Hydraulic stability of rubble mound breakwaters in breaking wave conditions: a comparative study of existing prediction formulas

Transcription

1 Hydraulic stability of rubble mound breakwaters in breaking wave conditions: a comparative study of existing prediction formulas Sander Franco Supervisor: Prof. Josep R. Medina Co-supervisor: Prof. dr. ir. Peter Troch Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering Laboratorio de Puertos y Costas Chair: Prof. Josep R. Medina Departamento de Ingeniería e Infraestructura de los Transportes Universidad Politecnica de Valencia Department of Civil Engineering Chair: Prof. dr. ir. Peter Troch Faculty of Engineering and Architecture Ghent University Academic year

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3 Hydraulic stability of rubble mound breakwaters in breaking wave conditions: a comparative study of existing prediction formulas Sander Franco Supervisor: Prof. Josep R. Medina Co-supervisor: Prof. dr. ir. Peter Troch Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering Laboratorio de Puertos y Costas Chair: Prof. Josep R. Medina Departamento de Ingeniería e Infraestructura de los Transportes Universidad Politecnica de Valencia Department of Civil Engineering Chair: Prof. dr. ir. Peter Troch Faculty of Engineering and Architecture Ghent University Academic year

4 Preface I would like to thank Prof. Josep R. Medina for giving me the opportunity to do all the research of my master thesis at the Laboratory of Ports and Coasts of the Polytechnic University of Valencia. Also his help on the moments I needed it were always helpful Next, I would like to thank Prof. dr. ir. Peter Troch for bringing me in in contact with Prof. Josep R. Medina. Also his help before, during and after my stay in Valencia is really appreciated. I also really appreciated the daily help and guidance by Mapi. When I had doubts, she always helped and encouraged me. Also a special thanks to Jorge, Ainhoa, Gloria and César for creating a nice atmosphere in the lab and answer my questions when I had some. Of course I can t forget my family. Not only for giving me the opportunity to go abroad and study, but also for the education and freedom they gave me which made me the person I am now. Last but not least I would like to thank my Erasmus friends in Valencia for the awesome weekends after a week of focusing on my thesis. Also my friends in Belgium have always been a great help during my whole study. The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the copyright terms have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation. Sander Franco January 18, 2016

5 Abstract Hydraulic stability of rubble mound breakwaters in breaking wave conditions: a comparative study of existing prediction formulas Supervisor: Prof. Josep R. Medina Co-supervisor: Prof. dr. ir. Peter Troch Master s dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering Laboratorio de Puertos y Costas Chair: Prof. Josep R. Medina Departamento de Ingeniería e Infraestructura de los Transportes Polytechnic University of Valencia Department of Civil Engineering Chair: Prof. dr. ir. Peter Troch Faculty of Engineering and Architecture Ghent University Academic year Summary Nowadays most breakwaters are built in shallow water. To enlarge the experimental basis of such a conditions, some tests are carried out in the 30mx1.2mx1.2m wave flume in the Laboratory of Ports and Coasts of the Polytechnic University of Valencia. Each wave series attacking the rubble mound breakwater scale model in shallow water consists of 1000 irregular waves with a fixed Iribarren number in a fixed water depth. After each wave series, the damage is measured. When total destruction occurs, the scale model is rebuilt and a new test series is initiated with different Iribarren number and/or in a different water depth. The measured damage is compared with the damage predicted by formulas based on the Shore Protection Manual (formula by Van der Meer (1988) and Medina (1994)) on the one hand and predicted by 3 sets of formulas based on Van der Meer (1998) and Van Gent (2004) on the other. Based on this comparison, some improvements to the original formulas are presented and verified by a repetition test. Besides this quantitative analysis of the damage, also a qualitative analysis is performed obtaining the different damage criteria. Keywords Rubble mound breakwater, damage, prediction formula, breaking conditions, irregular wave

6 Hydraulic stability of rubble mound breakwaters in breaking wave conditions: a comparative study of existing prediction formulas Sander Franco Supervisors: Prof. Josep R. Medina, Prof. dr. ir. Peter Troch Abstract-- Nowadays most breakwaters are built in shallow water. To enlarge the experimental basis of such a conditions, some tests are carried out in the 30mx1.2mx1.2m wave flume in the Laboratory of Ports and Coasts of the Polytechnic University of Valencia. Each wave series attacking the rubble mound breakwater scale model in shallow water consists of 1000 irregular waves with a fixed Iribarren number in a fixed water depth. After each wave series, the damage is measured. When total destruction occurs, the scale model is rebuilt and a new test series is initiated with different Iribarren number and/or in a different water depth. The measured damage is compared with the damage predicted by formulas based on the Shore Protection Manual (formula by Van der Meer (1988) and Medina (1994)) on the one hand and predicted by 3 sets of formulas based on Van der Meer (1998) and Van Gent (2004) on the other. Based on this comparison, some improvements to the original formulas are presented and verified by a repetition test. Besides this quantitative analysis of the damage, also a qualitative analysis is performed obtaining the different damage criteria. Keywords-- Rubble mound breakwater, damage, prediction formula, breaking conditions, irregular waves INTRODUCTION Most breakwaters nowadays are built in shallow water, but only little research has been done to improve the knowledge of predicting the damage of the armour layer caused by wave attack. One of the reasons of this lack of research is the fact that such a research is a challenging issue due to the uncertainties in the different irregular wave parameters in shallow water. The scope of this master thesis is to modify and improve existing damage prediction formulas for rubble mound breakwaters in breaking wave conditions based on the comparison between the measured and predicted damage. Two different approaches will be used. A first approach is based on the Shore Protection Manual (SPM) [1] with starting point the Hudson formula [2], another approach is based on the formulas developed by Van der Meer (VDM) [3] and Van Gent (VG) [4]. SPM-approach EXISTING STABILITY FORMULAS Both Van der Meer [3] and Medina [5] developed a formula to predict the damage of the armour layer of a rubble mound breakwater under wave attack. The starting point of these formulas is the Hudson formula [2]: W 50 = ρ rgh 3 K D 3 (1) cot α W 50 is the medium weight of an armour stone, ρ r the apparent rock density, g the gravitational acceleration, H the wave height, K D the stability coefficient, = ρ s /ρ w 1 the relative buoyant density and α the structure slope angle. Values for the stability coefficient K D can be found in the SPM [1]. Introducing the assumption stated by the SPM [1] that the highest of 10% of wave heights H 1/10 = 1.27H s should be used as the design wave height and by introducing the stability parameter N s = H s / D n50 with D n50 the nominal diameter of the armour stone and H s the significant wave height, this gives: H s = (K Dcotα) 1/3 (2) D n This formula only corresponds with a damage level of 0-5% (no damage). To overcome this limitation, table 7.9 of the SPM [1] was used by Van der Meer [3] and Medina [5] and by applying regression analysis, following formulas were developed:

7 Van der Meer: H s = 0.7(K D D cotα) 1/3 S SPM n50 Medina: 0.15 (3) H m0 = 1.15 (cot α) 1/3 0.2 S D SPM (4) n50 S is the damage parameter in which the SPM stands for the fact that table 7.9 from the SPM [1] was used to develop the formulas. H m0 is the significant wave height obtained from spectral analysis. VDM-approach The other approach used to predict the damage of the armour layer under wave attack is based on Van der Meer [3]. Van der Meer developed some formulas, based on tests in a wave flume, taking into account the number of waves N, the permeability of the breakwater P and the surf similarity parameter ξ m = tan α / 2πH s /(gt m 2 ) with T m the mean wave period. For deep water (h > 3H s,toe ), these formulas are: Plunging waves (ξ m < ξ crit ): H s = c D pl P 0.18 ( S 0.2 d n50 N ) 0.5 ξ m (5) Surging waves (ξ m > ξ crit ): H s = c D s P 0.13 ( S 0.2 d n50 N ) P cot α ξ m (6) The values for respectively c pl and c s are 6.2 and 1.0. The significant wave height H s is the incident wave height at the toe of the structure based on time domain analysis. The values of the permeability of the structure are given in Figure 1. It s recommended to use plunging conditions for cot α 4 irrespective of whether ξ m is smaller or larger than ξ crit. For shallow water (h < 3H s,toe ), Van der Meer [3] recommended to use H 2% instead of H s. With the know relation H 2% /H s = 1.4, the formulas become: Plunging waves (ξ m < ξ crit ): H s 0.2 H s = c D pl P 0.18 ( S d n50 N ) 0.5 ξ H m 2% (8) Surging waves (ξ m > ξ crit ): H s = c D s P 0.13 ( S d n50 N ) H s P cot α ξ H m 2% (9) Based on his own test, Van Gent [4] modified formulas (8) and (9) by using the spectral mean energy wave period T m 1,0 instead of the mean wave period T m for the calculation of the surf similarity parameter ξ s 1,0. Also the coefficients c pl and c s are modified, these equal respectively 8.4 and 1.3. These formulas are valid for both deep and shallow water: Plunging waves (ξ m < ξ crit ): H s = c D pl P 0.18 ( S d 0.2 n50 N ) H s 0.5 ξ H s 1,0 (10) 2% Surging waves (ξ s 1,0 > ξ crit ): H s D n50 = c s P 0.13 ( S d N ) 0.2 H s H 2% cot α ξ s 1,0 P (11) Van Gent also developed a new formula. It s a more simple formula, not taking into account the wave period and the difference between plunging and surging waves: H s = 1 D n (1 + D n50 core ) ( S 0.2 d D n50 armour N ) cot α (12) The permeability of the structure is taken into account by introducing the ratio of the nominal diameter of the stones in the core D n50 core and the nominal diameter of the armour layer D n50 armour. This formula can be used in both deep and shallow water. Figure 1: Values of permeability for different structures The distinction between plunging and surging waves is based on the critical breaker parameter ξ crit : ξ crit = [ c pl c s P 0.31 tan α] 1 P+0.5 (7) Cumulative damage All formulas according to the VDM-approach are limited to single storm events. To take into account subsequent storm events, Van der Meer developed a method which make directly use of the formulas of the VDM-approach based on the equivalence hypothesis (Figure 2) [6]. The procedure to assess the cumulative damage is as follows:

