ACTA MECHANICA SINICA, Vol.4, No.3 August, 1988

Science Press, Beijing, China

Allerton Press, INC., New York, U,S.A.

ISSN 0567--7718

DIFFRACTION OF A SOLITARY WAVE BY A THIN WEDGE

Chen Xuenong Liu Yingzhong

(Department of Naval Architecture and Ocean Engineering, Shanghai Jiaotong University)

ABSTRACr: The diffraction of a solitary wave by a thin wedge with vertical walls is studied when the

incident solitary wave is directed along the wedge axis. The method of multiple scales is extended to this

problem and reduces the task to that of solving the two-dimensional KdV equation with proper boundary and

initial conditions. The finite-difference numerical procedure is carried out with the fractional step algorithm

in which difference schemes are all implicit. Except the maximum run-up at the wall, the results in this paper

are found to corroborate the Melville's experiments not only qualitatively but also quantitatively. The

maximum run-up of our results agrees well with Funakoshi's numerical one but it is considerably larger than

that in Melville's experiment. An important reason for this discrepancy is believed to be the effect of viscous

boundary layer on the vertical side wall.

KEY WORDS: solitary wave, Mach reflexion, multiple-scale method, two-dimensional KdV equation,

finite-difference method

I. INTRODUCTION

It was observed in early experiments (Perroud (1957)/tl, Chen (1961) [21 and Wiegel (1964a,

b) t31[4l) that, for a solitary wave incident obliquely upon a straight wall, if the angle of incidence is

small (less than 45~ besides the incident and reflected waves, there is a third wave crest (called the

Mach stem) which intersects the wall normally; the incident wave, the reflected wave and the stem

meet at a point which is some distance away from the wall. Because of its geometrical resemblance to

the reflexion of shock waves in gas dynamics (see Whitham 1974lS1), the phenomenon in shallow

water waves has also been called Mach reflexion by Wiegel. Miles (1977a) 161 has conjectured that the

Mach reflexion of a solitary wave may be described asymptotically by the resonant interaction of

three Boussinesq solitary waves. Later on, Melville (1980) [sl made a more accurate experiment on

this problem to report experimental data and to examine the validity of Miles' model 161. It is

suggested that there is discrepancy between the measurements and the model, which may result

from the failure of the model. Funakoshi (1981) lgl studied this problem by solving the Boussinesq

equations numerically. His asymptotic results approach to the solution predicted by Miles but his

calculated maximum run,up at the wall is considerably larger than that in Melville's experiment. He

explained the reason for this discrepancy by a crude estimation of the effect of viscous laminar

boundary layer on the bottom using the decay law for an one.dimensional solitary wave.

In recent years, the parabolic approximation has been extended to many forward-scattering

problems in water waves. Mei & Tuck(1980) It ol have developed this approximation for studying the

problem of hnearized periodic shallow water waves at grazing incidence on a slender body. Yue &

Mei(1980) [111 further extended it to nonlinear Stokes waves and revealed their Mach reflexion

phenomenon qualitatively. Liu, Yoon & Kirby (1985) [1 z] have derived the parabolic approximation

Received 18 June 1987,

202 ACTA MECHANICA SINICA lg88

equation for refraction and diffraction of nonlinear periodic shallow water waves and studied several examples of refraction numerically.

In this paper, relying on the Boussinesq theory and using the same multiple-scale expansion as that in the parabolic approximation [11'12], we derive the two-dimensional KdV equation and its boundary conditions to describe the diffraction of nonlinear shallow water waves by a slender body with a side wall that is vertical throughout the water depth. This equation is formally analogous to one derived by Kadomtsev & Pelviashvili (1970) [13] or Oikawa, Satsuma & Yajima (1974) [14]. Solving it numerically is easier than solving the Bonssinesq equations. We solve it with the fractional step algorithm to calculate the problem of the diffraction of a solitary wave by a thin wedge. For each half-step, an implicit difference scheme is constructed. The results obtained in this paper are compared with Melville's experimental ones and Funakoshi's numerical ones. It is shown that the maximum run-up at the wall of our results agrees well with that of Funakoshi's and both of the tendencies are identical. The calculated profde of developed waves tallies quite well with that in Melville's experiment but the maximum run-up is considerably larger than Melville's. This discrepancy is same as one between Funakoshi's and Melville's. Based on the comparison of results with Melville, we consider that the effect of viscous boundary layer on the side wall of the wedge is one of main reasons for this discrepancy. This subject is likely to be more important and interesting and will be discussed in detail in a subsequent paper. The numerical data of surface prof'de are transformed into ones in sequence of time and the process of the diffraction is revealed visually with a series of 3-D graphs.

