**Applied Ocean Researcli 66 (2017) 23-31 **

**Contents lists available at ScienceDirect **

**A p p l i e d O c e a n R e s e a r c h **

**journal hornepagerwww.elsevier.corri/locate/apor**

**I m p r o v e d n u m e r i c a l solution of Dobrovol'skaya's b o u n d a r y integral**

** /**

**(**

**s **

**equations on s i m i l a r i t y flow for u n i f o r m s y m m e t r i c a l e n t r y of w e d g e s ^ **

**Jingbo Wang*, Odd M. Faltinsen **

**Centre/or Aufonomous Marine Operations and Systems, Norwegian University of Science and Technology, NO-749] Trondheim, Norway **

**CrossMark **

**A R T I C L E I N F O **

**Article history: **

**Received 14 November 2016 **
**Received in revised form 16 March 2017 **
**Accepted 8 May 2017 **

**Available online 20 May 2017 **

**Jfeywords: **
**Water entry **
**Wedge **

**Similarity solution **

**A B S T R A C T **

**Dobrovol'skaya 11J presented a siinilarity solution for the w a t e r entry of s y m m e t r i c a l wedges w i t h c o n **
**-stant velocity. T h e solution involves a n integral equation that becomes increasingly harder to n u m e r i c a l l y **
**solve as the deadrise angle decreases. Zhao and Faltinsen [2] w e r e able to present reliable results for d e a d **
**-rise angles d o w n to 4°. I n this paper, Zhao and Faltinsen's results are improved and reliable results for **
**deadrise angles d o w n to 1° are confirmed by comparing to the a s y m p t o d c solutions at small deadrise **
**angles and the soludons by the traditional boundary element method at relatively large deadrise angles. **
**The present similarity solution results provide a reference solution in theoretical studies of w a t e r entry **
**problems and in developing accurate n u m e r i c a l solvers for simulating strongly nonlinear w a v e - b o d y **
**interacrions, w h i c h flows are governed by Laplace equarion or Euler equarion. **

**© 2017 Elsevier Ltd. A l l rights reserved. **

**1. Introduction **

**Solid objects entering through a water (liquid) surface often **
**involves large unsteady hydrodynamlc loads and rapid deformation **
**of free surface and is therefore of great interest to the design of ship **
**bows, lifeboats, planning vessels, high-speed seaplanes, **
**surface-piercing propellers and offshore or coastal structures. Wagner [3] **

**studied water entry of wedges. He accounted for the local uprise of **
**the water and presented details ofthe flow at the spray roots, which **
**included predictions of maximum pressure. Wagner's first-order **
**outer-domain solution does not include the details at the spray **
**roots and overestimates the vertical hydrodynamlc forces. For finite **
**deadrise angles 9 as defined in Fig. 1, Wagner used a flat-plate ****approximation, w h i c h leads to pressure singularities at the plate **
**edges. Cointe and Armand [4] and Howison et al. [5] used matched **
**asymptotic expansions to combine Wagner's inner-flow-domain **
**solution at a spray root with an outer-flow-domain solution. In **
**that way, the pressure singularities at the spray roots are removed. **
**Cointe [6] studied also the jet domain and presented predictions **
**of the angle between the free surface and the body surface at the **
**intersection point for water entry of a wedge with constant entry **
**velocity. The theoretical model by Cointe and Armand [4] or **
**How-ison et al. [5] only gives satisfactory solution for small deadrise **
**angles. Faltinsen [7] studied water entry of wedges with larger **

*** Corresponding author. **

**E-mail address: jingbo.wangSntnu.no (J. Wang). ****http://dx.doi.org/10.1016/j.apor.2017.05.006 **
**0141 - 1 1 8 7 / ® 2017 Elsevier Ltd. All rights reserved. **

**deadrise angles. He accounted for the deadrise angle in constructing **
**the outer domain solution, which results in significant **
**improve-ment of the asymptotic solution for larger deadrise angles. The **
**previously mentioned theoretical models give approximate **
**analyt-ical solutions for two-dimensional problems. By neglecting gravity, **
**Dobrovol'skaya [1 ] presented the similarity solution, w h i c h exactly **
**represents the water entry of symmetrical semi-infinite wedges **
**with constant velocity within the framework of potential flow of **
**incompressible liquid. Semenov and lafrati [8] obtained similarity **
**solutions for the water entry of asymmetric wedges without flow **
**separation. For three-dimensional problems, Faltinsen and Zhao [9] **
**presented asymptotic solutions for water entry of axisymmetric **
**bodies; Scolan and Korobkin [10] presented exact analytical **
**solu-tions to the Wagner problem; W u and Sun (11 ] found the existence **
**of similarity solutions in the case of an expanding paraboloid **
**enter-ing water. **

**Dobrovol'skaya's similarity solution is applicable for any **
**dead-rise angle. Its existence and uniqueness has been proved by **
**Fraenkel and Keady [12]. The similarity solution has been widely **
**used for a reference solution in theoretical studies of water **
**entry problems and also in developing accurate numerical solvers **
**for simulating strongly nonlinear wave-body interactions, for **
**instance, by Mel et al. [13], S ö d i n g [14], Semenov and lafrati [8], **
**W u [15] and W a n g and Faltinsen [16]. W h e n used as a reference **
**solution, the similarity solution results should be accurate. The **
**similarity solution is represented by a nonlinear singular integral **
**equation, w h i c h is very difficult to solve. The challenges increase **
**with reducing the deadrise angle, because smaller deadrise angles **

**24 ** **]. Wang, O.M. Faltinsen/Applied Ocean Research 66 (2017) 23-31 **

**Fig. 1. Coordinate system and sketcii of a wedge symmetrically entering into calm **
**water ffo is lialf of the wedge angle; e is the deadrise angle; fio is the angle between ****the body surface and the water surface at the intersection point B. **

