Numerical Method for Calculation of the Solitary Waves on Water of the Arbitrary Depth

M. Yas'ko

Dnipropetrovsk University, Mechanical and Mathematical Faculty,
320625, Dnipropetrovsk-10, Ukraine

Abstract

A numerical method is developed for calculation steady nonlinear water waves travelling over the free surface of an inviscid, incompressible fluid under the assumption of potential flow. The problem is treated as two dimensional for stream function of disturbed flow. The numerical model uses a boundary element method for discretization in space. For definition of the unknown free boundary an iterative procedure is proposed. This procedure based on the solution of the third boundary problem on an each step. The features of the method is a comparatively small computational effort and enhance efficiency and precision. For illustration, some computational results are presented for the shape and integral properties of the solitary waves.

§1 Introduction

Solitary waves in shallow water have been preoccupied since the classical observations by Scott Russel in 1838. For reviews of modern numerical calculations of this type of wave see surveys of Miles [1] and Schwartz and Fenton [2]. After that solitary waves in water of finite depth have also been calculated by Hunter and Vanden-Broeck [3], and Tanaka [4]. The numerical calculations of the nonlinear waves are based on two techniques:
- methods of a transformed plane, in which physical domain is conformally mapped to a flat-bedded domain or to circle and conformal mapping have been used for solution, in works of Longuet-Higgins [5], Hunter and Vanden-Broeck[3], Tanaka[4];
- methods of a physical plane, in which boundary integral equations have been solved numerically in physical plane. For example, Rottman and Olfe [6] used boundary integral equation and Newton's iterative procedure for the solution of nonlinear algebraic system, Rienecker and Fenton [7] used Fourier expansion for this problem.

In the present work the direct boundary element method and new iterative procedure have been used for the solution of the solitary wave problem. The mathematical problem is formulated in §2, where an equation is written for a stream function ψ in a physical plane. In §3 the method of calculation of nonlinear problem is described. The numerical results for shape and integral properties of solitary waves are presented in §4.

§2 Problem definition

We consider a steady solitary wave, moving with speed U to the right, on the surface of a frictionless, irrotational, incompressible fluid of finite depth H, as in figure 1. The free surface is at constant pressure, and has zero surface tension.

Figure 1. The definition sketch of the solitary wave.

We choose Cartesian coordinates connected with wave with the x-axis parallel to the bottom and the z-axis directly vertically upwards. The coordinate z is measured from the level of the free surface at |x|=∞. There is a uniform gravitational field. Acceleration of the gravity g is acting in the negative z-direction.

We choose U as the unit of velocity and L=U2/2g as the unit of length. Thus the problem has one parameter - dimensionless depth h=2gH / U2.

The governing equation for the 2D flows in dimensionless variables is
Δψ = 0 (1)
where ψ is the disturbed stream function (stream function in fixed coordinate system); the velocity in moving coordinate system (x,z) is
Vx=1+ψ,z, Vz=-ψ,x

The boundary conditions can be described in the following way. As |x|–>∞, the flow is require to approach a uniform stream with constant velocity U in x-direction and uniform depth h. The kinematic condition on free surfaces (x(s), z(s)) and horizontal bottom is
Vn=-ψ,s=0, (2)
where n is the unit outward normal vector,s is the unit tangential vector, s represents the curvilinear abscissae of the point on a free streamline. Equation (2) indicates that the flux through the boundary is zero. This equation can be integrated over the free streamline with the condition ψ(∞)=0 and it becomes
ψ(s)=z(s), (3)
The horizontal bottom is a streamline also on which we require ψ=0.

On the free surface, where a pressure is constant, the Bernoulli's equation yields
Vs2 + z = 1, (4)
where Vs=ψ,n -x' is a tangential velocity on the free surface in the moving coordinate system. Then the dynamic boundary condition on the free boundary may be described in following manner
ψ,n = x' - (1-z)½, (5)
where s - is a curvilinear abscissae of the point on free streamline.

On the infinity
limx –>∞ ψ = 0 (6)

Thus, we have a classic free boundary problem: a location of the free streamline is unknown and two boundary conditions (3) and (5) are known on the free surface. For this problem, one parameter defines a unique solution: dimensionless depth h . Of course, we must denote that trivial solution ( ψ=0 and flat free surface) exists for any h.

§3 Numerical method

Let be initial location of the free surface (x0(s),z0(s)) is known. The boundary integral representation of solution of the equation
½ψ = ∫S (ψP,nGMP -ψPGMP,n) dSP (7)
where GMP=FMP-FLP is a Green's function for half-plane; M(x,z) and P(ξ,η) are points on the boundary, L(x,-2h-z) is a point symmetric M relatively bottom plane; FMP = -(2π)-1 ln rMP is a fundamental solution of the Laplace equation for infinity isotropic 2D medium; rMP is a distance between points M and P.

Consider on the free surface the boundary condition of the third kind
ψ,n = x'0 - ½(1-z0)-½(2-z0-ψ) = αψ+β, (8)
where α and β are coefficients, depending upon shape of free surface.

