## Abstract

The water entry problem of a wedge through free fall in three degrees of freedom is studied through the velocity potential theory for the incompressible liquid. In particular, the effect of the body rotation is taken into account, which seems to have been neglected so far. The problem is solved in a stretched coordinate system through a boundary element method for the complex potential. The impact process is simulated based on the time stepping method. Auxiliary function method has been used to decouple the mutual dependence between the body motion and the fluid flow. The developed method is verified through results from other simulation and experimental data for some simplified cases. The method is then used to undertake extensive investigation for the free fall problems in three degrees of freedom.

## 1. Introduction

Fluid/structure impact problems have a wide range of important applications. One such impact problem is that a body enters a free surface. Impact usually lasts for a very short period of time and the effects of gravity and the viscosity of the liquid are usually ignored (Batchelor 1967, p. 471). As a result, the velocity potential can be introduced, which satisfies the Laplace equation when the compressibility of the liquid is ignored. A major difficulty is, however, the boundary conditions on the free surface, which are fully nonlinear and are imposed on a surface which is unknown and is part of the solution itself. Dobrovol’skaya (1969) considered the problem of a two-dimensional symmetric wedge entering water at constant speed. Because there was no length scale, the problem became self-similar. In other words, the spatial variables and the time variable could be combined. She used the conformal mapping method and obtained the solution which satisfied the nonlinear free surface boundary conditions. The problem was further considered by Zhao & Faltinsen (1993). In particular, they solved the Laplace equation using the boundary element method together with the time matching method. Semenov & Iafrati (2006) considered the problem of vertical water entry of an asymmetric wedge using a kind of mapping method for the complex potential, while Xu *et al.* (2008) solved the problem of oblique water entry of an asymmetric wedge using the boundary element method for the complex potential. There are many other studies based on various simplified methods or asymptotic expansion, including those by Howison *et al.* (1991), Fraenkel & McLeod (1997) and Mei *et al.* (1999).

For the non-constant speed entry, an important work is that by Zhao *et al.* (1996). Wu *et al.* (2004) further considered the water entry problem of a symmetric wedge through free fall using both experiment and numerical simulation. They adopted fully nonlinear velocity potential theory in their simulation. As a result, the body motion and the fluid flow are fully coupled, or one requires the information from the other in advance before the solution procedure can proceed. To decouple this mutual dependence, Wu *et al.* (2004) adopted the auxiliary function method (Wu & Eatock Taylor 1996, 2003). This enabled them to obtain the body acceleration without the knowledge of the pressure distribution over the wedge surface. They compared their results with the experiment and found that the agreement was satisfactory.

The present paper considers water entry of an asymmetric wedge through free fall in three degrees of freedom. In other words, the body enters water with horizontal, vertical and rotational velocities. Such a case in fact is quite common for a surface ship, when its bow emerges from water and then re-enters water with heave, sway and roll velocities. In previous publications, there has been hardly any attempt to include the effect of rotation. The angular velocity in fact has several important effects. One of these is that the similarity solution no longer holds even when all the velocities are constants, because there is a length scale due to the ratio of the translational velocity and the rotational velocity. Another one is that, when the body rotates, the vertical velocity on some parts of the wedge may go upwards, which is a difficult problem for some of the methods purely aimed at downward velocity related to water entry. A further point is that the free fall motion has been considered only for vertical entry so far, or one degree of freedom. When there are three degrees of freedom, their coupling can have serious implications to the numerical accuracy and stability. Thus, the present work is of significance in terms of a new physical problem, mathematical modelling, numerical computation and practical applications.

