On some solitary and cnoidal wave diffraction solutions of the Gree-Naghdi equations

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Applied Ocean Researcli 47 (2014) 1 2 5 - 1 3 7

E L S E V I E R

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Applied Ocean Research

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A P P L IE D

O C E A N E

R E S E A R C H

On some solitary and cnoidal wave diffraction solutions of the

Green-Naghdi equations

R. Cengiz Ertekin^'*, Masoud Hayatdavoodi^ Jang Whan Kim''

= Dept. of Ocean and Resources Engineering. SOEST, University ofHawai'i at Manoa, 2540 Dole St., Holmes Hall 402, Honolulu, HI 96822, USA »Teclmip USA I / 700 Old Katy Rd., Suite ISO, Houston, TX 77079, USA

CrossMark A R T I C L E I N F O Article tiistory: Received 19 October 2013 Received i n revised f o r m 17 A p r i l 2014 Accepted 21 A p r i l 2014

Available online 20 May 2014 Keywords: Green-Naghdi equations Solitary wave Cnoidal wave Wave d i f f r a c t i o n Soliton fission A B S T R A C T

Nonlinear waves of the solitary and cnoidal types are studied in constant and variable water depths by use of the Irrotational Green-Naghdi (IGN) equations of different levels and the original Green-Naghdi (GN) equations (Level I). These equations, especially the IGN equations, have been established more recently than the classical water wave equations, therefore, only a handful applications of the equations are available. Moreover, their accuracies and the conditions under which they are applicable need to be studied. As a result, we consider a number of surface wave propagation and scattering problems that include soliton propagation and fission over a bump and onto a shelf, colliding solitons, soliton generation by an initial mound of water and diffraction of cnoidal waves due to a submerged bottom shelf, and compare the predictions with experimental data when available.

1. Introduction

Propagation of unidirectional long water waves is classically approximated assuming that the fluid is incompressible, homoge-nous and inviscid, and the f l o w is irrotational. In most of the problems, however, the resultant governing equations are sub-jected to the nonlinear boundary conditions on the unknown

free surface. Typically, an approximation of the solution to the equations is made by introducing some dimensionless parame-ters that are small, and expanding the boundary conditions and governing equations based on these parameters. As a result, the boundary conditions are approximated and the conservation laws are satisfied up to the order at which the expansions are consid-ered.

Beginning w i t h the last three decades of the 20th century, an attempt was made to approach the subject of water wave propa-gation in sheet-like fluids f r o m a different point of view, namely by use of the theory of directed fluid sheets or the Green-Naghdi theory. Unlike the classical approximations, the method does not require a priori assumptions on any scaling parameter nor f o l -lows a perturbation expansion; irrotationality of the flow is also not necessary. The result is a set of equations that satisfy the boundary conditions and the postulated integral conservation laws exactly.

* Corresponding author. Tel.: +1 8089566818; fax: +1 8089563498. E-mail address: ertekin@hawaii.edu (R.C. Ertekin).

The theory has its roots in the theory of plates and shells i n structural mechanics. A general theory of fluid sheets has been pre-sented by Green and Naghdi [13] for any type of homogenous and incompressible medium, i.e., viscous or inviscid fluid. The direct theory is based on a continuum model, namely directed or Cosserat surface, that is a deformable surface embedded in a Euclidean threedimensional space to every point of which a deformable vector -not necessarily along the normal to the surface - called a direc-tor, is assigned. The Cosserat surface, although three-dimensional i n character, only depends on two space dimensions and time, and it includes ƒ<• directors dj, d 2 d/f./C number of the directors define the Level of the theory. For the application to water waves (see [14], for instance), these directors prescribe the variation of the vertical component of the three-dimensional velocity along the water column. In Level 1, i.e., /<•= 1, the deformable medium is a body of sheet-like fluid consisting of a free surface and a sin-gle director attached to each point of the surface. Adopting the Cosserat surfaces, the mass, momentum and angular momentum of the deformable sheet-like (or shell-like) body are expressed i n such a way that a general set of equations of motion of the medium can be obtained.

Green and Naghdi [11 ] had shown that the Green-Naghdi equa-tions (the GN equaequa-tions, hereafter) result i n the same solitary wave solution as that attributed by Lamb [20] to Boussinesq and Rayleigh. They also showed that the equations of motion derived by the direct approach can be reduced to the KdV equations, prescribing appro-priate perturbation parameters. Ertekin [5] also showed, using a formal expansion parameter, that the GN equations reduce to the

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1 2 6 R.C. Ertekin etal./Applied Ocean Researci} 47(2014) 125-737

equations given by W u [29] and Boussinesq [2], for a flat seafloor, once the second and higher orders of the perturbation series are omitted. Choi and Camassa [3] used a scaling parameter to expand the Euler equations for internal jvaves, and have shown that in the shallow water limit (where fluid thickness is much smaller than the characteristic wavelength), their equations are equivalent to the GN equations for an inviscid, incompressible and homogenous fluid.

Although the GN equations are originally derived via a direct approach based on the Cosserat surfaces, over time, different meth-ods have been adopted to derive the same equations, usually w i t h the expense of some extra assumptions about the physical f l o w characteristics. For a special type of fluid sheet, that is incompress-ible and inviscid fluid. Green and Naghdi [12] showed that i t is possible to derive the partial differential equations in a systematic way from the exact three-dimensional equations of an incompress-ible, inviscid fluid by use of only a single approximation for the (three-dimensional) velocity field v*. The assumption is equivalent to the Level 1 assumption i n the direct approach, that is the vertical component of the velocity field is a linear function of the vertical coordinate ( i n an Eulerian system) and that the horizontal compo-nents are invariable in the vertical direction (see also, [6,23]). Such a velocity field allows for rotational flow on the horizontal surface, and the vorticity component on the horizontal plane does not need to be zero even though the shear flow on the vertical surfaces are ignored.

