## Abstract

The hydrodynamic problem of impact between a solid wedge and a liquid wedge is analysed. The liquid is assumed to be ideal and incompressible; gravity and surface tension effects are ignored. The flow generated by the impact is assumed to be irrotational and therefore can be described by the velocity potential theory. The solution procedure is based on the analytical derivation of the complex-velocity potential in a parameter plane and the function mapping conformally the parameter plane onto the similarity plane. The mapping function is found as a combination of the derivatives of the complex potential in the similarity and parameter planes, through the integral equations for mixed and homogeneous boundary-value problems in terms of the velocity modulus and the velocity angle with the fluid boundary, together with the dynamic and kinematic boundary conditions. These equations are solved through a numerical method. The procedure is first verified through comparisons with some known results. Simulations are then made for a variety of cases, and detailed results are presented in terms of the free surface shape, streamlines, pressure distribution on the wetted solid surface, and contact angles between the free surface and the body surface.

## 1. Introduction

Fluid/structure impact has a wide range of applications in many engineering problems. Impact usually lasts for a very short period of time, but extremely large hydrodynamic loads on structures can be generated. Green water impact on ship deck, slamming of ship bottom, and wave impact on offshore platforms or the coastline are well-known examples. In many cases, at the initial stage of fluid/structure impact, the flow can be considered as self-similar because there is no typical length scale in the problem. An example of these is a liquid wedge impacting a solid wall considered by Cumberbatch [1]. He used the self-similar variables to formulate the problem and obtained the mathematical solution in two matched forms valid at large and small distances from the wall, respectively.

Around two decades after Cumberbatch's work, numerical methods were developed for solving violent water impact onto a solid surface in the time domain. In particular, the boundary element method (BEM), together with a mixed-Eulerian–Lagrangian scheme [2,3], was adopted to predict magnitudes of the pressure maxima generated by the breaking wave on a rigid vertical wall and to study the motion of wedge-shaped breaking waves falling onto the free surface. If the wave crest approaches the vertical wall, the impact usually starts from a crest point and then develops along the wall as time progresses. The gravity effect is quite small owing to the short time scale of the impact and the flow is close to being self-similar. A numerical approach for this kind of problem based on the BEM using a stretched coordinate system has been proposed by Wu and co-workers [4–6]. Using this approach, Duan *et al.* [7] calculated the free surface shapes and pressure distribution on solid boundaries for oblique impact between solid and liquid wedges. In the case of asymmetric impact, they observed negative pressure near the wedge apex, which requires further investigation concerning the flow separation or formation of a vapour cavity in reality.

From the mathematical point of view, the problem of a liquid wedge impacting a solid wall belongs to the same class of water impact problems, which includes water entry of a wedge into a flat free surface. The latter was solved in a complete nonlinear self-similar formulation by Dobrovol'skaya [8] for the case of symmetric wedge entry, and by Chekin [9] for the specific case of oblique entry at which the stagnation point coincides with the wedge apex. Semenov & Iafrati [10] and Semenov & Yoon [11] considered oblique entry of a wedge into the free surface. All these methods are based on the theory of complex variables and reduce the problem to one or two integral equations, which are then solved by a numerical method. The solution of this kind of problem with emphasis on blunt bodies was also considered in the framework of matched asymptotic expansions in recent studies [12–16]. In this method, the order of magnitude of the deadrise angle between the body and the *x*-axis is used as a small parameter of expansion. In this study, we apply the integral hodograph method [17,18] to study oblique impact between liquid and solid wedges. A similar problem has been considered previously by others using the BEM [6,7]. The present method, however, provides some accurate detailed local results, such as the contact angles at the intersection points between the free surface and the body surface, and the length of the wetted surface. In addition, special attention is given to the limiting case of a liquid wedge of very small angle, for which the liquid wedge tends to a steady jet hitting a wall. The result is found to tend to the steady solution of a rectangular jet hitting a wall. Furthermore, for small deadrise angles in various cases, the high-pressure gradient is found to occur near the core of tip jets. The limiting conditions under which flow might separate from the wedge apex are also discussed.

The detailed solution method is based on the derivation of analytical expressions for two governing functions, which are the complex velocity and the derivative of the complex potential with respect to the coordinates of a chosen parameter plane. From these expressions, the complex potential and the function mapping the parameter plane into the similarity plane are obtained. Using the dynamic and kinematic boundary conditions, the problem is reduced to a system of an integral equation and an integro-differential equation in the parameter plane, in terms of the velocity magnitude and the velocity angle to the fluid boundary, respectively. The coupled equations are then solved through a numerical procedure based on the method of successive approximations.

The results are mainly presented in terms of streamline patterns, the contact angles at the intersection of the free surface with the solid boundary and the pressure distribution.