## 2. Mathematical model and numerical procedure

We consider the free fall water entry problem of a two-dimensional wedge with vertical, horizontal and rotational velocities. The wedge may be asymmetric with left deadrise angle *γ*_{1} and right deadrise angle *γ*_{2} as shown in figure 1. Here, we assume that the height of the wedge is sufficiently large so that its top will not be below any liquid during the impact. A Cartesian coordinate system *o*−*x*−*y* is defined, in which *x* is along the undisturbed free surface and *y* points upwards. At time *t*=0, the tip of the wedge is at the origin of the system. Let ** U**=

*U*

**−**

*i**V*

**be the translational velocity of the centre of the mass of the wedge, and**

*j***=**

*Ω**ω*

**be the rotational velocity, where**

*k***and**

*i***are the unit vectors in the**

*j**x*and

*y*directions, respectively, and

**=**

*k***×**

*i***. Here, the reason for the minus sign before**

*j**V*is that the vertical velocity is assumed to be positive when the body moves downwards.

*n*

_{1},

*n*

_{2},

*n*

_{3},

*n*in figure 1 are node numbers. The first two correspond to the intersections between the body surface and the free surface and the last two correspond to the intersection between the control surface and the free surface, which will be discussed further later in the numerical procedure.

*S*

_{0},

*S*

_{F}and

*S*

_{C}denote the body surface, free surface and control surface, respectively.

Impact in practical problems usually lasts for only a short period. Within such a short period, the viscosity of the fluid can usually be ignored (Batchelor 1967, p. 471). As a result, when the density of the fluid is assumed to be constant, the flow can be assumed to be irrotational and the velocity potential *ϕ* can be introduced, which satisfies the Laplace equation
2.1
in the fluid domain. On the wedge surface *S*_{0} we have
2.2
where ** n**=(

*n*

_{x},

*n*

_{y}) is the normal vector on the body surface pointing out of the fluid domain and

**=**

*X**X*

**+**

*i**Y*

**is the position vector from the centre of the mass of the body. The Eulerian form of the dynamic and kinematic boundary conditions on the free surface**

*j**S*

_{F}or

*y*=

*ς*can be written as 2.3 and 2.4

We may notice that the term of acceleration owing to gravity has been ignored in equation (2.3). This is because, when *t*≪|**U**|/*g*, this term is of higher order (Korobkin & Wu 2000). In the Lagrangian framework, the free surface boundary conditions above can also be written as
2.5
and
2.6

The above problem can be solved through the time stepping method. In the earlier work, this was usually started by assuming that a small part of the wedge is already in the water and the problem was solved in the physical domain (Zhao & Faltinsen 1993; Lu *et al.* 2000). The solution based on this method is clearly inaccurate at the initial stage. If the error is not controlled properly, it will accumulate and may lead to wrong results or even instability at the latter stage. Another issue of solving the problem directly in the physical domain is that, if the sizes of the elements remain more or less the same, there will be few elements initially but a large number of elements later on. Thus, a more rational approach is to use the stretched system (Wu *et al.* 2004; Wu 2007*a*).

We may define 2.7 where

Thus, when the wetted surface of a wedge starts from zero and then increases with time in the physical domain, its size can remain more or less the same in the *α*−*o*−*β* system together with the number of elements. In the stretched system, the Laplace equation will retain its form, while the free surface boundary conditions can be rewritten as
2.8
2.9
On the body surface, we have
2.10

To solve the boundary value problem derived above, we introduce the complex potential *w*=*φ*+i*ψ*, where *ψ* is the stream function. For an analytical function, Cauchy theorem then gives
2.11
when *ξ*′ is outside the fluid domain, where *ξ*=*α*+i*β*. In the numerical simulation, the fluid boundary is divided into *n* segments. Within each segment, *w* can be approximated by a linear function. We write
2.12
where
Let *ξ*′ approach *ξ*_{k} in equation (2.11), then substitution of equation (2.12) into equation (2.11) gives
2.13
where

Equation (2.13) can be further split into
2.14
based on the free surface and the rigid surface boundary conditions. On the free surface *S*_{F}, the potential is known through the time stepping technique. On the body surface *S*_{0}, the stream function can be written as based on the boundary condition in equation (2.2), where *D* is a constant. The control surface *S*_{C} is usually treated as a fixed and rigid boundary. It can therefore be treated as a stream line. As a result, *ψ* on *S*_{C} is a constant *C*. Here, only one of the two constants *D* and *C* can be chosen arbitrarily based on the definition of the stream function. In other words, if we take *D*=0, *C* has to be obtained through the solution. Thus, we move the known terms in equation (2.14) to the right-hand side. This leads to
2.15

