CHAPTER 180. Development of a Dune Erosion Model using SUPERTANK Data. Margery F. Overtoil 1 John S. Fisher 2 Kyu-Nam Hwang 3

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1 CHAPTER 180 Abstract Develpment f a Dune Ersin Mdel using SUPERTANK Data Margery F. Overtil 1 Jhn S. Fisher 2 Kyu-Nam Hwang 3 A mdel predicting dune ersin under strm cnditins has been develped using data frm SUPERTANK labratry experiments. In this mdel, the swash apprach is applied with the basic assumptin that the vlume erded frm the dune is a functin f the swash frce acting n the dune. The swash frce is characterized with swash parameters, specifically the swash height, the swash velcity and the swash perid. Fr given strm cnditins, the mdel predicts wave height variatin acrss-shre, swash height variatin n the beach face, swash velcity, swash perid, swash frces and finally vlume erded at the dune. Predictins f swash parameters and dune ersin are cmpared with measurements frm SUPERTANK labratry experiments. Intrductin A lt f effrt has been made t develp a predictin mdel fr dune ersin by strm events and the resulting beach prfile. An extensive research prgram fr the prcess f dune ersin due t wave impact was started by Fisher and Overtn (1984). Their apprach fr the prcess f dune ersin is based n the interactin f the wave swash and the dune. This apprach treats the dune ersin phenmenn as a time-dependent prcess in which a series f successive uprushes attack the dune face. Each individual uprush erdes a finite vlume f sand which in turn is depsited n the erding beach. Currently, the mst imprtant achievement in the develpment f a dune ersin mdel due t wave impact is the experimental bservatin f the linear 1 Assciate Prfessr, Department f Civil Engineering, Nrth Carlina State University, Bx 7908, Raleigh, NC Prfessr, Dept. f Civil Engineering, NCSU. 3 Graduate Research Assistant, Dept. f Civil Engineering, NCSU. 2488

2 DUNE EROSION MODEL 2489 relatinship between swash frce and dune ersin. This linear relatinship has been cnfirmed thrugh field measurements (Fisher et al., 1986) and labratry experiments (Overtn et al., 1988). Therefre, it is pssible t predict the dune ersin frm this relatinship if the swash frce is knwn fr the given strm cnditins. Hwever, limited knwledge f swash parameters, which are identified as swash height, swash velcity and swash perid, has discuraged the develpment f a quantitatively (r even qualitatively) reliable predictive mdel. It is therefre highly instructive t understand qualitatively the hydrdynamics f the swash zne in terms f waves generated by strms utside the surf zne. These waves cmmnly serve as an input fr the predictin f transfrmed waves in the surf zne. Fr this purpse, a set f experiments designed t simulate dune ersin under strm cnditins at prttype scale was cnducted in the large wave tank at the O. Ff. Ffinsdale Wave Research Labratry, Oregn State University as part f SUPERTANK Labratry Data Cllectin Prject. Thrugh the analysis f SUPERTANK labratry data fr the swash parameters, an attempt is made in this study t prduce useful infrmatin regarding hydrdynamics in the swash zne, and t develp a predictive mdel fr dune ersin by wave impact ft *N ft R.W.G. # f a WAVES OO CO 0000:0 GW.G 16 RW.G: 120' spacing C.W.G : 3'- 6'spacing + :RESIS.W.G :CAPAC. W.G N > ft ft Figure 1. Cnfiguratin f flume and lcatin f wave gages. Labratry Experiments The experiments were designed t simulate dune ersin under strm cnditins at prttype scale. Flume cnfiguratins with the lcatins f wave gages are shwn in Figure 1. A dune f 5 ft height was cnstructed with fine sands

