Cross-shore flow in waves breaking on a beach

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Cross-shore flow

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delft h/draulics

publication no. 395

Cross-shore flow in waves breaking

on a beach

Doctoral dissertation submitted at Delft University of Technology (reviewed by Prof. Dr. J.A. Battjes)

M.J.F. Stive

June 1988

CONTENTS ACKNOWLEDGEMENTS SAMENVATTING ABSTRACT page 1 INTRODUCTION 1.1 General 1 1. 2 Present contributions 3

2 INSTANTANEOUS SURFACE ELEVATIONS, VELOCITY FIELD AND PRESSURE FIELD

2.1 General discussion 7 2.2 Discussion of present contribution 11

3 RADIATION STRESS AND SET-UP

3.1 General discussion 15 3.2 Discussion of present contribution 16

4 ENERGY DISSIPATION AND WAVE HEIGHT DECAY

4.1 General discussion 19 4.2 Discussion of present contribution 21

5 UNDERTOW

5.1 General discussion 23 5.2 Discussion of present and other contributions 26

6 SUMMARY AND CONCLUSIONS 31

REFERENCES 35

-CONTENTS CONTINUED

APPENDIX A: PAPER REPRINTS

page A. 1 A scale comparison of waves breaking on a beach a. 7 A.2 Velocity and pressure field of spilling breakers a. 19 A. 3 A study of radiation stress and set-up in the nearshore region.a.43 A. 4 Energy dissipation in waves breaking on gentle slopes. a. 71 A. 5 Cross-shore mean flow in the surf zone a. 103

APPENDIX B: EXPERIMENTAL ARRANGEMENT AND PROCEDURES

B. 1 Introduction b. 5 B.2 Wave flume b.5 B. 3 Wave generation b. 5 B. 4 Wave conditions b. 6 B.5 Instrumentation b.7 B. 6 Measurement procedures b. 10 B.7 Data analysis b. 11 B. 8 References b.14 CURRICULUM VITAE

ACKNOWLEDGEMENTS

This thesis is based on a research study proposed by Prof. J.A. Battjes in the Coastal Research Programme of TOW-Kustonderzoek, funded by Rijkswaterstaat and jointly carried out by Rijkswaterstaat, Delft University of Technology and Delft Hydraulics. I am very grateful for the research opportunities offered by this programme and for the additional support by Delft Hydraulics to effectively realize this thesis.

My deep appreciation is for Prof. Battjes, who continues to teach me the basics of logic and science. During the study 1 was able to establish contacts with the international scientific world which I experienced as a necessary condition to conduct research. Of the many contacts established I would like to mention that with Prof. I.A. Svendsen, who

inspired me many times with his ideas and comments on Joint topics of interest. Nationally, I wish to thank my colleagues Prof. H.G. Wind and Dr. H.J. De Vriend for the many hours of fruitful discussions and resulting cooperations. Also, I wish to express my appreciation for the stimulating and helpful environment provided by all other colleagues at Delft Hydraulics.

A natural word of appreciation is due to my parents, who offered the basic opportunities for my education. The accomplishment of this thesis required family sacrifices, which were gracefully endured by Pauline and the children, Fionn, Daire and Niall.

-SAMENVATTING

In het laboratorium is een beknopte, maar gedetailleerde serie metingen verricht naar de vloeistofstroming in periodieke golven brekend op een talud. De vervolgens uitgevoerde analyse is gericht geweest op het begrijpen van de hydrodynamika van deze golven in de fase waarin het proces van breken als quasi-stationair kan worden beschouwd. Op flauw hellende stranden beslaat deze fase het grootste deel van de breedte van de brandingszone.

Waar de meeste voorgaande studies zich beperken tot de kenmerken van het wateroppervlak, omvat de onderhavige studie tevens de interne stromingskenmerken. Concreet is er een analyse gemaakt van het turbulente, het oscillerende en het golfgemiddelde stromingsveld in relatie tot integraal-eigenschappen als de golfspanning, de energie­ overdracht en de energie-dissipatie. De analyse van deze dynamische eigenschappen heeft zich beperkt tot de bewegingen dwars op de kust in de situatie van een in langsrichting uniforme kust en loodrecht invallende golven, zoals overeenkomend met de situatie in een laboratorium golfgoot.

