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ResearchCite this article: Semenov YA, Wu GX. 201Asymmetric impact between liquid and solidwedges. Proc R Soc A 469: 20120203.http://dx.doi.org/10.1098/rspa.2012.0203

Received: 3 April 2012Accepted: 31 October 2012

Subject Areas:applied mathematics, integral equations,mathematical modelling

Keywords:liquid/solid wedge, impact, complex potential,physical plane, similarity plane, parameterplane

Author for correspondence:G. X. Wue-mail: [email protected]

Asymmetric impact betweenliquid and solid wedgesY. A. Semenov and G. X. Wu

Department of Mechanical Engineering, University College London,LondonWC1E 6BT, UK

The hydrodynamic problem of impact between asolid wedge and a liquid wedge is analysed. Theliquid is assumed to be ideal and incompressible;gravity and surface tension effects are ignored. Theflow generated by the impact is assumed to beirrotational and therefore can be described by thevelocity potential theory. The solution procedure isbased on the analytical derivation of the complex-velocity potential in a parameter plane and thefunction mapping conformally the parameter planeonto the similarity plane. The mapping function isfound as a combination of the derivatives of thecomplex potential in the similarity and parameterplanes, through the integral equations for mixed andhomogeneous boundary-value problems in terms ofthe velocity modulus and the velocity angle withthe fluid boundary, together with the dynamic andkinematic boundary conditions. These equations aresolved through a numerical method. The procedureis first verified through comparisons with someknown results. Simulations are then made for avariety of cases, and detailed results are presented interms of the free surface shape, streamlines, pressuredistribution on the wetted solid surface, and contactangles between the free surface and the body surface.

1. IntroductionFluid/structure impact has a wide range of applicationsin many engineering problems. Impact usually lastsfor a very short period of time, but extremely largehydrodynamic loads on structures can be generated.Green water impact on ship deck, slamming of shipbottom, and wave impact on offshore platforms or thecoastline are well-known examples. In many cases, atthe initial stage of fluid/structure impact, the flow canbe considered as self-similar because there is no typicallength scale in the problem. An example of these isa liquid wedge impacting a solid wall considered byCumberbatch [1]. He used the self-similar variables to

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formulate the problem and obtained the mathematical solution in two matched forms valid atlarge and small distances from the wall, respectively.

Around two decades after Cumberbatch’s work, numerical methods were developed forsolving violent water impact onto a solid surface in the time domain. In particular, the boundaryelement method (BEM), together with a mixed-Eulerian–Lagrangian scheme [2,3], was adoptedto predict magnitudes of the pressure maxima generated by the breaking wave on a rigid verticalwall and to study the motion of wedge-shaped breaking waves falling onto the free surface. Ifthe wave crest approaches the vertical wall, the impact usually starts from a crest point and thendevelops along the wall as time progresses. The gravity effect is quite small owing to the shorttime scale of the impact and the flow is close to being self-similar. A numerical approach for thiskind of problem based on the BEM using a stretched coordinate system has been proposed by Wuand co-workers [4–6]. Using this approach, Duan et al. [7] calculated the free surface shapes andpressure 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, whichrequires further investigation concerning the flow separation or formation of a vapour cavityin reality.

From the mathematical point of view, the problem of a liquid wedge impacting a solid wallbelongs to the same class of water impact problems, which includes water entry of a wedgeinto a flat free surface. The latter was solved in a complete nonlinear self-similar formulationby Dobrovol’skaya [8] for the case of symmetric wedge entry, and by Chekin [9] for the specificcase 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 ortwo integral equations, which are then solved by a numerical method. The solution of this kindof problem with emphasis on blunt bodies was also considered in the framework of matchedasymptotic expansions in recent studies [12–16]. In this method, the order of magnitude of thedeadrise angle between the body and the x-axis is used as a small parameter of expansion. Inthis study, we apply the integral hodograph method [17,18] to study oblique impact betweenliquid and solid wedges. A similar problem has been considered previously by others using theBEM [6,7]. The present method, however, provides some accurate detailed local results, such asthe contact angles at the intersection points between the free surface and the body surface, andthe length of the wetted surface. In addition, special attention is given to the limiting case of aliquid 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 thecore of tip jets. The limiting conditions under which flow might separate from the wedge apex arealso discussed.

The detailed solution method is based on the derivation of analytical expressions for twogoverning functions, which are the complex velocity and the derivative of the complex potentialwith respect to the coordinates of a chosen parameter plane. From these expressions, the complexpotential 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 ofan integral equation and an integro-differential equation in the parameter plane, in terms ofthe velocity magnitude and the velocity angle to the fluid boundary, respectively. The coupledequations are then solved through a numerical procedure based on the method of successiveapproximations.

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

2. Theoretical formulation of the problemThe flow generated by the impact between a solid wedge of angle 2α and a wedge-shaped liquidcolumn of angle μ∞ is studied in a frame of reference fixed on the solid wedge. A sketch of the


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C

V•

V•g•

m•

dD•

mL

bL

gL

gR

bR

D¢•

D• D¢•

ih

A

BO

xn

t

y

s

B

B

O C

c x

z = x + ih

i

A

a

(a) (c)

2a

(b)

mR

Figure 1. Sketch for impact between a liquid wedge and a solid wedge: (a) similarity plane; (b) parameter plane; (c) definitionof various angles at the time of impact.

problem and the definitions of the geometric parameters are shown in figure 1a,c, respectively.The solid wedge is assumed to be symmetric about a vertical line. The bisectors of the solidand liquid wedges form a heel angle δ, which is positive when the symmetry line of the liquidwedge rotates in the counterclockwise direction; V∞ is the magnitude of the velocity and γ∞ isthe 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 problemthat the right-hand side of the solid wedge forms an angle γR = −π/2 + α with the horizontal axisx, while its left-hand side forms an angle γL = −π/2 − α. We extend the definition of a deadriseangle in our case as the angle between the undisturbed free surface and the wedge surface, asshown in figure 1c through βL and βR, respectively. The liquid is assumed to be inviscid andincompressible, and gravity and surface tension effects are neglected. The pressure on the freesurface is assumed to be constant and equal to the atmospheric pressure Pa.