3 2490 COASTAL ENGINEERING 1994 f median grain sizes, d 50, f 0.23 mm. A vibrating cmpactr was used t cnslidate the artificial dune. Sixteen resistance wave gages designed at OSU (Dibble and Sllitt, 1989) were used t btain wave data acrss-shre and ten capacitance type wave gages were placed n the beach in frnt f the dune t cllect swash data. The OSU data acquisitin system sampled the wave gages at a rate f 16 Hz. The pre- and pst-test beach and dune prfile were surveyed. Additinally, dune psitin was dcumented by a 35 mm camera befre, during and after each experiment. Each phtgraph f the dune included the prfile f the dune face and a 2 ft by 2 ft standard grid t determine scale. Thirteen experiments were successfully cnducted. The design cnditins f each test are given in Table 1. The design wave heights, H, ranged frm ft t ft while the wave perids, T, ranged frm 3 sec t 6 sec. The duratin f the tests, T d, ranged frm 10 minutes t 30 minutes and the water level (WL) varied frm 9.5 ft t 11.0 ft. The water level is defined as the distance frm the riginal bttm f the flume t the still water surface. Test# T d T H WL Test# T d T H WL (inin) (sec) (ft) (ft) (inin) (sec) (ft) (ft) Table 1. List f Experimental Design Parameters. Using a standard Furier Transfrm, frequency dmain analyses were perfrmed n the SUPERTANK hydrdynamic data cllected in the prebreaking and breaking zne. The wave parameters calculated frm frequency dmain analyses are rms and significant wave height. Bth time dmain analyses and frequency dmain analyses were perfrmed n the time histry f the water surface variatin in the swash zne. The swash parameters calculated frm time dmain analyses are mean value f swash height, swash peak height, swash velcity, swash perid. The representative swash heights calculated frm the pwer spectra are rms and significant swash height. Typical features f an individual swash recrded during the experiment are given in Figure 2. Height f swash is defined as the difference between the backgrund height and the height f the plateau befre the start f the backwash. The velcity f the leading edge f the swash (r swash velcity, V ) is determined by identifying the time at which the swash hits the first and the secnd prbes n

4 DUNE EROSION MODEL 2491 the beach. The swash perid T m between the initial swash hits. f an individual swash is defined as the time 1.2 Distance between tw gages, ds = 3 ft V = ds/dt c _ CD > 0) LU <U i C/>.CO Gage # 8 Gage # 9 Swash hits prbe Tim e (sec) Figure 2. Definitin f swash parameters. Survey data was used t determine the dune and beach prfiles fr each test. A plynmial smthing methd was applied t the survey data t btain the depth data equally spaced in the hrizntal directin. Specific vlume erded at the dune, Q e, was determined frm the befre and after prfiles f the dune face as recrded n 35 mm film, and checked with survey prfiles f the beach and dune. In rder t differentiate between beach ersin and dune ersin, the initial face f the dune was used t define the seaward extent f the dune. An individual swash frce acting n the dune is defined as F * = P V J H^ 0) where p is the density f the water, H swj is an individual swash height and V swi is the swash velcity. While quantifying the frce fr an individual swash is pssible using (1), it is nt always pssible t measure the amunt erded due t that lading. One pssible way is t use the "summing-up" methd by Overtn et al. (1988), in which the summatin f the frce in a given interval f time versus the vlume f erded in the same time interval becmes the quantity evaluated. Each individual swash frce was determined by (1) and summed up t btain the ttal swash frce, F sw, fr a given duratin f time. Linearity between dune ersin and swash frce was examined. Figure 3 shws the estimated data and the best linear

5 2492 COASTAL ENGINEERING 1994 fit line with a i?-squared value f between dune ersin and swash frce. An apparent linear relatinship exists Q=0.0024F i r O R= (kn/m) Figure 3. Linearity f dune ersin as a functin f swash frce. Mdeling f Swash Parameters Swash parameters required fr the predictin f dune ersin are the swash height, swash velcity, and swash perid. In the current stage f mdeling dune ersin due t wave impact, individual swashes (r waves) are nt cnsidered. Instead, statistically representative swash parameters (e.g., mean, rms and significant swash) are used t develp the predictin mdel. A beach prfile in the crss-shre directin is divided int three regins: the prebreaking zne, surf zne, and swash zne. In rder t predict the swash height in the swash zne with a given cnditin in deep water, it is necessary t predict the crss-shre variatin f wave height in the prebreaking and surf zne. Linear wave thery has been used t determine wave height acrss-shre frm deep water t the initial break pint f waves. Wave height transfrmatin in the surf zne has been calculated by applying the breaking wave dissipatin mdel by Thieke and Sbey (1990). Mdeling f Wave Height Several mdels have been develped t predict surf zne wave height variatin, based upn the cnservatin f energy equatin (Battjes and Janssen; 1978, Dally, et al.; 1985, and Stive; 1984). The steady-state, depth-integrated

