Optimum Shapes of Spatial Bodies with a Minimum Total Radiative Flux to the Surface

A solution of a variational problem of a slender body with a minimum total radiative heat flux, moving in a gas with a constant velocity, is constructed. It is found that there are three types of the transverse contour of the optimum body: a circumference, a starshaped contour, and a contour consisting of circle arcs and sectors of straight lines. The radiation parameter affects only the shape of the longitudinal contour and does not affect the optimum shape of the transverse contour. It is shown that the use of optimum spatial bodies allows a significant (more than 50%( decrease in the radiative heat flux to the body surface as compared to bodies of revolution with similar geometric characteristics.     

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.     

Strong viscous interaction on slender three-dimensional bodies

Hypersonic flow of a viscous gas past axisymmetric power-law bodies in the regime of strong interaction of the laminar boundary layer with the outer inviscid flow was studied in [1, 2]. In this paper the results obtained in [1, 2] are extended to the case of flow past a slender three-dimentional body.     

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.     

Analytical solution in the problem of constructing axisymmetric noses with minimum wave drag

An analytical procedure for determining the axisymmetric nose shapes that ensure minimum wave drag is developed. The solution is constructed as an improving variation on the conical shape. The target function is built up in an approximate form on the basis of the assumption that the relationship between the geometric and gasdynamic parameters is local in nature. It is shown that the optimal bodies are truncated power-law bodies with an exponent equal to 2/3. The bodies thus obtained are compared with those constructed in accordance with an exact formulation of the problem.     

Choice of optimal blunting radius of axisymmetric body with account for surface radiation

We consider the problem of determining the optimal blunting with respect to heat transfer of an axisymmetric body in supersonic gas flow at zero angle of attack with account for body surface radiation. Results are presented of a calculation of the optimal blunting radius of a cone with half-angle 10° for various values of the ratio of convective heat flux at the stagnation point to the radiative heat flux. It is shown that for small values of this ratio the small bluntings are optimal.     

One-dimensional and two-dimensional analogs for three-dimensional viscous flows in the neighborhood of the plane of symmetry of a blunt body

An approximate method of determining the heat transfer and friction stress in three-dimensional flow problems using the two-dimensional and one-dimensional solutions is proposed. This method is applicable over a wide range of Reynolds numbers — from low to high. On the basis of a theoretical analysis of the approximate analytic solution of the equations of a three-dimensional viscous shock layer it is shown that the problem of determining the heat flux in the neighborhood of the plane of symmetry of bodies inclined to the flow at an angle of attack can be reduced, firstly, to the problem of determining that quantity for an axisymmetric body and, secondly, to the problem of determining the heat transfer to an axisymmetric stagnation point. On the basis of an analysis of the results of a numerical solution of the problem it is shown that corresponding analogs can also be used for the friction stress. The accuracy of the similarity relations established is estimated by solving the problem by a finite-difference method. A similarity relation of the same kind was previously obtained in [1] for a double-curvature stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 117–122, January–February, 1990.     

Comparison of approximate analytical and numerical solutions for heat fluxes in viscous supersonic flow past a body

The axisymmetric supersonic laminar flow of a viscous perfect gas past a blunt body is studied over a wide range of Reynolds and Mach numbers. An approximate analytical solution for the distribution of the heat flux, divided by its value at the stagnation point, over the body surface is derived. The expression for the relative heat flux depends only on the body geometry and the pressure distribution over the body surface. To evaluate the accuracy and the range of validity of the formulas obtained, a comparison with numerical solutions of the complete Navier-Stokes equations was carried out for bodies of various shapes, such as spheres, ellipsoids, and paraboloids.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 125–132, January–February, 1996.     

Construction of Equivalent Axisymmetric Bodies for Calculating Heat Transfer and Viscous Stress on Three-Dimensional Bodies at Incidence

An effective approximate technique for calculating heat transfer, viscous stress, and species components on the windward side of three-dimensional bodies at incidence in hypersonic flow is developed. Using the similarity method, the solution of the three-dimensional problem is reduced to the solution of an axisymmetric problem. For determining the heat flux on a real body, modified two-dimensional equations are solved for equivalent axisymmetric bodies, specially constructed for meridional planes of the original body. For an arbitrary three-dimensional geometry and angle of attack formulas are derived and a conversion program is developed. These make it possible to determine all the parameters of the equivalent body corresponding to a given meridional plane of the original body; then these parameters are used as input data for calculating the viscous flow past the body. The solutions of the two-dimensional equations for the equivalent bodies are in good agreement with more exact solutions of the three-dimensional equations.     

