## Abstract

The two-dimensional motion of a rigid body with a smooth surface is studied during its oblique impact on a liquid layer. The problem is coupled: the three degrees of freedom of the moving body are determined together with the liquid flow and the hydrodynamic pressure along the wetted part of the body surface. The impact process is divided into two temporal stages. During the first stage, the wetted region expands at a high speed with jetting flows at both ends of the wetted region. In the second stage, the free surface of the liquid is allowed to separate from the body surface. The position of the separation point is determined with the help of the Brillouin–Villat condition. Calculations are performed for elliptic cylinders of different masses and with different orientations and speeds before the impact. The horizontal and vertical displacements of the body, as well as its angle of rotation and corresponding speeds are investigated. The model developed remains valid until the body either touches the bottom of the liquid or rebounds from the liquid.

## 1. Introduction

The two-dimensional unsteady problem of the oblique impact of a solid body onto a thin liquid layer is considered. The motivation for this study comes primarily from emergency landing of aircraft (ditching) and controlled landing of seaplanes in shallow waters [1]. Similar processes can be found in machinery engineering, where impacted surfaces are covered with a thin layer of oil [2], and in aircraft-icing applications, where ice crystals impact an aerofoil coated in a thin liquid layer as the result of flying through clouds [3]. In these applications, the component of the body velocity normal to the liquid surface is usually much smaller than the velocity component tangential to the liquid surface. For example, Boing 737 Flight Crew Training Manual [4] recommends to ‘maintain 200–300 fpm rate of descent’ during ditching, which means 1–1.5 m s^{−1} vertical component of velocity of the aircraft. In contrast, the horizontal speed of an aircraft before ditching can be from 15 to 50 m s^{−1}. To achieve a smooth landing on water, a pilot should follow special instructions reducing the touchdown speed and keeping the angle of attack within a certain range specific for the aircraft. An important concern in the problem of aircraft ditching is the suction force at the rear of the fuselage. The computed pressure distribution by Streckwall *et al.* [5] ‘shows a pronounced region of high pressure in the front area of the submerged part and negative pressures with respect to the ambient pressure in the convex curved part at the rear’.

During landing on water with a horizontal speed, the wetted area of the body surface initially expands in both directions. This stage of the landing is referred to as the impact stage. At this stage, spray jets are formed at both the leading and trailing edges of the wetted region, as shown in figure 1*a*. The hydrodynamic pressure in the wetted region has been calculated within the shallow-water model by Batyaev & Khabakhpasheva [6] together with the components of motion of the body in vertical and horizontal translation and rotation. It was shown that the duration of the impact stage is very short for a large horizontal component of the body speed before the impact. The present paper is concerned with the second stage of landing (figure 1*b*), when the jet at the trailing edge disappears and the free surface separates smoothly from the surface of the moving body. This stage is referred to as the planing stage. We assume that the free surface instantly separates from the body surface at the beginning of this stage. A similar assumption has been employed by Norkin & Korobkin [7] in the problem of sudden motion of a floating body. The separated part of the water surface and the subsequent motion of the separation point are determined by using the Brillouin–Villat condition, which requires the continuity of the pressure together with its tangential derivative at the separation point. This condition was used, for example, by Tuck & Simakov [8] and Semenov *et al.* [9] in the steady problem of a body moving along the free surface of liquid of infinite depth. Reinhard *et al.* [10] compared the solutions obtained with the Brillouin–Villat condition and some other possible conditions at the separation point. It was concluded that the correct condition can be selected only on the basis of experimental results.

The problem becomes even more complicated if the body is elastic and changes its shape during the interaction. This problem was studied by Reinhard *et al.* [11], where two following inequalities were used: (i) the free surface in the wake region behind the separation point cannot penetrate the surface of the plate, (ii) the pressure in the wetted part of the plate cannot be below the atmospheric pressure in the vicinity of the separation point. It was shown that these two inequalities lead to the Brillouin–Villat condition at the separation point. The approach developed by Reinhard *et al.* [12,13] is able to describe transition from separation at the trailing edge of the plate (governed by the Kutta condition) to separation at an inner point of the plate surface (governed by the Brillouin–Villat condition), and to the regime with jetting at the trailing edge of the wetted area of the plate (governed by the Wagner condition). The Wagner condition was named after Wagner [14] and studied by Howison *et al.* [15] and Korobkin [16].

