Applied Researcli 47 (2014) 125-137

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On some solitary and cnoidal diffraction solutions of the CrossMark 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

ARTICLE INFO ABSTRACT

Article tiistory: Nonlinear of the solitary and cnoidal types are studied in constant and variable water depths by Received 19 October 2013 use of the Irrotational Green-Naghdi (IGN) equations of different levels and the original Green-Naghdi Received in revised form 17 April 2014 (GN) equations (Level I). These equations, especially the IGN equations, have been established more Accepted 21 April 2014 recently than the classical water wave equations, therefore, only a handful applications of the equations Available online 20 May 2014 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 and scattering problems Keywords: that include propagation and fission over a bump and onto a shelf, colliding , soliton Green-Naghdi equations generation by an initial mound of water and diffraction of cnoidal waves due to a submerged bottom Solitary wave shelf, and compare the predictions with experimental data when available. Cnoidal wave Wave diffraction © 2014 Elsevier Ltd. All rights reserved. Soliton fission

1. Introduction The theory has its roots in the theory of plates and shells in structural mechanics. A general theory of fluid sheets has been pre­ Propagation of unidirectional long water waves is classically sented by Green and Naghdi [13] for any type of homogenous and approximated assuming that the fluid is incompressible, homoge­ incompressible medium, i.e., viscous or inviscid fluid. The direct nous and inviscid, and the flow is irrotational. In most of the theory is based on a continuum model, namely directed or Cosserat problems, however, the resultant governing equations are sub­ surface, that is a deformable surface embedded in a Euclidean three- jected to the nonlinear boundary conditions on the unknown dimensional space to every point of which a deformable vector - free surface. Typically, an approximation of the solution to the not necessarily along the normal to the surface - called a direc­ equations is made by introducing some dimensionless parame­ tor, is assigned. The Cosserat surface, although three-dimensional ters that are small, and expanding the boundary conditions and in character, only depends on two space dimensions and time, and governing equations based on these parameters. As a result, the it includes ƒ<• directors dj, d2 d/f./C number of the directors boundary conditions are approximated and the conservation laws define the Level of the theory. For the application to water waves are satisfied up to the order at which the expansions are consid­ (see [14], for instance), these directors prescribe the variation of ered. the vertical component of the three-dimensional velocity along Beginning with the last three decades of the 20th century, an the . In Level 1, i.e., /<•= 1, the deformable medium is attempt was made to approach the subject of water wave propa­ a body of sheet-like fluid consisting of a free surface and a sin­ gation in sheet-like fluids from a different point of view, namely gle director attached to each point of the surface. Adopting the by use of the theory of directed fluid sheets or the Green-Naghdi Cosserat surfaces, the mass, momentum and angular momentum theory. Unlike the classical approximations, the method does not of the deformable sheet-like (or shell-like) body are expressed in require a priori assumptions on any scaling parameter nor fol­ such a way that a general set of equations of motion of the medium lows a perturbation expansion; irrotationality of the flow is also can be obtained. not necessary. The result is a set of equations that satisfy the Green and Naghdi [11 ] had shown that the Green-Naghdi equa­ boundary conditions and the postulated conservation laws tions (the GN equations, hereafter) result in the same solitary wave exactly. 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 * Corresponding author. Tel.: +1 8089566818; fax: +1 8089563498. formal expansion parameter, that the GN equations reduce to the E-mail address: [email protected] (R.C. Ertekin).

