2. Boundary-value problem

A sketch of the physical domain is shown in figure 1(a). The body submerged below a calm free surface is symmetric about the $Y$-axis, and thus only half of the flow region is considered. Before the impact, $t=0$, the body and the liquid are at rest. At time $t=0^+$ the body is suddenly set into motion with acceleration $a$ directed downwards such that, during infinitesimal time interval $\Delta t \rightarrow 0$, the speed of the body reaches the value $U$. The problem of a rigid body moving in a fluid body is kinematically equivalent to the problem of a fluid body moving around a fixed rigid body with acceleration $a$ at infinity. We define a non-inertial Cartesian system of coordinates $XY$ attached to the body at point $A$, and an inertial system of coordinates $X^\prime Y^\prime$ in which the velocity of the liquid at infinity equals zero. The body is assumed to have an arbitrary shape, which can be defined by the slope of the body as a function of the arclength coordinate $S$, $\beta _b=\beta _b(S)$. The liquid is assumed to be ideal and incompressible. The flow starting from rest remains irrotational at subsequent times. Both gravity and surface tension are ignored.

Figure 1.

(a) Sketch of the physical plane, and (b) the parameter or $\zeta$-plane.

We can introduce complex potentials $W(Z)=\varPhi (X,Y)+\textrm {i}\varPsi (X,Y)$ and $W^\prime (Z)=\varPhi ^\prime (X,Y)+\textrm {i}\varPsi ^\prime (X,Y)$ with $Z=X+\textrm {i}Y$. In these systems, the velocity fields are related as follows:

(2.1)\begin{equation} \frac{\textrm{d}W}{\textrm{d}Z}=\frac{\textrm{d}W^\prime}{\textrm{d}Z} - \textrm{i}at, \end{equation}

where $a$ is the acceleration, $0 < t < \Delta t$ and $\Delta t \rightarrow 0$. Integrating (2.1), we can find

(2.2a–c)\begin{equation} W=W^\prime - \textrm{i}atZ, \quad \frac{\partial W}{\partial t}=\frac{\partial W^\prime}{\partial t} - \textrm{i}aZ, \quad \frac{\partial \varPhi^\prime}{\partial t}=\frac{\partial \varPhi}{\partial t} - aY. \end{equation}

By substituting the last equation into Bernoulli's equation

(2.3)\begin{equation} \frac{\partial \varPhi^\prime}{\partial t}+\frac{p}{\rho}+ \frac{|V^\prime|^2}{2}=\frac{p_a}{\rho}, \end{equation}

and integrating it over infinitesimal time interval $\Delta t\rightarrow 0$, one can obtain

(2.4)\begin{equation} P=\int_0^{\Delta t}p\,\textrm{d}t={-}\rho\varPhi+\rho UY, \end{equation}

where $p$ and $P$ are the pressure and impulsive pressure, respectively, and $U=a\Delta t$. Here, $|V^\prime |<\infty$ and $p_a$ are the velocity magnitude and the pressure on the free surface, respectively, whose integrals over time interval $\Delta t\rightarrow 0$ tend to zero.

We introduce dimensionless quantities normalized by $U$, $L$ and $\rho$: $v=|V|/U$, $x=X/L$, $y=Y/L$, $h=H/L$ and $\phi (s)=\varPhi (S)/(LU)$. The vertical impulse force $F_y$ is obtained by integrating the impulse pressure over the body surface,

(2.5)\begin{equation} F_y={-}2\rho L^2 U\int_{s_A}^{s_C}\phi(s)\cos(n,y) \,\textrm{d}s-2\rho UA=\rho mL^2 U, \end{equation}

where $s$ is the arclength coordinate along the body surface, $s_A$ and $s_C$ are the arclengths of points $A$ and $C$, $m$ is the added-mass coefficient and $A$ is the cross-sectional area. The factor ‘$2$’ accounts for the force acting on the part of the body symmetric about the $y$-axis. Owing to the symmetry of the body about the $y$-axis, the horizontal impulse force equals zero.

It is desired to determine the velocity potential $\phi (s)$ immediately after the impact.