## 2. Theoretical formulation of the problem

The flow generated by the impact between a solid wedge of angle 2*α* and a wedge-shaped liquid column of angle is studied in a frame of reference fixed on the solid wedge. A sketch of the problem and the definitions of the geometric parameters are shown in figure 1*a*,*c*, respectively. The solid wedge is assumed to be symmetric about a vertical line. The bisectors of the solid and liquid wedges form a heel angle *δ*, which is positive when the symmetry line of the liquid wedge rotates in the counterclockwise direction; is the magnitude of the velocity and is the angle between the velocity and the horizontal axis *x* of the Cartesian coordinate system *xy*, and is positive when it on the clockwise side of *x*. It follows from the geometry of the problem that the right-hand side of the solid wedge forms an angle *γ*_{R}=−*π*/2+*α* with the horizontal axis *x*, while its left-hand side forms an angle *γ*_{L}=−*π*/2−*α*. We extend the definition of a deadrise angle in our case as the angle between the undisturbed free surface and the wedge surface, as shown in figure 1*c* through *β*_{L} and *β*_{R}, respectively. The liquid is assumed to be inviscid and incompressible, and gravity and surface tension effects are neglected. The pressure on the free surface is assumed to be constant and equal to the atmospheric pressure *P*_{a}.

The problem under consideration is then self-similar. In fact, figure 1*a* is shown through the self-similar variables *x*=*X*/(*V* _{0}*t*) and *y*=*Y*/(*V* _{0}*t*), where *t* is the time started from the moment that impact occurs, and *V* _{0} is the velocity magnitude at the point *O*, which is one of the intersection points between the free surface and the body surface. As a result, the varying flow region in the physical plane *Z*=*X*+i*Y* is transformed into a time-independent region in the plane *z*=*x*+i*y*. We will represent the complex potential of the self-similar flow in the following form:
2.1where *ϕ* and *ψ* are the velocity potential and the stream function in the similarity plane *z*. The problem is to determine a function *w*(*z*) that conformally maps the stationary *z*-plane onto the complex-velocity potential plane *w*. We choose the first quadrant of the *ς*-plane as the parameter region corresponding to the flow region to derive expressions for the non-dimensionalized complex velocity, d*w*/d*z*, and the derivative of the complex potential, d*w*/d*ς*, as functions of the variable *ς*=*ξ*+i*η*. If these functions are found, the velocity field and the relation between the parameter region and the physical flow region can be determined as follows:
2.2where *v*_{x} and *v*_{y} are the *x*- and *y*-components of the non-dimensionalized velocity.

### (a) Expressions for the derivatives of the complex potential in the similarity and parameter planes

Conformal mapping allows us to fix three arbitrary points in the parameter region, which are *O*, *B* and *D* as shown in figure 1*b*. In this plane, the positive imaginary axis (*η*>0, *ξ*=0) corresponds to the free surface, and the positive real axis (*ξ*>0, *η*=0) corresponds to the wetted part of the solid wedge. The points *ς*=*a* and *ς*=*c* are the images of the stagnation point *A* and the wedge vertex *C* in the similarity plane, respectively. The parameters *a* and *c* are not known and have to be determined as part of the solution.

The boundary-value problem for the complex-velocity function can be formulated in the parameter plane as follows. At this stage, we may write the velocity modulus along the free surface, or along the positive part of the imaginary axis of the *ς*-plane as
2.3In the frame of reference fixed with respect to the solid wedge, the normal velocity component equals zero owing to the impermeability condition. This means that the argument *χ* of the complex velocity along the real axis of the parameter region is fixed and can be determined by the wedge orientation. Thus, we have
2.4The formula, derived by Semenov [17,18] using the Chaplygin [19] singular point method
2.5provides an integral form of the mixed boundary-value problem in the first quadrant of the complex *ς*-plane. In the equation, is the velocity at point *B* and . By evaluating the first integral in equation (2.5) with the step change of the function *χ*(*ξ*) given in equation (2.4), we obtain the expression for the complex velocity in the *ς*-plane in the form
2.6where *v*_{0}=*v*(*η*)|_{η=0}=1 is the velocity magnitude at point *O*.