IVIiles and Salmon [22] have used Hamilton's principal to derive the Level 1 GN equations for an inviscid and homogeneous fluid in a water of variable depth i n a Lagrangian form. They concluded that, for the case of flat seafloor, the Level 1 GN equations reduce to the Boussinesq equations i n which dispersion (but not nonlin-earity) is assumed to be weak. Later, Kim and Ertekin [18] and Kim et al. [16] used Hamilton's principle in Eulerian form to derive the irrotational version of the GN equations (IGN equations, here-after) to any Level K. Kim et al. [16] also showed that GN equations of any Level K can be derived f r o m the principle of virtual work. They also showed that the GN Level 1 equations are identical to the IGN Level 1 equations, if a flow starts f r o m a state of rest, even though no assumption of irrotationality is made to derive the equa-tions.

Shields and Webster [26] derived the classical GN equations for the motion of an inviscid fluid, for any number of directors K, using the Kantorovich method. They assumed a solution for the verti-cal velocity i n which a finite power-series form the variation of the velocity field in the vertical direction. Demirbilek and Webster [4] applied the Level 11 GN equations to some coastal-engineering problems. Recently, Zhao et al. [31] has applied the high-level GN equations derived by Webster et al. [27] w h o also used a power-series expansion of the particle velocities along the water column and made the high-level GN equations, involving lengthy and com-plicated equations, manageable.

Zhang et al. [30] followed the same assumption for the three-dimensional velocity field as that used in the derivation of the GN equations f r o m the exact three-dimensional equations of an incompressible, inviscid fluid given by Green and Naghdi [12], that is they prescribed a certain distribution for the vertical velocity component, and derived a set of nonlinear equations based on the scaling assumptions used i n Boussinesq-class equations. Under such assumptions, they were able to release the irrotationality con-dition used i n the Boussinesq equations. The final equations that they named Boussinesq-Green-Naghdi rotational wave equations, differ from the GN equations i n that the boundary conditions and conservation laws depend on the expansion level of the scaling parameter and thus are approximated.

In a study by Shields and Webster [26], convergence of the solitary and periodic solutions of the first three Levels of the GN

equations were examined and results were compared w i t h exact numerical solutions and other classical solutions. They concluded that the best convergence of the Level 1 results to higher Level results of the theory is achieved for flows around the critical speed. In a different study, Kim and Ertekin [18] evaluated the disper-sion relation of periodic solution of the first three Levels of the GN equations, along w i t h different Levels of IGN equations. It was shown that results of IGN equations are converged at Levels 111 or IV, depending on the problem.

Since no perturbation or scaling parameters are used in the derivation of the GN or IGN equations, there is no theoretical restriction on the l i m i t to which the theory is applicable. The only assumption made about the kinematics of the fluid flow is the distribution of the velocity field i n the vertical direction, which is varied w i t h the level of the theory. Therefore, applicability of different levels of the GN and IGN equations is to be discovered by numerical experiments. In this work, we shall make a direct comparison of the results of the basic GN equations (Level 1) and different levels of the IGN equations by applying the equations to a number of nonlinear wave diffraction problems concerning solitary and cnoidal waves. The main motivation for this work is then to compare the results obtained by different sets of equa-tions and w i t h the experimental data to assess the applicability of the GN Level 1 equations and IGN equations of different lev-els.

In Section 2 we w i l l review the basic idea and assumptions i n deriving the GN Level 1 equations via a direct method, also known as the restricted theory, in which the directors are assumed to be normal to the surface. The IGN equations, that are for an irrota-tional flow, are given in Section 3. The rest of the work (Sections 4 and 5) is mainly concerned w i t h the application of the equa-tions discussed i n Secequa-tions 2 and 3 to a number of nonlinear and unsteady problems of fluid sheets where solitary and cnoidal waves are used.

All of the cases considered here are two-dimensional. When referring to the interaction of t w o (or more) solitary waves, the waves are referred to as solitons.

2. The Level I Green-Naghdi equations

We use a rectangular Cartesian coordinate system (x,y, z) w i t h associated orthonormal base vectors e,-. In three-dimensions, the index i has a range of 1-3, and Greek subscripts take the value of 1, 2, and all lower case Latin subscripts designate partial differen-tiation w i t h respect to the indicated variable. The base vector 63 is vertically upward. The coordinate system is chosen such that the x-y plane is the still-water level (SWL). For simplicity, and w i t h i n the context of the application of the theory to water waves, we assume that the fluid has a constat mass density p* and is subject to the gravitational acceleration g i n the direction - 63. The fluid sheet is bounded below by the surface

z = a{x,y,t). (1)

Let us assume that the bottom surface is stationary, that is for all values of t

ff(x,y, t) = a ( x , y ) . (2) The normal pressure on this surface is p = p(x, y, t). The fluid sheet

is bounded above by the surface

z = fi{x,y,t), (3) and pressure on this surface is p = p(x, y , t) where

p { x , y , t ) = p Q - q { x , y , t ) , (4) and where po is the constant atmospheric pressure and q is the

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R.C. Ertekin et al. / Applied Ocean Research 47(2014) 125-137 1 2 7

