Numerical modeling of refraction and diffraction L. Balas, A. inan Civil Engineering Department, Gazi University, Turkey Abstract A numerical model which simulates the propagation of waves over a complex bathymetry where the bottom contours are not straight and parallel, has been developed. In the model, the combined effects of refraction and diffraction can be considered. It is assumed that waves are linear, harmonic, and irrotational, and the effects of currents and reflection on the wave propagation are negligible. Mild slope equation is modified, assuming that there is no energy propagation along the wave crests, however, the wave phase function changes to handle any horizontal variation in the wave height. In this manner, the disadvantage of the parabolic approximation that one grid coordinate should follow the dominant wave direction, which causes problems in complex bathymetries, has been overcome. The finite difference method has been selected as the solution method. Applied methodology allows the check for breaking. Model results are compared with those from laboratory experiments published in the literature, and model is applied to Marmara Sea. 1 Introduction The wave ray method and linear gravity wave theory were used in the early works of wave transformation. Berkoffll] solved transformations of linear waves considering the effect of both refraction and diffraction with an elliptic equation. This elliptic equation is known in the literature as mild slope equation. Radder[lO] simplified the mild slope equation with a parabolic approximation. The advantages of his model, are the validity for a non-homogenous media and the applicability to short waves in large coastal areas with complex bottom
186 Coastal Engineering V: Computer Modellit~g of Seas and Coastal Regions topography. Booij[3] solved the mild slope as a fimction of bottom slope. Since the waves are periodic, steady state solution was used. Model was proved to be applicable till a bottom slope of 113. Kirby&Darlymple[S] solved the parabolic equation for the Stokes waves by the multi-scale perturbation method. Copeland[5] solved the first order mild slope equation, including reflected waves. Chamberlain&Porter[4] used modified mild slope equations in the wave transformation. If sea bed is formed by ripples, modified mild slope equation is used, because normal mild slope equation does not give good results under these conditions. Tang&Quellet[l l] adapted nonlinear mild slope equation to the multi-frequent waves. As linear part of the equations includes mild slope equations, nonlinear part of them contains Boussinesq equation. The parabolic approximation has the main disadvantage that it requires one grid coordinate to follow the dominant wave direction (Ebersole[7]). When the bottom contours are not straight and parallel as in the case of complex bathymetries, this requirement causes problems. The model proposed by Ebersole[7] is an alternative approach to solve the open coast wave propagation problem in a more general way. It was based on the assumption that no energy was propagated along wave crests, however the wave phase hnction changed to accommodate any horizontal variation in wave height. 2 Theory The complex velocity potential has been chosen as (Ebersole[7]) ; + = aei' (1 in which, a(x,y): wave amplitude, s(x,y): phase function of the wave. If Eqn (1) is inserted to the equation that describes the propagation of harmonic linear waves in two horizontal dimensions, the following equation can be derived; in which V: horizontal gradient operator. To account the effect of diffraction, the wave phase fimction changes to consider any horizontal variation in the wave height. By the use of irrotationality of the gradient of the wave phase, function following equations can be derived; d in which i and j are the unit vectors in the X and y directions, respectively. The local wave angle, B(x,y) can be found from the following expression;
