2.1. Two-dimensional Wagner model for a symmetric body

Within the Wagner model, the fluid is assumed to be inviscid and incompressible. Gravity and surface tension effects are neglected. The flow is assumed to be irrotational and can thus be fully described through a velocity potential $\varphi$. Let us consider the water entry of a symmetric two-dimensional body into a semi-infinite liquid domain initially at rest as described in figure 1. The position of the body contour is defined by the equation $z=f(x)-h(t)$, where $f(x)$ is a function which describes the body shape and $h(t)$ corresponds to the depth of penetration after first contact with water. Variable $c(t)$ represents the half-width of the wetted surface, which is delimited by the turnover points or the root of the jets. Under Wagner's assumptions, the presence of the jet is neglected at leading order. The width of the wetted surface is determined using the so-called Wagner condition, which states that the displacement should be finite at $|x| = c$. Function $\eta (x,t)$ describes the elevation of the free surface induced by the water entry and the vertical velocity of the body is denoted $v(t)$. In the following, the upper-script notation $^{(w)}$ stands for quantities derived from the original Wagner model, i.e. neglecting gravity. In the original Wagner model, the velocity potential satisfies the following linear MBVP:

(2.1a)$$\begin{gather} \Delta\varphi^{(w)} =0, \quad z<0, \end{gather}$$

(2.1b)$$\begin{gather}\varphi^{(w)} = 0, \quad z=0, |x|\geqslant c^{(w)}(t), \end{gather}$$

(2.1c)$$\begin{gather}\varphi_{,z}^{(w)} ={-}v(t), \quad z=0, |x|< c^{(w)}(t), \end{gather}$$

(2.1d)$$\begin{gather}\varphi^{(w)} \rightarrow 0, \quad x^2+z^2\rightarrow\infty. \end{gather}$$

The velocity potential solution of (2.1) is given by the following expression on the body surface:

(2.2)\begin{equation} \varphi^{(w)}(x, z, t) ={-}v(t) \sqrt{c^{(w)}(t)^2-x^2},\quad \lvert x \rvert< c^{(w)}(t),\ z = 0. \end{equation}

Note however that the half-width of the wetted surface, $c^{(w)}$, in (2.1) is a parameter which must be determined using an additional condition, the so-called Wagner condition, which can be formulated as (Korobkin Reference Korobkin1996)

(2.3)\begin{equation} \int_{0}^{{\rm \pi}/2} f(c^{(w)} \sin \gamma)\,\textrm{d} \gamma = \frac{\rm \pi}{2}h(t). \end{equation}

The free-surface elevation, $\eta (x,t)$, can be obtained by integrating in time the vertical fluid velocity, $\varphi ^{(w)}_{,z}$, at the free surface, ($|x|>c^{(w)}(t)$, $z=0$), (Faltinsen Reference Faltinsen2005)

(2.4)\begin{equation} \eta^{(w)}(x,t) =\int_0^t \varphi^{(w)}_{,z}(x,0,\tau)\,\textrm{d}\tau = \int_0^t \left[\frac{v(\tau)x}{\sqrt{x^2-c^{(w)}(\tau)^2}}-v(\tau) \right]\textrm{d}\tau. \end{equation}

For a number of body shapes (e.g. when $f(x)$ is a polynomial function), a closed-form expression of $c^{(w)}$ can be derived using (2.3). As shown later in § 5, this expression can then be substituted into (2.4) to obtain a closed-form expression of the free-surface elevation.

2.2. Two-dimensional Wagner model with gravity

In the Wagner model the original nonlinear free-surface condition is first linearised assuming that there is a small deflection of the free surface and performing a Taylor expansion of the velocity potential in $z=0$. This first step leads to the well-known free-surface condition of linear surface gravity waves,

(2.5)\begin{equation} \varphi_{,t}(x, z, t) ={-}g \eta(x,t), \quad |x|>c(t),z=0. \end{equation}

Neglecting gravity and assuming that the fluid is initially at rest, i.e. $\varphi ^{(w)}(x,z,t=0$), one obtains the original free-surface condition $\varphi ^{(w)}=0$ in (2.1). Retaining the terms depending on gravity in the linear free-surface boundary condition, the MBVP satisfied by the velocity potential becomes

(2.6a)$$\begin{gather} \Delta\varphi =0, \quad z<0, \end{gather}$$

(2.6b)$$\begin{gather}\varphi(x,t) ={-}g \int_0^t\eta(x, \tau) \,\textrm{d}\tau, \quad z=0, |x|\geqslant c(t), \end{gather}$$

(2.6c)$$\begin{gather}\varphi_{,z}(x,t) ={-}v(t),\quad z=0, |x|< c(t), \end{gather}$$

(2.6d)$$\begin{gather}\varphi \rightarrow 0, \quad x^2+z^2\rightarrow\infty. \end{gather}$$

2.2.1. Derivation of the velocity potential accounting for gravity

Using the theory of analytic functions as detailed in Appendix A.1, the velocity potential solution of (2.6) on the wetted surface may be expressed as

(2.7)\begin{equation} \varphi(x, z, t) = \sqrt{c^2-x^2} \left\{{-}v+\frac{2}{\rm \pi}\int_{c}^{+\infty} \frac{\tau \varphi(\tau,t)}{\sqrt{\tau^2-c^2}(\tau^2-x^2)}\textrm{d}\tau \right\}, \quad |x|< c, z=0. \end{equation}

Note that the potential on the wetted surface depends on the potential on the free surface, which itself depends on the unknown free-surface elevation. The integrand in (2.7) is (weakly) singular at $\tau = c$ for $x< c$. In order to compute the velocity potential, and more importantly the time derivative of the velocity potential close to $|x|=c^-$, it is convenient to rearrange (2.7) by performing an integration by parts, leading to