The problem under consideration is then self-similar. In fact, figure 1a is shown throughthe self-similar variables x = X/(V0t) and y = Y/(V0t), where t is the time started from themoment that impact occurs, and V0 is the velocity magnitude at the point O, which is one ofthe intersection points between the free surface and the body surface. As a result, the varyingflow region in the physical plane Z = X + iY is transformed into a time-independent regionin the plane z = x + iy. We will represent the complex potential of the self-similar flow in thefollowing form:

W(Z, t)=Φ(Z, t)+ iψ(Z, t)= V20tw(z)= V2

0t[φ(z)+ iψ(z)], (2.1)

where 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-planeonto the complex-velocity potential plane w. We choose the first quadrant of the ς -plane asthe parameter region corresponding to the flow region to derive expressions for the non-dimensionalized complex velocity, dw/dz, and the derivative of the complex potential, dw/dς ,as functions of the variable ς = ξ + iη. If these functions are found, the velocity field andthe relation between the parameter region and the physical flow region can be determinedas follows:

vx − ivy = dwdz(ς) and z(ς)= z(0)+

∫ ς0

dw/dςdw/dz

dς , (2.2)

where vx and vy are the x- and y-components of the non-dimensionalized velocity.


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(a) Expressions for the derivatives of the complex potential in the similarityand 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 1b. In this plane, the positive imaginary axis (η > 0, ξ = 0) correspondsto the free surface, and the positive real axis (ξ > 0, η= 0) corresponds to the wetted part of thesolid wedge. The points ς = a and ς = c are the images of the stagnation point A and the wedgevertex C in the similarity plane, respectively. The parameters a and c are not known and have tobe determined as part of the solution.

The boundary-value problem for the complex-velocity function can be formulated in theparameter plane as follows. At this stage, we may write the velocity modulus along the freesurface, or along the positive part of the imaginary axis of the ς -plane as

∣∣∣∣dwdz

∣∣∣∣= v(η), 0<η <∞, ξ = 0. (2.3)

In the frame of reference fixed with respect to the solid wedge, the normal velocity componentequals zero owing to the impermeability condition. This means that the argument χ of thecomplex velocity along the real axis of the parameter region is fixed and can be determined bythe wedge orientation. Thus, we have

χ(ξ)= arg(

dwdz

)=

⎧⎪⎨⎪⎩

−γL, 0< ξ < a, η= 0,−π − γL, a< ξ < c, η= 0,−γR, c< ξ <∞, η= 0.

(2.4)

The formula, derived by Semenov [17,18] using the Chaplygin [19] singular point method

dwdz

= v∞ exp[

1π

∫∞

0

dχdξ

ln(ς + ξ

ς − ξ

)dξ − i

π

∫∞

0

d ln vdη

ln(ς − iης + iη

)dη + iχ∞

], (2.5)

provides an integral form of the mixed boundary-value problem in the first quadrant of thecomplex ς -plane. In the equation, v∞ = v(η)|η=∞ = vB is the velocity at point B and χ∞ =χ(ξ)|ξ=∞ = −γR. By evaluating the first integral in equation (2.5) with the step change of thefunction χ(ξ) given in equation (2.4), we obtain the expression for the complex velocity in theς -plane in the form

dwdz

= v0e−iγR

(ς − aς + a

)(ς + cς − c

)(1−2α/π)exp

[− iπ

∫∞

0

d ln vdη

ln(

iη − ς

iη + ς

)dη]

, (2.6)

where v0 = 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 alongthe surface increases in the direction for which the fluid region is on the left (figure 1a). With thisnotation,

dw = (vs + ivn)ds, (2.7)

where vs versus and vn are the tangential and normal velocity components, respectively. Let θ bethe angle between the velocity vector on the surface and τ , which means θ = tan−1(vn/vs). Thedefinition in equation (2.7) allows us to determine the argument of the derivative of the complexpotential dw/dς ,

ϑ(ς)= arg(

dwdς

)= arg

(dwds

dsdς

)= arg

(dwds

)+ arg

(dsdς

),


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..................................................


O+

BC

c

A

a

B

O

i

B–

O–, A+ B+, C+, A–

D•D¢•

D• D¢•

mR mL

p + m•

ih z

x

(a) (b)

Figure 2. (a) Variation of θ = tan−1(vn/vs) along the boundary of the fluid region; (b) the corresponding path in theparameter plane. The continuous changes of the function are shown by solid lines, whereas its step changes are shown bydashed lines.

which appears in equation (2.2). We notice that arg(dw/ds)= θ , arg(ds/dς)= 0 along the real axisof the parameter plane since ds> 0 and dς = dξ , and arg(ds/dς)= π/2 along the imaginary axissince ds< 0 and dς = idη. Then,

ϑ(ς)={θ , 0< ξ <∞, η= 0,θ + π/2, ξ = 0, 0<η<∞.

(2.8)

Now we determine the function θ(ς) along the whole fluid boundary, that is, along the half realand half imaginary axes of the ζ -plane. When moving along the free surface from point O topoint D∞, the function θ(ς) changes from the value μL at ξ = 0, η→ 0 to the value θD∞ = −γ∞ −π/2 + 1/2μ∞ + δ at ς = i. In order to get the left-hand side of the free surface away from thesolid wedge, we move along part of the circle of infinitely large radius from D∞ to D′∞, wherethe fluid is undisturbed and the known velocity direction gives arg(dw/dz)= −iγ∞. Thus, ϑ(ς)=arg(dw/dς)= arg(dw/dz)+ arg(dz/dς)= −iγ∞ + arg(dz/dς)will change in the same way as theslope to the free surface, providing at point D′∞ the value θD′∞ = −3/2π + δ − 1/2μ∞ − γ∞. Whenmoving in a counterclockwise direction along an infinitesimal semicircle centred at the point ς = icorresponding to the infinite radius in the z-plane, the function θ(ς) changes by �θD∞ = −(π +μ∞). The continuous changes of the function θ(ς) are shown in figure 2 by solid lines, while itsstep changes are shown by dashed lines. Furthermore, θ(ς) changes continuously when movingalong the free surface from point D′∞ to point B. On the interval, a< ξ <∞, η= 0, correspondingto the whole right-hand side of the solid wedge and the left-hand side between points C and A,θ(ς)≡ 0 because vn = 0 and vs > 0. On the interval 0< ξ < a, η= 0, θ(ς)≡ π because vn = 0 andvs < 0. Thus, when passing the point A (ς = a), as shown in figure 2b, θ(ς) takes a jump of�θA = π .The last jump �θO =μL − π occurs at point O when we move in the vicinity of the point ς = 0from the wedge surface, ξ > 0, η= 0, to the free surface, ξ = 0, η > 0, as shown in figure 2b. Thetotal jump of function ϑ(ς) at point ς = 0 taking into account equation (2.8) is �θO =μL − π/2.The subscripts ‘−’ and ‘+’ at points A, O andB in figure 2a denote sides that we meet before andafter 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:

θ(ς)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, a< ξ <∞, η= 0,�θA, 0< ξ < a, η= 0,λ(η)+�θA +�θO, ξ = 0, 0<η< 1,λ(η)+�θO +�θA +�θD, ξ = 0, 1<η<∞,

(2.9)


6


..................................................


where �θD = −(π + μ∞), �θO =μL − π , �θA = π , λ(0)= 0. The expression for the derivative ofthe complex potential can be obtained by applying the integral formula [17]

dwdς

= K exp

[1π

∫ 0

∞∂ϑ

∂ξln(ς2 − ξ2)dξ + 1

π

∫∞

0

∂ϑ

∂ηln(ς2 + η2)dη + iϑ∞

], (2.10)

which is derived from the Schwartz integral formula when the first quadrant of the ς -plane ischosen as the parameter region. Here, K is a real factor and ϑ∞ = θ(ς)|ς=ξ→∞ = 0.

By substituting equations (2.8) and (2.9) into the first integral in (2.10) when ς varies alongthe real axis and into the second integral when ς varies along the imaginary axis, and evaluatingthe integrals over the step changes of the function θ(ς), we finally obtain the expression for thederivative of the complex potential in the ς -plane as

dwdς

= Kς2μL/π−1 ς2 − a2

(ς2 + 1)(1+μ∞/π)exp

[1π

∫∞

0

dλdη

ln(ς2 + η2)dη]

, (2.11)

in which the integration over the step changes is done, for example, at point A (ς = a) as follows:

limε→0

∫ a+ε

a−εdϑdξ

ln(ς2 − ξ2)dξ = ln(ς2 − a2) limε→0

∫ a+ε

a−εdϑdξ

dξ = −�θA ln(ς2 − a2).

The minus sign here is due to the direction of the integration path being opposite to the jump�θA.From equations (2.6) and (2.11), the derivative of the mapping function and the

complex-velocity potential can be obtained as

dzdς

= Kv0ς2μL/π−1 (ς + a)2

(ς2 + 1)(1+μ∞/π)

(ς − cς + c

)(1−2α/π)

× exp[

1π

∫∞

0

dλdη

ln(ς2 + η2)dη + iπ

∫∞

0

d ln vdη

ln(

iη − ς

iη + ς

)dη + iγR

](2.12)

and

φ + iψ = w(ς)= w(0)+∫ ς

0

dwdς ′ dς ′, (2.13)

where 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 boundaryconditions. At infinity, the complex-velocity approaches the value v∞ exp(−iγ∞), where v∞ =V∞/V0. Thus letting ς = i in equation (2.6), the following condition is obtained:

− 1π

∫∞

0

d ln vdη′ ln

∣∣∣∣η − 1η + 1

∣∣∣∣dη +(

1 − 2απ

)(2 tan−1 1

c− π

)− 2 tan−1 1

a− α + π/2 + γ∞ = 0.

(2.14)

In the physical plane, the wetted lengths of the right- and left-hand sides of the solid wedge areV0t and VBt, respectively. Here, VB = vBV0 is the fluid velocity at point B in the physical plane,and vB = limη→∞ v(η) is obtained from the solution. The lengths of the segments OC and CB inthe similarity plane are then |zO| = 1 and |zB| = vB = VB/V0, respectively. Hence, the followingconditions are obtained:

∫ c

0

∣∣∣∣ dzdς

∣∣∣∣ς=ξ

dξ = 1 and∫∞

c

∣∣∣∣ dzdς

∣∣∣∣ς=ξ

dξ = vB. (2.15)


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(b) Dynamic and kinematic boundary conditionsThe Bernoulli equation in the physical plane linking point O and an arbitrary point in the flowgives

∂Φ

∂t

∣∣∣∣Z

+ V2

2+ Pρ

= ∂Φ

∂t

∣∣∣∣Z=ZO

+ V20

2+ Pa

ρ, (2.16)

where P and V are the pressure and velocity at an arbitrary point of the fluid domain, Pa is thepressure 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 theself-similar spatial coordinate of arc length defined previously, Semenov & Iafrati [10] reducedthis equation to the following:

c∗p + v2 = 1 + 2φ − 2

dφds

s, (2.17)

where at s = 0 or point O, the potential has been assumed to be zero (note, s> 0 on the sides ofthe solid wedge and s< 0 on the free surface OD). In equation (2.17), c∗

p = (P − Pa)/(0.5ρV20) is the

pressure coefficient. Along the free surface, the pressure is constant and equal to the atmosphericpressure Pa; therefore, c∗

p = 0. By taking the derivative of equation (2.17) with respect to s andaccounting for the relations dφ/ds = vs and vs = v cos θ , Semenov & Iafrati [10] obtained thefollowing differential equation:

dθds

= v + s cos θs sin θ

d ln vds

, (2.18)

which is valid on both the left and right free surfaces OD∞ and BD′∞.Multiplying both sides of equation (2.18) by ds/dη and taking into account that dθ/ds = dλ/ds,

0<η < 1, we obtain the following differential equation:

dλdη

= v + s cos θs sin θ

d ln vdη

, (2.19)

where the arc length coordinate s, which is a function of η, can be obtained by integratingequation (2.12) for the left free surface, or

s(η)= −∫ η

0

∣∣∣∣ dzdς

∣∣∣∣ς=iη′

dη′ = −K∫ η

0

η2μL/π−1

v(η)

η2 + a2

(1 − η2)(1+μ∞/π)exp

[1π

∫∞

0

dλdη′ ln |η′2 − η2|dη′

]dη,

(2.20)where 0 ≤ η < 1, and

s(η)=∫∞

η

∣∣∣∣ dzdς

∣∣∣∣ς=iη′

dη′, (2.21)

for the right free surface, for which 1<η <∞.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 thefollowing form:

1tan θ

d ln vds

= dds

[arg

(dwdz

)], (2.22)

which is valid on both the left and right free surfaces OD∞ and BD′∞. This equation is obtainedusing the fact that the acceleration of the fluid particle is orthogonal to the free boundary if thepressure along the free surface is constant.