8 -Calculate the damage S d1 for the first wave condition -Calculate for the 2 nd wave conditions how many waves would be required to give the same damage as under the 1 st wave conditions. This is denoted by N 1. -Add this number of waves, N 1, to the number of waves of the second wave conditions: N t = N 2 + N 1 -Calculate the damage under the 2 nd wave conditions S d2t with this increased number of waves N t. -Calculate for the 3th wave condition how many waves would be required to give the same damage as caused by the second wave condition etc. Figure 2: Cumulative damage approach by Van der Meer Test equipment EXPERIMENTAL SETUP The tests are carried out in the 1.2mx1.2mx30m wave flume in the Laboratory of Ports and Coasts of the Polytechnic University of Valencia (LPC-UPV). The wave generating system is controlled by an Active Wave Absorption Control System to take into account the reflected waves from the structure on the other side of the wave flume. Experimental design A foreshore slope is created to make the waves break. This foreshore consists of a first slope of 4% over a length of 625cm and a second slope of 2% over 1100cm on which the scale model is built on the end. The slope of the scale model (1:60) of the rubble mound breakwater is 3/2. The armour layer of the breakwater is composed out of 2 layers of quarry stone with nominal diameter equal to 3.18cm covering a filter layer and the core (Figure 3). Figure 3: Section of the rubble mound breakwater scale model, dimensions in cm Realized experiments Every test series is initiated with a wave series of 1000 waves with a wave height that doesn t produce breaking waves (8cm). By increasing the wave height with 1 cm every step and by keeping the Iribarren number constant, more and more waves are breaking until destruction of the model is observed. Because the Iribarren number is kept constant in one test series, a corresponding peak period T p can be calculated for each wave height. The different test series are performed for different Iribarren numbers (Ir = 3 and Ir = 5) and water depths (h model = 20, 30 and 40cm). One repetition test is carried out (Ir = 3 and h model = 20cm) to use later as a blind test to verify the improved and new developed formulas. Data analysis The wave characteristics (wave height, period, ) are obtained by SwanOne, a model developed by TUDelft in MatLab to simulate the evolution of a wave spectrum starting from deep water to shore. It s capable to simulate interactions and transformations of waves. The porosity has to be calculated to verify if the initial porosity is about 37% as prescribed [1] and also to calculate the damage caused by the wave action. It s calculated counting the number of stones using the formula below: p = A v = 1 ND 2 n50 (13) A tot A tot A v is the area of the voids, A tot is the total area of the structure slope in which the stones are counted and N is the number of stones. For the qualitative analysis of the damage, the damage criteria ([7] & [8]) are distinguished as follows: -Initiation of Damage: 5 or more units are displaced from the original position to a new one at a distance equal to or larger than a unit length -Initiation of Iribarren Damage: One stone and his surrounding stones of the 2 nd layer are visible -Initiation of Destruction: 2 or more stones are forced out of the lower armour layer. -Destruction: The filter layer is visible For the quantitative analysis, the damage is calculated using the eroded area A e which is calculated using the visual counting method. For this method, following formulas are used: S d = A e D n50 2 (14)

9 A e = N dd n50 ³ (1 p)r (15) S d is the damage level parameter, N d the number of eroded stones, p the porosity of the settled stones and R the width of the eroded area. Comparison measured and predicted damage The measured damage is compared by the predicted damage using the damage formulas mentioned before. For the SPM-approach, 2 formulas will be used: the Van der Meer formula (3) and the Medina formula (4). For the VDM-approach 3 sets of formulas will be used: the Van der Meer formulas ((5), (6), (8) & (9)), the Van der Meer formulas modified by Van Gent ((10) & (11)) and the Van Gent formula (12). For the VDM-approach, cumulative damage is taken into account. To evaluate the extent to which measured and predicted damage are similar, the relative mean squared error (rmse) will be calculated: rmse = (S d,predicted,i S d,measured,i ) 2 i (S d,measured S d,measured,i ) 2 i (16) For the SPM-approach, the K D coefficient for the Van der Meer formula is chosen to be 8 [9]. The damage will be predicted 5 times for each formula using 5 different wave heights. First of all, the significant wave height based on spectral analysis H m0 obtained from SwanOne will be used, because this is also the wave height proposed by the authors of the formulas. But because most of the test were carried out in breaking conditions, also the breaking wave height H b is calculated 3 times according to the method explained in the SPM [1] This method is based on the formula of Goda [10] (Figure 4). H b is calculated using once the mean wave period from time domain analysis T m, once the mean energy wave period T m 1,0 and once the peak period T p. Finally, the damage is predicted using the peak value of the significant wave height H 1/3 in the surf zone because an exact breaking wave height can t be calculated for irregular waves because the waves break over a wide zone [11], as Goda stated. This value can be read from Figure 5 Figure 4: Graph obtaining breaking wave height [10] Figure 5: Index curves for the maximum value of the significant wave height Also, for each used wave height a new formula will be proposed by calculating the coefficients C 1 and C 2 of the formula below obtaining the lowest rmse: H D n50 = C 1 (cot α) 1/3 S d C 2 (17) For the VDM-approach, the wave heights as explained in the formulas will be used. These are obtained from SwanOne. RESULTS Qualitative analysis By observing the photos taken after each wave series, the damage criteria of each test series can be distinguished. By calculating the average value of the each damage criteria, the following values are obtained: Table 1: Damage criteria Initiation of damage S d = 0.7 Initiation of Iribarren damage S d = 2.6 Initiation of destruction S d = 5.9 Destruction S d = 11.0

10 Predicted damage Sd[-] Predicted damage Sd[-] Predicted damage Sd[-] Quantitative analysis The values of the rmse of the comparison between the measured and predicted damage using the 5 different wave heights for the SPM-approach are shown in Table 2. Also the coefficients of formula (17) are shown. The new formulas were verified by a blind test using the repetition test, which resulted in low rmse. Table 2: rmse-values and values of coefficients new formulas SPM-approach rmse New VDM Medina C Formula 1 C 2 H m T m SPM T m 1, H b T p Goda For the VDM-approach, it can be observed that all breaking waves were plunging waves. The comparison between the measured damage and the damage predicted by the VDM formula and the VDM formula modified by VG are similar. As an example, the comparison using the VDM formula is given in Figure Comparison VDM formula R=0.78 rmse= Measured damage Sd[-] hs=20,ir=5 hs=30,ir=5 hs=20,ir=3 hs=30,ir=3 hs=40,ir=3 Figure 6: Comparison measured damage and predicted damage by VDM formula It can be observed that the predicted damage is almost always lower than the measured damage. It can also be observed that the lower the Iribarren number and the higher the water depth, the bigger the difference between the measured and the predicted damage. So by introducing the water depth at the toe of the structure h toe and the peak period T p (which influences the Iribarren number or surf similarity parameter), the prediction is improved, as can be seen on Figure 7. Also the 90% interval is drawn. For the VDM formula modified by VG the same conclusions can be made. The rmse decreases from respectively 0.77 to 0.09 for the improved VDM formula and from 0.49 to 0.09 for the improved VDM formula modified by VG. The improved formulas are given below: Improved VDM shallow water (c pl = 4.1): H s D n50 = c pl ( 2πh 0.2 toe gt2 ) P 0.18 ( S d 0.2 p N ) H s ξ 0.5 H m (18) 2% Improved VDM deep water (c pl = 2.9): H s D n50 = c pl ( 2πh 0.2 toe gt2 ) P 0.18 ( S 0.2 d p N ) 0.5 ξ m (19) Improved VDM formula modified by VG(c pl = 4.3): H s D n50 = c pl ( 2πh 0.2 toe gt2 ) P 0.18 ( S d 0.2 p N ) H s 0.5 ξ H s 1,0 (20) 2% Comparison improved VDM formula R=0.97 rmse= Measured damage Sd[-] hs=20,ir=5 hs=30,ir=3 hs=40,ir=3 hs=20,ir=3 hs=30,ir=5 Figure 7: Comparison measured and predicted damage by improved formula of VDM Finally, the VG formula can be improved by changing the coefficient (Figure 8 and Figure 9). The rmse decreases from 0.59 to The improved formula is given below: H s = 1 D n (1 + D n50 core ) ( S 0.2 d D n50 armour N ) cot α (21) R=0.94 rmse=0.59 Comparison VG formula Measured damage Sd[-] hs=20,ir=3 hs=30,ir=3 hs=30,ir=5 hs=40,ir=3 hs=20,ir=5 Figure 8: Comparison measured and predicted damage by formula of VG

11 Predicted damage Sd[-] Comparison improved VG formula R=0.94 rmse= Measured damage Sd[-] Figure 9: Comparison measured and predicted damage by the improved formula of VG CONCLUSIONS hs=20,ir=5 hs=40,ir=3 hs=30,ir=5 hs=30,ir=3 hs=20,ir=3 First of all, a qualitative analysis was executed determining the Initiation of Damage, Initiation of Iribarren damage, Initiation of destruction and Destruction. The average values based on the tests are given in Table 1. Next a quantitative analysis was carried out. For the SPM-approach, following conclusions can be made. If the (deep water) wave characteristics are not known, these can be obtained using SwanOne. The best formula to use these wave characteristics based on the executed tests is the new proposed formula (17) using the significant wave height H m0 with values for the coefficients C 1 and C 2 respectively 1.17 and If the deep water characteristics are known, the breaking wave height can be calculated and used to predict the damage according to the SPM approach using the new proposed formula with the mean period T p with values for the coefficients C 1 and C 2 respectively 1.32 and 0.17 or by using Goda s approach (Figure 5) and with values for the coefficients C 1 and C 2 respectively 1.26 and Using the VDM formula (3), the breaking wave height based on T m or T m 1 gives the best prediction of the damage while using the Medina formula (4) the significant wave height H m0 at the toe of the structure obtained by SwanOne gives the best approximation. For the VDM approach, the improved VDM formulas (18) & (19), the improved VDM formula modified by VG (20) and improved VG formula (21) are a very good improvement. If only a few wave characteristics are known (no wave period, H 2%,..), the improved VG formula can be used. In other cases, both the improved VDM formula and improved VDM formula by VG will give the best approximation. A really important remark should be made. The qualitative analysis, all improvements and new proposed formulas are based only a few test series. All formulas should be verified and tested with other tests and more repetition tests (also in other laboratories) as well as the qualitative analysis. These repetition tests are necessary because damage is a very sensitive parameter. The difference in measured damage for 2 equal tests can be 30% [11]. FURTHER RESEARCH Besides the repetition tests discussed above, other future research can be done by changing the parameters. For instance the nominal armour diameter, structure slope, foreshore slope, The conditions could also be changed in such a way that surging waves occur (instead of plunging waves as in these conditions). Finally also other armour units could be investigated. REFERENCES [1] CERC, Shore protection manual: Washington (D.C.) : Government printing office, [2] R. Y. Hudson, "Laboratory investigation of rubble-mound breakwaters," Journal of the Waterways and Harbors Division, vol. 85, pp , September [3] J. W. Van der Meer, "Rock slopes and gravel beaches under wave attack," PhD thesis, Delf University of Technology, [4] M. A. Van Gent, A. Smale, and C. Kuiper, "Stability of rock slopes with shallow foreshores," presented at the Coastal Structures 2003, Portland, [5] J. R. Medina, R. T. Hudspeth, and C. Fassardi, "Breakwater Armor Damage due to Wave Groups," Journal of Waterway, Port, Coastal, and Ocean Engineering, vol. 120, pp , [6] J. W. Van der Meer, "Design of concrete armour layers," in Proc 3rd dint con. coastal structures, Santander, Spain, 2000, pp [7] M. Losada, J. M. Desire, and L. M. Alejo, "Stability of blocks as breakwaters armour units," Journal of Structural Engineering, pp , [8] C. Vidal, M. Losada, and J. R. Medina, "Stability of mound breakwater's head and trunk," Journal of Waterway, Port, Coastal and Ocean Engineering, pp , [9] CIRIA, The rock manual : the use of rock in hydraulic engineering: London : CIRIA, [10] Y. Goda, "A synthesis of breaker indices," Transactions of the Japan Society of Japan Engineers, vol. 2, pp , [11] Y. Goda, Random seas and design of maritime structures: Singapore : World scientific, 2000.