**result in thinner and longer jet flows. Dobrovol'skaya [1 ] only **
**pre-sented results for deadrise angles equal to and larger than 3 0 ° . **
**Zhao and Faldnsen [2] pointed out that Dobrovol'skaya [1 I's results **
**for the deadrise angle of 3 0 ° are not accurate. They improved **
**Dobrovol'skaya's results and obtained results in a wider range of **
**deadrise angles (down to 4 ° ) . However, there w a s non-negligible **
**discrepancy in the pressure distribution on the wedge surface **
**w h e n compared w i t h the results by the boundary element method **
**(see [2], Fig. 6). Due to numerical challenges, results for deadrise **
**angles smaller than 4 ° have not been obtained yet. In this paper, a **
**nested iterative method based on quasi-dynamic under-relaxation **
**is proposed to derive accurate results of Dobrovol'skaya's similarity **
**solution. By employing this method, we successfully obtained **
**simi-larity solution results for deadrise angles down to 1 °. The numerical **
**error of the present similarity solution results has been estimated. **
**The accuracy of the results is further confirmed by comparing to **
**the asymptotic solutions at small deadrise angles and the **
**solu-tions by the traditional boundary element method at relatively **
**large deadrise angles. The present similarity solution results agree **
**well w i t h the asymptotic solutions at small deadrise angles and **
**the discrepancy between the two solutions tends to vanish w i t h **
**decreasing the deadrise angle, w h i c h are expected w h e n **
**com-paring well-developed asymptotic solutions to exact solutions. At **
**relatively large deadrise angles, the present results coincide with **
**those obtained by the tradidonal boundary element method, w h i c h **
**improve Zhao and Faltinsen [2]'s results. The assumptions of the **
**similarity solution must be kept in mind, such as a semi-infinite **
**wedge is considered. It is noted that, at small deadrise angles, the **
**airflow will cause the free surface to raise at the chines if the wedge **
**is rigid with a finite length. The consequence is that air cavities are **
**formed under the wedge bottom. However, there is more to it than **
**that. Hydroelasticity will in practice matter [17], Furthermore, **
**liq-uid compressibility can matter for small deadrise angles. Anyway, **
**it is important that numerical solvers are challenged in their **
**test-ing phase. For this purpose, the accurate similarity solution results **
**at small deadrise angles are good reference solutions to be used. **
**They also provide good reference solutions for theoretical studies **
**of water entry problems. **

**2. Mathematical model **

**2.J. Governing equation **

**To model the symmetrical entry of a semi-infinite wedge into **
**the initially calm water, a Cartesian coordinate system is **
**intro-duced: the X-axis is along the undisturbed water surface; they-axis **
**is along the body axis of symmetry and positive upwards. The **

**coor-dinate system and sketch of the water entry problem are shown in **
**Fig. 1. **

**The air flow is neglected. In case that the entry velocity is not **
**high enough to make acoustic effects relevant, it is appropriate to **
**assume that the water is incompressible after a very early stage **
**[18]. Because of the short duration of impact, viscous effects are **
**negligible provided that the Reynolds number is large. Further, the **
**flow is irrotational as there is no initial vorticity. Therefore, a **
**veloc-ity potential (p{x, y, t) of incompressible liquid satisfying Laplace's ****equation **

**3 ^ " ^ 3y2 **

**(1) **

**is introduced. The kinematic free-surface condition is that a water **
**particle remains on the free surface. The dynamic free-surface **
**condition is that the water pressure is equal to the constant **
**atmo-spheric pressure (surface tension is neglected). On the body surface, **
**the normal velocity of the water is equal to that of the wedge body. **

**2.2. Similarity solution **

**Dobrovol'skaya [1] has presented similarity solutions for the **
**water entry of symmetrical wedges w i t h constant velocity. In the **
**similarity flow, the velocity potential has the form **

**<plx,y,t) = vltt>{^,nl (2) **

**where i^o is the velocity of the wedge, ^ = x/vot, r, = y/vot and <I)(^, **

**r,) is a time-independent harmonic function. The function 0(^, rj) **

**has to satisfy the kinematic free-surface condtion, the dynamic **
**free-surface condition and the body-surface condtion, w h i c h are **
**expressed as **
**(3) **
**(4) **
**(5) **
**? - r ) ' ( ? ) ^ + ? ' ) ' ( ? ) - ' ) ( ^ ) = 0, **
**dr, a ? ****^ , 3 * ,,,di> i f d ^ \ \ i f d ^ y ****-a$> a * . **

**- 7 — cosojo - sinoJo = sinao **
**9? dr, **

**respectively. The flow under consideration is then represented as **
**a boundary-value problem. By using Wagner's h-function [3], the **
**boundary-value problem can be reduced and solved by finding the **
**solution of the nonlinear singular integral equation **

**^ ^ ^ i o j ; ' r - i ( l - T ) 4 « e x p [ - T / ; ^ d r ] d T ' ' **

**where O! = ao/jr (see Fig. 1) and **

**/ > - i ( l - r ) 4 « ( 2 r - i r e x p { - / ; , „ 5 ^ d r } d r **

**j ; n _ , r - ( 2 r - i r ' « e x p { / ; , P i ^ d r } d r • **

**Once the function^;s) is determined, the hydrodynamlc problem **
**can be considered as solved. Dobrovol'skaya [1 ] presented the **
**free-surface elevation in terms of}[s): **

**(6) **
**(7) **
**exp **
**t 2 ( 1 - t ) 2 + ****-****H****' ****(8) **

**1 0 " **

**1 0 " **

**Wans. O.M. FalHnsen/Applied Ocean Research 66 (2017) 23-31 **

**-5 **

**-5**

**10**

**10"**

**25**

**O**

**Ö****O O . . O O**. ; . o . : *

**10 -7**

**1 0 "**

**10"**

**10**

**10**

**10**

**10**

**10**

**10**

**-7**

**O**

**2**

**II****Fig. 2 . Relative eiror of numerical integration. The left subfigure shows the relative error of numerical integration ofthe integral ƒ ( - i + ° d t ; the right shows the maximum **
**relative error of numerical w„(t) over [0 1]; the symbol corresponds to = 1.0 x IO"'; the symbol corresponds to -5.0 x IO-". **

o | ^ r " § ,
**cos«o / r 2 ( l - r ) 2 " ^ " ( a r - l T ^ e x p **
**2 **
1** 3 1 **
**t " 2 ( l - t ) " 2 "^"exp **
**/ r [ r | 2 - ( l / r ) } - l **