Coastal Engineering V: Computer Modelling of Seas and Coastal Region5 187 The following energy equation is used to determine wave amplitude; Eqn (6) together with Eqn (2) and Eqn (7) result in the set of three equations that will be solved in terms of three wave parameters, H, 8 and / Vs / (Ebersole[7]). Eqns (6),(8) and (9) describe the refraction and diffraction phenomena. The basic assumptions are that the waves are linear, harmonic, irrotational, reflection is neglected and bottom slopes are small. 3 Numerical solution Solution method is a finite difference method that uses the mesh system shown in Figure (1). The fmite difference approximations can handle the variations in the horizontal mesh sizes. The horizontal mesh size Ax in the x-coordinate is orthogonal to the horizontal mesh size Ay in the y-coordinate. The horizontal mesh sizes Ax and Ay can be different from each other. Also, Ax can vary along the X coordinate and Ay can vary along the y coordinate. Figure 1. Finite difference mesh system
188 Coastal Engineering V: Computer Modelling of Seas and Coastal Regions Input model parameters are the deep water wave parameters, wave height (Ho), wave approach angle (80) and the wave period (T). Partial derivatives in the X-direction are expressed by forward finite differences of order O(Ax), and the partial derivatives in the y-direction are expressed by central finite differences of order 0(ay2) in equation (6) and in Equation (9), whereas partial derivatives in the X-direction are approximated with backward finite differences of order O(Ax), and partial derivatives in the y-direction are expressed by central finite differences of order 0(ay2) in Equation (8). Wave breaking is controlled during the computations. 4 Model applications Model predictions are compared with the results of a laboratory experiment (Whalin[l3]). The wave tank used in the experiments is shown in Figure 2. Deep water wave parameters are T=1.0 sec and H=0.019 m. Along the lateral boundaries, the gradient of wave height perpendicular to side walls is assumed to be zero, and wave approach angles are assumed to be in the X direction. Topography is symmetric about y=3.048m. Water depth changes from 0.4572m to 0.1524rn. Two different mesh sizes are used in the X-direction. For X values larger than x=15 m, the mesh size is Ax=0.762m, and it is equal to Ax=0.305m for X values smaller than x=15 m. The mesh size used in the y-direction is Ay=0.762m. Lineer waves were produced at the water depth of 0.4572 m. On the slope, there are semicircular steps that result in strong wave convergence. In this region, diffractive effects play an important role, and model differs from pure refraction models considerably. Comparisons of model predictions and measured data are shown in Figure 3. Results of study performed by Tsay&Liu[l2]) are presented in Figure 3 for comparison. Model simulation reflects well the effect of diffraction phenomenon and model predictions are in good agreement with the experimental results. Figure 2. The bathymetry of wave tank (water depths are in m) (Whalin, 1972)
Coastal Engineering V: Computer Modelling of Seas and Coastal Regions l89 - model -----Tsay and Liu, non Ihnear -Tsay and Liu, linear / *.Wholin / / / Figure 3. Comparison of model predictions where * experimental data (Whalin[l3]), solid line: model predictions, - - - numerical solution (nonlinear) of Tsay&Liu[l2], -- -- numerical solution (linear) of Tsay&Liu[l2] (T= l S, a=0.0195m and 8=0 ). In the second application, model predictions are compared with the results of wave tank experiment done by Berkoff et a1.[2]. The wave period of incoming waves is T=ls, and the wave height is H=0.01058 m. Wave approach angle is 18.5". Water depths in the tank decreases from 0.45m with a bottom slope of 1150. The bathymetry of the wave tank is given in Figure (4). In the numerical model grid sizes are selected as Ax=0.5m and Ay=O.Sm. Numerical model predictions along the cross section of x=l1 m and x=13 m are compared with the experimental data of Berkhoff et a1.[2], and presented in Figure (5) and in Figure (6), respectively. For comparison, numerical predictions of Kirby&Dalrymple[9] are depicted in the figures as well. Model predictions are in good agreement with the measurements. Model well reflects the experimental results near the shoal area.
190 Coastal Engineering V: Computer Modelling of Seas and Coastal Regions Figure 4. Wave tank bathymetry (water depths are in m) (Berkoff, 1982). - model +---. Kirby,non-linear.*.I.. Berkoff et al. Figure 5. Variation of relative wave height at x=l lm.