(2.8)\begin{align} \varphi(x, z, t) &={-}v\sqrt{c^2-x^2}+ \frac{2}{{\rm \pi} x} \left\{\textrm{sgn}(x)\frac{c \varphi(c^+, t) {\rm \pi}}{2}\right.\nonumber\\ &\quad \left.+\int_{c}^{+\infty} \arctan{\frac{\tau\sqrt{c^2-x^2}}{x\sqrt{\tau^2-c^2}}} (\varphi(\tau, t) +\tau \varphi_{,x}(\tau, t))\,{\rm d}\tau \right\}, \quad |x|< c, z=0. \end{align}

The time derivative of the velocity potential is obtained by differentiating (2.8) with respect to $t$, leading to

(2.9)\begin{align} \varphi_{,t}(x, z,t) &= \frac{1}{\sqrt{c^2-x^2}}\left\{{-}v c\dot{c} + \int_{c}^{+\infty} \frac{2\tau \dot{c}}{{\rm \pi} c\sqrt{\tau^2-c^2}}(\varphi(\tau,t) +\tau \varphi_{,x}(\tau,t))\,{\rm d} \tau \right\}\nonumber\\ &\quad +\frac{c \varphi_{,t}(c^+,t) }{|x|} + \frac{2}{{\rm \pi} x}\int_{c}^{+\infty} \arctan{\frac{\tau\sqrt{c^2-x^2}}{x\sqrt{\tau^2-c^2}}} (\varphi_{,t}(\tau,t) +\tau \varphi_{,tx}(\tau,t))\,{\rm d} \tau\nonumber\\ &\quad -\dot{v}\sqrt{c^2-x^2}, \quad |x|< c, z=0, \end{align}

where the overdot symbol stands for the total derivative with respect to $t$. The $x$-derivative of the velocity potential is obtained by solving the MBVP satisfied by the complex velocity ($\varphi _{,x}-{\rm i}\varphi _{,z}$) using the theory of analytic functions as detailed in Appendix A.2. It yields

(2.10)\begin{equation} \varphi_{,x}(x, t) = \frac{x v}{\sqrt{c^2-x^2}}-\frac{2x}{{\rm \pi} \sqrt{c^2-x^2}} \int_{c}^{+\infty}\frac{\sqrt{\tau^2-c^2}}{\tau^2-x^2}\varphi_{,x}(\tau, t)\,\textrm{d}\tau, \quad |x|< c. \end{equation}

Equations (2.7) to (2.10) all involve the unknown free-surface elevation through the velocity potential on the free surface. Although gravity may affect the free-surface elevation, one may expect that the deviation of the free-surface elevation from the free-surface elevation obtained with the original Wagner model remains small as far as the effect of gravity remains reasonably small. As a consequence, it seems reasonable to assume that the leading-order terms of the velocity potential on the free surface can be computed using the free-surface elevation obtained with the original Wagner model. Note however that, as observed in Zekri et al. (Reference Zekri, Korobkin and Cooker2021), gravity will tend to reduce the size of the wetted surface; therefore, we suggest to assume that for a similar size of the wetted surface, the free surface obtained with gravity is approximately similar to the free-surface elevation obtained with the original Wagner model. In other words, at time $t$, the half-width of the wetted surface with gravity, $c(t)$, is such that $c(t)=c^{(w)} (t^*)$, with $t^*< t$. The free-surface elevation which must be considered is thus $\eta (x,t) = \eta ^{(w)}(x,t^*)$. Denoting ${c^{(w)}}^{-1}$ the inverse function of ${c^{(w)}}$, we have $t^* = {c^{(w)}}^{-1}(c(t))$. The velocity potential on the free surface thus reads as

(2.11)\begin{equation} \varphi(x,t) ={-}g\int_0^t \eta(x,\tau) \,\textrm{d}\tau \approx{-}g\int_0^t \eta^{(w)}(x,{c^{(w)}}^{{-}1}[c(\tau)]) \,\textrm{d}\tau, \quad |x|>c, \end{equation}

The formal asymptotic analysis of Zekri et al. (Reference Zekri, Korobkin and Cooker2021) confirms that this approximation is reasonable. In Zekri et al. (Reference Zekri, Korobkin and Cooker2021) the free-surface shape is stretched to fit the new size of the wetted surface, but the idea is rather similar as the free-surface shape is assumed to be similar to the free-surface elevation given by the original Wagner model (in the stretched coordinates).

2.2.2. Modification of the wetted surface due to gravity

In order to take into account the effect of gravity on the wetted surface, it is convenient to reformulate the Wagner problem in terms of the displacement potential, $\phi$, defined as the time integral of the velocity potential,

(2.12)\begin{equation} \phi(x, z, t) = \int_0^t \varphi(x, z, \tau) \,\textrm{d} \tau. \end{equation}

The displacement potential satisfies the following MBVP:

(2.13a)$$\begin{gather} \Delta\phi =0, \quad z<0, \end{gather}$$

(2.13b)$$\begin{gather}\phi =\int_0^t \varphi(x,\tau) \,\textrm{d} \tau, \quad z=0, |x|\geqslant c(t), \end{gather}$$

(2.13c)$$\begin{gather}\phi_{,z} = f(x)-h(t),\quad z=0, |x|< c(t), \end{gather}$$

(2.13d)$$\begin{gather}\phi \rightarrow 0, \quad x^2+z^2\rightarrow\infty. \end{gather}$$