### dr

**,js = c / ( - 5 ( 1 - t ) - ^ " " e x p****sin[n/(t)Jdt.**

**dr**

**(9)**

**(10)**

**The pressure distribution on the wetted wedge surface was **
**expressed as **

**P - P o **

**- 2 R e [ V ( r ) - « r ) V / ( f ) | f „ { „ ] - lV,U)Vii^)\^.f(r)iy2 < r < 1), **

**where po is the pressure on the free surface, p is the density ****of water, V is the complex velocity potential of $ . The complex ****coordinate f of points on the wetted wedge surface is given by **
**formula **

**?(r) = ^(r) + ir,{r) =**** [^B -**** csinaoHo{r)] -hilns- ccosaoHo{r)],(12) ****where **

### /

**I **

**r i ( i - t ) - z + " ' ( 2 t - i ) - ' ' **

**- p [ ' j ( ^ [ ^ ( 2 4 i A ) ) - i ] ^ - . ** **(13) **

**The wedge apex fyi corresponds to r = 1 and the intersection point **
**fB to r = l . The functions [V/(f)V/(f)|f=f(,), Re[f(r)V(nif=f(r)] and **
**Re V(r) in Eq. (11) have the form **

[V'(f)V/(?)]f=«r) = **- - r sinaoGo(r) **
**-t **
**where **
**- ï ? B - l - - ^ c o s a o G o ( r ) **
**Go(r) = ƒ ( l - t ) - ' - " ( 2 t - l ) - i **
**t [ r { 2 - ( l / t ) ) - l ] ° ' **
**dt; ** **(15) **

**ReR(r)V/(f)lt=t,„I = [ f j ^ csinaoHo(r)llt8 - (c„Vc)sincïoGo(r)j **

**-Ins - ccosffoHo(r)](/)B - (c2/c)cosDfoGo(r)|; (16) **

**„1 3 1 **

**R e V ( r ) =1(^1 + ^ 2 ) + / r 2 ( i _ t ) ^ 2 + V - i ) - ' ' **

**x e x p **

**T [ T { 2 - ( l / t ) ) - l ] **

**dr **

**lCoGo(t)-c(^B sinao -I-/)B cosao] **

**(17) **

**dt. **

**(11) 2.3. Numerical method **

**We present a nested iterative method for accurately solving Eqs. **
**(6) and (7). By introducing the functions **

**w(t) **

**r f{r)-m **

dr,
**r f{r)-m**

**W ( 0 = e x p [ w ( t ) - w ( 0 ) ] **

**in the domain t 6 [0 1], Eq. (6) can be converted to **
**.2 rs ( i - t ) - i - « + / W t - / ( O v v ( t ) ****(18) **
**(19) **
**dt. (20) **
**(14) 9 ( 0 **

**r^m-i(i - r ) - ï + « - ^ ( ^ ' w - i ( T ) d T **

**Over [0 1],/(s) is a monotone increasing function, w h i c h has the **
**following representation [1 ] **

**m ) = i / 2 + a - p , (21) ****f(s) = 0[s^^^)when (22) **

**/ ( s ) = / ( l ) - y ( l - s ) ' / ' - 2 ^ w h e n s ^ l . (23) **

**Further, we introduce the functions **

**) / 2 t ' ****(2(t) = e x p [ g ( t ) - w ( 0 ) ] , **
**(24) **
**(25) **
**Jo ^ - ( f + 1 ****exp[g(t) - w( **
**W o ( 0 = ( 2 j [(i + t r 2 + ^ ' " Q - ' ( t ) l r " ( l - t ) - i + ^ d t , (26) **
**Co{s) = Q ) " / f^"* "^ f)--^''^Q(t)]t-i+''(l - 0-2~^dt. (27) **

**26 ** **J. Wang, OM. Faltinsen /Applied Ocean Research 66(2017)23-31 **

**It is easy to verify that Ho(s) and Go(s) are consistent w i t h Ho(r) and **
**Go(r), respectively. Furthermore, c^/c^ = Ho(0)/Go(0). **

**To evaluate Ho(0) and Go(0), numerical integration should be **
**used. Directly applying a standard quadrature rule for singular **
**inte-grals may result in significant errors. To overcome this drawback, **
**w e adopt analytical methods for the integration in the vicinity of **
**singularities. For example, to evaluate t - ° d t , we splitthe domain **
**[0 1] into [0 e] and [e 1 ](e is a small number). The integral is ****evalu-ated analytically over [0 e] and by a numerical quadrature rule over **
**[e 1 ]. In the present study, the singular integrals have the form **

**i: **

**i:**

**g(t)t-"(i**

**t)"'dt,**

**where g(t) is a smooth function. For integration of this kind of **
**integrals, w e propose a quadrature method: the domain [0 1) is **
**divided into three subdomains, i.e. [0 1 ] = [0 e] U [e 1 - £] U [1 - e **
**1]; the subdomains [0 e] and [1 - e 1] are divided into a number of ****smaller elements; the function Z.(t)=g(t)(l - t ) " ' ' and R{t)=g{t)t-'' ****are assumed to have a linear variation over all elements in [0 e] **
**and [1 - e 1 ]. respectively and then the analytical integration is **
**per-formed; the trapezoidal rule is adopted for numerical integration **
**over [e 1 - e]. As long as element sizes over [0 e] and [1 - e 1 ] are **
**small enough, the assumption of a piecewise linear variation of I ( t ) **
**over [0 e] and K(t) over [1 - e 1] results in a negligible error The **
**numerical integration over [e 1 - fi] dominates the error of the **
**inte-gration over [01]. Once Ho(0) and Go(0) are obtained, all numerical **
**node values of the functions Ho(s) and Go(s) become known. **