Coastal Engineering 1': Computer Modelling of Seas and Coastal Regions 19 1 - model e - - -. Kirby,non-lineor... Berkoff et al. Figure 6. Variation of relative wave height at x=13m. 5 Application to Marmara Sea In the Sea of Marmara, Marmara New Port Breakwater will be constructed between the city of Tekirdag and Marmara Ereglisi (DLH[6]). For the area shown in Figure 7, the numerical model has been applied to simulate the wave transformations. Here, wave transformation from the dominant wave direction which is the SSW direction, is presented. The deep water wave parameters are used to specify the offshore boundary conditions and zero gradient boundary conditions are applied for wave heights and wave angles along the lateral boundaries. Deep water parameters are wave period T=6 S., wave height H=3 m. and approach angle 8=20. Model predictions are presented in Figure 8. Model provides reasonable estimations for the area. Waves converge on the shoal, conveyance of energy onto shoal results in the decrease of wave heights. Model can be used successfully for the areas having complicated bathymetries.
192 Coastal Engineering V: Computer Modelling of Sea5 and Coastal Regions X (m) Figure 7. Batymetry of the computational area. Figure 8. Wave heights(m) in the computational area.
Coastal Engineering V: Computer Modelling of Seas and Coastal Regions 193 6 Conclusions A numerical model has been developed to simulate the wave transformation of monochromatic linear waves as they propagate over irregular bathymetries. Model predictions are in good agreement with the experimental results. Model successfull application to a real coastal water body has been demonstrated. Model can simulate the effect of pure refraction or effects of refraction together with diffraction which is important over complex bathymetries. There is no assumption regarding the curvature of the wave height in any direction in the model. Only one computational domain is enough to simulate the transformation of waves from different directions with different approach angles. Developed model is a reliable tool for simulating the transformation of linear waves over complicated bathymetries. References 1. Berkhoff, J. C. W., Computation of combined refraction-diffractionl Proceedings of 13th International Conference on Coastal Engineering, ASCE, I, pp. 472-490, 1972. 2. Berkhoff, J.C.W., Booy, N. & Radder, A.C., Verification of numerical wave propagation models for simple harmonic linear water waves, Coastal Engineering, 6, pp.255-279, 1982. 3. Booij, N.: A note on the accuracy of the mild slope equation', Coastal Engineering, 7, pp. 19 1-20:, 1983. 4. Chamberlain, P.G. & Porter, D., The Modified Mild-Slope Equation, Journal offluid Mechanics, 291, pp. 393-407, 1995. 5. Copeland, G.J.M., A practical alternative to the mild-slope wave equation, Coastal Engineering, 9, pp. 125-149, 1985. 6. DLH, Results of Marmara Xew Port Breakwater Stability Experiments, Ministry of Transportation, General Directorate of Construction of Railways, Ports and Airports, Technical Report No:4 (In Turkish), 1999. 7. Ebersole, B. A., Refraction-Diffraction Model For Linear Water Waves, Journal of Waterway, Port, Coastal and Ocean Engineering, 11 1, pp. 939-953, 1985. 8. Kirby, J.T. & Dalrymple,R.A., A parabolic equation for the combined refraction- diffraction of Stokes waves by mildly varying topography, Journal of Fluid Mechanics, 136, pp.453-456, 1983.
194 Coastal Engineering V: Computer Modelling of Seas and Coastal Regions 9. Kirby, J.T. & Dalrymple, R.A. Verification of a parabolic equation for propagation of weakly- nonlinear waves, Coastal Engineering, 8, pp. 219-232, 1984 10. Radder, A.C., On the parabolic equation method for water-wave propagation, Journal offluid Mechanics, 95, pp. 159-176, 1979. 11. Tang, Y. & Ouellet Y., A new kind of nonlinear mild-slope equation for combined refraction- diffraction of multifrequency waves, Coastal Engineering, 31, pp.3-36, 1997. 12. Tsay, T.K. & Liu, P.L.F., Refraction- diffraction model for weakly nonlinear water waves, Jouranal of Fluid Mechanics, 141, pp: 265-274, 1984. 13. Whalin,.R. W., Wave refraction theory in a convergence zone, Proc. of the 13th Coastal Engineering Conference, Vol.1, pp:45 1-470, 1972.