TONG Fei-fei

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China

School of Civil and Resource Engineering, The University of Western Australia, Crawley, WA 6009, Australia, E-mail: tongf.fei@gmail.com

SHEN Yong-ming, TANG Jun, CUI Lei

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China

WATER WAVE SIMULATION IN CURVILINEAR COORDINATES USING A TIME-DEPENDENT MILD SLOPE EQUATION*

TONG Fei-fei

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China

School of Civil and Resource Engineering, The University of Western Australia, Crawley, WA 6009, Australia, E-mail: tongf.fei@gmail.com

SHEN Yong-ming, TANG Jun, CUI Lei

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China

(Received May 6, 2010, Revised October 3, 2010)

The purpose of this article is to model the detailed progress of wave propagation in curvilinear coordinates with an effective time-dependent mild slope equation. This was achieved in the following approach, firstly deriving the numerical model of the equation, i.e., Copeland’s hyperbolic mild-slope equation, in orthogonal curvilinear coordinates based on principal of coordinate transformation, and then finding the numerical solution of the transformed model by use of the Alternative Directions Implicit (ADI) method with a space-staggered grid. To test the curvilinear model, two cases of a channel with varying cross section and a semi-circular channel were studied with corresponding analytical solutions. The model was further investigated through a numerical simulation in Ponce de Leon Inlet, USA. Good agreement is reached and therefore, the use of the present model is valid to calculate the progress of wave propagation in areas with curved shorelines, nearshore breakwaters and other complicated geometries.

curvilinear coordinates, mild slope equation, water wave, analytical, numerical modeling

1. Introduction

Most of the numerical models of nearshore wave propagation based on the mild slope equation have been solved by the finite difference method with rectangular grids[1-6]. However, this kind of grids is not suitable for domains with complicated topography with curved boundaries, and it is better to

adopt varying grids according to the water depth

and boundaries. The boundary-fitted curvilinear coordinates have been brought in the simulation of nearshore wave and current for around thirty years, however, until the end of last century, attention had been mostly paid to the effect of different kinds of curvilinear coordinates and finding out analytical solutions[7,8]. Recently, interests about curvilinear wave models have been numerically reignited as Beji and Nadaoka[9]proposed a fully dispersive nonlinear wave model in curvilinear systems, and Yuan et al.[10], Shi and Kirby[11]and Zhang et al.[12,13]introduced curvilinear meshes to their parabolic mild slope equations, respectively. On the other hand, the use of arbitrary curvilinear grids in current simulations in rivers has been investigated intensively by, among others, Bai et al.[14], Zhao et al.[15], Zhang and Shen[16]and Wu and Wang[17]. Significant progress has been made in curvilinear wave models not only because covariant-contravariant tensor was widely used[9,11], but also because several kinds of curvilinear transformation methods were detailedly summarized

* Project supported by the National Natural Science Foundation of China (Grant Nos. 50839001, 50979036), the National Science and Technology Major Special Project of China on Water Pollution Control and Management (Grant No. 2009ZX07528-006-01).

Biography: TONG Fei-fei (1983-), Male, Master

However, it is known that the parabolic approximation of mild slope equation ignores wave reflection in the main propagation direction of waves[1]. Although subsequent researches have made advances to relax this limitation as it might otherwise be, this inferior position still cannot be ignored. The hyperbolic mild slope equation firstly developed by Copeland[1]in the form of a pair of first order partial equations, could describe water wave propagation with a combination of refraction, reflection and diffraction. Madsen and Larsen[2], Lee et al.[3]and Oliveira[4]successively developed Copeland’s model by introducing an efficient numerical methods, enlarging its application areas and summarizing boundary conditions. Zheng et al.[5]brought nonlinear items to the hyperbolic approximation of the mild slope equation and presented an even more efficient finite difference scheme to find out the solution. Interestingly, no literature published has applied this efficient wave model in curvilinear coordinates yet.

This article presents a numerical model of the Copeland hyperbolic mild slope equation[1]in orthogonal coordinates based on basic coordinatetransformation definition. The transformed model is discretized with space-staggered grids and solved with the Alternative Directions Implicit (ADI) method proposed by Zheng et al.[5]in Cartesian coordinates. Model results are validated by three cases involving a linear channel with varying cross section[9], a semi-circular channel[8]and lastly, the Ponce de Leon Inlet[18]. The agreement of numerical solutions to analytical results available suggests that the current model is an efficient tool for wave simulation in shoaling water with complicated geometry.