In order to analyse the behaviour of the velocity potential along the free surface, it is useful to introduce the unit vectors **n** and ** τ** in the normal and tangential directions of the fluid boundary, respectively. The normal vector points out of the fluid region, while the spatial coordinate

*s*along the surface increases in the direction for which the fluid region is on the left (figure 1

*a*). With this notation, 2.7where

*v*

_{s}versus and

*v*

_{n}are the tangential and normal velocity components, respectively. Let

*θ*be the angle between the velocity vector on the surface and

*τ*, which means . The definition in equation (2.7) allows us to determine the argument of the derivative of the complex potential d

*w*/d

*ς*, which appears in equation (2.2). We notice that , along the real axis of the parameter plane since d

*s*>0 and d

*ς*=d

*ξ*, and along the imaginary axis since d

*s*<0 and d

*ς*=id

*η*. Then, 2.8Now we determine the function

*θ*(

*ς*) along the whole fluid boundary, that is, along the half real and half imaginary axes of the

*ζ*-plane. When moving along the free surface from point

*O*to point , the function

*θ*(

*ς*) changes from the value

*μ*

_{L}at

*ξ*=0,

*η*→0 to the value at

*ς*=

*i*. In order to get the left-hand side of the free surface away from the solid wedge, we move along part of the circle of infinitely large radius from to , where the fluid is undisturbed and the known velocity direction gives . Thus, will change in the same way as the slope to the free surface, providing at point the value . When moving in a counterclockwise direction along an infinitesimal semicircle centred at the point

*ς*=

*i*corresponding to the infinite radius in the

*z*-plane, the function

*θ*(

*ς*) changes by . The continuous changes of the function

*θ*(

*ς*) are shown in figure 2 by solid lines, while its step changes are shown by dashed lines. Furthermore,

*θ*(

*ς*) changes continuously when moving along the free surface from point to point

*B*. On the interval, ,

*η*=0, corresponding to the whole right-hand side of the solid wedge and the left-hand side between points

*C*and

*A*,

*θ*(

*ς*)≡0 because

*v*

_{n}=0 and

*v*

_{s}>0. On the interval 0<

*ξ*<

*a*,

*η*=0,

*θ*(

*ς*)≡

*π*because

*v*

_{n}=0 and

*v*

_{s}<0. Thus, when passing the point

*A*(

*ς*=

*a*), as shown in figure 2

*b*,

*θ*(

*ς*) takes a jump of

*Δθ*

_{A}=

*π*. The last jump

*Δθ*

_{O}=

*μ*

_{L}−

*π*occurs at point

*O*when we move in the vicinity of the point

*ς*=0 from the wedge surface,

*ξ*>0,

*η*=0, to the free surface,

*ξ*=0,

*η*>0, as shown in figure 2

*b*. The total jump of function

*ϑ*(

*ς*) at point

*ς*=0 taking into account equation (2.8) is

*Δθ*

_{O}=

*μ*

_{L}−

*π*/2. The subscripts ‘−’ and ‘+’ at points

*A*,

*O*and

*B*in figure 2

*a*denote sides that we meet before and after we pass the point when we walk along the boundary in the clockwise direction.

By introducing a continuous function *λ*(*ς*), we can write *θ*(*ς*) as follows:
2.9where , Δ*θ*_{O}=*μ*_{L}−*π*, Δ*θ*_{A}=*π*, *λ*(0)=0. The expression for the derivative of the complex potential can be obtained by applying the integral formula [17]
2.10which is derived from the Schwartz integral formula when the first quadrant of the *ς*-plane is chosen as the parameter region. Here, *K* is a real factor and .

By substituting equations (2.8) and (2.9) into the first integral in (2.10) when *ς* varies along the real axis and into the second integral when *ς* varies along the imaginary axis, and evaluating the integrals over the step changes of the function *θ*(*ς*), we finally obtain the expression for the derivative of the complex potential in the *ς*-plane as
2.11in which the integration over the step changes is done, for example, at point *A* (*ς*=*a*) as follows:
The minus sign here is due to the direction of the integration path being opposite to the jump Δ*θ*_{A}.

From equations (2.6) and (2.11), the derivative of the mapping function and the complex-velocity potential can be obtained as
2.12and
2.13where *w*(0) is the velocity potential at point *O*.

Equations (2.6) and (2.11) contain the parameters *a*, *c*, *K* and the functions *v*(*η*) and *λ*(*η*), which are to be determined from physical considerations and the dynamic and kinematic boundary conditions. At infinity, the complex-velocity approaches the value , where . Thus letting *ς*=*i* in equation (2.6), the following condition is obtained:
2.14In the physical plane, the wetted lengths of the right- and left-hand sides of the solid wedge are *V* _{0}*t* and *V* _{B}*t*, respectively. Here, *V* _{B}=*v*_{B}*V* _{0} is the fluid velocity at point *B* in the physical plane, and is obtained from the solution. The lengths of the segments *OC* and *CB* in the similarity plane are then |*z*_{O}|=1 and |*z*_{B}|=*v*_{B}=*V* _{B}/*V* _{0}, respectively. Hence, the following conditions are obtained:
2.15

### (b) Dynamic and kinematic boundary conditions

The Bernoulli equation in the physical plane linking point *O* and an arbitrary point in the flow gives
2.16where *P* and *V* are the pressure and velocity at an arbitrary point of the fluid domain, *P*_{a} is the pressure on the free surface and *ρ* is the density of the liquid.