II. APPROXIMATION OF THE DIFFRACTION In the Boussinesq theory, there are two small parameters,

e = A * / h * I~ = k ' h * O(e) ='O(p 2) << 1 (2.1) where .4* is a charecteristic wave amplitude, h* is the quiescent water depth, k* is a charecteristie wave number. The nondimensionalized Boussinesq equations! 151191 for shallow water waves over

even bottom are p2

~, + V25 + By. (~V~) - ~ v2v2| ffi 0(82) l (2.2)

~2 +-~8(Vq~) 2 0(~2) !

~, - ~-V2~, + ~ ffi

where oxyz is a motionless rectangular coordinate system in which z-- 0 lies on the quiescent free

surface and the axis x is fixed at the longitudinal axis of the body; V = ( ~-~, ~ ) ; ~x ay V~ denotes the

velocity on the bottom; ~ denotes the elevation of free surface. The transformation relations between dimensionless variables and corresponding dimensional variables denoted by superscript "*" are

(x,y) fk* (x* ,y* ) z = z * / h * t=k*(g*h*) 1/2t* /

A* I (2.3) = ~*/A* ~ = ~ * / [ ~ - , - ~ (g'h*) '/2]

where g* is the acceleration of gravity. A slender body with a side wall being vertical throughout the water depth stands in the shallow water and the incident shallow water waves propagate to the body in the direction of axis x. On the side wall y - - f • (x), the velocity of fluid satisfies the boundary condition,

Vol.4, No.3 Chen & Liu: Diffraction of a Solitary Wave by a Thin Wedge 203

~y~* /~'~ (z+l)'V' (~_y0*) (~*= -~x g~2 (z + I)'V' a-;~-~ ~ + 0 6 u ' ) z ox Iax (2.4,

Since the slope of the side of the body varies slowly, the order of dr+ (x) is assumed to be �9 dx

O(st/z). The order of the change of the angle of the wave direction is also O(8t/2). Hence we adopt following scales to describe the problem.

~ f f i x - t } x = ax (2.5) Y = el/2y

These scales are same as ones used in [1t] [12]. Substituting differential relations,

Ot O~

= t3--~ + a~-~ (2.6)

0-~ ----" 8~/2

into (2.2), we obtain

- ; r + ~r + 8(2~cx + ~rr + ;~r162 + ;r162 - ~ r 1 6 2 1 6 2 ffi O(8=)

~ -- Oc + ' [ ~ c C ~ + 2(q)c)' ] = O(")

Then we get, from (2.8),

(2.7)

(2.8)

~/6s (2.12) 2 /~ /~

-- ------~-v~ F+ -- ~ - ~ b + 3,/6

Thus the equations (2.9) (2.10) and the boundary condition (2.11) become, respectively, u -- v~ (2.13)

1 u, + uu~ + u ~ + ~vs~ ffi 0 (2.14)

d v~ = v~,-~b+(~) ~ ffi b+(z) for ~>_.0 (2.15)

= @r + 0(8) (2.9)

Adding (2.7) and ~ (2.8) and using (2.9), we have

2Ocx + ~r r "F 3OcOcc + ~ c r = O(8) (2.10)

The approximation of boundary condition (2.4) to the fLrSt order is

a| OY dX cgq + 0(,) Y = F+ (X) for X>_.0 (2.11)

where F+(X) = 81!2f+ (x) In order to get the concise form like the KdV equation in [16], we take the transformation

204 ACTA MECHANICA SINICA 1988

Equation (2.14) may also be written as 1

(u, + uu~ + u~o~)~ + ~ u ~ = 0 (2.16)

which is formally identical with the corresponding equation derived in [13] or 1"14] and is called two- dimensional KdV equation or KP equation. If u is independent of ~, equation (2.16) is reduced to

ordinary KdV equation

u, + uuo + ur = 0 (2.17) For an initial sohtary wave propagating in the direction of T, the solution of solitary wave Qf