12 Table of Contents CHAPTER 1: INTRODUCTION... 1 CHAPTER 2: LITERATURE STUDY Wave reflection... 2 a. Reflection coefficient... 2 b. Separation methods... 3 c. LASA method Breaking waves... 5 a. General... 5 b. Wave breaking criteria... 6 Regular waves... 7 Irregular waves Damage a. Failure modes breakwater b. Armour damage Qualitative analysis Quantitative analysis Stability formulae a. Hudson b. Van der Meer Deep-shallow water Deep water conditions Shallow water conditions c. Van der Meer modified by Van Gent d. Van Gent e. Cumulative damage Melby method Van der Meer method CHAPTER 3: EXPERIMENTAL SETUP Test equipment a. Wave generator b. Energy dissipation system c. Wave measurement... 28

13 d. Cameras Experimental design a. Foreshore slope b. Model Core Filter Armour layer Realized experiments Data analysis a. Wave analysis SwanOne Measurements by wave gauges in canal without a model Measurements by wave gauges in canal with model Comparison between SwanOne and measurement in canal without model b. Porosity measurement c. Damage calculation d. Comparison measured damage with predicted damage SPM-approach VDM-approach CHAPTER 4: RESULTS Wave data Measured damage and porosity Comparison measured damage with predicted damage a. SPM-approach b. VDM-approach CHAPTER 5: CONCLUSIONS REFERENCES APPENDIX A: SUMMARY DATA SWANONE APPENDIX B: PHOTOS MODEL AFTER WAVE ACTION... 60

14 List of Figures Figure 1: Breaker types... 5 Figure 2: Limiting steepness in deep water (CERC, 1984)... 6 Figure 3: Graph showing formula by Goda (1970)... 8 Figure 4: Graph showing formula by Weggel (1972)... 8 Figure 5: Breaker index as a function of the Iribarren number... 9 Figure 6: Index curves for the maximum value of the significant wave height Figure 7: Different failure modes according to Bruun (1978) Figure 8: Different breakwater failure modes according to Burcharth (1992) Figure 9: Eroded area Figure 10: Values of permeability for different structures Figure 11: Sensitivity analysis for damage level parameter and permeability Figure 12: Cumulative damage approach by Van der Meer Figure 13: Wave flume at LPC-UPV Figure 14: Wave generator (Herrera, 2013) Figure 15: Energy dissipation system and schematically overview of the frameworks Figure 16: Position of the sensors, dimensions in meter Figure 17: Section of the rubble mound breakwater, dimensions in cm Figure 18: Sieve curve of the core Figure 19: Distribution of the filter layer stones Figure 20: Distribution of the armour layer stones Figure 21: Rubble mound model constructed with an initial porosity of approximately 37% Figure 22: Screenshot of LPCLab Figure 23: Comparison between significant wave height obtained from SwanOne and measurements in the canal without the model Figure 24: Marked stones by AutoCAD Figure 25: Comparison formulas based on Figure 26: Comparison formulas based on the (Goda, 2000) Figure 27: Comparison formulas based on Figure 28: Comparison formulas based on Figure 29: Comparison formulas based on Figure 30: Verification new formulas (blind test) Figure 31: Comparison measured and predicted damage by formula of VDM Figure 32: Comparison measured and predicted damage by improved formula of VDM Figure 33: Comparison measured and predicted damage by formula of VDM modified by VG... 50

15 Figure 34: Comparison measured and predicted damage by improved formula of VDM modified by VG Figure 35: Comparison measured and predicted damage by formula of VG Figure 36: Comparison measured and predicted damage by the improved formula of VG Figure 37: Verification new formulas... 51

16 List of Tables Table 1: Values surf similarity parameter for different breaker types... 6 Table 2: Design values of for a two-diameter thick armour layer Table 3: as a function of armour layer damage and armour type Table 4: Range of parameters VDM formula Table 5: Range of parameters VDM formula modified by VG and VG formula Table 6: Range of parameters Melby method Table 7: Test data for Ir=3 and h=20cm Table 8: Test data for Ir=5 and h=20cm Table 9: Test data for Ir=3 and h=30cm Table 10: Test data for Ir=5 and h=30cm Table 11: Test data for Ir=3 and h=40cm Table 12: Summary formulas VDM-approach Table 13: Summary of the SwanOne data at the toe of the structure ( Table 14: Summary of the SwanOne data at the toe of the structure ( ) Table 15: Summary of the SwanOne data at the toe of the structure ( ) Table 16: Measured damage, porosity and qualitative damage analysis for Table 17: Measured damage, porosity and qualitative damage analysis at Table 18: Measured damage, porosity and qualitative damage analysis for and Ir=3. 45 Table 19: Measured damage, porosity and qualitative damage analysis for the repetition test (, Ir=3) Table 20: Proposed new formulas based on SPM-approach Table 21: of all formulas Table 22: Qualitative damage analysis... 52

17 List of Symbols Symbol Description/Explanation Unit Eroded area Height of one strip Width of one strip Damage Particle size for which 15% of the particles are smaller Particle size for which 95% of the particles are smaller Damage according to the virtual net method Damage of one strip Nominal block diameter or equivalent cube size Median nominal diameter or equivalent cube size Median nominal diameter of the armour stones Median nominal diameter of the core material Incident wave energy Reflected wave energy Gravitational acceleration Wave height Wave height in deep water conditions Significant wave height based on time domain analysis equal to the average of highest 1/3 of all waves heights Average of highest 1/10 of all wave heights Wave height exceeded by 2% of the waves Breaking wave height Incident wave height Critical wave height over which all waves are breaking (irregular waves) Mean wave height Significant wave height based on spectral analysis Root-mean-square wave height Significant wave height Significant wave height at the toe of the structure Reflected wave height

18 Wave height at toe of the structure Total wave height Water depth Breaking water depth Water depth at the toe of the model Water depth at the wave paddle of the wave generating system Water depth at toe of the structure Stability coefficient Shoaling coefficient Wave number Wave length Wave length in deep water condition Slope of the beach (gradient) Number of stones Number of waves Number of armour units in 1 strip Damage Number of eroded stones Stability parameter Damage Time counter Permeability of the structure Porosity of settled stones Initial porosity Porosity strip i after wave attack Width of the eroded area Crest freeboard, level of crest relative to SWL Reflection coefficient Relative mean squared error Damage level parameter Damage level parameter based on table 7.9 in the SPM (CERC, 1984) Wave steepness

19 Wave period Significant wave period based on time domain analysis Mean wave period based on time domain analysis Mean energy wave period Spectral peak period Duration time of storm to reach damage level Duration time of additional storm Medium weight of an armour stone Structure slope angle Slope of the beach JONSWAP spectral shape parameter Breaker depth index Relative buoyant density Spectral shape parameter Surf similarity parameter/iribarren number Critical breaker parameter Surf similarity parameter using the mean wave period Surf similarity parameter using the mean energy wave period Mass density of stone Bulk density of material as laid on slope Apparent rock density Density of water Effective friction angle between rock Breaker height index Angular frequency of waves

20 List of Abbreviations Abbreviation AWACS CEM De DFT ESCOLIF FFT HeP IDa Ide IIDa JONSWAP LASA LPC SPM SWL TUDelft UPV VDM VG Explanation Active Wave Absorption Control System Coastal Engineering Manual Destruction Discrete Fourier transform Estabilidad hidráulica de los mantos de escollera, cubos y Cubípodos frente a oleaje limitado por el fondo/hydraulic stability of cube, Cubipod and rubble mound breakwaters in depth limited conditions Fast Fourier Transform Heterogeneous packing Initiation of damage Initiation of destruction Initiation of Iribarren damage Joint Northsea Wave project Local approximation using simulated annealing Laboratory of Ports and Coasts/Laboratorio de Puertos y Costas Shore Protection Manual Still water level Delft University of Technology Polytechnic University of Valencia/Universidad Politécnica de Valencia Van der Meer Van Gent

21 Chapter 1: Introduction Rubble mound breakwaters are structures protecting a coastal area from excessive wave action as there are ports, port facilities and coastal installations. They consist of mainly quarried rock. Generally armour stone or artificial concrete armour units are used for the outer armour layer which ensures the protection against wave attack Most breakwaters nowadays are built in shallow water, but only little research has been carried out to improve the knowledge of predicting the damage of the armour layer caused by wave attack. One of the reasons of this lack of research is the fact that such a research is a challenging issue due to the uncertainties in the different irregular wave parameters in shallow water. The scope of this master thesis is to modify and improve existing damage prediction formulas for rubble mound breakwaters in breaking wave conditions based on the comparison between the measured and predicted damage. Some model tests in the wave flume in the Laboratory of Ports and Coasts of the Polytechnic University of Valencia were carried out for this purpose. Two different approaches will be used to predict the damage. A first approach is based on the Shore Protection Manual. The starting point of this approach is the Hudson formula, another approach is based on the formulas developed by Van der Meer. In chapter two, the theoretical background of these approaches will be given together with a resume of the existing literature dealing with the hydraulic stability of rubble mound breakwaters including wave reflection, wave breaking and (armour layer) damage. Chapter three describes the test equipment together with an explanation of the experimental design. Also data analysis will be extensively discussed. In chapter four, the results are presented. The best existing damage prediction formulas are chosen and some improvements of these formulas are presented. Chapter 5 finally contains the conclusions of the realized work. 1

22 Chapter 2: Literature study In this chapter, an overview of the existing literature about the stability of rubble mound breakwaters in breaking conditions will be given. First of all, an important process that influences all experiments will be discussed, i.e. wave reflection. Also some methods to overcome this problem will be given. Next the phenomenon of wave breaking will be discussed. The different breaker types and the methods to calculate the breaking wave height in both regular and irregular conditions will be defined. Further also the damage that waves cause on the armour layer of a breakwater is treated by explaining the different failure modes that can damage a breakwater. Also the qualitative and quantitative analyses of armour damage will be reviewed. Finally the heart of the matter will be discussed, i.e. the different stability formulas. A historical overview is given and the most important formulas will be discussed. 1. Wave reflection a. Reflection coefficient When waves attack a structure, 3 different processes will take place: Dissipation of energy on the permeable medium and transmission of a part of the energy through the structure to the other side of the structure Breaking of the waves over the slope of the structure Reflection of the waves on the structure The breaking of the waves will be introduced in calculations by adjusting coefficients or taking into account the maximum breaking wave height. Also the permeability will be taken into account by different coefficients in empirical formulas. The amount of the wave energy that will be reflected on a sloping structure depends on the slope, permeability and roughness of the structure. Also the wave steepness and angle of wave attack influence the reflected wave energy. A measure to take these influences into account is the reflection coefficient which is equal to the ratio of the reflected wave height and the incident wave height. Because of the proportional relationship between the wave energy and the square of the wave height: (1) The reflected, incident and total wave height and energy are related as follows: (2) This gives following formulas for the incident and reflected wave height: 2

23 (3) (4) b. Separation methods The problem is that this reflection coefficient is unknown. To obtain the incident or reflected wave height, some methods are developed. Most of these methods are based on the linear wave theory: it s assumed that an irregular wave train can be represented as a superposition of a finite number of regular waves of different amplitude, emphasis and frequency (Mansard and Funke, 1980). These methods will be discussed briefly and only the characteristics of these methods will be given. An extensive discussion would lead us too far. If more information is needed, the references can be consulted. The classic method to separate the incident waves and reflected used in most laboratories is the Two point method by Goda and Suzuki (1976) generalized using 3 sensors as the Three least squares method by Mansard and Funke (1980) and Gaillard et al. (1980). Each lab can have his own variations and adaptations on these methods. The method of Goda and Suzuki (1976) is based on earlier work of Kajima (1969) and Thornton and Calhoun (1972). Goda & Suzuki introduce the Fast Fourier transform (FFT) in the method of Thornton & Calhoun to identify the linear wave component in each signal. This two point method has some assumptions that limit its functionality: Linear dispersion: Only stationary waves are considered Linear superposition is assumed to calculate the irregular wave components No noise: the components of high and low frequency are eliminated previously Global estimation: It s necessary to use the complete registration of the waves to estimate the incident and reflected wave. Later, the tow point method of Goda and Suzuki was extended to the Three point method. As mentioned before, each laboratory has its own variation and adaptation on these methods. The method that will be discussed more in detail is the one used in the Laboratory of Ports and Coasts (LPC), i.e. the LASA method c. LASA method The LASA method (Local Approximation using Simulated Annealing) is a method developed by Medina (2001) and improved by Figueres and Medina (2004) to separate reflected and incident waves. The method is intended to overcome some of the limitations of previous methods, i.e. the limitations of linearity and stationarity. To do so, the LASA method (Medina, 2001) is based on a local approximation model considering linear and Stokes II non-linear components and uses simulated annealing to calculate and optimize the model parameters of the local approximation model. The main characteristics of this method are the possibility to take into account the data of n sensors (n 2), the analysis of both non-stationary regular and irregular waves and the 3