3. Conformal mapping

To find the complex potential $w(z)$ directly is a challenge; therefore, we introduce an auxiliary parameter plane, or $\zeta$-plane, as suggested by Michell (Reference Michell1890) and Joukovskii (Reference Joukovskii1890). We formulate boundary-value problems for the complex velocity function, $\textrm {d}w/\textrm {d}z$, and for the derivative of the complex potential, $\textrm {d}w/\textrm {d}\zeta$, both defined in the $\zeta$-plane. Then the derivative of the mapping function is obtained as $\textrm {d}z/\textrm {d}\zeta = (\textrm {d}w/\textrm {d}\zeta )/(\textrm {d}w/\textrm {d}z)$, and its integration provides the mapping function $z = z(\zeta )$ relating the coordinates in the parameter and physical planes.

We choose the first quadrant of the $\zeta$-plane in figure 1(b) as the region corresponding to the fluid region in the physical plane (figure 1a). The conformal mapping theorem allows us to arbitrarily choose the locations of three points, which are points $O$ at the origin ($\zeta =0$), $D$ ($D^\prime$) at infinity and $B$ at $\zeta = 1$ (see figure 1b). The position of points $A$ ($\zeta =a$) and $C$ ($\zeta =c$) has to be determined from the solution of the problem and physical considerations.

3.1. Expressions for the complex velocity and the derivative of the complex potential

The body is considered to be fixed; therefore, the velocity direction and the slope of the body coincide. Besides, at this stage, we assume that the velocity magnitude on the free surface is a known function of the parameter variable, $v(\eta )$. Then the boundary-value problem for the complex velocity in the first quadrant of the parameter plane can be written as follows:

(3.1)$$\begin{gather} \chi (\xi ) = \arg \left(\frac{\textrm{d}w}{\textrm{d}z} \right) = \left\{{ \begin{array}{@{}ll} -\beta_b(a) + \beta_0, & 0 \leq \xi \leq a, \\ - \beta_b(\xi), & a \leq \xi \leq c, \\ - \beta_b(c) - \beta_0, & c \leq \xi < \infty. \end{array}} \right. \end{gather}$$

(3.2)$$\begin{gather}v(\eta ) = \left| {\frac{\textrm{d}w}{\textrm{d}z}} \right|_{\zeta=\textrm{i}\eta}, \quad 0\leq\eta<\infty, \end{gather}$$

where $\beta _0={\rm \pi} /2$, $\beta _b(a)={\rm \pi}$ and $\beta _b(c)=0$. Equation (3.1) satisfies the conditions: $\chi (\xi )=-{\rm \pi} /2$ along the intervals $OA$ and $CD$ on the symmetry line, and $\chi (\xi )=-\beta _b(\xi )$ along the body. The argument of the complex velocity exhibits jumps $\varDelta =-{\rm \pi} /2$ at points $A$ and $C$ when we move along the boundary in the physical plane from point $O$ to point $D$.

This boundary-value problem can be solved by applying the following integral formula (Semenov & Yoon Reference Semenov and Yoon2009):

(3.3)\begin{align} \frac{\textrm{d}w}{\textrm{d}z} = v_{\infty}\exp \left[ {\frac{1}{{\rm \pi} }\int_0^\infty {\frac{\textrm{d}\chi }{\textrm{d}{\xi }}\ln \left( {\frac{\zeta + {\xi }}{\zeta - {\xi }}} \right)\textrm{d}{\xi } - \frac{\textrm{i}}{{\rm \pi} }\int_0^\infty {\frac{\textrm{d}\ln v}{\textrm{d}{\eta }}\ln \left( {\frac{\zeta - \textrm{i}{\eta }}{\zeta + \textrm{i}{\eta }}} \right)\textrm{d}{\eta } + \textrm{i}\chi_{\infty}}}} \right], \end{align}

where $v_\infty =v(\eta )_{\eta \rightarrow \infty }$ and $\chi _\infty =\chi (\xi )_{\xi \rightarrow \infty }$. Substituting (3.1) and (3.2) into (3.3) and evaluating the first integral over the step change at points $\zeta =a$ and $\zeta =c$, we obtain