(2.17) remains undisturbed by the slender body when ~ ~< ~o = 0, / U •1/2 U

u = Useeh z ~ ~ ) (a - ~- z) T ~< ~o = 0 (2.18) %

where U is the initial wave amplitude of u, a positive real number. The initial condition of the diffraction of solitary wave for (2.13) and (2.14) is the expression (2.18) when r = ~o = 0. For a finite value of ~, u and v in this problem satisfy the followings,

u = 0 at tt--, + o o (2.19)

v = 0 at a ~ + o v i.e. t - - , - o o (2.20)

vx = 0 (when ~ is large enough or at a straight side wall ~ = const) (2.21) For the sake of explicitness, the relations between u, v, o, ~, ~ and corresponding dimensional

variables are listed, by using (2.1)(2.3)(2.5) and (2.12), as follows 2 2 /h* = /(h*(g*h*) in) =

x * / h * - t* ( g * h * ) l n / h * = tr x * / h * = ~/(8(6e) 1/2) (2.22)

y* I h* = ~ I (e6 'n) If a = A~ / h* where A~ denotes the initial wave amplitude of ~*,

2 a = ~ U~ (2.23)

HI. NUMERICAL PROCEDURE

We now solve equations (2.13) and (2.14) with boundary conditions (2.15) (2.19) (2.20) (2.21) and initial condition (2.18) numerically by using finite-difference method for the problem of diffraction of solitary waves by a thin wedge. Since the wedge is symmetric about axis x or z, the calculation is carried out only in the half region bounded by the oblique side of the wedge ~ ffi T" r

and another straight side ~ = (J=-2)AA, where T -- tg~l/el/2, ~b i is the half apex angle of the wedge, and J~ is a sufficiently large positive integer. Let AA = n" T" Ar where n is a positive integer.

Define a net function as u}k = u((i - 1)AT, q--1)A~, kAa)

i=1 , , . , I J=Yoo), "", Jm k = - K , ".', K (3.1) The other variables also have such defined functions.

(2.13) with the condition (2.20) is approximated by the trapezoid-integration,

1 i V}k" ----" -- h=K2~ ]-(u}~+ 1 + u}h)Aa = ~ + , A a -- ~u i~Aa (3.2)

where

Vol.4, No.3 Chen & Liu: Diffraction of a Solitary Wave by a Thin Wedge 205

= - ,Z = -

(2.14) with boundary conditions (2.15) (2.21) and (2.19) is calculated with a fractional step 1

algorithm which separates (2.14) into two parts, u~ + uu# + u,~,e = 0 and u~ + ~ v ~ = O. For first

1 1 2 half step, the Crank-Nicolson scheme is constructed for u, + -~ uu~, + ~(u )~ + ur = 0 according

to the suggestion in [17].

l l l 1

+ -: + .:+1)(~+, - ~_,)/(2A~)] 1 + ~(--~-~p [ ( ~ + ~ - 2 ~ + , + 2 ~ _ ~ - ~ - 2 ) . J

+ (4+~ - 2 . :+ , + 2,4_, - ,4-~)3 = o

J = jro,o, "", j r . ~ = - K + 2 , ..., K - 2 (3.~) where the subscript j is omitted; ~ is the intermediate value of u. The boundary condition (2.19) is approximated as

/ U \x/2 U

J = Jro(o, "", jr. k --- - K , - - K + I , K - l , K

For another half step, the full "~mplicit scheme is constructed for ut + I O, vAA Z

(3.5)

Form (2,21),

(u ~+1-~)IAz+ 1 Ae J+,

1 AO" |+, o.i+t _~+t x f~ - ~. (A,;L)2 ( u ) + . ~ . . , . . .~ + u j _ . . ~ f f i , ,

j = I~,+,~ + 1, ..., 1=-I k= -K, ..., K !3.6)

+, _ +t -~+' ~,,~+' kffi-K, K (3.7)

Because the oblique side a-- TT is not right at a node, the Taylor expansion of (2.15) in powers of

RA~ at the node Jo(~+,) + 1 being nearest to the side is taken as

v~ + RA,t v~ = T(v. + RA,tvo~) + O ( R ~ ) 2 ;t ffi Jo ,+ .~A~. (3 .8)

where RA~ is the displacement from the side to the node ]o~l+ ,) + 1 in direction of 4; IRI ~< 1/2. Let 1 1

v. ffi ~ (v~+, - vJ ffi ~ (u,+, + u~)

Thus the difference scheme of (3.8) is

~ [ ( v ~ + , v~_I)/2 + R(v j+ , + v j . , - - 2v~)]~+,'+'

AG 2A~ [ (u~+, - u~_,)/2 + R(u#+, + uj-1 - 2u~)]: +'