Substituting the complex velocity in equation (2.6) into equation (2.22) and multiplying bothsides of the result by ds/dη= |dz/dς |ς=iη, the following integral equation in terms of the functiond ln v/dη is obtained:

− 1tan θ

d ln vdη

+ 1π

∫∞

0

d ln vdη′

2η′

η′2 − η2 dη′ = 2aa2 + η2 +

(2απ

− 1)

2cc2 + η2 , (2.23)

for 0<η≤ 1. A similar equation can be obtained for 1 ≤ η <∞ corresponding to BD. As a result,equation (2.23) is found to be valid along the whole imaginary axis of the parameter plane.


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The integral equations (2.19) and (2.23) along with (2.14) and (2.15) make it possible todetermine the functions θ(η) and v(η), and the parameters a, c, K. Once these functions andparameters are found, the contact angles between the wedge sides and the free surface, μR andμL, can be determined as follows (figure 2): μL = limη→0 θ(η), μR = π − limη→∞ θ(η).

Taking into account that between points O and A dφds = vs = −v, while on the rest of thewetted part of the solid wedge dφds = vs = v, the pressure coefficient along the left and rightsides of the solid wedge can be obtained from equation (2.17) as follows:

cp(ξ)= 2(P − Pa)

ρ V2ref

= c∗p(ξ)

v20

v2ref

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−2(φ + sv)+ (1 − v)2

(v∞ sin γ∞)2, 0< ξ < a,

−2(φ − sv)+ (1 + v)2

(v∞ sin γ∞)2, a< ξ <∞,

(2.24)

where φ, v, s are determined from equations (2.12), (2.6) and (2.2) and can be written as follows:

ϕ(ξ)= Re[w(ς)]ς=ξ , v(ξ)=∣∣∣∣dw

dz

∣∣∣∣ς=ξ

, s(ξ)=∫ ξ

0

∣∣∣∣ dzdς

∣∣∣∣ς=ξ ′

dξ ′, v∞ = v(η)η=1

and vref is the magnitude of the projection of the impact velocity onto the bisector of the liquidwedge, or vref/v∞ = sin γ∞.

3. Flow analysis

(a) Numerical approachThe integral equations (2.19) and (2.23) are solved numerically by iteration through the methodof successive approximations. Equations (2.14) and (2.15) are used in each iteration. In discreteform, the solution is sought on two sets of points. The first, 0<ηj < 1, j = 1 . . .N, corresponds tothe segment OD∞ of the free surface and the points are distributed in such a way that the segmentsize increases geometrically away from O. The second set of points corresponding to D′∞B ischosen to be the inverse reflection of the first set of points about η= 1, i.e. ηj = 1/η2N−j+1, j = N +1 . . . 2N. The numerical approach used in the present study is based on the method of successiveapproximations 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 owingto the singularity occurring here [8]. In the present solution, a similar singularity can be seen inthe expression for the derivative of the complex potential in equation (2.11) at point ς = 0 when2μL/π − 1< 0 and due to the improper integral with upper limit at η= ∞. The singularities at thetwo intersection points depend on the values of the contact angles μL and μR, respectively, andthe range of variation of the function λ(η), which determines the order of singularity at ς = ∞. Fora given discretization along the η-axis discussed earlier, the corresponding arc length coordinatess1 = s(η1) and s2N = s(η2N) nearest to contact points O and B in the similarity plane can be obtainedusing equations (2.20) and (2.21), respectively, as

s1 = −Ka2 exp

(2π

∫ 1/η1

η1

dλdη

ln ηdη

)η

2μL/π

12μL/π

(3.1)

and

s2N = KvB

η2μR/π

12μR/π

. (3.2)

Numerical integration of the function ds/dη near the contact point O over the interval 0 ≤ η≤ η1requires an extremely small step of order 10−15, owing to the singularity η2μL/π−1 for 2μL/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.


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..................................................


(a)3

2

1

0

y

–3 –2 –1 0x

1 2 3 –3 –2 –1 0x

1 2 3–1

(b)

Figure 3. Streamline patterns for a symmetrical liquid wedge of angleμ∞ = 60◦ impacting a solid flat surface at obliqueangles: (a) γ∞ = 90◦ and (b) γ∞ = 120◦.

If we choose a typical value η1 = 10−5, then s1 = s2N ≈ 0.5 is obtained for deadrise angle βL =βR ≈ 5◦, i.e. the length of the first node in the similarity plane is about half of the wetted wedgelength. The arc length coordinates s1 = s(η1) and s2N = s(1/η1) affect all other spatial coordinatessi and sj on the left and right free surfaces,

sj = sj−1 +∫ ηj

ηj−1

dsdη

dη, j = 2, . . . , N,

sj−1 = sj +∫ ηj

ηj−1

dsdη

dη, j = N + 2, . . . , 2N,

and, correspondingly, the numerical evaluation of the function s = s(η), which appears in theintegral equation (2.19).

(b) Validation of the numerical approachThe numerical approach for solving a system of integral equations (2.19)–(2.23) is similar to thatin 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 theliterature. For further validation and verification purposes here, we consider a case of liquidwedge impacting on a flat solid wall. Physically, the velocity of the liquid along the direction ofthe wall should have no effect on the shape of the free surface and the pressure on the wetted partof the wall. This is also obvious enough if the mathematical model is established in the systemmoving with the liquid [4–6]. However, in the present model, the system is fixed on the solidwedge. Thus, when the liquid wedge hits the wall, the presence of the velocity along the wallleads to a different mathematical problem, which has to be solved separately. As a result, thiscould be a test case for verification of the method.

Figure 3 shows the streamlines for a liquid wedge of angle μ∞ = 60◦ and a heel angle δ = 0hitting a flat solid wall (α= 90◦). For the case of γ∞ = 90◦ shown in figure 3a, the component ofthe impact velocity along the direction of the wall surface is equal to zero, while for the case ofγ∞ = 120◦ shown in figure 3b, the velocity component along the wall surface is vx = −vy cot γ∞,where vy is the velocity component perpendicular to the wall and is positive when it is oppositeto y. Although the streamlines may look different in these two cases, the free surface shapes arethe same.


