MAST III / PROVERBS
Probabilistic Design Tools
for Vertical Breakwaters
MAS3 - CT95 - 0041
FINAL REPORT
VOLUME IIa
HYDRODYNAMIC ASPECTS
Edited by N.W.H. Allsop April 1999 co-sponsored by Commission of the European UnionPrinted at:
CONTENTS OF VOLUME II
1. VOLUME IIa – HYDRAULIC ASPECTS
Chapter 1:ALLSOP, N.W.H. (1999): Introduction.
Chapter 2.1: ALLSOP, N.W.H.; DURAND, N. (1999): Influence of steep seabed slopes on breaking waves for structure design. 28 pp.
Chapter 2.2: MCCONNELL, K.J. (1999): Derivation, validation and use of parameter map. 5 pp.
Chapter 2.3: CALABRESE, M.; VICINANZA, D. (1999): Estimation of proportion of impacts. 15 pp.
Chapter 3.1: VOORTMAN, H.G.; HEIJN, K.M. (1999): Wave transmission over vertical breakwaters. 9 pp.
Chapter 3.2:ALLSOP, N.W.H.; BESLEY, P.; FRANCO, L. (1999): Wave overtopping discharges. 8 pp.
Chapter 3.3:ALLSOP, N.W.H. (1999): Wave reflections. 13 pp.
Chapter 4.1: VOORTMAN, H.G.; VAN GELDER, P.H.A.J.M.; VRIJLING, J.K. (1999): The Goda model for pulsating wave forces. 5 pp.
Chapter 4.2: FLOHR, H.; MCCONNELL, K.J.; ALLSOP, N.W.H. (1999): Negative or suction forces on caissons: development of improved prediction methods. 17 pp., 1 Annex.
Chapter 4.3: BURCHARTH, H.F.; LUI, Z. (1999): Force reduction of short-crested non-breaking waves on caissons. 17 pp., 3 Annexes.
Chapter 4.4: VRIJLING, J.K.; VAN GELDER, P.H.A.J.M. (1999): Uncertainty analysis of non breaking waves. 12 pp.
Chapter 4.5: VAN GENT, M.R.A.; TORENBEEK, R.V.; PETIT, H.A.H. (1999): VOF model for wave interaction with vertical breakwaters. 11 pp.
Chapter 4.6: LÖFFLER, A.; KORTENHAUS, A. (1999): Non breaking waves - pressures on berms. 23 pp.
Chapter 5.1: KORTENHAUS, A.; OUMERACI, H.; ALLSOP, N.W.H.; MCCON-NELL, K.J.; VAN GELDER, P.H.A.J.M.; HEWSON, P.J. ET AL. (1999): Wave impact loads - pressures and forces. 39 pp.
Chapter 5.2: WALKDEN, M.; WOOD, D.J.; BRUCE, T.; PEREGRINE, D.H. (1999): Seaward impact forces. 25 pp.
Chapter 5.3: ALLSOP, N.W.H.; CALABRESE, M. (1999): Impact loadings on vertical walls in directional seas. 19 pp.
Chapter 5.4: VAN GELDER, P.H.A.J.M.; VRIJLING, J.K.; HEWSON, P.J. (1999): Uncertainty analysis of impact waves and scale corrections due to aeration. 12 pp.
Chapter 5.5: LÖFFLER, A.; KORTENHAUS, A.; WOOD, D.J. (1999): Wave impact loads - pressures on a berm. 22 pp.
Chapter 6.1: WALKDEN, M.; MÜLLER, G. (1999): Strongly depth limited waves. 4 pp. Chapter 6.2: MARTíN, F.L.; LOSADA, M.A. (1999): Wave loads on crown walls. 36 pp. Chapter 6.3: MUTTRAY, M.; OUMERACI, H. (1999): Wave loads on caisson on high
mounds. 28 pp.
Chapter 7: CRAWFORD, A.R.; HEWSON, P.J. (1999): Field measurements and database. 4 pp.
Chapter 8.1: DE GERLONI, M.; COLOMBO, D.; BÉLORGEY, M.; BERGMANN, H.; FRANCO, L.; PASSONI, G.; ROUSSET, J.-M.; TABET-AOUL, E.H. (1999): Alter-native low reflective structures - perforated vertical walls. 41 pp.
Chapter 8.2: KORTENHAUS, A.; OUMERACI, H. (1999): Alternative low reflective structures - other type of structures. 29 pp.
2. VOLUME IIb –GEOTECHNICAL ASPECTS
Chapter 1: DE GROOT, M.B. (1999): Introduction.
Chapter 2: KVALSTAD, T.J. (1999): Soil investigations and soil parameters. 20 pp.
Chapter 3: LAMBERTI, A.; MARTINELLI, L.; DE GROOT, M.B. (1999): Dynamics. 56 pp. Chapter 4: DE GROOT, M.B. (1999): Instantaneous pore pressures and uplift forces. 38 pp. Chapter 5: KVALSTAD, T.J. (1999): Degradation and residual pore pressures. 37 pp.
Chapter 6: IBSEN, L.B.; JAKOBSEN, K.P. (1999a): Limit state equations for stability and deformation. 20 pp., 3 Annexes.
Chapter 6, Annex B: IBSEN, L.B.; JAKOBSEN, K.P. (1999b): Permanent deformations due to impact loading. 9 pp.
Chapter 6, Annex A: JAKOBSEN, K.P.; SØRENSEN, J.D.; BUCHARTH, H.F.; IBSEN, L.B. (1999): Failure modes - limit state equations for stability. 26 pp.
Chapter 6, Annex C: LAMBERTI, A. (1999): Combined effect of dilatancy in rubble mound and caisson inertia. 9 pp.
Chapter 7: KVALSTAD, T.J.; DE GROOT, M.B. (1999): Uncertainties. 30 pp.
Chapter 8: GOLÜCKE, K.; PERAU, E.; RICHWIEN, W. (1999): Influence of design parameters - stability analysis on feasibility level. 31 pp.
3. VOLUME IIc –STRUCTURAL ASPECTS
Chapter 1: CROUCH, R.S. (1999): Introduction.
Chapter 2: MARTINEZ, A.; KOVARIK, J.-B.; BERDIN, D. (1999): Structural design of vertical breakwaters - limitations of current practice and existing design codes. 37 pp. Chapter 3: VROUWENVELDER, A.W.C.M.; BIELECKI, M. (1999): Caisson reliability
during transport and placing. 36 pp.
Chapter 4: CROUCH, R.S. (1999a): In-service behaviour of cellular reinforced concrete caissons under severe wave impact. 39 pp.
Chapter 5: CROUCH, R.S. (1999b): Some observations on the durability and repair of concrete structures in a marine environment. 30 pp.
4. VOLUME IId –PROBABILISTIC ASPECTS
Chapter 1: VRIJLING, J.K. (1999): Introduction.
Chapter 2: VRIJLING, J.K. (1999): Fault tree analysis of a vertical breakwater. 9 pp.
Chapter 3: SØRENSEN, J.D.; BURCHARTH, H.F. (1999): Limit state equations including uncertainties. 26 pp., 1 Annex.
Chapter 4.1: VOORTMAN, H.G.; KUIJPER, H.K.T.; VRIJLING, J.K. (1999): Economic optimal design of vertical breakwaters. 17 pp.
Chapter 4.2: SØRENSEN, J.D.; BURCHARTH, H.F. (1999): Partial safety factor system. 24 pp.
Chapter 5.1: LAMBERTI, A.; MARTINELLI, L.; DE GROOT, M.B.; GOLÜCKE, K.; VAN HOVEN, A.; ZWANENBURG, C. (1999): Hazard analysis of Genoa Voltri breakwater. 40 pp.
Chapter 5.2: VOORTMAN, H.G.; VRIJLING, J.K. (1999): Reliability analysis of the Easchel breakwater. 29 pp.
Chapter 5.3: SØRENSEN, J.D.; BURCHARTH, H.F. (1999): Other representative structures: Mutsu-Ogawara, Niigata East and West. 19 pp.