(3.4)\begin{align} \frac{\textrm{d}w}{\textrm{d}z} &= v_\infty\left(\frac{\zeta - a}{\zeta + a}\right)^{{1}/{2}} \left(\frac{\zeta - c}{\zeta + c}\right)^{{1}/{2}} \nonumber\\ &\quad \times \exp\left[\frac{1}{\rm \pi}\int_a^c{\frac{\textrm{d}\beta_b}{\textrm{d}{ \xi}}\ln \left(\frac{\zeta - \xi}{\zeta + \xi}\right) \textrm{d}{\xi}} - \frac{\textrm{i}}{\rm \pi}\int_0^\infty {\frac{\textrm{d}\ln v}{\textrm{d}{\eta }}\ln \left( {\frac{\zeta - \textrm{i}{\eta }}{\zeta + \textrm{i}{\eta }}} \right)\textrm{d}{\eta}} - \textrm{i}\beta_0 \right]. \end{align}

It can be easily verified that for $\zeta =\xi$ the argument of the right-hand side of (3.4) is the function $-\beta _b(\xi )$, while for $\zeta =\textrm {i}\eta$ the modulus of (3.4) is the function $v(\eta )$, i.e. the boundary conditions (3.1) and (3.2) are satisfied. It can also be seen that the complex velocity function has zeros of order $1/2$, which correspond to the flow around a corner of angle ${\rm \pi} /2$ at points $A$ and $C$.

In order to derive the derivative of the complex potential, we have to analyse its behaviour. Before the impact, the free surface is flat, and it coincides with the $x$-axis (see figure 1a). The pressure along the free surface is constant. Then, as follows from the Euler equation, the velocity generated by the impact is perpendicular to the free surface where the pressure is constant, or the velocity is directed in the $y$-direction. Therefore, the $x$-component of the velocity is zero, and $\textrm {d}w=(\textrm {d}w/\textrm {d}z) \,\textrm {d}z=(v_x-\textrm {i}v_y )\,\textrm {d}x=-\textrm {i}v\,\textrm {d}x$. Thus, the free surface corresponds to the interval $(-\infty ,0)$ on the imaginary axis $\psi$ in the $w$-plane. We have $\textrm {Im}(w)=0$ along the line $OABCD$ due to the impermeability condition on the body $ABC$ and the intervals $OA$ and $CD$ of the symmetry line. At the same time, $\textrm {Re}(w)$ changes from zero at point $O$ to $-\infty$ at infinity $D$. Thus, the flow region in the physical plane corresponds to the third quadrant in the $w$-plane. They are related as $w=-K\zeta$, where $K$ is a positive real number. Then one can obtain

(3.5)\begin{equation} \frac{\textrm{d}w}{\textrm{d}\zeta}={-}K. \end{equation}

The simple form of the complex potential, $w(\zeta )=-K\zeta +w_O$, allows one to exclude the parameter $\zeta$ and obtain the solution in Kirchhoff's form, for which the complex velocity is a function of the complex potential. Here, $w_O$ is the complex potential at point $O$.

The derivative of the mapping function is obtained by dividing (3.5) by (3.4), i.e.

(3.6)\begin{align} \frac{\textrm{d}z}{\textrm{d}\zeta} &={-}K\left(\frac{\zeta + a}{\zeta - a}\right)^{{1}/{2}}\left(\frac{\zeta + c}{\zeta - c}\right)^{{1}/{2}}\nonumber\\ &\quad \times \exp\left[\frac{1}{\rm \pi}\int_a^c{\frac{-\textrm{d}\beta_b}{\textrm{d}{\xi}}\ln \left(\frac{\zeta - \xi}{\zeta + \xi}\right) \textrm{d}{\xi}} + \frac{\textrm{i}}{\rm \pi}\int_0^\infty {\frac{\textrm{d}\ln v}{\textrm{d}{\eta }}\ln \left( {\frac{\zeta - \textrm{i}{\eta }}{\zeta + \textrm{i}{\eta }}} \right)\textrm{d}{\eta}} + \textrm{i}\beta_0 \right]. \end{align}

The integration of this equation yields the mapping function $z = z(\zeta )$ relating the parameter and the physical planes. Equations (3.4) and (3.5) include the parameters $a$, $c$ and $K$ and the functions $v(\eta )$ and $\beta _b(\xi )$, all to be determined from physical considerations and the kinematic boundary condition on the free surface and on the solid boundary $OABCD$.

