Zhan Wang, Bin-bin Zhao, Wen-yang Duan, R. Cengiz Ertekin,2, Masoud Hayatdavoodi,3, Tian-yu Zhang

1. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

2. Department of Ocean and Resources Engineering, University of Hawaii, Honolulu, USA

3. Civil Engineering Department, School of Science and Engineering, University of Dundee, Dundee, UK

Abstract: In this paper, steady solutions of solitary waves in the presence of nonuniform shear currents are obtained by use of the high-level Green-Naghdi (HLGN) model. We focus on large-amplitude solitary waves in strong opposing shear currents. The linear-type currents, quadratic-type currents and cubic-type currents are considered. In particular, the wave speed, wave profile,velocity field, particle trajectories and vorticity distribution are studied. It is demonstrated that presence of the nonuniform shear current modifies the velocity field and vorticity field of the solitary wave.

Key words: Solitary wave, nonuniform shear current, velocity field, particle trajectories, vorticity field

Solitary waves have been an important topic in nonlinear water wave field for many decades. Dutykh and Clamond[1]proposed an effective method to calculate the profile and the velocity field of the solitary waves (H /d≤0.79, where H is the wave amplitude and d is the water depth) by solving Euler’s equations. Recently, Duan et al.[2]calculated steep solitary waves, and even a limiting-amplitude solitary wave with H /d=0.833199 by using the high-level irrotational Green-Naghdi (HLIGN) model.Zhong and Wang[3]derived a strongly nonlinear weakly dispersive wave model to study some solitary wave transformation problems. Tong et al.[4]used a harmonic polynomial cell method to study the solitary wave collisions problems.

Meanwhile, wave-current interaction is universal in coastal regions, and the wave profile, speed and particle trajectories are different when compared with wave field with no current. Thus, it is important to study the effect of wave-current interaction of the flow field.

Steady solutions of solitary waves in the presence of linear shear currents have been studied by some researchers. Choi[5], Pak and Chow[6]used asymptotic method and third-order solution, respectively, to study the solitary-wave profile, wave speed and streamlines for a solitary wave in linear shear current. Duan et al.[7]used the high-level Green-Naghdi (HLGN) model to obtain more accurate results for large-amplitude solitary waves in linear shear currents to compare with the results of Pak and Chow[6].

In this paper, we use the high-level Green-Naghdi (HLGN) model to investigate solitary waves in nonuniform shear currents. The fluid is inviscid and incompressible. The water depth is constant.

A two-dimensional Cartesian coordinate system is used where origin is at the SWL, and where x is the horizontal axis and positive to the right and z is the vertical axis and positive up.

where ρis the mass density,p is the pressure,g is the gravitational acceleration and t is time.

Meanwhile,the kinematic boundary conditions are written as:where ( , )x tη is the surface elevation measured from the SWL.

In the HLGN model,only a single assumption on the velocity variation in vertical direction is introduced as:

where unand wnare the unknown velocity coefficients that are obtained aspart of the solution.

We then substitute Eq.(4)into Eq.(1)and Eq.(3b)to eliminatewn(n =0,1,…, K).We substitute Eq.(4)into Eq.(2),multiply each term byznand integrate from - dtoηand eliminate pressure terms.With these steps,we obtain the HLGN equations.The unknowns are un( n=0,1,…, K-1)and η .We refer the reader to Webster et al.[8]for more detailson thederivation of the HLGN equations.

In this paper,we consider a solitary wave with amplitude H propagating from left to right with a constant speed c.Three types of background currents are considered,namely linear-type currents,quadratictype currents and cubic-type currents.Sketch of the physical problem is shown in Fig.1.

Fig.1 (Color online)Sketch of a solitary wave propagating in shear currents

We note that for these three type currents,the current velocity at the bottom isuc( -d )=0and it is uc(0)=U at the SWL.

