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

Abstract

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,τw, and the local heat flux, qw, 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.