CHAPTER 1: INTRODUCTION
N.W.H. ALLSOP
Professor (associate) University of Sheffield, c/o HR Wallingford, Howbery Park, Walling-ford, Oxon OX10 8BA, UK,
e-mail: w.allsop@sheffield.ac.uk
This volume is part of the final report of the MAST III project PROVERBS, PRObabilistic design tools for VERtical BreakwaterS (February 1996 – January 1999) under contract no. MAS3-CT95-0041. The various parts of the final report are as follows (this volume in bold letters):
Volume I
OUMERACI, H.; KORTENHAUS, A.; ALLSOP, N.W.H.; DE GROOT, M.B.; CROUCH, R.; VRIJLING, J.K.; VOORTMAN, H.G (1999): Prob-abilistic design tools for vertical breakwaters. Balkema, Rotterdam, ca 350 pp.
Volume IIa
ALLSOP, N.W.H. (ed) (1999): Probabilistic design tools for vertical breakwaters – Hydrodynamic aspects. MAST III – PROVERBS – pro-ject. Technische Universität Braunschweig, Braunschweig, Germany, 400 pp.
Volume IIb
DE GROOT, M.B. (ed) (1999): Probabilistic design tools for vertical breakwaters –Geotechnical aspects. MAST III – PROVERBS – project. Technische Universität Braunschweig, Braunschweig, Germany, 250 pp. Volume IIc
CROUCH, R. (ed) (1999): Probabilistic design tools for vertical breakwa-ters – Structural aspects. MAST III – PROVERBS –project. Technische Uni-versität Braunschweig, Braunschweig, Germany, 140 pp.
Volume IId
VRIJLING, J.K.(ed) (1999): Probabilistic design tools for vertical breakwa-ters – Probabilistic aspects. MAST III – PROVERBS – project. Technische Universität Braunschweig, Braunschweig, Germany, 170 pp.
This volume deals with hydrodynamic aspects of vertical breakwaters and seawalls. It is pro-duced by the PROVERBS Task 1 team which consists of 13 institutes from 6 European coun-tries. The chapters of this volume have the same titles as the sections of Chapter 2 of Vol-ume I (Hydrodynamic Aspects) in order to facilitate the study of the different aspects. The key issues addressed in this volume are as follows:
identification of wave conditions at the structure including breaking type identifi-cation and probability of occurrence of breaking and non breaking waves;
description of hydraulic responses of the structure such as wave transmission, breaking criteria at the structure (parameter map), and estimation of probability of impacts;
description of the wave loads on vertical breakwaters under pulsating loads includ-ing pressures and forces, negative forces, 3-d effects, uncertainties and scale ef-fects, and pressures on berms;
description of the wave loads on vertical breakwaters under impact loads including pressures and forces, seaward forces due to overtopping waves, 3-d effects, uncer-tainties and scale effects, and pressures on berms;
description of broken wave loads on vertical breakwaters including pressures and forces, wave loads on crown walls of rubble mound breakwaters and loading of high mound composite breakwaters;
description and background information for field measurements under PROV-ERBS and development of data base of these full-scale measurements;
overview of alternative low reflection structures such as perforated structures and horizontally composite structures;
The various chapters of this volume cover the aforementioned aspects and are briefly summa-rised in the following.
Chapter 2 describes the behaviour of waves in front of the structure and at the wall under depth limited conditions. The influence of steep sloping berms on the breaking of waves is discussed in Chapter 2.1 and the importance of the local wave height at the structure is high-lighted.
Chapter 2.2 discusses the development of a parameter map which helps to select the kind of breaking or non breaking waves which are to be expected at the vertical breakwater. Three simple relative input parameters are described and the data support for this map is discussed. Nevertheless, the map cannot predict the percentage of breaking and non breaking waves at the structure. Therefore, a method has been summarised in Chapter 2.3 which describes the estimation of the proportion of impacts hitting the structure.
tended to describe wave transmission over different caisson shapes. Chapters 3.2 and 3.3 go on and discuss wave overtopping discharges and wave reflections from vertical breakwaters. These Chapters are updates from the previous MAST II MCS-project under contract MAS2-CT92-0047 and no additional research has been spent on these subjects within PROVERBS. Chapter 4 discusses PROVERBS results for non breaking waves. First, the Goda model is summarised in Chapter 4.1. Chapter 4.2 contains a new approach on how to deal with neagtive or suction forces which might be relevant for seaward structure stability under some special conditions. Sainflou’s formulae were revisited and updated to an engineering design approach for this type of forces. Chapter 4.3 goes on and discusses the effects of 3-dimensionality on non breaking waves using further 3-dimensional hydraulic model tests. Chapter 4.4 briefly introduces the problems of uncertain-ties associated with the input design parameters and the Goda method. The use of numerical modelling and applications of highly sophisticated numerical models like the VOF model and pressure-impulse modelling have been investigated and results were compared to hydraulic model tests in Chapter 4.5. A new simple engineering approach has been developed in Chapter 4.6 to describe pressures on the berm in front of vertical breakwaters.
Chapter 5 deals with the same subjects already discussed by Chapter 4 but for impact waves. Chapter 5.1 introduces the new PROVERBS method to design for impacting pressures and forces at the structure. It gives relevant background information for these methods and data sources. Chapter 5.2 summarises new results on seaward impact forces which are due to over-topping water masses on the back of the caissons. These forces were investigated in hydraulic model tests and supporting numerical models. Chapter 5.3 discusses the effect of 3-dimensionality for impact waves supported by 3D hydraulic model tests in two different wave basins. Some ideas on uncertainties and a new scaling approach for impact waves are consid-ered in Chapter 5.4 which includes the effect of aeration on breaking waves. Chapter5.5 even-tually summarises results for pressures on berms under impact conditions which have been found from hydraulic model tests.
Chapter 6 concentrates on broken wave loads on vertical breakwaters, high mound compos-ite breakwaters and crown walls of rubble mound breakwaters. Chapter 6.1 summarises the design method as proposed by the British Standards which was also used in PROVERBS. Chapter 6.2 continues to describe wave loads on crown walls of rubble mound structures where results from field measurements and hydraulic model tests were used. Chapter 6.3 dis-cusses results for an innovative high mound composite breakwaters which was first developed in Japan. Results from large-scale model tests were used to derive wave pressures and forces for this type of structures.
Chapter 7 discusses the data base which has been developed in PROVERBS and contains data from five field investigations all over Europe. The breakwaters at each site were dis-tinctly different in terms of construction, wave climate and tidal range, providing a wide cross section of field results and different types of breakwaters. Field sites were Dieppe at the Northern coast of France, Porto Torres on Sardinia in Italy, Gijon in Northern Spain, Las Palmas on the Canary Islands and Alderney on the Channel Islands.
Chapter 8 reports on wave loadings on alternative low reflection breakwaters. In Chapter 8.1 perforated breakwaters are described which were intensively investigated within PROVERBS by means of field investigations and corresponding hydraulic model tests. Various approaches were tested to develop a new and simple design approach for this type of structures. Chap-ter 8.2 concentrates on horizontally composite breakwaChap-ters which are very familiar in Japan and which have proved to effectively reduce the total wave loads on the structure. Results from large-scale model tests have been used to derive simple design formulae for these reduc-tions.
Each chapter contains a list of references which points the reader to further and more detailed information on the respective subject. It is hoped that the information provided in Volume I and the chapters herein will assist coastal engineers in practical design tasks of vertical breakwaters, sea walls and similar types of structures.
CHAPTER 2.1: INFLUENCE OF STEEP SEABED SLOPES ON
BREAKING WAVES FOR STRUCTURE DESIGN
N.W.H. Allsop1) & N. Durand2) 1)
Professor (associate) University of Sheffield, c/o HR Wallingford, Howbery Park, Wallingford, Oxon OX10 8BA, UK, e-mail: w.allsop@sheffield.ac.uk
2)
Visiting Researcher at HR Wallingford, Howbery Park, Wallingford, Oxon OX10 8BA, UK
ABSTRACT
Prediction of limiting wave heights in conditions of depth-induced breaking is subject to considerable uncertainties, yet the (local) wave height is probably the most important input variable in design of coastal, harbour or shoreline structures subject to wave action.
