Homotopy-based analytical approximation to nonlinear short-crested waves in a fluid of finite depth*

WANG Ping （王苹）, LU Dong-qiang （卢东强）

1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

2. School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China E-mail: pingwang2003@126.com

3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China

Homotopy-based analytical approximation to nonlinear short-crested waves in a fluid of finite depth*

WANG Ping （王苹）1,2, LU Dong-qiang （卢东强）1,3

1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

2. School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China E-mail: pingwang2003@126.com

3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China

(Received January 8, 2014, Revised May 18, 2015)

A nonlinear short-crested wave system, consisting of two progressive waves propagating at an oblique angle to each other in a fluid of finite depth, is investigated by means of an analytical approach named the homotopy analysis method (HAM). Highly convergent series solutions are explicitly derived for the velocity potential and the surface wave elevation. We find that, at every value of water depth, there is little difference between the kinetic energy and the potential energy for nonlinear waves. The nonlinear short-crested waves with a larger angle of incidence always contain the more potential wave energy. With the aid of the HAM, we obtain the dispersion relation for nonlinear short-crested waves. Furthermore, it is shown that the wave elevation tends to be smoothened at the crest and be sharpened at the trough as the water depth increases, and the wave pressure crests and troughs become steeper with increasing incident wave steepness.

nonlinear short-crested waves, finite water depth, homotopy analysis method (HAM), wave energy, wave profile

Introduction

A short-crested wave system, made of two progressive wave trains propagating at an angle to each other, occurs in some realistic coastal regions, such as in front of a vertical seawall or an offshore platform where the oblique wave reflection appears, behind an obstacle of finite width where diffracted waves intersect, and on the ocean surface where two swell waves cross with each other. The main motivation for this topic is to study how to design some coastal structures safely. One of the important problems concerned in this field appears to be an accurate prediction of the characteristics of short-crested waves in a fluid of finite depth.

For a short-crested wave system, considerable work has been done since the first theoretical model proposed by Fuchs[1]in 1952. However, the previous researches on the short-crested waves are mainly within the scope of linear theory which is only valid for the small-amplitude waves. Such models are not appropriate to describe nonlinear short-crested waves of arbitrary amplitude. Furthermore, due to the limitation of mathematical tools, most analytical studies on the nonlinear short-crested wave often follow the wellknown perturbation methods which can be applied to weakly nonlinear problems only. For example, Fuchs[1]derived a second-order solution in dimensional form by means of a perturbation parameter related to the ratio of the wave height to the wavelength parallel to the wall. Subsequently, Chappelear[2]further extended the pioneering work of Fuchs[1]to the third-order perturbation approximation.

However, the solutions of Fuchs[1]and Chappelear[2]were not applicable for the limiting condition of the short-crested waves, i.e., the progressive and standing waves. In 1979 Hsu et al.[3]optimized the previous analytical method, in the frame of perturbationtechnology, by introducing the wave steepness as the small perturbation parameter, and derived the thirdorder solution in non-dimensional form, including progressive and standing waves limits. Following the framework of Hsu et al.[3], Roberts[4]obtained the third-order expansion based on a three-dimensional generalization of the mapping from the velocity-potential-stream function space to the physical space, and Fenton[5]further studied the wave forces and pressures on a vertical wall due to short-crested waves. Recently, Jian et al.[6]investigated the short-crested capillarygravity wave patterns with a uniform current and obtained the third-order approximate solution through a perturbation expanding technique.

Numerical perturbation approximations have been widely applied to the nonlinear short-crested wave system in recent years. Among these, Roberts[7]investigated the highly nonlinear properties of steadily propagating short-crested waves in deep water. The 27th order perturbation expansion in terms of the wave steepness was derived numerically. Then the effects of several parameters including the water depth and the incident angle on the wave energy were studied in detail.

In the present study, we will apply the homotopy analysis method (HAM) developed by Liao[8]in 1992, which has been demonstrated as a powerful analytical approach for weakly/highly nonlinear problems, in order to investigate analytically nonlinear short-crested waves in a fluid of finite depth. The convergent series solutions for the velocity potential and the surface wave elevation are obtained by introducing a convergence control parameter c0. It is found that the nonlinear short-crested wave energy is greatly affected by the oblique angle of incidence. Furthermore, the effect of the water depth on the wave energy and the wave elevation is studied in detail. All the results obtained will help enrich our basic understanding on the characteristics of nonlinear short-crested waves in a fluid of finite depth.