By taking advantage of self-similarity of the flow defined in equation (2.1), and introducing the self-similar spatial coordinate of arc length defined previously, Semenov & Iafrati [10] reduced this equation to the following:
2.17where at *s*=0 or point *O*, the potential has been assumed to be zero (note, *s*>0 on the sides of the solid wedge and *s*<0 on the free surface *OD*). In equation (2.17), is the pressure coefficient. Along the free surface, the pressure is constant and equal to the atmospheric pressure *P*_{a}; therefore, *c**_{p}=0. By taking the derivative of equation (2.17) with respect to *s* and accounting for the relations d*ϕ*/d*s*=*v*_{s} and , Semenov & Iafrati [10] obtained the following differential equation:
2.18which is valid on both the left and right free surfaces and .

Multiplying both sides of equation (2.18) by d*s*/d*η* and taking into account that d*θ*/d*s*=d*λ*/d*s*, 0<*η*<1, we obtain the following differential equation:
2.19where the arc length coordinate *s*, which is a function of *η*, can be obtained by integrating equation (2.12) for the left free surface, or
2.20where 0≤*η*<1, and
2.21for the right free surface, for which .

The kinematic boundary condition derived by Semenov & Iafrati [10] in terms of the velocity magnitude *v* and velocity angle *θ* with the free surface for any self-similar flow problem has the following form:
2.22which is valid on both the left and right free surfaces and . This equation is obtained using the fact that the acceleration of the fluid particle is orthogonal to the free boundary if the pressure along the free surface is constant.

Substituting the complex velocity in equation (2.6) into equation (2.22) and multiplying both sides of the result by d*s*/d*η*=|d*z*/d*ς*|_{ς=iη}, the following integral equation in terms of the function is obtained:
2.23for 0<*η*≤1. A similar equation can be obtained for corresponding to *BD*. As a result, equation (2.23) is found to be valid along the whole imaginary axis of the parameter plane.

The integral equations (2.19) and (2.23) along with (2.14) and (2.15) make it possible to determine the functions *θ*(*η*) and *v*(*η*), and the parameters *a*, *c*, *K*. Once these functions and parameters are found, the contact angles between the wedge sides and the free surface, *μ*_{R} and *μ*_{L}, can be determined as follows (figure 2): .

Taking into account that between points *O* and *A* d*ϕ*d*s*=*v*_{s}=−*v*, while on the rest of the wetted part of the solid wedge d*ϕ*d*s*=*v*_{s}=*v*, the pressure coefficient along the left and right sides of the solid wedge can be obtained from equation (2.17) as follows:
2.24where *ϕ*, *v*, *s* are determined from equations (2.12), (2.6) and (2.2) and can be written as follows:
and *v*_{ref} is the magnitude of the projection of the impact velocity onto the bisector of the liquid wedge, or .

## 3. Flow analysis

### (a) Numerical approach

The integral equations (2.19) and (2.23) are solved numerically by iteration through the method of successive approximations. Equations (2.14) and (2.15) are used in each iteration. In discrete form, the solution is sought on two sets of points. The first, 0<*η*_{j}<1, *j*=1…*N*, corresponds to the segment of the free surface and the points are distributed in such a way that the segment size increases geometrically away from *O*. The second set of points corresponding to is chosen to be the inverse reflection of the first set of points about *η*=1, i.e. *η*_{j}=1/*η*_{2N−j+1}, *j*=*N*+1…2*N*. The numerical approach used in the present study is based on the method of successive approximations used by recent studies [10,11] for solving self-similar water-entry problems.

The solution at the intersection of the free surface and body surface is very challenging owing to the singularity occurring here [8]. In the present solution, a similar singularity can be seen in the expression for the derivative of the complex potential in equation (2.11) at point *ς*=0 when 2*μ*_{L}/*π*−1<0 and due to the improper integral with upper limit at . The singularities at the two intersection points depend on the values of the contact angles *μ*_{L} and *μ*_{R}, respectively, and the range of variation of the function *λ*(*η*), which determines the order of singularity at . For a given discretization along the *η*-axis discussed earlier, the corresponding arc length coordinates *s*_{1}=*s*(*η*_{1}) and *s*_{2N}=*s*(*η*_{2N}) nearest to contact points *O* and *B* in the similarity plane can be obtained using equations (2.20) and (2.21), respectively, as
3.1and
3.2Numerical integration of the function d*s*/d*η* near the contact point *O* over the interval 0≤*η*≤*η*_{1} requires an extremely small step of order 10^{−15}, owing to the singularity *η*^{2μL/π−1} for 2*μ*_{L}/*π*=1. A similar example is in Zhao & Faltinsen [20] who used the step of integration up to order 10^{−25} when they solved Dobrovol'skaya's [8] integral equation.