This paper presents selected results from laboratory experiments to measure depth-limited wave breaking over steep bed slopes (1:50, 30, 20, and 1:10) in fully random wave conditions. Experimental measurements are compared with predictions for Hrms, Hs and Hmax under shoaling and breaking. The effects of shoaling and breaking on the wave height distributions are explored. An alternative empirical method to predict H1/3 and Hmax is suggested.
1. INTRODUCTION
1.1. Is there a problem?
Many coastal structures and some harbour breakwaters are constructed in relatively shallow water depths where the larger wave heights that constitute the primary input parameters in structure design are significantly influenced by depth-limited breaking. Prediction methods to calculate hydraulic or stability responses of these structures generally use the incident significant wave height (Hs) as primary input variable, often defined in the water depth at the seaward toe of the structure (hs). Where wave breaking has significant influence on design wave heights, this approach therefore requires that prediction methods for depth-limiting must be robust and reliable.
Design methods for wave overtopping, armour movement and related responses require values of the incident significant wave height, H1/3. In contrast, calculations of wave forces using Goda's method often use an upper limit estimate of Hmax such as H1/250. Neither of these values are reliably derived from wave breaking prediction methods which use spectral measures.
A few design methods use offshore wave heights in deeper water, say Hso, and use empirical methods to predict the response directly, calibrated for a range of simple bed slopes. The best-known examples are methods developed by Goda (1975, 1985) to predict overtopping or forces on vertical walls which use a single equivalent sea bed slope. Such methods assume that each approach bathymetry may be represented by a simple bed slope, and that the empirical prediction methods fully represent the effects of different wave transformations on the response of interest.
Most experimental studies on wave breaking have been on bed slopes shallower than 1:30, typically 1:50 or 1:100. On these slopes, wave shoaling is relatively mild, and wave breaking reasonably well understood, but there is growing evidence that steep bed slopes transform waves differently and give more severe hydraulic and structure responses. Jones & Allsop (1994), Southgate & Stripling (1996), Hamm & Peronnard (1997), Nelson (1997) and McConnell & Allsop (1998) demonstrate not only that many methods to predict wave conditions under breaking suffer from significant limitations, but that some responses seem to be particularly influenced by local sea bed slopes, in some instances to an extent not covered by using local wave heights in the calculations. These seem to be especially noticeable for slopes steeper than 1:50.
1.2. Research studies
New or improved methods to predict wave behaviour and breaker heights are needed to improve safety of structures constructed in the surf zone. In the first instance, a series of hydraulic model studies were completed by HR Wallingford for the Flood & Coast Defence division of the Ministry of Agriculture, Fisheries and Food (MAFF) to provide more information on wave breaking behaviour. Co-operation with Alkyon Consultancy & Research and TUDelft in the Netherlands to share laboratory and field data on wave breaking is intended to ensure as large and reliable a dataset as possible. It should however be noted that the results presented in this particular paper represent only part of the data, and relatively simple levels of analysis.
Other studies on wave breaking at vertical or composite breakwaters have been conducted as part of the EU PROVERBS project, see particularly Calabrese & Allsop (1998). Those studies have concentrated on whether wave conditions, depth and geometry will cause wave
impacts on the wall, and are not intended to predict breaking / broken wave heights, so will not be addressed further here.
2. PREDICTION METHODS FOR WAVE BREAKING
This paper does not attempt to give an overall review of prediction methods for wave breaking, but draws on the review by Southgate (1995). Additional papers or reports are cited where they amplify or up-date that review. It may be noted that many of the methods cited by Southgate give information only at a single point, but in design of realistic structures, it is important that predictions be valid over a wide range of relative depths / breaking regions. Predictions of wave breaking are tested here against measurements from the onset of breaking onwards.
The primary cause for wave breaking in deep water is that the wave steepness exceeds the fundamental limit given for individual waves by:
(H / L)max = 0.142 (1)
In shallow water, the main processes of interest in wave breaking may be divided into two. The first processes are those of wave transformations up to, but not beyond, the point of breaking. These include refraction and diffraction (neither discussed further here), and shoaling (here of considerable interest). These processes involve no significant loss of energy and are essentially reversible. The second set of processes are those which occur from breaking onwards. These processes involve significant loss of energy and are not reversible. It is noted that in some analysis, effects of shoaling and then breaking have been somewhat confused. Confusions may also have arisen from differences between regular waves (where all waves behave the same) and random waves (in which breaking positions and other features vary with period and height of each wave). These differences are particularly evident where methods to predict wave breaking have been developed using regular waves only, but are then applied to "real sea" cases where waves are random.
A final source of confusion is the use of significant wave height Hs in design methods, and numerical models to predict wave conditions, without more careful definition of its derivation, be it spectral (Hs = Hm0) or statistical (Hs = H1/3). Differences between H1/3 and Hm0 were first highlighted by Thompson & Vincent (1984) and described more recently by Hamm & Peronnard (1997). The main problem arises where one method has been used to define wave conditions in model testing used to derive empirical design methods, and then a different method is used to derive design wave conditions. Comparisons in this study have shown that differences between H1/3 and Hm0 are greatest for low wave steepnesses when non-linear shoaling is pronounced.
2.1. Flat bed slopes
For very shallow bed slopes, usually taken as flatter than 1:100, it is often assumed that a simple limit to the individual wave height relative to local water depth may be given by:
Hmax / hs = 0.78 (2a)
Later researchers showed that this limit, suggested by McCowan based on solitary wave theory (see Southgate, 1995), might be increased to Hmax/hs = 0.83. Perversely, Le Mehaute appears to give a much lower limit of individual wave height relative to local water depth:
Hmax / hs = 0.55 (2b)
2.2. Sloping seabeds
The methods most frequently used in practical design calculations for structures are those by Weggel (1972), Goda (1975) and Owen (1980). Weggel used regular wave test data to derive simple empirical expressions to predict maximum wave heights in depth hs:
Hmax / hs = b / (1 + a hs /(gT2) ) (3a)
where coefficients a and b are defined in terms of the seabed slope m:
a = 43.75 (1 – exp (-19m)) (3b)
b = 1.56 /(1 + exp (-19.5m)) (3c)
Goda (1975) developed a prediction method for irregular wave breaking, even suggesting a method to transform the wave height distribution under breaking conditions. For shoaling, Goda used Shuto's method instead of simple linear wave methods to give the shoaling coefficient Ks.
For wave breaking where hs/Lpo 0.2:
H1/250 = 1.8 Ks Hso (4a) For hs/Lpo < 0.2: H1/250 = min { (0* Hso + 1* hs), max* Hso, 1.8 Ks Hso } (4b) where: 0* = 0.052 (Hso/Lpo)-0.38 exp (20m1.5) (4c) 1* = 0.63 exp (3.8m) (4d)
max* = max { 1.65, 0.53(Hso/Lpo)-0.29 exp (2.4m) } (4e) Where H1/3 is needed in design, Goda suggested a similar method for H1/3.
