Theory of the collapsing axisymmetric cavity

We investigate the collapse of an axisymmetric cavity or bubble inside a fluid of small viscosity, like water. Any effects of the gas inside the cavity as well as of the fluid viscosity are neglected. Using a slender-body description, we show that the minimum radius of the cavity scales like $h_0 \propto t'^{\alpha}$, where $t'$ is the time from collapse. The exponent $\alpha$ very slowly approaches a universal value according to $\alpha=1/2 + 1/(4\sqrt{-\ln(t')})$. Thus, as observed in a number of recent experiments, the scaling can easily be interpreted as evidence of a single non-trivial scaling exponent. Our predictions are confirmed by numerical simulations

Buckling instability of crown sealing

Despite the scholarly fascination with water entry of spheres for well over a century,1 we present a new observation, namely, the crown-buckling instability. This instability is characterized by striations appearing near the top of the crown walls just prior to the surface seal, as shown in Fig. 1(a). The crown wall collapses inward due to the pressure differential across the wall created by the moving air in the wake of the sphere and surface tension within the crown. Since the rate of collapse is faster than that at which fluid drains out from the neck region, fluid collects into the striations and the crown buckles. The wall is slightly thicker along these striations than in between where the films are more susceptible to air flow and get drawn inward into the crown interior, thereby developing into bag-like structures (Figs. 1(a) and 1(b)) that ultimately atomize, causing a fine spray inside the crown. Under atmospheric conditions, this typically occurs within 5 ms after impact

Local properties of extended self-similarity in 3D turbulence

Using a generalization of extended self-similarity we have studied local scaling properties of 3D turbulence in a direct numerical simulation. We have found that these properties are consistent with lognormal-like behavior of energy dissipation fluctuations with moderate amplitudes for space scales $r$ beginning from Kolmogorov length $\eta$ up to the largest scales, and in the whole range of the Reynolds numbers: $50 \leq R_{\lambda} \leq 459$. The locally determined intermittency exponent $\mu(r)$ varies with $r$; it has a maximum at scale $r=14 \eta$, independent of $R_{\lambda}$.Comment: 4 pages, 5 figure