Nonregular reflection of a ring shock from the axis of symmetry

Abstract

NONREGULAR REFLECTION OF A RING SHOCK FROM THE AXIS OF SYMMETRY E. M. Barkhudarov, M. O. Mdivnishvili, I. V. Sokolov, M. I. Taktakishvili, and V. E. Terekhin UDC 533.6.011.72 INTRODUCTION The convergence of one-dimensional spherical or cylindrical shock waves is accompanied by an unbounded increase in the flow parameters as the shock wave approaches the center (axis) of s w~netry (unbounded c%unulation) [i, 2]. The cumulation effect is also char- acteristic of nonone-dimensional convergent axisymmetric shock waves. It has been shown experimentally that a ring shock wave is amplified as it moves towards the axis [3]. A theoretical analysis of the problem within the framework of Whitham's method indicates that the cumulation of the ring shock is unbounded [4] in the absence of lower azimuthal harmonics of the perturbation of the shape of the shock front [5]. Here we investigate the pattern of the flow that develops after the arrival of the ring shock at the center of the ring. The principal result is the formation of a Mach shock wave observed on reflection of the ring wave from the axis of symmetry at small values of the axial coordinate z, reckoned from the center of the ring, nonregular reflection being predicted at arbi- trarily small distances from the center. As shown theoretically [6], the nonregular reflection of the ring shock at small values of z is due to the increase in the angle of inclination of the incident wave front to the axis in accelerated convergence. This conclusion is consistent with the well- known hypothesis of Courant and Friedrichs [7] to the effect that in any case an axisym- metric ("conical") convergent shock wave undergoes nonregular reflection as distinct from a plane wave, which may be regularly reflected. This hypothesis was recently [8] proved for a conical shock wave in a perfect gas. I. The shock wave investigated was created, as in [3], as a result of the surface breakdown of a large number (~i00) of spark gaps on the inner surface of a ring facing the axis of symmetry. The radius and thickness of the ring were R K = 5 cm and h = 1 cm, respectively. The energy introduced into the discharge E ~ 1.2 kJ. The average value of the Mach number of the shock wave traveling towards the axis was M 0 ~ 2. The shock wave was visualized by the shadow method using a laser light source with illumination in a direction perpendicular to the axis of the ring. The sequence of shadowgrams shown in Fig. 1 was obtained for different realizations of the process by varying the time delay of the laser radiation pulse. The space scale and the time in- tervals between the photographs are indicated in the figure. The coordinates correspond- ing to the position of the Mach wave on the axis are equal to: z I = 1.4 cm, z= = 2.5 cm, z 3 = 4 cm. The source of the ring shock is on the left. In the shadowgrams the formation of the discontinuity configuration typical of Mach reflection (Fig. 2) is clearly visible. In Fig. 2 the letters AI denote the incident wave, AR the reflected wave, AM the Mach wave, and AE the contact discontinuity. In the first shadowgram because of the poor spatial resolution the Mach wave is not visible, but the strong distortion of the incident wave front near the axis as a result of the acceleration of the wave during cumulation is obvious. In Fig. 3 we have plotted the experimental values of the radius r M (curve 2) and the Mach wave velocity M M (curve 3) together with the value of the angle v (curve 4) between the incident wave front and the axis of symmetry measured at the point A (see Fig. 2). If there were no cumulation effect, the angle v would be given by the expression ~,=arctg