[4) is a statement of tlie exact dynamic free surface condition. Let V* = ViCi be the fluid velocity at time t. For an incompressible fluid, we have the continuity equation

Vi,i = 0. (5) The following kinematic boundary conditions are to be enforced on

the surfaces given by (1) and (3):

dt dx da da V2 ys = 0 on z : dy f 3 = 0 on 2 (6a) (6b) Further, we define the thickness of the fluid sheet and its midplane by

li + a

e[x,y,t) = li-a, S ( x , y , t ) :

₍₇₎

Note that, up to this point, no assumption about the dynamics of the fluid sheet is made, and the boundary conditions (4) and (2) are exact. Here, we make a single assumption that the directors remain perpendicular to the horizontal surface (but may deform along their lengths), which results in the Level 1 equations. Such an assumption is equivalent to the vertical component of velocity being linear i n z, and consequently, the incompressibility condi-tion (5) is satisfied exactly as long as the horizontal components of velocity are independent of z.

The velocity v* of the middle surface z=5(x, y, t) and the director velocity take the following forms:

v* = ue-i + ve2 + >-e2, w = \ve2, where

u=k, v = y, X = S, w = 6,

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(9) and where superposed dot denotes the material time derivative so that if fix, y, t) is any function of x, y and t, then

f = f t + llf>c + vfy- (10) A double superposed dot denotes the second material time deriva-tive. From (8) the fluid acceleration v* and director acceleration w follows:

V* = iie-[ + ve2 + ics, w = wej. ₍₁₁₎ In the final step, the integral balance equations of mass conserva-tion, momentum, moment of momentum (or director momentum) and energy can be postulated w i t h no further assumptions [13], Such restricted model results i n the equations of incompressibility and the motion of the fluid via a direct approach. For an inviscid fluid, they are given as

0{iix + Uy) + w = 0, p'Oii = -px + PPx - pax, P*ei> = -Py + pPy - pay, p"OX = -p +p - gp*6,

i p * ö w = H - i ( p + p ) , (12) where p is the integrated pressure, per unit area of the middle

surface z = S, defined by

P = p*dz, (13)

and p*(x, y, z, t) is the three-dimensional pressure. Ertekin [5] obtained rather a classical f o r m of the governing equations (12) as

^t + V.{(li + ^-a)V] = 0, (14a)

H ^ = - l { [ 2 f

+

« ] , ö + [ 4 f - « a

+ {li + ^-a)[ö! + 2l]x}, (14b)

v + g^y + ^ = - l m + a]ya+m-a]yl

+ ih + ^ - a ) [ a + 2l] }, (14c)

where /i is the water depth, f is the free surface elevation measured f r o m the SWL, V = uei -i- ve2 is the two-dimensional velocity, and V is the gradient vector operator, V =(9/9x) ei +(9/9y) 62. Ertekin [5] coined these equations the Green-Naghdi equations.

The vertical component of particle velocity is then w r i t t e n as [6] z - a

^ li + ^ --ai^-') (15)

An analytic solitary wave solution of the GN Level I equations (2) can be found i n Ertekin [5], who has studied a number of constrained domain problems in shallow water involving solitons. The solution is given by

f ( x ) = A s e c r 3A

4lil{ho+A) (16)

where A is the amplitude of the solitary wave measured f r o m the SWL and is given by

A = ^ - h o , (17)

where U is the speed (critical or supercritical) of the wave, ho is the constant water depth and x = x XQ Ut, where x q is the m i d -point of the solitary wave at time t = 0. Periodic shallow-water wave solutions, i.e., cnoidal waves, can only be obtained in a semi-closed f o r m when the water depth is constant at the vicinity of the wave-maker. Such a solution of (2) is given by Ertekin and Becker [7] i n two-dimensional Galilean coordinates.

The nonlinear GN Level 1 equations, (14a) and (14b) for the t w o -dimensional case, are solved in this work. At a given time, Eq. (14a) can be solved explicitly for f t once u is known. Next, f t can be sub-stituted into (14b) to solve for Ut. We note that i n solving the GN Level 1 equations, (2), we have used the dimensionless form of the equations by use of the dimensionally independent set of p*,g and ho.

The final equations are solved numerically by the central-difference method, second-order accurate i n space, and w i t h the Modified Euler Method for time integration. We approximate the continuous variables f(x, f ) and u(x, t) by the discrete variables f ( i , n) and u(!, n) where i is the mesh point i n the spatial domain and

n indicates a mesh point on the time axis. At each given time, such

discretization results i n an NT x NT coefficient matrix, where Nj is the total number of mesh points i n the computational domain. In a two-dimensional domain, w h i c h is our interest in this study, the coefficient matrix is a tridiagonal one and can be solved exactiy by the Thomas algorithm (see e.g., [1]), which eliminates any matrix operations and thus is a very efficient method. A convergence test of the solution of the GN Level 1 equations can be found i n Ertekin et al.[8].

We recall that the GN Level 1 equations satisfy the free-surface and bottom boundary conditions (2) exactiy. As such, we are basi-cally dealing w i t h an initial-value problem, that is only the initial value of the variables ( f and u i n two-dimensions) should be spec-ified except at the two ends of the domain. This is accomplished by the numerical wavemaker located at one end of the domain (upwave end), which is capable df creating solitary or cnoidal

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1 2 8 R.C. Ertekin et al. / Applied Ocean Research 47(2074) 125-137

waves. To minimize tlie size of the computational domain, how-ever, we have to predict the values of the variables at the downwave end (open boundary) of the numerical domain. Previous works [28,5] have shown that the relatively simple Orianski's condition w i t h constant phase speed c = ±^/gh^, where HQ is the water depth at the location where the open-boundary condition is being applied, prevents significant reflections f r o m the open-boundary. We also use this open-boundary condition here which reads

at + ci2x = 0, (18)

where Q may be f or u.