**The proposed quadrature method is also applied to evaluate **

### /

**I**

**[7/(r)(l _ r ) f ( i ) - / ( ' " ) w - i ( T ) ] T ^ i ( l - T)-'+^dT, (28) **

**w h i c h is equal to the denominator of the integrand in (20). It is **
**noticed that /2(0)=oo and the first integrand of (20) has only one **
**singularity (which locates at t= 1). So the trapezoidal rule is suitable **
**for the quadrature over [0 1 - e ] and the analytical method over **
**[ 1 - fi 1 ]. By using the expression **

**/2(t) = a ( t ) ( l - t ) ^ w i t h a ( l ) = ^ W - i ( l ) , (29) **

**Eq. (20) can be integrated analytically over[l - f i i ] . **

**Before the integration of Eq. (20), Ho(s) and Go(s), the functions **
**W(t) and Q(t) should be evaluated. On a given grid system, **

**0 = tl < t2 < . . . < tw < tw+i = 1, **
**/ 2 ( t ) : **

**So w e set the initial guess of f i t ) to be **

**/o(t) = ( 3 / 8 - f a ) t . (32) **

**A nested iterative method is proposed to solve Eqs. (6) and (7) for **

**At)-(1) W(t) and Qlt) are evaluated based on the k-th approximadon of **

**m,i-e.fkit). **

**(2) The right-hand side of Eq. (20) is denoted as f(/"(t), W(t), Qlt)). **
**An iteration is performed: Mt)=F{fk{t), Wit), Q{t)). **

**(3) Compute the error Cout: =maxo<t<i \f-{t) -fk{t)\. If Cout is smaller ****than the prescribed value, ^;t) is approximated as/^(t) and the **
**iteration is completed. Otherwise, set/^^'(t) = (1 - **
**u)/k(t)-i-"ƒ* (t). where u is a under-relaxation factor. Go to inner **
**iter-ations. **

**(4) Perform the i-th inner iteradon: /i"(t) = F(fl'\t), W(t), Q.{t)). ****Compute the error e,,, := maxo<t<il/"i''{t)-/^''(t)l ** **If **
**ei„<min{0.1eout, 0.01}, set A + i ( t ) =/^"(t). exit the **
**inner iterations and go to step (1). Otherwise, set **
**/i^'+"(t) = ( l - u ) / ^ ' ' ( t ) + u/i'''(t) and perform the (i + l ) - t h **

**inner iteradon. **

**Through the iteradon process, a constant under-relaxadon **
**fac-tor may result in divergent results. So it should be changing **
**(dynamically) during iterations. Initially, the under-relaxation **
**fac-tor is set to a prescribed value UQ, which should be less than 1. **
**W e denote the present approximation of J[t) asfoidit), the ****inter-mediate a p p r o x i m a t i o n / . ( f ) : = f ( / ' o i d ( t ) . W(t), Qlt)) and the new ****approximation/new(t): = (1 - uY„u{t) + uMt). Further, we introduce ****the function **

**Bit) = 1/2+ a - f i t ) . ****(33) **

**Boid(f). B'(t) and Bnevv(t) correspond to/o,d(t),/.(t) and/new(t) **
**respec-tively. Physically speaking, B(t) denotes the angle between the **
**wedge surface and the free surface. The dynamic under-relaxation **
**factor is determined by the following criterion: **

**<t>1/2 **

**I B n e w ( t ) - B o i d ( t ) **
**Bold(t) **

### = r ,

**(34)**

**where F is a prescribed value less than 1. Immediately, the dynamic **
**under-relaxation factor can be obtained: **

**r L . v / * ( t ) - / o i d ( 0 **
**" = ^ r " ' > ^ / ^ l / 2 + a - / „ , d ( 0 **

**(35) **

**f i t ) is assumed to have a linear variation over any element [t, tf+i ]. **

**Then, the function w ( t ) can be expressed as **

**w(t) ** **/ ( f , ) - / ( t ) + ****f(ti+l)-fit,) ****ti+1 ** **( t - f l ) **

**Accounting for the asymptotic behavior of (23) over [tn tjv+i ], w e **
**modify w ( l ) : **

**w ' ( l ) = w ( l ) + [^^/2)-2p " ^] ^^^^-•'^f^N)) **_{. ( 1 / 2 ) - 2 ^ }

**q(t) is evaluated in the similar w a y for t > 5 and it is evaluated by **
**the trapezoidal rule for t < ^. It should be noted that W( 1) is equal **
**t o Q ( l ) . **

**Dobrovol'skaya [1] showed t h a t / ( I ) should satisfy the **
**condi-tion: **

**l + a < / ( l ) < i + a . **

**This criterion represents that the change of the angle between the **
**wedge surface and the free surface should be small after an **
**itera-tion. For large deadrise angles, T = 0.1 is used for the present study. **
**For small deadrise angles, smaller V is used. It should be noted that ****/ (.^^ _ (\ frequently changing the under-relaxation factor w i l l also result in **
**( " ' ^ ^ ) +/(fi+i'--''(f|' (3'') divergent results. In the present study, w e use a quasi-dynamic **
**under-relaxation factor: if the intermediate under-relaxation **
**fac-tor u- determined by (35) is close to the present under-relaxation ****factor**** UoM**** (for instance**** HUOM -**** U')/"oidl < 0-2). the under-relaxation **
**factor remains unchanged; if u. is significantly smaller than **
**(for instance u-\Uo\i < 0.2), the under-relaxation factor is changed to ****Unew = O.SUoid; once n (for instance 1000) inner iterations, the **
**under-relaxation factor is updated as u„ew = niax{0.8Uow, min{1.2UoH, **
**u - } } ; the under-relaxadon factor has a upper limit, w h i c h is set **

**tobeuo-(31) **

**2.4. Grid and accuracy **

**In the present study, w e use the grid system, w h i c h is **

**symmet-1.1000 elements are uniformly distributed over [0 **

**Wang, O.M. Faltinsen /Applied Ocean Research 66 (2017) 23-31 ****27 **

**Cp^ Pressure **

**Cp^ Pressure**

**Fig. 3. Definitions of parameters ctiaracterizing slamming pressure during water **
**entry of a wedge. C, = pressure coefficient. **