0141-1187/$ - see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.apor.2014.04.005 126 R.C. Ertekin etal./Applied Ocean Researci} 47(2014) 125-737 equations given by Wu [29] and Boussinesq [2], for a flat seafloor, equations were examined and results were compared with exact once the second and higher orders of the perturbation series are numerical solutions and other classical solutions. They concluded omitted. Choi and Camassa [3] used a scaling parameter to expand that the best convergence of the Level 1 results to higher Level the Euler equations for internal jvaves, and have shown that in the results of the theory is achieved for flows around the critical speed. shallow water limit (where fluid thickness is much smaller than In a different study, Kim and Ertekin [18] evaluated the disper­ the characteristic ), their equations are equivalent to sion relation of periodic solution of the first three Levels of the the GN equations for an inviscid, incompressible and homogenous GN equations, along with different Levels of IGN equations. It was fluid. shown that results of IGN equations are converged at Levels 111 or Although the GN equations are originally derived via a direct IV, depending on the problem. approach based on the Cosserat surfaces, over time, different meth­ Since no perturbation or scaling parameters are used in the ods have been adopted to derive the same equations, usually with derivation of the GN or IGN equations, there is no theoretical the expense of some extra assumptions about the physical flow restriction on the limit to which the theory is applicable. The only characteristics. For a special type of fluid sheet, that is incompress­ assumption made about the kinematics of the fluid flow is the ible and inviscid fluid. Green and Naghdi [12] showed that it is distribution of the velocity field in the vertical direction, which possible to derive the partial differential equations in a systematic is varied with the level of the theory. Therefore, applicability of way from the exact three-dimensional equations of an incompress­ different levels of the GN and IGN equations is to be discovered ible, inviscid fluid by use of only a single approximation for the by numerical experiments. In this work, we shall make a direct (three-dimensional) velocity field v*. The assumption is equivalent comparison of the results of the basic GN equations (Level 1) and to the Level 1 assumption in the direct approach, that is the vertical different levels of the IGN equations by applying the equations component of the velocity field is a linear function of the vertical to a number of nonlinear wave diffraction problems concerning coordinate (in an Eulerian system) and that the horizontal compo­ solitary and cnoidal waves. The main motivation for this work is nents are invariable in the vertical direction (see also, [6,23]). Such then to compare the results obtained by different sets of equa­ a velocity field allows for rotational flow on the horizontal surface, tions and with the experimental data to assess the applicability and the vorticity component on the horizontal plane does not need of the GN Level 1 equations and IGN equations of different lev­ to be zero even though the shear flow on the vertical surfaces are els. ignored. In Section 2 we will review the basic idea and assumptions in IVIiles and Salmon [22] have used Hamilton's principal to derive deriving the GN Level 1 equations via a direct method, also known the Level 1 GN equations for an inviscid and homogeneous fluid as the restricted theory, in which the directors are assumed to be in a water of variable depth in a Lagrangian form. They concluded normal to the surface. The IGN equations, that are for an irrota­ that, for the case of flat seafloor, the Level 1 GN equations reduce tional flow, are given in Section 3. The rest of the work (Sections to the Boussinesq equations in which (but not nonlin- 4 and 5) is mainly concerned with the application of the equa­ earity) is assumed to be weak. Later, Kim and Ertekin [18] and tions discussed in Sections 2 and 3 to a number of nonlinear and Kim et al. [16] used Hamilton's principle in Eulerian form to derive unsteady problems of fluid sheets where solitary and cnoidal waves the irrotational version of the GN equations (IGN equations, here­ are used. after) to any Level K. Kim et al. [16] also showed that GN equations All of the cases considered here are two-dimensional. When of any Level K can be derived from the principle of virtual work. referring to the interaction of two (or more) solitary waves, the They also showed that the GN Level 1 equations are identical to waves are referred to as solitons. the IGN Level 1 equations, if a flow starts from a state of rest, even though no assumption of irrotationality is made to derive the equa­ 2. The Level I Green-Naghdi equations tions. Shields and Webster [26] derived the classical GN equations for We use a rectangular Cartesian coordinate system (x,y, z) with the motion of an inviscid fluid, for any number of directors K, using associated orthonormal base vectors e,-. In three-dimensions, the the Kantorovich method. They assumed a solution for the verti­ index i has a range of 1-3, and Greek subscripts take the value of cal velocity in which a finite power-series form the variation of 1, 2, and all lower case Latin subscripts designate partial differen­ the velocity field in the vertical direction. Demirbilek and Webster tiation with respect to the indicated variable. The base vector 63 is [4] applied the Level 11 GN equations to some coastal-engineering vertically upward. The coordinate system is chosen such that the problems. Recently, Zhao et al. [31] has applied the high-level GN x-y plane is the still-water level (SWL). For simplicity, and within equations derived by Webster et al. [27] who also used a power- the context of the application of the theory to water waves, we series expansion of the particle velocities along the water column assume that the fluid has a constat mass density p* and is subject and made the high-level GN equations, involving lengthy and com­ to the gravitational acceleration g in the direction - 63. The fluid plicated equations, manageable. sheet is bounded below by the surface Zhang et al. [30] followed the same assumption for the three- z = a{x,y,t). (1) dimensional velocity field as that used in the derivation of the GN equations from the exact three-dimensional equations of an Let us assume that the bottom surface is stationary, that is for all incompressible, inviscid fluid given by Green and Naghdi [12], that values of t is they prescribed a certain distribution for the vertical velocity ff(x,y, t) = a(x,y). (2) component, and derived a set of nonlinear equations based on the scaling assumptions used in Boussinesq-class equations. Under The normal pressure on this surface is p = p(x, y, t). The fluid sheet such assumptions, they were able to release the irrotationality con­ is bounded above by the surface dition used in the Boussinesq equations. The final equations that z = fi{x,y,t), (3) they named Boussinesq-Green-Naghdi rotational wave equations, differ from the GN equations in that the boundary conditions and and pressure on this surface is p = p(x, y, t) where conservation laws depend on the expansion level of the scaling parameter and thus are approximated. p{x,y,t) = pQ-q{x,y,t), (4) In a study by Shields and Webster [26], convergence of the and where po is the constant atmospheric pressure and q is the solitary and periodic solutions of the first three Levels of the GN pressure due to surface tension (which we take as zero here). Eq. R.C. Ertekin et al. / Applied Ocean Research 47(2014) 125-137 127

