The Transition to Turbulence in a Boundary Layer on a Blunt Cone in Supersonic Flow

Abstract

/ N A BLUNT CONE placed symmetrically in a supersonic stream, ^^^ the fluid near the surface has passed through a normal shock and that away from the surface through a nearly conical shock. Thus near the tip the Mach number, Mi, and the Reynolds number per inch, piUi/m, at the edge of the boundary layer are smaller than on a sharp cone. Further downstream, the edge of the boundary layer is farther from the wall and the values of Mi and PiUi/m are more nearly those on a sharp cone. Moeckel has pointed out that, if transition to turbulence occurs at a critical value of Rx = piUix/m, where x is the distance downstream from the tip of the cone, then it will occur for a larger value of x—i.e., further downstream—on a blunt cone than on a sharp cone in the same stream. This is, of course, a simplified argument in that it takes no account of the way in which the boundary layer develops in the laminar regime; it is the purpose of the present note to attempt a more detailed description of this development. Experiments were carried out at RAE (Farnborough) on a series of cones, each having an included angle of 15°, and having tip radii varying from less than 0.001 in. (sharp) to 0.49 in. These cones were placed in streams whose undisturbed Mach numbers were, respectively, 3.12 and 3.81. The position of transition to turbulence was observed using a shadowgraph technique: the results, for the given upstream conditions, are shown in Fig. 1. I t is immediately apparent that for small values of the tip radius, r, the trend predicted by Moeckel is observed. For somewhat larger values, however, the trend is reversed. Some traverses of the boundary layer were made with a pitot tube, and it was found that (within experimental error) the Reynolds number based on the momentum thickness, 82, instead of the distance x, was a more appropriate parameter, and this does take some account of the way in which the boundary layer develops, since this affects the value of 52. The boundary layer is, in fact, developing under a flow which already possesses a velocity shear normal to the wall. The simple case of a linear shear in two-dimensional incompressible flow has been discussed theoretically by Li and Glauert, who take the velocity profile to be