For hs/Lpo < 0.2:
H1/3 = min { (0Hso + 1hs), maxHso, KsHso } (4g) where:
0 = 0.028 (Hso/Lpo)-0.38 exp (20m1.5) (4h)
1 = 0.52 exp (4.2m) (4i)
max = max { 0.92, 0.32(Hso/Lpo)-0.29 exp (2.4m) } (4j) Noting that for steep bed slopes, waves may shoal substantially before breaking starts, Owen (1980) developed a simple method to provide first-estimates of the upper limit to the (significant) wave height Hsb in any water depth hs for each of five bed slope. The method was derived as a part-way point in predicting wave overtopping of seawalls, and was not itself validated against any data on breaking wave heights. Owen's simple curves were derived graphically, see Figure 1, but were later described by empirical equations relating breaker index Hsb/hs to relative depth hs/gTm2:
Slope Breaking limit, Hsb/hs
1/100 Hsb/hs = 0.58 - 2 (hs/gTm2) (5a)
1/50 Hsb/hs = 0.66 - 10.583 (hs/gTm2) + 229.17 (hs/gTm2)2 (5b) 1/30 Hsb/hs = 0.75 - 20.083 (hs/gTm2) + 479.17 (hs/gTm2)2 (5c) 1/20 Hsb/hs = 0.95 - 38.417 (hs/gTm2) + 895.83 (hs/gTm2)2 (5d) 1/10 Hsb/hs = 1.54 - 97.83 (hs/gTm2) + 2541.67 (hs/gTm2)2 (5e) In analysing laboratory and field data for slopes up to 1:20, Battjes & Stive (1984) did not detect any systematic dependence of wave conditions on slope, but did find an influence of wave steepness. They developed an expression for a breaking coefficient, taken by Southgate (1995) to give the limiting r.m.s. wave height:
Hrms / hs = 0.5 + 0.4 tanh (33 Hrms,o / Lpo) (6) Figure 1 Simple breaking curves, after Owen
The literature gives relatively little advice on changes to wave height distributions with breaking, but Simm (1991) cites equations for extreme wave heights H0.1% and H1% probably originating from Klopman & Stive (1989):
H1% = 1.517 Hs / (1+(Hs/hs))1/3 (7a)
H0.1% = 1.859 Hs / (1+(Hs/hs))1/2 (7b)
2.3. Numerical models
During this study, many of the wave measurements were also compared with two numerical models: WENDIS and COSMOS2D. WENDIS is a Wave ENergy DISsipation model, designed to estimate near-shore wave conditions at coastal structures where shoaling (linear shoaling theory), bed friction (Hunt and Bretschneider & Reid), and wave breaking (Weggel) may be significant. COSMOS2D is a numerical model of near-shore hydrodynamics, sediment transport and morphological changes which includes wave transformation by
refraction, shoaling (linear wave theory), bed friction and wave breaking (Battjes & Stive and
Weggel). Some of these comparisons were discussed by Durand & Allsop (1997).
3. EXPERIMENTAL STUDY
Wave conditions were measured using statistical (and later spectral) methods for 2 or 3 water levels over five different bed slopes:
1:50, indicative of shallow sand beach slopes; 1:30 and 20, indicative of steeper sand beaches;
1:10 and 1:7, indicative of rock coasts and shingle beaches.
The results discussed in this paper were mostly derived from supplementary tests conducted by the visiting researcher on a 1:30 slope, with some comparisons with data from the main series of tests on bed slopes of 1:50, 1:20 and 1:10. Additional data for a 1:10 slope by Allsop (1990) and on a 1:20 slope by Southgate & Stripling (1996) has not yet been included in these analyses.
3.1. Outline of experiments
The main tests used uni-modal or bi-modal seas, 30 wave conditions divided into six sequences of five tests, described by Hawkes et al (1998) and Coates et al (1998):
a) Wind-sea only
b) More wind-sea (80%) than swell (20%) c) Equal wind-sea and swell
d) More swell (80%) than wind-sea (20%) e) Swell only
Swell and wind sea conditions used standard JONSWAP spectra (=3.3) defined by Hs and Tp. Bi-modal conditions combined two JONSWAP spectra, each defined by Hs and Tp. Supplementary tests of wave breaking included a few regular wave conditions; some modified spectra of different spectral peakedness (=1 to 7), and a few rectangular spectra. Other measurements in tests reported by Hawkes et al (1998) and McConnell & Allsop (1998) included wave overtopping discharges, rock armour stability, and wave pressures / forces on vertical walls. 1:30 slope W a v e B a s i n 1 10 1 30 SPENDING BEACH PADDLE 0.0 SWL 0.7m 0.6m 9m 0.5m
Probe 15 Probe 3 Probe 0
2m 0.2m
2m 2m
Figure 2 Configuration for supplementary tests on 1:30 bed slope
Most of the tests on wave breaking used the Absorbing Flume at Wallingford with a working length of 36m, and equipped with a random wave generator with computer-controlled absorption system. In each instance, the sea bed slope was terminated in a horizontal section below water, behind which was a gravel beach to absorb remaining wave energy. The supplementary tests on 1:30 slope used a 6m wide flume within a wave basin with two mobile piston paddles, and the 1:30 slope itself was slightly unusual as it featured a short steep (1:10) approach ramp at the toe, Figure 2.
Up to 16 wave probes were placed along the test sections, usually 1 or 2 probes in deep water near the wave paddle, 3 or 4 being placed along the horizontal section at the top of the slope, and the remaining probes up the slope. Each test was run for 500 waves or longer, sampled at 20 Hz (model) giving an average of 20-60 points per wave. In general, wave measurements were analysed statistically using a zero down-crossing definition for each wave. Selected data files were later also analysed spectrally, and some of the results of this analysis are discussed by Hurdle et al (1998) in an accompanying paper.
4. RESULTS
4.1. Parameters derived
Most wave measurements discussed here are presented as significant wave heights, H1/3. Both spectral and statistical methods have been used to derive wave heights during these studies, but most weight in this paper will be given to statistical measures of wave height, particularly Hs = H1/3. This should avoid problems of confusion between Hm0 and H1/3 highlighted originally by Thompson & Vincent (1984), and most recently by Hamm & Peronnard (1997).
Two other measures of wave heights may be used. The maximum wave height Hmax will depend on sample size, and is itself relatively unstable. In this study, Hmax is generally given by H99.8% or H99.9%. A more stable measure of wave height favoured by morpho-dynamic researchers is root mean square wave height, Hrms, defined spectrally as Hrms = Hm0/2.
The main measures of wave period used here are the spectral peak period, Tp or mean period Tm. The peak period is more stable than the mean period measured either spectrally or statistically, and is less susceptible to distortion by measurement or calculation errors. Sea states have been categorised by the (fictional) steepness, spo = Hso/Lpo, where Lpo = gTp2/2.
4.2. General shoaling and breaking
The complex nature of breaking under random waves precludes any presentation of all features in a single graph. The initial form of presentation by Durand & Allsop (1997) is used here for the first few examples with measurements of local significant wave height against distance along the test flume, see Figure 3.
This presentation allows the effect of shoaling to be identified as waves react to the rising seabed. The onset of breaking occurs at the peak of the wave height, although in some tests this onset was quite difficult to assess. Breaking continues as the waves move into shallower water or, for some tests, over the horizontal bed at the top of the slope, even if at a slower rate. For most of these tests, rates of breaking appear to be relatively constant for a particular
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
d istan ce fro m to p o f slo p e (m )
H1/
3
H s o = 0.110m , s po = 0.031 H s o = 0.190m , s po = 0.053
Figure 3 Effect of wave height on breaking for constant wave period, 1:30 slope
bed slope, but breaking starts at different points along the slope and at different depths, depending upon offshore conditions.
The lowest steepness waves in Figure 4, spo = 0.007, shoal more than low steepness waves, spo = 0.018, or than the moderate steepness waves, spo = 0.034. The conditions shown in Figure 4 show little increase in wave height due to shoaling for wave steepnesses of spo 0.034, but breaking starts quite far up the approach slope. Lower wave steepnesses show more shoaling, reach a greater wave height and breaking starts earlier.
Durand & Allsop (1997) compared transformations of the same (offshore) wave condition for bed slopes of 1:10, 1:20, and 1:30. Wave breaking on the 1:10 and 1:20 slopes was delayed compared with the 1:30 slope, as might be expected, but the breaking appeared to occur in similar water depths. Over the steeper slopes, the process of breaking and energy dissipation was compressed into rather shorter distances.
4.3. Wave height distributions
Distributions of individual wave heights are not necessarily of great interest to sediment and wave transformation modellers, but for designers of coastal / harbour or shoreline structures, changes to the distribution of wave heights, particularly to the probabilities of larger individual heights, are of considerable interest.