24 discretization of the wave analysis. The LASA method has been verified with the two point method of Goda and Suzuki (1976), Kimura (1985) and others. It became clear that the LASA method is very robust and consistent method in numerical and physical test both for regular as irregular waves. The general process of the LASA method to realize the separation of incident waves and reflected waves can be divided in three steps: Elimination of the noise Establishment of the frames for the estimation of the central points Definition of a local approximation model Figueres and Medina (2004) optimized the original LASA method, based on Stokes II components, and developed LASA-V, a method that is based on a local approximation model considering non-linear Stokes-V components. The LASA-V method can be used for waves with a high steepness. This model allows the analysis of tests with non-linear and non-stationary waves. 4

25 2. Breaking waves a. General When waves arrive in the surf zone, they start to break. Simply said, this is because when a wave approaches a beach, its length starts to decrease and the wave height increases. Galvin (1968) classified wave breaking in four categories: Spilling, plunging, collapsing and surging breakers (Figure 1). Spilling breakers produce a foamy water surface because of an unstable wave crest. Spilling breakers are also characterized by their symmetrical wave contours. This type of breaker is typical for very gentle beach slopes. Plunging waves produce a high splash, coming from the crest that curls over the shoreward face of the wave. The wave front is first very vertical, starts to curl and finally falls. A lot of energy is dissipated during this process. This kind of breaking is observed on gentle to intermediate beach slopes. In very steep beaches, surging waves occur: the wave will not break. The front of the wave arrives on the beach with minor breaking. The wave goes up and down on the slope only forming a little bit of foamy water. Collapsing waves are classified somewhere between surging and plunging waves. The crest is not breaking, but the lower part of the shoreward face steepens up and falls. An irregular turbulent water face is created. Figure 1: Breaker types The classification of the different breaking waves is based on the surf similarity parameter or Iribarren number (Iribarren and Nogales, 1949). This number is proportional to the tangent of the slope of the beach and inversely proportional to the root of the wave steepness: (5) With Slope of the beach [] Wave height in deep water conditions [] Wavelength in deep water conditions [] Wave period Gravitational acceleration 5

26 The values for the surf similarity parameter for the different breaker types (Battjes, 1974) are given in Table 1. Table 1: Values surf similarity parameter for different breaker types Surging/collapsing Plunging Spilling b. Wave breaking criteria The used criteria for wave breaking in coastal engineering are mostly semi-empirical formulas obtained by laboratory tests. A wave remains stable and only doesn t break if the velocity of the water particles is lower than the wave celerity (Stokes, 1880). Michell (1893) observed that the limiting wave steepness, which is the ratio of the wave height and the wavelength, is in deep water. This occurs when the crest angle is equal to 120 (Figure 2). Figure 2: Limiting steepness in deep water (CERC, 1984) When a wave is moving into shoaling water, the limiting steepness which it can attain, will decrease (CERC, 1984). Miche (1944) observed that the limiting wave steepness for waves in depths less than equal is to, with the water depth. A wave that moves from deep water to shoaling water will move towards the shore until it breaks. The wave height at breaking is commonly defined by the breaker (depth) index defined as the maximal wave height to depth ratio, with the subscript standing for the breaking point. (6) A second parameter, the breaker height index, is also widely used: (7) However, the latter induces a greater uncertainty in the prediction of according to present authors (Camenen and Larson, 2007). 6

27 Regular waves The first value of the breaker depth index for regular waves was estimated by McCowan (1891) and equal to 0.78 for a solitary wave traveling over a horizontal bottom. Munk (1949) defined the breaker height index of a solitary wave as: (8) and the breaker depth index: (9) Subsequent observations and investigations by Iversen (1952), Galvin (1968), Goda (1970), Weggel (1972), and others have shown that and depend on incident wave steepness and beach slope. Thus for regular waves and uniform beach slope, next formulas are proposed: Goda (1970) (10) Weggel (1972) (11) Battjes (1974) (12) Ostendorf and Madsen (1979) (13) Singamsetti and Wind (1980) (14) Smith and Kraus (1990) (15) 7

28 Rattanapitikon and Shibayama (2000) (16) Those formula were compared with a compiled data set (Camenen and Larson, 2007), but there was not one best formula. The best results for the existing formulas and use of all data were given by Battjes (1974), Goda (1970) and Ostendorf and Madsen (1979). For steep beaches, the Weggel (1972) formula gave the best prediction. Also some general remarks were made. Weggel (1972) and Singamsetti and Wind (1980) overestimate the breaker depth index and produce considerable dispersion of the results while Smith and Kraus (1990) underestimate the breaker depth index. A disadvantage of the Ostendorf and Madsen (1979) and Battjes (1974) formulas are that no value can be calculated for a beach with a slope equal to zero. A semi-empirical relationship for the breaker height index is derived from linear wave theory by Gaughan and Komar (1974) and equals: (17) Finally another criterion is given by Rattanapitikon and Shibayama (2000) and Rattanapitikon et al. (2003), with the wavelength when the wave breaks calculated using the linear wave theory: (18) Both Goda (1970) and Weggel (1972) included their formulas in a graph. These are given in Figure 3 and Figure 4. Figure 3: Graph showing formula by Goda (1970) Figure 4: Graph showing formula by Weggel (1972) 8

29 To calculate the breaking wave height from the breaker height index, The deep water wave height has to be calculated if this is not known. This can be done using the shoaling formula (Goda, 2000): (19) With Shoaling coefficient by small amplitude wave theory [] wavenumber Irregular waves For irregular waves, the breaking point as well as the breaking wave height can start over a wide zone, in contrast to the case of regular waves (Goda, 2000). In the zone where more or less all waves are breaking (saturated breaking zone), the root-mean-square breaking wave height and the zero-moment wave height depend on the local depth (Thornton and Guza, 1983): (20) (21) CIRIA (2007) states that the typical values for the breaker depth index are 0.5 to 0.6. These values mainly depend on the Iribarren number, and they can reach 1.5 for individual waves. Data from different authors are shown in Figure 5. Figure 5: Breaker index as a function of the Iribarren number The critical wave height over which all the waves are broken can be found by applying an energy flux balance and is given by (Battjes and Stive, 1985): (22) 9

30 This formula was later modified by Nairn (1990): (23) A linear relationship between the product of the local wave number and the water depth was found by Ruessink et al. (2003): (24) Goda (2010) changed his formula (10) for irregular waves: (25) The deep water wavelength is calculated using the period obtained from time-domain analysis. This formula was verified starting with regular breaking waves for 6 different slopes (including 0%). As final remark, it can be stated that, as said before, the breaking point can t be defined clearly. Therefore, Goda (2000) proposes to use the peak value of the significant wave height within the surf zone as an alternative to the breaker height. These values are depicted on Figure 6. Figure 6: Index curves for the maximum value of the significant wave height 10

31 3. Damage a. Failure modes breakwater To be able to talk about the stability of a mound breakwater, it s necessary to consider both the individual stability of an armour unit as the stability of all units together. The loss of stability can have different reasons which should be understood. Bruun (1978) is one of the authors who has worked with it. He stated 11 different principal failure modes. They are summarized as follows (Vanhoutte, 2009): I. Loss of armour units from the principal armour layer (increasing porosity) II. III. IV. Rocking of the armour units; breaking occurs due to fatigue Damage of the inner slope by wave overtopping Sliding of the armour layer due to a lack of friction with the layer below V. Lack of compactness in the underlying layers, causing transmission of energy to VI. VII. VIII. IX. the interior of the breakwater; this might lift the cap and the interior layers. Undermining of the crone wall Breaking of the armour units caused by impact, simply by exceeding its structural resistance or by slamming into other units Settlement or collapsing of the subsoil Erosion of the breakwater toe or the breakwater interior X. Loss of the mechanical characteristics of the materials XI. Construction errors These failure modes are depicted in Figure 7. Figure 7: Different failure modes according to Bruun (1978) All these 11 different failure modes can be classified in 5 groups (Gomez-Martin, 2002): Unit stability: refers to the capacity of each armour unit to resist movement subjected to the wave action. (I,II,III) Global stability: it s the stability of the complete breakwater or more specific the complete armour layer. (IV,V,VI) Structural stability: refers to the capacity of each unit to resist (without breaking) the tensions caused by transport, construction, wave action, used granular and movements caused by currents. (II, VII) Geotechnical stability: stability of the underground. It includes the carrying capacity and the sensitivity to erosion of the breakwater toe.(viii, IX) Construction errors (X,XI) 11

32 The global and unit stability are the stability that will be considered here. Together they can be called the hydraulic stability. Burcharth (1992) also enumerate the different failure modes of a breakwater (Figure 8). Figure 8: Different breakwater failure modes according to Burcharth (1992) In this thesis, most attention will go to the hydraulic stability of the armour units. This is also the main mode of failure. It s classified in the failure group of Unit Stability according to Gomez- Martin (2002). This failure mode can be caused by 2 different reasons: the simple extraction of the units under wave action and the settlement caused by the heterogeneous packing. The heterogeneous packing (HeP) failure mode is a failure mode that is significant in the case of regular armour units (Gomez-Martin, 2006). The HeP failure mechanism reduces the packing density of the armour layer near the SWL without extracting elements, generating zones with low porosity and corresponding zones with high porosity. The impact of the HeP failure mode depends on: I. Type of armour unit II. III. IV. Difference between the initial porosity and the minimum porosity Slope of the armour layer Friction coefficient between the armour layer and the secondary layer b. Armour damage There are two ways to quantify the damage: the qualitative and the quantitative way. Qualitative analysis To do a qualitative analysis of the damage, several stages of damage should been distinguished. Losada et al. (1986) created three hydrodynamic criteria: incipient damage, Iribarren s damage and destruction. Vidal et al. (1991) further developed these criteria by adding a fourth criteria: Initiation of destruction. Hence the four criteria are: Initiation of damage (IDa): certain number of units are displaced from their original position to a new one at a distance equal to or larger than a unit length. 12

33 Initiation of Iribarren damage (IIDa): wave action may extract armour units placed on the lower armour layer. This can be defined as the moment when one stone and his surrounding stones of the 2 nd layer are visible. Initiation of destruction (IDe): small number of units (two or three) in the lower armour layer are forced out. Destruction (De): more pieces of the secondary layer are removed and the filter layer is visible. A disadvantage of these criteria, is the fact that HeP is not taken into account, because only the units extracted are considered. Quantitative analysis The damage in the quantitative analysis is measured by counting the number of displaced units or by measuring the eroded surface profile of the armour slope (USACE, 2002). If the damage is measured by counting the displaced units, which is mostly done in case of (complex structures of) concrete armour units, the damage can be given as a percentage displaced units within a reference area: (26) The damage can also be given as a dimensionless parameter: (27) Another way to calculate the damage is making use of the eroded area. This is mostly done in case of rock armour. One of the first who used this way of calculating were Iribarren (1938) and Hudson (1959). Hudson defined the damage as the percent erosion of original volume: (28) Thompson and Shuttler (1975) defined another damage parameter : (29) With Erosion area in a cross-section Bulk density of material as laid on the slope Mass density of stone Diameter of stone that exceeds the 50% value of sieve curve The advantage of these formulas is the fact that the damage is independent of the size of the armour layer, compared to a percentage of damage. The disadvantages of the last formula are the measurement of the bulk density and the use of the sieve diameter instead of the actual 13

