Improved numerical solution of Drobovol'skaya's boundary integral equations on similarity flow for uniform symmetrical entry of wedges

Applied Ocean Researcli 66 (2017) 23-31

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

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

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

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

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:

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

/

[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 ,

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

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 '

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

- The present similarity solution -The asymptotic solution

lOOOOr 0.565 0.57 0.575 1

(a) 0 = 1°

- 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=

- 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°

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

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

1 2

-2

C

2

-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 =

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 )

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)

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

_-

Zhao Asym. Disc.

_

₀₃₇₀₈

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)

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.

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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.