When 1≤*k*<*n*_{1} or *n*_{2}<*k*<*n*_{3}, the real part of this equation will be taken, and when *n*_{1}<*k*<*n*_{2} or *n*_{3}<*k*<*n* the imaginary part will be taken, where point *n* is the same as point 0 in figure 1. This is to improve the quality of the matrix. It ought to be pointed out that at the intersections of the rigid surface and the free surface, or *j*=*n*_{1},*n*_{2},*n*_{3},*n* the potential functions have been moved to the right-hand side of the equation. Furthermore, we can choose a node on *S*_{C} and take the real part of equation (2.15). This gives an extra equation to obtain *C*. The procedure is similar to that used by Wu (2006) for the twin wedges, where further details can be found.

When a jet develops, which is a thin fluid layer along the wedge surface, the numerical accuracy can be a problem. The accuracy could be improved if the size of the element in the jet zone is comparable to the thickness of the jet. This is, however, not practical because of the large number of elements required. We therefore need to introduce special treatment. We rotate the coordinate system *α*−*o*−*β* to form *ξ*−*o*−*η*. Let the *η* axis be parallel to the wedge surface. As the jet is very thin, say a 10th of the size of a typical element, the potential in the jet zone can be written as
2.16
where the subscript *b* indicates the value on the body surface. Following the procedure of Wu (2007*b*), if we apply the equation on the free surface, where the potential is known, the value of *φ*_{b} can then be obtained. A similar procedure can be for the stream function. There is then no unknown in the jet zone and the accuracy of the solution can be improved.

As the time step advances in the simulation, a saw-tooth free surface profile may appear in the jet zone. Smoothing is then necessary. Here, we adopt the five-point algorithm, which has been used by Maruo & Song (1994) and Sun (2007). If the element nodes are numbered sequentially from 1 to *m*, we then have
2.17
2.18
2.19
2.20and
2.21
where *y*_{i} is the original value of the parameter to be smoothed, which can be the node coordinates or the potential, and *f*_{i} is its value after smoothing. We can apply this algorithm every several time steps.

## 3. Equation of motion

Based on Newton’s second law, equation of motion for the wedge can be written as
3.1
where
and the dot indicates the derivative with respect to time. *m* in the above equation is the mass of the two-dimensional wedge of unit length, and *I*_{zz} is the corresponding rotational inertial about the centre of the mass. The hydrodynamic force (*F*_{1},*F*_{2})=(*F*_{x},*F*_{y}) and the moment *M* can be obtained from the Bernoulli equation, or
3.2
and
3.3
where *ρ* is the density of the liquid.

When the potential is found, ∇*φ* can be obtained from the spatial derivatives. The ∂*ϕ*/∂*t* term, however, has to be found from the temporal derivative, which can be obtained only from results at different time steps. This could, in theory, be achieved through the finite difference method, but there are several difficulties in reality. One of the issues is that, when the re-meshing has been applied, the same node required for calculating ∂*ϕ*/∂*t* may have disappeared. One way to overcome this difficulty is to treat ∂*ϕ*/∂*t* as another harmonic function. Its boundary condition on the free surface can be obtained from the Bernoulli equation. On the body surface, Wu (1998) has shown
3.4

This can be further expanded as 3.5

The direct solution of *ϕ*_{t} is still not straightforward because the acceleration terms on the right-hand side of equation (3.5) are unknown. To decouple the mutual dependence of the body motion and the fluid flow, we adopt the auxiliary function method (Wu & Eatock Taylor 1996, 2003). Thus, we introduce functions *χ*_{i} (*i*=1,2,3), which satisfy the Laplace equation
3.6
in the fluid domain. On the body surface, we require that
3.7
and on the free surface
3.8

