Modeling of random wave transformation with strong wave-induced coastal currents

Water Science and Engineering, Mar. 008, Vol. 1, No. 1, 18 6. DOI: 10.388/j. issn. 1674-370.008.01.003 ISSN 1674 370, http://kk.hhu.edu.cn, e-mail: wse@hhu.edu.cn Modeling of random wave transformation with strong wave-induced coastal currents Zheng Jinhai* 1, H. Mase, Li Tongfei 1 1. College of Ocean, Hohai University, Nanjing 10098, P. R. China. Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto 611-0011, Japan Astract: The propagation and transformation of multi-directional and uni-directional random waves over a coast with complicated athymetric and geometric features are studied experimentally and numerically. Laoratory investigation indicates that wave energy convergence and divergence cause strong coastal currents to develop and inversely modify the wave fields. A coastal spectral wave model, ased on the wave action alance equation with diffraction effect (WABED), is used to simulate the transformation of random waves over the complicated athymetry. The diffraction effect in the wave model is derived from a paraolic approximation of wave theory, and the mean energy dissipation rate per unit horizontal area due to wave reaking is parameterized y the ore-ased formulation with a reaker index of 0.73. The numerically simulated wave field without considering coastal currents is different from that of experiments, whereas model results considering currents clearly reproduce the intensification of wave height in front of concave shorelines. Key words: random wave; coastal current; spectral wave model; numerical simulation 1 Introduction In some coastal areas with complicated athymetry, the convergence and divergence of wave energy inversely create a Doppler shift, influence wave refraction, reflection, and reaking, and significantly modify the overall redistriution of wave fields, causing strong coastal currents (Castelle et al. 006; MacMahan et al. 006). Under such circumstances, the effects of amient currents on wave transformations should e taken into account in coastal wave prediction. Reliale wave predictions in coastal areas are crucial to coastal engineering applications associated with shore protection, coastal morphological evolution, haror construction, navigation channel maintenance and maritime disaster reduction. The modeling of coastal wave transformation has therefore een a suject of considerale interest in the field of haror, coastal and offshore engineering, and has advanced a great deal in the past few decades. Several models that can predict comined refraction, diffraction, reflection and dissipation are now used in practice (Berkhoff 197; Nwogu 1993; Resio 1993; Kiry and Dalrymple 1994; Booij et al. 1999; Panchang and Demirilek 001; Lin et al. 008). However, each of these models is accompanied y its own set of modeling difficulties. The purpose of this study is to test the aility of the WABED wave model to predict the transformation of uni-directional and multi-directional random waves over a athymetrically and geometrically complicated coast where wave-induced coastal currents are well developed This work was supported y the Program for New Century Excellent Talents in Universities (Grant No. NCET-07-055). *Corresponding author (e-mail: jhzheng@hhu.edu.cn) Received Jan. 07, 008; accepted Mar. 1, 008

and affect the overall distriution of wave energy. It is a spectral wave model ased on the wave action alance equation, with a diffraction term formulated from a paraolic approximation of wave theory (Mase et al. 005). The mean energy dissipation rate per unit horizontal area due to wave reaking is parameterized y the ore-ased formulation of Battjes and Janssen (1978) with a reaker height of 0.73 times of the water depth (Zheng et al. 006). Experimental setup Experimental studies were carried out in the Research and Development Department of the Kansai Electric Power Company, Inc., Japan. The wave asin was 0 m long and 38 m wide, and a model each was uilt in the middle. The nearshore ottom topography is plotted in Figure 1. The coastal configuration is a concave curve with two headlands. An offshore rip current was therefore formed y the assemling of two longshore currents. Uni-directional and multi-directional Bretschneider-Mitsuyasu type wave spectra were generated y an irregular wave-maker with 60 wave paddles, each of which had a width of 30 cm. The wave directional spreading function was of the Mitsuyasu type with S max =5. The model-to-prototype scale was 1:15. Using the electrical capacitance wave gauges and the electromagnetic type current meters, oth wave and current data were measured at 17 locations in six regions, as shown in Figure. The data were collected for 400 s after wave paddles started to move, and the data of first 60 s were discarded in the analysis to avoid transient effects. Tale 1 shows the test conditions with different return periods. Tale 1 Test conditions Return period (years) Significant wave height (cm) Significant wave period (s) 1 1.86 0.68 5 5.14 1.00 10 5.75 1.06 0 6.6 1.11 100 7.6 1.1 Figure 3 displays experimental results of the transformation of multi-directional and uni-directional waves with a return period of 100 years, in which the contours denote the value of the local measured significant wave height eing normalized y the incident one. The maximum measured current velocity induced y uni-directional waves in the prototype was.7 m/s. Measurements reveal that the wave-induced nearshore currents are very strong, resulting in wave height intensification in the vicinity of the concave coasts. On the other hand, the wave heights in front of the headland do not increase due to the wave-induced coastal currents moving in the same direction of wave transformation. It should e noted that the mass transport flux is not measured in the offshore area, since it is too small to affect the wave field. In the multi-directional wave case, there is a region around (9.0, 6.0) in which the wave height is 1.05 times of that of the incident wave. In the uni-directional wave case, there is a Zheng Jinhai et al. Water Science and Engineering, Mar. 008, Vol. 1, No. 1, 18 6 19