This MBVP, in which the displacement potential is not equal to zero on the free surface, is solved in Moore, Ockendon & Oliver (Reference Moore, Ockendon and Oliver2013). A more detailed derivation of the solution is presented in Appendix B. Making a slightly different change of variable, the Wagner condition becomes

(2.14)\begin{equation} \int_{0}^{{\rm \pi}/2} \left( f(c \sin \gamma)-h(t) + \frac{\phi_{,x}\left(\dfrac{c}{\sin \gamma},t\right)}{\sin \gamma}\right)\textrm{d} \gamma = 0. \end{equation}

In the general case, $c(t)$ cannot be extracted from the last term in (2.14) to express $c$ as a function of the other parameters because the term $\phi _{,x}$ on the free surface derives from successive integrals with respect to time. In order to solve this equation, it is convenient to differentiate (2.14) with respect to time, which leads to the following first-order differential equation:

(2.15)\begin{equation} \dot{c} ={-}\frac{\displaystyle\int_{0}^{{\rm \pi}/2} \left({-}v(t) +\phi_{,xt}\left.\left(\dfrac{c}{\sin \gamma},t\right)\right/\sin \gamma \right)\textrm{d} \gamma}{\displaystyle\int_{0}^{{\rm \pi}/2}\left( f_{,x}(c \sin \gamma)\sin \gamma + \phi_{,xx}\left.\left(\dfrac{c}{\sin \gamma},t\right)\right/\sin^2 \gamma\right)\textrm{d} \gamma}. \end{equation}

In the following this equation is solved numerically using the first-order explicit Euler method. Note that the wetted surface at a time instant $t$ is fully determined by the position of the body $h(t)$ at this time instant through (2.3) when gravity is ignored. However, this is no longer true when gravity is considered because the displacement potential which appears in the second term of (2.14) is defined by a double integral over time. Similarly to § 2.2, we suggest using the free-surface elevation obtained from the original Wagner model in order to simplify the computation of the displacement potential in (2.15) as follows:

(2.16)\begin{align} \phi(x, t) & ={-}g\int_0^t \int_0^\nu \eta(x,\tau) \,\textrm{d}\tau \,\textrm{d}\nu \nonumber\\ &\approx{-}g\int_0^t \int_0^\nu \eta^{(w)}(x,{c^{(w)}}^{{-}1}(c(\tau)) \, \textrm{d}\tau \,\textrm{d}\nu, \quad |x|>c. \end{align}

Once the wetted surface and the partial derivatives of the velocity potential with respect to $t$ and $x$ are known, it is possible to compute the pressure acting on the body.

2.2.3. Computation of the pressure and force acting on the body

In the following the pressure distribution is obtained using the MLM proposed by Korobkin (Reference Korobkin2004). This model allows us to take into account the shape of the body and was shown to be accurate in terms of hydrodynamic force prediction (Tassin et al. Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2010). Writing the MLM under the form given in Tassin et al. (Reference Tassin, Piro, Korobkin, Maki and Cooker2013) and adding the hydrostatic component of Bernoulli's equation, the pressure acting on the body surface, $P$, can be expressed as

(2.17)\begin{equation} P(x,t) ={-}\rho\left(\varphi_{,t} + \frac{\varphi_{,x}^2 } {2(1+f_{,x}^2)}+(f-h)\dot{v}+\frac{v^2}{2} + (f-h)g\right), \end{equation}

where $\varphi _{,t}$ and $\varphi _{,x}$ are given by (2.9) and (2.10), respectively. This equation presents a non-integrable singularity at $|x| = c(t)^-$. In order to compute the force acting on the body, we follow the procedure proposed by Tassin et al. (Reference Tassin, Piro, Korobkin, Maki and Cooker2013) which consists in splitting (2.17) into two terms, $P_v$ and $P_a$, in the integration

(2.18)\begin{equation} F(t) = \int_{{-}c^*(t)}^{c^*(t)}P_v(x,t)\, \textrm{d} x + \int_{{-}c(t)}^{c(t)}P_a(x,t) \, \textrm{d} x, \end{equation}

where $c^*(t)$ corresponds to the closest point $x$ to $c(t)$ where $P_v(x,t)$ is positive and $P_a$ comprises only terms depending on acceleration,

(2.19)\begin{equation} P_a ={-}\rho(\dot{v}\sqrt{c^2-x^2}+ (f-h) \dot{v}). \end{equation}

Here $P_v$ comprises the remaining pressure terms and includes all the terms resulting from the action of gravity.

3.1. Axisymmetric Wagner model without gravity

The Wagner model also applies to the case of an axisymmetric body penetrating water where the problem is now expressed in the cylindrical coordinates $(r,\theta,z)$. In the original axisymmetric Wagner problem, the velocity potential satisfies the following MBVP:

(3.1a)$$\begin{gather} \Delta\varphi^{(w)} =0, \quad z<0, \end{gather}$$

(3.1b)$$\begin{gather}\varphi^{(w)} = 0, \quad z=0, r\geqslant c^{(w)}(t), \end{gather}$$

(3.1c)$$\begin{gather}\varphi^{(w)}_{,z} ={-}v(t), \quad z=0, r< c^{(w)}(t), \end{gather}$$

(3.1d)$$\begin{gather}\varphi^{(w)} \rightarrow 0, \quad r^2+z^2\rightarrow\infty. \end{gather}$$

The velocity potential solution of (3.1) is given by the following expression on the body surface:

(3.2)\begin{equation} \varphi^{(w)}(r, \theta,z,t) = \frac{-2v}{\rm \pi} \sqrt{{c^{(w)}}^2-r^2}, \quad z = 0. \end{equation}

