Axisymmetric Buckling of Uniformly Loaded Spherical Caps Undergoing Plastic Deformation

Fig. 2 Width of the entropy wake behind a sphere; comparison of schlieren experiments (A), present theory applied to measured shock shape (B) and to Korkan's shock profile (C), and theory of Webb evaluated with y = 1.2, and y = 1.3 (D). i.e. for a sphere, due to an empirical formula reported by Korkan.5 In either case the procedure is to express the local shock angle as a function of y; s(y) is then found from the oblique shock relations with inclusion of real gas effects. With Eqs. (2) and (4), and s(y) derived from the shock profile, sufficient information is available to evaluate the integral (1), and to determine the position, rmax, of the extreme values of s. Experiments have been performed in a hypersonic free-flight range using a light-gas gun of 15 and 20 mm bore diameter. Schlieren photographs of the wake behind blunt bodies have been recorded with a high-speed movie camera. The film length covered a wake regime up to several hundred diameters behind the body, which permitted a determination of the final position of the two bands, designating the "width" of the entropy wake (Fig. 1). Spheres and cylinders flying in the direction of their axes were launched in a Mach number range of 7 ^ M^ ^ 17. The distance between the two bands was measured from the schlieren records by means of a densitometer. An essential requirement was to establish a laminar wake. This was achieved by lowering the pressure in the range to a value below 20 torr. The very low density level associated with this pressure required the use of an extremely sensitive double pass schlieren system. Shadowgraphs of the flying body were taken simultaneously with each schlieren record in order to determine the shape of the bow shock. In evaluating Eq. (1), s(y) has been derived from observed shock shapes and also, in the case of the sphere, from Korkan's shock profiles. The results presented in Figs. 2 and 3 yield a quantitative interpretation of the schlieren pattern produced by the entropy wake. The "width" of the entropy wake of a sphere is shown in Fig. 2 as a function of the flight Mach number M^ ; rmax, which is here a dimensionless quantity, increases with increasing Mach number. The experimental values of r max, as measured from the schlieren photographs, are in relatively good agreement with the calculated points determined by using an experimental shock shape. The experimental results are also well described by Webb's theory, if a value of y = 1.2 and a drag coefficient of CD = 1.0 are assumed for the numerical evaluation of this theory. At lower Mach numbers, i.e. at lower wake temperatures, the experimental points approach values given by Webb's theory evaluated with y = 1.3. A similar result is obtained in comparing experimental and calculated entropy wake radii for the case of the cylindrical flight model (Fig. 3). Only four experimental shock profiles for the numerical evaluation of the present theory were available here. The absolute values of r max are higher than for a sphere flying with equal Mach number, thus indicating an influence of body drag. This is in agreement with the calculations of Webb who found, that the entropy wake radius (or "width") increases with the square root of the drag coefficient.