If we choose a typical value *η*_{1}=10^{−5}, then *s*_{1}=*s*_{2N}≈0.5 is obtained for deadrise angle *β*_{L}=*β*_{R}≈5^{°}, i.e. the length of the first node in the similarity plane is about half of the wetted wedge length. The arc length coordinates *s*_{1}=*s*(*η*_{1}) and *s*_{2N}=*s*(1/*η*_{1}) affect all other spatial coordinates *s*_{i} and *s*_{j} on the left and right free surfaces,
and, correspondingly, the numerical evaluation of the function *s*=*s*(*η*), which appears in the integral equation (2.19).

### (b) Validation of the numerical approach

The numerical approach for solving a system of integral equations (2.19)–(2.23) is similar to that in Semenov & Iafrati [10] for the vertical entry of an asymmetric wedge into a flat free surface. It was validated there through comparisons of the obtained results with those available in the literature. For further validation and verification purposes here, we consider a case of liquid wedge impacting on a flat solid wall. Physically, the velocity of the liquid along the direction of the wall should have no effect on the shape of the free surface and the pressure on the wetted part of the wall. This is also obvious enough if the mathematical model is established in the system moving with the liquid [4–6]. However, in the present model, the system is fixed on the solid wedge. Thus, when the liquid wedge hits the wall, the presence of the velocity along the wall leads to a different mathematical problem, which has to be solved separately. As a result, this could be a test case for verification of the method.

Figure 3 shows the streamlines for a liquid wedge of angle and a heel angle *δ*=0 hitting a flat solid wall (*α*=90^{°}). For the case of shown in figure 3*a*, the component of the impact velocity along the direction of the wall surface is equal to zero, while for the case of shown in figure 3*b*, the velocity component along the wall surface is , where *v*_{y} is the velocity component perpendicular to the wall and is positive when it is opposite to *y*. Although the streamlines may look different in these two cases, the free surface shapes are the same.

The streamlines in the figure have been obtained in two steps. At the first step, the line in the parameter plane corresponding to the *i*th streamline in the similarity plane is determined from the following equation:
3.3where *ψ*_{i} is the value of the stream function for the *i*th streamline in the figure, and *ς*_{0i}=i*η*_{i} is the point in the parameter plane corresponding to the intersection of the free surface and the *i*th streamline. At the second step, the streamline in the similarity plane is determined by integrating equation (2.12) along the contour .

The pressure distributions along the wall in the above two cases are shown in figure 4*a*, together with the one for . It can be seen that the pressure distribution is only shifted relative to the point *x*=0, where the liquid wedge touches the wall at the initial time, while the curve shapes remain the same. These results confirm the obvious physical fact that the component of the liquid wedge velocity along the flat rigid solid wall during the impact does not have any effect on the shape of the free surface and the pressure distribution on the wetted wall surface.

In figure 4*b*, we also compare the obtained pressure distribution (solid line) with the results of Duan *et al.* [7] (dashed line) and Zhang *et al.* [21] (dotted line), which have been taken manually from their figures. The present result is indistinguishable graphically with that of Duan *et al*. [7], while there is some discrepancy between our result and that from Zhang *et al*. [21]. This is because some approximation for the free surface shape was adopted in Zhang *et al*. [21], but not in Duan *et al*. [7]. It should be noted that for this impact problem, which starts with a single contact point, an effective numerical method in the time domain is to use the stretched coordinate system developed by Wu [4] and subsequently used in recent studies [5,6]. Special treatment was introduced in the jet zone in these publications, which led to accurate results for the pressure. In the present study, we have evaluated analytically the spatial coordinates *s*_{1} and *s*_{2N} closest to the contact points *O* and *B*, which has provided accurate evaluation of the function *s*=*s*(*η*) along the whole free surface and, consequently, we have obtained overall accuracy of the solution. This may be part of the reason for the good agreement between the present result and that of Duan *et al.* [7]. As discussed in §1, we note that although the problem can be solved by the BEM, it is difficult for the method to provide some of the detailed results accurately, such as the intersection angle and wetted length, which, on the other hand, can be achieved by the present method.

### (c) Symmetric impact between liquid and solid wedges

Having verified the method, we shall consider various cases. The pressure distribution and free surface shapes during the symmetric water impact between the liquid and solid wedges (or water entry of a solid wedge, and a liquid wedge hitting a solid flat wall) have been investigated by many authors [1,5–7,22,23]. They used either approximate analytical methods or the BEM to solve the integral equation. In particular, recent studies [5,6,23] adopted the BEM for the complex-velocity potential coupled with the stretched coordinate system [4], and provided detailed results for wedges of deadrise angles larger than 20^{°}. At smaller angles, the time step has to be further reduced. Results could be obtained, but the computational effort would increase significantly.