In deep water, individual wave heights, Hi, generally conform to a Rayleigh distribution. Such distributions plot as straight
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
distance from top of slope (m)
H1/3 Hso = 0.107m Tp = 1.11s , spo = 0.054 Tp = 1.41s , spo = 0.034 Tp = 1.97s , spo = 0.018 Tp = 3.20s , spo = 0.007
Figure 4 Effect of wave period on breaking for constant offshore wave height, 1:30 slope
0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 -ln(1-P) (Hi/H 1/3 o) 2 0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 P (%) Hi/H 1/3 o Probe 14 Probe 12 Probe 8 Probe 5 Probe 3 1.41 1.73 1.00 0.00 2.24 2.00 99.9 99.8 99.3 98.2 95.0 86.5 63.2 0.0
Figure 5 Individual wave heights, spo = 0.054, 1:30 slope
lines in the format used in Figures 5-8, where individual wave heights are presented as (Hi/Hso)2, and non-exceedance probabilities as -ln(1-P). On these scales, -ln(1-P) = 2.0 corresponds to H1/3 for a true Rayleigh distribution, H98% is given by -ln(1-P) = 3.91, H99% is given by -ln(1-P) = 4.61, and H99.6% by -ln(1-P) = 5.52.
In Figures 5-8, wave probe 14 is in deepest water, probe 12 is about 2m (model) up the 1:30 slope, probes 8 and 5 are over the slope, and probe 3 is at the top of the slope.
Under breaking, the largest waves in the distribution break first, reducing towards the breaking limit. After some breaking, it is likely that a proportion of the energy will be
re-distributed, perhaps combining with waves that have
shoaled further. In still shallower depths, further breaking will then apply over a greater part of the wave height distribution.
Some of these processes are illustrated for a typically steep storm sea state (spo = 0.054) in Figure 5. These results do not support the suggestion that individual waves above the breaking limit (Hi>Hb) will fall to that limit whilst waves smaller than the breaking limit are unaffected.
Even in the relatively shallow water in these test facilities, wave heights at the gauge in deepest water (probe 14) give only a slight curve away from the theoretical Rayleigh distribution. The effect of breaking for this sea state is relatively uniform, shown by the steady decrease of wave heights at each successive probe from that in deepest water (probe 14) to the shallowest (probe 3). 0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 -ln(1-P) (Hi/H 1/ 3 o) 2 0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 P (%) Hi /H 1/ 3 o Probe 14 Probe 12 Probe 8 Probe 5 Probe 3 99.9 99.8 99.3 98.2 95.0 86.5 63.2 0.0 1.41 1.73 1.00 0.00 2.24 2.00
Figure 6 Individual wave heights, spo = 0.034, 1:30 slope 0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 -ln(1-P) (Hi/H 1/3 o) 2 0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 P (%) Hi/H 1/3 o Probe 14 Probe 12 Probe 8 Probe 5 Probe 3 1.41 1.73 1.00 0.00 2.24 2.00 99.9 99.8 99.3 98.2 95.0 86.5 63.2 0.0
Results for a smaller wave height and lower wave steepness (spo = 0.034) shown in Figure 6 more closely match normal expectations. Individual heights up to Hs (given here by H1/3 at -ln(1-P)=2.0) are hardly affected by wave breaking up to probe 8. There are however noticeable reductions in wave heights above 98%, reducing H99.4% by perhaps 20%. This behaviour will be important for any response that is more strongly influenced by the largest waves in the distribution. Further inshore at probes 5 and 3, the effect of wave breaking again gives more uniform reductions of wave height across the full range of exceedance values. Retaining the same offshore significant wave height, but further reducing the wave steepness to spo = 0.018 in Figure 7, and spo = 0.007 in Figure 8 gives more surprising results, although some of these effects might have been anticipated from the shoaling behaviour shown by some of the results examined by Durand & Allsop (1997).
In Figure 7, the largest waves occur over the start of the approach slope (probes 12 and 8), particularly noticeable for wave heights above H1/3. For very low steepnesses in Figure 8, wave heights at H1/3 and higher non-exceedance levels increase markedly from the seaward point (probe 14) up the early part of the approach slope (probes 12 and 8). At this last position (probe 8), breaking only reduces individual wave heights below their offshore value above about 98% non-exceedance. Breaking only starts to reduce H1/3 by probe 5, well up the approach slope.
4.4. Comparison with prediction methods
As discussed above, random waves do not give a single breaking point, so the identification of onset of breaking under random waves is difficult and may give inaccurate results. For simple prediction methods, more useful comparisons are with wave heights measured after the onset of breaking. These comparisons are however complicated by different definitions of wave height given by the different prediction methods. Weggel's and Goda's methods give estimates of wave heights close to the maximum, Hmax and H1/250, Owen's simple method gives estimates of Hsb. 0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 -ln(1-P) (Hi/H 1/3 o) 2 0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 P (%) Hi/H 1/3 o Probe 14 Probe 12 Probe 8 Probe 5 Probe 3 1.41 1.73 1.00 0.00 2.24 2.00 99.9 99.8 99.3 98.2 95.0 86.5 63.2 0.0
Figure 8 Individual wave heights, spo = 0.007, 1:30 slope
Considering first maximum wave heights, measurements of Hmax shown relative to the local water depth as Hmax/hs are compared with predictions using Weggel's method for a bed slope of 1:10 in Figure 9, for a slope of 1:30 in Figure 10, and for a slope of 1:50 in Figure 11. The results have been taken from the point of breaking (indicated by the first reduction of Hs, see for example Figures 3 & 4). Weggel's method does not predict shoaling, so no comparison is made for positions seaward of the breaking point.
For the 1:10 slope in Figure 9, Weggel's prediction gives relative breaking heights Hmax/hs up to 1.4 at the lowest values of relative depth. Measured maximum wave heights significantly exceed the predicted values for hs/gTp2 < 0.005, suggesting that the prediction method may give unsafe results in this region. Safer predictions for the 1:10 slope are given for 0.005 < hs/gTp2 < 0.015. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp 2 ) Hmax / h s
generic W eggel's method for 1:10 slope experimental Hmax/hs
unsafe
safe
Figure 9 Weggel's Hmax prediction, 1:10 slope
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp 2 ) Hmax / h s
generic W eggel's method for 1:30 slope experimental Hmax/hs
unsafe
safe
For the 1:30 slope in Figure 10, Weggel's prediction gives relative breaking heights Hmax/hs up to about 1.0 at the lowest values of relative depth. Measured maximum wave heights significantly exceed predicted values for hs/gTp2 <0.002, suggesting that the prediction method may give unsafe results in this region. Safer predictions for the 1:30 slope are given for 0.002 < hs/gTp2 < 0.02.
For the 1:50 slope in Figure 11, Weggel's prediction gives relative breaking heights Hmax/hs up to about 0.9 at the lowest values of relative depth measured here. Maximum wave heights however significantly exceed predicted values for hs/gTp2 < 0.007, suggesting that Weggel's method may give unsafe results in this region. Safer predictions for the 1:50 slope are given for 0.007 < hs/gTp2 < 0.02, but the prediction method does still not seem to reproduce well the general effect of increasing breaker index with decreasing relative depth hs/gTp2.
Measurements of Hmax from tests on the 1:30 slope were compared with predictions by Weggel's and Goda's methods by Durand & Allsop (1997). It was noted that only a few design methods for coastal structures / breakwaters use the maximum wave height, or H1/250 as used by Goda for wave forces on caissons. Most such methods require predictions of H1/3. Goda's predictions for H1/3 would therefore be potentially more useful, but these predictions agreed less well with measurements of H1/3 than the agreement with Hmax.