[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, H^=-l{[2f + «],ö+[4f-«a we have the continuity equation + {li + ^-a)[ö! + 2l]x}, (14b) Vi,i = 0. (5) The following kinematic boundary conditions are to be enforced on the surfaces given by (1) and (3): v + g^y + ^=-lm + a]ya+m-a]yl

ys = 0 on z: V2 (6a) + ih + ^-a)[a + 2l] }, (14c) dt dx dy

da da where /i is the water depth, f is the free surface elevation measured f3 = 0 on 2 (6b) from 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 Further, we define the thickness of the fluid sheet and its midplane [5] coined these equations the Green-Naghdi equations. by The vertical component of particle velocity is then written as [6]

li + a z-a e[x,y,t) = li-a, S(x,y,t): (7) (15) ^ li + ^- -ai^-') Note that, up to this point, no assumption about the dynamics of An analytic solitary wave solution of the GN Level I equations (2) can the fluid sheet is made, and the boundary conditions (4) and (2) be found in Ertekin [5], who has studied a number of constrained are exact. Here, we make a single assumption that the directors domain problems in shallow water involving solitons. The solution remain perpendicular to the horizontal surface (but may deform is given by along their lengths), which results in the Level 1 equations. Such an assumption is equivalent to the vertical component of velocity 3A being linear in z, and consequently, the incompressibility condi­ f(x)=Asecr (16) 4lil{ho+A) tion (5) is satisfied exactly as long as the horizontal components of velocity are independent of z. where A is the of the solitary wave measured from the The velocity v* of the middle surface z=5(x, y, t) and the director SWL and is given by velocity take the following forms: (17) v* = ue-i + ve2 + >-e2, w = \ve2, (8) A=^-ho, where where U is the speed (critical or supercritical) of the wave, ho is the constant water depth and x = x - XQ - Ut, where xq is the mid­ u=k, v = y, X = S, w = 6, (9) point of the solitary wave at time t = 0. Periodic shallow-water wave and where superposed dot denotes the material time so solutions, i.e., cnoidal waves, can only be obtained in a semi-closed that if fix, y, t) is any function of x, y and t, then form 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] in f=ft + llf>c + vfy- (10) two-dimensional Galilean coordinates. A double superposed dot denotes the second material time deriva­ The nonlinear GN Level 1 equations, (14a) and (14b) for the two- tive. From (8) the fluid acceleration v* and director acceleration w dimensional case, are solved in this work. At a given time, Eq. (14a) follows: can be solved explicitly for ft once u is known. Next, ft can be sub­ stituted into (14b) to solve for Ut. We note that in solving the GN V* = iie-[ + ve2 + ics, w = wej. (11) Level 1 equations, (2), we have used the dimensionless form of the In the final step, the integral balance equations of mass conserva­ equations by use of the dimensionally independent set of p*,g and tion, momentum, moment of momentum (or director momentum) ho. and energy can be postulated with no further assumptions [13], The final equations are solved numerically by the central- Such restricted model results in the equations of incompressibility difference method, second-order accurate in space, and with the and the motion of the fluid via a direct approach. For an inviscid Modified Euler Method for time integration. We approximate the fluid, they are given as 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 in the spatial domain and 0{iix + Uy) + w = 0, p'Oii = -px + PPx - pax, n indicates a mesh point on the time axis. At each given time, such discretization results in an NT x NT coefficient matrix, where Nj is P*ei> = -Py + pPy - pay, p"OX = -p +p - gp*6, the total number of mesh points in the computational domain. In a two-dimensional domain, which is our interest in this study, the ip*öw = H-i(p + p), (12) coefficient matrix is a tridiagonal one and can be solved exactiy by where p is the integrated pressure, per unit area of the middle the Thomas algorithm (see e.g., [1]), which eliminates any matrix surface z = S, defined by operations and thus is a very efficient method. A convergence test of the solution of the GN Level 1 equations can be found in Ertekin et al.[8]. p*dz, (13) P = We recall that the GN Level 1 equations satisfy the free-surface and bottom boundary conditions (2) exactiy. As such, we are basi­ and p*(x, y, z, t) is the three-dimensional pressure. Ertekin [5] cally dealing with an initial-value problem, that is only the initial obtained rather a classical form of the governing equations (12) value of the variables (f and u in two-dimensions) should be spec­ as ified except at the two ends of the domain. This is accomplished by the numerical wavemaker located at one end of the domain ^t + V.{(li + ^-a)V] = 0, (14a) (upwave end), which is capable df creating solitary or cnoidal 128 R.C. Ertekin et al. / Applied Ocean Research 47(2074) 125-137

waves. To minimize tlie size of the computational domain, how­ where ever, we have to predict the values of the variables at the downwave end (open boundary) of the numerical domain. Previous works ^111 = / fni[y)fn{y)dY, [28,5] have shown that the relatively simple Orianski's condition Jo, with constant speed c = ±^/gh^, where HQ is the water depth Bmn Yfm{y)fn(Y)dY, at the location where the open-boundary condition is being applied, UY)f;MdY, (26) prevents significant reflections from the open-boundary. We also

use this open-boundary condition here which reads Cmn=/ aY)mY)dy, Ci„=mn — I YUY)my)dy, •^°i Jo at + ci2x = 0, (18) C^n = ƒ y^UY)f,[{Y)dY.

