1d modelling of infragravity wave propagation on fringing reef using swash

Nội dung text: 1d modelling of infragravity wave propagation on fringing reef using swash

1. BÀI BÁO KHOA HỌC 1D MODELLING OF INFRAGRAVITY WAVE PROPAGATION ON FRINGING REEF USING SWASH Phạm Lan Anh1 Abstract: This paper presents wave propagation process on an idealized fringing reef profile using numerical model, in which infra-gravity wave motion is of major concern. The results show that SWASH is capable to accurately predict infra-gravity wave initiated at the reef face and reef crest for the case of high reflection coefficient at back reef with the model skill in range between 0.94 and 0.98. Over 24 scenarios, SWASH tends to slightly underestimate the spectra wave height in compared to measured i one. Moreover, for low relative reef flat submergence, Hm0 /d, SWASH does not account correctly the breaking position which affects the wave energy dissipation. Keywords: Fringing reefs, infra-gravity waves, wave breaking, wave set-up, SWASH. 1. INTRODUCTION * hydrostatic model using shallow water equation is A typical fringing reef is characterized by an rational solution in terms of time and expenses. abruptly steep face slope and a shallow reef flat SWASH (Simulating WAves till Shores) is a connecting to the shoreline. Sea swell wave non-hydrostatic wave-flow model takes its starting energy mostly dissipates, only infra-gravity wave point as Navier Stoke equation to calculate the energy dominates in the mid and back reef. Infra- surface elevation and currents. To simplify the gravity waves (IG) generated on the reef flat by problem, the free surface is described by a single the breaking-point mechanism (Longuet-Higgins value function that allows non-hydrostatic models and Stewart 1962) with period ranging from 20s to efficiently compute free surface flow. Zijlema up to 250s. They are responsible for several performed model validation based on hydrodynamic effects such as infra-gravity wave experimental data from Demirbilek et al. (2007) resonance, entrapped energy on the flat, swash with a fringing reef of 6m length (corresponding waves which make infra-gravity waves an 120m in prototype) and mild back reef slope important aspect for coastal engineering. (1/12). Rinjdorp (2012) compared the model There is a need to have an accurate prediction results in case of bi-chromatic wave propagate on with proper description of the generation and mild and steep beach slope. In recent study, model nearshore transformation of IG waves on the validation was carried out for a fringing reef of fringing reef. Several types of numerical models 10m width and fore reef slope 1/20 and no back have been applied in studying IG waves such as reef slope with model scale 1:40 (Pham Lan Anh “surf beat model” (Roelvink et al. 2009, Dongeren et al. 2020). In fact, the reef dimensions (fore reef et al. 2012), Boussinesq model (Madsen et al. slope, reef flat length, beach slope) have a great 1991, Nwogu 2010, Yao et al. 2012, Lin and Liu impact on the wave transformation across reef flat. 1998). However, it is computationally more The back reef slope steepness can range from expensive and time consuming to apply for minimum value of 1/12 up to maximum of vertical complex bathymetry. Alternatively, a non- (Buckley 2018). Most of recent studies focus on 1 Khoa Công Trình, Trường Đại học Thủy lợi the variation of flat length or fore reef slope, KHOA HỌC KỸ THUẬT THỦY LỢI VÀ MÔI TRƯỜNG - SỐ ĐẶC BIỆT (12/2021) 171