The prediction of water impact flows for small deadrise angles is also rather challenging in the present methodology. It is partly caused by the singularities occurring at the intersection of the free surface and the solid body. A similar problem is in the problem of water entry of a wedge. Dobrovol'skaya [8] converted the problem into an integral equation for a function *f*(*t*) over 0≤*t*≤1. Her numerical solution is, however, not entirely accurate. Zhao & Faltinsen [20] resolved this integral using an extremely small step. In fact, the smallest step in one case is of order 10^{−25}. Their results are then in good agreement with those from other methods.

The model derived by Semenov & Iafrati [10] has led to a different integral equation. It makes it possible to solve wedge impact problems with small deadrise angles using standard arithmetic tools and reasonable computation time. It is the same for the present study, which enables us to calculate some extreme cases.

It is well known that for water entry of a wedge into a flat free surface, a peak pressure will appear near the root of the jet at small deadrise angles [20]. We can expect a similar effect for the impact between a liquid and solid wedge at the small deadrise angles, marked as *β*_{L} and *β*_{R} in figure 1*c*. Figure 5 shows several configurations of the liquid and solid wedges forming a deadrise angle of 10^{°}. The flow patterns in figure 5*a*,*b* change only slightly when the impact problem changes from a liquid wedge hitting a flat wall to a solid wedge hitting a flat liquid surface. In figure 5*c*, the streamlines correspond to an acute liquid wedge impacting on a solid reflex corner and in figure 5*d*, a liquid wedge with an obtuse angle impacting an acute solid corner.

The pressure distributions corresponding to these cases are shown in figure 6. In all the cases, there is a peak pressure and after that the pressure drops to the ambient pressure. For cases (*a*) and (*b*), the pressure distributions are very close to each other. For cases (*c*) and (*d*), the pressure becomes much lower. The reason for that may be because of the smaller area of contact with the solid surface in (*c*), due to the acute angle of the liquid wedge. In (*d*), the further reduction of the pressure looks reasonable owing to the possibility of the liquid wedge extending in the transverse direction.

In figure 7, the dependence of the contact angle (between the free surface and the wedge surface) on the angle of the liquid wedge and of the solid wedge for the symmetric flow configuration are shown. As mentioned previously, this is usually very challenging numerically, owing to the singularity of the solution at these points, which can be tentatively seen from equation (2.19) as *s*→0. In the present solution, in order to achieve sufficient accuracy, with an error not larger than 1 per cent, we use a set of points *η*_{j}, *j*=1…2*N* along the axis of the parameter plane, distributed in the manner described in §3*a* and place the first point at *η*_{1}=10^{−6} and the last at *η*_{2N}=1/*η*_{1}=10^{6}. Our numerical results seem to suggest that the contact angle is approximately equal to the half-angle of the liquid wedge when the angle of the liquid wedge tends to zero, i.e. , as .

The contact angle *μ*_{L} reaches its maximum value at some angle , which depends on the angle of the solid wedge *α*. For *α*→0, the contact angle *μ*_{L}/*π*=*μ*_{R}/*π*→0.25 at corresponding to the deadrise angle *β*_{L}=*β*_{R}≈135^{°}. The same maximum value of the contact angle at the same deadrise angle was obtained for the water-entry problem of a flat plate [24]. It is worth noting that the maximum value of the contact angle for the vertical entry of a symmetric wedge *α*→0 is *μ*_{L}/*π*=*μ*_{R}/*π*≈0.1 for the case , which agrees with that obtained by recent studies [8,25].

During the impact, the liquid between the free surface and the body surface near the intersection is wedge shaped with angle *μ*_{L}, (or *μ*_{R}), moving along the wall. In a marine context, this ‘new’ liquid wedge may produce a secondary impact on the lower side of the deck of offshore platforms or ships, etc. In the case of a wall, i.e. *α*=90^{°}, the obtained maximum value of the contact angle is *μ*_{L}/*π*≈0.0278, at .

It is found that the impact problem between the symmetric liquid wedge and the solid wall in the limiting case tends to the steady problem of a rectangular jet impacting on the wall perpendicularly. Indeed, as can be seen from figure 7, the contact angle *μ*_{L}→0 when . By equation (3.1), the location of the contact point *O* tends to infinity. Moreover, the order of singularity at point , (*ς*=*i*) in equation (2.11) for becomes 1/(*ς*−*i*), corresponding to the logarithmic singularity in the complex potential occurring for a steady jet of finite flow rate. For , the normal component of the velocity *v*_{n}→0 along the whole free surface, and therefore the function ∥*θ*(*η*)∥→0. In this case, from equation (2.23), it follows that . By substituting d*θ*/d*η*=0, and *μ*_{L}=0 into equations (2.6) and (2.12), we obtain the following expressions for the derivative of the complex potential and for the complex velocity corresponding to the steady flow of the jet spreading along the wall:
and
in which we have used the asterisk to indicate that this is the steady solution corresponding to the rectangular jet hitting the wall.