where Q may be f or u. The IGN equations are completed by the relation between the sur­ face potential ^(x, t) and , which is given by 3. Irrotational Green-Naghdi equations 9 9r ^ 9r . .... (27) + TTT— =/m(l)0x- The irrotational version of the Green-Naghdi equations can be 9x dfm, dfrr derived from Hamilton's principle, as shown by Kim et al. [16], As numerical implementation, the canonical equations (3) are We seek for the solution in a vector function space that satisfies solved by the Runge-Kutta 4th-order method after discretizing 0 kinematic constraints of divergence free in the fluid domain and and f by the finite-element method. After the discretization, Eq. no-leak condition on the floor. In two dimensions, the velocity (27) becomes algebraic equations for the discretized ij/m and 0, field (u, w) that satisfies the kinematic constraints are given by the which are solved by a banded-matrix solver. Details of the numer­ stream function !/f(x, z, t), i.e.. ical scheme can be found in Kim et al. [17,19].

dif{x,z, t) 9iA(x,z, t) u(x,z, t) = W(X, Z, f) : fix, a, t) = 0. 4. Solitary wave dz ' 9x ' (19) We have selected several numerical experiments on the diffrac­ tion of solitary waves when we used the GN Level 1 and IGN In Level KIGN equations, the stream function is given by odd-order equations. Comparisons are made with other numerical solutions polynomials in a normalized vertical coordinate y: and physical experiments when available. The cases studied here are as follows. (1) A solitary wave propagating from deep water to shallow water (and vice versa), with a linear transition between iA(x, z, t) = Y^i^mix, t)MY), i,2,...,/<:, (20) m •- the two depths. (2) A solitary wave propagating over a submerged m=l curved bump. (3) Reflection of a soliton from a vertical wall. (4) where Interaction (collision and overtaking) of two solitons over a flat seafloor. (5) Waves produced by an initial mound of water (dam z-o- break problem). In all the solitary wave cases, the GN Level I and the Y (21) I - a IGN Level III equations are solved. Due to the absence of perturba­ tion terms or assumptions on certain scaling factors, application of For the restricted case, where the lower boundary of the fluid the GN and IGN equations to different problems should be assessed domain is stationary, i.e., computationally.

^(x, f) = -/i(x), (22) Here, we make an initial assumption that the IGN Level 111 equations are the converged solution within the IGN models. This and where we have constant pressure on the free surface, assumption is made partially based on the study by Kim and Ertekin [18]. If there is disagreement between the GN Level I p(x,f) = po=0, (23) and IGN Level III results, then different levels of the IGN equa­ tions, including lower and higher levels (until convergence is (the atmospheric pressure is set to zero without loss of generality), achieved) are studied. Kim et al. [16] have shown that at the the IGN equations are given by two canonical equations^for the free flrst level, IGN equations are identical to the GN Level I equa­ surface elevation f (x, t) and surface velocity potential 4>ix, t), tions if a motion starts from a state of rest. Nevertheless, in the following test cases, and when different levels of the IGN equa­ ft + «Ax = 0, (24a) tions are presented, results of the IGN Level I equations are also given. X d dT dT ^ We note that by convergence of different Levels of the IGN (24b) '^'^-TxW.^d^-'^' equations, we refer to the convergence between interpolation func­ tions of different levels used to interpolate the velocity field in the where f = tj/{x, f, t) is the stream function on the free surface and vertical direction, i.e., when converged, higher order of the interpo­ T is the kinetic energy given by lation functions do not provide any better description of the fluid flow. Next, the numerical solution of these problems are discussed 1 separately.