3.2. Body boundary conditions

The arclengths between points $A$ and $B$, $s_{AB}$, and between points $B$ and $C$, $s_{BC}$, and the depth of submergence are determined as follows:

(3.7a–c)\begin{equation} \int_a^1\left|\frac{\textrm{d}z}{\textrm{d}\zeta} \right|_{\zeta=\xi}=s_{AB}, \quad \int_1^c\left|\frac{\textrm{d}z}{\textrm{d}\zeta} \right|_{\zeta=\xi}=s_{BC}, \quad \int_0^a\left|\frac{\textrm{d}z}{\textrm{d}\zeta} \right|_{\zeta=\xi}=h. \end{equation}

The unknown function $\beta _b (\xi )$ is determined from the following integro-differential equation:

(3.8)\begin{equation} \frac{\textrm{d}\beta_b}{\textrm{d}\xi} = \frac{\textrm{d}\beta_b}{\textrm{d}s} \left|\frac{\textrm{d}z}{\textrm{d}\zeta} \right|_{\zeta=\xi}, \end{equation}

where $\beta _b(s)$ is the given function. Equation (3.8) is solved by iteration using $\textrm {d}\beta _b/\textrm {d}\xi$ in (3.6) known at the previous iteration.

3.3. Free surface boundary conditions

An impulsive impact is characterized by infinitesimally small time interval $\Delta t\rightarrow 0$ such that the position of the free surface does not change during the impact. From the Euler equations, it follows that the velocity generated by the impact is perpendicular to the free surface ($p=p_a$):

(3.9)\begin{equation} \arg \left(\left. \frac{\textrm{d}w}{\textrm{d}z}\right|_{\zeta=\textrm{i}\eta} \right)={-}\beta_0, \quad 0 \le \eta \le \infty. \end{equation}

Taking the argument of the complex velocity from (3.4), we obtain the following integral equation in the function $\textrm {d}\ln v/\textrm {d}\eta$:

(3.10)\begin{align} &\int_0^\infty \frac{\textrm{d}\ln v}{\textrm{d}\eta }\ln \left| \frac{\eta^\prime-\eta}{\eta^\prime+\eta} \right|\textrm{d}\eta^\prime+\tan^{{-}1}\left(\frac{\eta}{a}\right) + \tan^{{-}1}\left(\frac{\eta}{c}\right)\nonumber\\ &\quad + \frac{2}{\rm \pi}\int_a^c \frac{\textrm{d}\beta_b}{\textrm{d}\xi} \tan^{{-}1}\left(\frac{\eta}{\xi}\right)\textrm{d}\xi =0. \end{align}

Equation (3.10) is a Fredholm integral equation of the first kind with a logarithmic kernel. Its solution takes the form (see Appendix A)

(3.11)\begin{equation} v(\eta)=\sqrt{\eta^2+a^2}\sqrt{\eta^2+c^2}\exp\left(\frac{1}{\rm \pi}\int_a^c\frac{\textrm{d}\beta_b}{\textrm{d}\xi}\ln (\eta^2+\xi^2)\,\textrm{d}\xi\right). \end{equation}

Equations (3.7a–c), (3.8) and (3.10) form a closed system of equations in the parameters $a$, $c$ and $K$ and the functions $\beta _b(\xi )$ and $\ln v(\eta )$.

5. Upward impulsive impact

Here, we analyse how the direction of an impact on a submerged body affects the velocity field. In the system of coordinates attached to the body, a change of the impact direction results in a reversal of the velocity direction of the liquid at infinity and on the whole solid boundary, including the body and the symmetry line. Equation (3.1) will keep its form if we set values $\beta _0=-{\rm \pi} /2$, $\beta _b (a)=0$ and $\beta _b(c)=-{\rm \pi}$, or subtract ${\rm \pi}$ from $\chi (\xi )$ in (3.1). In this case, the expression for the complex velocity (3.4) keeps its form. The derivative of the complex potential is obtained from (3.5) if we change the sign of the expression. Thus, all the equations of the problem keep their form. Therefore, the velocity magnitude on the free surface (3.11) remains the same for both upward and downward impact directions.

In the system of coordinates related to the liquid, a change of the impact direction results in a reversal of the velocity direction on the free surface, while the magnitude of the velocity and the added-mass coefficients are identical to the case of downward impact. Figure 5 illustrates the velocity distribution on the free surface corresponding to an upward impact of the plate for the same depths of submergence as shown in figure 2 for a downward impact. The velocity distribution on the free surface near the plate edges predetermines flow separation from the plate edges in later times.

Figure 5.

The same as in figure 2 but opposite (upward) direction of impact.