10


..................................................


(a)3

2

1

0

cp

–3–4 –2 –1 0x

1 2 3 –3 –2 –1 0x

1 2 3

(b)

Figure 4. Pressure distribution on the wetted wall surface during the impact of the liquid wedge of angle μ∞ = 60◦: (a)solid line: γ∞ = 90◦, dotted line: γ∞ = 120◦, dashed line: γ∞ = 110◦; (b) comparison of the present results (solid line)with results of Duan et al. [7] (dashed line) and Zhang et al. [21] (dotted line).

The streamlines in the figure have been obtained in two steps. At the first step, the line Γi(ς̄)

in the parameter plane corresponding to the ith streamline in the similarity plane is determinedfrom the following equation:

Im(∫ ς̄

ς0i

dwdς

dς)

=ψi, (3.3)

where ψi is the value of the stream function for the ith streamline in the figure, and ς0i = iηi isthe point in the parameter plane corresponding to the intersection of the free surface and the ithstreamline. At the second step, the streamline in the similarity plane is determined by integratingequation (2.12) along the contour Γi(ς̄).

The pressure distributions along the wall in the above two cases are shown in figure 4a,together with the one for γ∞ = 110◦. It can be seen that the pressure distribution is only shiftedrelative to the point x = 0, where the liquid wedge touches the wall at the initial time, while thecurve shapes remain the same. These results confirm the obvious physical fact that the componentof the liquid wedge velocity along the flat rigid solid wall during the impact does not have anyeffect on the shape of the free surface and the pressure distribution on the wetted wall surface.

In figure 4b, we also compare the obtained pressure distribution (solid line) with the results ofDuan et al. [7] (dashed line) and Zhang et al. [21] (dotted line), which have been taken manuallyfrom 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 isbecause some approximation for the free surface shape was adopted in Zhang et al. [21], butnot in Duan et al. [7]. It should be noted that for this impact problem, which starts with asingle contact point, an effective numerical method in the time domain is to use the stretchedcoordinate system developed by Wu [4] and subsequently used in recent studies [5,6]. Specialtreatment was introduced in the jet zone in these publications, which led to accurate results forthe pressure. In the present study, we have evaluated analytically the spatial coordinates s1 ands2N closest to the contact points O and B, which has provided accurate evaluation of the functions = s(η) along the whole free surface and, consequently, we have obtained overall accuracy of thesolution. This may be part of the reason for the good agreement between the present result andthat of Duan et al. [7]. As discussed in §1, we note that although the problem can be solved bythe BEM, it is difficult for the method to provide some of the detailed results accurately, suchas the intersection angle and wetted length, which, on the other hand, can be achieved by thepresent method.


11


..................................................


(a)420

–2

8

4

–8 –4 0

x

4 8

0

–16 –12 –8 –4 0 4 8 12 16

(b)

(c) (d)

y

y

4

0

–4

–8 –4 0x

4 8

–8

420

–2–4

–16 –12 –8 –4 0 4 8 12 16

Figure 5. Streamline patterns for head-on impact between symmetric liquid and solidwedgeswithβL = βR = 10◦: (a)α=90◦,μ∞ = 160◦; (b)α = 80◦,μ∞ = 180◦; (c)α= 140◦,μ∞ = 60◦; (d)α= 30◦,μ∞ = 280◦.

(c) Symmetric impact between liquid and solid wedgesHaving verified the method, we shall consider various cases. The pressure distribution and freesurface shapes during the symmetric water impact between the liquid and solid wedges (orwater entry of a solid wedge, and a liquid wedge hitting a solid flat wall) have been investigatedby many authors [1,5–7,22,23]. They used either approximate analytical methods or the BEM tosolve 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 resultsfor wedges of deadrise angles larger than 20◦. At smaller angles, the time step has to be furtherreduced. 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 thepresent methodology. It is partly caused by the singularities occurring at the intersection of thefree 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] resolvedthis 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 makesit possible to solve wedge impact problems with small deadrise angles using standard arithmetictools and reasonable computation time. It is the same for the present study, which enables us tocalculate some extreme cases.

It is well known that for water entry of a wedge into a flat free surface, a peak pressurewill appear near the root of the jet at small deadrise angles [20]. We can expect a similar effectfor the impact between a liquid and solid wedge at the small deadrise angles, marked as βLand βR in figure 1c. Figure 5 shows several configurations of the liquid and solid wedgesforming a deadrise angle of 10◦. The flow patterns in figure 5a,b change only slightly when theimpact problem changes from a liquid wedge hitting a flat wall to a solid wedge hitting a flatliquid surface. In figure 5c, the streamlines correspond to an acute liquid wedge impacting ona solid reflex corner and in figure 5d, a liquid wedge with an obtuse angle impacting an acutesolid corner.


12


..................................................


80

60

40

20

0

–16 –12 –8 –4 0x

4 8 12 16

cp

Figure 6. Pressure distribution on the wetted surface of the solid wedge for cases (a, solid line), (b, dashed line), (c, dashed-dotted line) and (d, dotted line) in figure 5.

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 pressurebecomes much lower. The reason for that may be because of the smaller area of contact withthe solid surface in (c), due to the acute angle of the liquid wedge. In (d), the further reductionof the pressure looks reasonable owing to the possibility of the liquid wedge extending in thetransverse direction.

In figure 7, the dependence of the contact angle (between the free surface and the wedgesurface) on the angle of the liquid wedge and of the solid wedge for the symmetric flowconfiguration 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 fromequation (2.19) as s → 0. In the present solution, in order to achieve sufficient accuracy, with anerror not larger than 1 per cent, we use a set of points ηj, j = 1 . . . 2N along the axis of the parameterplane, distributed in the manner described in §3a and place the first point at η1 = 10−6 and the lastat η2N = 1/η1 = 106. Our numerical results seem to suggest that the contact angle is approximatelyequal to the half-angle of the liquid wedge when the angle of the liquid wedge tends to zero, i.e.μL =μR ≈μ∞/2, as μ∞ → 0.