Figure 8 shows the streamlines and pressure distribution for the liquid wedge of half-angle 1^{°} impacting the solid. It can be seen from the figure that the streamlines become almost parallel to each other and the maximum pressure reaches unity, i.e. the maximum pressure for the steady flow. Thus, the solution obtained for the liquid wedge impact problem continuously tends to the solution for the steady jet flow. The problem of a steady jet falling from a vertical pipe and hitting a horizontal plate including the gravity effect has recently been considered by Christodoulides & Dias [26].

### (d) Asymmetric/oblique impact

The effect of asymmetry is investigated by performing numerical calculations for fixed angles of solid and liquid wedges, but different heel angles, *δ*, and velocity directions at infinity, .

The streamlines corresponding to asymmetric/oblique impact of a liquid wedge of angle are shown in figure 9 for both the vertical ((*a*) and (*b*)) and oblique ((*c*), (*d*) and (*e*)) cases. Figure 9*c*–*e* has been re-scaled to provide the same vertical component of the incoming velocity as in the vertical impact. For all the cases shown in figure 9, the stagnation point approaches the root of the jet on the side with the smaller deadrise angle. The contact angle becomes larger on the side with a larger deadrise angle and smaller on the opposite side. The density of streamlines increases with the magnitude of the velocity because the flux of the liquid between the two nearest streamlines is constant. The streamline patterns clearly show an increase in the velocity near the root of the jets and a decrease in the velocity magnitude near the stagnation point. The magnitude of coordinates of the intersection points at the left- and right-hand sides in figure 9 shows the ratio between the velocity of the tips and the vertical component of the incoming component. The comparison between the pairs (*a*) and (*c*), and (*b*) and (*d*) shows the symmetric configuration of each pair with respect to the *y*-axis, which should be obvious as the horizontal velocity merely reverses its direction but retains its magnitude.

For case (*b*), the angle *θ* on the right-hand side increases from value *π*−*μ*_{R} at point *B* to *π* at point where the normal and tangential components of the velocity equal *v*_{n}=0 and , respectively. For the case with angle *δ*>30^{°}, the normal component of the velocity at point becomes negative and the angle *θ* becomes larger than *π*. In our solution procedure, the range of the angle is defined in the interval −*π*≤*θ*≤*π*. In order to predict cases with *δ*>30^{°}, a horizontal component of the velocity could be added to provide −*π*≤*θ*≤*π*. This case is shown in figure 9*e*.

The pressure distributions along the wall are shown in figure 10. In the figure, the solid line corresponds to case (*a*) in figure 3, the dashed and dotted lines with the peak pressure at *x*<0 correspond to cases (*a*) and (*b*) in figure 9 for vertical impact, respectively. The same types of lines with the peak pressure at *x*>0 correspond to the cases (*c*) and (*d*). In the cases of asymmetric vertical impact ((*a*) and (*b*) in figure 9), the pressure peak near the roots of the jet appears to be similar to that observed in a symmetric wedge entering into a flat free surface [20]. The peak of the pressure increases with decreasing deadrise angle. This behaviour of the pressure distribution along the wall was also observed by Duan and co-workers [7,23] using the BEM. We note that the *y*-axis in figure 10 has a break to show the peak value of the pressure for the case shown in figure 9*f*, which results in a discontinuity of the dash-dotted line.

Figure 11 shows streamlines corresponding to an impact between a liquid wedge of angle and a solid wedge of half-angle 45^{°}. The wedge orientation and the impact velocity may form condition , for which the liquid at infinity runs away from the right-hand side of the solid wedge. Under such a condition, flow separation from the wedge vertex may occur, as discussed by Xu *et al*. [23], thus changing the flow topology and requiring another mathematical formulation for the problem. In this case, only one side of the wedge would be in contact with the liquid, resulting in a problem corresponding to that of a liquid wedge impact on a flat plate. The initial separation/ventilation in the case of water entry of a wedge into a flat free surface was also studied by Judge *et al*. [27] experimentally and theoretically. They found that the flow separation would occur suddenly at some limiting combination of the heel angle and the angle of the impact velocity, which provides condition , which is similar to the case shown in figure 11*c*. We note that case (*c*) in figure 11 is the limiting case for which the convergence of the numerical procedure based on successive approximations could be obtained. However, we ought to point out that this could the limit of the mathematical model. Flow separation in real situations is highly complex, which could be affected by gas entrainment, surface tension, fluid viscosity, etc. These effects are evidently beyond the scope of this study, and will need further investigation through experiment and improved mathematical model.