34 mass of the stone. To improve this formula, Broderick (1983) deleted the bulk density and defined the damage level parameter as follows : With Eroded area around SWL [ ] (30) Medium weight of an armour stone [] By introducing the median nominal diameter the damage can be written as: (31) The eroded area is depicted in Figure 9. It takes into account both settlement and displacement. The eroded area can be seen as the number of squares with a side that fits into the erosion area. Another physical description of the damage of is the number of cubic stones with a side of eroded within a -wide strip of the structure. The actual number of stones eroded within this strip can be more or less than, depending on the porosity, the grading of the armour rocks and the shape of the rocks. Generally the actual number of rocks eroded in a - wide strip is equal to 0.7 to 1 times the damage.(van der Meer, 1998) Figure 9: Eroded area The slope angle of the structure has a big influence on the limits of. These limit values are characterized as follows: Start of damage/initial damage: corresponding to no damage in the Hudson formula (see later) Intermediate damage Failure, when the filter layer is visible 14

35 For design purposes of a double layer armour stone breakwater, these values are given in Table 2. For S-values higher than 15-20, the deformation of the structure results in an S-shaped profile (Van der Meer, 1998). Table 2: Design values of for a two-diameter thick armour layer Slope Start of Intermediate Failure (under damage damage layer visible) 1: : : : : There are different ways to measure the eroded area. It can be measured by a surface profiler (mechanic/laser profiler), by computing the planar eroded area on the outer layer of the armour, using a digital image processing technique or by counting the removed armour stones settled over the original armour layers (Vidal et al., 2006). If the latter is used, the eroded area is given by: (32) With the number of eroded stones, the porosity of the settled stones and the width of the eroded area. This method is called the visual counting method. The disadvantage of all formulas above is that they do not take into account the heterogeneous packing. Therefore, a new method was necessary that takes into account the changes in porosity. This method is called the new Virtual Net Method (Gómez-Martín and Medina, 2006). In this method, the armour layers are divided into strips with each a width of times the equivalent cube size and a length. The number of armour units in every strip is counted and with this number, the porosity of every strip after wave attack is calculated using the formula below. (33) Next, the dimensionless damage in each strip can be calculated taking into account the initial porosity : (34) By summing up the different damages over the different strips, the equivalent dimensionless armour damage could be obtained: (35) 15

36 4. Stability formulae Until 1933 no methods to calculate a rubble mound breakwater existed. The breakwaters were build using the experience obtained from the construction of older ones. Obviously, this qualitative knowledge wasn t sufficient for the construction of breakwaters. Also, the complexity of the phenomena that were involved (wave characteristics, wave behavior, ) impeded the development of this qualitative knowledge. The first formula for the calculation of rubble mound breakwaters was formulated by Castro (1933). Castro showed that the wave forces are the reason of the destruction of the breakwaters and that the waves push the rock over the breakwater. However, this almost never happens Iribarren (1938) published a new formula for the weight of the rocks resisting a certain wave height for the principal layer of the breakwater. This formula would be the starting point for the later developed formula by Hudson. From 1949, a big development in the knowledge and of the formulas started. Also the phenomena related with the water flow over the slope of the breakwater were studied. The discussion of all the different formulas and modified formulas would lead us so far, so only the most important and the formulas that have most to do with this master thesis will be discussed. a. Hudson One of the most known and used stability formula is Hudson`s formula (Hudson, 1959). It s based on model tests with regular waves on non-overtopped rock structures with a permeable core and based on the pioneering work of Iribarren (1938) and Hudson and Moore (1951): Iribarren (1938) (36) With Medium weight of armour stone [] Apparent rock density [ ] =0.015 & (rock-fill & concrete blocks randomly dumped) Wave height [] [-] Density of water [ ] Hudson and Moore (1951) Structure slope angle [rad] (37) With Wave height at the toe of the breakwater [m] constant between and , depends on the slope Effective friction between rock [-] 16

37 Hudson (1959) (38) The parameters are equal to the ones stated before. The Hudson formula can be applied for structures with a slope from 1:1.5 to 1:5. The values of the stability coefficient depend on many characteristics: Shape of the element of the armour layer Nature of the element of the armour layer (concrete-rock) Number of armour layers Roughness of the element Degree of interlocking obtained in placement Water depth near the structure (breaking or non-breaking) Part of the mound breakwater (head or body) Angle of incident wave Porosity of the core Size of the core Width of the crest In the Shore Protection Manual (SPM) (CERC, 1977), the values of the stability coefficient for rough angular quarry stone armour units are and for respectively breaking waves on the foreshore and non-breaking waves on the foreshore, corresponding with a damage level of 0-5% (no damage condition). These values are valid for rough, angular, randomly placed armour stone in two layers on a breakwater trunk. Also it is suggested to use the significant wave height in equation (37). In addition other values were suggested for a wide range of armour units and other conditions. In SPM (CERC, 1984) some modifications to the use of Hudson s formula are made. One of these modifications is the use of the average of the highest 10% of wave heights as the design wave height which is equal to 1.27 instead of. Also the value for breaking waves is changed from 3.5 to 2.0; the value for non-breaking waves remains the same. The original Hudson formula (37) can be rewritten, in terms of the stability parameter : (39) With Stability parameter [-] Significant wave height [] Mean nominal diameter of the armour stone [] 17

38 This gives: (40) The formula of Hudson has its limitations. First, it can only be used for regular waves. Also the wave period and storm duration are not taken into account. Besides these limitations, Hudson s formula can only be used for non-overtopped and permeable structures only. Finally there is no description of the damage level, because the formula only corresponds with a damage level of 0-5% (no damage). To overcome this last limitation, higher damage percentages have been determined as a function of the wave height for several types of armour units. The values for armour stone are given in Table 3 (CERC, 1984). Table 3: as a function of armour layer damage and armour type Armour Type Relative wave height Damage with corresponding damage level [-] [-] Smooth armour stone Angular armour stone The notation instead of is used in this case, because it s not clarified in the SPM how the damage is measured and the conversion from the damage to the damage level parameter is only an assumption. Also a slightly different conversion could be made. Van der Meer (1988) modified equation (40) by using Table 3 (angular armour stone) and applying regression analysis. This modified formula for the stability number equals: (41) According to the Rock Manual (CIRIA, 2007), the values for an impermeable and permeable core for both breaking and non-breaking conditions are respectively 1 and 4, accepting that 5% of the data will lead to a higher damage level than predicted. values of respectively 4 and 8 can be used to describe the main trend. This was concluded after comparing data used by Van der Meer (1988) and Van Gent et al. (2004) with the equation above using different values. Medina et al. (1994) proposes another empirical formula to include the data in Table 3 in equation (40): 18

39 (42) One of the reasons of the difference between this formula and formula (41) is because Medina used a different conversion between the damage and the damage level parameter. But because the formula is still based on the data from the SPM (CERC, 1984), the notation is still used. b. Van der Meer Deep-shallow water While the classification according to wave propagation of water waves (deep water waves, transitional water waves and shallow water waves) is based on the relative depth criterion (respectively, and ) (USACE, 2002), the definition of shallow water is defined by for the limit of the field of application of the Van der Meer formulae (CIRIA, 2007). More recently, Van Gent developed even another classification, based on the ratio of the significant wave height at the structure to the one observed offshore. If the ratio is larger than 0.9 or smaller than 0.7, the structure is in respectively deep and very shallow water (with a considerable amount of wave breaking). If the ratio is between 0.7 and 0.9, the structure is said to be in shallow water conditions and some (limited) wave breaking will already occur (Van Gent et al., 2004). Deep water conditions Besides the modified Hudson formula, Van der Meer (1988) also developed also some formulas, He based his study on the earlier work of Thompson and and Shuttler (1975). An extensive series of model tests was conducted at Delft Hydraulics. This series include structures with a wide range of core/underlayer permeabilities and a wide range of wave conditions. The Van der Meer (VDM) formula takes, in contrast to the Hudson formula, into account the effects of storm duration, wave period, the structure s permeability and a clearly defined damage level. VDM considers 2 types of breaking waves: plunging and surging waves. The transition from plunging conditions to surging conditions is given by the critical breaker parameter and depends on the structure slope : (43) For plunging waves ( ): (44) 19

40 For surging waves ( ): (45) With Significant wave height [m], of the incident waves at the toe of the structure P Permeability of the structure [-] N Number of incident waves at the toe of the structure [-] surf similarity parameter based on the mean wave period from time domain analysis and significant wave height It is recommended to assume plunging conditions for irrespective of whether the surfsimilarity parameter is smaller or larger than. The values for the permeability of different structures are given in Figure 10. The lower boundary is equal to 0.1. This is the case when the armour layer has a thickness of 2 diameters of the armour unit and the layer under it is impermeable. This is often the case for seawalls and revetment. The upper boundary is given by a homogeneous structure without filter and core. It only consists of rock. The permeability is then equal to 0.6. The considered range of parameters by Van der Meer (1988b) is given in Table 4. Figure 10: Values of permeability for different structures 20

41 Table 4: Range of parameters VDM formula Parameter Symbol Range Stability number 1-4 Wave steepness Surf similarity parameter Damage as a function of the number of waves Ratio deep water wave height and water depth at toe < Armourstone gradation Permeability Slope angle 1:1.5-1:6 Spectral shape parameter Crest height -1-2 The VDM formulas are related to a single storm event. The maximum number of waves is equal to 7500 because after this number of waves, the damage reaches more or less an equilibrium This means that the damage for more than 7500 waves is found by using. For a number of waves smaller than 1000, the formulas give a slight overestimation. The limits of the damage level parameter mainly depend on the slope of the structure If the Van der Meer formula is used to for design proposes, it s required to do a sensitivity analysis for all parameters. Also a sensitivity analysis of the constants should be done. Two graphs are given as an example of such an analysis (Figure 11). The influence on the changes of the damage level parameter and permeability on the design wave height for different breaker parameters is analysed. Figure 11: Sensitivity analysis for damage level parameter and permeability 21

42 Shallow water conditions Some of the tests carried out by Van der Meer were in shallow water. Van der Meer (1988) recommends to use the wave height exceeded by 2% of the waves instead of the significant wave height. This because the distribution of the wave heights deviate from the Rayleigh distribution (due to wave breaking) so the stability of the armour layer in depth limited waters is better described by the higher characteristic value of the wave height distribution than by. The known ratio is equal to 1,4 for a Rayleigh distribution, so the deep water values of the coefficients and should be changed to respectively 8.7 and 1.4 for shallow water. The transition between plunging and surging waves is the same as equation (43) using the coefficients and valid for shallow water. For plunging waves ( ): (46) For surging waves ( ): (47) In conclusion, a remark should be made. A safer approach for design purposes is to use formulas (44) and (45) with. In that case the truncation of the wave height exceedance curve due to wave breaking is not taken into account (Van der Meer, 1988). The equations (44) and (45) will give the same results as (46) and (47) if the waves are Rayleigh distributed due to the know Rayleigh-distribution-based ratio that was introduced before. To do the correct calculation, the actual ratio should be obtained, which is not always possible because often only the wave heights based on energy ( is based on the zero-est moment of the spectrum ) are obtained. This means that (actually ) is different from and so it s difficult to find a good estimation for. To overcome this issue, a good approximation for based on is given by Battjes and Groenendijk (2000) who proposed a composed Weibull-Rayleigh distribution for the waves. Van Gent et al. (2004) concluded that both wave heights and can be used, leading to almost the same accuracy on average. c. Van der Meer modified by Van Gent Van Gent et al. (2004) modified, based on his own tests, the Van der Meer formula for shallow foreshore. First, he uses the spectral mean energy wave period instead of the mean wave period obtained from time-domain analysis. Also the coefficients and are recalibrated and have a value of respectively 8.4 and 1.3. Thus the final formulas are: For plunging waves ( ): (48) 22