**e], where s is set to 1 0 " ' ^ From e to i , the element length is ****geo-metrically increasing, i.e. the ratio between successive elements **
**is constant. The size of the smallest element, /Q, is approximately **
**lO-^e. The largest element is next to 1 and its size is denoted as **

**Im. Through Eq.**** (21), w e know that the accuracy of J[t] directly **
**influences that of fi. Cointe [6] has proposed **

**(36) **

**for small deadrise angles. For example, 6 equal to ^ corresponds to ****the asymptotic value of equal to 1.54 x 10"^. The numerical error **
**of less than 1% of fi requires that the numerical error of f [ t ) should **

**be less than 10"'' approximately. The numerical error o f f { t ) can be ****estimated by assessing the accuracy of the numerical integration. **

**To estimate the accuracy of numerical integration of the integral **

**t - i + ° l ( t ) d t w i t h 0 < a < l , **

**w e assume that I ( t ) can be expanded in the Taylor series **

**L{t) = bo + bit+b2fi+--: It is possible to assess the accuracy of **

**numerical integration of **

**. 1 / 2 **

### i '

**r i + ' ' t " d t w i t h 0 < a < l and n = 0 , 1 , 2 ,**

**because this integral can be evaluated analytically. In practice w e **
**only need to assess a few lowest order terms, since they are usually **
**dominant. **

**Similarly, to estimate the accuracy of the numerical w(t), we **
**assess the numerical integration of **

**ƒ rnzm^ry^ith f„{t)=7(1^(1 - tf **

**ƒ rnzm^ry^ith f„{t)=7(1^(1 - tf**

**o r < equivalently f„{t) = Vtt".****10000**

**8000**

**6000**

**E5.**

**I 4000**

**2000**

**- The present similarity solution **
**-The asymptotic solution **

**lOOOOr **
**0.565 0.57 0.575 **
**1 **

**(a) 0 = 1° **

**1000**

**800**

**600**

**I 400**

**200**

**- The present similarity solution **
**- T h e asymptotic solution **
**I 10001 • • **
**800 **
**600 **
**400 **
**200 **
**0 **
**0.565 0.57 0.575 **
**1 **
**y/vQt **

**(a) 0 = 3= **

**2500**

**2000**

**^ 1 5 0 0**

**I 1000**

**500**

**600**

**500**

**400**

**^ 3 0 0**

**I**

**3 200**

**100**

**0**

**- The present similarity solurion **
**-The asymptotic solution **

**2500 r **
**2000 **
**1500 **
**1000 **
**500 **
**0 **
**0.565 0.57 0.575 **
**0 1 2 **
**y/vot **

**(b) 0 = 2° **

**yhot****4° **

**^The present similarity solution **
**The asymptotic solution **

**600 **
**600 **
**500 **
**400 **
**300 **
1 **200 **
**IOC **
**0 **
**IOC **
**0 **
**0.565 0.57 0.575 **
**0 ** **1 ** **2 ** **3 **

**Fig. 4. Comparion between the similarity and asymptotic solutions of the pressure coefficient on the wetted wedge surface. The embedded figures display the pressure **
**distribution around the maximum pressure. **

**28 ** **/ Wang. O.M. Faltinsen /Applied Ocean Researcli 66(2017)23-31 **

**2 0 **

**15 **

**1 **

**1 0**

**-The present similarity solution **
**The B E M solution **
**I , 4 **

### 1 2

**3**

### -2

**C**

### 2

**1.5**

**0 . 5**

**- 0 . 5**

**-The present similarity solution **
**-The B E M solution **

**-1 - 0 . 5 0 0 . 5 1 1.5 **

### I

'

### I

**The present similarity solution **
**The B E M solution **

**- 0 . 5 ** **0.5 ** **1.5 **

**The present similarity solution **
**The B E M solution **

**3 **

**- The present similarity solution **
**-The B E M solution **

**- 0 . 5 ** **0 . 5 **

**y/vat **

**1.5 **

**The present similarity solution **
**The B E M solution **

**(c)**

** 9 =**

**9 =**

** 40= **

**Fig. 5 . Comparison of the pressure distribution on the wetted wedge surface and the free surface elevation for wedges with deadrise angles of 20°, 30- and 40° symmetrically **
**entering into calm water. **

**J, Wang, O.M. Faltinsen / Applied Ocean Research 66(2017)23-31 ****29 **

**- The boundary eiement method solution **
**-The present similarity solution **

**Zhao & Faldnsen (1993)'s similarity solution **

**0 . 0 5 ** **0.1 0.15 0.2 ** **0.25 **
**16 **
**15.5 **
**15 **
**145 **
**J , 14 **
**•^"135 **
**13 **
**12.5 **
**12 **

**- — The bonadary-clemeot-method solution **
**The present similarity solution **

**Zhao & Faltinsen (1993)'s similarity solution **

**0 . 0 5 ** **0.1 0.15 **

rot

**0.2 ** **0.25 **

**Fig. 6. Convergence history of the maximum pressure and total vertical hydrodynamlc force for the numerical simulation of a wedge with the deadrise angles of 30° **
**symmetrically entering into calm water by the boundary element method. **

**In the above expression, w e have used Eq. (23) and the fact that p **
**is much smaller than 1 /2. It can be shown that **

**^ / T T " - ^ / t t " **

**w „ ( t ) = / d r _{r -}_{ t }**

**n-1 **

**= 2t" l - V t l n ( v ^ - h l ) l +Y,t"-<-'/{i + 3/2). ****(37) **

**The numerical tests are performed for the integrals J^^'^ f-'+^dt ****and Wn(t) on the proposed grid system, where /„, = 1.0 x I Q - ^ and **
**5 . 0 x 1 0 - ^ are used. The accuracy is represented as the relative **
**error, er, w h i c h is calculated by comparing w i t h the exact **
**solu-tion. Cmax denotes the maximum relative error of Wn(t) over [0 1]. **
**The results are shown in Fig. 2. For a < 1 and a = 3, the relative error **

**of 0(10"'') is obtained. W e note that the trapezoid rule gives exact **
**results for a = 1 and 2. Therefore, the relative error of the numerical **
**integration of the three lowest terms, **