6 DUNE EROSION MODEL 2493 equatin gverning the energy balance fr waves prpagating directly twards shre is simply ^- = -D (2) 3c where F is the wave energy flux, x is the distance alng the prpagatin path, and D is the rate f energy dissipatin per unit plan area due t breaking. Thieke and Sbey (1990) established a frm f a predictive equatin fr the breaking wave dissipatin as D = f bw we (3) where f bw is a dimensinless breaking wave dissipatin cefficient, c is the spectral peak frequency and E is the wave energy. Adpting a simple direct partitin estimatr yields an expressin fr f bw. #11 ( Hi A»=«1+ IJ2 exp (4) rms ^ \ rms where the cefficient a is f rder \jn (r perhaps H b /(nh)), H m, a lcal limiting wave height, is rder f 0.83/z and H rms is the rt-mean-square wave height I T#12 H b = 1.55 ft h b = 2.91ft Measurements Mdified Thieke and Sbey x Distance frm SWL=0, x^swl c W Figure 4. Measured and predicted wave height variatin in the prebreaking and breaking zne. Predictins by Thieke's mdel fr wave height variatin in the surf zne are shwn in Figure 4 as a dtted line. The measured rms wave height has been cmputed by 2v2cr ( is a spectral estimate f the standard deviatin f the water surface). While the trend predicted by Thieke's mdel is physically reasnable, the mdel underpredicts significantly the wave height near the swash zne. In rder t btain mre accurate predictins f the wave height near the swash zne, a mdificatin t Thieke's mdel has been made t decrease the breaking wave

7 2494 COASTAL ENGINEERING 1994 dissipatin near the swash zne. Calibrating the predicted wave heights (by Thieke's mdel) t the measurements, the quantity f the cefficient a in (4) was cmputed fr all 13 tests. By scaling arguments it can be shwn that Thieke's mdel is imprved when the cefficient a is related t H b, h b, and H 0 by the fllwing expressin : a = 0.8^ ^ 5.79 (5) K H where H b, h b, and H 0 are the initial breaking wave height, the crrespnding depth, and deepwater wave height, respectively. Nte that predictin f wave height in the middle f the surf zne (beach prfile sectin between -50 and -30 ft f xswl) was sacrificed t btain the best predictin near the swash regin. It was necessary since the wave height predicted at the end f the surf zne is used as an initial value fr the predictin f swash height in the swash zne. The mdel predictins with the mdified cefficient a are shwn as a slid line in Figure 4. Near the swash zne, agreement between predicted and measured wave height is excellent. 2 -> Elevatin abve SWL, y, (ft) Figure 5. Significant swash height variatin n a beach face fr Test #5. Mdeling f Swash Height Since the swash phenmena in nature is s cmplicated, n attempt has been made t express, in a simple manner, the swash height variatin n a beach face. Figure 5 shws the measured significant swash height (H ms ) variatin versus beach face elevatin (y) fr Test #5. The elevatin f beach face represents the psitive elevatin abve the still water level (SWL) and the negative elevatin belw it. The mst imprtant experimental bservatin is that the swash height decreases linearly with beach face elevatin, which is apparent in all 13 tests. This