3. Irrotational Green-Naghdi equations

The irrotational version of the Green-Naghdi equations can be derived f r o m Hamilton's principle, as shown by K i m et al. [16], We seek for the solution i n a vector function space that satisfies kinematic constraints of divergence free i n the fluid domain and no-leak condition on the sea floor. In two dimensions, the velocity field (u, w ) that satisfies the kinematic constraints are given by the stream function !/f(x, z, t), i.e..

u(x,z, t) = dif{x,z, t)

dz ' W ( X , Z , f ) :

9iA(x,z, t)

9x ' fix, a, t) = 0. (19) In Level KIGN equations, the stream function is given by odd-order polynomials in a normalized vertical coordinate y:

iA(x, z, t) = Y^i^mix, t)MY), m •- i , 2 , . . . , / < : ,

m = l where z o -Y I - a (20) (21)

For the restricted case, where the lower boundary of the fluid domain is stationary, i.e.,

^(x, f) = -/i(x), (22) and where we have constant pressure on the free surface,

p ( x , f ) = p o = 0 , (23) (the atmospheric pressure is set to zero without loss of generality),

the IGN equations are given by t w o canonical equations^for the free surface elevation f (x, t) and surface velocity potential 4>ix, t), f t +

«Ax =

0,

X d dT dT ^

' ^ ' ^ - T x W . ^ d ^ - ' ^ '

(24a)

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where f = tj/{x, f, t) is the stream function on the free surface and

T is the kinetic energy given by

1 171=1 n = l +2 | - ( f -I- h\Bl„ + hxBmn]i^mxfn + ^ ( ( l + ''x) Qnn - 2 ( f + h)xhxCl^n + + h x f c l , } f„,i,n] , (25) where ^ 1 1 1 = / fni[y)fn{y)dY, Jo, Bmn UY)f;MdY, Yfm{y)fn(Y)dY, mn — I Jo (26)

C m n = / aY)mY)dy, Ci„= YUY)my)dy,

•^°i

C^n = ƒ y^UY)f,[{Y)dY.

The IGN equations are completed by the relation between the sur-face potential ^(x, t) and stream function, which is given by

9 9 r ^ 9 r . . . . .

+ T T T — = / m ( l ) 0 x

-(27) 9x d f m , dfrr

As numerical implementation, the canonical equations (3) are solved by the Runge-Kutta 4th-order method after discretizing 0 and f by the finite-element method. After the discretization, Eq. (27) becomes algebraic equations for the discretized ij/m and 0, which are solved by a banded-matrix solver. Details of the numer-ical scheme can be found i n Kim et al. [17,19].

4. Solitary wave

We have selected several numerical experiments on the diffrac-tion of solitary waves when we used the GN Level 1 and IGN equations. Comparisons are made w i t h other numerical solutions and physical experiments when available. The cases studied here are as follows. (1) A solitary wave propagating f r o m deep water to shallow water (and vice versa), w i t h a linear transition between the two depths. (2) A solitary wave propagating over a submerged curved bump. (3) Reflection of a soliton f r o m a vertical wall. (4) Interaction (collision and overtaking) of t w o solitons over a flat seafloor. (5) Waves produced by an initial mound of water (dam break problem). In all the solitary wave cases, the GN Level I and the IGN Level III equations are solved. Due to the absence of perturba-tion terms or assumpperturba-tions on certain scaling factors, applicaperturba-tion of the GN and IGN equations to different problems should be assessed computationally.

Here, we make an initial assumption that the IGN Level 111 equations are the converged solution w i t h i n the IGN models. This assumption is made partially based on the study by K i m and Ertekin [18]. If there is disagreement between the GN Level I and IGN Level III results, then different levels of the IGN equa-tions, including lower and higher levels (until convergence is achieved) are studied. Kim et al. [16] have shown that at the flrst level, IGN equations are identical to the GN Level I equa-tions i f a motion starts f r o m a state of rest. Nevertheless, i n the following test cases, and when different levels of the IGN equa-tions are presented, results of the IGN Level I equaequa-tions are also given.

We note that by convergence of different Levels of the IGN equations, we refer to the convergence between interpolation func-tions of different levels used to interpolate the velocity field i n the vertical direction, i.e., w h e n converged, higher order of the interpo-lation functions do not provide any better description of the fluid flow.

Next, the numerical solution of these problems are discussed separately.

4. ?. Solitary wave propagating over a submerged shelf

Propagation of a solitary wave over a submerged shelf is studied by use of the GN Level I and IGN equations of different

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R.C Ertekin et al./Applieti Ocean Research 47 (2014) 125-137 1 2 9

s

O Pi '

.2

I

t o -Gauge! — G N L I - - -IGN L III ° Experimental (Goring 1979) -Gauge! 10 •Gauge 3 = 3:56 n •Gauge 3 = 3:56 n /p \ fb Time (s)

Fig. 1 . Solitary wave propagating f r o m deep to shallow w a t e r over a linear slope, GN Level I (solid line) and IGN Level III (dashed line) vs. laboratory experimental o f Goring |101 (circles). Gauge 1 is located at the b o t t o m o f the slope and Gauge 2 is on top o f the slope. The schematic of the wave t a n k is not t o scale.

levels. The equations are used to make a comparison w i t h the laboratory experiments of Goring [10]. In the laboratory experi-ments, the deep-water depth {/iq = 31.08 cm), shallow-water depth (/ii = 15.54cm) and incident-wave height (>l = 3.1cm) are kept constant, while the length of the submerged shelf (L) is varied. Transmission of the solitary wave over the slope recorded by three wave gauges (L = 300cm case) is shown i n Fig. 1. Excellent

agreement between the GN Level I and IGN Level 111 equations w i t h the laboratory measurements is observed.