The discretization of (7), which leads to the classic "boundary element method" technique (see, Brebbia and others [8] is described below. In the boundary element method, the above integral equation is solved numerically by dividing the boundary S into N boundary elements, in each of which ψ and ψ,n are approximated by constants. We denote these values by ψi and ψi,n, i=1,…N; and apply equation (8) at one nodal point Mi in the center of each boundary element to obtain
½ψi =Aij ψj,n+Bij ψj (9)
Aij=∫SjGij,ndSP Bij=-∫SjGijdSP
where ∫Sj denotes integration over the j-th boundary element. Here and what follows the summation over repeating indices is assumed. Coefficients Aij and Bij of the linear system of equations (9) are integrated analytically over boundary elements using the analitical formulas.

Eliminating the ψi,n from each elements on the free surface by applying the boundary condition (8) in each nodal point, we thus obtain a system of N linear algebraic equations with N unknown ψi
(Aij αj + Bij - ½δij) ψj= - Aij ψj (10)
where δij is the Kroneker delta. The system of N linear algebraic equations (10) was solved by the direct Gaussian elimination method.

In order to solve the problem of the free boundary, the shape of the free streamline must be calculated by successive iterations. The new location of the free surface is calculated by formula
z = ½ (z0 + ψ), (10)
after that iterative cycle continued. If iteration process is coincided, then formula (8) translates in dynamic boundary condition (5). The iterative procedure is continued until a converge criterion is satisfied. Usually, 15÷30 iterations demanded for the coincidence. Number of iterations increased for highest waves to 40÷50. On the free boundary, the velocity component normal to the free surface will be zero only at the convergence of the iterative process. After finishing the iterative process , the maximum height of wave and other characteristics have been defined as a function of dimensionless depth h.

§4 Numerical results

Number of boundary elements for the half of the wave was N=80 for all computations. Initial approximation of the free streamline calculated by formula
z = a sech² cx,
where a=2.0-h and c=0.667a½. To demonstrate the numerical scheme developed above, we consider a 2D solitary wave. The computations showed that solitary waves exist for h in range from 2.0 to 1.1895. The computed profiles of the free surface are presented in figure 2 at the values of h=1.8, 1.6, 1.4 and 1.19 (curves 1, 2, 3 and 4 accordingly). In case h=1.8 we have a gently sloping solitary wave. In case h=1.19 we have a solitary wave nearly to maximum height. In this case the numerical method gives sharper angle at the crest ~120°.

Figure 2. Profiles of the solitary waves.

The highest wave had been computated at h=1.1895 and her amplitude was Zmax=0.981. Extrapolation for Zmax=1 gave value hmin=1.189. In table 1 the ratio Zmax compared with data of other authors.

Table 1. Comparison of maximum wave height with data of other authors.

workszmax/hmin
Lenau [9]0.827
Longuet-Higgins and Fenton [10]0.827
Yamada and others [11]0.8262
Witting [12]0.8332
Cooker [13]0.83
present work0.841
Thus, we can say that velocity of solitary waves may be in range (gH)½ ÷ 1.3(gH)½.
The trajectory of point on the free boundary are presented in figure 3 for solitary waves shown in figure 2. To calculate the trajectory of a surface fluid particle we find its velocity and integrate by the time-stepping algorithm of two order. Denote that this curves are very similar cycloids.

Figure 3. Trajectories of the point on the free surface.

Now we shall consider some integral quantities of the solitary wave relative to fixed coordinate system. Firstly, we may define net displacement, or excess mass
S = ∫S z dx.
We may define the dimensionless gravitational potential energy
Ep =½ ∫S z² dx.
and the kinetic energy of the relative motion, which can be written using Green's theorem as follows
Ek =½ ∫S ψ ψ,n dx.

The computed values of Zmax, Ek, Ep and S are shown in table 2.

Table 2. Calculated integral properties of solitary waves.

hzmaxEKEP S
1.900.10410.03390.06681.9805
1.800.20470.09110.17572.6380
1.700.30620.15950.30093.0266
1.600.40920.23350.43043.2461
1.500.51430.30840.55493.3524
1.400.62430.37950.66583.3495
1.300.74380.44100.75253.2433
1.220.86370.47470.78893.0505
1.210.88380.47670.78913.0131
1.200.90930.47730.78662.9658
1.190.95750.47300.77452.8835
The wave amplitude Zmax, potential and kinetic energy and also displacement S are presented in figure 4 as a functions of dimensionless depth h (curves 1, 2, 3 and 4 accordingly).

Figure 4. Integral propeties of the solitary waves.

If h is decreased than amplitude Zmax of the solitary wave is increased right up to 1 for h=1.189. In range 1.3 < h < 2 the amplitude Zmax change nearly linearly and solitary waves can be accurately approximated by solutions to the Kortveg-de Vries equation. Other functions is changed nonmonotonically with h, but the potential energy always exceeds the kinetic.

§5 Conclusions

The basic principle of the boundary element method has been presented in this paper for the calculation of the solitary waves in nonlinear formulations. The advantage of this method is the simplicity of the boundary calculations in the physical plane. Comparisons with analytical and numerical data of other authors suggest that the solutions obtained by the present numerical method are quite accurate. The flow is especially well modeled in the case of the highest waves. Similar numerical technique could be used to investigation of the gravity-capillary and periodic nonlinear waves.

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