De analyse van het oscillerende en het turbulente stromingsveld levert een kwantitatieve bevestiging van eerder gemaakte hypothesen over het zog-type karakter van de quasi-statlonalre, watersprongachtige brekende-golf beweging. Het hiermee verkregen inzicht in de momentane eigenschappen vormt de basis voor de analyse van de

i ntegraal-e i genschappen.

Uit een onderzoek naar de convectieve bijdragen aan de golfgeinduceerde impulsoverdracht wordt gevonden dat de gebruikelijke benaderingen bij het afleiden van het gemiddeld waterstandsverhang uit de veranderingen van golfspanning tot bevredigende resultaten leiden. Echter, het wordt aangetoond dat de gebruikelijke theorieën voor de voorspelling van het verlies aan golfspanning belangrijke beperkingen kennen. De analyse van het verlies van energie-overdracht, als gevolg van de door breken

geïnduceerde dlsslpatle, en de hieraan gerelateerde golfhoogte-afname bevestigt de zwakte van de nog veel gebruikte gelijkvormigheids-benadering én de kracht van de energiebalansgelijkvormigheids-benadering. Deze laatste benadering volgend, zijn enige suggesties gegeven voor de modellering van de afname van de golfhoogte en de gerelateerde waterstandsvariatie.

Een analyse van de onderstroom tenslotte bevestigt eerder gedane vooronderstellingen dat deze stroming wordt aangedreven door het verschil tussen de over de vertikaal niet-unlforme convectieve

impulsoverdracht en de over de vertikaal uniforme drukgradiënt als gevolg van de waterstandsopzet. Een voorspellend model voor de onderstroom wordt afgeleid, waarbij blijkt dat het belangrijk is rekening te houden met de sterke impulsafname boven het nivo van het golfdal.

-ABSTRACT

A limited, but detailed programme of measurements of the fluid flow in periodic waves breaking on a beach has been conducted in the laboratory. The subsequent analysis has been directed towards the understanding of the hydrodynamics of these waves in the quasi-steady breaking mode. It is in this mode that waves travel a substantial distance through the surf zone on a gently sloping beach.

While most previous studies on wave breaking are restricted to water surface characteristics, this study includes the internal flow characteristics. Specifically, an analysis is made of the turbulent, the oscillatory and the wave-mean flow field in relation with integral properties as radiation stress, energy flux and energy dissipation. The analysis of these dynamics is restricted to the cross-shore motions in the situation of alongshore uniformity of the beach and of the normally incident wave field, as encountered in a laboratory wave flume.

The analysis of the oscillatory and turbulent flow field yields quantitative confirmation of earlier hypotheses on the wake flow character of the quasi-steady, borelike breaking wave motion. The understanding of the instantaneous properties forms the basis for the analysis of the integral properties.

From an investigation of the convective contributions to the wave-induced momentum flux it is found that the common approximations made to derive the mean water level set-up from the loss of radiation stress lead to satisfactory results. However, common theories to predict the radiation stress decay are shown to have important shortcomings. The analysis of the loss of energy flux and associated wave height decay due to the breaking-induced dissipation confirms the weakness of the still common similarity approach and the strength of the energy balance method. Following the latter method, some modelling suggestions for the wave height decay and related set-up variation are made.

Finally, an analysis of the undertow confirms the proposition made earlier that this flow is driven by the imbalance between the vertically nonuniform convective momentum flux and the vertically uniform pressure gradient due to the set-up. A predictive model for the undertow is derived, which indicates that it is important to account for the strong wave momentum decay above the wave trough level.

-CHAPTER 1

INTRODUCTION

1.1 GENERAL

The breaking of wind generated waves in the surf zone of a natural beach is an often impressive spectacle. Seaward of the surf zone the waves have gently sloping, smooth surfaces, but as the water becomes shallower the waves develop a steep front face and finally an overturning crest leading to wave breaking. As the waves progress further into shallower water, the unsteady, initial wave breaking motion transforms into a more organized, quasi-steady breaking motion, during which the waves have turbulent white fronts.