The problem under consideration in this paper is coupled: the liquid flow, motions of the body during its interaction with the liquid, position of the leading edge of the wetted area, where the jet is formed, and the position of the separation point should be determined simultaneously. The motions of the body are governed by the equations of momentum conservation in the horizontal and vertical directions and the conservation of angular momentum of the body [3]. The external loads in these equations are calculated by integration of the hydrodynamic pressure along the wetted part of the body surface. The aerodynamic forces and viscous drag from the water are neglected in the present study (see also [3]). The hydrodynamic pressure in the wetted region and the position of this region on the body surface are determined within the shallow-water model. This model was used by Korobkin [2] for the vertical impact of a symmetric two-dimensional body, by Oliver [17] for three-dimensional bodies and by Hicks & Smith [3] for skimming impact. The model provides the leading-order solution of the hydrodynamic problem, where the horizontal dimension of the body *L* is much greater than the depth of the liquid layer *H*. The flow caused by the oblique impact is described by the method of matched asymptotic expansions with *ε*=*H*/*L* being a small parameter of the problem. The flow is divided into four regions: I, the region beneath the body surface; II, the jet root region; III, the spray jet; IV, the outer region, see figure 2. The jet root regions are placed at both the leading and trailing edges of the wetted part of the body surface during the impact stage but only at the leading edge during the planing stage. The size of the jet root region is of the order of the liquid depth *H*, which is much smaller than the size of region I, which is estimated to be , and the size of the outer region IV. The shallow-water equations describe the flows in the regions I and IV. The one-dimensional flows in these regions are matched through the jet root regions, where the flows are two-dimensional and quasi-steady in the leading order. During the planing stage, the matching conditions through the jet root region II^{−} at the trailing edge are replaced by the Brillouin–Villat condition at the separation point.

Numerical calculations are performed for elliptic cylinders of different masses, different orientations and different speeds before impact. Several scenarios of the body interaction with the liquid layer are revealed. It is shown that the jetting is mainly responsible for the reduction of the kinetic energy of the body during impact.

The governing equations of the body motions and the flow in the thin liquid layer are derived in §2. These equations are transformed in §3 to a nonlinear system of ordinary differential equations in time and a nonlinear algebraic equation with respect to the position of the separation point. Numerical scheme and numerical results are presented in §4. The conclusions are drawn and future work is discussed in §5.

## 2. Governing equations

The two-dimensional unsteady problem of a body impact on a thin layer of ideal and incompressible liquid is considered. The surface of the body is smooth without corner points and edges. The rigid body motions and the liquid flow are described in the Cartesian coordinate system (*x*,*y*) shown in figure 1*a*. The line *y*=0 corresponds to the flat bottom of the liquid layer, and the line *y*=*H* to the initial undisturbed position of the free liquid surface. The liquid initially is at rest. At the initial instant of time, *t*=0, the body surface tangentially touches the free liquid surface at a single point *x*=0, *y*=*H*. The shape of the rigid body is described within the local coordinate system *ξOη* with the origin at the centre of mass of the body, see figure 1*a*. The lower part of the body surface, which potentially can be in contact with the liquid during the body motion, is given by the equation *η*=*f*(*ξ*), where *f*(*ξ*) is a smooth function. The motions of the body are described by the coordinates of its centre of mass *x*=*x*_{0}(*t*), *y*=*y*_{0}(*t*) and the angle *α*(*t*) of the body rotation with respect to the positive *x*-axis. The initial value of this angle, *α*(0), and the initial velocities of the body motions, , , are given.

The local and global coordinates are related by the equations
2.1Equations (2.1), where *η*=*f*(*ξ*), provide the position of the surface of the moving body in the global coordinate system in the parametric form *x*=*X*(*ξ*,*t*),*y*=*Y* (*ξ*,*t*). We assume that the shape of the body is such that the first equation *x*=*X*(*ξ*,*t*) can be inverted, *ξ*=*ξ*(*x*,*t*), and then the second equation gives *y*=*y*_{b}(*x*,*t*), where the function *y*_{b}(*x*,*t*) is smooth and depends on the body motions. At the point of the initial contact between the liquid and the body surface, we have *x*=0, *y*=*H* and (∂*y*_{b}/∂*x*)(0,0)=0. These conditions provide three equations with respect to *x*_{0}(0), *y*_{0}(0) and the local coordinate *ξ*_{0} at which the first contact occurs.

