EXTREMAL PROBLEMS OF HEAT-TRANSFER TO 3-DIMENSIONAL BODIES AT HYPERSONIC SPEEDS

The design of shuttle-like hypersonic spacecraft [70, 77] gives rise to the problem of investigating the spatial configurations that are optimum from the point of view of thermal heating and other characteristics, which enable the weight of the required thermal protection to be reduced. The problem of optimizing the weight of thermal protection depends on many parameters and has not yet been solved in a rigorous mathematical formulation. Approximate formulations of the optimization problem have therefore been considered, the solution of which has enabled axisymmetrical optimum shapes of bodies to be obtained with the minimum convective [43, 61, 731 and radiation [35-37, 58, 60) heat fluxes. It is known from attempts to solve variational problems of a body with minimum drag [29-31, 51-53, 621, that the transition to essentially three-dimensional configurations enables a reduction in the drag to be achieved compared with axisymmetrical bodies. A similar situation should obviously also occur when optimizing the shape of a body for heat flux. In this paper we present for the first time variational problems for finding the optimum shape of three-dimensional bodies of minimum overall thermal heating when moving along an incoming trajectory. In papers by other authors [21, 29-31, 48, 51-53, 62] the problems of determining the three-dimensional optimum aerodynamic shapes from the point of view of the minimum wave or total drag were considered. A brief review is given of research which has been done to determine the convective and radiation heating of three-dimensional bodies and the fundamental formulas for the wave drag, the fraction drag, and the convective and radiation fluxes to three-dimensional bodies moving in dense layers of planetary atmospheres are presented. The formulas depend explicitly on the conditions of entry into the atmosphere of the planet and on the geometry of the body, which enables variational problems to be formulated on determining the three-dimensional shape of the body from the conditions for minimum overall heating (convective and radiation) of the surface along the trajectory of motion.