2. Numerical model

2.1 Governing equations

The time-dependent mild slope equation, introduced by Booij[19]can be expressed as

where η is the surface elevation, C and Cgthe phase velocity and the group velocity, ω the angular frequency, and k the local wave number, which can be determined by the linear wave dispersion relation

where h is the water depth and g the acceleration due to gravity.

where i is the imaginary unit, thus,

Then, substitution of Eq.(4) to Eq.(1) gives,

Based on this euation, Copeland[1]derived the following hyperbolic mild slope equation consisting of a pair of first order partial difference equations.

2.2 Coordinates transformation

Assume that the coordinates in Cartesian systemsare (x,y) and the curvilinear coordinates are (ξ1,ξ2). The aim of coordinates transformation is to find an arbitrary relationship, expressed by, ξ1=(x,y), ξ2=(x,y), to change the chosen boundaries and grids in the physical domain (x,y) to become much more ordered in the image domain (ξ1,ξ2), as shown in Fig.1.

The principle of coordinates transformation can be easily found in literature. For the integrity of this article we simply give the course below, and details could be referred to Tong et al.[20]and Shi et al.[18]In curvilinear coordinates, for a certain function φ,

J is Jacobian, in orthogonal curvilinear coordinates being simplified to J=?gξ2gξ1, and gξ1, gξ2are the Lamé coefficients

Equations (11), (12) and (13) constitute the numerical model of Copeland’s mild slope equation in curvilinear coordinates. For convenience, the superscripts of variables in these transformed equations is omitted hereafter.

3. Numerical approach and boundary conditions

After coordinate transformation, the curvilinear grids can be regarded as rectangular ones. The numerical approach and boundary conditions are similar to that for the Madsen and Larsen’s mild slope equation[20]. Since each term in the derived hyperbolic mild-slope Eqs.(11), (12) and (13) has its counterpart compared to original Eqs.(6), (7) and (8), the ADI method used by Zheng et al.[5]in Cartesian coordinates can be directly applied in the present model.

It should be noticed that only the boundary condition for η needs to be given as the equations to be solved here are first-order ones. Assume that the reflected waves are negligible at the incident boundary, where0x,0A and0θ are the position of initial boundary, wave amplitude and wave direction (based on x-direction in Cartesian coordinates) of incident waves.

Lateral boundary conditions are expressed as[20]

where φ is a certain scalar function at the boundary, n the unit vector normal to the boundary and α the absorption coefficient of the boundary.

where θ is the wave direction at the right boundary.

In order to minimize the effect of wave reflection due to unknown wave direction, the so-called sponge layer might be needed to place at the computationalboundary[2,4].

4. Validations and results analysis

The transformed model is validated by three tests, and the numerical results are compared to available analytical results. The first case is about wave propagation in a liner channel with varying width, the second example describes wave movement in a semi-circular channel, and the last one gives an engineering application of the model to the Ponce de Leon Inlet, USA. Analysis of the numerical results is also presented in this part.

When regular waves propagate in a channel with slowly varying cross section, the wave amplitude changes according to the following formula[9]

where a0and b0are wave amplitude and the breadth of the channel at x=0, a(x) and b(x) are the corresponding values of an arbitrary section of the channel.

Figure 3 shows the surface variation of waves 105 s after its entering the basin. The data are normalized to the wavelength and local breadth of the channel in the x and y directions, respectively. Figure 4 illustrates the comparison of numerical and analytical wave amplitude along the mid-section of the channel. They agree perfectly well with each other. Probably as supposed, the wave amplitude increases when the channel converges, and vice versa, which could be demonstrated by both numerical and analytical data.