171=1 n=l 4. ?. Solitary wave propagating over a submerged shelf +2 |-(f -I- h\Bl„ + hxBmn]i^mxfn + ^((l + ''x) Qnn Propagation of a solitary wave over a submerged shelf is -2(f + h)xhxCl^n + + hxfcl,} f„,i,n] , (25) studied by use of the GN Level I and IGN equations of different R.C Ertekin et al./Applieti Ocean Research 47 (2014) 125-137 129

s O -Gauge! — GNLI Pi ' - - -IGN L III .2 ° Experimental (Goring 1979) I

to 10

•Gauge 3 = 3:56 n /p \ fb

Time (s)

Fig. 1. Solitary wave propagating from deep to shallow water over a linear , GN Level I (solid line) and IGN Level III (dashed line) vs. laboratory experimental of Goring |101 (circles). Gauge 1 is located at the bottom of the slope and Gauge 2 is on top of the slope. The schematic of the wave tank is not to scale.

levels. The equations are used to make a comparison with the agreement between the GN Level I and IGN Level 111 equations with laboratory experiments of Goring [10]. In the laboratory experi­ the laboratory measurements is observed. ments, the deep-water depth {/iq = 31.08 cm), shallow-water depth Next, we compare the results of the GN Level 1 and IGN (/ii = 15.54cm) and incident- (>l = 3.1cm) are kept equations. The comparisons for two solitary wave constant, while the length of the submerged shelf (L) is varied. (7l//io =0.12 and 0.3) are shown in Figs. 2 and 3. All the numbers Transmission of the solitary wave over the slope recorded by in these figures are dimensionless with respect to the deep water three wave gauges (L = 300cm case) is shown in Fig. 1. Excellent depth (ho). The length of the shelf is kept constant (L/ho = 10) and

-GNLI 0.2 hGauge 1 k36' •IGNLffl

0.1

0

-0,1, 20 30 40 50 60 70 80 90 100

I \ I ! I 1 '—' 0.2 a o 0.1

E 0 0) O -O.lJ i i i i i i 10 20 30 40 60 90 100 urf a OT

I 1_ 10 20 30 40 50 60 70 80 90 100 tylgjhil

Fig. 2. GN Level I vs. IGN Level I , solitary wave propagating over a submerged shelf,/\//io =0.12, /!i//io = 0.451,I/;io = 10. At time t=D, the soliton is atx//io=30. The slope starts atxlho=40. 130 R.C. Ertekin et al. /Applied Ocean Research 47(2014) 125-137

0.6 1 1 Gauge 1 - 36 GNLI 0.4 •—•IGN LI 0.2 •••••••IGNLII ---IGNLIII 0 /V ; 1 — IGN LIV

1 ! 1 1 0.4 Gauge 3 = 96 , L 0.2 _ 1 : 1 : 0 III^ I ft A 40 60 100 120 140

1 0,4 Gauge 4= 125

0,2 j ti ^..i: : 0 i 1 1 1 1 1 20 40 80 100 120 140

Fig. 3. GN Level 1 vs. IGN equations, solitary wave propagating from shallow (ho) to deep (h,) water over a slope,/\/ho = 0.3, h,/ho = 0.451, L/ho = 10. At time t=0, the soliton is at xjho = 30. The slope starts at xjho =40. IGN LII and III plots cannot be seen as they are under the plot of Level IV results. The diffracted wave calculated by the IGN Level I is almost identical to that of the GN Level L

it Starts atx/fto =40. IGN equations of Levels I, II, III and IV are used 4.2. Solitary wave propagating over a submerged bump in Fig. 3, and results of these different levels of IGN equations con­ verge starting from Level III. As the solitary wave propagates onto Diffraction of a solitary wave propagating over a submerged the shelf, the front of the wave begins to steepen, indicating the curved-bump is studied by solving the GN Level I and IGN equations. strengthening of nonlinear effects compared with the dispersive To obtain a continuous Uxxx, which appears in the GN momentum effects. It then splits up into a number of smaller amplitude soli­ equation (2), the profile of the submerged bump is modeled by use tons. For the relatively large ratio of A//ii(= 0.665) shown in Fig. 3, of an eight-order polynomial given by although the tail waves are in close agreement, the IGN Level III equations predict a higher amplitude wave (and therefore a faster- a{x)=fg[4ix'^-fi)Y -ho, -l

Fig. 3. ^ 1 0.2 Johnson (1972) Comparison of the first (Ai), second (A2) and third (A3) succes­ < n sive solitons with the results of Johnson [15], Schember [25] and -A, 0.15 . v,-Si:llcmber(1982) Madsen and Mei [21] is given in Fig. 4. As /ii decreases, the GN i Level 1 underestimates the wave amplitude of the first successive V • Madsen & Mci (1969); 0.1 : wave in comparison to IGN Level III. The second and third successive —1 Theory waves, however, show a better agreement even for small values -A; A Madsen & Mei (1969); Experiment 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

;ii=0.5. ) 0 2 D 4 0 6 Propagation of a solitary wave from shallow to deep water h,/h<, (/ii//io = 2) over a submerged shelf (L/;io = 20) is also studied and the results are shown in Fig. 5. The slope starts from x/ho=40. A Fig. 4. Comparison of the amplitude of the first (Ai), second (A2) and third (A3) close agreement is observed between the IGN Level III and the GN successive solitons generated due to the propagation of the solitary wave from deep to shallow water over a linear slope, A/ho = 0.12, L/ho = 10. In the experiments of Level I equations. Madsen and Mei [21], results are given only for the first two successive solitons. R.C. Ertekin et al./Applied Ocean Research 47 (2014) 125-137 131