The minimum curvature *L* of the body surface *η*=*f*(*ξ*) is assumed to be much greater than the liquid depth *H*. At the leading order as *H*/*L*→0, the liquid flow beneath the body, *c*_{−}(*t*)<*x*<*c*_{+}(*t*), is described by the following equations [2,3,6]
2.2where *u*(*x*,*t*) is the horizontal speed of the flow, *p*(*x*,*t*) the hydrodynamic pressure, the total pressure is equal to *p*(*x*,*t*)+*ρg*(*H*−*y*), *ρ* the liquid density and *g* the acceleration due to gravity. During the impact stage, the wetted region, *c*_{−}(*t*)<*x*<*c*_{+}(*t*), expands quickly in both directions with , and the liquid outside the interval *c*_{−}(*t*)<*x*<*c*_{+}(*t*) remains at rest. The equations of mass, momentum and energy conservation at the boundaries between the outer regions, *x*>*c*_{+}(*t*) and *x*<*c*_{−}(*t*), and the flow region beneath the body, *c*_{−}(*t*)<*x*<*c*_{+}(*t*), provide the speeds of the leading and trailing edges and the pressures *p*_{±}(*t*)=*p*(*c*_{±}(*t*),*t*) at these edges
2.3The matching conditions (2.3) were used in the past in different forms in some earlier studies [2,3,6,18,19].

At the end of the impact stage, *t*=*t*_{*}, the derivative tends to zero and the jet at the trailing edge disappears. During the subsequent planing stage, the conditions (2.3) at the trailing edge of the wetted region are replaced by the Brillouin–Villat conditions
2.4where *c*_{−}(*t*) now is the coordinate of the separation point for *t*>*t*_{*} with *c*_{−}(*t*_{*}+0)≠*c*_{−}(*t*_{*}−0) in general case. The position of the separation point is governed by the second condition in (2.4). This condition is not well established but reasonable (see §1 for discussion).

The body motions are governed by the equations
2.5where *m* is the mass of the body per unit length, *J* the moment of inertia of the body, *F*_{x}(*t*) and *F*_{y}(*t*) are the horizontal and vertical components of the hydrodynamic force acting on the body surface in the contact region and *M*(*t*) the moment of the hydrodynamic force.

The forcing terms in (2.5) are calculated by integrating the hydrodynamic pressure along the wetted portion of the body surface 2.6 2.7 and 2.8Note that the buoyancy force is included in the hydrodynamic loads (2.6)–(2.8).

The system of equations and matching conditions (2.2)–(2.8) is coupled: the body motions and the hydrodynamic loads should be determined simultaneously and together with the coordinates *c*_{+}(*t*), *c*_{−}(*t*) of the leading and trailing edges of the wetted region.

## 3. Motions of the body during both impact and planing stage

This section is concerned with the hydrodynamic loads (2.6)–(2.8) and the body motions during both stages of the body interaction with the liquid layer. There is no flow in front of the moving body, *x*>*c*_{+}(*t*), owing to the condition that the speed of the leading edge is greater than the critical shallow-water wave speed. Therefore, the hydrodynamic pressure *p*(*x*,*t*) should be determined only in the interval *c*_{−}(*t*)<*x*<*c*_{+}(*t*) by using equations (2.2) and the conditions (2.3) during the impact stage and conditions (2.3) for the leading edge and (2.4) for the trailing edge during the planing stage. During the planing stage, the flow in the wake behind the body, *x*<*c*_{−}(*t*), does not affect the hydrodynamic loads acting on the body in the present model. This situation is different from a similar problem of oblique impact onto a deep water, where the flow in the wake gives an important contribution to the hydrodynamic loads [10]. The equations of the flow (2.2) and equations of the body motions (2.5) are nonlinear and can be solved only numerically. Similar equations were derived by Hicks & Smith [3] for the skimming impact on a shallow liquid layer, where the penetration of the plate into the layer is small compared with the layer thickness. The nonlinearity of the equations (2.2) does not significantly increase the difficulties of solving the problem. A main difficulty here is connected with the implementation of the Brillouin–Villat condition (2.4). Having in mind that the problem will be analysed numerically, we are only concerned with the reduction of the governing equations from §2 to a form which allows us to integrate the resulting equations in time.

Differentiating the equation of the body surface *y*=*y*_{b}(*x*,*t*) in time and using (2.1) we find which makes it possible to integrate the continuity equation in (2.2)
3.1where *C*(*t*) is a function to be determined. Here dot stands for the time derivative.

The derivative , which is required to calculate the pressure distribution *p*(*x*,*t*) from the momentum conservation equation in (2.2), is obtained from equation (3.1) in the form
3.2where ** z**(

*t*)=(

*x*

_{0},

*y*

_{0},

*α*) is the vector of unknowns and the function is independent of the second derivative .