Next, we compare the results of the GN Level 1 and IGN equations. The comparisons for t w o solitary wave amplitudes (7l//io =0.12 and 0.3) are shown i n Figs. 2 and 3. All the numbers in these figures are dimensionless w i t h respect to the deep water depth (ho). The length of the shelf is kept constant (L/ho = 10) and

0.2 0.1 0 - 0 , 1 , hGauge 1 k 3 6 ' - G N L I • I G N L f f l 2 0 30 40 50 60 7 0 80 90 100 '—' 0.2 a o 0.1 0.1 E 0 0) O - O . l J urf a - O . l J

OT

I \ I ! I 1 i i i i i i 10 20 30 40 60 90 100 I 1_

Fig. 2. GN Level I vs. IGN Level I starts atxlho=40.

10 2 0 30 40 50 60 70 80 90 100 tylgjhil

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130 R.C. Ertekin et al. /Applied Ocean Research 47(2014) 125-137 0.6 0.4 0.2 0 1 Gauge 1 - 36 1 G N L I •—•IGN L I •••••••IGNLII - - - I G N L I I I G N L I •—•IGN L I •••••••IGNLII - - - I G N L I I I

/ V ;

G N L I •—•IGN L I •••••••IGNLII - - - I G N L I I I 1 — IGN L I V 1 ! 1 1 Gauge 3 = 96 , L

_ 1 :

1 :

^ ft A

I I I I 0.4 0.2 0 0,4 0,2 0 40 60 100 120 140 1 Gauge 4 = 125

j t i ^ . . i : :

i

1 1 1 1 1 20 40 80 100 120 140

Fig. 3. GN Level 1 vs. IGN equations, solitary wave propagating f r o m s h a l l o w (ho) to deep ( h , ) w a t e r over a slope,/\/ho = 0.3, h,/ho = 0.451, L/ho = 10. A t t i m e t = 0 , the s o l i t o n is at xjho = 30. The slope starts at xjho = 4 0 . IGN L I I and III plots cannot be seen as t h e y are under the plot o f Level IV results. The d i f f r a c t e d w a v e calculated by the IGN Level I is almost identical to that o f the GN Level L

it Starts atx/fto =40. IGN equations of Levels I , II, III and IV are used in Fig. 3, and results of these different levels of IGN equations con-verge starting f r o m Level III. As the solitary wave propagates onto the shelf, the front of the wave begins to steepen, indicating the strengthening of nonlinear effects compared w i t h the dispersive effects. It then splits up into a number of smaller amplitude soli-tons. For the relatively large ratio of A//ii(= 0.665) shown i n Fig. 3, although the tail waves are in close agreement, the IGN Level III equations predict a higher amplitude wave (and therefore a faster-moving one) compared w i t h the one predicted by the GN Level I equations. A complete list of different cases of solitary wave prop-agating over a submerged shelf by use of the Level 1 equations is given by Ertekin and Wehausen [9],

In this study, the IGN equations are solved by the Runge-Kutta 4th-order method after discretizing the domain by use of the finite-element method, while the GN Level I equations are solved by a central-difference method, second order accurate i n space, and w i t h the Modified Euler Method for time integration. Such differ-ence between the numerical solutions of the equations leads to a slight difference between the results of the Level 1 GN equations and the Level I IGN equations for large wave amplitudes as seen in Fig. 3.

Comparison of the first (Ai), second (A2) and third (A3) succes-sive solitons w i t h the results of Johnson [15], Schember [25] and Madsen and Mei [21] is given i n Fig. 4. As /ii decreases, the GN Level 1 underestimates the wave amplitude of the first successive wave in comparison to IGN Level III. The second and third successive waves, however, show a better agreement even for small values of h ] . It is of interest that all computational results (GN and oth-ers) are above the experimental results of Madsen and Mei [21 ] for ; i i = 0 . 5 .

Propagation of a solitary wave f r o m shallow to deep water (/ii//io = 2) over a submerged shelf (L/;io = 20) is also studied and the results are shown in Fig. 5. The slope starts f r o m x/ho=40. A close agreement is observed between the IGN Level III and the GN Level I equations.

4.2. Solitary wave propagating over a submerged bump

Diffraction of a solitary wave propagating over a submerged curved-bump is studied by solving the GN Level I and IGN equations. To obtain a continuous Uxxx, w h i c h appears i n the GN momentum equation (2), the profile of the submerged bump is modeled by use of an eight-order polynomial given by

a{x)=fg[4ix'^-fi)Y -ho, -l<x<l, (28)

where L = 2/ is the length (at the bottom of the bump) and B is the height of the bump. A set of wave amplitude (A//10 = 0.6) and bump amplitudes {Bjho = 0.2, 0.6) and a fixed bump length (L/;io = 10) is considered here. Locations of the wave gauges are fixed. Gauge 2

0.25 0.2 <

i

0.1 0.15 \ n 1 ^ 1 n - A , : ^ 1 V — 1 - A ; ) 0 2 D 4 0 6 - • G N L I - I G N L i n Johnson (1972) . v,-Si:llcmber(1982)

• Madsen & Mci (1969); Theory

A Madsen & Mei (1969); Experiment

h,/h<,

Fig. 4. Comparison o f the a m p l i t u d e o f the first ( A i ) , second (A2) and t h i r d ( A 3 ) successive solitons generated due to the propagation o f the solitary wave f r o m deep to shallow water over a linear slope, A/ho = 0.12, L/ho = 10. In the experiments o f Madsen and Mei [21], results are given o n l y for the first t w o successive solitons.