The radius of the wetted surface is governed by the Wagner condition which may be expressed as (see Korobkin & Scolan Reference Korobkin and Scolan2006)

(3.3)\begin{equation} \int_{0}^{{\rm \pi}/2}\sin \gamma {\cdot} f(c^{(w)}(t) \sin \gamma)\,\textrm{d} \gamma = h(t). \end{equation}

The vertical velocity at the free surface is given by Scolan & Korobkin (Reference Scolan and Korobkin2001) and reads as

(3.4)\begin{equation} \varphi_{,z}^{(w)}(r,\theta,z,t) = \frac{-2 v(t)}{\rm \pi}\left\{\arcsin\left(\frac{c^{(w)}(t)}{r}\right)-\frac{c^{(w)}(t)} {\sqrt{r^2-c^{(w)}(t)^2}}\right\}, \quad z = 0. \end{equation}

The free-surface elevation, $\eta (r,t)$, can be obtained by integrating (3.4) with respect to time.

3.2. Axisymmetric Wagner model with gravity

Taking into account the effect of gravity in the axisymmetric Wagner problem, the velocity potential satisfies the following MBVP:

(3.5a)$$\begin{gather} \Delta\varphi =0, \quad z<0, \end{gather}$$

(3.5b)$$\begin{gather}\varphi ={-}g \int_0^t \eta(r,\tau)\,\textrm{d}\tau, \quad z=0, r\geqslant c(t), \end{gather}$$

(3.5c)$$\begin{gather}\varphi_{,z} ={-}v(t), \quad z=0, r< c(t), \end{gather}$$

(3.5d)$$\begin{gather}\varphi \rightarrow 0, \quad r^2+z^2\rightarrow\infty. \end{gather}$$

As the Laplace equation is linear, the velocity potential, $\varphi$, solution of (3.5) may be sought in the form $\varphi =\varphi ^{(w)}+\varphi ^{(g)}$, with $\varphi ^{(g)}$ solution of the following MBVP:

(3.6a)$$\begin{gather} \Delta\varphi^{(g)} =0,\quad z<0, \end{gather}$$

(3.6b)$$\begin{gather}\varphi^{(g)} ={-}g \int_0^t \eta(r,\tau)\,\textrm{d}\tau, \quad z=0, r\geqslant c(t), \end{gather}$$

(3.6c)$$\begin{gather}\varphi^{(g)}_{,z} = 0, \quad z=0, r< c(t), \end{gather}$$

(3.6d)$$\begin{gather}\varphi^{(g)} \rightarrow 0, \quad r^2+z^2\rightarrow\infty. \end{gather}$$

The normal derivative of the velocity potential being equal to zero over the disc $r< c(t)$, the velocity potential on the wetted surface can be computed using the following equation given in Iafrati & Korobkin (Reference Iafrati and Korobkin2008) and referred to as Sneddon's formula:

(3.7)\begin{equation} \varphi^{(g)} (r, z ,t)= \frac{2}{\rm \pi} \int_0^{+\infty} \frac{\varphi^{(g)}(\sqrt{(c^2-r^2) \tau ^2 + c^2},z=0,t)}{\tau^2+1} \textrm{d} \tau, \quad r<c, z = 0. \end{equation}

Similarly to § 2, we suggest to approximate the free-surface elevation when computing the velocity potential over the free surface on the right-hand side of (3.7) as

(3.8)\begin{equation} \varphi^{(g)}(r, z,t) \approx{-}g\int_0^t \eta^{(w)}(r,{c^{(w)}}^{{-}1}(c(\tau)) \,\textrm{d}\tau, \quad r>c,z=0, \end{equation}

where $\eta ^{(w)}(r,t)$ is obtained by integrating (3.4) with respect to time.

3.3. Modification of the wetted surface due to gravity

Similarly to the two-dimensional case, it is necessary to take gravity into account when computing the wetted surface radius (in the Wagner condition).

Moore & Oliver (Reference Moore and Oliver2014) proposed an analytical formulation to derive the wetted surface radius for an axisymmetric body considering a non-zero potential on the free surface. However, this paper contains an error and, for this reason, the formulation was not applied here. An erratum to the formulation should be published soon. Therefore, we suggest to adapt the numerical method based on the BEM proposed in Tassin et al. (Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2012) in order to solve the modified Wagner problem with gravity.

3.3.2. Computation of the free-surface elevation

In order to calculate the free-surface elevation for a given value of the wetted surface radius, $c(t)$, it is convenient to reformulate the Wagner problem in terms of the displacement potential, $\phi$, defined as the time integral of the velocity potential and which satisfies the following MBVP:

(3.9a)$$\begin{gather} \Delta\phi =0, \quad z<0, \end{gather}$$

(3.9b)$$\begin{gather}\phi =\int_0^t \varphi(r,\tau) \,\textrm{d} \tau,\quad z=0, r\geqslant c(t), \end{gather}$$

(3.9c)$$\begin{gather}\phi_{,z} = f(r)-h(t), \quad z=0, r< c(t), \end{gather}$$

(3.9d)$$\begin{gather}\phi \rightarrow 0, \quad r^2+z^2\rightarrow\infty. \end{gather}$$

Following Tassin et al. (Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2012), the displacement potential at a point $\boldsymbol {p}$ located at $z=0$ may be expressed as follows using Green's third identity:

(3.10)\begin{equation} \tfrac{1}{2}\phi(\boldsymbol{p},t) = \iint_{WS}\phi_{,z}(\boldsymbol{q},t)G(\boldsymbol{p},\boldsymbol{q}) \,\textrm{d}S_{\boldsymbol{q}} + \iint_{FS}\phi_{,z}(\boldsymbol{q},t)G(\boldsymbol{p},\boldsymbol{q}) \,\textrm{d}S_{\boldsymbol{q}}. \end{equation}

