Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
186 Coastal Engineering V: Computer Modellit~gof 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;
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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 ( a y 2 ) 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)
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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.
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
190 Coastal Engineering V: Computer Modelling of Seas and Coastal Regions
Figure 4. Wave tank bathymetry (water depths are in m) (Berkoff, 1982).
+---.
.*.I..
model
Kirby,non-linear
Berkoff et al.
Figure 5. Variation of relative wave height at x=l lm.
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
Coastal Engineering 1:' Computer Modelling of Seas and Coastal Regions
191
--.
-
model
Kirby,non-lineor
Berkoff et al.
......
e
-
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.
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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.
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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, 111 , pp. 939953, 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.
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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. 219232, 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.