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R.C. Ertekin et al./Applied Ocean Research 47 (2014) 125-137 131 0.3 CJ H 0 CO 0.3 0.2 0.1 0' -0.1 10 : y »\ ; i i Q a u g k 2 = . . 6 f i _

, / '\

i i 1 i i i i 20 40 50 60 90 1 ( ( 1 -i i 1 i i 20 30 40 50 60 70 80 90

Fig. 5. GN Level I vs. IGN Level 111 predictions, soliton passing f r o m shallovi' w a t e r over a submerged shelf to deeper water, /I//10 = 0.3 and hi //lo = 2. A t t i m e t = 0 , the soliton is at x/ho = 30. The slope starts at x\ha = 4 0 .

is on top of the bump. The comparisons are given in Figs. 6 and 7. All numbers i n these figures are dimensionless w i t h respect to the water depth (/iq). In all the cases, amplitude of the main solitary wave is reduced and i t is followed by a train of dispersive waves of decreasing amplitude and wave length. Overall, close agreement is observed, particularly for the smaller A/B ratio. Unlil<e the case of a solitary wave passing a submerged shelf the GN Level 1 results predict a slightly faster moving wave. This is of interest as one would expect a slower-moving wave by the GN Level 1 equations.

as the wave amplitude is slightly smaller. As the bump ampli-tude increases, however, the difference becomes more significant (shown in Fig. 7), particularly in the amplitude of the trailing disper-sive solitons that are underestimated by the GN Level 1 equations compared w i t h the IGN equations of different levels. The differ-ences between the GN and IGN Ll solutions i n the trailing wave region are again due to the use of different numerical method, how-ever, the prediction of the wave profile in the main wave area is very close. The amplitude of the dispersive waves and the reflected

0.6 0.4 0.2 0 I 0.6 0.4 G N L I - - - I G N L I I I G N L I - - - I G N L I I I A \ iauge 1 = 24 1 1 1 1 1 1 1 i 10 20 25 30 35 40 45 50 1 1 ! ! i ! ! ! ll \ 11 \ \ Gauge 2 =i 41.04 i i 1 1 1 1 1 1 & 0.2 K 0 3 10 20 25 30 35 40 45 50 0.6 0.4 0,2 0 T i I 1 1 r 20 25 30 t\fgJho 35 40 45 50 55

Fig. 6. GN Level I vs. IGN Level III, soliton passing over a submerged b u m p . A/ho = 0.6, B/ho = 0.2. Gauge 2 is located o n top o f the b u m p . A t t i m e t = 0, the soliton is located at x/ho = 20,4.

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132 R.C. Ertekin et ai. / Applieti Ocean Research 47(2014) 125-137 0.6 0.4 0.2 0 • :Gaugefl•=24•• - G N L I " I G N L I • I G N L n • I G N L f f l - I G N L I V 0 10 15 20 25 30 35 40 45 50 55 ^ 0 . 6 Ö .3 0.4 0.2 0 0 I I ! • ! I I I I i i i i -^T'"'"'-"' i i i i 10 15 20 25 30 35 40 45 50 55 I I I I l l l l l - \ 1 i i i i i i 0.6 0.4 0.2 0 0 10 15 20 35 40 45 50 55

Fig. 7. GN Level I vs. IGN, s o l i t o n passing a submerged b u m p , Ajho = 0.6, B/ho = 0.6. Gauge 2 is located on t o p o f the b u m p . At t i m e t = 0 , the soliton is located at x/ho=20.4. IGN L11 and III plots cannot be seen as they are under t h e p l o t of Level IV results.

waves seem to depend more on the bump amplitude rather than the initial wave amplitude.

4.3. Reflection of a soliton from a vertical wall

The problem of a soliton impacting a vertical wall is also studied. The reflection of a solitary wave f r o m a vertical wall is equivalent to the case of two solitons of equal amplitudes traveling i n oppo-site directions and colliding. A set of two solitary wave amplitudes

(A/iio = 0.3, 0.6) are considered and the results are shown i n Figs. 8 and 9. There is good agreement between the GN Level 1 and IGN Level 111 results for the smaller amplitude wave case. For the larger amplitude case (shown i n Fig. 9), the higher levels of the IGN equa-tions predict a wave w i t h a higher amplitude at Gauge 3, where the vertical wall is located. IGN Level IV equations predict some oscilla-tions at the location of the wall (Gauge 3 of Fig. 9) immediately after the soliton reflects f r o m the wall. Despite the differences between the results at the location of the wall, the reflected waves, farther

0.6 0.4 0.2 0 ( -0.6 0,4 Gauge 1 = 15.6 : — G N L I • - - I G N L I I I 20 25 30 35 40 ₄₅ 50 1 1 1 ! 1 ! ! 1 Gauge 2 = 30 i i \ \ \ -10 20 30 35 40 45 50 1 _{• / N • - • •} Gaugs ' 3 = 45.6