It is in the region of wave breaking that important morphological processes shaping our coasts take place. There is only limited knowledge of the mechanisms behind these processes. From the viewpoint of the deductive research method detailed observations and analyses should lead to such knowledge and eventually to reliable process prediction models. Much of the early research on wave breaking has been devoted to the onset of the breaking process: many experiments were conducted to determine at which location and with which height and form a monochromatic wave train starts to break on a beach of constant slope (see Galvin, 1972, for a review). Theoretical studies of the initiation of wave breaking are less frequent. For the problem of finding where waves break on a gentle beach there are two long-known theoretical approaches (see the review by Peregrine, 1983): shallow water wave steepening and wave growth towards limiting steepness. However, both approaches are of limited validity in the context of waves breaking on beaches, so that they only provide theoretical starting points for studies of wave breaking.

The initial breaking process involves nearly without exception the overturning of the wave crest. In the last decade a series of papers has been published demonstrating numerical solutions for overturning waves

(see Peregrine, 1983, for a review). It turns out that for both deep water and finite water depth and irrespective of the physical mechanism used to initiate breaking all numerical overturning motions are similar. The importance of these studies lies in the fact that they provide measures of the accelerations and pressures of the highly unsteady flow in those fluid regions where experimental observations are very difficult.

As the jet formed by the overturning crest hits the wave's front face or preceding trough it seems at first sight that the resulting unsteady fluid motion is chaotic. However, careful observations (see Basco, 1985, for a review) indicate a surprising amount of order amongst the main flow features, notably the vortex motions. The overturning jet entraps a tube of air and sets off a primary vortex motion. The splash-up process of the Jet as it hits the water surface may attain several possible modes, but commonly results in a new jet generating a secondary vortex motion.

The unsteady process of the overturning wave crest and the associated vortex motions persist only over a horizontal distance of several times the initial breaking wave height. Soon, this initial mode of wave breaking transforms into the relatively well organized, quasi-steady breaking wave mode. At this stage the waves have turbulent white fronts of spray and bubbles and their heights diminish roughly in proportion to the mean water depth. This quasi-steady state is maintained until a bar crest is passed or -in case of continuously decreasing depth- until the shoreline is reached. Natural surf zones thus formed may have widths from tens of meters to several hundreds of meters depending on the bottom topography and the severeness of the wave conditions. The larger widths are usually found on sandy, dissipatlve-type beaches. Especially on these beaches the horizontal scale of the initial breaking process is small compared to the total surf zone width. The decaying waves then have a relatively long life travelling through the surf zone as

-quasi-steady, bore-like breaking waves.

The above discussion has focussed on the cross-shore breaking wave motions, which are also central in this study. For a review of surf zone dynamics in the more general horizontally two-dimensional case reference

is made to Battjes (1988).

1.2 PRESENT CONTRIBUTIONS

The present study was undertaken with the aim to contribute to an understanding of the cross-shore internal flow field in breaking waves, with an emphasis on the quasi-steady breaking wave mode. The empirical basis is formed by a series of laboratory experiments In a flume. This comprised the measurement of the fluid flow properties of periodic wave trains breaking on a plane beach. While most previous studies on wave breaking are restricted to water surface characteristics, the present study includes the internal flow characteristics.

The simplified situation of normally incident, periodic waves breaking on a plane slope was chosen for the following reasons. Firstly, the emphasis on the cross-shore quasi-steady breaking motions more or less Justifies the restriction to alongshore uniformity of the beach and normally incident waves, as encountered in a laboratory wave flume. Secondly, the periodicity of the waves allows the application of ensemble-averaging and other phase-coupling techniques to reveal the structure in space and time of the internal properties. Thirdly, on the plane slope the breaking strength of the bore-like waves remains nearly constant, so that the quasi-steady character can be well studied.