The contact angle μL reaches its maximum value at some angle μ∞, which depends on theangle of the solid wedge α. For α→ 0, the contact angle μL/π =μR/π → 0.25 at μ∞/(2π)≈ 0.25corresponding to the deadrise angle βL = βR ≈ 135◦. The same maximum value of the contactangle at the same deadrise angle was obtained for the water-entry problem of a flat plate [24]. Itis worth noting that the maximum value of the contact angle for the vertical entry of a symmetricwedge α→ 0 is μL/π =μR/π ≈ 0.1 for the case μ∞/(2π)≈ 0.5, which agrees with that obtainedby recent studies [8,25].

During the impact, the liquid between the free surface and the body surface near theintersection is wedge shaped with angle μL, (or μR), 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 offshoreplatforms or ships, etc. In the case of a wall, i.e. α = 90◦, the obtained maximum value of thecontact angle is μL/π ≈ 0.0278, at μ∞/(2π)≈ 0.11.

It is found that the impact problem between the symmetric liquid wedge and the solid wall inthe limiting case μ∞ → 0 tends to the steady problem of a rectangular jet impacting on the wallperpendicularly. Indeed, as can be seen from figure 7, the contact angle μL → 0 when μ∞ → 0.


13


..................................................


0.30

0.250.2°1°9°20°40°60°90°130°

0.20

0.15

0.10

0.05

0 0.2 0.4m•/(2p)

a

m R/p

0.6 0.8 1.0

Figure 7. Contact angle against the half-angle of the liquid wedge for various angles of the solid wedge.

0.101.0

0.5

0

0.05

0

–0.15 –0.10 –0.05 0x

0.05 0.10 0.15

ycp

Figure 8. Streamlines and pressure distribution for the liquid wedge of half-angleμ∞/2= 1◦ impacting the solid wall.

By equation (3.1), the location of the contact point O tends to infinity. Moreover, the order ofsingularity at point D∞, (ς = i) in equation (2.11) for μ∞ = 0 becomes 1/(ς − i), correspondingto the logarithmic singularity in the complex potential occurring for a steady jet of finite flowrate. For μ∞ → 0, the normal component of the velocity vn → 0 along the whole free surface, andtherefore the function ‖θ(η)‖ → 0. In this case, from equation (2.23), it follows that d ln v/dη→ 0.By substituting dθ/dη= 0, d ln v/dη= 0 and μL = 0 into equations (2.6) and (2.12), we obtain thefollowing expressions for the derivative of the complex potential and for the complex velocitycorresponding to the steady flow of the jet spreading along the wall:

dw∗

dς= K

ς2 − a2

ς(ς2 + 1)

anddw∗

dz= e−iγR

(ς − aς + a

),

in which we have used the asterisk to indicate that this is the steady solution corresponding tothe 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 toeach other and the maximum pressure reaches unity, i.e. the maximum pressure for the steadyflow. Thus, the solution obtained for the liquid wedge impact problem continuously tends to the


14


..................................................


3(a) (b)

(c)

(e)

(d)

2

1y

0

–1–4 –3 –2 –1 0 1 2 3

3

2

1y

y

x

0

–1–3 –2 –1 0 1 2 3 4

x–3 –2 –1 0 1 2 3 4 5

3

2

1

x

0

–1–2 0 2 4 6

–5 –4 –3 –2 –1 0 1 2 3

Figure 9. Streamlines for an asymmetric liquid wedge of angleμ∞ = 60◦ impacting the solid wall: vertical impact (γ∞ =90◦): (a) δ= 20◦ and (b) δ = 30◦; oblique impact: (c) γ∞ = 110◦, (d) γ∞ = 120◦, (e) γ∞ = 130◦.

solution for the steady jet flow. The problem of a steady jet falling from a vertical pipe and hittinga horizontal plate including the gravity effect has recently been considered by Christodoulides &Dias [26].

(d) Asymmetric/oblique impactThe effect of asymmetry is investigated by performing numerical calculations for fixed angles ofsolid and liquid wedges, but different heel angles, δ, and velocity directions at infinity, γ∞.

The streamlines corresponding to asymmetric/oblique impact of a liquid wedge of angleμ∞ =60◦ are shown in figure 9 for both the vertical ((a) and (b)) and oblique ((c), (d) and (e)) cases.Figure 9c–e has been re-scaled to provide the same vertical component of the incoming velocityas in the vertical impact. For all the cases shown in figure 9, the stagnation point approaches theroot of the jet on the side with the smaller deadrise angle. The contact angle becomes larger onthe side with a larger deadrise angle and smaller on the opposite side. The density of streamlinesincreases with the magnitude of the velocity because the flux of the liquid between the two neareststreamlines is constant. The streamline patterns clearly show an increase in the velocity near theroot of the jets and a decrease in the velocity magnitude near the stagnation point. The magnitudeof coordinates of the intersection points at the left- and right-hand sides in figure 9 shows theratio between the velocity of the tips and the vertical component of the incoming component.


15


..................................................


13

12

6

cp 4

2

0

–4 –2 0x

2 4 6

Figure 10. Pressure distributions along the solidwall corresponding to the cases shown infigures 3a and9: solid line (symmetricimpact, case shown in figure 3a); dashed and dotted lines with the peak pressure at x < 0 (asymmetric vertical impact, cases9a and 9b, respectively); dashed, dotted and dashed-dotted lines with the peak pressure at x > 0 (oblique impact, cases 9c,9d, 9e, respectively).

The comparison between the pairs (a) and (c), and (b) and (d) shows the symmetric configurationof each pair with respect to the y-axis, which should be obvious as the horizontal velocity merelyreverses its direction but retains its magnitude.

For case (b), the angle θ on the right-hand side increases from value π − μR at point B to π atpoint D′∞ where the normal and tangential components of the velocity equal vn = 0 and vs = −v∞,respectively. For the case with angle δ > 30◦, the normal component of the velocity at point D′∞becomes negative and the angle θ becomes larger than π . In our solution procedure, the rangeof the angle θ = tan−1(vn/vs) is defined in the interval −π ≤ θ ≤ π . In order to predict cases withδ > 30◦, a horizontal component of the velocity could be added to provide −π ≤ θ ≤ π . This caseis shown in figure 9e.