Substituting (2.2), (2.6)–(2.8), (3.1) and (3.2) in equations (2.5), we obtain the following system of three linear equations (*j*=1,2,3) with respect to the second derivatives , , and
3.3The coefficients *A*_{ji} and the right-hand-side functions *f*_{j} in (3.3) are given in the electronic supplementary material, appendix. Note that the coefficients *A*_{ji}, 1≤*j*≤3, 1≤*i*≤4 in (3.3) depend on the current position and orientation of the body, and on the position of the wetted region on the body surface, but not on the velocities and accelerations of the body motion. The right-hand side in (3.3) additionally depend on the body velocities but not on the body accelerations.

By integrating the momentum conservation equation in (2.2) along the wetted region from *x*=*c*_{−}(*t*) to *x*=*c*_{+}(*t*) and using (3.1) and (3.2) and the matching conditions (2.3) and (2.4), we obtain the fourth equation (*j*=4) which has the same form as equations (3.3). To compute the coefficients *A*_{ji}, 1≤*j*,*i*≤4, and the right-hand sides *f*_{j}, we need to evaluate 13 integrals *I*_{k}(** z**,

*t*) at each time step. These integrals are listed in the electronic supplementary material, appendix. There are eight unknown functions of time

*x*

_{0}(

*t*),

*y*

_{0}(

*t*),

*α*(

*t*),

*C*(

*t*),

*γ*

_{+}(

*t*),

*γ*

_{−}(

*t*),

*p*

_{+}(

*t*) and

*p*

_{−}(

*t*) in the derived system of differential equations. During the impact stage, we have four differential equations (3.3) and four equations (2.3) to find these eight unknown functions by numerical integration. The equations (2.3) are transformed to the corresponding equations with respect to

*γ*

_{+}(

*t*) and

*γ*

_{−}(

*t*) by using , which follow from (2.1).

To start the numerical integration of equations (2.3) and (3.3), we need the initial values and first derivatives of the unknown functions. The derivatives and , which are singular at *t*=0, and the pressures *p*_{+}(0) and *p*_{−}(0) are evaluated by using asymptotic methods. This difficulty with the initial conditions is specific for bodies with smooth surfaces and is not presented for shapes with sharp trailing edge [3]. At the point of first contact, we have *γ*_{+}(0)=*γ*_{−}(0)=*γ*_{0}, where *γ*_{0} is a solution of the equation (∂*y*_{b}/∂*x*)(*γ*_{0},0)=0. This equation together with equation (2.1) provides where prime stands for the derivative with respect to the parameter *γ* and *α*_{0} is the initial angle of the body inclination. The equations for the coordinates of the impact point *y*(*γ*_{0},0)=*H* and *x*(*γ*_{0},0)=0 provide the initial values of the coordinates of the centre of mass of the body Initial asymptotics of *γ*_{±}(*t*) are obtained as , and the initial value of the function *C*(*t*) is . The initial values of the functions *p*_{±}(*t*) are
3.4Equations (2.3) and (3.3) lead to the system of nine nonlinear ordinary differential equations for the functions *x*_{0}(*t*), , *y*_{0}(*t*), , *α*(*t*), , *C*(*t*), *γ*_{+}(*t*) and *γ*_{−}(*t*) (compare with the system derived by Hicks & Smith [3], §4). Nine initial conditions for these functions follow from the asymptotic formulae derived above and prescribed values , , *α*(0), . The resulting initial value problem is integrated up to the time instant *t*_{*} at which .

In the present model, the liquid is allowed to separate instantly at *t*=*t*_{*} from the body surface. The differential equations (3.3), where *j*=1,2,3,4, and (2.3) for *c*_{+}(*t*) and *p*_{+}(*t*) are still valid during the planing stage with *p*_{−}(*t*) set to zero in (3.3) in accord with the Brillouin–Villat condition (2.4). The second condition, *p*_{x}(*c*_{−}(*t*),*t*)=0, in (2.4) and the momentum conservation equation in (2.2) provide *u*_{t}+*uu*_{x}=0 at *x*=*c*_{−}(*t*). By using (3.2), this condition can be written as
3.5where the function *U*(*γ*,*t*) is defined in the electronic supplementary material, appendix. Then, the linear system of four algebraic equations (3.3) is solved with respect to , , , and the result is substituted in (3.5), which gives the nonlinear equation for the unknown function *γ*_{−}(*t*). The system of eight ordinary differential equations, which follow from (2.3) and (3.3) and the algebraic equation for *γ*_{−}(*t*) are solved numerically starting from *t*=*t*_{*}. The matching conditions at *t*=*t*_{*} assume that the unknown functions *C*(*t*), **z**(*t*), and *γ*_{+}(*t*) are continuous at *t*_{*} but *γ*_{−}(*t*_{*}+0)≠*γ*_{−}(*t*_{*}−0). This implies also that the body accelerations are not continuous at *t*=*t*_{*}.