2. howbeit, neglecting the variation of back reef (4) slope, otherwise leaving it in a mild steepness. The friction term is added via the bottom shear Hence, this paper aims to demonstrate the stress following the quadratic friction law: robustness of SWASH in simulating wave (5) propagation on an idealized fringing reef flat (10m c is the friction coefficient, U the depth in model scale) in case of the steeply inclined f average flow velocity, h=d+ is the total water back reef slope (1/5). Therefore, SWASH depth including wave set-up. capabilities in simulating IG waves are To capture wave breaking a condition of the investigated by comparing model predictions with wave front is applied to initiate the breaking experimental data of report of Pham Lan Anh process m > √gh in which, represents the (2020) for infra-gravity wave characteristics on t maximum surface steepness and determines the fringing reef. onset of breaking process. To display persistence This paper is structured as followed: The first of wave breaking the condition m > √gh is part focuses on SWASH basic equation and its t applied with < and  is the threshold to stop sensitive parameters. It is followed by the model breaking. setup description in SWASH based on the 3. MODEL SETUP laboratory setup in a wave flume (Pham Lan Anh Laboratory experiments was carried out in 2020) to research on infra-gravity wave Holland flume in Thuyloi university which is characteristics. The third part displays the results 45m long 1.2m high and 1.0 m wide. The and discussion on the wave spectrum, sea-swell geometry of an idealized fringing reef is shown waves and infra-gravity waves. in figure 1 characterizing a 1:40 scale model. 2. FUNDAMENTAL EQUATIONS Six waves gauges were installed to measure Swash derived from incompressible Navier water surface elevation at the shoreline, along Stokes equations that describe conservation of the reef flat and in deep water. A paddle wave mass and momentum. In this study unidirectional maker is located 22m from the location of waves are considered in 2 dimensional plain. The model structure creating irregular wave field free surface z=(x,t) and the bottom z= d(x), t is based on JONSWAP spectrum with peak the time, x and z are the Cartesian coordinates enhancement factor  = 1.25. Tested wave with z defines downward and z=0 located in the heights (0.06m – 0.15m) and periods (1.0s-1.6s) still water level. is found most suitable to Viet Nam storm wave (1) condition in East Sea. For further experiment (2) details, it can be made a reference to Pham Lan Anh (2020). (3) Offshore Reef face Crest Mid-reef Back reef Where u(x,z,t) and w(x,z,t) are the horizontal -7.5 forereef +1.0 +4.0 +7.0+8.5+9.9 Shoreline W6 W5 W4 W3 W2 W1 Wave and vertical velocities respectively; ph(x,z,t) and paddle 1/5 Pnh(x,z,t) are the hydrostatic and non-hydrostatic d pressure respectively; xx,xz, ,zx ,zz are the 1/5 O x turbulent stresses and is the density of water. 22m 2.5m 10m 3.5m 45m Considering a unit water column, an equation of surface water elevation is derived by balancing the Figure 1. Experiment setup for an idealized mass conservation equation: fringing reef 172 KHOA HỌC KỸ THUẬT THỦY LỢI VÀ MÔI TRƯỜNG - SỐ ĐẶC BIỆT (12/2021)
3. The input wave conditions for simulating wave transformation in SWASH are listed below. Table 1. Experiment scenarios Flat depth d (m) Back reef Model Prototype Wave i i Model Prototype steepness Hm0 (m) Tp (s) Hm0 (m) Tp (s) steepness 0.06 1.35 1.2 8.54 0.01 0.05 2.0 0.07 1.50 2.8 9.50 0.02 0.1 4.0 0.09 1.40 3.6 8.85 0.03 1/5 0.15 6.0 0.09 1.60 3.6 10.12 0.02 0.2 8.0 0.12 1.60 4.8 10.12 0.03 0.15 1.80 6.0 11.38 0.03 i Hm0 incoming spectrum wave height; Tp peak period of incoming waves Numerical simulation was performed with a offshore boundary was minimized by a weakly computational domain ranging within 21m, reflected boundary. which is narrower than the experimental 4. RESULTS AND DISCUSSION domain. The purpose of domain restriction is to There are 24 tests chosen from the whole keep the hydraulic boundary as close to the reef data experiment in the report (Pham Lan Anh face as possible. To ensure spatial resolution, a 2020) which represent the variation in water grid size of 1cm was chosen which was meant depth, in coming wave height and period. Figure that, on average, there were from 285 grid cells 2 shows the transformation water surface water to 506 grid cells per wavelength. On the west elevation from deep water WG6 to reef crest boundary, time series of wave gauge 6 in deep WG5, mid-reef WG2, WG3, WG4 and at the water was applied. This satisfied the condition shoreline WG1. that area of interest should be kept at least two It can