43 For surging waves ( ): With (49) The formulas were developed using tests in the ranges given in Table 5 and are valid in both deep and shallow water d. Van Gent Another (more simple) formula was developed by Van Gent et al. (2004), based on a earlier series of tests with a 1:100 foreshore (Smith et al., 2002) and additional tests with a considerably steeper foreshore (1:30). The structure slopes were 1:2 and 1:4. Both test series were combined and analysed to develop the new formula. In the new formula the permeability is directly related to a structure parameter (mean nominal diameter of the core material ). The influence of the period isn t taken into account. On the one hand the wave period influences the damage, but on the other this influence is small compared to the amount of scatter in the data. For the same reason, there is also no difference between plunging and surging waves. Besides this, the ratio also influences the damage, but again, these influence is considered small. The Van Gent formula is given by: (50) The range of tests used to develop the formula is the same as mentioned above in Table 5. The formula is valid in both shallow and deep water. Table 5: Range of parameters VDM formula modified by VG and VG formula Parameter Symbol Range Surf similarity parameter 1-5 Ratio deep water wave height and water depth at toe Ratio 2% and significant wave height Ratio significant wave height and water depth at toe Ratio significant and deep water wave height Armour diameter 0.022m-0.035m Slope angle Slope structure 1:2 & 1:4 Crest height

44 e. Cumulative damage All previous formulas are limited to single storm events. Some studies were performed to take into account the phenomenon of progressive damage due to subsequent storm event. 2 different methods taking into account subsequent storm events will be discussed: the method developed by Melby (2001) and the method developed by Van der Meer (2000). Melby method The cumulative damage is evaluated by (51) With Damage level at time [] Damage level at time [] Duration time of additional storm [] Duration time of storm to reach a damage level [] The stability number, based on the significant wave height from time domain analysis [] Mean wave period [] Time counter [] Coefficient determined in experiments, The range of validity of the parameters of the laboratory tests on which the formula is based is limited and is given in Table 6. Also the depth-limited wave conditions and the wave conditions of the subsequent events are relatively constant. Table 6: Range of parameters Melby method Parameter Symbol Range Surf similarity parameter 2-4 Structure slope 0.5 Ratio of armour and filter stone sizes 2.9 Permeability <0.4 Van der Meer method The approach of Van der Meer to take into account cumulative damage makes directly use of the Van der Meer deep water stability formula (44) and (45). Nevertheless this approach can also be applied to the Van der Meer formula in shallow water ((46) and (47)), the Van der Meer formula modified by Van Gent (48) and (49) and the Van Gent formula (50). It s based on the equivalence hypothesis. The procedure to assess the cumulative damage is as follows: 24

45 Calculate the damage for the first wave condition Calculate for the second wave conditions how many waves would be required to give the same damage as under the first wave conditions. This is denoted by. Add this number of waves, to the number of waves of the second wave conditions: Calculate the damage under the second wave conditions with this increased number of waves Calculate for the third wave condition how many waves would be required to give the same damage as caused by the second wave condition etc. Figure 12: Cumulative damage approach by Van der Meer 25

46 Chapter 3: Experimental setup In this chapter, the test equipment and experimental model will be discussed. Also the realized experiments and calculations will be explained. First of all, the wave flume will be described including the wave generating system, the system that dissipates the energy, the sensors that measure the wave elevations and the visual equipment. Next the design of the foreshore and model will be set out together with the properties of the used quarry stone. Furthermore, the characteristics of the realized experiments will be given. Finally this chapter will be concluded by the different steps taken to analyse the data. The different methods to obtain the wave characteristics will be explained, the porosity measurement method will be discussed together with the damage calculation method. Also it will be explained how the original formulas will be improved. 1. Test equipment All tests are performed in the 2D wave flume in the Laboratory of Ports and Coast of the Polytechnic University of Valencia(LPC-UPV)(Figure 13). The length of the wave flume is 30 meters, the height is 1.2 meters and the width is 1.2 meters. There is a false bottom of 25 cm for the circulation of the water. The wave generator is installed at one side of the wave flume. The system to dissipate the energy of the waves is installed at the other side. The breakwater model is placed in front of the energy dissipater. A part of the walls of the wave flume is transparent to give the researchers the possibility to see what happens inside the wave flume. Figure 13: Wave flume at LPC-UPV 26

47 a. Wave generator As mentioned before, the wave generator is installed at one end of the wave flume. It consists of and metal plate that is installed vertical and connected to a piston which generates the translation of the plate. This piston is driven by an electrical servomotor (Figure 14). Figure 14: Wave generator (Herrera, 2013) The wave generating system is controlled by an Active Wave Absorption Control System (AWACS). This is a digital control system that absorbs the reflection of the waves and generates the desired waves. This is needed to take into account the reflected waves from the structure on the other side of the wave flume. Since incident waves on the model are reflected, they could re-reflect on the wave maker, which results in an uncontrollable, and undesirable nonlinear distortion of the desired waves impinging on the test structure, because the wave generator keeps on having the same movement. So the reflected waves from the breakwater are taken into account by the AWACS. The principle of the AWACS is as follows. It measures the surface elevation by 2 wave gauges integrated in the paddle front. This measured surface elevation is in fact the superposition of the desired wave and the reflected wave. The reflected wave is identified by the digital recursive filter of the AWACS and absorbed by the wave maker. The specific type of AWACS installed in LCP-UPV is DHI AWACS2, from Denmark. The wave making system has 3 options. First of all, it can reproduce previous generated waves. Secondly, it can produce regular waves (Stokes 1 st order). Last it can also generate irregular waves. The parameters that has to be given in into the system are (depending if regular or irregular waves will be generated): Scale: it s used to make the conversion from the parameters of the prototype to the model Water depth: water depth in front of the wave paddle. Wave height and period Spectrum(only for irregular waves) and (only for irregular waves) Skewness (only for irregular waves) Duration 27

48 b. Energy dissipation system The energy dissipation system is placed at the end of the wave flume, on the opposite side of the wave generating system. It consists of five groups of three grooved metal frameworks and a plastic plate which is perforated. The metal frameworks have 3 different porosities: 70%, 50% and 30%. The frameworks are placed in such a way so the framework with the highest porosity is the closest to the approaching wave. The first group of 3 metal frameworks with a porosity of 70% is followed by 2 groups with a porosity of 50%. The porosity of the last 2 groups is 30%. The voids between the 3 frameworks of this last group are filled with quarry stone (Figure 15). Figure 15: Energy dissipation system and schematically overview of the frameworks c. Wave measurement The wave flume is equipped with a series of wave gauges and run-up sensors. Because there is no run-up during the test carried out for this master thesis, only the wave gauges will be discussed. The wave gauges measure the surface elevation. They consist of two vertical electrodes. When they are submerged in the water, the sensors measure the conductivity of the volume of water between the electrodes. The conductivity changes as a function of the surface elevation between the electrodes. The gauges has to be calibrated every time before the tests to overcome any errors caused by for instance changes in water level, caused by leaks and evaporation. The wave gauges are placed in different groups: One group close to the wave paddle to measure the wave height generated by the wave generating system in deep water (S1, S2, S3 and S4), one group close to the model to measure the wave height at the toe of the structure (S11 and S12). Between these 2 groups, there are placed some other sensors along the slope of the beach (Figure 16). 28

49 Figure 16: Position of the sensors, dimensions in meter Mansard and Funke (1980) defined some criteria for the distance between 3 wave gauges, with and the distances between the 3 wave gauges: (52) Using these criteria, a distance between the waves gauges is chosen for all the wave periods that will be used, so it wouldn t be necessary to change them every time after one test. d. Cameras The wave flume is equipped with different hardware for visual registration. Both digital pictures and digital videos are made. They are not only used to have a good view of them in the office during the test, but also to take the pictures used to calculate the damage. 29

50 2. Experimental design a. Foreshore slope A foreshore slope is created to make the waves break (Figure 16). The slope of the first one located 545 cm in front of the wave paddle and with a length of 625cm is 4%. Behind this slope, another foreshore slope of 2% is constructed. Its length is 1100 cm and the model is built on the end of this slope. The water level varies according to the test series between 61.7 and 81.7 in front of the wave paddle with steps of 10 cm. This corresponds with a water level between 20 and 40 cm at the toe of the breakwater model. b. Model The used model (scale 1:60) is a rubble mound breakwater model with a slope of 3/2 at the side exposed to the waves (Figure 17). The armour layer of the breakwater is composed out of 2 layers of quarry stone with a porosity of approximately 37% without a toe berm. The model has 3 different parts: The core, a filter layer and an armour layer. The characteristics of the 3 will be briefly discussed. Core Figure 17: Section of the rubble mound breakwater, dimensions in cm The characteristics of the core are not really relevant. But for the completeness, the sieve analysis is given in Figure 18. Figure 18: Sieve curve of the core 30

51 Filter The size of the stones of the filter used in the model have a nominal diameter of. The distribution for a sample of 20 filter layer stones is given in Figure 19. Figure 19: Distribution of the filter layer stones Armour layer The mean nominal diameter of the stones used is equal to. The distribution of a sample of 25 stones is given by Figure 20. Figure 20: Distribution of the armour layer stones The initial porosity of the armour stones has to be approximately 37% (CERC, 1984) before the test series start. The model with this initial porosity is depicted on Figure 21. Figure 21: Rubble mound model constructed with an initial porosity of approximately 37% 31

52 3. Realized experiments The realized experiments on the rubble mound breakwater model are part of a 2 years long ongoing project called ESCOLIF (Estabilidad hidráulica de los mantos de escollera, cubos y Cubipodos frente a oleaje limitado por el fundo/ Hydraulic stability of cube, Cubipod and rubble mound breakwaters in depth limited conditions) which is carried out at the LPC-UPV. The goal of the project ESCOLIF is to improve the knowledge of the hydraulic stability of single- and doublelayer Cubipod armours (an armour unit developed by LPC-UPV) in depth limiting conditions. Another goal of ESCOLIF is to increase the experimental basis of double-layer cubes and quarry stone in depth limiting conditions: The initial testing conditions (foreshore slope of 0, 2 and 4%) were modified by adding also a foreshore slope of 10% to the tests because a high influence of the slope of the foreshore was observed. The experiments discussed in this master thesis are carried out with 1000 irregular waves with a JONSWAP spectrum ( ). The foreshore for all test is equal to 2%. The typical breaker type under such a foreshore slope is spilling breakers. Every test series is initiated with a wave series of 1000 waves with a wave height that doesn t produce breaking waves (). By increasing the wave height with 1 cm every step and by keeping the Iribarren number constant, more and more waves are breaking until destruction of the model is observed. Because the Iribarren number is kept constant in one test series, a corresponding peak period can be calculated for each wave height. The different test series are performed for different Iribarren numbers ( and ) and in different water depths ( and ). One repetition test is carried out ( and ) to use later as a blind test to verify the improved and new developed formulas. An overview of the input data for the different test series is given in Table 7 to Table 11 Table 7: Test data for Ir=3 and h=20cm Model scale Prototype scale

53 Table 8: Test data for Ir=5 and h=20cm Model scale Prototype scale Table 9: Test data for Ir=3 and h=30cm Model scale Prototype scale