On other rigid boundaries, we impose 3.9

Making use of *χ*_{i} and Green’s identity, we have
3.10
*χ*_{i} in the above equations can also be solved in the stretched system. In fact, this can be achieved by simply defining *χ*_{i}(*x*,*y*,*t*)=*sμ*_{i}(*α*,*β*,*t*). Substituting equations (3.2)–(3.4) and (3.10) into equation (3.1), we have (Wu & Eatock Taylor 1996, 2003)
3.11
where [*C*] is a matrix with
3.12
and
with
3.13

It should be noted that, when the functions defined in *x*−*o*−*y* in the above equations are replaced by those in *α*−*o*−*β*, the integration is over the surface defined in the corresponding coordinate system.

## 4. Numerical results and discussions

Before the simulation is made, we define
4.1
which is in fact the horizontal displacement of the centre of the mass of the wedge. We further define *θ* as the angle between the centre line of the wedge and the *y* axis. When the body rotates, we have
4.2
where *θ*_{0} is the initial value of *θ* when the wedge touches the water. Also we assume that the mass centre is located at the centre line of the wedge, and *l* is the distance between the mass centre and the apex of the wedge.

We first use the present method developed for a body in free fall with three degrees of freedom to solve the problem of a wedge entering water at constant translational speed. This problem becomes self-similar when the effect of the gravity on the flow is ignored. It means that *φ* in equation (2.7) is no longer an explicit function of time and its dependence on time is through *α* and *β*. As discussed in the introduction, the similarity solutions based on these non-dimensional variables have been obtained by Dobrovol’skaya (1969) for vertical entry of a symmetric wedge, Semenov & Iafrati (2006) for vertical entry of an asymmetric wedge and Xu *et al.* (2008) for oblique entry of an asymmetric wedge. Here, we shall use the similarity solution of Xu *et al.* (2008) as the initial solution and then use the time stepping method for the solution at the later stage. The typical size of an element on or near the wedge is 0.04 in the stretched coordinate system. The element size increases gradually based on the distance to the impact zone. This is to reduce the memory and CPU requirement within the desired accuracy. The control surface is chosen at , (*i*=1,2) and *β*=−10, and the largest element far away from the body is about five times those near the body. The typical time step Δ*t*=4×10^{−4} s, which is reduced when necessary to ensure that the fluid particle will travel no more than half of the element size within one step. Figure 2 gives the results for an asymmetric wedge with left and right deadrise angles *γ*_{1}=*π*/3, *γ*_{2}=*π*/4, respectively, entering calm water obliquely at constant velocity *U*_{0}=0.3 m s^{−1}, *V*_{0}=1.0 m s^{−1}. Here and in the other figures, the subscript 0 indicates the initial value. The simulation started at *s*=5×10^{−4} m with the similarity solution as the initial solution. The time stepping method is then used and the results given in figure 2 correspond to *s*=0.47973 m. Figure 2 shows that non-dimensionalized free surface profiles and pressure distributions from the time domain simulation and the similarity solution are in good agreement. In figure 2, *x*′ is measured from the apex of the wedge and thus .