## 4. Numerical results

The ordinary differential equations derived in §3 are integrated in time by the second-order predictor–corrector method. The nonlinear equation is solved by the bisection method at each step of the integration. Details of the numerical procedure can be found in the electronic supplementary material.

Calculations are performed in dimensional variables for an elliptic cylinder *ξ*^{2}/*a*^{2}+*η*^{2}/*b*^{2}=1 with the semi-axis *a*=0.5 m and *b*=0.125 m. The mass of the cylinder per unit length *m* is varied from 150 to 1500 kg m^{−1}. Results are presented for the water layer of density *ρ*=1000 kg m^{−3} and depth *H*=0.05 m. The critical shallow-water wave speed is equal to 2.32 m s^{−1}. The initial angle *α*(0) is varied from 0^{°} up to 15^{°} and in all calculations, presented here. The time step of integration *Δt* is equal to 10^{−4}*s*. The elliptic cylinder in parametric form is given by , , where the parameter *γ* varies from zero to 2*π* (*γ*=0 at the point *ξ*=*a*, *η*=0).

Numerical calculations were performed to distinguish three cases of the body interaction with a thin layer of the liquid: (i) a light body which exits liquid shortly after impact, (ii) a body of moderate mass which interacts with the liquid for longer time but finally exits water, (iii) a heavy body which penetrates the liquid and approaches the bottom at the end of calculations. More results of numerical analysis can be found in the electronic supplementary material.

Numerical calculations revealed that the duration of the impact stage is very short compared with the duration of the subsequent planing stage. However, the hydrodynamic pressure is very high at the beginning of the impact stage. To resolve details of the hydrodynamic loads shortly after the initial impact and explain the transition from impact to planing stage, computations were performed for elliptic cylinders of masses *m*=150 and 500 kg m^{−1} inclined initially at *α*(0)=12^{°} and entering the thin liquid layer with the initial velocity and . The duration of the impact stage was found to be *t*_{*}=4.48 ms for *m*=150 kg m^{−1} and *t*_{*}=5.46 ms for *m*=500 kg m^{−1}. It was shown that pressure *p*_{−}(*t*), horizontal speed of the flow at the trailing edge *u*_{−}(*t*) and the speed of the trailing edge approach smoothly to zero at the end of the impact stage.

The distribution of the pressure along the wetted part of the body surface for *m*=500 kg m^{−1} is shown in figure 3*a* at different time instants in the global coordinates. The pressure *p* is in kPa and *x* in centimetres. The figure also shows the positions of the wetted area in time. The dot in figure 3 indicates the pressure (3.4) at *t*=0, *p*_{±}(0)=56.64 kPa. The pressure is positive everywhere in the wetted region for 0≤*t*<3.5 ms and becomes negative close to the trailing edge starting from *t*=3.5 ms (figure 3*b*). The region of negative pressures expands in both directions. The pressure in this region drops down to −12 kPa at the end of the impact stage. Note that the atmospheric pressure is 100 kPa. This value should be added to the computed pressure to obtain the physical pressure. Figure 3*b* proves that the physical pressure in the wetted region does not drop below 88 kPa. The latter value is much higher than the critical pressure at which water starts to cavitate. The critical pressure of pure water can be estimated as the vapour pressure which is about 2.3 kPa at a temperature of 20^{°}*C*. This implies that the observed reduction of pressure near the trailing edge does not lead to cavitation.

At *t*=*t*_{*}, the region of negative pressure reaches the trailing edge which is decelerating due to the horizontal motion of the body. At this time instant the pressure at the trailing edge is zero and negative on the rear of the wetted region. This means that the hydrodynamic pressure close to the trailing edge is lower than the ambient pressure. Although the air flow is not included in the present model, we can argue that this pressure difference forces air to flow beneath the body surface through the trailing edge separating the liquid from the surface along a certain interval. The length of this interval is determined by the Brillouin–Villat condition. Starting from *t*_{*} the body motions and the liquid flow are computed together with the position of the separation point (figure 1*b*).

*Case 1.* This case is for the light body, *m*=150 kg m^{−1}. The initial impact conditions are those shown above. Note that the impact occurs behind the centre of mass. Calculations are performed for both impact and planing stages starting from the impact instant and till the complete exit of the body from the liquid. Positions of the elliptic cylinder at different time instants are shown in figure 4. It is seen that the penetration depth is very small which suggests that the skimming theory by Hicks & Smith [3] can be employed. Figure 4 also shows that at the middle of the interaction period the body is above the initial water level but still in contact with the liquid.

