Cavity formation on the surface of a body entering water with deceleration

Abstract

The two-dimensional water entry of a rigid symmetric body with account for cavity formation on the body surface is studied. Initially the liquid is at rest and occupies the lower half plane. The rigid symmetric body touches the liquid free surface at a single point and then starts suddenly to penetrate the liquid vertically with a time-varying speed. We study the effect of the body deceleration on the pressure distribution in the flow region. It is shown that, in addition to the high pressures expected from the theory of impact, the pressure on the body surface can later decrease to sub-atmospheric levels. The creation of a cavity due to such low pressures is considered. The cavity starts at the lowest point of the body and spreads along the body surface forming a thin space between a new free surface and the body. Within the linearised hydrodynamic problem, the positions of the two turnover points at the periphery of the wetted area are determined by Wagner’s condition. The ends of the cavity’s free surface are modelled by the Brillouin–Villat condition. The pressure in the cavity is assumed to be a prescribed constant, which is a parameter of the model. The hydrodynamic problem is reduced to a system of integral and differential equations with respect to several functions of time. Results are presented for constant deceleration of two body shapes: a parabola and a wedge. The general formulation made also embraces conditions where the body is free to decelerate under the total fluid force. Contrasts are drawn between results from the present model and a simpler model in which the cavity formation is suppressed. It is shown that the expansion of the cavity can be significantly slower than the expansion of the corresponding zone of sub-atmospheric pressure in the simpler model. For forced motion and cavity pressure close to atmospheric, the cavity grows until almost complete detachment of the fluid from the body. In the problem of free motion of the body, cavitation with vapour pressure in the cavity is achievable only for extremely large impact velocities.

Appendix: A solution of the MBVP

In this Appendix, we present the solution of the problem (20)–(26) such that inequalities (15) and (16) are satisfied. The problem (20)–(24) can be reformulated as a Dirichlet problem for the analytic function

$$\begin{aligned} \omega _0^*(z)=(\varphi _x-\mathrm{i}\varphi _y-\mathrm{i}\ddot{h})g(z),\quad z=x+\mathrm{i}y\quad (y<0), \end{aligned}$$

(78)

where $g(z)$ is the characteristic function defined in (27). Equations (21)–(26) imply that $\omega _0^*(z)$ is bounded at $z=\pm c$, has simple poles at $z=\pm d$ and its real part is zero on the $x$-axis. Such a function has the form:

$$\begin{aligned} \omega _0^*(z)=\mathrm{i}\frac{Q(z)}{z^2-d^2}, \end{aligned}$$

(79)

where $Q(z)=\sum _{k=0}^4 \alpha _k z^k$ is a polyomial with real coefficients. This polynomial is of order four since $\omega _0^*(z)=O(z^2)$ as $z\rightarrow \infty$. As the flow is symmetric, it can be shown that $\omega _0^*(z)$ satisfies the symmetry condition $\overline{\omega _0^*(z)}=-\omega _0^*(-\overline{z})$ where the bar denotes the complex conjugate of a complex number. It follows from this symmetry condition that $\alpha _1=\alpha _3=0$. We find the following formula for the complex acceleration $\omega _0(z)=\varphi _x-\mathrm{i}\varphi _y$ by using the definition of $\omega _0^*(z)$ and Eq. (79):

$$\begin{aligned} \omega _0(z)=\mathrm{i}\ddot{h}\left( 1+\frac{\beta _0+\beta _1(z^2-c^2)+\beta _2(z^2-c^2)^2}{(z^2-d^2)^{3/2}\sqrt{z^2-c^2}}\right) \end{aligned}$$

(80)

with real coefficients $\beta _i$. Since $\omega _0(z)$ decays in the far field, we have $\beta _2=-1$.

We investigate the pressure behaviour at the detachment point $z=c$. The pressure gradient along the contact region $c<x<d$, $y=0$, is given by the real part of $\omega _0(z)$ in (80) together with the linearised Bernoulli’s equation (4). The asymptotic behaviour of $p_x(x,0,t)$, $c<x<d$, at $x=c$, $y=0$, is

$$\begin{aligned} p_x(x,0,t)\sim -\frac{\beta _0\ddot{h}}{\sqrt{2}c(d^2-c^2)^{3/2}}(x-c)^{-1/2}\quad (x\rightarrow c). \end{aligned}$$

(81)

If $\beta _0\ddot{h}>0$, the pressure is below the cavitation pressure $p_\mathrm{c}$ in the vicinity of the detachment point $c$, which is a contradiction to the inequality (16). Therefore,

$$\begin{aligned} \beta _0\ddot{h}\le 0. \end{aligned}$$

(82)

Now we analyse the free-surface elevation $y=\eta (x,t)$ in the cavity region $-c<x<c$ close to the separation point $x=c$. Kutta’s condition guarantees that the fluid leaves the body tangentially (see [19]). The imaginary part of (80) together with the kinematic boundary condition (12) gives us the asymptotic behaviour of the vertical fluid acceleration on the free surface at $x=c$:

$$\begin{aligned} \eta _{tt}(x,0,t)\sim -\frac{\beta _0\ddot{h}}{\sqrt{2}c(d^2-c^2)^{3/2}}(c-x)^{-1/2}\quad (x\rightarrow c). \end{aligned}$$

(83)

The value $\beta _0\ddot{h}$ in (83) is responsible for whether or not the fluid intersects the body at the separation point. For a smooth body, it follows from (15), (82) and (83) that both conditions on the behaviours of the hydrodynamic pressure and the cavity thickness are satisfied only with $\beta _0=0$.

The value of $\beta _1$ determines the value of the acceleration potential $\varphi (x,y,t)$ in the cavity region $|x|<c$, $y=0$. Equation (23) and the far-field condition (24) imply that $\int _{-\infty }^0\varphi _{y}(0,y,t)\,\hbox {d}y=-p_\mathrm{c}/\varrho$, where $\varphi _y$ is given by the imaginary part of $\omega _0(z)$ in (80). Hence,

$$\begin{aligned} \beta _1=2d^2-c^2-\frac{1}{E(q)}\left( \frac{p_\mathrm{c}}{\varrho \ddot{h}}d+c^2K(q)\right) ,\quad q=\sqrt{1-\frac{c^2}{d^2}}, \end{aligned}$$

(84)

where $K(q)$ and $E(q)$ are the complete elliptic integrals of the first and second kinds, respectively.