The laboratory experiments which initiated this study were conducted in October and November 1978. The limited, but detailed measurement programme involved measurements of the fluid flow properties of periodic waves breaking on a plane, 1 in 40 sloping beach for two different incident conditions. The data analysis was done in phases, all focussing on the understanding of the internal hydrodynamics of the waves in the

quasi-steady breaking mode. Specifically, analyses were made of the turbulent, the oscillatory and the wave-mean flow field in relation with integral properties as radiation stress, energy flux and energy dissipation. Also, the relevance of these small scale laboratory measurements compared to natural conditions was investigated. This was done by a scale comparison of the present measurement set with a similar set in a large scale wave flume. It appears that in the scale range investigated (wave heights of 0.1 m to 1.5 m) there are no significant deviations from the Froude scaling. This implies that dynamically no restrictions are to be expected. These results are described in a paper. Also, each of the above mentioned phases of the analysis was concluded by a paper, resulting in another four papers. Each of these papers contributes to a further understanding of some specific cross-shore features of breaking waves on a beach and together they contribute to a coherent picture of these features. In this thesis the separate contributions are presented collectively and the wider perspective is outlined in which the subjects of the papers must be placed.

The form chosen to present the results of the study in this thesis is as follows. Chapters 2 to 5 deal with the respective phases of analysis, as treated by the four papers. Each of these Chapters opens with a general discussion of the topic, describing the state-of-the-art and indicating the interrelation with the other topics, and ends with a description of the specific contributions of the paper. The reader may obtain a quick impression of the total subject of cross-shore features of quasi-steady breaking waves by reading these Chapters. For details of the present contributions reference is made to the paper reprints in Appendix A, where Paper 1 concerns the scale comparison and Papers 2 to 5 correspond to Chapters 2 to 5.

The analysis of the instantaneous surface elevations, velocity fields and pressure fields was tackled first and is discussed in Chapter 2. The idea behind this is that an understanding of the instantaneous quantities is required before a thorough analysis can be made of the time- and depth-integrated properties, notably radiation stresses and

-energy fluxes.

The first integral quantity addressed is the wave-induced momentum flux or radiation stress. This is done in Chapter 3. Specific attention is given to the common approximations in the mean horizontal momentum balance. Furthermore, the radiation stress decay and associated set-up are discussed.

The second group of integral properties addressed concerns the wave energy density, the energy flux and the dissipation. This is done in Chapter 4. The still common similarity approach is compared with different approaches to model the energy dissipation source term and associated wave height decay, which have been proposed since 1962.

Chapter 5 discusses the driving mechanisms behind the undertow or returnflow. The internal momentum balance is Investigated and a method is proposed to take account of the strong decay of wave momentum above wave trough level. This method is used in a predictive model for the undertow.

Finally, a summary and conclusions are given in Chapter 6.

Although each paper describes the relevant aspects of the experimental set-up, it is considered appropiate to present an overall, complete description of the experimental arrangement and procedures. This is done in Appendix B.

CHAPTER 2

INSTANTANEOUS SURFACE ELEVATIONS, VELOCITY FIELD AND PRESSURE FIELD

2.1 GENERAL DISCUSSION

The present Chapter addresses the instantaneous flow characteristics of the waves as they go through the quasi-steady breaking process. An understanding of these instantaneous features is required before time-and depth- integrated properties can be investigated. Here we may think of physical properties like mean densities and fluxes of momentum and energy which play an important role in the understanding and modelling of surfzone phenomena. These properties are discussed in the following Chapters.

Apart from the above reason, knowledge of the instantaneous variations of surface elevations and internal velocity field in breaking waves is of importance for investigations on the problem of sediment transport modelling in the surf zone or that of wave forces on structures in the surfzone.

Surface elevations and periodic velocity field

In the shoaling region the features of a periodic wave surface profile are those of a Stokes or a cnoidal wave, i.e. asymmetrical about the horizontal plane and nearly symmetrical about the vertical plane. After a violent transition in the outer breaking region the wave profile in the inner breaking region appears to transform into a more or less stable, vertically asymmetric wave profile, in which the front faces of the waves are steeper than the back faces.

Mathematically, the situation may be described as follows. At a fixed point the wave elevation, considered as a function of time f(t), is

-approximated as a Fourier series:

C = a.cos wt + a_ cos (2ut + 6n) + . ... + a cos (nwt + è ) (2.1) _{1 2 2 n n}

Stokes and cnoidal waves are characterized by a phase-lock between the harmonics such that <f> = 0. Both in the laboratory (Flick et al., 1981)

and in the field (Elgar and Guza, 1985, and Doering and Bowen, 1986) it is found that the relative phase of -for instance- the second harmonic in the quasi-steady breaking zone smoothly increases towards TT/2. This results in the characteristic sawtooth bore-shape found in this zone.

