Numerical and theoretical investigation on three-dimensional trajectory characteristics of skipping stones

An in-depth study of the complicated trajectory characteristics of skipping stones is carried out in the present work. A three-dimensional numerical simulation validated by acquiring good agreement with experimental results is established. It is devoted to illustrating five different types of motion responses after the stone impacts the water surface in the o-xy plane, including “dive”, “hydroplaning trout”, “hydroplaning skip”, “stable skip”, and “skipping trout”. Then, the lateral deviations are investigated quantitatively based on dimensionless parameter sin(α+β)cosα in the o-yz plane. Steady interval and linear interval are divided for lateral deviation Z₁/D and Z₂/D based on the values of (α+β) and α, respectively. The results reveal that (1) Z₁/D increases almost linearly with the increasing sin(α+β)cosα at different slopes in different (α+β) intervals; (2) Z₂/D increases almost linearly with the increasing sin(α+β)cosα at different slopes in different α intervals; (3) in linear interval, numerical lateral deviations are much larger than the fitting values at points β/α≥4.5, and much smaller than the fitting values at points 2.3 ≤β/α<4.3. Finally, a theoretical approach is proposed to predict the maximum immersion depth of trailing edge point P.