The duration of the body interaction with the liquid layer is about 0.1 s. During this time, the body travels 1 m along the liquid surface. It is shown that the total reduction of the horizontal velocity does not exceed 2 per cent but the vertical velocity varies significantly. It varies from −1 m s^{−1} at *t*=0 to −0.2 m s^{−1} at *t*=0.05 s and remains negative, , at the end of the planing stage (see the electronic supplementary material for more details). The latter result is difficult to recognize looking at figure 4. The exit of the body occurs due to the body rotation. The inclination angle *α*(*t*) of the body drops monotonically from 12^{°} down to a negative value at the end of the interaction. Therefore, the body rotation plays an important role in the impact.

To demonstrate details of the interaction, the positions of the trailing and leading edges are shown in local coordinates in figure 5*a* in terms of the parameter *γ*. In contrast to the global motions of the edges, the motions in the local coordinate system, *γ*_{+}(*t*) and *γ*_{−}(*t*), indicate the position of the wetted region on the surface of the body. Note that *γ*=1.5*π* corresponds to *ξ*=0, *η*=−*b* and *γ*=*π* to *ξ*=−*a*, *η*=0. It is seen that during the planing stage the wetted area does not vary significantly and shrinks at the end of the stage, when the body exits from the liquid. Also the motion of the lowest point of the body surface is shown by the dashed line. The end of the impact stage, *t*_{*}=4.46 ms, is easy to recognize in figure 5. At *t*=*t*_{*}, the free surface instantly separates from the body surface and the trailing edge jumps from *γ*≈1.1*π* to *γ*≈1.25*π*. The elevation of the free surface at the leading edge, *y*_{+}(*t*), and at trailing edge, *y*_{−}(*t*), as well as the vertical coordinate of the lowest point of the body surface, , are shown in figure 5*b*.

Figures 5*a*,*b* make it possible to subdivide the planing stage into five phases. The first phase, 4.46 ms <*t*<12.4 ms, follows just after the impact stage. At this phase , the leading edge is above the initial liquid level, the trailing edge is below this level and the lowest point of the body is inside the wetted region (figure 1*b*). During the second phase, 12.4 ms <*t*<71.3 ms, the lowest point of the body is behind the trailing edge (figure 6*a*). The wetted area is above the equilibrium level of the liquid starting from 42 ms. At the end of the second phase, the lowest point approaches the trailing edge and penetrates into the wetted region. During the third phase, 71.3 ms <*t*<88.8 ms, the lowest point travels from the trailing edge towards the leading edge. Relative positions of the body and the liquid surface are those in figure 1*b*. At the beginning of the fourth phase, the lowest point appears in front of the leading edge, which is due to strong rotation of the body. The position of the body with respect to the liquid region during the fourth phase, 88.8 ms <*t*<98 ms, is sketched in figure 6*b*. At *t*=98 ms, the lowest point touches the liquid surface in front of the original wetted region with formation of the secondary wetted region. Shortly after the secondary contact occurs, the original wetted region disappears at *t*=102 ms. For *t*>98 ms, we should solve the problem with the wetted region consisting of two intervals, as shown in figure 7. This regime of interaction is not considered in this paper. Note that the body is above the initial liquid level in this regime and the inclination angle of the body is small. This implies that we are within the skimming impact theory by Hicks & Smith [3]. A sketch of the positions of the wetted parts of the body in time is shown in figure 7. The wetted region 1 may co-exist with the secondary region 2. The body is in the air after the region 2 has disappeared, but the next contact of the body with the liquid layer through a region 3 has not yet started.

The hydrodynamic pressure in the wetted region is shown in figure 8*a* during both impact and planing stages. It is seen that the pressure at the leading edge, *p*_{+}(*t*), increases initially but then decays to zero at *t*=102 ms. The pressure distribution changes suddenly at the beginning of the planing stage. Figure 8 also shows the positions of the wetted region at different times. It is clear that the pressure decreases with time.