We solve this physical problem in wave coordinates,whereX =x - ctand Z =z Some boundary conditions should be considered to solve un( n=0,1, …, K-1)and η .If we set the solitary-wave crest at X =0,based on the symmetry characteristics,surface elevation ηand velocity coefficientsunare:

When X →∞since there is no wave-current interaction,the surface elevation =0η .Meanwhile,the boundary conditions should describe the shear currentsexactly.Thus,for the quadratic shear currents discussed in the present study,we have

The Newton-Raphson method is used to obtain the travelling solution. The solitary-wave solution with no currents of the HLGN model is used as the initial values, see Zhao et al.[9]. Then we gradually increase U from 0 to the desired value. See Duan et al.[7]for more details who took a similar approach.

Next, we will show the numerical results of the steady solutions of the solitary wave in nonuniform shear currents. We nondimensionalize all the parameters by g and d. The bar over the following quantities means they are dimensionless.

Fig. 2 (Color online) Solitary-wave profile for the quadraticcurrent case,H=0.1,U=-1.2

Fig. 3 (Color online) Relationship between the wave speedc and current strength U for different currents,H=0.5

The solitary-wave profiles are shown in Fig. 4.From Fig. 4, we see that the solitary-wave profiles are much wider for the opposing-current cases than that for the no-current case. Also, the solitary-wave profiles for the linear-current case, quadratic-current case and cubic-current case show very little differences in Fig. 4.

Fig. 4 (Color online) Profiles of the solitary wave under presence of different shear currents,H=0.5

Fig.5 (Color online) Horizontal velocity along the water column at wave crest of the solitary wave under presence of different shear currents,H =0.5,U = -1.0

The velocity fields are shown in Fig.6.Shown in Fig.6,in the presence of strong opposite current,the horizontal particle velocity near the free surface is negative.Compared with the no-current case,we observe an obvious vortex under the wave crest in other three cases.In the linear-current case,the vertical position of vortex is near Z = -0 .34,while it is near Z =0and Z =0.13for the quadraticcurrent case and cubic-current case,respectively.Note that there is no assumption of irrotationality in the present theory.

Fig.6 Velocity field of the solitary wave under presence of different shear currents,H =0.5

The vorticity along the water column at different horizontal positions(X =0,3,6 and 40)and the vorticity field for the solitary wave under the presence of different shear currents are shown in Figs.8,9,respectively.For the no-current case,the vorticity is zero in the entire domain shown in Figs.8(a),9(a).For the linear-current case,the wave-current interaction does not change the vorticity field as shown in Figs.8(b),9(b).For the quadratic-current case,nonlinear vorticity distribution is observed under the wave crest shown in Fig.8(c).Vorticity is zero at the seafloor for the quadratic-current and cubic-current case.The maximum values are observed at the free surface,which isω= 2and ω= 3for the quadratic-current case and cubic-current case,respectively.The change of vorticity with water depth is different in the two cases of quadratic and cubic currents.Near the seafloor,vorticity changes faster under the quadratic-current case.Near the free surface,however,it changes faster under the cubic-current c ase,see Figs.8(c),8(d).

Next,we consider the particle trajectories.Suppose at initial time t0a given particle is at( x0, z0).We can obtain the new position ( x , z) of the particle at time t as:

Fig.7 (Color online) Particle trajectories at different vertical positions of the solitary wave under presence of different shear currents,H =0.5

In this paper,we focus on a strongly nonlinear solitary wave with H =0.5propagating in the presence of opposing nonuniform shear currents with U = - 1.The linear-type currents,quadratic-type currents and cubic-type currents are studied.Conclusionsare reached below:

(1)The wave-current interaction affects the velocity field,particle trajectories and vorticity field significantly.

(2)While no vortex is observed under the solitary wave alone,presence of opposing shear currents results in the formation of a vortex under the wave.The vertical position of the vortex increases under the linear,quadratic and cubic current profiles,respectively.

(3)Variation of vorticity along the water columns is different under different current profiles.Near the free surface,cubic-current case shows the largest change of vorticity when compared with other cases.Near the seafloor,vorticity changes faster along the vertical axis,under the quadratic-current case.

Fig.8 (Color online)Vorticity distribution ω( Z)at different horizontal positions of the solitary wave under presence of different shear current H =0.5

Fig.9 (Color online) Vorticity field of the solitary wave under presence of different shear currents,H =0.5