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

Figure 11 shows streamlines corresponding to an impact between a liquid wedge of angleμ∞ = 60◦ and a solid wedge of half-angle 45◦. The wedge orientation and the impact velocitymay form condition γ∞ < γR, for which the liquid at infinity runs away from the right-hand sideof the solid wedge. Under such a condition, flow separation from the wedge vertex may occur, asdiscussed by Xu et al. [23], thus changing the flow topology and requiring another mathematicalformulation for the problem. In this case, only one side of the wedge would be in contact withthe 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 surfacewas also studied by Judge et al. [27] experimentally and theoretically. They found that the flowseparation would occur suddenly at some limiting combination of the heel angle and the angle


16


..................................................


(a) (b)

(c)

3

2

1

0

–1

–2–3 –2 –1 0 1 2 3

x–3 –2 –1 0 1 2 3

x

y

3

2

1

0

–1

–2–3 –2 –1 0 1 2 3

x

y

Figure 11. Oblique impact between a solid wedge of half-angleα= 45◦ and a liquid wedge of angleμ∞ = 60◦: (a) δ= 0,(b) δ = 20◦ and (c) δ= 40◦.

4

3

2

1

0

–1–3 –2 –1 0

s1 2

cp

Figure 12. Pressure distribution along the wetted surface of the solid wedge for cases (a) (dotted line), (b) (dashed line) and(c) (solid line) shown in figure 11; s< 0 from thewedge apex corresponds to the horizontal side of the solid wedge, while s> 0corresponds to the vertical side.


17


..................................................


(a) (b)

(c)

3

2

1

0

–1

–2–2 –1 0 1 2 3

3

4

2

1

0

–1

–2–2 –1 0 1 2 3 4

x x

y

3

2

1

0

–1

–2–2 –1 0 1 2 3

x

y

Figure 13. Oblique impact between a solid wedge half-angleα= 45◦ and a liquid wedge of angleμ∞ = 60◦ with the heelangle δ = 0 and velocity direction: (a) γ∞ = 90◦, (b) γ∞ = 100◦ and (c) γ∞ = 25◦.

of the impact velocity, which provides condition γ∞ ≈ γR, which is similar to the case shown infigure 11c. We note that case (c) in figure 11 is the limiting case for which the convergence of thenumerical procedure based on successive approximations could be obtained. However, we oughtto point out that this could the limit of the mathematical model. Flow separation in real situationsis 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 investigationthrough 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 wedgeapex. It occurs because the stagnation point coincides with the wedge apex. For case (b) on thelower side of the wedge, there is a maximum of the pressure. It appears because the pressurecoefficient 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 tothe impact. The pressure is then obviously larger than that at the contact point and at the wedgeapex, as reflected by the maximum in the curve.

Figure 13 shows the effect of the direction of the incoming velocity on the streamlines. Thestagnation point moves very little from the wedge vertex along the lower wedge side as thehorizontal component of the velocity increases. The wetted length of the vertical side of thewedge is somewhat smaller than that of the horizontal side. It can also be seen from the pressure


18


..................................................


cp

2

1

0

–1–2.0 –1.5 –1.0 –0.5 0.5 1.0 2.01.50

s

Figure 14. Pressure distribution along thewetted surface of the solidwedge for cases (a) (dotted line), (b) (dashed line) and (c)(solid line) in figure 13; s< 0 corresponds to the horizontal side of the solid wedge and s> 0 corresponds to the vertical side.

distributions in figure 14 that the higher pressure occurs on the vertical side while its wettedlength is smaller. The behaviour of the pressure distribution is qualitatively the same for variousangles of impact velocity. The pressure distribution on the wedge sides is similar to that shownin figure 12. The difference is only that the distance between the stagnation point and the wedgeapex is much smaller than those in the cases shown in figure 12. Our obtained numerical resultalso show that the pressure varies sharply near the apex of the wedge and seems to tend to minusinfinity at the apex.

4. ConclusionsWe have presented a mathematical procedure for the fully nonlinear problem of impact betweena solid wedge and a liquid wedge, on the basis of incompressible velocity potential theory. Thismethodology has made it possible to derive analytical expressions for the complex velocity andthe derivative of the complex potential in the parameter plane. They are found through integralformulae for the mixed and homogeneous boundary-value problems in terms of the velocitymodulus and the angle between the velocity and the fluid boundary. The mapping functionbetween the similarity plane and the parameter plane is obtained from these expressions. Itexplicitly contains the singularities, including those at the intersection points of the body and freesurfaces. 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 theintegral 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 steadyjet 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 ofthe free surface shape, streamlines, pressure distribution on the wetted solid surface and contactangles between the free surface and the body surface.

Numerical results are presented for a wide range of angles for both the liquid and solidwedges, including reflex ones. The capability of the method to predict various configurationsincluding blunt solid and ‘blunt’ liquid wedges enables us to show that the pressure distributionalong the wetted surface is very similar for the case of a blunt solid wedge entering a flat freesurface and for the case of a ‘blunt’ liquid wedge impacting a solid wall. The major parameter


19


..................................................


determining the pressure distribution for these cases is the deadrise angle. When the solid orliquid wedge has an acute angle, the difference in the pressure distributions appears even for thesame small deadrise angle.

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

Dobrovol’skaya [8] and Fraenkel & Keady [25] showed analytically that the contact angle atthe intersection could be no larger than μL/π = 0.25. From the numeric results for water entry ofa symmetric wedge onto a flat free surface, they found that the largest contact was μL/π = 0.1,which has been confirmed by the present work (see figure 7, which shows that the largest μL/π isabout 0.1 for the case of μ∞/(2π)= 0.5). We have then found numerically that the largest contactangle of μL/π ≈ 0.25, which occurs in the case of symmetric impact between a thin solid wedgeand a water wedge of μ∞/(2π)≈ 0.25, at deadrise angles βL = βR ≈ 135◦. For a liquid wedgeof small angle, it has been found from the presented analytical solution that the result smoothlytends to that corresponding to the steady flow of a rectangular jet hitting a solid wall. We have alsofound that in this case, the maximum value of the contact angle is about μL/π =μR/π ≈ 0.0278,which occurs at μ∞/(2π)≈ 0.11. If the angle of the liquid wedge μ∞ → 0, then the contact anglealso tends to zero.

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

References1. Cumberbatch E. 1960 The impact of a water wedge on the wall. J. Fluid Mech. 7, 353–374.