The energy conservation law states
4.1where is the kinetic energy of the body motion, *P*_{b}(*t*)=*mg*(*y*_{0}(*t*)−*y*_{0}(0)) is the potential energy of the body, *E*_{j}(*t*) is the kinetic energy of the jets and *E*_{l}(*t*) and *P*_{l}(*t*) are the kinetic and potential energies of the flow in the main flow region. The kinetic energy of jets is calculated by the formula
4.2Note that the amount of energy which leaves the main flow region through a jet of thickness *h*_{j} is equal to , where is the speed of the liquid in the jet and is the mass of the liquid which leaves the flow region through the jet. The second integral in (4.2) corresponds to the contribution from the jet at the trailing edge. This jet exists only during the impact stage, 0<*t*<*t*_{*}. The terms in equation (4.1) as functions of time are shown in figure 8*b*. In case 1, the kinetic energy of the body *E*_{b}(0) at the impact instant, *t*=0, is equal to 7575 kg m s^{−2}. This constant value is shown with the thin line. The evolution of the kinetic energy of the horizontal motion of the body is shown with curve 1. It is seen that this part of the total energy of the body–liquid system provides a main contribution to the energy conservation law (4.1). Other terms in (4.1) give less than 4 per cent of the total energy. The total energy of the body, *E*_{b}(*t*)+*P*_{b}(*t*), as a function of time (curve 2) and together with the jet energy, *E*_{b}(*t*)+*P*_{b}(*t*)+*E*_{j}(*t*), (curve 3) indicate that the energy of the flow in the liquid layer, *E*_{l}(*t*)+*P*_{l}(*t*), is negligible compared with other energy components. We may conclude that about 4 per cent of the initial energy of the body is spent on jetting during both the impact and planing stages in this case.

*Case 2.* The conditions of the impact are the same as in case 1 but now the body is more than three times heavier, *m*=500 kg m^{−1}. As a result, the elliptic cylinder is in contact with the liquid for a much longer time (figure 9). It travels more than 9 m before exiting the liquid. The duration of the interaction is longer than 1 s, which is more than 10 times longer than in case 1. The ellipse image in figure 9 is disturbed by a vertical scale exaggeration of five times. Calculations revealed that during the planing stage the horizontal speed of the body reduces by 18 per cent (see the electronic supplementary material) and the vertical speed (figure 10*a*) oscillates being positive for substantial periods. At the end of the planing stage, when the wetted region shrinks, the centre of mass of the body moves down but the body surface in the wetted region moves upwards from the liquid. The inclination angle of the cylinder varies significantly (figure 10*a*). Starting from the initial angle of 12^{°} it drops down to 4^{°} and increases up to 20^{°} shortly before the end of the planing stage. The separation point *γ*_{−}(*t*) was found to be always behind the centre of mass and the leading edge *γ*_{+}(*t*) is mainly in front of it, except the time interval 0.4 s<*t*<0.6 s and the end of the planing stage, *t*>0.9 s. The vertical coordinates of the leading edge *y*_{+}(*t*) and the separation point *y*_{−}(*t*) are shown in figure 10*b*. The free-surface elevation in front of the planing body is found to be 12 cm at time *t*=0.88 s which is more than twice the liquid depth *H*=5 cm. The distance between the body's lowest point and the bottom can be as small as 3 cm with the penetration depth being 40 per cent of the initial depth of the liquid. We cannot expect the skimming theory to be valid for such deep penetration. More results and discussions can be found in the electronic supplementary material.

*Case 3.* It was not easy to find impact conditions which lead to contact of the body with the bottom of liquid layer. Keeping the dimensions of the body and the thickness *H* of the liquid layer as in cases 1 and 2, and the initial horizontal speed 10 m s^{−1}, we found that the body does not touch the bottom and finally exits the liquid if either the inclination angle *α*(0) is not small, or the body is light, or the initial vertical velocity is small. Only increasing the mass of the body up to 600 kg m^{−1}, the initial vertical speed up to 3 m s^{−1} and decreasing *α*(0) down to 3^{°}, we arrived at conditions which lead to the impact between the body and the bottom (figure 11). The interaction time is very short in this case, *t*<45 ms. Correspondingly, the horizontal speed does not change significantly, but the vertical speed of the centre of mass increases monotonically from −3 to 0.5 m s^{−1} during 45 ms. The average deceleration of the vertical motion is estimated as 74 m s^{−2} and the hydrodynamic force as 44 kN m^{−1}.

At the time of impact with the bottom, the centre of mass of the body is moving upwards, but the inclination angle increases from 3^{°} to 9^{°} and the lowest point of the body is behind the centre of mass. As a result, the lowest point of the body surface approaches the bottom and the impact with the bottom occurs at *t*≈45 ms with the vertical speed of centre mass of 1 m s^{−1}. The free-surface elevation at the leading edge, *y*_{+}(*t*), is three times larger than the liquid depth *H* and the separation point approaches the point of impact with bottom at the end of the calculations. At *t*=45.3 ms, the body surface to the left from the impact point is dry.