0.3

: y »\ ; i i Qaugk2=..6fi _ CJ ,/ '\

i i 1 i i i i H 0 10 20 40 50 60 90

1 ( ( 1 - CO

0.3

0.2

0.1 i i 1 i i 0' 20 30 40 50 60 70 80 90

-0.1 Fig. 5. GN Level I vs. IGN Level 111 predictions, soliton passing from shallovi' water over a submerged shelf to deeper water, /I//10 = 0.3 and hi //lo = 2. At time t=0, the soliton is at x/ho = 30. The slope starts at x\ha =40.

is on top of the bump. The comparisons are given in Figs. 6 and 7. as the wave amplitude is slightly smaller. As the bump ampli­ All numbers in these figures are dimensionless with respect to the tude increases, however, the difference becomes more significant water depth (/iq). In all the cases, amplitude of the main solitary (shown in Fig. 7), particularly in the amplitude of the trailing disper­ wave is reduced and it is followed by a train of dispersive waves of sive solitons that are underestimated by the GN Level 1 equations decreasing amplitude and wave length. Overall, close agreement is compared with the IGN equations of different levels. The differ­ observed, particularly for the smaller A/B ratio. Unlil

0.6 GNLI ---IGNLIII 0.4 A \ iauge 1 = 24 0.2 0 1 1 1 1 1 1 1 i I 10 20 25 30 35 40 45 50

0.6 1 1 ! ! i ! ! ! 0.4 ll \ 11 \ \ Gauge 2 =i 41.04 & 0.2

K 0 i i 1 1 1 1 1 1 10 20 25 30 35 40 45 50 3

T i I 1 1 r 0.6

0.4

0,2

0 20 25 30 35 40 45 50 55

t\fgJho

Fig. 6. GN Level I vs. IGN Level III, soliton passing over a submerged bump. A/ho = 0.6, B/ho = 0.2. Gauge 2 is located on top of the bump. At time t= 0, the soliton is located at x/ho = 20,4. 132 R.C. Ertekin et ai. / Applieti Ocean Research 47(2014) 125-137

0.6 • :Gaugefl•=24•• -GNLI "IGN LI 0.4 •IGNLn •IGNLffl 0.2 -IGNLIV 0

0 10 15 20 25 30 35 40 45 50 55

I I ! • ! IIII ^0.6 Ö .3 0.4 0.2 0 iiii -^T'"'"'-"' iiii 0 10 15 20 25 30 35 40 45 50 55

IIII lllll 0.6

0.4 - \ 1 i i 0.2 0 iiii 0 10 15 20 35 40 45 50 55

Fig. 7. GN Level I vs. IGN, soliton passing a submerged bump, Ajho = 0.6, B/ho = 0.6. Gauge 2 is located on top of the bump. At time t=0, the soliton is located at x/ho=20.4. IGN L11 and III plots cannot be seen as they are under the plot of Level IV results.

waves seem to depend more on the bump amplitude rather than (A/iio = 0.3, 0.6) are considered and the results are shown in Figs. 8 the initial wave amplitude. 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 4.3. Reflection of a soliton from a vertical wall amplitude case (shown in Fig. 9), the higher levels of the IGN equa­ tions predict a wave with a higher amplitude at Gauge 3, where the The problem of a soliton impacting a vertical wall is also studied. vertical wall is located. IGN Level IV equations predict some oscilla­ The reflection of a solitary wave from a vertical wall is equivalent tions at the location of the wall (Gauge 3 of Fig. 9) immediately after to the case of two solitons of equal amplitudes traveling in oppo­ the soliton reflects from the wall. Despite the differences between site directions and colliding. A set of two solitary wave amplitudes the results at the location of the wall, the reflected waves, farther

— GNLI 0.6 •--IGNLIII Gauge 1 = 15.6 : 0.4

0.2

0 ( 20 25 30 35 40 45 50

111! 1 ! ! 1 -0.6 Gauge 2 = 30 i i 0,4 \ \ \ -

10 20 30 35 40 45 50

1 • /N • - • • 0.6 Gaugs ' 3 = 45.6 0.4

0.2 : .:

0 10 20 25 30 35 40 45 50

Fig. 8. Soliton reflecting from a vertical wall, GN Level 1 vs. IGN Level 111, A/ho = 0.3. At time t=0, the soliton is at Gauge 1. The wall is at x//io = 45.6. R.C. Ertekin et al./Applieti Ocean Research 47(2014) 125-137 133

lllll 1 ! I "1.5 - Gauge'2'= 30 i ; \ \ a .2 1

M 0.5

o 0 i i i 1 1 1 1 1 10 15 20 25 30 35 40 45 50 OT

Fig. 9. Soliton reflecting from a vertical wall, GN Level 1 vs. lGN,/l//!o = 0.6. At time t=0, the soliton is at Gauge 1. The wall is at x//io=45.6. IGN L 111 plot cannot be seen as it is under the plot of Level IV. away from the wall, are in good agreement, see the second wave at a slightly faster-moving soliton at Gauge 3. After passing each other, Gauge 2 in Fig. 9. solitons recover their original form and the agreement is closer for the smaller amplitude wave seen in the Gauge 3 results. 4.4. Interacting solitons Another case of a soliton of amplitude Af//!o = 0.6 overtaking another soliton of smaller amplitude, As//io = 0.1 is also studied. Two colliding solitons of different amplitudes are modeled in The results are shown in Fig. 11. There is a very close agree­ this section. The amplitude of the right-moving soliton on the ment between the GN Level 1 and IGN Level 111 results, although left (Ai/ZiQ = 0.6) is six times larger than the amplitude of the left- GN Level 1 predicts a slightly faster-moving soliton. The smaller- moving soliton on the right (4r//io =0.1). The results are shown in amplitude wave predicted by different models is in complete Fig. 10. A close agreement is observed, although GN Level 1 predicts agreement.

jGauge | = ,2l —GNLI - - -IGN L III

25 30 35 40 -0 "5; 1 1 iGauge 2 = 4i -0.6: \ 04 • p/ \ \ '• /' *\ '•/' A 7' ^\

1 1 10 20 25 30 35 40 3 CO

! ! \ bauge 3 = 6^ 0.6

0.4 11 \ / * " 0.2 /' / *

0 - •-^^^^•i— ! \ 1 1 10 15 25 30 35 40

Fig. 10. Two colliding solitons of different amplitudes, GN Level I vs. IGN Level 111, At/ho = 0.6 and An/ho = 0.1. At r=0, the left soliton is at Gauge 1, and the right soliton is at Gauges. 134 R.C. Ertekin et al. / Applied Ocean Research 47 (2014) 125-137

Fig. 11. Soliton of amplitude /1F//IO = 0.6 overtaking a soliton of amplitude,/\s/lio = 0.1. Both solitons are moving in the same direction. At time 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 number of solitons are generated, traveling at supercritical speeds. Although there is a perfect agreement between different models for Waves generated by an initial rectangular mound of water is the first two waves at wave Gauge 1, close to the initial location of considered in this section. Initial height of the mound is A/ho = 0.4, the mound, there is a disagreement beginning from the third wave and the initial length is L/ho = 12, and at time t = 0 the water is at where Level 111 results predict faster moving waves. The ampli­ rest. Calculations start from this state, without the need of any tude of waves predicted by both models are in agreement. Farther smoothing or filtering of the surface elevation or velocities. As the downwave, at Gauge 2, a better agreement is achieved, particularly initial hump of water is released in Fig. 12 (dam-break problem), a for the first three solitons. The best agreement is seen at Gauge 4,

Fig. 12. Waves generated by an initial mound of 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 of the wave gauges are dimensionless with respect to tlie water depth. 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 Wave length Wave height Phase speed Steepness

(Ty^) (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

with the experimental data of Ohyama et al. [24]. Four test 205. 1 209. 5 208. 1 cases (wave conditions) are considered and these are shown in Incident Wave II II II X X X Table 1. The shelf remains unchanged in all cases. A schematic V W UJl2 üati4 of the numerical wave tank is shown in Fig. 13. The results calculated at Gauges 3 and 5 are given in Figs. 14 and 15, respec­ tively. / IT Overall, good agreement with 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

Fig. 13. Cnoidal waves propagating over a submerged shelf, a schematic of the of Level IV is perfectly under the Level V results in all cases. GN numerical wave tank and dimensionless (with respect to water depth) location of Level 1 predicts lower trough in Cases 2 and 4 in comparison to the the wave gauges. experimental measurements at Gauge 3. The crest is also underes­ timated. IGN Levels 111, IV and V results, on the other hand, are in better agreement with the experimental data, except that the wave the farthest gauge downwave, where the solitons are almost fully crest is slightly underestimated in Cases 1, 2 and 4. developed. The number of solitons generated by different models is in excellent agreement. 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 5. Cnoidal waves and 4). IGN Levels 111, IV and V equations, show a very close agree­ ment with the laboratory experiments even for the steepest case The problem of cnoidal waves passing over a submerged shelf (Case 2). The only disagreement in this case is the slight difference is presented in this section. GN Level 1 and IGN Levels 1-V equa­ in the phase and amplitude of the higher harmonics. Once again, tions are used for this study. For this test cases, a five-point the small differences between GN Level 1 and IGN Level 1 is due to formula, similar to that used in Ertekin et al. [8], is adopted the difference in numerical solution of the two set of equations, and to smooth velocity (u) and surface elevation f of the GN Level the smoothing process used in the GN Level 1 equations. 1 equations at every five time steps. The results are compared