As discussed in the introduction, the work on a wedge entering water has been almost exclusively for a body in translational motion. There has hardly been any work for a body with rotation. We consider such a case for a wedge of *γ*_{1}=7*π*/36, *γ*_{2}=11*π*/36, entering water with constant translational velocity *U*_{0}=0, *V*_{0}=5 m s^{−1} and constant rotational velocity *ω*_{0}=−2.5 rad s^{−1}. One of the major differences when *ω*≠0 is that the flow is no longer self-similar, as there is a length scale due to the ratio of the translational speed and the rotational speed. Nevertheless, we still choose the similarity solution based on the translation motion as an initial solution. However, as *s*→0, the stream function on the body surface given after equation (2.14) can be written as *ψ*=(*U*_{0}−*ω*_{0}*Y*)*β*+(*V*_{0}−*ω*_{0}*X*)*α*+*D*. We also notice that, as *s*→0, only the tip of the wedge is in the water. Thus, (*X*,*Y*) in this equation can be taken from the tip of the wedge relative to the body mass centre. This means that the vertical and horizontal velocities have been modified as , and for the initial similarity solution. Figure 3*a* shows that the jet on the left-hand side of the wedge is quite thin and long at smaller *s*. This is due to the smaller deadrise angle on the left. As the wedge rotates clockwise, the deadrise angle on the left will increase. As a result, figure 3 shows that the jet on the left becomes thicker. On the right-hand side of the wedge, water is pushed by the wedge owing to rotation. The free surface profile near the wedge deforms significantly. As *s* increases further, figure 3 shows that the free surface elevation on the right-hand side is not single-valued, even in the coordinate system perpendicular and parallel to the wedge surface. Figure 3*b* shows that the pressure on the left-hand side of the wedge is higher than that on the right at smaller *s*. Once again, this is clearly due to the effect of the smaller deadrise angle on the left. As the wedge rotates clockwise, the pressure on the right-hand side increases significantly and becomes much bigger than that on the left.

The above examples are for forced motion, or the velocity of the body is prescribed. We now consider a case of a symmetric wedge entering water vertically through free fall motion. The problem was investigated by Wu *et al.* (2004) through both experiment and numerical simulation. In the numerical simulation, they have set *U*=0 and *ω*=0 in advance. Here, we shall consider all the three degrees of freedom. This means that *U* and *ω* are part of the solution. We shall then show that our results in the vertical direction agree well with those of Wu *et al.* (2004) and the results in the other two modes remain negligibly small or virtually zero. This is to demonstrate that our method based on the three degrees of freedom is stable and the results obtained are accurate. Figure 4 gives a comparison for a symmetric wedge with deadrise angles *γ*_{1}=*π*/4, *γ*_{2}=*π*/4 and *l*=0.25 m, *I*_{zz}=45 kg m^{2}. *g*_{e} in figure 4 is the acceleration due to gravity and the frictional resistance in the experiment. The results from the present computation are graphically the same as those obtained by Wu *et al.* (2004). The experimental data by Wu *et al.* (2004) are also included in figure 4. Figure 4 further gives results for the horizontal and rotational motions. The accelerations obtained in these two modes are of the order of 10^{−2} or smaller. They are sufficiently close to zero when they are compared with the vertical acceleration.

Having verified the methodology, we now apply our algorithm to a wedge in free fall through three degrees of freedom. The wedge that we choose is initially symmetric about the vertical line and has initial deadrise angles *γ*_{1}=*γ*_{2}=*π*/4. The initial entry velocity is *U*_{0}=5 m s^{−1}, *V*_{0}=5 m s^{−1}, *ω*_{0}=0. The mass of the wedge is set as *m*=100 kg m^{−1}, and *l*=0.25 m, *I*_{zz}=45 kg m^{2}. The similarity solution is used as an initial solution. Figure 5*a* gives the free surface profile as the wedge enters water. Figure 5*b*,*c* presents the displacements in three degrees of freedom against time *t*, in which the vertical displacement is taken as positive when the wedge moves downwards. Because the initial horizontal and vertical entry speeds are the same, the slopes of the translational displacements are the same. On the other hand, the initial acceleration in the vertical direction is *g*, while it is zero in the horizontal direction. Thus, the vertical displacement increases faster than the horizontal one. For the forced constant speed entry, these curves would be straight lines. The curvature of these lines reflects the effect of acceleration in the free fall motion. When the wedge touches the water, the centre of the mass is above the still water level and the pressure on the most wetted part on the right-hand side of the wedge will create a negative moment. This is demonstrated by the negative rotational displacement in figure 5*c*. When the wedge goes down further, more and more pressure on the right-hand side will create a positive moment. This will change the direction of the rotation, as shown in figure 5*c*. Figure 6 gives the pressure distribution along the wedge surface at different penetration distances. At the earlier stage of the impact, because of the horizontal velocity, the pressure on the right-hand side is much higher than that on the left. When the direction of the rotation changes, the pressure on the right-hand side is reduced significantly. We may also notice that the pressure near the tip of the wedge is negative. This means that air may be sucked into this region and an air pocket may be formed. In fact, this has been discussed by Judge *et al.* (2004) based on their experimental study. However, the coupled aero/hydrodynamic analysis for such a problem is beyond the scope of the current work. Thus, the effect of the trapped air near the tip is ignored.