Buoyancy-induced flow of non-Newtonian fluids over a non-isothermal body of arbitrary shape in a fluid-saturated porous medium

The buoyancy-induced flows of non-Newtonian fluids over non-isothermal bodies of arbitrary shape within saturated porous media have been treated using the boundary layer approximations and the power-law model to characterize the non-Newtonian fluid behavior. Upon introducing a general similarity transformation which considers both the geometrical effect and the wall temperature effect on the development of the boundary layer length scale, the governing equations for a non-isothermal body of arbitrary shape have been reduced to those for a vertical flat plate. The transformed equations reveal that a plane or axisymmetric body of arbitrary shape possesses its corresponding family of the wall temperature distributions which permit similarity solutions. Numerical integrations were carried out using the Runge-Kutta-Gill method, and the results of the heat transfer function were presented once for all plane and axisymmetric bodies. As illustrations, local wall heat flux distributions were discussed for wedges, cones, spheres, circular cylinders and other geometries. Furthermore, an approximate formula based on the Karman-Pohlhausen integral relation has been presented for speedy and sufficiently accurate estimation of heat transfer rates.     

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.     

Axisymmetric analogy for three-dimensional viscous flow problems

A method of solving three-dimensional flow problems with the aid of two-dimensional solutions, which can be used for any Reynolds numbers, is proposed. The method is based on the use of similarity relations obtained in the theoretical analysis of the approximate analytic solution of the equations of a three-dimensional viscous shock layer. These relations express the heat flux and the friction stress on the lateral surface of a three-dimensional body in terms of the values on the surface of an axisymmetric body. The accuracy is estimated by comparing the results with those of a numerical finite-difference calculation of the flow past bodies of various shapes. Similar similarity relations were previously obtained in [1] for the plane of symmetry of a blunt body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–118, November–December, 1991.The authors are grateful to G. A. Tirskii for his interest in their work.     

Conical and Quasi-Conical Incompressible Fluid Flows

The solution of the problem of fluid flow inside a cone with a small vertex angle is obtained in closed form. The conditions of occurrence of singular separation are considered within the framework of conical flow theory. A class of conical flows in which the vorticity is transported along streamlines of the potential velocity component is detected.Quasi-conical incompressible fluid flow, i.~e. a flow inside and outside an axisymmetric body with power-law generators is defined by analogy with supersonic compressible fluid flow. The conditions under which the effect of vorticity and swirling is significant are found as a result of an inspection analysis. An approximate solution of the problem of fluid flow inside a zero corner is found.A coordinate expansion representing a plane analog of conical flow is constructed in the neighborhood of the separation point of a creeping flow on a smooth surface.     

Acoustic radiation from slender bodies

A general approach is proposed to compute the acoustic field radiated by axisymmetric slender bodies at medium (Ka 1) and low (Ka 1) frequencies. A multiple-scales technique is developed to determine the sound pressure in the near field. The far field solution is built up as a continuous distribution of sectorial spherical harmonics and their densities are deduced by matching with the near field solution. Some applications are presented to support the analytical results.     

Vibrations of a body in a bounded volume of viscous fluid

The problem of the vibrations of a body in a bounded volume of viscous fluid has been studied on a number of occasions [1–4]. The main attention has been devoted to determining the hydrodynamic characteristics of elements in the form of rods. Analytic solution of the problem is possible only in the simplest cases [2]. In the present paper, in which large Reynolds numbers are considered, the asymptotic method of Vishik and Lyusternik [5] and Chernous' ko [6] is used to consider the general problem of translational vibrations of an axisymmetric body in an axisymmetric volume of fluid. Equations of motion of the body and expressions for the coefficients due to the viscosity of the fluid are obtained. It is shown that in the first approximation these coefficients differ only by a constant factor and are completely determined if the solution to the problem for an ideal fluid is known. Examples are given of the determination of the “viscous” added mass and the damping coefficient for some bodies and cavities. In the case of an ideal fluid, general estimates are obtained for the added mass and also for the influence of nonlinearity. Ritz's method is used to solve the problem of longitudinal vibrations of an ellipsoid of revolution in a circular cylinder. The hydrodynamic coefficients have been determined numerically on a computer. The theoretical results agree well with the results of experimental investigations.     

