Periodic Discontinuities in the Acceleration of Spheres in Free Flight

Abstract

The trajectories of unrestrained spheres being accelerated from rest behind a normal shock wave in a shock tube have been recorded by high-speed stroboscopic photography, and precise measurements of the particle positions made with an optical comparator. Data of sufficient precision were obtained to permit the accurate determination of the second time derivative of the particle trajectories. The experiments were conducted in air with plastic spheres ranging in diameter from 0.7 to 2.4 mm and with specific gravities from 0.7 to 1.5. Particle Reynolds numbers from 1 × 104 to 5 × 104 were obtained, and absolute particle accelerations from 1 × 106 to 5 × 106 ± 0.16 × 106 cm/s2 were measured. The relative velocities were all subsonic; relative Mach numbers ranged from 0.3 to 0.6. The values of the particle acceleration, when plotted as a function of time, were found to contain periodic abrupt changes similar to a mathematical discontinuity. It is postulated that the associated abrupt changes in the forces exerted on the particles are caused by periodic detachment of low-pressure centers of vorticity which form behind the spheres. Values of the drag coefficients determined from the data similarly exhibit periodic abrupt changes. These changes appear, in the way they consistently increase during each cycle of the period and fall at the point of “discontinuity,” to be compatible with the postulate that the periodic behavior is caused by formation and abrupt detachment of low-pressure vortices in the wake. Values of the drag coefficients were found to vary from 10 percent to 30 percent above steady-state values at comparable Reynolds numbers—the range of the computed values of the drag coefficient being caused by the periodic nature of the quantity.