**r1/2 **

**I . • **

**r i + ° t " d t ( 0 < a < l , n = 0, 1 and 2), **

**is estimated to be 0(10"''). Because the lowest order terms of L(t) **
**are regarded to be dominant, the relative error of the **
**numeri-cal integration of /^^^^ r i + ° L ( t ) d t ( 0 < a < l ) is also estimated to **
**be 0 ( 1 0 - 7 ) ^Yte given grid systems. Similarly, the relative error ****of the numerical w(t) is estimated to be 0(10"''). It is noted that **
**the accuracy can be controlled or improved by modifying the grid **
**system. **

**2.5. Asymptotic formula for small deadrise angles **

**P-Po ****| T | V 2 **
**( l H r | i / 2 ) **
**-for x> c(t), ****d c . **
**p - p o = p v o c ^ { ( c 2 ** **•1/2 ** **[ 2 c ( c - x ) ] - V 2 ) **

### + 2 p

### ( § ) V l

**^ / 2 ( l + | r | ' / 2 ) **

**-(39)**

**(40)**

**for 0 < X < c(t),**

**where y^ is they-coordinate of the intersection point between the **
**free surface and the wedge body, c(t) = ^Trvof cot 0 and |T| is related **
**to X by **

**X - c = [S/nX-In |T| - 4|T|i/2 - | T H - 5) ****(41) **

**5 is thejet root thickness and it is expressed as 5 = 7n/^2c[4dc/dtr^. **
**Eqs. (39) and (40) represent the pressure distribution on the **
**wet-ted wedge surface. W h e n |T| = 1, i.e. x=c, the m a x i m u m value of p **
**occurs: **

**Pmax — Po **

**(42) **

**The correspondingy-coordinate is **

**ymax = f o t { j r / 2 - l ) . (43) **

**By integrating the pressure along the wetted wedge surface, w e can **
**obtain the vertical hydrodynamlc force **

**Fy = i j r V i ^ t c o t ^ e [^/{jTcote] -f 7r/2 - V2] , (44) **

**Fy = i j r V i ^ t c o t ^ e [^/{jTcote] -f 7r/2 - V2] , (44)**

**w h e r e t c is the root of the equation l n T + 4 T ^ / 2 + T - 5 = 2 7 r 2 c o t 2 0 **
**and corresponds to x = 0 . Korobkin [20] also presented the pressure **
**formulations, w h i c h are asymptotically equivalent to the present **
**ones. **

**For small deadrise angles, the hydrodynamlc problem can be **
**solved by matched asymptotic expansions (Cointe and Armand [4], **
**Cointe [6] and Howison et al. [5]). The detailed theories will not be **
**repeated here. Our attention is to present the asymptotic **
**formu-las, w h i c h w i l l be used in the following section to compare w i t h **
**numerical results of the similarity solution. **

**Cointe [6] has proposed Eq. (36) for the contact angle between **
**the free surface and the wetted water surface at small deadrise **
**angles. Based on the similar asymptotic analysis, Zhao and Faltinsen **

[2]** gave that **

**y B = V o t ( 7 r - l ) , (38) **

**3. Numerical results **

**By the present method, w e successfully obtained numerical **
**results of the similarity solution for deadrise angles down to 1 ° . **
**The largest element Im in the numerical integration procedure ****is as follows. Im = 1.0 X 10-3 is used for deadrise angles > 4 ° and **

**Im = 5.0 X 10-"* for deadrise angles <4°.**** Table 1 shows the slamming **
**parameters predicted by the present method, Zhao and Faltinsen **

**[2]'s method and the asymptotic method. The discrepancy between **
**the present similarity solution and the asymptotic solution tends to **
**vanish with decreasing the deadrise angle. At small deadrise angles, **
**the asymptotic solution of the pressure distribution on the wedge **

**3U ** **J. Wang, O.M. Faltinsen/Applied Ocean Research 66 (2017) 23-31 ****Table 1 **

**Comparison of slamming parameters during water entry of a wedge with constant vertical velocity Vo. 9 = deadrise anj **
**ymax =y-coordinate of maximum pressure; ASj = spatial extent of slamming pressure (see Fig. 3); c = O.SHVot cot d; **

**= angle between the free surface and the wedge surface at the intersection point (see Fig. 1); y j =y-coordinate of **
**Zhao = Zhao and Faltinsen (2]'s similarity solution; Asym. = asymptotic solution; Disc. = jPres. - Asym. 1/Asym.. **

**ble; Cp„„ - pressure coefficient at maximum pressure; **
**F,-total vertical hydrodynamlc force on the wedge; **
**intersection point; Pres.-present similarity solution; **

**1 **
**2 **
**3 **
**4 **
**10 **
**2 0 **
**30 **
**40 **
**1 **
**2 **
**3 **
**4 **
**10 **
**20 **
**30 **
**40 **
**Pres. **
**8089.759 **
**2018.211 **
**894.490 **
**501.441 **
**77.699 **
**17.735 **
**6.895 **
**3.253 **
**ASs/c **
**0.000966 **
**0.003806 **
**0.008450 **
**0.014843 **
**0.089093 **
**0.411347 **
**Zhao **
**503.03 **
**77.847 **
**17.774 **
**6.92 **
**3.266 **
**0.01499 **
**0.09088 **
**0.4418 **
**Asym. **
**8098355 **
**2023355 **
**898356 **
**504.606 **
**79360 **
**18.626 **
**7.402 **
**3.504 **
**0.000982 **
**0.003935 **
**0.008878 **
**0.015842 **
**0.103891 **
**0.557516 **
**Disc. **
**0.11% **
**0.25% **
**0.43% **
**0.63% **
**2.09% **
**4.78% **
**6.86% **
**7.19% **
**1.66% **
**3.28% **
**4.82% **
**631% **
**1424% **
**26.23% **
**N'T ****1 ** **0.0000152 ** **-** **0.0000154 ** **1.65% **
**2 ** **0.0000597 **