8 DUNE EROSION MODEL 2495 linearity f the swash height n the beach face plays a key rle in establishing a mdel fr swash height variatin. Applying bundary cnditins f i)// sms = 0 at y = y mla and ii)# sl = H sws at y = 0, this linearity yields a predictin mdel fr the swash height: 1- H y (6) H., y m where H sm represents a significant swash height at j=0 and y m is the beach face elevatin where a significant swash height becmes zer. Nte that if y max is knwn fr a given deep water wave cnditin, (6) becmes an initial value prblem and can be used directly t cmpute the swash height fr a given elevatin. In rder t quantify y max fr each test, the left side f (6) was cmputed with the measured swash height and pltted with y. The best linear fit lines (crssing the rigin) t the measured data allw determinatin f ^max f r eacn test - Tne significant swash height at j=0, H s s, fr each test has been determined by the interplatin f the measured swash height. * 5 CO en V 4 CO t» m * :s TJ r re <i> re 2 2 n >. ^-> I i i I O s^ - 9) O rm ^ - ~ - tr i I I I Ymax («) Figure 6. Cmparisn fjw with the mean runup height by Mase et al. (1984). Since j; max can be physically interpreted as a representative runup height fr irregular waves climbing n the beach, it wuld be valuable t examine y m!a with respect t the existing mdels fr runup heights. Mean run-up heights fr each test were cmputed frm the runup height equatin prpsed by Mase and Iwagaki (1984). The cmparisns between j max and the cmputed runup heights are shwn in Figure 6. It is nticed that y max appears t matches rughly with mean runup heights R m by Mase et. al (1984), as y max deviate by at mst a factr f 2 frm./?,.

9 i i 2496 COASTAL ENGINEERING 1994 The swash height variatin in the swash zne was predicted using (6). The cmparisns between the measured and the predicted swash height variatin in the swash zne fr Test # 5 are shwn in Figure 7. The slid line represents the predicted swash height variatin using R m cmputed frm runup equatin by Mase et. al (1984). 1.5 I i T#5 in x ^^-Q \ X SWL '*v Figure 7. Cmparisn between predicted and measured swash heights fr Test #5. Swash Velcity Mdel In rder t develp a predictin mdel fr swash velcity, it is assumed that it is pssible t idealize the water particle as a slid particle which retains its identity. Thus, a particle f swash height, H m, is cnsidered t mve up and dwn the beachface as a slid particle wuld. If the nrmally incident waves are cnsidered, a frce balance implies m^ dv = -mgsm6-j-j?-\v f m,, m \V sw (7) at 8 H m where m is a mass f the water particle, V m is a swash velcity, g is the gravitatinal acceleratin, 6 is the beach slpe and / is a frictin factr. Fr simplicity, a frictinless planar beach is assumed. Eliminating cmmn terms and applying initial cnditins f V^ = V^ at x=y=0 (where x is the distance in the water particle translatin directin frm the lcatin f SWL=0), simple integratin f this equatin yields v^=pg{y^-y) (8) where y is the elevatin psitive frm SWL.

10 DUNE EROSION MODEL 2497 The swash velcity fr each test has been predict frm (8). The cmparisns between the measured and the predicted velcity are given in Figure 8. Nte that the predicted velcity matches well with the measured significant velcity. _ CD > c CD C/5 D CD G 9> Measured Significant Swash Velcity, V sws (ft/s) Figure 8. Cmparisn f the predicted swash velcity t the measured swash velcity. Mdeling f Swash Perid In rder t develp a predictin mdel fr swash perid, a pssible simple methd is t apply a mdel which predicts the prbability f wave vertpping fr ffshre structures. Since the prbability f runup can be expressed &st sw /T BW, the runup predictin mdel becmes 71, = 71. exp assuming that the runup height n natural beaches has a Rayleigh distributin. In (9), T sx, and T sw are the initial swash perid at_y=0, and the swash perid at any elevatin f y, respectively, and R rms is the rms value f runup height. Nte that in rder t apply (9) fr the predictin f swash perid, it is necessary t express bth T. 0 and R rms in terms f knwn deepwater wave cnditins r swash variables. It is pssible t express 71 with T 0 using the labratry data by Mase and Iwagaki (1984). They indicated that the rati f the number f deep water waves t the number f runup waves, N/N 0 (r TJT m<^, varies nnlinearly with the deep water wave cnditin, expressed as the Iribarren number. Fitting a secnd rder plynmial t their experimental data yields y, 2~ (9)