Aerodynamic Characteristics of Slender Star-Shaped Bodies with Supersonic Leading Edges

A theory of the flow around slender conical bodies with a star-shaped cross-section in regimes with shock attachment along the length of the leading edges is developed. The aerodynamic characteristics of the slender star body, a single cycle of which represents a V-shaped wing, are obtained for different numbers of reflections of the disturbances proceeding from the leading edges of the cycles from the wing cantilevers, using superposition of the solutions of linear problems of flow around the V-wing.     

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.     

The blunt-body problem in a nonuniform hypersonic viscous gas flow

The laws of heat transfer associated with the interaction of underexpanded supersonic gas jets and obstacles or blunt bodies have been investigated, for example, in [1–3]. Similar problems of nonuniform flow occur when bodies move in the wake behind other bodies; however, in this case the laws of heat transfer have so far received little attention [4–8]. It has been established that for a certain Reynolds number and flow nonuniformity parameters a zone of reverse-circulatory flow develops near the front of the blunt body. However, the conditions of transition to separated flow have not been determined. This paper presents a self-similar solution of the equations of the viscous shock layer near the stagnation line in supersonic flow past an axisymmetric blunt body located behind another body. On the basis of this solution a separationless flow criterion is proposed. The effect of the nonuniformity and the Reynolds number on the shock standoff distance, the convective heat flux and the friction drag of the blunt body is investigated. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–125, November–December, 1986. In conclusion the authors wish to thank I. G. Eremeitsev for useful suggestions and G. A. Tirskii for discussing their work.     

Multiobjective optimization for design of multifunctional sandwich panel heat pipes with micro-architected truss cores

A micro-architected multifunctional structure, a sandwich panel heat pipe with a micro-scale truss core and arterial wick, is modeled and optimized. To characterize multiple functionalities, objective equations are formulated for density, compressive modulus, compressive strength, and maximum heat flux. Multiobjective optimization is used to determine the Pareto-optimal design surfaces, which consist of hundreds of individually optimized designs. The Pareto-optimal surfaces for different working fluids (water, ethanol, and perfluoro(methylcyclohexane)) as well as different micro-scale truss core materials (metal, ceramic, and polymer) are determined and compared. Examination of the Pareto fronts allows comparison of the trade-offs between density, compressive stiffness, compressive strength, and maximum heat flux in the design of multifunctional sandwich panel heat pipes with micro-scale truss cores. Heat fluxes up to 3.0 MW/m² are predicted for silicon carbide truss core heat pipes with water as the working fluid.     

GENERAL REYNOLDS ANALOGY RELATION ON BLUNT-NOSED BODIES$^{\bf 1)}$

钝头体壁面的摩阻和热流分布规律不同,平板流动中的雷诺比拟关系在钝头体壁面失效. 文章在前期高超声速广义雷诺比拟理论研究工作的基础上,利用数值仿真的方法对不同外形和来流参数条件下的钝头体广义雷诺比拟关系开展进一步研究. 通过建立钝头体绕流边界层的理论分析模型,得到了钝头体壁面雷诺比拟系数的线性分布预示公式. 采用数值求解 N-S 方程的方法,计算了圆柱和幂次体壁面的摩阻和热流以及二者之间的比拟系数. 通过与前期数值和理论结果对比,以及计算收敛性和网格无关性检验,对数值方法进行了验证. 通过在不同雷诺数 ($Re_\infty = 3.98\times 10^2 \sim 1.59\times 10^6$) 和马赫数 ($M_\infty = 3\sim 12$) 条件下的计算结果对比分析雷诺比拟系数的分布,总结了钝头体中广义雷诺比拟关系受外形和来流条件的影响,评估了广义雷诺比拟理论的适用性. 研究发现,在较高雷诺数条件下,离驻点较远的下游 ($\theta > 60^\circ$) 部位,雷诺比拟系数的分布不同程度地偏离理论预示的线性规律. 相比于圆柱外形,幂次体壁面的雷诺比拟系数分布的线性规律相对较好,其分布斜率略低于圆柱壁面的结果. 研究表明,如果针对实际外形和雷诺数进行适当修正,可以提高广义雷诺比拟关系的预示精度.