This article chooses the largest channel from Ref.[8] for study as the wave pattern in this case is much more complicated. The channel lies between two circular vertical wall, with the inner radius r=75m and the outer radius R=200m, and covers a 180oarc, as shown in Fig.5. The depth of the channel is of constant value of 4 m. The coordinate transformation can be expressed as

The time-dependent hyperbolic mild-slope equation is adopted in this article, which makes our calculations show clearly the progress of waves propagating through the flume. Figures 7 and 8 illustrate the transient wave surface distribution after 30 s and 60 s of incidence. It is seen that waves basically propagate straight forward at the beginning, but as the channel bend, wave diffraction phenomenon appears near the inner wall, Fig.8 shows that as waves spread the role of reflection by the outer wall significantly is enhanced, resulting in the disturbance of wave distribution around the bend between 60oand 90o, where reflected wave and diffraction wave produce a strong superposition pattern. Then, wave surface variation seems to be more regular. After 120 s of incidence, waves have spread throughout the channel. Near the outlet boundary, reflection of the outer wall is re-strengthened as shown in Fig.9.

The calculated wave heights are compared with analytical results in Fig.10 along the 35oand 60oradiuses, respectively. Figure 11 shows a comparisonof numerical and analytical solutions for the normalized surface displacement along the inner and outer walls. It can be easily seen that the results of this article are in good agreement with the theoretical solution. Disagreement can only be found near the outlet boundary, mainly due to the wave angle there being difficult to determine. This indicates that the numerical model of Copeland’s mild-slope equation in curvilinear coordinates is accurate. In addition, compared to the analytical model[8], the present model can clearly exhibit the process of wave propagation in the channel, while the numerical computation is much more efficient than the Boussinesq model in curvilinear coordinates[18].

4.3 An engineering application of the present model

Shi et al.[18]carried out numerical simulation of wave propagation in the Ponce de Leon Inlet, USA, using Boussinesq equation in curvilinear coordinates. The Ponce de Leon Inlet consists of two estuaries, a breakwater and complex terrain and shorelines, as shown in Fig.12. We choose the sea area for study and the orthogonal curvilinear grids for computation are shown in Fig.13. The grids resolution near shorelines and the breakwater are a little higher so that waves in these domains could be better simulated.

The reflection of the incident boundary has not been taken into account in the calculation. Regular incident waves enter the domain from offshore boundary with a small amplitude and 15 s in wave period. The time step is set to be 1.0 s. For this case the model needs approximately 300 iterations to yield the desired results as shown in Fig.14. It is clear that the breakwater has a significant effect on the distribution of water waves. Strong reflection of water waves occurs in the vicinity of upward side of the breakwater, while there seem to be small wave distributing in the other side, mainly generated by the wave diffraction. Near the center of the calculation region, wavelengths change to be much shorter than in most other areas, due to the rapid shoaling water depth. All of these is in line with the actual situation. For the same reasons as Shi et al.[18], i.e., the present model does not consider wave breaking, therefore the present article does not compare the numerical results with field data; but this example shows that the numerical model in this article can be further applied to geometries with complicated topography and boundary conditions in engineering.

5. Conclusions

Compared to elliptic and parabolic mild-slope equation, the hyperbolic mild-slope equation can be applied to large sea areas and considers a combination of wave reflection, diffraction and refraction. Thus it is a more efficient tool of nearshore wave model. In order to meet the engineering requirements, i.e., to calculate wave propagation in areas with complicated topography and curved shorelines in most engineering cases, this article has derived a numerical model ofCopeland hyperbolic mild-slope equation[1]in orthogonal curvilinear coordinates. The transformed model is solved by a finite difference scheme using the efficient ADI method. Three numerical tests are conducted to validate the accuracy of the present model. The agreement between numerical results and available analytical solution indicates that this numerical model can effectively simulate the wave transmission in complex regions. At the same time, compared to the analytical result[8], the present numerical model show a clear process of the wave propagation and deformation, while it is much more efficient than the Boussinesq model in curvilinear coordinates[18]in its numerical computation. In addition, the application of the model to the Ponce de Leon Inlet, USA, suggests that it is applicable in large sea areas.

However, the present model needs to be further improved by introducing the recent advancement of mild slope equation as discussed in Section 1, for example to include wave breaking items and to consider irregular incident wave, in order to be applied more effectively in engineering.

[1] COPELAND G. J. M. A practical alternative to the mild-slope wave equation[J]. Coastal Engineering, 1985, 9(2): l25-249.

[2] MADSEN P. A., LARSEN J. An efficient finitedifference approach to the mild-slope equation[J]. Coastal Engineering, 1987, 11(4): 329-351.