11 2498 COASTAL ENGINEERING 1994 a = - =0.69^ Slnfl (10) where T 0 is the deepwater wave height and S, is the Iribarren number (r surf similarity parameter). Frm the analysis f the labratry measurements, Hwang (1995) als gives a linear relatinship between R^ and y max : ^ax=1.65/c (11) Frm (10) and (11), the swash perid is given as -exp 165^ y m A cmparisn between the predictins and the measurements fr the swash perid is shwn in Figure 9. The predictins fr the swash perid are in gd agreement with the measurements. (12) * 100 r h- T3 _ O 'l_ <D Q_ sz V) r 5 C/3 CD Measured Mean Swash Perid, T swm (s) Figure 9. Cmparisn f the predicted t the measured swash perid. Mdeling f Swash Frce Previusly, the summing-up methd has been used t relate the swash frce t the vlume erded at the dune. Hwever, applying this methd directly t the predictin mdel fr dune ersin is cmputatinally intensive since each swash has t be cmputed and summed up fr each time step (usually 20 minutes). An alternative apprach is t use statistically representative swash parameters, such as mean, rms and significant values. As an example, a swash frce by mean parameters may be denned as

12 DUNE EROSION MODEL 2499 F_ -V H (13) 71 where subscript m represent the mean values f the swash height, the swash velcity, and the swash perid. Nte that the rati T/Ts^,, in (13), which represents the number f swash hit at the dune face, is intrduced t scale the ttal swash frce fr a given duratin f test. 150 z F sw (kn/m) Figure 10. Relatinship between F^ and F^ The swash frce by mean parameters was cmputed with the measured mean swash parameters fr each test. Cmparisns f the swash frce using (13) with ttal swash frce (F ) btained by summing-up methd are shwn in Figure 10. It is interesting t ntice that swash frce defined by mean swash parameters is linearly related with the swash frce by the summing-up methd. The linear relatinship btained by the best fit line is F^ = 0.69F, (14) Predictive Mdel fr Dune Ersin The dune ersin mdel predicts the vlume erded at the dune due t wave impact. The mdel cnsists f an input data rutine, prebreaking zne wave height rutine, surf zne wave height rutine, swash zne hydrdynamics rutine, dune ersin rutine, and an utput rutine. As initial input data, the mdel requires the deepwater wave cnditins and initial beach and dune prfile data. Linear wave thery is used t determine the wave height frm deepwater r a specified water depth ffshre t the breaking pint. Shreward f the breaking

13 2500 COASTAL ENGINEERING 1994 pint, a mdified frm f Sbey's predictin mdel is used t cmpute the wave height. The swash zne hydrdynamics rutine cmputes each statistically representative swash height, swash velcity, and swash perid. Once the swash parameters are determined, the crrespnding swash frce is cmputed and used t calculate the vlume erded at the dune. In rder t determine the wave heights acrss-shre, the mdel uses an explicit slutin scheme in which quantities knwn at a specific grid pint are used t determine crrespnding quantities at the next grid pint. Prpagatin f individual waves is nt described by the mdel. ^ 0 E CO E ^s c "55 p S UJ i a 9> _ - ^O ? Figure Measured Dune Ersin, Q e (m 3 /m) 11. Cmparisn f predicted dune ersin t the measured dune ersin using predicted R m frm runup equatin. Dune ersin was simulated fr all 13 tests using the data frm SUPERTANK labratry experiments. Data used fr each predictin were the significant wave height at Gage #14, peak spectral wave perid at Gage #14, mean wave perid at Gage #14, beach and dune prfile, test duratin, and dune te lcatin. Assuming that y max can be replaced by runup height, the runup height equatin prpsed by Mase and Iwagaki (1984) was used fr the cmputatin f y m ax which was required fr the predictin f all swash parameters (swash height, swash velcity and swash perid). Cmparisns f mdel predicted dune ersins with measurements fr all 13 tests are shwn in Figure 11. It is easily nticed that the predictin f dune ersin by mdel is nt successful. The mdel predictins deviate by as much as a factr f 4 frm the measured dune ersin.