206 A C T A M E C H A N I C A S IN ICA 1988

1 R 1 R ,'11+ 1 = ~ T[u~ + ~(uj§ -- uj-l)]~ +1 + ~ TEuj + ~(uj+l - uj-1H~+l

J ffi You+ 1~+ 1 (3.9) ~ 0 itl+ 1 ~ �9 . In above equation, the terms denoted by subscript k + I are known. ~oo + 1 ).k can De expressea m

terms of u t+ 1 and u ~+ 1 JO(l+ 1)+ 1 'k Jo(i+ 1)+ 2 'k" The initial condition (2.18)is written as

~_ U\x/2 ]

The numerical procedure is conducted step by step along with the increase of/. When i = 1, u~ is determined by (3.10). At the step of i+1 , assume u~t to be known. For every f'Lxed j, (3.4) constitutes a linear algebraic equations of a quin-diagonal matrix for the unknown ujh. Substituting (3.5) into (3.4), the equations for.ujt are solved out with double sweep method of a quin-diagonal matrix. For every fLxed k, (3.6) constitutes a linear algebraic equations ofa tri-diagonal matrix for the unknown u~ 1. Substituting (3.7) and (3.9) into (3.6), the equations~for uJ~" 1 are solved out with the double sweep method ofa tri-diagonal matrix. Repeating the above procedure, all uJ~ (i-- 1, "", I) are obtained.

The full implicit scheme (3.6)is originally unconditionally stable but the introduction of boundary condition (3.9) may arouse instability of (3.6). Analysis of boundary stability of the scheme is difficult. Here we only give a stability condition, obtained from numerical experiments of solving (3.6) with (3.7) and (3.9),

In actual calculation, take U= 12, A~=0.15. For the convenence of transforming data of u~ into that in sequence of time, take Az -- 8A~r, where 8 = 3~ / (2 U). In order to satisfy the condition (3.11), AI -- n" T" Az, where n is a suitable positive integer.

IV. NUMERICAL BESULTS Calculations are made for g ffi0.05, 0.1 and 0.15. The results are compared with Funakoshi's

numerical ones and" Melville's experimental ones. Define the coordinate system used by Melville Is3 as ox~y*, shown in Fig.1. The relation

between ox*y* and ox*y* is

z~ = z* cosr + y ' s ina i / y* --x 's inai + y*cos~l , (4.1)

l l l [ l l l l J I 1 I ~ J I

/ l I I 0 X "

Fig.1. Coordinate systems and the Mach-reflexion pattern, where o~uy~ and o x * f are used in ['8] and in this paper

respectively, I is the incident wave, R is the reflected wave and M i s the Maeh stem.

Vol.4, No.3 Chen & Liu: Dif fract ion of a Sol i tary Wave by a Thin Wedge 207

Fig.2.

F~.3.

2A.

~%,. 2.c

, ' ~ ! F . I ~ . _ , , , , l I I . I . I , I �9 ""r0 ..~ 40 60 80 I00 120 140 160 180

Time evolution of maximum run-up at the wall for a :0.06. Results in this paper: - - , ~ ' : ~ / 4 0 , ~/20, ~/13.

Results in Funakoshi's calculation[9]: x , ~ ,=~ /40 ; -, ~,=~/20;, + , ~ j=a /13 .

2 . 0

1"9 f 1.8 1.7 ~ 2 0 1.61 ~

1.4 15 ~

1 . 2 1 0 ~

I . ] ~ e ~ L " ' " ~ ' o �9 �9 - ~ f .

1 . 0 i i I i l , I I i !

0 1O 2 0 3

x ~ l h "

Mazimum run-up at the wall varying with ~ / h* for a : 0.1. Results in this paper: - - , ~, -- 10 ~ 15 ~ 20 ~ . Results

in Melville's experimentlsl: @, ~i : 10~ [], ' ~ i : 15~ A, ~kl = 20 ~

~.4.