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp2) Hsb / h s , H max / h s Experimental H(1/3)/hs Best Fit for H(1/3)/hs Experimental Hmax/hs Best Fit for Hmax/hs
Figure 12 New prediction, 1:10 slope 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp 2 ) Hmax / h s
generic W eggel's method for 1:50 slope experimental Hmax/hs
unsafe
safe
An alternative approach was therefore sought in which a revised empirical method was fitted to results for bed slopes of 1:10, 1:30 and 1:50, see Figures 12-14. In deriving the new empirical method, it was important that the new method will:
a) reproduce the general form of breaking with respect to hs/gTp2; b) reproduce reliably the asymptote at high relative depths;
c) give better description of the breaking limits at low values of hs/gTp2; d) describe limits for Hs and Hmax using equations of the same form.
It was also intended that the new method would be simple to apply in desk study calculations.
A form of equation was sought to relate both Hs/hs and Hmax/hs to relative depth hs/gTp2. Each potential method was tested against data from these studies for Hs and Hmax, and for bed slopes of 1:10, 1:30 and 1:50. The equation was:
y = y + (y0 - y)·ec (8)
where y = Hmax/hs or H1/3/hs x = hs/gTp2 and c = -b·x0.78
Values of the coefficients were derived by minimising total errors in the fit to measured wave heights from these studies. Initial values of (y0 - y) and y were given in an earlier analysis, Allsop et al (1998), but those values gave anomalous compa-risons between different bed slopes.
A further re-analysis by Melito (1998) for PROVERBS Task 1 generated a more consistent 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp2) Hsb / h s , H m ax / h s Experimental H(1/3)/hs Best Fit for H(1/3)/hs Experimental Hmax/hs Best Fit for Hmax/hs
Figure 13 New prediction, 1:30 slope
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp 2 ) Hsb / h s , H max / h s Experimental H(1/3)/hs Best Fit for H(1/3)/hs Experimental Hmax/hs Best Fit for Hmax/hs
coefficients (y0 - y) and y were predicted as function of the bed slope, m. That analysis gave for H1/3:
(y0 - y) = 0.62 + 7·m y = 0.53 – 0.0012/m (9) and for Hmax:
(y0 - y) = 0.98 + 9.13·m y = 0.72 – 0.0019/m (10) Values of these coefficients are summarised in Table 1.
Table 1: Coefficients for eqn. 8
Bed slope
Coefficients for H1/3 1:50 1:30 1:10
(y0 - y) = 0.62 + 7·m 0.760 0.853 1.320
b 93.20 93.20 93.20
y = 0.53 – 0.0012/m 0.470 0.494 0.518
Coefficients for Hmax
(y0 - y) = 0.98 + 9.13·m 1.163 1.284 1.893
b 80.26 80.26 80.26
y = 0.72 – 0.0019/m 0.626 0.664 0.701
5. CONCLUSIONS
These studies have demonstrated that present methods to predict inshore wave conditions on steep bed slopes for structure design suffer some important limitations.
Wave shoaling is particularly important in increasing (statistical) wave heights on steep bed slopes, but requires the use of non-linear methods such as Shuto's to give the full increase of wave height over steep bed slopes.
Weggel's method is widely used in many numerical models to represent wave breaking, but this method tends to under-estimate breaking for low relative depths. Owen's method is convenient to use, but was not ever directly validated against measurements of wave breaking.
A new set of empirical equations (eqns. 8-10) and coefficients (Table 1) have been developed from the data sets described here to predict breaking limits for Hmax and H1/3 on steep bed slopes using the bed slope, m. The new methods are intended to be self-consistent, which has
given slightly less good agreement between data and prediction than if each bed slope had been treated differently.
The ranges of data used here suggest that the new method will be valid over the ranges: 1:10 slope 0.002< hs/gTp2 <0.011;
1:30 slope 0.002< hs/gTp2 <0.02, but likely to be conservative for 0.002< hs/gTp2 <0.005;
1:50 slope 0.003< hs/gTp2 <0.02.
ACKNOWLEDGEMENTS
This paper is based on studies by the Coastal Group of HR Wallingford supported by the UK Ministry of Agriculture, Fisheries, and Food under Research Commissions FD02 and FD07, and by the EU's MAST project PROVERBS under MAS3-CT95-0041. Additional support was given by the University of Sheffield and HR Wallingford. The authors of this paper gratefully acknowledge assistance in processing data by Kirsty McConnell and Rob Jones. Particular thanks for help in deriving the new empirical method are due to Ivano Melito and Sebastien Bourban.
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REVISED BREAKING COEFFICIENTS AND VALIDATION
AGAINST MEASUREMENTS
I. MELITO ‡ Annex to
INFLUENCE OF STEEP SEABED SLOPES ON BREAKING WAVES FOR STRUCTURE DESIGN
(N.W.H. Allsop & N. Durand)
1. INTRODUCTION
The principal purpose of this study is to provide a more consistent method for wave breaking on steep bed slopes (1:50, 1:30 and 1:10).
In the earlier version of this paper, Allsop et al (1998), the experimental data for H1/3/hs and Hmax/hs were analysed separately for each bed slope and Hmax was generally given by H99.9%. The fitting function was the following:
y = y + (y0 - y)·ec (A)
where y = Hmax/hs or H1/3/hs x = hs/gTp2 and c = -b·x0.78
Values of the coefficients were found by minimising total error in the fit. In fact, given a set of experimental data (xi , yi) such that yi is dependent on xi (for i=1, n), it may possible to find a relationship yi = f(xi) for all i. For any experimental point, the vertical distance between the points and the fitting function gives the measurement error Ei. The numerical method minimises the total error Et, where Et is the sum of the squares of the error quantities Ei
The data are now re-analysed by taking in account all the points and the bed slope m (but separately for H1/3 and Hmax). The fitting function is always given by equation A, but the coefficients (y0 - y) and y are defined in terms of the bed slope m. In fact:
(y0 - y) = ao + a·m y = do + d/m (B)
In this case, the numerical method minimises the global error Eg, where Eg is the sum of the total error Et for each bed slope taken in account.
‡
2. ANALYSIS DATA RESULTS
For each bed slope and for H1/3/hs and Hmax/hs data sets, the fitting coefficients are summarised in Table 1a. Values of the previous fitting coefficients are given in Table 1 of the earlier version of this paper, Allsop et al (1998). In that analysis, the data for H1/3 and Hmax were taken in account separately for each bed slope.
Table 1a: Coefficients for eqn. A (free solution)
Bed slope
Coefficients for H1/3 1:50 1:30 1:10
(y0 - y) = 0.62 + 7·m 0.760 0.853 1.320
b 109.0 109.0 109.0
y = 0.53 – 0.0012/m 0.470 0.494 0.518
Coefficients for Hmax
(y0 - y) = 0.98 + 1.53·m 1.011 1.031 1.133
b 80.26 80.26 80.26
y = 0.72 – 0.0019/m 0.626 0.664 0.701
For each bed slope, Hsb/hs and Hmax/hs versus hs/gTp2 are plotted; each graph shows the experimental points for H1/3/hs and Hmax/hs, both the earlier fitting curves [Eqn. 8, Table 1 of Allsop et al (1998)] and the fitting curves obtained by the free solution (Eqn. A, Table 1a) for such points, see Figures 1-3. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp 2 ) Hsb / h s , H m ax / h s Experimental H(1/3)/hs Earlier Best Fit for H(1/3)/hs Best Fit for H(1/3)/hs (free) Experimental Hmax/hs Earlier Best Fit for Hmax/hs Best Fit for Hmax/hs (free)
These fitting curves (free solution) show consistent values of the asymptote at high relative depths, but they do not give a good description of the breaking limits at low values of hs/gTp2. So, further checking was performed by forcing the values of the fitting coefficients: “b” for H1/3 (from 109.0 to 93.20) and “a” for Hmax (from 1.53 to 9.13).
In this manner, we have a different solution, here in after called forced solution (Eqn. A, Table 2a).
Values of the fitting coefficients derived by the forced solution are summarised in Table 2a and the results shown for bed slopes 1:10, 1:30 and 1:50 in Figures 4-6.