1. Mathematical formulation

A system of nonlinear short-crested waves in front of a vertical seawall in a fluid of finite depthd is considered. Cartesian coordinates (x, y, z )are taken in such a way that the x -and y -axes are fixed on the undisturbed surface while the z -axis points vertically upwards. The vertical wall coincides with y =0. Assuming that the fluid is inviscid and incompressible, and the flow is irrotational, there exists a velocity potential φ(x, y, z, t)which satisfies the Laplace equation

where ζ(x, y, t) is the surface wave elevation andt the time variable. The bottom boundary condition reads

In addition, at the wall boundary it holds that

The nonlinear kinematic condition is modeled as

The nonlinear dynamic boundary condition is[9]

whereg is the gravitational acceleration andE is a Bernoulli constant.

There was some confusion regarding whether or not the Bernoulli constant is included in the dynamic boundary condition of periodic traveling waves in the literature. Recently, Vasan and Deconinck[9] showed that the two boundary conditions (namely E = 0 or E ≠ 0 ) were equivalent up to the addition of a uniform horizontal velocity. In other words, eliminating the Bernoulli constant corresponds to changing the speed of the traveling wave. To make a comparison between the solution obtained by Hsu et al.[3] and the thirdorder series solutions presented here, we also assume E = 0 as made in Ref.[3].

Following Hsu et al.[3], we put all equations into dimensionless form to simplify the formulation. Let ε=ka, wherekanda are the wave number and the amplitude of short-crested waves of the first order, respectively. Then the following dimensionless quantities are adopted

wherew is the angular frequency of the incident orreflected waves.

Here we consider a system of nonlinear shortcrested waves which are formed by two progressive waves propagating at an oblique angle to each other. Letθbe the angle of incidence measured from the normal to the wall. Based on the traveling-wave method, we introduce the following independent variable transformations

Then, we can express the potential functionand the traveling wave profileThus the short-crested waves can be considered in the coordinates (X*,Y*,Z*). With the transformation (7) and dimensionless quantities (6), the governing Eq.(1) and boundary conditions (2)-(3) and (5) are transformed into

With Eqs.(7) and (6), we give a combination of Eqs.(4) and (5) on Z=εζ

So the unknownφand ζ are governed by Eqs.(8)-(12).

In accordance with Roberts[7], we also quote two quantities which measure how much energy there is in the waves. Similarly, let KEand PEbe dimensionless mean kinetic and potential energy densities per unit area in the(X, Y)-plane, respectively, which can be written as

The pressure in the wave motion is given by Bernoulli’s theorem in terms of dimensionless quantities as

where p*=p/(ρg/ k). The dimensionless pressure will be written without the asterisk for convenience throughout the rest of the paper.

2. Analytical approach based on the HAM

As is well known, the perturbation methods have been widely applied to solve nonlinear water wave problems. Unfortunately, the perturbation methods depend extremely on the small/large physical parameters in general, and then are valid only for weakly nonlinear problems. To solve the weakly/fully nonlinear problems effectively and accurately, Liao[8]proposed the HAM based on the concept of homotopy in algebraic topology. In recent years, the HAM has been applied to many highly nonlinear problems in science, finance and hydrodynamics[10-21]. In this section, the HAM is employed to solve the nonlinear partial differential equations (PDEs) (8)-(12) for two unknown variablesφandζ.

2.1 Solution formulas and initial guesses

Under the assumption that the short-crested waves are propagating without a change of shape, we consider that the wave profile ζ is periodic inX andY[7]. From physical point of view, it is clear that the wave profile can be expressed as

where βi,jis a constant to be determined. According to the governing Eq.(8), the boundary conditions (9) and (10),φshould be in the form

where αi,jis a constant to be determined. Equations (16) and (17) are called the solution expressions ofζ andφrespectively, which play important roles in the frame of the HAM. We construct the initial guess for the potential function as

where A is an unknown coefficient. We choose ζ0=0 as the initial guess forζ to simplify the subsequent solution procedure[20,21]. It should be emphasized that the high-order terms can sufficiently provide the corrections for the analytical series solutions due to the nonlinearity inherence in Eqs.(11) and (12) although the initial guessζ0is zero.

2.2 Continuous variation

The nonlinear boundary-value problem governed by the PDEs (8)-(12) is solved by means of the HAM. Instead of solving these nonlinear PDEs directly, we construct two homotopiesΦandη. The homotopy Φsatisfies the Laplace equation

and the boundary conditions

The homotopiesΦandηare governed by a new PDES, namely the so-called zeroth-order deformation equations

where q∈[0,1]is an embedding parameter,c0denotes a nonzero convergence-control parameter,L1[·]is an auxiliary linear operator with the property L1[0]= 0,N1[·]and N2[·]are two nonlinear differential operators defined by

where

It is noted that the definitions for N1[·]and N2[·] are based on the two nonlinear boundary condition (12) and (11), respectively. Whenq increases from 0 to 1, Φandηvary continuously from their initial guess φ0and ζ0to the exact solutionsφandζ of the original problems, respectively. The Taylor series for Φandηat q =0are

where

Assuming that c0is so properly chosen that the series in Eq.(23) converge at q= 1, we have the socalled homotopy-series solutionsφ = Φq=1and ζ = ηq=1. Furthermore, since the HAM provides extremely large freedom in the choice of auxiliary linear operators, we can only choose linear operators ofΦ in N1[·]as the auxiliary linear operator L1[·]by means of the solution expression (17) which is obtained under the physical considerations as