(doi:10.1017/S002211206000013X)2. Dold JW, Peregrine DH. 1986 An efficient boundary-integral method for steep unsteady water

waves. On some problems of similarity flow of fluid with a free surface. In Numerical methodsfor fluid dynamics II (eds KW Morton, MJ Baines), pp. 671–679. Oxford, UK: Oxford UniversityPress.

3. Cooker MJ, Peregrine DH. 1995 Pressure-impulse theory for liquid impact problems. J. FluidMech. 297, 193–214. (doi:10.1017/S0022112095003053)

4. Wu GX, Sun H, He YS. 2004 Numerical simulation and experimental study of water entry of awedge in free fall motion. J. Fluids Struct. 19, 277–289. (doi:10.1016/j.jfluidstructs.2004.01.001)

5. Wu GX. 2007 Fluid impact on a solid boundary. J. Fluids Struct. 23, 755–765. (doi:10.1016/j.jfluidstructs.2006.11.002)

6. Wu GX. 2007 Liquid column and liquid droplet impact. Q. J. Mech. Appl. Math. 60, 497–511.(doi:10.1093/qjmam/hbm020)

7. Duan WY, Xu GD, Wu GX. 2009 Similarity solution of oblique impact of wedge-shaped watercolumn on wedged coastal structures. Coastal Eng. 56, 400–407. (doi:10.1016/j.coastaleng.2008.09.008)

8. Dobrovol’skaya ZN. 1969 Some problems of similarity flow of fluid with a free surface. J. FluidMech. 36, 805–829. (doi:10.1017/S0022112069001996)

9. Chekin BS. 1989 The entry of a wedge into incompressible fluid. Prikl. Matem. Mekhan. 53,300–307.

10. Semenov YuA, Iafrati A. 2006 On the nonlinear water entry problem of asymmetric wedges.J. Fluid Mech. 547, 231–256. (doi:10.1017/S0022112005007329)

11. Semenov YA, Yoon BS. 2009 Onset of flow separation at oblique water impact of a wedge.Phys. Fluids 21, 112 103–112 111. (doi:10.1063/1.3261805)

12. Howison SD, Ockendon JR, Wilson SK. 1991 Incompressible water-entry problems at smalldeadrise angles. J. Fluid Mech. 222, 215–230. (doi:10.1017/S0022112091001076)

13. Iafrati A, Korobkin AA. 2004 Initial stage of flat plate impact onto liquid free surface. Phys.Fluids 16, 2214–2227. (doi:10.1063/1.1714667)

http://dx.doi.org/doi:10.1017/S002211206000013X

http://dx.doi.org/doi:10.1017/S0022112095003053

http://dx.doi.org/doi:10.1016/j.jfluidstructs.2004.01.001



http://dx.doi.org/doi:10.1093/qjmam/hbm020

http://dx.doi.org/doi:10.1016/j.coastaleng.2008.09.008

http://dx.doi.org/doi:10.1016/j.coastaleng.2008.09.008



http://dx.doi.org/doi:10.1063/1.3261805




20


..................................................


14. Howison SD, Ockendon JR, Oliver JM. 2004 Oblique slamming, planning and skimming.J. Eng. Math. 48, 321–337. (doi:10.1023/B:engi.0000018156.40420.50)

15. Howison SD, Ockendon JR, Oliver JM, Purvis R, Smith FT. 2005 Droplet impact on a thin fluidlayer. J. Fluid Mech. 542, 1–23. (doi:10.1017/S0022112005006282)

16. Oliver JM. 2007 Second-order Wagner theory for two-dimensional water-entry problems atsmall deadrise angles. J. Fluid Mech. 572, 59–85. (doi:10.1017/S002211200600276X)

17. Semenov YA. 2003 Complex potential of an unsteady flow with a free boundary. VestnikHersonskogo universiteta, Herson, Ukraine 2, 384–387 (in Russian).

18. Semenov YA. 2004 Method for solving nonlinear problems on unsteady free-boundary flows.In XXI Int. Congress Theoretical and Applied Mechanics, Warsaw, Poland, 15–21 August 2004. Seehttp://ictam04.ippt.gov.pl.

19. Chaplygin SA. 1910 About pressure of a flat flow on obstacles. On the airplane theory. Moscow,Russia: Moscow University.

20. Zhao R, Faltinsen O. 1993 Water-entry of two-dimensional bodies. J. Fluid Mech. 246, 593–612.(doi:10.1017/S002211209300028X)

21. Zhang S, Yue DKP, Tanizawa K. 1996 Simulation of plunging wave impact on a vertical wall.J. Fluid Mech. 327, 221–254. (doi:10.1017/S002211209600852X)

22. Faltinsen OM, Greco M, Landrini M. 2002 Green water loading on a FPSO. ASME J. Mech.Arctic Eng. 122, 97–103. (doi:10.1115/1.1464128)

23. Xu GD, Duan WY, Wu GX. 2008 Numerical simulation of oblique water entry of anasymmetrical wedge. Ocean Eng. 35, 1597–1603. (doi:10.1016/j.oceaneng.2008.08.002)

24. Faltinsen OM, Semenov YA. 2008 Nonlinear problem of flat-plate entry into an incompressibleliquid. J. Fluid Mech. 611, 151–173.

25. Fraenkel LE, Keady G. 2004 On the entry of a wedge into water: the thin wedge andan all-purpose boundary-layer equation. J. Eng. Math. 48, 219–252. (doi:10.1023/B:engi.0000018157.35269.a2)

26. Christodoulides P, Dias F. 2010 Impact of a falling jet. J. Fluid Mech. 657, 22–35. (doi:10.1017/S0022112010001291)

27. Judge C, Troesch AW, Perlin M. 2004 Initial water impact of a wedge at vertical and obliqueangles. J. Eng. Math. 48, 279–303. (doi:10.1023/B:engi.0000018187.33001.e1)

http://dx.doi.org/doi:10.1023/B:engi.0000018156.40420.50



http://ictam04.ippt.gov.pl




http://dx.doi.org/doi:10.1016/j.oceaneng.2008.08.002

http://dx.doi.org/doi:10.1023/B:engi.0000018157.35269.a2

http://dx.doi.org/doi:10.1023/B:engi.0000018157.35269.a2



http://dx.doi.org/doi:10.1023/B:engi.0000018187.33001.e1


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