! Casejl I \ \

-0, i iiii 1, 1 ! CaseÖ I I 1 1 I a 0, ^ i I—t

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. [24] (circles). Refer to Fig. 13 forthe location of the wave gauges. Wave conditions for different cases are given in Table 1. IGN Llll plot is hidden behind the Level IV and Level V plots most of the time. 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 wave Gauge 5, GN Level 1 and IGN equations vs. laboratory experiments of Ohyama et al. [241 (circles). Refer to Fig. 13 for the location of the wave gauges. Wave conditions for different cases are given in Table 1. IGN L III plot is hidden behind the Level IV and Level V plots most of the time.

6. Concluding remarks periodic wave cases, although the IGN Level III results are in excel­ lent agreement with the IGN Level IV results, we conclude that the Our main objective has been to use the GN Level 1 and IGN Level IV is the converged level of the IGN equations. equations of different levels in a number of existing numerical and physical experiments and examine their applicability and accuracy. For small amplitude wave interactions, the GN Level 1 Aclcnowledgements and IGN Level III equations are in very close agreement in all of the cases. We choose IGN Level III as the converged model for The works of RCE and MH are partially based on funding from the problems that we have considered here. As the nonlineari- the State of Hawaii's Department of Transportation (HDOT) and ties increase, that is the local wave height becomes large in a the Federal Highway Administration (FHWA) through the HDOT given experiment, the GN Level I equations seem to underesti­ Research Branch, Grant numbers DOT-08-004 and TA 2009-1R. This mate the higher harmonics compared to the higher levels of the funding to support research on and hurricane-generated IGN equations, although the models are in good qualitative agree­ wave loads on coastal bridges is gratefully acknowledged. The ment overall. Almost in all of the cases, IGN Level IV results are authors would like to also thank Professor W.C. Webster of U.C. on top of the Level III results, confirming the convergence. The Berkeley and Dr. BinBin Zhao of Harbin Engineering University for assumption of the linear change of vertical velocity over the water their helpful comments on the presentation of the IGN equations column made in the GN Level I equations seems to be debatable of different levels. The work of M.H. is also supported by the Link in these cases. In the case of a solitary wave, we found that the Foundation's Ocean Engineering and Instrumentation Fellowship. GN Level I equations provide satisfactory results for wave ampli­ Any findings and opinions contained in this paper are those of the tudes up to about A//10 f» 0.4. The agreement with the IGN Level III authors and do not necessarily reflect the opinions of the funding and laboratory experiments appear to diverge as the wave ampli­ agencies. 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. References For periodic waves, the dimensionless Ursell number (C/r = HX^/lig) appears to be a better gauge for the applicability of the |1] Ames W, Lee S, Zaiser J. Non-linear vibration of a traveling threadline. Int J Non-Linear Mech 1968;3:449-56. GN Level I equations. Generally, the GN Level I equations are in [2] Boussinesq J. Théorie de I'intumescence liquide appelée onde solitaire ou de better agreement with the IGN equations and the laboratory exper­ translation. Comptes Rendus Acad Sci Paris 1871;72:755-9. iments for smaller values of Ur, that is when the nonlinearity is [3] Clioi W, Camassa R. Fully nonlinear internal waves in a two-fluid system.] Fluid Mech 1999:396:1-36. rather smaller. Since the small differences between the results of [4] Demirbilek Z, Webster W. Application of the Green-Naghdi theory of fluid the GN Level I and IGN Level I models is due to the differences in sheets to shallow water wave problems. Technical report CERC-92-11. Vicks- numerical solutions of the two set of equations, and the smoothing burg, Mississippi: US Army Corps of Engineers; 1992. [5] Ertekin RC. Soliton generation by moving disturbances in shallow water: the­ process used in the GN Level I equations, it would be necessary to ory, computation and experiment University of California at Berkeley; 1984. use exactly the same methods of solution in solving the two sets of May, v+352 pp. (TC 172.E74 1984) [Ph.D. thesis). equations at even high-levels of these equations in making compar­ [6j Ertekin RC. Nonlinear shallow water waves: the Green-Naghdi equations. In: isons in the future. The IGN Levels IV and V results are in excellent Proc Pacific congress on marine sci and techno, PACON'88.1988, p. 0ST6/42-52. [7] Ertekin RC, Becker JM. Nonlinear diffraction of waves by a submerged shelf in agreement with the laboratory measurements. In the case of the shallow water. J Offshore Mech Arct Eng 1998;120:212-20. R.C Ertekin etal./Applied Ocean Research 47(2014) 125-137 137

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