Turbulent Boundary Layer on a Cone in a Supersonic Flow at Zero Angle of Attack

Abstract

A simple rule is given for transforming local heat-transfer coefficients from flat plates to cones at zero angle of attack for fully turbulent boundary layers in supersonic flow. The rule is that the cone heat-transfer coefficient is the flat-plate coefficient for one-half the Reynolds Number on the cone, the Mach Number and wall-to-free-stream temperature ratio remaining the same. INTRODUCTION AND SUMMARY T E CALCULATION OF T H E TEMPERATURE of a COne in supersonic flight requires a knowledge of the heat-transfer coefficient of the boundary layer of the cone. Hantzsche and Wendt solved the problem for laminar boundary layers. They found that the local laminar heat-transfer coefficient on a cone at zero angle of attack can be obtained by merely multiplying by v 3 the local laminar heat-transfer coefficient on a flat plate. Since heat transfer and skin friction are proportional, this rule applies also to local skin-friction coefficients for cones. Because laminar heat transfer and skin friction vary inversely with the square root of the Reynolds Number, another way of formulating the transformation rule for conical laminar layers is to state that the cone solution is the flat-plate solution for one-third of the Reynolds Number on the cone. The present report offers a simple transformation rule for fully turbulent boundary layers on cones, similar to that for laminar layers. I t is found that, for turbulent boundary layers, the cone solution for local heat transfer is the flat-plate solution for one-half of the Reynolds Number on the cone, the Mach Number and wall-to-free-stream temperature ratio remaining the same. The effect is an increase over the local turbulent heat-transfer or skin-friction coefficient of flat plates of about 10 to 15 per cent. The work is an extension to compressible flow of an analysis by Gazley, who neglected the effect of compressibility and assumed a power law for velocity distribution; his results approach those obtained when the above general rule for turbulent boundary layers is applied to incompressible flow at high Reynolds Numbers. Received July 26, 1951. * Aerodynamics Engineer, Aerophysics Laboratory. MOMENTUM EQUATION FOR THE BOUNDARY LAYER ON A CONE AT ZERO ANGLE OF ATTACK The von Karman momentum equation for a boundary layer in steady state on a body of revolution at zero angle of attack is 5 r dp r — / rpu(um u)dy = — — / r dy rwrw (1) ox Jo ox Jo In this relation, x is a coordinate distance measured along the body from the forward most point, y is the other coordinate measured from the surface along a normal to the surface, u is the flow velocity relative and parallel to the surface at the point (x, y)y p is the density of the fluid at (x, y), r is the normal distance from the body axis to (x, y), p is the pressure at (x), TW is the shear stress at the surface, and 8 is the boundary-layer thickness measured normal to the surface. Subscripts w and oo refer to the wall and outer edge of the boundary layer, respectively. With a cone, when the flow is supersonic and the shock wave is attached, the pressure along the rays is constant for in viscid fluids. Therefore, since the boundary layer is thin ( 8 rw r 8 ~ ^T I pu(Uco u) dy = -TW (4) rw Ox Jo But for the cone (l/rw)(brw/bx) = 1/x (5) so that, finally, from Eq. (4), the momentum equation for thin boundary layers on a cone in supersonic flight at zero angle of attack is