Here $G(\boldsymbol {p},\boldsymbol {q})$ is the free-space Green function defined as $G(\boldsymbol {p},\boldsymbol {q}) =[4{\rm \pi} |\boldsymbol {p}-\boldsymbol {q}|]^{-1}$, $\boldsymbol {q}$ is a point located at $z=0$, $WS$ denotes the wetted surface and $FS$ denotes the free surface. Similarly to Tassin et al. (Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2012), we use (3.10) and a collocation method to compute numerically the free-surface elevation, $\eta =\phi _{,z}$. The collocation method consists in enforcing (3.10) in a number of points $\boldsymbol {p}_i$ located on the free surface. The free surface is discretized in $N_e$ intervals over which the free-surface elevation is assumed to be piecewise linear,

(3.11)\begin{equation} \eta(\xi) = \frac{\eta(\xi_{k+1})-\eta(\xi_k)}{\xi_{k+1}-\xi_{k}} (\xi-\xi_k) +\eta(\xi_k), \quad \xi \in [\xi_k, \xi_{k+1}], \end{equation}

where $\xi =r/c(t)$. The free-surface elevation is assumed to be small enough beyond a threshold value $\xi _{max}=5$ so that it has a marginal contribution to the second integral on the right-hand side of (3.10). Making use of this discretization, the integral on the free surface in (3.10) is split into $N_e+1$ terms corresponding to the coefficient of influence of each unknown, $\eta _k=\eta (\xi _k)$. The evaluation of (3.10) at $N_e+1$ collocation points $(\boldsymbol {p}_i)$ allows us to write the following linear system of equations:

(3.12)\begin{equation} \boldsymbol{A}\boldsymbol{X} = \boldsymbol{B}+ \boldsymbol{B}^{(g)}. \end{equation}

Here $\boldsymbol {X}={(\eta _1,\ldots, \eta _{N_e+1})^{\rm T}}$, $\boldsymbol {A}$ results from the integral over the free surface in (3.10) and $\boldsymbol {B}$ results from the integration over the wetted surface. Their expression can be found in Tassin et al. (Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2012). In contrast to Tassin et al. (Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2012), an additional term $\boldsymbol {B}^{(g)}=1/2[\phi (\boldsymbol {p}_1),\ldots, \phi (\boldsymbol {p}_{N_e+1})]^{\rm T}$ appears because the displacement potential is not equal to 0 over the free surface when gravity is included. Similarly to (2.16), the displacement potential on the free surface may be approximated as

(3.13)\begin{align} \phi(\boldsymbol{p}_i,t) ={-}\int_0^t\int_0^\nu g \eta(\xi_{\boldsymbol{p}_i} c(\tau),\tau) \,\textrm{d}\tau \,\textrm{d}\nu \approx{-}\int_0^t\int_0^\nu g \eta^{(w)}(\xi_{\boldsymbol{p}_i} c(\tau),{c^{(w)}}^{{-}1}(c(\tau)))\,\textrm{d}\tau \,\textrm{d}\nu, \end{align}

where $\eta ^{(w)}$ is obtained as the time integral of the vertical velocity (3.4). With this approximation, the $i$th coefficient of $\boldsymbol {B}^{(g)}$ at a time instant $t_{n+1}$ reads as

(3.14)\begin{equation} \boldsymbol{B}^{(g)}_i(t_{n+1}) ={-}\tfrac{1}{2}\int_0^{t_n}\int_0^\nu g \eta^{(w)}(\xi_{\boldsymbol{p}_i} c(\tau),{c^{(w)}}^{{-}1}(c(\tau)))\,\textrm{d}\tau \,\textrm{d}\nu, \end{equation}

where we only integrate with respect to time until $t_n$ to avoid numerical issues at the first iteration (the free-surface elevation becomes singular when we set $c(t_{n+1})=c(t_n)$). The term $\boldsymbol {B}^{(g)}$ could be evaluated using the time history of the wetted surface elevation computed through the BEM. However, in order to obtain results valid under the same assumptions as in the two-dimensional case, it is evaluated using the analytical expression of the free-surface elevation, which also simplifies the computation. The resolution of the linear system (3.12) allows us to obtain the free-surface elevation at time $t_{n+1}$ assuming that $c(t_{n+1})=c(t_{n+1})_m$.

3.3.3. Enforcing the Wagner condition

Once the free-surface elevation has been computed, its value at the contact point is known and the corresponding error on the Wagner condition can be computed as

(3.15)\begin{equation} e(c(t_{n+1})_m) = \eta(\xi_0,t_{n+1})_m - f(c(t_{n+1})_m) + h(t_{n+1}). \end{equation}

For the next iteration, the value of $c(t_{n+1})_m$ is updated as

(3.16)\begin{equation} c(t_{n+1})_{m+1} = c(t_{n+1})_m+ \alpha_m\,e(c(t_{n+1})_m), \end{equation}

where $\alpha _m$ is a dynamic relaxation coefficient which is initialized and updated in a similar way as in Tassin et al. (Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2012). The error resulting from the updated value $c(t_{n+1})_{m+1}$ can now be computed. This process is repeated until the error reaches an acceptable value. In our simulations, this procedure was initialized with $c(t_{n+1})_0=c(t_n)$, $c(t_n)$ being the converged value of $c(t)$ at $t=t_n$.