region around (9.0, 6.0) in which the wave height is 1.0 times of that of the incident wave. The wave height intensification area in the uni-directional wave case is far greater than that in the multi-directional wave case. Theoretically, the wave height should e smaller due to energy divergence in this region if no current occur, therefore, the wave height intensification here is oviously caused y the strong opposing current, as shown in Figure 3. Figure 1 Bank line and nearshore water depth contour of physical model Figure Locations of wave gauges and current meters (a) Multi-directional wave case () Uni-directional wave case Figure 3 Experimental results of normalized significant wave heights and wave-induced currents 3 Numerical simulations 3.1 Description of the WABED wave model In order to take the effect of amient currents into account, the WABED wave model uses the action density rather than the energy density, since the action density is conservative in the presence of amient currents whereas the energy density is not (Bretherton and Garrett 1968). 0 Zheng Jinhai et al. Water Science and Engineering, Mar. 008, Vol. 1, No. 1, 18 6

The Doppler shift is considered in the solution of intrinsic frequency calculated y the wave dispersion equation. The action density against the asolute frequency is calculated directly. The wave action alance equation with diffraction in terms of three variales is used in the development of a practice-oriented random wave model for coastal engineering studies of inlets, navigation projects, and wave-structure interactions. In these applications, wave reaking, dissipation, reflection, diffraction, and wave-current interaction are important processes that need to e represented accurately for reliale estimation of wave properties in engineering design, maintenance, and operations. The governing equation of WABED wave model is ( CN) ( CN x y ) ( CN θ ) κ 1 + + = ( CCgcos θ N y) CCgcos θn y N x y y y ε θ σ (1) where N is the wave action density, defined as the wave energy-density E divided y the relative angular frequency σ (with respect to a current), (x, y) are the horizontal coordinates, and θ is the wave direction measured counterclockwise from the x-axis. The first term on the right-hand-side of Eq. (1) is the wave diffraction term formulated from a paraolic approximation of wave theory, in which the coefficient κ is a free parameter to e optimized to tune the diffraction effect. The recommended value is.5 (Mase 001). C and C g are wave celerity and group velocity, respectively, ε is the parameterization of wave reaking energy dissipation rate per unit horizontal area, and C, C, and C x y θ are the characteristic velocity with respect to x, y and θ, respectively, which can e expressed as C cos x = Cg θ + u () C sin y = Cg θ + v (3) σ h h u u v v C = sinθ cosθ cosθsinθ cos θ sin θ sinθcosθ θ + + (4) sinh kh x y x y x y where u and v are current velocity components in the x and y directions and k is the wave numer. The relationships etween the relative angular frequencyσ, the asolute angular frequencyω, the wave numer vector k, the current velocity vector U, and the water depth h are as follows: σ = g k tanh k h (5) σ = ω ku (6) A forward-marching, first order upwind finite-difference method is used to solve the aove wave-action alance equation with diffraction (Mase 001; Mase et al. 005). With given values at the offshore oundary, solutions in the direction of wave propagation are otained at each forward marching step, and statistical quantities are calculated at each row efore moving to the next row. If the seaward-reflection option is activated, the model will perform ackward marching for seaward-reflection after forward-marching calculations are completed. The parameterization of the mean energy dissipation rate per unit horizontal area due to Zheng Jinhai et al. Water Science and Engineering, Mar. 008, Vol. 1, No. 1, 18 6 1

wave reaking, ε, is derived from Battjes and Janssen s formulation (1978). The dissipation was calculated for a ore of the same height, and the proaility of occurrence of reaking waves was estimated from the Rayleigh distriution, with an upper cut-off determined y local depth in shallow water. Their formula is αρg D = Q fh (7) 4 where D is the ulk energy dissipation due to all reaking waves, ρ is the density of water, g is the gravitational acceleration, α is a constant of order one, f is the mean frequency, H is the reaker height, and Q is the proaility of the fraction of roken waves passing at a point, which can e estimated as the area of the truncated Rayleigh wave height proaility distriution function under the Delta function at H. Q is determined y 1 Q H rms = (8) ln Q H where H rms is the root-mean-square wave height. Battjes and Janssen (1978) used a constant reaker parameter of 0.8 in Eq. (8) to determine the maximum possile individual wave height at the local water depth. The reaker parameter in SWAN version 40.01 and previous versions has een considered exclusively dependent on the local water depth, and expressed either as a constant value or as eing ottom-slope dependent. With SWAN version 40.11, the ottom-slope dependency has een removed and improved predictions have een otained with a constant value of 0.73 (Booij et al. 1999). Chen et al. (005) also included this reaking criterion in their finite element coastal/haror wave model ased on an extended mild-slope wave-current equation. The parameterization of wave-reaking energy dissipation rate per unit horizontal area can e descried as D ε = ρgh σ (9) 3. Numerical modeling ( 8 rms ) In the numerical simulations, the grid size was set at 0. m in oth along-shore and cross-shore directions. The input wave spectrum was divided into 10 frequency ins and 36 direction ins. The velocity of the current in each cell was estimated y interpolation of the measured data so that the wave model would not e influenced y computation error in the numerical nearshore circulation model. Figures 4 and 5 show calculated wave height fields with and without currents for multi-directional and uni-directional incident waves. With currents included in the simulations, simulated results showed wave height intensification in front of the concave shoreline. This feature was asent in simulations made without accounting for currents. Figure 6 provides a measured and computed data comparison of the normalized significant wave heights. The predicted values are within 0% of the measured values. Figures Zheng Jinhai et al. Water Science and Engineering, Mar. 008, Vol. 1, No. 1, 18 6