: .:

0.6 0.4 0.2 0 10 20 25 30 35 40 45 50

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R.C. Ertekin et al./Applieti Ocean Research 47(2014) 125-137 133 " 1 . 5 a .2 1

M

0.5 o 0

OT

l l l l l - Gauge'2'= 30 i ; \ \ 1 ! I i i i 1 1 1 1 1 10 15 20 25 30 35 40 45 50

Fig. 9. Soliton reflecting f r o m a vertical w a l l , GN Level 1 vs. lGN,/l//!o = 0.6. At t i m e t = 0 , the soliton is at Gauge 1. The w a l l is at x//io=45.6. IGN L 111 plot cannot be seen as i t is under the plot o f Level IV.

away from the wall, are in good agreement, see the second wave at Gauge 2 i n Fig. 9.

4.4. Interacting solitons

Two colliding solitons of different amplitudes are modeled i n this section. The amplitude of the right-moving soliton on the left (Ai/ZiQ = 0.6) is six times larger than the amplitude of the left-moving soliton on the right (4r//io =0.1). The results are shown in Fig. 10. A close agreement is observed, although GN Level 1 predicts

a slightly faster-moving soliton at Gauge 3. After passing each other, solitons recover their original form and the agreement is closer for the smaller amplitude wave seen i n the Gauge 3 results.

Another case of a soliton of amplitude Af//!o = 0.6 overtaking another soliton of smaller amplitude, As//io = 0.1 is also studied. The results are shown i n Fig. 11. There is a very close agree-ment between the GN Level 1 and IGN Level 111 results, although GN Level 1 predicts a slightly faster-moving soliton. The smaller-amplitude wave predicted by different models is i n complete agreement. jGauge | = ,2l — G N L I - - - I G N L I I I -0 "5; 25 30 35 40 1 1 _{iGauge 2 = 4 i} \

/ \

• p \ '• /' *\ '•/' A 7 ' ^ \ 1 1 -0.6: 0 4 3 CO 10 20 25 30 35 40 ! ! \ _{b a u g e 3 = 6^} 11 \ / * " / ' - • - ^ ^ ^ ^ • i — ! \ 1 / * 1 0.6 0.4 0.2 0 10 15 25 30 35 40

Fig. 10. T w o c o l l i d i n g solitons of d i f f e r e n t amplitudes, GN Level I vs. IGN Level 111, At/ho = 0.6 and An/ho = 0.1. A t r = 0 , the l e f t s o l i t o n is at Gauge 1, and the right soliton is at Gauges.

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134 R.C. Ertekin et al. / Applied Ocean Research 47 (2014) 125-137

Fig. 11. Soliton o f amplitude /1F//IO = 0.6 o v e r t a k i n g a soliton o f amplitude,/\s/lio = 0.1. Both solitons are m o v i n g i n the same direction. A t t i m e t = 0 , the slovi' soliton is at xjho = 34.92, and the fast one is at x//io = 14.88.

4.5. Initial mound of water

Waves generated by an initial rectangular mound of water is considered i n this section. Initial height of the mound is A/ho = 0.4, and the initial length is L/ho = 12, and at time t = 0 the water is at rest. Calculations start f r o m this state, w i t h o u t the need of any smoothing or filtering of the surface elevation or velocities. As the initial hump of water is released i n Fig. 12 (dam-break problem), a

number of solitons are generated, traveling at supercritical speeds. Although there is a perfect agreement between different models for the first t w o waves at wave Gauge 1, close to the initial location of the mound, there is a disagreement beginning f r o m the third wave where Level 111 results predict faster moving waves. The ampli-tude of waves predicted by both models are i n agreement. Farther downwave, at Gauge 2, a better agreement is achieved, particularly for the first three solitons. The best agreement is seen at Gauge 4,

Fig. 12. Waves generated by an initial m o u n d o f water, GN Level I vs. IGN Level 111, A/iio = 0.4, L//!o = 12. The schematic of the wave tank is not to scale. Location o f the wave gauges are dimensionless w i t h respect to t l i e w a t e r depth.

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R.C. Ertekin et ai. /Applied Ocean Researcli 47(2014) 125-137 135

Table 1

periodic wave conditions used in the experiments.

Case Wave period W a v e length Wave height Phase speed Steepness Ursell n u m b e r

( T y ^ ) (mo) (H//io) ic/^/glk:) (H/X) Ur=.(Hk-')/hl

1 5.94 4.72 0.05 0.795 0.0106 0.236 2 5.95 4.72 0.1 0.793 0.0212 0.472 4 8.92 8.16 0.1 0.915 0.0123 0.816 6 11.88 11.35 0.1 0.956 0.0088 1.135 Incident Wave 205. 1 208. 1 209. 5 II X II II X X J l V W U2 üati4

/ I T

Fig. 13. Cnoidal waves propagating over a submerged shelf, a schematic o f the numerical wave tank and dimensionless ( w i t h respect to w a t e r depth) location of the w a v e gauges.

the farthest gauge downwave, where the solitons are almost fully developed. The number of solitons generated by different models is in excellent agreement.

5. Cnoidal waves

The problem of cnoidal waves passing over a submerged shelf is presented i n this section. GN Level 1 and IGN Levels 1-V equa-tions are used for this study. For this test cases, a five-point formula, similar to that used in Ertekin et al. [8], is adopted to smooth velocity (u) and surface elevation f of the GN Level 1 equations at every five time steps. The results are compared

w i t h the experimental data of Ohyama et al. [24]. Four test cases (wave conditions) are considered and these are shown i n Table 1. The shelf remains unchanged i n all cases. A schematic of the numerical wave tank is shown i n Fig. 13. The results calculated at Gauges 3 and 5 are given in Figs. 14 and 15, respec-tively.

