Numerical Calculation of Sharp Flat Plate Transitional and Turbulent Skin Friction

Abstract

Even in the early stages of the process the photographs show a high density gradient immediately behind the shock in proximity to the wall. This compression wave becomes better defined as the shock progresses until at about a = 2.15 in. it becomes a reflected shock and a Mach reflection results. This Mach reflection is of the inverted type in that the triple point moves towards the wall and the reflection eventually becomes regular.The shock-shock shown in Fig. 4b was obtained from geometrical considerations by knowing that the wall shock has a Mach number of 2.2 as predicted from the simple wave theory. On the other hand Whitham's shock-shock theory 3 does not permit the shock-shock to approach the wall as this would permit rays that come from each side of the start of the shockshock, to intersect. This is inadmissable in the theory. There is thus an anomaly in the end conditions given by the simple wave theory and the conditions required by the shock-shock theory.The surprising result shown by these tests is that notwithstanding the difference in the starting point of shock curvature nor the noncentred nature of the wave, Whitham's theory gives good agreement as to the position where the shock-shock starts, and as to the position of the shock-shock established on the basis of simple geometrical considerations.A further interesting point to notice is that the shock curvature is mainly confined to the lower portion of the shock; to such an extent that at a = 2.45 in. the shock cannot be distinguished from being plane. Changes in Mach number carried by the upper characteristics of the fan are thus evidently small.A comparison between the theoretical and experimental shock profiles shows similar effects to previous comparisons for a shock diffracting on plane walls. 5 The experimentally obtained curvature is less than the theoretical so that the part of the shock in contact with the wall is weaker than expected from the theory. These previous tests also showed that the agreement is better for stronger shocks (M 0 > 3.0) and the same would be expected for shocks shaped in the manner described here. Nomenclature C f = local skin-friction coefficient based on boundary-layer edge conditions, 2tjp e U e 2 M e = local edge Mach number Re e x = local Reynolds number based on boundary-layer edge conditions, p e U e x/u e T aw = adiabatic wall temperature T w = wall temperature U e = streamwise velocity at edge of boundary layer x = surface distance measured from apex of plate H e = viscosity evaluated at boundary-layer edge temperature p e = density evaluated at boundary-layer edge pressure and temperature T W = wall shearing stress