 6J. In [7] a variational problem was formulated to determine the shape of an axisymmetric body from the condition that the radiative heat flux to the body at a given trajectory point should be a minimum for the entire surface, and an analytical solution was found for this in limiting cases. The present paper investigates a more general variational problem: determination of the shape of an axisymmetric body from the condition that the total radiative influx of heat to the body along its atmospheric trajectory should be a minimum. A solution has been obtained for a class of slender bodies for different isoperimetric conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–100, March–April, 1978.     

Method of meridional sections in problems of a three-dimensional boundary layer

The difficulties and clumsiness of problems of calculating the heat transfer distribution over the surface of a body in a three-dimensional flow are well known. It is shown that this problem can be considerably simplified where the influence of the three-dimensionality of the flow, which in certain applications it is important to take into account, is only weak. In this case the three-dimensional problem can be reduced to a set of two-dimensional problems along the lines of meridional sections of the body. This has been demonstrated in detail with reference to the method of effective length or local similarity, which is widely used in engineering practice and is particularly justified in the the case of turbulent heat transfer law. However, in the three-dimensional case it is complicated by the need to calculate the distribution of the streamlines over the surface of the body [1–4]. In the presence of slight asymmetry of the flow the problem can be substantially simplified, mainly as a result of the demonstrated possibility of replacing (with quadratic accuracy) the streamlines by the lines of meridional sections. The possibility of an independent solution of the exact boundary layer equations along each meridional plane is demonstrated for the above-mentioned approximation (rule of meridional sections).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–73, May–June, 1986.     

Nonmonotonic dependence of the convective heat flux on the rate of blowing of opaque admixtures into a radiating H-He shock layer

The results are given of calculations of radiative and convective heat transfer in a radiating H-He shock layer in the neighborhood of the stagnation point of a blunt body when graphite ablation products are blown from the surface. It is found that under the conditions in the shock layer characteristic for motion of the body in the atmosphere of Jupiter [3] the dependence of the convective flux on the blowing rate is essentially nonmonotonic. The maximal value is comparable with the radiative flux to the surface under these conditions. It is shown that a decisive part in the mechanism which increases the convective flux is played by the presence near the surface of particles which effectively absorb radiative energy in the spectral regions in which an appreciable radiation flux reaches the boundary layer; the difference between the transport properties of the blown and the oncoming gases is also important.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 106–113, May–June, 1981.     

Formulas for the heat and diffusion fluxes to a catalytic surface in a chemically nonequilibrium multicomponent boundary layer

On the basis of an asymptotic expansion of the solution of the equations of a multicomponent chemically nonequilibrium boundary layer for large Schmidt numbers, formulas are obtained for the heat flux and the diffusion fluxes of the reaction products and chemical elements on a surface with arbitrary catalytic activity. The results are compared with well-known analytic and numerical solutions. The comparison reveals the high accuracy of the formulas proposed. The results of calculating the diffusional separation of the mixture due to the selectivity of the catalytic properties of the surface with respect to recombination of oxygen and nitrogen atoms are presented. Values of the reduction of the convective heat fluxes due to the catalytic properties of the surface are obtained over a wide range of conditions in the free stream.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 171–176, March–April, 1996.     

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     

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.     

Three-dimensional turbulent boundary layer on a body of complex shape

Flow and heat transfer problems associated with three-dimensional compressible gas flow past a body of complex shape at a small angle of attack are investigated on the basis of a finite-difference calculation. The results of a numerical solution of the equations of the three-dimensional turbulent boundary layer are presented. The effect of the leading parameters on three-dimensional flow development and heat transfer is analyzed. The characteristic flow regions in the boundary layer are found: lines of divergence and convergence on the surface, separation zones and flow interfaces. The location of the maximum values of the heat flux and friction on the surface is determined, the behavior of the limiting streamlines on the body is described, and the intensity of the secondary flows in the boundary layer is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–35, September–October, 1986.     

Calculation of supersonic equilibrium-dissociating air-xenon mixture flow past a blunt body

The flow of an equilibrium-reacting multicomponent three-element air-xenon mixture is numerically investigated. The effect of multicomponent diffusion on the convective heat transfer to the body surface is examined. The dependence of the convective heat transfer to the body surface and the total shock-layer spectral radiation flux P_m on the xenon concentration is obtained. A comparison of the calculated data for P_m and the experimental data of [2] gives good agreement. A simple approximation for the convective heat flux at the stagnation point as a function of xenon concentration is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 156–164, July–August, 1991.In conclusion the authors wish to thank I. A. Sokolova for supplying data on the resistance coefficients of the various mixtures and S. A. Yunitskii for discussing the numerical method.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

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.     