Overall, good agreement w i t h the laboratory measurements is observed, especially by the IGN Levels 111, IV and V equations. In most of the cases, the IGN Level 111 appears to be the converged solu-tion. Shown in Fig. 15, very small difference is observed between IGN Level 111 and Level IV at wave gauge 5, farther downwave. The IGN Level IV, however, is clearly the converged solution, as the plots of Level IV is perfectly under the Level V results i n all cases. GN Level 1 predicts lower trough i n Cases 2 and 4 in comparison to the experimental measurements at Gauge 3. The crest is also underes-timated. IGN Levels 111, IV and V results, on the other hand, are i n better agreement w i t h the experimental data, except that the wave crest is slightly underestimated i n Cases 1, 2 and 4.

Farther downwave, at Gauge 5 in Fig. 15, the GN Level 1 equa-tions seem to miss the higher harmonics generated due to the shelf effects. This becomes more obvious for higher steepness (Cases 2 and 4). IGN Levels 111, IV and V equations, show a very close agree-ment w i t h the laboratory experiagree-ments even for the steepest case (Case 2). The only disagreement i n this case is the slight difference in the phase and amplitude of the higher harmonics. Once again, the small differences between GN Level 1 and IGN Level 1 is due to the difference i n numerical solution of the two set of equations, and the smoothing process used i n the GN Level 1 equations.

! _{C a s e j l} I \ \ ! i i i i i -0, 1, 0, I—t 1 a ! C a s e Ö I I 1 1 I ^ i i i i i i i i i

Fig. 14. Cnoidal waves propagating over a submerged shelf at wave Gauge 3, GN Level 1 and IGN equations vs. laboratory experiments of Ohyama et al. [ 2 4 ] (circles). Refer to Fig. 13 f o r t h e location o f the wave gauges. Wave conditions f o r d i f f e r e n t cases are given i n Table 1. IGN L l l l p l o t is hidden b e h i n d the Level IV and Level V plots most o f the t i m e .

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136 R.C. Ertekin et al. / Applied Ocean Research 47(2014) 125-137

Time [t/T]

Fig. 15. Cnoidal waves propagating over a submerged shelf at w a v e Gauge 5, GN Level 1 and IGN equations vs. laboratory experiments o f Ohyama et al. [241 (circles). Refer to Fig. 13 f o r the location o f the wave gauges. Wave conditions for d i f f e r e n t cases are given i n Table 1. IGN L III plot is hidden behind the Level IV and Level V plots most of the time.

6. Concluding remarks

Our main objective has been to use the GN Level 1 and IGN equations of different levels i n a number of existing numerical and physical experiments and examine their applicability and accuracy. For small amplitude wave interactions, the GN Level 1 and IGN Level III equations are i n very close agreement in all of the cases. We choose IGN Level III as the converged model for the problems that we have considered here. As the nonlineari-ties increase, that is the local wave height becomes large in a given experiment, the GN Level I equations seem to underesti-mate the higher harmonics compared to the higher levels of the IGN equations, although the models are in good qualitative agree-ment overall. Almost i n all of the cases, IGN Level IV results are on top of the Level III results, confirming the convergence. The assumption of the linear change of vertical velocity over the water column made in the GN Level I equations seems to be debatable in these cases. In the case of a solitary wave, we found that the GN Level I equations provide satisfactory results for wave ampli-tudes up to about A//10 f» 0.4. The agreement w i t h the IGN Level III and laboratory experiments appear to diverge as the wave ampli-tude increases to Ajho^O.G or higher. In all solitary wave cases studied here, IGN Level III is the converged level of the IGN equa-tions.

For periodic waves, the dimensionless Ursell number (C/r = HX^/lig) appears to be a better gauge for the applicability of the GN Level I equations. Generally, the GN Level I equations are i n better agreement w i t h the IGN equations and the laboratory exper-iments for smaller values of Ur, that is when the nonlinearity is rather smaller. Since the small differences between the results of the GN Level I and IGN Level I models is due to the differences i n numerical solutions of the two set of equations, and the smoothing process used in the GN Level I equations, it would be necessary to use exactly the same methods of solution i n solving the two sets of equations at even high-levels of these equations in making compar-isons i n the future. The IGN Levels IV and V results are in excellent agreement w i t h the laboratory measurements. In the case of the

periodic wave cases, although the IGN Level III results are i n excel-lent agreement w i t h the IGN Level IV results, we conclude that the Level IV is the converged level of the IGN equations.

Aclcnowledgements

The works of RCE and MH are partially based on funding f r o m the State of Hawaii's Department of Transportation (HDOT) and the Federal Highway Administration (FHWA) through the HDOT Research Branch, Grant numbers DOT-08-004 and TA 2009-1R. This funding to support research on tsunami and hurricane-generated wave loads on coastal bridges is gratefully acknowledged. The authors would like to also thank Professor W.C. Webster of U.C. Berkeley and Dr. BinBin Zhao of Harbin Engineering University for their helpful comments on the presentation of the IGN equations of different levels. The work of M.H. is also supported by the Link Foundation's Ocean Engineering and Instrumentation Fellowship. Any findings and opinions contained i n this paper are those of the authors and do not necessarily reflect the opinions of the funding agencies.

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