~ } . 0 -

I . g ~-- - o

!.4 ~ x i ~.. II Ko

1,3 x

l . o . . . . ' , : , . I I I | i | I I I I I " I I 1 J" I i 1

5 10 15 L~

Distribution of maximum elevation of waves at the enaJs-section ~ / k = 16.7 for a--0.15. Results in this paper:

, ~i--I0 ~ 20~ 30~ Results in Melville's experimenttS]: + , ~ j : I0~ x , ~t = 20~ ", ~ i : 30~

208 ACTA MECHANICA SINICA 1988

The comparision of max imum run-up at the wall uw with Funakoshi 's results is shown in Fig.2,

for ~ = 0.05 and ~i = ~ / 40, lr / 20, ~r / 13. T h e y are rather accordant with each other. But when ~i

increases, the discrepancy between them increases accordingly. The reason for this is thought to be

that the model in this paper is suitable only for ~i "" O(dl/Z) �9 For ~ =0.1 and ~ = 10 ~ 15 ~ 20 ~ the

maximum run-up uw in our results is compared with Melville's one, as shown in Fig.3. Our

numerical values of uw are rather larger than the experimental ones. Morever, the numerical values

still increase monotonically with x~/h* even when the experimental ones tend to a constant

approximately. This discrepancy is same as that between Funakoshi 's and Melville's.

Let 's turn to investigate the distribution of maximum elevation of waves at a cross-section.

Fig.4 shows the distributions of the max imum elevation of our results and Melville's at the cross-

section x~r / h* = 16.7 for ~ = 0.15 and ~l = 10~ 20~ 30~ Near the wall (y* / h* --- 0), the values of our

(a) t*~/g*/h* = 0 (b) t*~/g*/h*ffi8.94

the Mach s t e m

(c) t*,/g* / h* = 17.s9 (d) t* , /g ' /h* = 26.83

rig.5.

e depressmn

re) t* , / g / h* =35.7s

Evolution of diffraction of a solitary wave by a thin wedge for ~t =0.15 and ~1=20 ~

Vol.4, No.3 Chen & Liu: Diffraction of a Solitary Wave by a Thin Wedge 209

results are larger than those of Melville's, but beyond a distance away from the wall, the

experimental values are larger than those of ours.

We consider that the discrepancy between the numerical calculations and the experiment is

mainly due to the effect of viscous boundary layer on the oblique side wall, This viscous boundary

layer can attenuate the run-up at the wall owing to the fact that the velocity of fluid arrives at the

largest value near the wall where the largest difference of the maximum ekvation between

calculations and the experiment takes place? as shown in Fig.4. On the other hand, the thickness of

this boundary layer can make the maximum elevation beyond a distance away from the wall in the

experiment larger than that in calculations. Certainly, the boundary layer on the bottom can also

affect the attenuation of waves. But this effect is so small as to be negligible, because the

experiment lsl shows that the amplitudes of incident waves have no obvious changes as the waves are

propagating through a distance.

In this paper, original calculated data are transformed into data in sequence of time, which are

displayed step-by-step with a series of 3-D graphs to simulate the actual processes visually. The five

moments of the case of a=0.15, ~ i=20~ are selected to be shown in Fig.5 a e. From these 3-D

graphs, we can clearly see the development of the Mach stem and the reflected wave. In Fig.5 d, e we

indicate Mach stem, reflected crest, incident crest and others. Because the angle o f incidence is

small, the reflected wave in Fig.5 d, e is weak. I f the angle becomes smaller, the reflected crest will

disappear, leaving apparently only the incident crest and the stem. Behind Mach stem, there is a

depression where the elevation is less than zero. This depression also exists in the Melville's

experiment (see Fig.9 in [8]) and the Funakoshi's calculation [91. In order to investigate the nature of

Mach stem further, 3-D grephs of another view angle are used to display the wave profiles at two

moments t*(g*h*) t/2 =22.36 and 26.83 in the same process as in Fig.5, shown in Fig.6a, b. It is

shown that the crest line of the Mach stem tends to be perpendicular to the wall side and the stem

propagates along the side.

(a) t* ~S"/h* =22.~ 0~) t*,/g*lh*=26.83 Fig.6. Evolution of the diffraction for a = 0.15 and ~l = 20~ in stereographs of another view angle.

V. CONCLUSION

The two-dimensional KdV equation and its boundary conditions describing the diffraction of

Boussinesq shallow water waves by a stender body with a vertical wall are derived in this paper. The

problem of diffraction of solitary waves by a thin wedge (the Mach reflexion) is numerically solved

with finite-difference method. The calculated results are compared with results o f Melville and

Funakoshi. It is shown that the two-dimensional KdV equation and its boundary conditions are

appropriate to describe the problem and the numerical procedure is suitable and concise. If we

expect to carry out numerical simulation of Maeh reflexion more accurately, the mathematical

model must be improved by taking account of the effect of viscous boundary layer on the vertical

210 ACTA MECHANICA SINICA lg88

side wall.

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