Table 2a: Coefficients for eqn. A (forced solution)
Bed slope Coefficients for H1/3 1:50 1:30 1:10 (y0 - y) = 0.62 + 7·m 0.760 0.853 1.320 b 93.20 93.20 93.20 y = 0.53 – 0.0012/m 0.470 0.494 0.518 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp 2 ) Hsb / h s , H max / h s Experimental H(1/3)/hs Earlier Best Fit for H(1/3)/hs Best Fit for H(1/3)/hs (free) Experimental Hmax/hs Earlier Best Fit for Hmax/hs Best Fit for Hmax/hs (free)
Figure 2 Fitting curves (free solution), 1:30
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp2) Hsb / h s , H m ax / h s Experimental H(1/3)/hs Earlier Best Fit for H(1/3)/hs Best Fit for H(1/3)/hs (free) Experimental Hmax/hs Earlier Best Fit for Hmax/hs Best Fit for Hmax/hs (free)
Coefficients for Hmax
(y0 - y) = 0.98 + 9.13·m 1.163 1.284 1.893
b 80.26 80.26 80.26
y = 0.72 – 0.0019/m 0.626 0.664 0.701
3. CHECKING WITH OTHER MEASURED DATA
Further analyses were performed to check the new method. In fact, predictions for H1/3 and Hmax derived by using both free (Table 1a) and forced (Table 2a) solution were compared against measured wave heights during calibration of model studies for a reef breakwater.
3.1. Test conditions and wave calibration
Tests were performed at scale 1:40 and with a bed slope generally 1:30 using wave heights and water levels representing different return period events. Offshore wave conditions used in the physical model and local depth hs (at probes 2, 3 and 4) are summa-rised below: (a) hs = 5.7m, Hs = 4.2m, Tp = 11.5s, Tm = 9.0s 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp 2 ) Hsb / h s , H max / h s Experimental H(1/3)/hs Earlier Best Fit for H(1/3)/hs Best Fit for H(1/3)/hs (forced) Experimental Hmax/hs Earlier Best Fit for Hmax/hs Best Fit for Hmax/hs (forced)
Figure 4 Fitting curves (forced solution), 1:10 slope
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp2) Hsb / h s , H m ax / h s Experimental H(1/3)/hs Earlier Best Fit for H(1/3)/hs Best Fit for H(1/3)/hs (forced) Experimental Hmax/hs Earlier Best Fit for Hmax/hs Best Fit for Hmax/hs (forced)
(b) hs = 6.7m, Hs = 4.8m, Tp = 12.5s, Tm = 9.8s (c) hs = 7.7m, Hs = 5.3m, Tp = 13.9s, Tm = 10.8s (d) hs = 7.7m, Hs = 5.8m, Tp = 14.6s, Tm = 11.4s
It may be noted that any increase in offshore waves simply produced more breaking inshore, resulting in a smaller significant wave height. Probably, there is an effect of transversal waves on the mean period.
Two wave probes were located in front of the wave paddles (probes 0 and 1); three wave probes (2, 3 and 4) were positioned along the centerline of the reef during calibration, see Figure 7. A repeating sequence of waves, defined by a JONSWAP spectrum and programmed on the computer controlling the wave paddles, created a calibration test length of 1000 waves. The output from wave probes was analysed spectrally to estimate Hm0 and also using statistical methods to derive Hs and Hmax. In this study, Hmax is generally given by H99.9%. The mean period Tm, derived both statistically and spectrally, is given in Table 3a. The wave heights obtained by analysis (spectral and statistical) and prediction methods are summarised in Table 4a.
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 0.005 0.01 0.015 0.02 0.025 hs / (gTp2) Hsb / h s , H max / h s Experimental H(1/3)/hs Earlier Best Fit for H(1/3)/hs Best Fit for H(1/3)/hs (forced) Experimental Hmax/hs Earlier Best Fit for Hmax/hs Best Fit for Hmax/hs (forced)
Figure 7 Model basin Table 3a: Mean periods
Pb. 0 (Offshore) Pb. 0 (Offshore) Wave Pb. 2,3,4 Stat. Spect. Stat. Spect.
cond. hs Tp Tm Tm Tm Tm Tm
(m) (sec) (sec) (sec) (sec) (sec) (sec)
(a) 5.7 11.5 9.0 8.8 9.7 10.2 10.7
(b) 6.7 12.5 9.8 9.3 10.2 10.6 11.6
(c) 7.7 13.9 10.8 10.4 11.6 11.8 12.7
(d) 7.7 14.6 11.4 11.0 11.8 12.3 12.9
Pb. 2 Pb. 3 Pb. 4
Wave Pb. 2,3,4 Stat. Spect. Stat. Spect. Stat. Spect.
cond. hs Tp Tp Tm Tm Tm Tm Tm Tm Tm
(m) (sec) (sec) (sec) (sec) (sec) (sec) (sec) (sec) (sec)
(a) 5.7 11.5 11.5 9.0 8.9 9.4 9.1 9.4 9.4 9.7
(b) 6.7 12.5 12.5 9.8 9.4 10.2 9.5 11.0 9.8 10.3
(c) 7.7 13.9 13.9 10.8 10.2 12.1 10.4 12.1 10.5 11.9
Table 4a: Wave heights (measured and predicted)
Pb. 0 (Offshore) Pb. 1 (Offshore) Stat. Spect. Stat. Spect. Tp Hs Hmax Hm0 Hs Hmax Hm0 (sec) (m) (m) (m) (m) (m) (m) 11.5 4.34 7.03 4.44 4.11 7.49 4.25 12.5 4.74 8.04 4.85 4.79 8.75 4.84 13.9 5.82 9.98 5.81 6.41 12.23 6.36 14.6 6.57 11.22 6.61 7.11 13.63 6.95 Pb. 2 Pb. 3 Pb. 4
Stat. Spect. Stat. Spect. Stat. Spect. Tp Hs Hmax Hm0 Hs Hmax Hm0 Hs Hmax Hm0
(sec) (m) (m) (m) (m) (m) (m) (m) (m) (m) 11.5 4.13 5.88 3.31 4.30 5.79 3.37 4.43 5.98 3.49 12.5 4.54 6.79 3.62 5.01 7.01 3.82 5.17 7.73 4.03 13.9 5.46 7.88 4.13 5.82 9.58 4.53 5.65 8.84 4.35 14.6 5.54 8.15 4.33 5.90 8.77 4.70 6.01 10.27 4.72 Average Pb. 0, 1 Average Pb. 2, 3, 4 Stat. Spect. Stat. Spect. Tp Hs Hmax Hm0 Hs Hmax Hm0 (sec) (m) (m) (m) (m) (m) (m) 11.5 4.23 7.26 4.35 4.29 5.88 3.39 12.5 4.77 8.40 4.85 4.91 7.18 3.82 13.9 6.12 11.11 6.09 5.64 8.77 4.34 14.6 6.84 12.43 6.78 5.82 9.06 4.58 New Method (Pb. 2, 3, 4) (Pb. 2, 3, 4) free (Table 1a) forced (Table 2a) Goda Rayleigh Tp H1/3 Hmax H1/3 Hmax H1/3 H1/250 H99.9% (sec) (m) (m) (m) (m) (m) (m) (m) 11.5 3.82 5.62 4.07 6.07 4.00 5.16 5.31 12.5 4.49 6.62 4.80 7.15 4.68 6.04 6.22 13.9 5.29 7.77 5.65 8.42 5.46 7.09 7.30 14.6 5.46 7.99 5.83 8.70 5.55 7.26 7.48
3.2. Comparisons for H1/3
Measured wave heights (Hs) for probes 2, 3 and 4, Goda's and new method's predictions derived by using both free (Table 1a) and forced (Table 2a) solution are plotted against hs/gTp2 in Figure 8.
Goda's method seems to match measurements for probe 2 for all the wave conditions, except for the lowest wave height; for probes 3 and 4 is under measurements.