54 Table 10: Test data for Ir=5 and h=30cm Model scale Prototype scale Table 11: Test data for Ir=3 and h=40cm Model scale Prototype scale

55 4. Data analysis a. Wave analysis The wave characteristics (wave height, period ) have to be obtained to estimate the damage using the different prediction formulas. These values can be obtained using different methods. Their registered values can t be used directly because these values consist of 2 different influences, namely the incident wave and the reflected wave by the structure. The characteristics of the incident wave are the characteristics that will be used to do all calculations. There are 3 different methods to obtain the characteristics of the incident waves with each its advantages and disadvantages: SwanOne, measurements by wave gauges in canal without model and measurements by wave gauges in canal with model. Each one will be discussed. SwanOne SwanOne is a model developed by TUDelft in MatLab to simulate the evolution of the wave spectrum starting from deep water to shores. The model is capable to simulate interactions and transformations of waves (TUDelft, 2015). The different input parameters are: Bottom profile Current Wave direction Water level Wind velocity and direction Boundary conditions: arbitrary spectrum file or definition input diameters (significant wave height based on spectral analysis [m] and the peak period [s]). Also the output locations should be provided, i.e. the locations along the profile where you want to obtain the output parameters. The output parameters are: Significant wave height [m] Root-mean-square wave height [m] Peak period [s] Mean absolute period [s] Mean period [s] Mean energy wave period [s] Significant wave height calculated using the Battjes and Groenendijk method [m] Wave height exceeded by 2% of the waves [m] Mean wave height of highest 1/10 fraction of waves [m] No breakwater model is used in SwanOne thus also no reflection by the model is possible. Therefore the obtained parameters are the incident characteristics. 35

56 Measurements by wave gauges in canal without a model A second method to measure and calculate the wave characteristics is using directly the surface elevations measured by the wave gauges in the wave flume. To avoid reflection by the model, no model is placed in the canal. Theoretically the waves will not be reflected by the end side of the canal since the energy of the waves is dissipated there by the energy dissipating system. The wave characteristics are obtained by the software developed in the LPC and is called LPCLab 2.0. LPCLab 2.0 calculates different wave heights and period spectra and moments for later calculations using the surface elevations measured by the wave gauges. The wave characteristics are analysed by LPCLab 2.0 in both the time-domain and frequency domain and generates information about all relevant parameters and gives also some graphs. i. Time-domain analysis In the time-domain, each individual wave is defined by the downward crossing of the zero-line by the surface elevation (zero down-crossing). The mean wave height is calculated from time-series of individual waves. ii. Frequency domain analysis In addition to the time domain analysis, the wave spectrum of the realized test is calculated using the discrete Fourier transform (DFT) of the measured surface elevation. The total registered waves are divided in different time-windows. The fast Fourier transform (FFT) is done over the different windows and the results are summed at the end. The width of the time-window, the number of points to represent the spectrum and the percentage of overlap between the different windows can be chosen by the user. Another important parameter to put in LPCLab 2.0 is a range of frequencies so useless frequencies can be eliminated. A screenshot of LPCLab 2.0 is given in Figure 22. Figure 22: Screenshot of LPCLab

57 Hm0, t (SWAN)[cm] Measurements by wave gauges in canal with model. LPCLab 2.0 can also be used to process the surface elevations that are measured by the wave gauges in the wave flume when the model is present. The only problem is that the model will reflect the waves so LPCLab 2.0 won t generate the incident wave characteristics, but only the registered wave characteristics. To overcome this problem, the LASA method can be used. However, the LASA or LASA-V method can t be used in breaking wave conditions, so the LASA method can only be applied on the wave gauges in front of the wave paddle, where the waves aren t breaking. Hence the incident and reflected waves are known for the sensors in front of the wave paddles and only the total registered waves are known for the rest of the sensors. Starting from the calculated reflection coefficient in front of the wave paddle (mean value of the 3 wave gauges in front of wave paddle) obtained from the LASA method, the height of the incident and reflected waves can be calculated using equations (3) and (4) starting from the total measured wave height by the other wave gauges where the waves are breaking. The problem with this method is that the assumption is very rough (it s assumed that the reflection coefficient in front of the wave paddle is equal for the whole canal). Of course this is not the case. Comparison between SwanOne and measurement in canal without model A comparison between the significant wave height obtained from SwanOne and the significant wave height obtained from the measurements in the canal without the model is depicted in Figure 23. It can be stated that the values of the significant wave height obtained from SwanOne are slightly higher than the ones obtained from the measured water elevations in the canal without model. For this master thesis, the wave characteristics are obtained from SwanOne. 20,00 15,00 10,00 5,00 0,00 0,00 5,00 10,00 15,00 20,00 Hm0,t (without model) [cm] Figure 23: Comparison between significant wave height obtained from SwanOne and measurements in the canal without the model 37

58 b. Porosity measurement The porosity has to be measured for 2 purposes. First, the initial porosity has to be measured to verify if this one is about 37% as prescribed (CERC, 1984). Also, the porosity after each test should be measured, because this porosity is used to calculate the damage. The porosity, equal to the ratio of the area of the voids and the total area, is calculated counting the number of stones. The counting of the stones is done using the software AutoCAD. First a virtual net is drawn over the picture of the breakwater. Different strips are drawn. This is done because the Virtual Net Method requires this (which is used for other test in the ESCOLIF project, but doesn t has a real meaning for this master thesis). Then each stone is marked (Figure 24). Finally, by the command PRICAPAXYZ, developed by the LPC-UPV, the marks are counted () and saved. Finally the porosity can be calculated: (53) Figure 24: Marked stones by AutoCAD c. Damage calculation The damage of the armour layer is calculated according to the visual counting method. This method is discussed in the literature study and makes use of equations (31) and (32). The number of eroded stones is calculated by counting the difference of stones between the initial state and the state after wave attack. To do so, again AutoCAD is used as described above. To do the qualitative analysis of the damage, the damage criteria are distinguished as follows: Initiation of damage (IDa): 5 or more units are displaced from their original position to a new one at a distance equal to or larger than a unit length Initiation of Iribarren damage (IIDa): One stone and his surrounding stones of the 2 nd layer are visible Initiation of destruction (IDe): 2 or more stones in the lower armour layer are forced out Destruction (De): The filter layer is visible 38

59 d. Comparison measured damage with predicted damage Once the damage is calculated, the measured damage will be compared with the predicted damage. This predicted damage will be calculated using 2 approaches: The SPM-approach and the VDM approach. SPM-approach The first approach is the one based on the SPM (CERC, 1984) by Van der Meer (1988) and Medina et al. (1994): Van der Meer (1988) Medina et al. (1994) (54) (55) All parameters are explained in the literature study. For the Van der Meer (1988) formula, the value for the stability coefficient is taken to be equal to 8 because the goal is to predict the damage so the value for an permeable core describing the main trend is chosen. The damage will be predicted using both prediction formulas for 5 different wave heights. First the significant wave height obtained from spectral analysis will be used because this is also the wave height proposed by the authors of the formulas. But because the tests are performed in breaking conditions, the damage will also be calculated using the breaking wave height. There are different methods to calculate the breaking wave height for regular waves, but for irregular waves the waves can break in a wide range as discussed before so there is not one right value. Because of this, the breaking wave height is calculated in 4 different ways. The first 3 are based on the method described in the Shore Protection Manual(CERC, 1984). For this method, Figure 3 based on Goda (1970) can be used. It s not clearly defined which period should be used to calculate the deep water wave steepness, so the breaking wave height is calculated 3 times based on the mean absolute period, the mean energy wave period and the peak period with all periods obtained from SwanOne. As 5 th used wave height, the wave height proposed by Goda (2000) for irregular conditions is used, i.e. the peak value of the significant wave height in the surf zone which can be read from Figure 6. Next the predicted and measured damage will be compared for each formula and each used wave height. Furthermore 5 new formulas will be presented using the 5 different wave heights based on all the tests except for the repetition test ( and ). 39

60 The new formula will be developed by calculating the values of the constant coefficients and of: (56) The coefficients will be calculated using the Solver function in excel obtaining the lowest relative mean squared error. The is used to evaluate the extent to which measured and predicted damage are similar. It s equal to: (57) Also the correlation coefficient will be calculated, but this is only a measure of the linear correlation between the measured and predicted damage, so won t be really used in the discussion of the comparison between the predicted and measured damage.. The repetition test will be used to verify the new formulas as a blind test. In conclusion the best of the 5 different formulas will be chosen to predict the damage. No cumulative damage is taken into account because the formulas from the SPM-approach do not include the number of waves. So it can be assumed that the damage of a lower wave height which is already present before the initiation of the wave series with a higher wave height also will be caused by a certain (lower) number of waves with the higher wave height than the previous waves. VDM-approach The second approach is based on Van der Meer (1988) and Van Gent et al. (2004). The damage for each test will be predicted (taking into account cumulative damage as discussed before) and compared to the measured damage (the data points with a measured damage higher than 15 were deleted, because it doesn t make sense to take them into account; destruction will already have occurred at such a high value). This is also shown in Table 2. The prediction of the damage will be done 3 times. Once according to the Van der Meer formula (in deep or shallow water according to the conditions), once according to the Van der Meer formula modified by Van Gent and finally once according to the formula by Van Gent. A summary of all formulas is given in Table

61 VDM Deep water Plunging Table 12: Summary formulas VDM-approach (58) VDM Deep water Surging (59) VDM Shallow water Plunging (60) VDM Shallow water Surging (61) VDM (mod by VG) Shallow and deep water Plunging (62) VDM (mod by VG) Shallow water Surging (63) VG Shallow and deep water (64) The 3 formulas will be compared using again the and in a smaller extent the correlation coefficient and some improvements to the original formulas will be presented. Finally the modified original formulas will be verified with a blind test. 41

62 Chapter 4: Results In this chapter the results of the comparison between the measured and predicted damage are presented. The comparison is based on all test but the repetition test series. This repetition test is used later as a blind test to verify the new and improved formulas. The comparison is done for the 2 approaches as discussed before, i.e. The SPM-approach and the VDM-approach. To quantify the error between the measured and predicted damage, the relative mean squared error ( is calculated for each case. Finally a new formula or improvement to the original formula is proposed based on the comparison between measured and predicted damage. The repetition test is used to verify these new and improved formulas (blind test). 1. Wave data As mentioned before, the used wave data is obtained from SwanOne at the toe of the model (unless otherwise mentioned). The most important parameters are listed in Table 13, Table 14 and Table 15. All data from SwanOne at the toe of the structure is listed in Appendix A. Table 13: Summary of the SwanOne data at the toe of the structure ( Theoretical values SwanOne (at toe of structure) Theoretical values SwanOne (at toe of structure)

63 Table 14: Summary of the SwanOne data at the toe of the structure ( ) Theoretical values SwanOne (at toe of structure) Theoretical values SwanOne (at toe of structure) Table 15: Summary of the SwanOne data at the toe of the structure ( ) Theoretical values SwanOne (at toe of structure)

64 2. Measured damage and porosity The measured damages and porosities for the initial state and after wave series are listed in Table 16 to Table 19 The initial porosity of each test series was 36% or 37%. So this is in line with the requirements (CERC, 1984). Also the different damage criteria are identified. This is based on the photos taken after each wave series. The photos are listed in Appendix B. The average damage parameter for each damage criteria is calculated and the following values are obtained: Initiation of damage: Initiation of Iribarren damage: Initiation of destruction: Destruction: Table 16: Measured damage, porosity and qualitative damage analysis for Q.A. Q.A. initial 0 37% initial % % % % % IDa % % IIDa % IDa % % % IDe % IIDa % % % % % De % IDe % De % 44