[3] LEE C., PARK W. S. and CHO Y. et al. Hyperbolic mild-slope equations extended to account for rapidly varying topography[J]. Coastal Engineering, 1998, 34(3-4): 243-257.

[4] OLIVEIRA F. S. B. F. Improvement on open boundaries on a time dependent numerical model of wave propagation in the nearshore region[J]. Ocean Engineering, 2000, 28(1): 95-115.

[5] ZHENG Yong-hong, SHEN Yong-ming and QIU Da-hong. New numerical scheme for simulation of hyperbolic mild-slope equation[J]. China Ocean Engineering, 2001, 15(2): 185-194.

[6] TANG J., SHEN Y. M. and ZHENG Y. H. et al. An efficient and flexible computational model for solving the mild slope equation[J]. Coastal Engineering, 2004, 51(2): l43-154.

[7] KIRBY J. T., DALRYMPLE R. A. and KAKU H. Parabolic approximations for water waves in conformal coordinate systems[J]. Coastal Engineering, 1994, 23(3-4): 185-213.

[8] DALRYMPLE R. A., KIRBY J. T. and MARTIN P. A. Spectral methods for forward-propagating water waves in conformally-mapped channels[J]. Applied Ocean Research, 1994, 16(5): 249-266.

[9] BEJI S., NADAOKA K. Fully dispersive nonlinear water wave model in curvilinear coordinates[J]. Journal of Computational Physics, 2004, 198(2): 645-658.

[10] YUAN D. K., LIN B. L. and XIAO H. Parabolic wave propagation modeling in orthogonal coordinate systems[J]. China Ocean Engineering, 2004, 18(3): 445-456.

[11] SHI F. Y., KIRBY J. T. Curvilinear parabolic approximation for surface wave transformation with wave-current interaction[J]. Journal of Computational Physics, 2005, 204(2): 562-586.

[12] ZHANG H. S., ZHU L. S. and YOU Y. X. A numerical model for wave propagation in curvilinear coordinates[J]. Coastal Engineering, 2005, 52(6): 513-533.

[13] ZHANG Hong-sheng, YOU Yun-xiang and ZHULiang-sheng et al. Mathematical models for wave propagation in curvilinear coordinates and comparisons among different models[J]. Journal of Hydrodynamics, Ser. A, 2005, 20(1): 106-117(in Chinese).

[14] BAI Yu-chuan, XU Dong and LU Dong-qiang. Numerical simulation of two-dimensional dam-break flows in curved channels[J]. Journal of Hydrodynamics, Ser. B, 2007, 19(6): 726-735.

[15] ZHAO Ming-deng, LI Tai-ru and HUAI Wen-xin et al. Finite proximate method for convection-diffusion equation[J]. Journal of Hydrodynamics, 2008, 20(1): 47-53.

[16] ZHANG Ming-liang, SHEN Yong-ming. Threedimensional simulation of meandering river based on 3-D RNG k-ε turbulence model[J]. Journal of Hydrodynamics, 2008, 20(4): 448-455.

[17] WU Zhao-chun, WANG Dao-zeng. Numerical solution for tidal flow in opening channel using combined MacCormack-finite analysis method[J]. Journal of Hydrodynamics, 2009, 21(4): 505-511.

[18] SHI F. Y., DALRYMPLE R. A. and KIRBY J. T. et al. A fully nonlinear Boussinesq model in generalized curvilinear coordinates[J]. Coastal Engineering, 2001, 42(4): 337-358.

[19] BOOIJ N. Gravity waves on water with non-uniform depth and current[D]. Ph. D. Thesis, Delft, The Netherlands: Delft University of Technology, 1981.

[20] TONG Fei-fei, SHEN Yong-ming and TANG Jun et al. Numerical modeling of the hyperbolic mild-slope equation in curvilinear coordinates[J]. China Ocean Engineering, 2010, 24(4): 585-596.

[21] LI B. An evolution equation for water waves[J]. Coastal Engineering, 1994, 23(3-4): 227-242.

10.1016/S1001-6058(09)60118-9

SHEN Yong-ming,

E-mail: ymshen@dlut.edu.cnand curvilinear coordinates were employed to the nonlinear Boussinesq model[18]. All of these studies proved that curvilinear numerical models are applicable to practical problems of simulating current and water wave propagation in complicated geometries, such as meandering rivers, estuaries and areas near islands.