The new method, derived with the forced solution (Table 2a),
slightly over-predicts measurements for probe 2,
except for the lowest wave height; for probe 3 and 4 gives under-predictions for all the wave conditions, but matches measurements for probe 4 for wave condition (c).
More stable comparisons are given in Figure 9 where measurements at probes 2, 3 and 4 are averaged, and these compared with predictions by Goda and the new method using both free (Table 1a) and forced (Table 2a) solution. The forced solution (Table 2a), gives an excellent prediction for higher wave heights, but slightly under-predicts the remaining wave conditions; Goda's method under-predicts breaking for all the wave conditions.
3.3. Comparisons for Hmax
In these tests, Hmax is generally given by H99.9%. In fact, Hmax is the measured maximum wave height on 1000 waves; hence, we can assume Hmax corresponding to H99.9%.
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 0.0035 0.0037 0.0039 0.0041 0.0043 0.0045 hs/(gTp 2 ) H (m )
New Method's Prediction H(1/3) (forced) New Method's Prediction H(1/3) (free) Goda's Prediction H(1/3) Hs (probe 2) Hs (probe 3) Hs (probe 4) d c b a
Figure 8 New method: test for H1/3
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 0.0035 0.0037 0.0039 0.0041 0.0043 0.0045 hs/(gTp2) H (m)
New Method's Prediction H(1/3) (forced) New Method's Prediction H(1/3) (free) Goda's Prediction H1/3
Average of Hs (Probes 2, 3, and 4)
d c
b a
Figure 9 New method: test for H1/3 (averaged measurements)
Measured wave heights (Hmax) for probes 2, 3 and 4 and predictions of H99.9% are plotted against hs/gTp2 in Figure 10. Predictions of H99.9% were obtained by the new method using both free (Table 1a) and forced (Table 2a) solution, and also considering a Rayleigh distribution. In fact, from Goda's prediction of H1/250 it is possible to derive prediction of H99.9%: assuming a Rayleigh distribution for the wave heights, H99.9% / H1/250 = 1.03.
Calculated values of H99.9% assuming a Rayleigh distribution under-predict measurements for all the wave conditions. The new method, derived with the forced solution (Table 2a), seems to match measurements for probe 3 for all the wave conditions, but for the wave condition (c) gives a strong under-prediction for the same probe; for probe 2 it slightly over-predicts measurements for all the wave conditions; for probe 4 it under-predicts measurements except for wave condition (a).
More stable comparisons are given in Figure 11 where measurements at probes 2, 3 and 4 are averaged, and these compared with predictions mentioned above. The forced solution (Table 2a) gives a good result for lower wave heights, but slightly under-predicts the remaining wave conditions; calculated values of H99.9% assuming a Rayleigh distribution under-predict breaking for all the wave conditions. 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 0.0035 0.0037 0.0039 0.0041 0.0043 0.0045 hs/(gTp2) H (m)
New Method's Prediction Hmax (forced) New Method's Prediction Hmax (free) Prediction of H99.9% Hmax (probe 2) Hmax (probe 3) Hmax (probe 4) d c b a
Figure 10 New method: test for Hmax
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 0.0035 0.0037 0.0039 0.0041 0.0043 0.0045 hs/(gTp2) H ( m )
New Method's Prediction Hmax (forced) New Method's Prediction Hmax (free) Prediction of H99.9%
Average of Hmax (Probes 2, 3 and 4)
d c
b a
Figure 11 New method: test for Hmax (averaged measurements)
4. CONCLUSIONS
The curves shown in the previous figures have been obtained by calculating values of the fitting coefficients (Figures 1-3) and by forcing some of these values (Figures 4-6). The curves obtained with the forced solution (Table 2a) seem relatively reliable and match the previous curves given in the earlier version of this paper, Allsop et al (1998), for bed slopes 1:50 and 1:10, both for H1/3/hs and Hmax/hs. However, for 1:30 bed slope there were strong over-predictions for low values of relative depth.
The new method, derived with the forced solution (Table 2a) and tested for slope 1:30, gives a good prediction of measured wave heights. For lower wave heights, prediction of H1/3 in Figure 9 is closely matched by Goda's method, but slightly below measurements. For higher wave heights, prediction of H1/3 is rather higher than Goda's prediction, but matches measurements. For all wave conditions, predictions of Hmax exceed predictions of H99.9% derived by assuming a Rayleigh distribution, but matches measurements for lower wave heights and slightly under-predicts them for higher wave heights, see Figure 11.
CHAPTER 2.2: DERIVATION, VALIDATION AND USE OF
PARAMETER MAP
K. McCONNELL1) 1)
Engineer, Coastal Structures, HR Wallingford, Howbery Park, Wallingford, UK, OX10 8BA e-mail: kmcc@hrwallingford.co.uk
ABSTRACT
This short note discusses the derivation and validation of a parameter response map for use within PROVERBS in the prediction of impact forces. The note presents the final parameter map upon which the probabilistic methods of Task 4 are based.
Key words: Parameter map, Impact prediction, Wave loading
1. INTRODUCTION
A parameter response map for prediction of the type of wave loading on vertical and vertically composite breakwaters based on structure geometry and wave conditions was first suggested by Allsop et al (1996a), Figure 1. This was based on data from a series of small-scale model tests completed at HR Wallingford for vertical and composite breakwaters, termed the HR94 tests.
Moderate waves "Vertical"
Breakwater hb/hs < 0.3
Pulsating loads No impact, broken waves Composite
Breakwater 0.3 < hb/hs < 0.9
Crown Walls Rubble Mound Breakwater
hb/hs > 0.9 Low Mound Breakwater 0.3 < hb/hs < 0.6 High Mound Breakwater 0.6 < hb/hs < 0.9 Beq/Lpi > 0 4 Impact loads Transition 0.55 < Hsi/d < 0.65 Small waves 0.3 < Hsi/d < 0.55 Small waves Hsi/d < 0.35 Large waves 0.65 < Hsi/d < 1.3 Small waves 0.3 < Hsi/d < 0.55 Large waves 0.65 < Hsi/d < 1.3 Moderate waves 0.55 < Hsi/d < 0.65 Large waves 0.35 < Hsi/d Narrow berm 0.08 < Beq/Lpi < 0.14 Moderate berm 0.14 < Beq/Lpi < 0.4 Wide berm Beq/Lpi > 0.4 Fig. 1: Parameter map, after Allsop et al (1996)
Further development of this parameter map, investigation of the dimensionless parameters used and definitions of types of wave loading are discussed in Allsop et al (1995), Allsop et al (1996b) and Calabrese et al (1996) with reference to the HR94 data.
structure are the relative berm height, hb/hs, the relative wave height, Hsi/d, and the relative berm width, Beq/Lpi. The wave parameters Hsi and Lpi are determined in the water depth hs, and Lpi is determined by linear wave theory. Geometric parameters Hb, Hs and d are defined in Figure 2. The equivalent berm width, defined halfway up the berm, Beq =Bb +hb/2tan. These parameters are based on the standard PROVERBS notation, Kortenhaus (1997).
Fig. 2: Definition of geometric parameters
Firstly the relative berm height, hb/hs determines the type of structure – whether a simple vertical wall, a composite structure with a low mound, a composite structure with a high mound or a rubble mound with a crown wall.
The relative wave height, Hsi/d, then determines whether wave impacts will occur, based on the depth of water at the toe of the caisson, which in the case of a composite structure will be the water depth on the berm.
For high mound breakwaters (0.6<hb/hs<0.9) exposed to large waves (0.65<Hsi/d<1.3) a further sub-division is made, based on the relative width of the berm, Beq/Lpi,, where Beq is the equivalent berm width, defined halfway up the berm, Beq =Bb +hb/2tan.
This parameter map indicates that wave impacts will occur for three categories of conditions: vertical walls – large waves ( Hsi/d>0.35)
low mound breakwaters – large waves (0.65<Hsi/d<1.3)
high mound breakwaters - large waves (0.65<Hsi/d<1.3), moderate berms (0.14<Beq/Lpi <0.4)