14 DUNE EROSION MODEL 2501 Since the runup height equatin has been used fr the predictin f y mm, a slight mdificatin t the mdel was made t check the effect f the runup height n the predictin f dune ersin. Instead f calculating the runup height and using it as y max, the measured y max was used fr the simulatin f swash frce and dune ersin fr each test. Predictins f dune ersin using the measured >Wc are shwn in Figure 12 and cmpared with the measured dune ersin. It is bvius that predictins are much imprved, as the mdel predictins deviate by at mst a factr f 2 frm the measured values. 0.6 E CO*- E, a 0.4 V) i- UJ <D c 3 Q i i I I O y/ ys - ~ - 6 i I I I 3F Figure O,3., Measured Dune Ersin, Q e (m/m) 12. Cmparisn f predicted dune ersin t the measured dune ersin using the measured jw Clearly, y max is a significant factr and plays a imprtant rle in the predictin f dune ersin. Cnsequently, it is cncluded that quantitatively gd predictin fr dune ersin can be btained by imprving the ability f predictin f ymax. Cnclusins A simulatin mdel has been develped using SUPERTANK experimental data t predict dune ersin due t wave impact. The analysis f the labratry data cnfirmed an apparent linearity between swash frce and dune ersin. With a mdificatin f Sbey's mdel, predictins f wave height variatin in the surf zne were imprved, especially near the swash zne. Swash height, swash velcity, and swash perid predicted by the mdel agreed well with the labratry measurements. Hwever, the mdel predicted dune ersin with limited success. An uncertainty in the value f y max is the primary cause which led t the failure f ' 0.6

15 2502 COASTAL ENGINEERING 1994 the predictin f dune ersin. The results indicate that imprvement in the dune ersin mdel will depends upn mre reliable estimates f runup height. References Battjes, J. A., and Janssen, J. P. F. M "Energy Lss and Set-up Due t Breaking f Randm Waves", Prceedings f the 16th Castal Engineering Cnference, ASCE, Vl. 1, pp Dally, W. R., and Dean, R. G, and Dalrymple, R. A "Wave Height Variatin Acrss Beaches f Arbitrary Prfiles", Jurnal f Gephysical Research. Vl. 90, N. 6, pp Dibble, T. L., and Sllitt, C. K "New Designs fr Acustic and Resistive Wave Prfiles", Prceedings, Wrkshp n Instrumentatin fr Hydraulic Labratries, IAHR. Fisher, J. S., and Overtn, M. F "Numerical Mdel fr Dune Ersin Due t Wave Uprush", Prceedings f the 19th Castal Engineering Cnference, ASCE, pp Fisher, J. S., Overtn, M. F., and Chishlm, T "Field Measurements f Dune Ersin", Prceedings f the 20th Castal Engineering Cnference. ASCE, pp Hwang, K "Predictive Mdel fr the Dune Ersin and Swash Hydrdynamics", Ph.D. Dissertatin in preparatin, Nrth Carlina State University, Raleigh, NC. Mase, H., and Iwagaki, Y "Run-Up f Randm Waves n Gentle Slpes", Prceedings f the 19th Castal Engineering Cnference. ASCE, pp Overtn, M. F., Fisher, J. S., and Yung, M. A "Labratry Investigatin f Dune Ersin", Jurnal f Waterway, Prt. Castal and Ocean Engineering. ASCE, Vl. 114, N. 3, pp Stive, M. J. F "Energy Dissipatin in Waves Breaking n Gentle Slpes", Castal Engineering, Vl. 8, pp Thieke, R. J., and Sbey, R. J "Crss-shre Wave Transfrmatin and Mean Flw Circulatin", Castal Engineering, Vl. 14, pp