To give some further insights into the free fall motion of the wedge under these conditions, figure 7 shows the acceleration and the velocity components in three directions. When the body touches the water, the force on the body will increase rapidly because of the increase of the wetted surface and relatively larger velocity. This force will act as a resistance to the body motion and the translation velocity will therefore decline sharply. As it happens, the rate of its decline will eventually decrease because of the reduced magnitude of the acceleration. In fact, it can be observed that the accelerations in figure 7*a* tend to zero as time increases. Based on Newton’s Law, it is expected that the horizontal motion would stop together with the translation eventually. The vertical motion will continue. However, the speed will decline while the wetted surface area of the wedge increases. The total hydrodynamic force will balance the weight of the body. This may take a considerable period of time and is a result of neglect of the effect of the acceleration owing to gravity on the fluid. As the interest in impact is mainly over a short period of time, such a long time simulation is not included here. The behaviour of rotation follows the discussion given previously. As the body moves down, the direction of the rotational acceleration could change. As time progresses further, it tends to zero.

Here, we continue to consider a case of a wedge entering water with an initial rotational velocity. The wedge is initially asymmetric about the vertical line and has initial deadrise angles *γ*_{1}=7*π*/36, *γ*_{2}=11*π*/36 or *θ*_{0}=−0.1745. The initial entry speeds are *U*_{0}=0, *V*_{0}=5 m s^{−1} and *ω*_{0}=−2.5 rad s^{−1}. The mass, the centre of its gravity and the rotational inertial are the same as those corresponding to figure 5. The initial solution is obtained in the manner corresponding to figure 3. The free surface elevations at several different time steps are given in figure 8*a*. Figure 8*b* gives the displacement of the wedge. When the wedge touches the water, its deadrise angle on the left-hand side is smaller and a horizontal force to push the wedge to move in the direction of the *x* axis should be expected. However, before long, owing to the effect to rotation, the wedge is pushed to move in the opposite direction.

Figure 9 presents the pressure distribution on the wedge surface. The evolution pattern of the pressure with the time step is similar to that discussed previously. Here, we ought to emphasize that the general trend is that pressure will decrease as *t* increases further. This is clearly due to the decline of the translational and rotational velocities as a result of the hydrodynamic force, which can be seen in figure 10*d*–*f*. Further in figure 10, all the accelerations change rapidly at the initial stage of the impact. They all reach a peak at different time steps and then decline in the opposite direction. There are further oscillations in the curves of horizontal and rotational accelerations. However, it is expected that they will decrease eventually as time progresses.

To investigate the effect of the initial rotational velocity more closely, we further consider free fall motion of a wedge with deadrise angles *γ*_{1}=*γ*_{2}=*π*/4, entering water at *U*_{0}=2.0 m s^{−1}, *V*_{0}=5.0 m s^{−1}, and three rotational velocities *ω*_{0}=−2.0, 0 and 2.0 rad s^{−1}, respectively. The mass, the centre of the gravity and the rotational inertial are the same as the wedge discussed in figures 5 and 8. Figure 11*a* gives the horizontal displacement. At *ω*_{0}=−2.0 rad s^{−1}, the clockwise rotational velocity will increase the pressure on the right-hand side of the wedge and reduce the pressure on the left. This reduces its speed in the *x* direction. As a result, the displacement is reduced. At *ω*_{0}=2.0 rad s^{−1}, the effect is opposite and the horizontal displacement is larger. The effect of the initial rotational velocity on the vertical displacement is much smaller. In figure 11*c*, we can see that the curves corresponding to *ω*_{0}=−2.0 and 2.0 rad s^{−1} are not symmetric. This is clearly due to the effect of *U*_{0}. All these show that the interactions between the three degrees of freedom are important during the free fall motion.