Numerical analysis of natural convection in a thin-walled cylindrical vessel with hemispherical end-plates completely filled with liquid

The results are given of calculations of convection in a thin-walled cylindrical vessel with hemispherical end-plates over the range of Rayleigh and Fourier numbers that correspond to the main regimes of unsteady laminar convection in the presence of a given heat flux.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 204–203, September–October, 1984.     

A nonequilibrium viscous shock-wave layer in the vicinity of the critical point with the associated heat exchange taken into account

Results are presented in this article of numerical calculations of a viscous shock layer with the associated heat exchange in the vicinity of the critical point of a spherical blunt body taken into account in the presence of nonequilibrium chemical processes in the shock layer and on the surface of the body about which the flow takes place. A number of papers [1–4], in which specification of the surface temperature of the obstacle was utilized, have been devoted to the numerical investigation of a nonequilibrium viscous shock layer. At the same time the surface temperature of a body varies in actual flight due to heating, and together with this there is catalytic activity of the material, which appreciably complicates the problem and necessitates the simultaneous treatment of the course of processes in the gaseous and solid phases. The use of a separate formulation is difficult in this case, since the formulas for the thermal flux from the gaseous phase are of an estimative nature [5] when a volume nonequilibrium chemical reaction is present for a surface having an arbitrary catalytic activity. Taking account of the associated heat exchange has been done before for a number of problems of boundary-layer theory [6, 7], and in this case it has permitted determining the characteristics which are most important from the practical standpoint under conditions of flight along a specified trajectory, as well as under specified time-independent conditions of flight at altitudes at which the approximation of a viscous shock layer is valid. The effect of catalytic activity is discussed for a number of surface materials, and it is shown that the use of the formulas of boundary layer theory can appreciably distort the behavior of the surface temperature as a function of time for a certain altitude range.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 108–114, May–June, 1979.     

Convective heating of a blunt-nosed body in a nonuniform hypersonic gas stream

An investigation is made into the convective heating of a blunt-nosed body in an expanding stream of heated gas. The gas parameters at the outer edge of the boundary layer are determined on the basis of a solution obtained earlier by the authors [3]. Expressions are obtained which make it possible to convert the convective heat flux to a body in a uniform gas stream to one in a nonuniform stream. Dimensionless numbers are found and their influence on the convective heat flux to the body is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 127–133, July–August, 1981.     

Laminar boundary layer of infinitely long elliptic cylinders for an arbitrary yaw angle

Results are presented of the calculation of the laminar boundary layer on infinitely long elliptic cylinders in a supersonic perfect gas flow at an arbitrary angle of attack. It was assumed that the Prandtl number is constant and equal to 0.7, the dynamic viscosity coefficient follows a power-law variation ( T^0.76) with temperature, and there is high heat transfer at the body surface (H_1w=0.05).The calculations showed that a change of the body shape—the ellipticity coefficient =b/a—has a significant effect on the nature of the distribution and the magnitude of the local heat flux.In evaluating the thermal fluxes at the blunt leading edges, swept wings are usually considered as infinitely long yawed cylinders. In studying heat transfer at the surface of bodies of small aspect ratio at high angles of attack, wide use is made of the hypothesis of plane sections, when each section, orthogonal to the longitudinal axis of the body, is considered equivalent to a corresponding yawed infinite cylinder.By now quite detailed studies have been made of the behavior of the boundary layer on an infinitely long yawed circular cylinder with both the laminar and turbulent flow regimes for a compressible gas [1, 2]. However, there are no data on the heat transfer at the surface of a yawed infinite cylinder with arbitrary cross section, although the availability of such data is urgently needed, for example, for the proper selection of the form of the leading edges of the swept wing.This article presents the results of the calculation of the characteristics of the laminar boundary layer on the surface of infinite elliptic cylinders in a supersonic perfect gas flow. The calculations were made over a quite wide range of flight Mach number M, yaw angle , and ellipticity factor . The data presented on the distribution of the relative heat flux along the cylinder directrix may be used also for estimating the heat flux with account for the real properties of air if we know the corresponding value of the heat flux in the vicinity of the stagnation line.