_{-}

**0.0000617**

**3.21%**

**3**

**0.0001324**

**-**

**0.0001389**

**4.70%**

**4**

**0.0002318**

**0.0002329**

**0.0002469**

**6.13%**

**10**

**0.0013336**

**0.001337**

**0.0015432**

**13.59%**

**20**

**0,0047738**

**0.004783**

**0.0061728**

**22.66%**

**30**

**0.0098928**

**0.009913**

**0.0138889**

**28.77%**

**40**

**0.0166148**

**0.01663**

**0.0246914**

**32.71%**

**Pres.**

**y m a / f O t**

**0.5702**

**0.5694**

**0.5685**

**0.5673**

**0.5544**

**05078**

**0.4254**

**0.2897**

**FjA/Jfot')**

**25071.865**

**6164311**

**2694351**

**1489.933**

**212.959**

**42.272**

**14.020**

**5.447**

**ys/vot****2.1401**

**2.1378**

**2.1350**

**2.1315**

**2.0976**

**1.9930**

**1.8329**

**1.6234**

**Zhao ** **Asym. ** **Disc. **

**_ **

_{03708 }

_{03708 }**0.10%**

**_**

**05708**

**0.24%**

**_**

**0.5708**

**0.40%**

**0.5695**

**0.5708**

**05556**

**0.5708**

**2.87%**

**0.5087**

**0.5708**

**11.03%**

**0.4243**

**0.5708**

**25.47%**

**0.2866**

**0.5708**

**49.24%**

**25261387**

**0.74%**

**_**

**6266.478**

**1.63%**

**_**

**2762352**

**2.46%**

**1503.638**

**1540.474**

**3.28%**

**213.980**

**231S78**

**8.20%**

**42.485**

**50.640**

**16.53%**

**14.139**

**18.748**

**25.22%**

**5.477**

**8322**

**34.56%**

**2.1416**

**0.07%**

**2.1416**

**0.17%**

**2.1416**

**031%**

**2.1363**

**2.1416**

**0.47%**

**2.1004**

**2.1416**

**2.06%**

**1.9955**

**2.1416**

**6.94%**

**1.8363**

**2.1416**

**14.41%**

**1.6253**

**2.1416**

**24.20%**

**M ! ^ u m non-dhnensional curvature ofthe free surface during water entry of a wedge with constant vertical velocity. S - deadrise angle; - maximum non-dimensional **
**curvature of the free surface. **

**ei') ****1 **
**237.783 **
**2 **
**120.845 **
**3 **
**81.841 **
**4 **
**62317 **
**5 **
**50.588 **
**6 **
**42.755 **
**7 **
**37.148 **
**8 **
**32.934 **
**9 **
**29.646 **
**ICmzx ****10 **
**27.008 **
**15 **
**19.012 **
**20 **
**14.918 **
**25 **
**12392 **
**30 **
**10.655 **
**40 **
**8393 **
**50 **
**6.982 **
**60 **
**6.033 **
**70 **
**5.683 **

**is compared to the present similarity solution, w h i c h is shown in **
**Fig. 4. Good agreement is obtained, except that there is a small **
**dis-crepancy (also indicated in Table 1) in the position of m a x i m u m **
**pressure. These verify the asymptotic theories, since the **
**asymp-totic solution should approach the exact solution w h e n the deadrise **
**angle goes to zero. **

**At relatively large deadrise angles, the present similarity **
**solu-tion is checked againt the boundary element method [ 16,19], which **
**solves the boundary integral equation transformed from Eq. (1) **
**and tracks the evolution of the free surface by the second order **
**Runge-Kutta method. At the start of the numerical simulation by **
**the boundary element method, a small penetration vofo of the **
**wedge into the water is given, the velocity potential is set to zero **
**and the free-surface elevation is prescribed by the Wagner's **
**outer-domain solution **

**y(x) = f o t o - T ^ arcsin [^1 - voto, (45) **
**c(roJ L X J **

**where c(to) = ^nvotoCotO is the x-coordinate of the intersection ****point between the free surface and the wedge surface. As time **
**increases, the solution by the boundary element method should **
**approach the exact solution. Fig. 5 compares the present **
**similar-ity solution to the converged boundary-element-method solution. **
**Except for the free surface of jet flow, perfect agreement has been **
**obtained between the two methods. The discrepancy in the free **
**surface of jet flow is due to the cut-off of jet flow used in the **
**boundary element method for stabilizing the numerical solution. **

**Zhao and Faltinsen [2] also compared their similarity solution w i t h **
**a boundary element method. It is observed that there are some **
**discrepancies between their two solutions (see [2], Fig. 6). It is **
**indi-cated that Zhao and Faltinsen |2]'s similarity solution can probably **
**be improved. Further, the convergence history of the **
**boundary-element-method solution is illustrated in Fig. 6, w h i c h shows the **
**results of a wedge w i t h the deadrise angles of 3 0 ° symmetrically **
**entering into calm w a t e r It can be seen that the **
**boundary-element-method solution converges to the present similarity solution and **
**Zhao and Faltinsen [2 ]'s results slightly overestimate the **
**boundary-element-method solution, w h i c h has been observed before (see **
**[21], Fig. 15). It confirms that the present similarity solution **
**improves Zhao and Faltinsen [21's results. **

**It is observed that the m a x i m u m curvature of the free surface, **

**Kmax. occurs at the root of the jet. This parameter increases as the **

**deadrise angle decreases that brings an attention to the effect of **
**surface tension w h i c h may matter locally at small deadrise angles. **
**Based on ^ = x/vot and r] = y/vot, the curvature of the free surface ****can be expressed as **

**K = k/vot, (46) **

**i< = \^tnti-n'^"m^ + 1'^f^- (47) **

**i< = \^tnti-n'^"m^ + 1'^f^- (47)**

**Inserting equation (8) into (47), w e can show **

**Wang. O.M. Faltinsen / Applied Ocean Researcli 66(2017)23-31 ****31 **

**where, s = 1 corresponds to the intersection point between the free **
**surface and the wedge body and s = 0 corresponds to the free surface **
**at infinity. Then, the maximum non-dimensional curvature of the **
**free surface is **