In figure 12, we show the pressure distributions corresponding to the flow patterns in figure 11. For symmetric impact (case (*a*)), the pressure is positive on both sides, including at the wedge apex. It occurs because the stagnation point coincides with the wedge apex. For case (*b*) on the lower side of the wedge, there is a maximum of the pressure. It appears because the pressure coefficient is zero at the contact point and it is negative at the wedge apex. On the other hand, one could expect that the pressure on some part of the surface would be positive in reaction to the impact. The pressure is then obviously larger than that at the contact point and at the wedge apex, as reflected by the maximum in the curve.

Figure 13 shows the effect of the direction of the incoming velocity on the streamlines. The stagnation point moves very little from the wedge vertex along the lower wedge side as the horizontal component of the velocity increases. The wetted length of the vertical side of the wedge is somewhat smaller than that of the horizontal side. It can also be seen from the pressure distributions in figure 14 that the higher pressure occurs on the vertical side while its wetted length is smaller. The behaviour of the pressure distribution is qualitatively the same for various angles of impact velocity. The pressure distribution on the wedge sides is similar to that shown in figure 12. The difference is only that the distance between the stagnation point and the wedge apex is much smaller than those in the cases shown in figure 12. Our obtained numerical result also show that the pressure varies sharply near the apex of the wedge and seems to tend to minus infinity at the apex.

## 4. Conclusions

We have presented a mathematical procedure for the fully nonlinear problem of impact between a solid wedge and a liquid wedge, on the basis of incompressible velocity potential theory. This methodology has made it possible to derive analytical expressions for the complex velocity and the derivative of the complex potential in the parameter plane. They are found through integral formulae for the mixed and homogeneous boundary-value problems in terms of the velocity modulus and the angle between the velocity and the fluid boundary. The mapping function between the similarity plane and the parameter plane is obtained from these expressions. It explicitly contains the singularities, including those at the intersection points of the body and free surfaces. This fluid/structure impact problem is then reduced to a system of integral and integro-differential equations after the dynamic and kinematic boundary conditions are imposed in the integral formulae. These equations are solved through the method of successive approximation. For a liquid wedge of small angle, the obtained solution continuously tends to that of the steady jet hitting a solid wall. The procedure is first verified by comparing with some known results. Simulations are then made for a variety of cases, and detailed results are presented in terms of the free surface shape, streamlines, pressure distribution on the wetted solid surface and contact angles between the free surface and the body surface.

Numerical results are presented for a wide range of angles for both the liquid and solid wedges, including reflex ones. The capability of the method to predict various configurations including blunt solid and ‘blunt’ liquid wedges enables us to show that the pressure distribution along the wetted surface is very similar for the case of a blunt solid wedge entering a flat free surface and for the case of a ‘blunt’ liquid wedge impacting a solid wall. The major parameter determining the pressure distribution for these cases is the deadrise angle. When the solid or liquid wedge has an acute angle, the difference in the pressure distributions appears even for the same small deadrise angle.

For asymmetric impact between the liquid and solid wedges, there are some limiting configurations within which a solution could be obtained by the present method. For such configurations, it is found that the pressure over the whole wetted surface becomes less than the pressure on the free surface.

Dobrovol'skaya [8] and Fraenkel & Keady [25] showed analytically that the contact angle at the intersection could be no larger than *μ*_{L}/*π*=0.25. From the numeric results for water entry of a symmetric wedge onto a flat free surface, they found that the largest contact was *μ*_{L}/*π*=0.1, which has been confirmed by the present work (see figure 7, which shows that the largest *μ*_{L}/*π* is about 0.1 for the case of ). We have then found numerically that the largest contact angle of *μ*_{L}/*π*≈0.25, which occurs in the case of symmetric impact between a thin solid wedge and a water wedge of , at deadrise angles *β*_{L}=*β*_{R}≈135^{°}. For a liquid wedge of small angle, it has been found from the presented analytical solution that the result smoothly tends to that corresponding to the steady flow of a rectangular jet hitting a solid wall. We have also found that in this case, the maximum value of the contact angle is about *μ*_{L}/*π*=*μ*_{R}/*π*≈0.0278, which occurs at . If the angle of the liquid wedge , then the contact angle also tends to zero.

## Acknowledgements

This work is supported by the Lloyd's Register Educational Trust (LRET) through the joint centre involving University College London, Shanghai Jiao Tong University and Harbin Engineering University. The LRET is an independent charity working to achieve advances in transportation, science, engineering and technology education, training and research worldwide for the benefit of all.

- Received April 3, 2012.
- Accepted October 31, 2012.

- © 2012 The Author(s) Published by the Royal Society. All rights reserved.