Figure 12*a* gives the free surface profiles at *s*=0.3 m. Figure 12*b*–*d* presents the pressure distribution at *s*=0.1, 0.2 and 0.3 m, respectively. At *ω*_{0}=−2.0 rad s^{−1}, the clockwise rotation increases the peak of the pressure on the right-hand side of the wedge. However, the pressure on the other part of the same side can be reduced. The pressure on the left-hand side of the wedge is reduced almost everywhere by the clockwise rotation. At *ω*_{0}=2.0 rad s^{−1}, the counterclockwise rotation increases the peak of the pressure on the left-hand side of the wedge.

Figure 13 gives the acceleration and the velocity of the wedge. At small *s*, the effect of the initial rotational velocity on the horizontal acceleration is small. As *s* increases, the combination of the horizontal velocity and clockwise rotational velocity leads to a much larger negative acceleration. In fact, this can further lead the wedge to move in the opposite direction, as shown in figure 13*d*. The effect of the initial angular velocity on the vertical acceleration and velocity is much smaller. Owing to the initial horizontal velocity, the rotational acceleration and velocity curves corresponding to *ω*_{0}=−2.0 and 2.0 rad s^{−1} do not show symmetry about the *x* axis. This once again shows the importance of coupling.

Finally, to ensure that the results corresponding to the free fall motion in three degrees of freedom are sufficiently accurate, we can check the mass conservation and energy conservation during the simulation. We use a case with deadrise angles *γ*_{1}=*γ*_{2}=*π*/4, and initial velocities *U*_{0}=2.0 m s^{−1}, *V*_{0}=5.0 m s^{−1}, *ω*_{0}=−2.0 rad s^{−1}. The mass conservation requires that *S*′, which is the area occupied by the liquid above the original liquid surface, is equal to *S*, which is the area displaced by the wedge below the calm water surface. The accuracy is checked through *ε*_{mr}=|*S*′−*S*|/*S*. Similarly, the accuracy of the energy conservation is checked through *ε*_{Er}=|*E*−*E*_{0}|/*E*_{0}, where *E* is the total energy of the system during the impact and *E*_{0} is the initial kinetic energy of the wedge when it touches the water surface. It should be noted that *E* includes both kinetic energies of the wedge and water. It also includes the variation of the potential energy of the wedge as its weight has been treated as an external force in the equation of motion. It, however, does not include the potential energy of the liquid as the effect of the gravity on the flow has been ignored. Thus, the total energy of the system can be written as
4.3

The results of mass conservation and energy conversation are shown in figure 14. The largest error in the former is about 0.5 per cent, while the largest error in the latter is about 5 per cent over the impact period. This should give us sufficient confidence in the accuracy of the results provided. It ought to show that, towards the end of the simulation, the errors in both mass and energy conservation increase much faster. If we were to carry on the simulation, the accumulated error could become significant. However, we could use finer mesh and smaller time steps to reduce the error in mass conservation. The results given here are to show the accuracy during the simulation period that we have made.

## 5. Conclusions

The water entry problem of a wedge through free fall in three degrees of freedom has been solved using the velocity potential theory. The stretched coordinate system method is clearly the key to the success of the simulation. It has been helped by a suitable choice of the initial solution. The method developed has been verified though comparison with published results and experimental data for some simplified cases and good agreement has been found. It is further verified through the mass conservation and energy conversation. Detailed simulations show that there are some strong couplings between the three degrees of freedom. These results have some important applications for practical problems such as ship slamming, which will be further investigated in our future work.

## Acknowledgements

This work is supported by the National Science Foundation of China (no. 50779008) and the Programme of Introducing Talents of Discipline to Universities of China, to which the authors are most grateful. The third author also wishes to thank the Cheung Kong and Long Jiang Visiting Professorship Schemes hosted by HEU.

## Footnotes

- Received November 19, 2009.
- Accepted January 25, 2010.

- © 2010 The Royal Society