The pressure distribution along the wetted part of the body surface is shown in figure 12*a* with time step 5 ms starting from *t*=5 ms. Two lines with the intervals of negative pressures correspond to the impact stage (*t*=5 ms and *t*=10 ms). Note that the pressure at *t*=0 is very high, *p*_{±}(0)=1016.7 kPa, for these impact conditions. It is seen that at the planing stage the pressure has a peak inside the wetted region due to the decreasing distance between the body and bottom. At the last time instant, *t*=40 ms, shown in figure 12*a* the pressure peak is approximately 170 kPa. This instant is about 5 ms before the contact of the body with the bottom. The pressure distribution after *t*=40 ms is shown in figure 12*b*. The pressure at *t*=42 ms (curve 1) is positive in the wetted region and has the peak there where the distance of body surface from the bottom is minimum. The pressure peak increases with time. The maximum pressure at *t*=44.85 ms is about 1034 kPa (curve 3) and comparable with the initial value of the pressure *p*±(0). It is interesting to notice that at *t*=44 ms the pressure becomes negative in front of the peak. The region of negative pressure grows both towards the leading edge and the pressure peak. The minimum pressure is −20 kPa at *t*=44 ms and −80 kPa at *t*=44.85 ms. Note that the atmospheric pressure is about 100 kPa. The pressure at the leading edge approaches zero by the time of the body impact with the bottom. These results imply that cavitation may start in the wetted region shortly before the body touches the bottom. We may speculate that after the impact between the body and bottom (figure 13*a*), the body bounces back and the liquid instantly separates from the body surface over a large interval, as sketched in figure 13*b*.

In the case under consideration, the energy conservation law (4.1) provides that 12 per cent of the initial energy of the body is spent on jetting and 4 per cent on generation of flow in the liquid layer. In total, the body lost 16 per cent of its energy in 45 ms due to its interaction with the liquid layer.

## 5. Conclusion

A model of oblique impact of a smooth rigid body with a horizontal shallow liquid layer was presented. The problem under consideration is coupled: motions of the body were determined together with the liquid flow and the hydrodynamic pressure along the wetted part of the body surface. The position of the wetted region was also determined as part of the solution.

Two stages of the body interaction with the liquid layer were distinguished. During the first stage, the wetted region expands at a high speed with jetting flows at both ends of the wetted region. At the end of this stage, the hydrodynamic pressure was found to be below the atmospheric pressure along a significant part of the wetted region, which is due to the horizontal velocity of the body. During the second stage, the jet at the trailing edge disappears and the free surface separates smoothly from the surface of the moving body. The position of the separation point was determined with the help of the Brillouin–Villat condition, which requires the continuity of the pressure together with its tangential derivative at the separation point.

Equations of the body motions, the jump conditions at the periphery of the wetted region and the momentum conservation equation for the liquid flow were used to derive the system of nine nonlinear ordinary differential equations describing the body–liquid interaction during the first stage. During the second stage, the same system of ordinary differential equations was used, except for the equation governing the position of the trailing edge. The latter differential equation was replaced with an algebraic equation which followed from the Brillouin–Villat condition. The system of governing equations was solved numerically.

The horizontal and vertical displacements of the body, as well as its angle of rotation and corresponding speeds were studied as functions of time. The developed model was integrated in time until the body either touched the bottom of the liquid or rebounded from the liquid.

Calculations were performed for elliptic cylinders of different masses and with different orientations and speeds before the impact. It was shown that a light body exits the liquid shortly after impact. A body of moderate mass interacts with the liquid for a longer period with oscillations of its vertical displacement and the angle of rotation. A heavy body penetrates the liquid layer and hits the bottom at the end of the calculations.

The evolutions of both kinetic energy of the body and the energy of the generated flow were analysed. It was shown that the jets at both the leading and trailing ends of the wetted region are the main beneficiaries of the reduction of body energy during the impact.

## Acknowledgements

The authors would like to thank Prof. F. T. Smith for guidance and discussions, and Dr M. Cooker and Dr A. Tassin and Mr. M. Reinhard for valuable help and criticism. The work was supported by the grant RFBR 10-08-00076 and by the FP7 project SMAES—Smart Aircraft in Emergency Situations 266172. Preliminary results of this work were presented at the 10th All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics (Nizhny Novgorod, Russia, 24–30 August 2011) and at the 27th International Workshop on Water Waves and Floating Bodies (Copenhagen, Denmark, 22–25 April 2012).

- Received October 23, 2012.
- Accepted December 18, 2012.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.