65 Table 17: Measured damage, porosity and qualitative damage analysis at Q.A. Q.A. initial % initial 0 36% % % % % IDa % % % IDa % IIDa % % IDe % IIDa % De % IDe % % % % De initial % Table 18: Measured damage, porosity and qualitative damage analysis for and Ir=3 initial % % % % Q.A % IDa % % IIDa % IDe % De % 45

66 Table 19: Measured damage, porosity and qualitative damage analysis for the repetition test (, Ir=3) Q.A. initial % % % % % IDa % % % IIDa % % % IDe % % De 3. Comparison measured damage with predicted damage a. SPM-approach The comparisons between the measured damage and the damage predicted by the Medina and VDM formula using the 5 different wave heights are shown in Figure 25 to Figure 29. The proposed new formulas are also depicted on each graph, and are presented in Table 20. The verification of these new formulas is done using the repetition test (blind test) which is depicted in Figure 30. The blue data points represent the comparison between the measured damage and the predicted damage calculated using the Medina formula, the grey data points the comparison between the measured damage and predicted damage by the VDM formula while the orange data points represent the comparison between the measured damage and the predicted damage according to the new formula. Of course this formula will give the best comparison because the coefficients of this formula were chosen in such a way that the measured and predicted damage agree in the best way. 46

67 Predicted damage S SPM^X[-] Predicted damage S SPM^X[-] Predicted damage S SPM^X[-] Predicted damage S SPM^X[-] Predicted damage S SPM^X[-] Predicted damage S SPM^ [-] Comparison Medina, VDM and New formula based on Hm0 (SwanOne) 2,0 Medina(X=0.2) Comparison Medina, VDM and New formula based on Hbr (Goda, 2000) 2,0 Medina (X=0.2) 1,6 VDM (X=0.15) New Formula (X=0.14) 1,6 VDM (X=0.15) New formula (X=0.17) 1,2 1,2 0,8 R=0.96 rmse=0.18 R=0.97 0,4 rmse=0.73 R=0.97 rmse=0.07 0,0 0,0 0,4 0,8 1,2 1,6 2,0 Measured damage Sd^X[-] Figure 25: Comparison formulas based on 0,8 R=0.92 rmse=0.21 R=0.93 0,4 rmse=0.25 R=0.99 rmse=0.14 0,0 0,0 0,4 0,8 1,2 1,6 2,0 Measured damage Sd^X[-] Figure 26: Comparison formulas based on the (Goda, 2000) Comparison Medina, VDM and New formula based on Hbr (Tm) (CERC, 1984) 2,0 Medina (X=0.2) VDM (X=0.15) 1,6 New Formula (X=0.17) Comparison Medina, VDM and New formula based on Hbr (Tm-1) (CERC, 1984) 2,0 Medina (X=0.2) VDM (X=0.15) 1,6 New Formula (X=0.17) 1,2 1,2 0,8 R=0.91 rmse=0.34 R=0.92 0,4 rmse=0.19 R=0.99 rmse=0.17 0,0 0,0 0,4 0,8 1,2 1,6 2,0 Measured damage Sd^X[-] Figure 27: Comparison formulas based on 0,8 R=0.92 rmse=0.49 0,4 R=0.92 rmse=0,19 R=0.99 rmse=0,16 0,0 0,0 0,4 0,8 1,2 1,6 2,0 Measured damage Sd^X[-] Figure 28: Comparison formulas based on Comparison Medina, VDM and New formula based on Hbr (Tp) (CERC, 1984) 2,0 Medina (X=0.2) VDM (X=0.15) 1,6 New Formula(X=0.17) 1,2 2,0 1,5 Verification new formulas by repeated test Hm0 SWAN Hb (Tm), SPM Hb (Tm-1), SPM Hb (Tp), SPM Hb, Goda 0,8 R=0.92 rmse=0.91 R=0.93 0,4 rmse=0.33 R=1.00 rmse=0.14 0,0 0,0 0,4 0,8 1,2 1,6 2,0 Measured damage Sd^X[-] Figure 29: Comparison formulas based on 1,0 rmse=0.05 rmse=0.10 rmse=0.12 rmse=0.15 rmse=0.16 0,5 0,5 1,0 1,5 2,0 Measured damage Sd^X[-] Figure 30: Verification new formulas (blind test) 47

68 Table 20: Proposed new formulas based on SPM-approach Used wave height New formula (65) (66) based on (67) based on (68) based on (69) The of all formulas using all the different wave height are given in Table 21. It can be concluded that for the existing formulas the Medina formula using the significant wave height is the best formula to predict the damage (. Also the VDM formula using breaking wave height gives a low, which is equal to 0.19 using both or. The best fitting proposed new formula is equation (65) using. The equals All proposed new formulas were verified by a blind test (Figure 30), and it can be concluded that also here, the best fitting formula is equation (65) with a of Table 21: of all formulas VDM Medina New Formula Blind test SwanOne CERC (1984) Goda (2000)

69 Predicted damage Sd[-] Predicted damage Sd[-] b. VDM-approach The comparison between the predicted and the measured damage calculated by the original Van der Meer formula is depicted in Figure 31. As can be seen on the figure, it s clear that all predicted damage is lower than the measured damage. It can also be observed that the lower the Iribarren number and the higher the water depth, the bigger the difference between the measured and the predicted damage. That is why the original formula will be improved by introducing the water depth at the toe of the structure and the peak period which influences the Iribarren number. All waves are plunging waves so only the formulas for plunging waves can be improved. The power to which the new dimensionless parameter is raised and the value of the constant are calculated using the solver function in excel obtaining the smallest : For shallow water: For deep water: (70) (71) By introducing the dimensionless parameter that takes into account the influence of the water depth and Iribarren number, the decreases from 0.77 to 0.09 which is a very big improvement (Figure 32). Also the correlation coefficient improves. The interval in which 90% of the values are located is also indicated on Figure Comparison VDM formula R=0.78 rmse= Measured damage Sd[-] hs=20,ir=5 hs=30,ir=5 hs=20,ir=3 hs=30,ir=3 hs=40,ir=3 Figure 31: Comparison measured and predicted damage by formula of VDM Figure 32: Comparison measured and predicted damage by improved formula of VDM The same improvement was executed for the VDM formula modified by VG. The same observations and conclusions can be made: Almost everywhere the measured damage is higher than the predicted damage, and the lower the Iribarren number & higher the water depth, the higher the difference between measured and predicted damage (Figure 33). Again the same improvement is made by introducing a dimensionless parameter Comparison improved VDM formula R=0.97 rmse= Measured damage Sd[-] hs=20,ir=5 hs=30,ir=3 hs=40,ir=3 hs=20,ir=3 hs=30,ir=5 49

70 Predicted damage Sd[-] Predicted damage Sd[-] (72) Again the comparison is improved by introducing the dimensionless parameter. The decreases from 0.49 to The interval in which 90% of the values are located is indicated on the figure Comparison VDM formula (modified by VG) R=0.77 rmse= Measured damage Sd[-] hs=20,ir=5 hs=30,ir=5 hs=20,ir=3 hs=30,ir=3 hs=40,ir=3 Figure 33: Comparison measured and predicted damage by formula of VDM modified by VG Comparison improved VDM formula (modified by VG) R=0.97 rmse= Measured damage Sd[-] hs=20,ir=5 hs=40,ir=3 hs=30,ir=5 hs=20,ir=3 hs=30,ir=3 Figure 34: Comparison measured and predicted damage by improved formula of VDM modified by VG If the comparison between the measured and predicted damage by the VG formula is observed, there can t be really made a distinction between the different test series. The only conclusion that can be made is that the predicted damage is almost always lower than the measured damage (Figure 35). That s the reason why only the constant coefficient is changed: (73) By changing the coefficient from 0.57 to 0.67, the results are improved from a of 0.59 to 0.12 (Figure 36). The interval in which 90% of the values are located is indicated on the figure. 50

71 Predicted damage Sd [-] Predicted damage Sd[-] Predicted damage Sd[-] R=0.94 rmse=0.59 Comparison VG formula hs=20,ir=3 hs=30,ir= Comparison improved VG formula R=0.94 rmse=0.12 hs=20,ir=5 hs=40,ir=3 6 hs=30,ir=5 6 hs=30,ir=5 3 hs=40,ir=3 3 hs=30,ir= Measured damage Sd[-] hs=20,ir=5 Figure 35: Comparison measured and predicted damage by formula of VG Measured damage Sd[-] hs=20,ir=3 Figure 36: Comparison measured and predicted damage by the improved formula of VG Finally, the improved formulas of VDM, VDM modified by VG and VG can be verified by a blind test using the repetition test. The comparison between the measured damage and the predicted damage by the 3 new formulas is given in Figure 37. To make the improvement clearer also the predicted values by the original formulas are depicted in the graph Comparison of old and new formulas with repeated test VDM Measured damage S[-] Figure 37: Verification new formulas VDM (mod by VG) VG Improved VDM Improved VDM (mod by VG) Improved VG It can be concluded that the new formulas give a good improvement of the original formulas. 51

72 Chapter 5: Conclusions The aim of this thesis was to compare and improve different prediction formulas. 2 different approaches were used: the approach based on the SPM (CERC, 1984) and the approach based on Van der Meer (1988). First of all, a qualitative analysis was executed determining the Initiation of damage, Initiation of Iribarren damage, Initiation of destruction and destruction. The average values based on the tests are given in Table 22. Table 22: Qualitative damage analysis Initiation of damage Initiation of Iribarren damage Initiation of destruction Destruction If the (deep water) wave characteristics are not known, these can be obtained using SwanOne. The best formula to use these wave characteristics based on the executed tests is the new proposed formula using the significant wave height : If the deep water characteristics are known, the breaking wave height can be calculated and used to predict the damage according to the SPM approach using the new proposed formula with the mean period : Also the new proposed formula using the peak value of the significant wave height in the surf zone can be used if the deep water characteristics are known: Using the VDM formula, the breaking wave height based on or gives the best prediction of the damage while using the Medina formula the significant wave height at the toe of the structure obtained by SwanOne gives the best approximation. For the approach based on Van der Meer (1988), the improved VDM formula (deep & shallow water) and improved VDM formula modified by VG for plunging waves and improved VG formula are given respectively by: 52

73 The red parameters are the introduced ones to improve the prediction. If only a few wave characteristics are known (no wave period,,..), the improved VG formula can be used. In other cases, both the improved VDM formula and improved VDM formula by VG will give the best approximation. A really important remark should be made. The qualitative analysis, all improvements and new proposed formulas are based only a few test series. All formulas and qualitative analysis should be verified and tested with other test series and more repetition tests (also in other laboratories). These repetition tests are necessary because damage is a very sensitive parameter. The difference in measured damage of 2 equal tests can be 30% (Van der Meer, 2000). Other future research can be done by changing the parameters, for instance the nominal armour diameter, structure slope, foreshore slope, The conditions could also be changed in such a way so surging waves occur (instead of plunging waves as in these conditions). Finally also other armour units should be investigated. 53

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78 Vidal, C., et al. (2006). Wave Height Parameter for Damage Description of Rubble-Mound Breakwaters, Elsevier B.V. Weggel, J. R. (1972). "Maximum Breaker Height." Journal of the Waterways, Harbors and Coastal Engineering Devision 98(WW4):

79 Appendix A: Summary data SwanOne 59

80 Appendix B: Photos model after wave action h s =20cm and Ir=3 h s =20cm and Ir=5 60

81 h s =30cm and Ir=3 61

82 h s =30cm and Ir=5 h s =40cm and Ir=3 62

83 h s =20cm and Ir=3 (repetition test) 63

84 64