Theoretical and Numerical Approach to “Magic Angle” of Stone Skipping

Abstract

We investigate the condition for the bounce of circular disks which obliquely impacts on fluid surface. An experiment [ Clanet, C., Hersen, F. and Bocquet, L., Nature 427, 29 (2004) ] revealed that there exists a "magic angle" of 20• between a disk's face and water surface in which condition the required speed for bounce is minimized. We perform three-dimensional simulation of the diskwater impact by means of the Smoothed Particle Hydrodynamics (SPH). Futhermore, we analyze the impact with a model of ordinal differential equation (ODE). Our simulation is in good agreement with the experiment. The analysis with the ODE model gives us a theoretical insight for the "magic angle" of stone skipping.PACS numbers: 45.50. Tn, 47.11.+j, 47.90.+a Problem of impacts and ricochets of solid bodies against water surface have been received a considerable amount of attention [1,2,3,4,5,6]. In the early stage, the problem was of importance in naval engineering concerning the impacts of canon balls on sea-surface [7]. Investigations then revealed that there exists a maximum angle of incidence θ max for impacts of spheres, above which the rebound does not occur [8]. Besides, it was empirically found that the θ max relates to specific gravity of sphere σ as θ max = 18/ √ σ. This relation was theoretically explained using a simple model of an ordinal differential equation (ODE) [8,9]. In military engineering today, the problem of water impacts may be not as important as that of a century ago, however, recently it attracts renewed interest under the studies of locomotion of basilisk lizards [10] and stone-skip [11].This study is motivated by experimental study of stone-skip -bounce of a stone against water surfaceby C. Clanet et. al. [12]. They investigated impacts of a circular disk (stone) on water surface and found that an angle about φ = 20• between the disk's face and water surface would be the "magic angle" which minimizes required velocity for bounce. In this paper, we study theoretically and numerically the oblique impact of disks and water surface. Our simulation successfully agrees with the experiment. Moreover, we apply an ODE model [17] to the disk-water impact and obtain an analytical form of the required velocity v min and maximum angle θ max as a function of initial disk conditions.To perform a numerical simulation of the disk-water impact, we solve the Navier-Stokes equation using the technique of Smoothed Particle Hydrodynamics (SPH) [13,14]. Fig. 1 is the snapshots of our simulation. The SPH method is based on Lagrangian description of fluid and has an advantage to treat free surface motion. Several representation of the viscous term have been proposed for this method. In this work, we adopt an artificial viscous term [15] which is simple for computation * Electronic address: nagahiro@cmpt.phys.tohoku.ac.jp and sufficiently examined with Couette flow [16]. In our simulation, we neglect surface tension and put the velocity of sound of the fluid, at least, 25 times larger than the incident velocity of the disk.In ...

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