be seen that within 200 sec capturing wave length from the boundaries. On the east most of wave gauges show a common tendency boundary, wave naturally reflected at a hard of fluctuation between SWASH calculation and beach slope of 1/5 without a sponge layer. Two measurement. Near the reef face and move vertical layers were chosen which is the best toward mid-reef (WG5, WG4) there is several choice in increasing the frequency dispersion difference at peak due to phase lag or magnitude and an initial time step is 1ms. The friction difference, however, this is linearly minimized coefficient is 0.01 for Manning option. This once waves moves toward shoreline (WG3, value is not known beforehand and must be WG2). In WG6, wave shape stays in symmetry obtained after calibration. It was a similar form which is typical for deep water waves, implementation for the threshold of wave whereas from WG5 to WG1 waves transform in starting breaking =0.6 and the threshold of asymmetric shape with high positive peaks but wave stopping breaking =0.3. Reflection from low and flat negative peaks. KHOA HỌC KỸ THUẬT THỦY LỢI VÀ MÔI TRƯỜNG - SỐ ĐẶC BIỆT (12/2021) 173
4. Shoreline 0.15 0.15 WG1 WG2 0.10 0.10 0.05 0.05 (m)  0.00 0.00 -0.05 -0.05 -0.10 -0.10 500 550 600 650 700 500 550 600 650 700 Mid-reef 0.15 0.15 WG3 WG4 0.10 0.10 0.05 0.05 (m)  0.00 0.00 -0.05 -0.05 -0.10 500 550 600 650 700 500 550 600 650 700 Offshore 0.20 0.20 0.15 WG5 0.15 WG6 0.10 0.10 0.05 0.05 (m)  0.00 0.00 -0.05 -0.05 -0.10 -0.10 -0.15 500 550 600 650 700 500 550 600 650 700 Figure 2. Measured and SWASH calculated water elevation under d=0.2m, i Hm0 =15cm, Tp=1.8s; in black represent SWASH calculated, in red represent measured. Figure 3 displays the wave spectra evolution Moreover, wave height variation across the from reef face (WG5) to the shore line (WG1) reef flat indicates that the model reproduces within low frequency band (fIG 1/2 fp, fIG is the properly the wave dissipation process via IG frequency, fp is the peak frequency of incident breaking at the reef crest and reef face. Waves waves). The spectra evolution, in general, agree rapidly attenuate at the reef crest which is around well with the measured data, except for the very half of the incident wave height. The low IG frequency. The computed energy (black) measurement and predicted model agree very underestimates at high harmonics and well to this point (figure 4), except D70H07T150 i overestimates at lower harmonics which seems (d=0.2m, Hm0 =7cm, Tp=1.5s) and D70H09T140 i compensate each other and still lead to equally (d=0.2m, Hm0 =9cm, Tp=1.4s). The general correlative in Hm0IG (figure 6a, b, c). These tendency of wave height distribution for these could be corrected by refining the grid size two scenarios is almost reverse to the decreasing down to 0.5cm to improve the spatial resolution tend of wave height. This seems matching well in the model. with set-up distribution across the reef flat in The spectra wave height in SWASH and in D70H07T150 and D70H09T140 (figure 5). measurement is estimated as follow: Explanation is presented based on table 2 and (5) figure 6d below. 174 KHOA HỌC KỸ THUẬT THỦY LỢI VÀ MÔI TRƯỜNG - SỐ ĐẶC BIỆT (12/2021)
5. Shoreline 0.0025 WG1 WG2 0.0020 0.002 /Hz) 2 0.0015 0.0010 0.001 E(f) E(f) (m 0.0005 0.0000 0.00 0.05 0.10 0.15 0.20 0.25 0.300.0000.00 0.05 0.10 0.15 0.20 0.25 0.30 Mid-reef 0.0020 0.0020 WG3 WG4 0.0015 0.0015 /Hz) 2 0.0010 0.0010 E(f) E(f) (m 0.0005 0.0005 0.0000 0.0000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Offshore + reef crest 0.0009 0.006 WG5 WG6 0.0006 0.004 /Hz) 2 E(f) E(f) (m 0.0003 0.002 0.0000 0.000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 f (Hz) f (Hz) Figure 3. Comparison of measured and calculated wave spectra at the IG wave frequency band i under d=0.05m, Hm0 =9cm, Tp=1.6s; in black represent SWASH, in red represent measured. Figure 4. Wave height (Hm0) distribution in deep Figure 5. Set-up along the reef flat, black water and along the reef flat, black and red and red represent SWASH and measured, represent SWASH and measured, respectively respectively KHOA HỌC KỸ THUẬT THỦY LỢI VÀ MÔI TRƯỜNG - SỐ ĐẶC BIỆT (12/2021) 175