7 and 8 show comparisons of the calculated and measured normalized significant wave heights along longitudinal and transverse transects for multi-directional and uni-directional waves, respectively. Results from simulations that incorporate the effects of currents are in good agreement with data on wave height intensification caused y strong opposing coastal currents. (a) Without currents () With currents Figure 4 Simulated wave fields for the multi-directional wave case (a) Without currents () With currents Figure 5 Simulated wave fields for the uni-directional wave case Two statistical parameters were used to evaluate the overall performance of the WABED wave model. The first was the mean value of asolute relative errors for the normalized significant wave height, defined as H1/3 H1/3 1 ( H ) ( H ) R = N H ( H ) N 1/3 0 ci 1/3 0 mi 100% (10) i= 1 1/3 1/3 0 mi Zheng Jinhai et al. Water Science and Engineering, Mar. 008, Vol. 1, No. 1, 18 6 3

where N is the total numer of wave height data availale in each experiment condition, (H 1/3 ) 0 is the incident significant wave height, H 1/3 is the significant wave height at the locations of wave gauges, the suscripts c and m denote the computed and measured normalized significant wave height, respectively. Smaller values of R are indicators of agreement etween computed and measured data. A value of zero implies a perfect match etween computations and measurements. The second statistical measure was the correlation coefficient etween computed and measured normalized wave heights. Figure 6 Computed versus measured normalized wave height Figure 7 Normalized wave height comparisons along different transects for multi-directional wave case Figure 8 Normalized significant wave height comparisons along different transects for uni-directional wave case 4 Zheng Jinhai et al. Water Science and Engineering, Mar. 008, Vol. 1, No. 1, 18 6

The statistics indicate that the mean values of asolute relative errors in wave height prediction for uni-directional and multi-directional incident waves are, respectively, 5.4% and 7.67%, and the correlation coefficients are 0.94 and 0.9. This demonstrates that the WABED wave model is capale of predicting random wave transformation with strong wave-induced coastal currents when an accurate current field is given. Additional studies are underway to develop a quasi-3d wave-induced coastal current model and couple it with the present WABED wave model. 4 Concluding remarks Laoratory investigations and numerical simulations were carried out to study the transformation of random waves over a coast with complicated athymetric and geometric features. Wave energy convergence and divergence were found to cause strong nearshore currents and, inversely, to significantly modify the overall redistriution of wave fields in the experiments. The wave field generated y the WABED wave model is different from that of experiments in which current is excluded. The simulated results that consider currents accurately capture the phenomenon of wave height intensification in front of concave shorelines. The mean values of asolute relative errors in wave height prediction are 5.4% and 7.67%, and the correlation coefficients are 0.94 and 0.9, for uni-directional and multi-directional incident waves, respectively. The satisfactory performance of numerical simulations indicates that the effects of currents are indeed important in predictions of random wave transformation over complicated coastal athymetry and geometry. Acknowledgements The first author is grateful to the China Scholarship Council and the Ministry of Education, Culture, Sports, Science and Technology Japan for their financial support during his stay (Oct. 005 Sept. 006) as a visiting scholar at the Disaster Prevention Research Institute of Kyoto University in Japan. References Battjes, J. A., and Janssen, J. P. F. M. 1978. Energy loss and set-up due to reaking of random waves. Proceedings of 16th International Conference on Coastal Engineering. Hamurg: ASCE, 569 587. Berkhoff, J. C. W. 197. Computation of comined refraction and diffraction. Proceedings of 13th International Conference on Coastal Engineering. Vancouver: ASCE, 745 747. Booij, N., Ris, R. C., and Holthuijsen, L. H. 1999. A third-generation wave model for coastal regions: 1. Model description and validation. Journal of Geophysical Research, 104 (C4), 7649 7666. Bretherton, F. P., and Garrett, C. J. R. 1968. Wave trains in inhomogeneous moving media. Proceedings of Royal Society, Ser. A, 30, 59 554. Castelle, B., Bonneton, P., Senechal, N., Dupuis, H., Butel, R., and Michel, D. 006. Dynamics of wave-induced currents over an alongshore non-uniform multiple-arred sandy each on the Aquitanian Coast, France. Continental Shelf Research, 6(1), 113 131. Chen, W., Panchang, V., and Demirilek, Z. 005. On the modeling of wave-current interaction using the elliptic mild-slope wave equation. Ocean Engineering, 3 (17 18), 135 164. Zheng Jinhai et al. Water Science and Engineering, Mar. 008, Vol. 1, No. 1, 18 6 5

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