4.1. Hybrid BEM approach

Similarly to the Wagner theory, the water impact problem is here faced under the assumptions of an inviscid and incompressible fluid. The flow is assumed irrotational, which allows us to formulate the problem in terms of the velocity potential $\varphi$. The governing equations, written in an earth-fixed frame of reference, with $y$ the horizontal axis coinciding with the still water level, and $z$ the vertical axis oriented upwards, are

(4.1a)$$\begin{gather} {\nabla^{2}\varphi} = 0, \quad \varOmega, \end{gather}$$

(4.1b)$$\begin{gather}{\varphi,_{n}} ={-}\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{n}, \quad WS, \end{gather}$$

(4.1c)$$\begin{gather}{\frac{{\rm D}\varphi}{{\rm D}t}} = {\frac{\left| \boldsymbol{\nabla} \varphi \right|^{2} }{2}-gz}, \quad FS, \end{gather}$$

(4.1d)$$\begin{gather}{\frac{{\rm D}\boldsymbol{x}}{{\rm D}t}} = \boldsymbol{u}, \quad FS, \end{gather}$$

where $\boldsymbol {v}$ is the body vertical velocity (positive downwards), $\varOmega$ is the fluid domain, $WS$ and $FS$ are the wetted surface and the free surface, respectively, $\boldsymbol {n}$ is the unit vector normal to the boundary of the fluid domain oriented inwards, ${{\rm D}}/{{\rm D}t}$ is the total derivative with respect to time $t$, $\boldsymbol {x}$ is the position of a particle lying at the free surface and $\boldsymbol {u}$ is the fluid velocity at $\boldsymbol {x}$. The problem is solved through a mixed Eulerian–Lagrangian approach inspired from Longuet-Higgins & Cokelet (Reference Longuet-Higgins and Cokelet1976) and the velocity potential is written in terms of a boundary integral representation. Enforcing it at the boundary of the fluid domain, a boundary integral equation of mixed first and second kind, for the velocity potential and its normal derivative on the free surface is obtained,

(4.2) \begin{equation} \frac{1}{2}\varphi(\boldsymbol{p})=\int_{WS\bigcup FS}{\left( \frac{\partial \varphi (\boldsymbol{p})} {\partial \boldsymbol{n}} G(\boldsymbol{p},\boldsymbol{q}) -\varphi(\boldsymbol{q}) \frac{\partial G(\boldsymbol{p},\boldsymbol{q})}{\partial \boldsymbol{n}}\right) {\rm d}S_{\boldsymbol{q}}}, \end{equation}

where $G$ is the free-space Green function for the Laplace operator. In two dimensions, the Green function is defined by $G(\boldsymbol {p},\boldsymbol {q}) = {\log (|\boldsymbol {p}-\boldsymbol {q}|)}/{2{\rm \pi} }$, and in three dimensions $G(\boldsymbol {p},\boldsymbol {q}) =[4{\rm \pi} |\boldsymbol {p}-\boldsymbol {q}|]^{-1}$. The solution is arranged in two steps. First, the boundary value problem formed by (4.1a), (4.1b) and the current value of the velocity potential at the free surface is solved. This provides the velocity potential on the wetted surface and its normal derivative on the free surface, along with the tangential velocity which is obtained through differentiation of the velocity potential along the boundary. Once the velocity field along the free surface is completely determined, the second step of the solution scheme is performed: the dynamic and kinematic free-surface conditions, (4.1c) and (4.1d), are integrated in time to compute the new free-surface shape and the velocity potential on it. The solution is achieved numerically through a zero-order discretization in space, thus approximating the variables with a piecewise constant distribution on straight panels, and a second-order Runge–Kutta scheme for the integration in time.

The hydrodynamic load acting on the impacting body is computed through the integration of the pressure distribution over the wetted surface, which is calculated by using the unsteady Bernoulli equation,

(4.3)\begin{equation} p-p_{inf}={-}\rho\left(\varphi_{,t}+\frac{\left| \boldsymbol{\nabla} \varphi \right|^{2} }{2}+gz \right), \end{equation}

where $p_{inf}$ is the pressure on the undisturbed fluid. In (4.3) the time derivative of the velocity potential along the body, $\varphi _{,t}$, is unknown and it is evaluated by formulating a boundary value problem similar to (4.1) which is solved numerically by the BEM (see Battistin & Iafrati Reference Battistin and Iafrati2003).

4.2. Jet model

During the impact, the water rises along the body and a thin jet is formed due to the flow singularity at the intersection between the free surface and the body surface (Greenhow Reference Greenhow1987). An accurate modelling of this thin jet with the BEM approach would require a very fine local discretization and, owing to the local high speed of the fluid, a quite small time step would be necessary to guarantee the stability and the accuracy of the time integration scheme. Consequently, the computational effort would increase substantially. Commonly, for limiting the computational effort, the jet is cutoff and replaced by a panel orthogonal to the body contour where appropriate boundary conditions are applied. This procedure allows us to obtain, despite the approximation, an accurate estimation of the hydrodynamic load (Zhao & Faltinsen Reference Zhao and Faltinsen1993; Battistin & Iafrati Reference Battistin and Iafrati2003). However, an evident drawback is the loss of details in the description of the fluid motion inside the thin jet, which are important in some phenomena (e.g. flow separation). The jet model presented in Battistin & Iafrati (Reference Battistin and Iafrati2004) is adopted here to preserve a detailed description of the flow in the jet. This model is based on the discretization of a part of the jet region in control volumes, which are bounded by the wetted surface and the free surface (see figure 2), where the velocity potential is written in the form of a harmonic polynomial series, up to second order, about the corresponding centroid ($x^*,z^*$),

(4.4)\begin{align} \varphi_{i}^j(x,z)&=A_i+B_i(x-x^*)+C_i(z-z^*)+\tfrac{1}{2}D_i[(x-x^*)^2 -(z-z^*)^2]\nonumber\\ &\quad +E_i(x-x^*)(z-z^*). \end{align}

Figure 2. Sketch of the jet region.