6. Table 2. Comparison of wave breaking position between experiment observation and SWASH calculation with 10 consecutive waves Breaking position Type of breaking Scenarios SWASH Lab observation SWASH Lab observation d=0.05m; H i=6cm; Reef crest Reef crest - D55H06T135 m0 Spilling Tp=1.35s d=0.05m; H i=9cm; Reef face Reef face - D55H09T160 m0 Plunging Tp=1.60s d=0.1m; H i=12cm; T =1.6s Reef face Reef face Spilling & D60H12T160 m0 p - & reef crest plunging i D70H07T150 d=0.2m; Hm0 =7cm; Tp=1.5s No breaking Reef crest - Spilling i D70H09T140 d=0.2m; Hm0 =9cm; Tp=1.4s No breaking Reef crest - Spilling i D70H15T180 d=0.2m; Hm0 =15cm; Tp=1.8s Reef face Reef face - Plunging In order to explain further the above difference, short spectra wave height Hm0SS (Yao 2019) ten consecutive incoming waves to the reef were generated by varying breaking point mechanism considered after 500 sec elapsed to investigate the on the flat are estimated as position of wave breaking. It can be seen that , (6) SWASH assesses incorrectly the breaking position of scenarios with relative small spectra waves. In S(f) variance wave energy density spectrum. Wave this case relative spectra wave height is made spectra is divided into short wave-high frequency i (f>fc) and IG wave - low frequency (f≤ fc) relied on dimensionless to reef flat water depth, Hm0 /d. The i the demarcating frequency fc, where fc = 0.5fp and fp is ratio Hm0 /d reflects the shallowness of the water depth or the reef flat submergence which affects the peak frequency of the incident wave. the depth-induced wave breaking. Based on table In figure 6 a, b, c the results show good i agreement between measurement and prediction. 2, figure 6d, if Hm0 /d 0.45 SWASH does not account for breaking occurrence on the reef crest, There is minimal scatter in relation between while in fact it happens in laboratory observation measured and calculated IG wave height (figure i 6c). These high harmonics frequencies have (gray shading). In cases Hm0 /d>0.45 the breaking position is determined correctly in comparison underestimates in calculated energy spectra, hence with laboratory observation. Hence, although the leading to some skew points at the shoreline and dispersive relation has been improved by applying mid-reef. the vertical layers, SWASH still has short-coming Finally, the error estimator for SWASH and in energy dissipation calculation in different wave measurements has been calculated to confirm such breaking conditions (figure 6d). good agreement. The model skill was determined by bias and scatter index (SI) (table 3) Infra-gravity spectra wave height Hm0IG and ) (7) (8) Xcalculated and Xmeasured are the quantities that are brought to compare, corresponding values for SWASH and namely the spectra wave height Hm0, the sea- measured parameters; N is the total number swell (short) wave height Hm0SS and low of data set considered. In table, there are two frequency wave height Hm0IG. 176 KHOA HỌC KỸ THUẬT THỦY LỢI VÀ MÔI TRƯỜNG - SỐ ĐẶC BIỆT (12/2021)
7. 0.18 0.18 0.16 a. 0.16 b. 0.14 0.14 0.12 0.12 0.10 0.10 (m) (m) 0.08 m0,M 0.08 Shoreline Shoreline m0SS,M H H Mid-reef Mid-reef 0.06 0.06 Reef crest Reef crest 0.04 Offshore 0.04 Offshore 0.02 0.02 0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20 H (m) H (m) m0,S 1.0 m0SS,S 0.040 0.8 SWASH c. d. LAB 0.035 0.6 0.4 0.030 0.2 0.025 0.0 (m) -0.2 0.020 m0IG,M -0.4 H Shoreline 0.015 -0.6 Mid-reef Breaking position X(m) position Breaking -0.8 0.010 Reef crest Offshore -1.0 0.005 -1.2 0.00 0.01 0.02 0.03 0.04 0.05 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 H (m) H i/d m0IG,S m0 Figure 6. Calculated spectra wave height (subscript S) versus measured spectra wave height (subscript M) (a). the total spectra wave height (a) short wave height (c) IG wave height i (d) wave breaking position versus flat submergence Hm0 /d. Table 3. Error estimators for calculated and measured wave height and wave set-up Gauge Bias Hm0 (m) SI Hm0 Bias Hm0SS (m) SI Hm0SS Bias Hm0IG (m) SI Hm0IG Bias set-up (m) SI set-up WG1 0.00363 0.94 -0.0004 0.95 0.003 0.931 -0.0014 0.96 WG2 0.0029 0.96 -0.0004 0.96 0.0021 0.94 -0.00003 0.96 WG3 -0.0007 0.96 -0.0026 0.96 -0.0016 0.94 -0.001 0.96 WG4 0.0016 0.96 -0.0017 0.96 0.002 0.94 -0.002 0.97 WG5 -0.0036 0.98 -0.0067 0.96 0.0014 0.95 0.0018 0.98 WG6 0.0018 0.98 0.0016 0.96 0.0019 0.95 -0.002 0.98 Overall the comparison between SWASH runs robustly capture the vertical steepness in and measurement shows a good agreement at all decaying wave height distribution over reef flat locations under 24 scenarios. The maximum happening at the reef face. Moreover, results average difference is approximately 4mm at WG1 also show that the model slightly underestimates and WG5 which is totally small compared to the calculated wave height. The difference typical wave height ranging from 6cm to 15cm. ranges between 2mm to 4mm in compared to For the scatter index, the lowest value happens at typical wave height ranging from 6cm to 15cm WG1 under low frequency wave height (93.1%) and the skill model ranges between 0.94 and which is highly acceptable. 0.98 which is highly reliable. However, for low i 5. CONCLUSION Hm0 /d ( 0.45), SWASH does not account SWASH are, in general, capable to reproduce correctly the breaking position which affects the sufficiently and correctly the infra-gravity wave wave energy dissipation. Therefore, it is highly transformation across the fringing reef in case of recommended to simulate the model with i steeply inclined back reef slope. For relatively high Hm0 /d (>0.45) to gain a proper complicated breaking process, SWASH can result. It is worth to say that 0.45 is a relative KHOA HỌC KỸ THUẬT THỦY LỢI VÀ MÔI TRƯỜNG - SỐ ĐẶC BIỆT (12/2021) 177
8. value inferred from 24 runs of this modelling. ACKNOWLEDGEMENT Further study should be taken to determine Authors would like to express the gratitude to correctly this transitional value which reflects reviewers who read and correct concretely to the relative submergence. complete the paper. REFERENCE Baldock, T. E. (2012). Dissipation of incident forced long waves in the surf zone—Implications for the concept of ‘‘bound’’ wave release at short wave breaking. Coastal Engineering, 60, 276–285. Longuet-Higgins, M. S., & Stewart, R. W. (1962). Radiation stress and mass transport in gravity waves, with application to ‘surf beats. Journal of Fluid Mechanics, 13, 481–504. Nwogu, O., & Demirbilek, Z. (2010). Infragravity wave motions and runup over shallow fringing reefs. Journal of Waterway, Port, Coastal, and Ocean Engineering, 136, 295–305. Marcel Zijlema (2012) Modelling wave transformation across a fringing reef using SWASH. Coastal Engineering 2012 Dirk Rijndorp & M. Zijlema (2012) Non-hydrostatic modelling of infra-gravity waves using SWASH. Coastal Engineering prceeding 2012 DOI: 10.9753/icce.v33.currents.27 Yu Yao (2019) Effects of reef morphology variations on wave processes over fringing reefs. Applied ocean research 82 (2019) 52-62 Pham Lan Anh (2020). Validation of experiment data set over fringing reef by Swash 1D Proceedings of the annual conference of Thuyloi university Phạm Lan Anh (2020) Báo cáo đề tài nghiên cứu cơ sở “Nghiên cứu xác định đặc trưng sóng ngoại trọng lực trên đảo nổi” Trường đại học Thủy lợi Tóm tắt: MÔ PHỎNG QUÁ TRÌNH TRUYỀN SÓNG NGOẠI TRỌNG LỰC TRÊN THỀM ĐẢO NỔI BẰNG MÔ HÌNH SWASH 1D Bài báo trình bày mô phỏng quá trình biến đổi sóng, đặc biệt là sóng ngoại trọng lực trên thềm đảo nổi xa bờ bằng mô hình SWASH 1D. Kết quả cho thấy SWASH mô phỏng tương đối chính xác sự hình thành và biến đổi sóng ngoại trọng lực với độ chính xác mô hình SI từ 0.94 tới 0.98. SWASH có xu hướng cho kết quả chiều cao sóng nhỏ hơn thí nghiệm một chút. Với những giá trị độ nông tương đối trên thềm i thấp (Hm0 /d nhỏ), SWASH chưa tính đến chính xác vị trí sóng vỡ, vì vậy ảnh hưởng tới tiêu hao năng sóng và độ lớn nước dâng, chiều cao sóng ở trường hợp này. Từ khóa: Đảo nổi, sóng ngoại trọng lực, sóng vỡ, nước dâng, SWASH. Ngày nhận bài: 09/10/2021 Ngày chấp nhận đăng: 09/11/2021 178 KHOA HỌC KỸ THUẬT THỦY LỢI VÀ MÔI TRƯỜNG - SỐ ĐẶC BIỆT (12/2021)