2.3 High-order deformation equations

The linear PDEs for the unknown φmandζmcan be derived directly from the zeroth-order deformation equations. Substituting the homotopy-Maclaurin series (23) into Eqs. (19)-(20), and equating the like-power ofq, we have the so-called mth-order deformation equations

where Hm=H (m-2), and H(·)is the Heaviside step function. The explicit expression and detailed derivation forare given in Appendix A.

One can see that the unknown termsφmandζmare governed by linear PDEs (25) and (26). It should be noted that at this stage the sub-problems forφmand ζmare not only linear but also decoupled. Thus, these high-order deformation equations can easily be solved.

2.4 First-order approximation and dispersion relation

Substituting the initial guesses (18) andζ0=0 into Eq.(26), we can get

where C= cothd . The coefficient α0,1in the initial approximation ofφ0in Eq.(18) is still unknown. An additional equation which can be used to determine the value ofAis

where e is an even integer,o an odd integer, and H the dimensionless wave height to first order based on the HAM. Then, using the inverse linear operator L1in Eq.(25), we can get the solutions for Aand φ1as follows

where αi,jis the (i, j)-th unknown coefficient.

We find the common solution φ1has one unknown coefficientα1,1which can be determined by avoiding the “secular” termsin Xin φ2. We note all subsequent functions occur recursively. By continuing the above approach, we can obtain higher-order functionsφmandζm. It should be mentioned that, based on the HAM, the dimensionless dispersion relation for nonlinear short-crested waves in a fluid of finite depth can be written as

It is noted that the convergence control parameter c0still retains in Eq.(31).

2.5 Optimal convergence-control parameter

Once we fix all parameters in the approximate series solutions, there is still an unknown convergence control parameter c0which is used to guarantee the convergence of the series approximation. According to Liao[21], the optimal value ofc0is determined by the minimum of the total squared residuefor the nonlinear problem, which is defined by

where

Fig.1 Residual squares ofversus c0

Table1 The total residual square errorfor different approximation order withm with c0=-1.0

Table1 The total residual square errorfor different approximation order withm with c0=-1.0

mTmε 2 4.093×10-94 3.104×10-96 7.771×10-108 8.142×10-1110 2.089×10-12

Fig.2 Wave profiles for d= 1.5π. The wave propagates in the X direction

3. Discussion and results

To choose an optimal value for the convergence control parameter c0in such a way that all the series solutions forφand ζ converge quickly, we firstly consider the case with d= 0.2π,ε= 0.05,θ= π/4, H= 0.15and take these data hereinafter for computation unless otherwise stated. The curves of the total residual square errorat several orders versus the convergence-control parameterc0are shown in Fig.1. We find that at every orderhas the smallest value

Fig.3 Comparison of our present dimensionless 3th-order surface profile ζ with those of Hsu et al.[3]for different incident wave steepnessesε.θ= π/4,d/ Ly= 0.1, t= 0

which corresponds to the optimal c0. For example, asm = 4and, the smallest value ofis 3.104×10-9. So we choose, and the total residual square errordecreases quickly as the order m increases, as shown in Table 1. It is also found thatcan decrease to 32.089×10-12at the 10th-order approximation, which indicates the convergence of the series solutions. It is demonstrated that the analytical solutions obtained here by means of the HAM are highly accurate. Figure 2 is the free surface profile for the short-crested waves, showing that the short-crested waves are periodic not only in the propagating direction but also in the perpendicular direction.

Fig.4 Wave profile ζ versus y/ Lyfor different water depths d

We compare the present solution for the wave profileswiththat obtained by Hsu et al.[3]who used the perturbation techniques. Figure 3 is graphical representations for the 3rd-order solution in Eqs.(26) and (16) and the asymptotic solution in Eq.(70) in Ref.[3], in whichis a function offor different wave steepnessεandis the wave length in they direction. It can be seen from Fig.3 that the homotopy-series solution for the surface elevation at the crest near the wall y=0is in good agreement with the perturbation-series approximation[3], even in cases whereεis not small (ε>0.2). It is clear that the trough of surface elevation is steeper than those predicted by Hsu et al.[3], especially in the cases with ε > 0.2 . Thus the variations of the short-crested waves at the trough will be underestimated if the 3rdorder perturbation solution[3] is used for the wave defection. We can also note that, as ε increases, the discrepancy between the two theories will become large for a finiteε, especially at the wave trough. The possible reason for this discrepancy is that the perturbation technology is strongly dependent on small physical parameters and then it usually breaks down as nonlinearity becomes strong (i.e.,εincreases), while the HAM is independent of any small parameters and thus it may be feasible for highly nonlinear problems. Physical observation or experimental data are required in the future to verify the theoretical predictions.