The coefficients $A_{i},\ldots,E_{i}$ are additional unknowns, so five additional equations, for each control volume, are required. As the vertices of each volume correspond to the panel centroids ($\bar {P}_{i-1}$, $\bar {P}_{i}$ on the body side, and $\hat {P}_{i-1}$, $\hat {P}_{i}$ on the free-surface side), four of them are derived by enforcing the boundary conditions at the body and free-surface sides,

(4.5a–d)\begin{align} \varphi_{i,n}^j(\bar{P}_{i-1})=\varphi_{,n}(\bar{P}_{i-1}), \quad \varphi_{i,n}^j(\bar{P}_{i})=\varphi_{,n}(\bar{P}_{i}), \quad \varphi_{i}^j(\hat{P}_{i-1})=\varphi(\hat{P}_{i-1}), \quad \varphi_{i}^j(\hat{P}_{i})=\varphi(\hat{P}_{i}), \end{align}

where $\varphi _{i,n}^j$ is the normal derivative of the velocity potential provided by the harmonic expression. The right-hand side of the first two equations of (4.5a–d) are known from the body boundary condition (4.1b), whereas the right-hand side of the last two equations of (4.5a–d) are known from the time integration of the dynamic boundary condition (4.1c), as explained in § 4.1. The fifth condition is the following matching condition which ensures the continuity of the normal derivative of the velocity potential (on the free-surface side) at adjacent control volumes,

(4.6)\begin{equation} \varphi_{i,n}^j(\hat{P}_{i-1})=\varphi_{(i-1),n}^j(\hat{P}_{i-1}). \end{equation}

It is worth noting that at the first edge on the free-surface side of the first control volume, the condition (4.6) enforces the matching of the normal derivative of the velocity potential computed by the simplified model with that resulting from the solution of the boundary integral problem.

By introducing the jet model, i.e. when the thin jet has already developed, the discretizated form of the boundary integral equation (4.2), used in the remaining part of the fluid domain, is coupled with the set of (4.5a–d) and (4.6). In this way, the unknown coefficients of the expansions and the unknown velocity potential and its normal derivative are computed simultaneously. This hybrid BEM approach is also adopted to compute numerically the time derivative of the velocity potential $\varphi _{,t}$ which is needed to evaluate the pressure acting on the body contour.

5.1. Expression of the problem for the case of a wedge

The body shape of an infinite wedge is defined by the function

(5.1)\begin{equation} f(x) = \tan \beta |x|. \end{equation}

Substituting (5.1) into (2.3), the expression of the half-width of the wetted surface is obtained as

(5.2)\begin{equation} c^{(w)}(t) = \frac{{\rm \pi} h(t)}{2\tan \beta}. \end{equation}

Substituting (5.2) into (2.4) and integrating with respect to $t$, we obtain the following expression for the free-surface elevation under the assumptions of the original Wagner model,

(5.3)\begin{equation} \eta^{(w)}(x,t) = \frac{2 \tan\beta}{\rm \pi}\left[x\arcsin{\frac{c^{(w)}(t)}{x}} -c^{(w)}(t)\right]. \end{equation}

Substituting (5.3) into (2.11), one obtains the following approximation for the velocity potential on the free surface:

(5.4)\begin{equation} \varphi(x, z , t) \approx{-}\frac{2 g \tan \beta}{\rm \pi} \int_{0}^{t} \left( x \arcsin{\frac{c(\tau)}{x}}- c(\tau) \right)\textrm{d} \tau, \quad |x| >c,z =0. \end{equation}

The expression of the velocity potential and its derivatives at the free surface can now be substituted into (2.9) and (2.10) in order to compute the pressure on the wetted surface and into (2.15) in order to compute the width of the wetted surface.

5.2. Water entry of a $15^\circ$ deadrise-angle wedge at constant velocity and with gravity

In this section we investigate the water entry of a wedge with a deadrise angle of $15^\circ$. The wedge enters water at a constant vertical velocity $v = 0.5\ {\rm m}\ {\rm s}^{-1}$. The evolution of the wetted surface half-width obtained from the SAM and the FNPF solver is depicted in figure 3(a). The results obtained from the two approaches neglecting gravity ($g=0\ {\rm m}\ {\rm s}^{-2}$) are also depicted for comparison. One can see that gravity tends to reduce the size of the wetted surface and that the results obtained from the two approaches are in very good agreement. The latter is confirmed by the relative difference between the two approaches shown in figure 3(b). In the first instants, the relative difference is high because the FNPF simulation starts with a non-zero initial penetration depth which induces a short transitory stage at the beginning of the simulation. Then, the relative difference between the two approaches quickly decreases to $0.01\,\%$ for the case without gravity and to $0.5\,\%$ for the case with gravity. The pressure distributions obtained for different Froude numbers are depicted in figure 4(a). Note that the Froude number is defined as $Fr = V/\sqrt {g h(t)}$ and, therefore, the different instantaneous Froude numbers correspond to different time instants. Owing to the effect of gravity, the pressure is no longer self-similar. In fact, the peak pressure decreases in time (i.e. when the Froude number decreases) and its position changes. For small values of $Fr$, the maximum value of the pressure is no longer at the jet root but at the centre of the wedge where the hydrostatic contribution is maximum. The pressure distribution obtained by adding the term ‘$-\rho g z$’ to the original MLM formulation, i.e. neglecting the effect of gravity on the wetted surface width and on the velocity potential, is also depicted in figure 4(a). This latter comparison shows how the new semi-analytical formulation improves the prediction of the pressure distribution compared with the more simplistic approach consisting in adding the hydrostatic contribution to the original MLM pressure model.