**/fmax =niaxo<s<iK{s). (49) **
**Table 2 shows the maximum non-dimensional curvature versus the **

**deadrise angle. The maximum curvature of the free surface, **

**f m a x = Umix/Vot, (50) ****can be large at the very intial stage and/or at small deadrise angles, **

**w h i c h implies that surface tension may matter locally. **

**The present similarity solution results of the pressure **
**distribu-tion on the wetted wedge surface, the free surface configuradistribu-tion **
**and the curvature of the free surface are given in Appendix A. **

**4. Conclusions **

**A reliable and accurate method is developed for solving the s i m **
**-ilarity flow of a wedge symmetrically entering into calm water w i t h **
**constant velocity. By using the present method, exact solutions of **
**the similarity flow for deadrise angles less than 4 ° are first obtained. **
**The accuracy of the numerical results is estimated. The **
**numeri-cal results show that the asymptotic theories proposed by Cointe **
**and Armand [4], Howison et al. [5] and Zhao and Faltinsen [2] are **
**consistent with the present method at small deadrise angles: w i t h **
**decreasing the deadrise angle, the discrepancy between the two **
**solutions of slamming parameters tends to vanish; the pressure **
**distribution on the wetted wedge surface obtained by the present **
**method agrees well w i t h the asymptotic solution. At relatively large **
**deadrise angles, the present similarity solutions agree almost **
**per-fectly with the traditional boundary element method and improve **
**Zhao and Faltinsen [2]'s results. All these demonstrate that the **
**present similarity solution is accurate. It can be used as a **
**refer-ence solution in theoretical studies of water entry problems and **
**in developing accurate numerical solvers for simulating strongly **
**nonlinear wave-body interactions. Finally, the curvature of the free **
**surface has been investigated. The m a x i m u m curvature occurs at **
**the root of the jet, where the surface tension matters at the very **
**initial stage and/or at small deadrise angles. **

**Adtnowledgements **

**Appendix A. Supplementary data **

**Supplementary data associated w i t h this article can be found, **
**in the online version, at http://dx.doi.Org/10.1016/j.apor2017.05. **
**006. **

**References **

**[1] Z.N. Dobrovotskaya, On some problems of similarity flow of fluid with a free **
**surface,]. Fluid Mech. 36 (1969) 805-829. **

**[2] R Zhao, O.M. Faltinsen, Water entry of two-dimensional bodies,]. Fluid Mech. **
**246(1993)593-612. **

**[3] H. Wagner, Ober StoB- und Gleitvorgange an der Oberflache von **
**Flüssigkeiten, Z. Angew. Math. Mech. 12 (1932) 193-215. **

**[4] R. Cointe, J.-L. Armand, Hydrodynamlc impact analysis of a cylinder, ASMEJ. **
**Offshore Mech. Arc. Eng. 109 (1987) 237-243. **

**[5] S.D. Howison, J.R. Ockendon, S.K. Wilson, Incompressible water entry **
**problems at small deadrise angles,]. Fluid Mech. 222 (1991) 215-230. **
**[6] R. Cointe. Free surface flows close to a surface-piercing body, in: T. Miloh **

**(Ed.), Mathematical Approaches in Hydrodynamics, SIAM, Philadelphia, USA, **
**1991, pp. 319-334. **

**[7] O.M. Faltinsen, Water entiy of a wedge with finite deadrise angle,]. Ship Res. **
**46(2002)39-51. **

**[8] Y.A. Semenov, A. lafrati. On the nonlinear water entry problem of asymmetric **
**wedges,]. Fluid Mech. 547 (2006) 231-256 **

**[9] O.M. Faltinsen, R. Zhao, Water entry of ship sections and axisymmetric bodies. **
**AGARD Report 827. High Speed Body Morion in Water, 1998. **

**[10] Y.-M. Scolan, A A Korobkin, Three-dimensional theory of water impact. Part **
**1. Inverse Wagner problem, ] . Fluid Mech. 440 (2001) 293-326. **

**[11] G.X. Wu, S.L Sun, Similarity solution for oblique water entry of an expanding **
**paraboloid,]. Fluid Mech. 745 (2014)398-408. **

**[12] L E . Fraenkel, G. Keady, On the entry of a wedge into water: the thin wedge **
**and an all-purpose boundary-layer equation,]. Eng. Math. 48 (2004) 219-252. **
**[13] X. Mei, Y. Liu, D.K.P. Yue, On the water impact of general two-dimensional **

**sections. Appl. Ocean Res. 21 (1999) 1-15. **

**[14] H. Söding, Flow computations for ship safety problems, Ocean Eng. 29 (2002) **
**721-738. **

**[15] G.X. Wu, Fluid impact on a solid boundary,]. Fluid Struct. 23 (2007) 755-765. **
**[16] ]. Wang, O.M. Faltinsen, Numerical investigation for air cavity formation **

**during tiie high speed water entry of wedges. ]. Offshore Mech. ArcL 135 **
**(2013)ni. **

**[17] O.M. Faltinsen, The eflfect of hydroelasticity on ship slamming, Proc. R. Soc. A **
**355(1997)575-591. **

**[18] A A Korobkin, D.H. Peregrine. The energy distribution resulting from an **
**impact on a floating body,]. Fluid Mech. 417 (2000) 157-181. **

**|19] ]. Wang. C. Lugni, O.M. Faltinsen, Experimental and numerical investigation of **
**a fi-eefall wedge vertically entering the water surface, Appl. Ocean Res. 51 **
**(2015)181-203. **

**[20] AA. Korobkin, Analytical models of water impact, Eur. ]. Appl. Math. 15 (2004) **
**821-838. **

**[211 J. Wang, C. Lugni, O.M. Faltinsen, Analysis of loads, motions and cavity **
**dynamics during freefall wedges vertically entering the water surface, Appl. **
**Ocean Res. 51 (2015)38-53. **

**The research activity has been supported by the Centre of **
**Autonomous Marine Operations and Systems (AMOS) whose main **
**sponsor is the Norwegian Research Council (Project number **
**223254-AMOS). We thank the reviewers for their valuable **
**com-ments and suggestions. **