Table2 KEand PEfor Eqs.(13) and (14) with various water depthsd

Fig.5 Angular frequencyw versus the incident wave steepness εfor different water depths d

The effect of the water depthdon the short-crested wave elevationζ is studied, as shown in Fig.4. We can see that the crest of wave elevation increases greatly with an increasing dand the steeper trough is formed. Most importantly, with a different viewpoint from all the research objectives in Refs.[1-7], we consider the effects of the water depthdon the mean kinetic energy density KEand the mean potential energy density PEof the nonlinear short-crested waves in detail. Different values of KEand PEfor various water depthsd are shown in Table 2. It is found that KEand PEare almost equal at every value of water depthd . A graphical representation for the nonlinear dispersion relation given by Eq.(31) with H= 1is shown in Fig.5. With the increase of incident wave amplitudeε, the angular frequencyw increases for d< 0.2π, while for smalld,wdecreases slightly asεincreases from0to0.5. We can also find that the angular frequencywbecomes large with the increment of the water depth. Figure 6 shows the kinetic energy density PEof short-crested waves for different angles of incidenceθin various water depths. It can be seen from Fig.6 that the short-crested waveswith the largerθalways contain larger PEin different water depths. Figure 7 illustrates the variation of the dimensionless wave pressurepacting on the vertical wall as a function ofX for different small parameterε. It can be seen from Fig.7 that, asεincreases, the crests and the troughs of the pressurep in the 3rd-order approximation become steeper and steeper.

Fig.6 Kinetic energy density PEversus the water depthd for different incident anglesθ

Fig.7 Wave pressure p versusX for differentε

4. Conclusions

A system of nonlinear short-crested waves in a fluid of finite depth is studied analytically using the homotopy analysis method. Instead of solving the nonlinear PDEs for the waves directly by means of other analytical techniques, we reconstruct a new family of nonlinear PDEs as the zeroth-order deformation equations. Then the nonlinear PDEs are further transferred into an infinite number of linear PDEs, namely the higher-order deformation equations on the known boundary, which can easily be solved. More importantly, the highly convergent series solutions for the fully nonlinear wave equations are obtained by choosing the optimal convergence control parameter. Besides, the homotopy-series approxima-tions showing the behavior of nonlinear short-crested waves are compared with the previously reported asymptotic solutions, and the agreement is very good for weakly nonlinear waves especially at the wave crest. These conclusions demonstrate the accuracy and validity of the HAM for the nonlinear short-crested wave problems.

The homotopy-based dispersion relation shows that the wave frequency of short-crested wave system is affected by incident wave steepness and water depth. The frequencyw becomes large with the increment of the water depth. Incident wave steepness also causes analogous influence on the frequency except that water depthd (d<0.2π）is small.

The effects of the water depth and the incidence angle on the wave energy including the kinetic and the potential energies for nonlinear short-crested waves are analyzed detailedly. At every value of water depth, the values of the kinetic and the potential energies are nearly equal. A particularly interesting feature of the results here is that, in various water depths, the larger incidence angles is, the more potential energy is. Furthermore, we study the influence of the water depth on the wave elevation and the effect of incident wave steepness on the wave pressure. It is found that, as the water depth increases, the wave elevation is smoothened at the crest and is sharpen at the trough. And the wave pressure exerted on the vertical wall will become larger with the increasing wave steepness. All these results demonstrate that the angle of incidence, the water depth and incident wave steepness have important effects on the energy, the pressure and the profile of nonlinear short-crested waves in a fluid of finite depth.

Acknowledgement

The authors thank the anonymous reviewers for their constructive comments.

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Appendix A: The detailed derivation of the Eqs.(25) and (26) and the expression for φmand ζm

For anyZ , we have a Maclaurim series

For Z=η, it follows from Eqs.(A1) and (A2) that

where

Thus we have, for Z=η

where

Substituting the series expansions (A1) and (A4) into the boundary conditions (19) and (20), then equating the like-power of the embedding parameter q, we have two linear boundary conditions (25) and (26) respectively. The explicit expressions forin these two conditions are given by

where

Appendix B: Expressions of the coefficients

where

* Project Supported by the National Key Basic Research Development Program of China (973 Program, Grant No. 2014CB046203), the National Natural Science Foundation of China (Grant No. 11472166) and the Natural Science Foundation of Shanghai (Grant No. 14ZR1416200).

Biography: WANG Ping (1975-), Female, Ph. D. Candidate

LU Dong-qiang,

E-mail: dqlu@shu.edu.cn