Figure 3. Comparison of the wetted surface half-width predicted by the SAM and the FNPF solver for a $15^\circ$ deadrise-angle wedge entering water at constant velocity with and without gravity. (a) Evolution of the wetted surface half-width as a function of time. (b) Evolution of the relative difference between the two approaches, $(c_{FNPF} - c_{SAM})/c_{FNPF}$, as a function of time.

Figure 4. Comparison of the pressure coefficient, $C_p = 2p/({\rho v^2})$, distribution along a $15^\circ$ deadrise-angle wedge entering water at constant speed. (a) Semi-analytical results accounting for the modification of the wetted surface due to gravity. (b) Semi-analytical results obtained using the wetted surface predicted by the original Wagner model. For comparison purposes, a dashed-dotted line computed as the sum of the hydrostatic term and the MLM has been added.

In order to illustrate the consequences of the wetted surface reduction under the action of gravity on the pressure distribution, the pressure distribution obtained at the same time instants with the SAM but neglecting the effect of gravity on the wetted surface are plotted in figure 4(b). This comparison clearly shows that taking into account gravity in the Wagner condition not only improves the prediction of the wetted surface width but also improves the overall prediction of the pressure distribution, and, in particular, the prediction of the pressure peak value. The force coefficient predictions obtained from the two approaches are depicted in figure 5(a). The relative difference between the two approaches is depicted in figure 5(b). A good agreement between the two models is obtained in terms of force, the relative difference remaining under $4\,\%$. Most importantly, it is very interesting to note that the SAM remains very accurate in terms of force prediction even when the gravity contribution to the total force becomes of the same order of magnitude as the slamming force at infinite Froude number (figure 5a). The force prediction obtained by adding the hydrostatic force to the original MLM, which is also depicted in figure 5(a), is less accurate than the SAM with gravity. Note that adding the hydrostatic pressure term to the original MLM model leads to an underestimation of the pressure in the centre of the wedge and to an overestimation of the wetted surface (see figure 4), but by integrating the pressure over the wetted surface these two errors of opposite sign compensate partially each other and the force prediction appears to be reasonably good (see figure 5a).

Figure 5. Non-dimensional force acting on a wedge represented as a function of the inverse of the Froude number squared. (a) Force coefficient. (b) Relative difference $(F_{SAM}-F_{FNPF})/F_{FNPF}$ between the semi-analytical results and the numerical results.

The results presented in this section showed that the SAM proposed in § 2 gives accurate results, both in terms of force and pressure distribution, for a wedge with a deadrise angle of $15^\circ$ penetrating water vertically with a constant velocity. It appears that it is important to take into account the effect of gravity on the wetted surface to obtain an accurate prediction of both the pressure distribution and the resulting force. In the next section the effect of gravity during the water entry of wedges with different deadrise angles is investigated.

5.3. Relative influence of gravity during the water entry of a wedge at constant speed depending on the deadrise angle

In the previous section we observed that the action of gravity on the water entry of a $15^\circ$ deadrise-angle wedge increased as the Froude number decreased. However, as shown by Yan & Liu (Reference Yan and Liu2011) in the case of a cone, one may expect that the importance of gravity at a given value of $Fr$ will differ for wedges of different deadrise angles. Indeed, based on the developments presented in Appendix C, we expect that the effect of gravity on the non-dimensional force coefficient and on the non-dimensional wetted surface width is governed by the non-dimensional parameter $Fr^* = Fr/\sqrt {\tan \beta }$, which we may refer to as the ‘effective Froude number’. Simulations of wedges entering water at constant velocity have been performed for deadrise-angle values ranging from 10 to $30^\circ$ with the SAM in order to confirm the primary role played by this parameter. The evolution of the force coefficient and of the gravity contribution to the non-dimensional force coefficient as a function of the inverse of the effective Froude number squared are depicted in figures 6(a) and 6(b), respectively. Comparisons with numerical results obtained from the FNPF solver for $\beta =10$, 15, 20 and $30^\circ$ are also shown. One can see that the semi-analytical results almost overlap and that the numerical results are in very good agreement with the semi-analytical ones, hence demonstrating that the importance of gravity during the water entry of a wedge is governed by the effective Froude number, $Fr^*$. The evolution of the non-dimensional half-width of the wetted surface is depicted in figure 7 as a function of the inverse of the effective Froude number squared. One can see that almost all the results overlap, hence confirming that the modification of the non-dimensional wetted surface half-width due to gravity is governed by the effective Froude number. Considering the effective Froude number thus allows us to account for the geometry of the impactor when estimating the relative effect of gravity.

Figure 6. Evolution of the non-dimensional force coefficient during the water entry of a wedge at constant velocity as a function of the inverse effective Froude number squared for different deadrise angles. (a) Total force coefficient and (b) gravitational component of the force coefficient. Here $F_\infty$ denotes the force computed without including gravity, which is to say that the Froude number is infinite.

Figure 7. Evolution of the non-dimensional half-width of the wetted surface during the water entry of a wedge at constant velocity as a function of the inverse effective Froude number squared for different deadrise angles. (a) Non-dimensional half-width and (b) gravitational component of the half-width. Here $c_\infty$ denotes the half-width of the wetted surface computed without including gravity, which is to say that the Froude number is infinite. The analytical results are so close that they are indistinguishable.