The periodic horizontal velocity field undergoes an identical phase change between first and second harmonic. An important consequence of this change of horizontal asymmetry into vertical asymmetry is found when considering the cross-shore wave-induced sediment transport. Adopting a formulation in which the sediment transport responds instantaneously to the near-bottom water velocity it is found (Stive, 1986) that the generally shoreward directed transport outside the surfzone disappears as the waves become sawtooth-shaped inside the surfzone.

The turbulent flow field

A qualitative understanding of the flow field in quasi-steady breaking waves or nearly periodic bores is a necessary step towards further analysis. An important contribution was made by Peregrine and Svendsen

(1978), who visualized the breaking wave flow by adding a low concentration of detergent to the water in their flume. They describe how a turbulent, aerated wedge is initiated at the toe of the turbulent front face of the bores. The turbulence decays as it spreads out in a region from the free surface to depths increasing with distance downstream from the toe until it eventually reaches the bottom or rather the bottom boundary layer. Longuet-Hlggins and Turner (1974) treat the typical region of recirculation on the bore front (or surface roller) as separated from the turbulent wedge region. Peregrine and Svendsen (1978) on the other hand hypothesize that the roller is not an isolated flow

region and that it merely acts as a trigger to initiate the turbulence. They suggest that initially the turbulent flow resembles a mixing layer and that further downstream the turbulent flow resembles a wake. Battjes and Sakai (1981) compare the flow with that in one half of a symmetric wake. In all these suggestions the bore flow is primarily considered as a free turbulence shear flow. It is interesting to note that Rouse

(19S9) suggested the same to hold for the hydraulic Jump, of which it is well known that its flow closely resembles that in a bore or a quasi-steady breaking wave.

It is only recently with the development of instruments like the laser-doppler velocimeter that the turbulence structure of the internal velocity field in breaking waves has been revealed experimentally. Reference is made to Stive (1980, see Paper 2 in Appendix A ) , Flick et al. (1981), Nadaoka and Kondoh (1982) and Sakai et al. (1982), who consider surf zones on rather gently sloping beaches, and Battjes and Sakai (1981), who consider a steady flow with a breaking surface induced by a hydrofoil. More recent examples -all originating from Japanese investigation programmes- are Sakai et al. (1984), Nadaoka (1986), Mlzuguchi (1986), Nadaoka et al. (1986) and Okayasu et al. (1986). These studies all confirm more or less the above suggestions of the structure of the turbulence shear flow. As pointed out in the review by Battjes (1988) the later studies improve our quantitative knowledge of the breaking induced motion, e.g. by the technique used by Nadaoka (1986) to separate the rotational from the lrrotational motion.

An interesting result of Nadaoka's study is the revelation of the large coherent turbulent flow structures in the form of vortices. This result seems to be in good accordance with Svendsen (1987), who recently made a more detailed analysis of the present data-set with respect to turbulent kinetic energy. The conclusions concern especially the depth variation and the time variation (over a wave period) of the total turbulent kinetic energy. His study confirms that the depth variation of the latter is relatively small outside the relatively thin bottom boundary layer suggesting strong vertical mixing by the large scale turbulent vortices.

-Theoretical bore models

Several theoretical treatments of the flow in bores or steady breaking waves have been attempted. Important examples are Longuet-Higgins

(1973), Johns (1980) and Madsen (1981). As brought forward by Madsen the approach of Longuet-Higgins is unrealistic since the shear flow is assumed to be restricted to the recirculation region near the free surface. Also, the approach of Johns is unrealistic in the sense that no turbulence is generated at the surface yielding nearly depth-uniform flow profiles only disturbed by the bottom boundary layer. A more realistic inclusion of phenomena is found in Madsen's approach, which adopts a vertically nonuniform velocity distribution with turbulence features , while maintaining the hydrostatic pressure assumption.

Since its stationarity makes the hydraulic jump more simple to handle, Madsen (1981, see also Madsen and Svendsen, 1983) decided to model the hydraulic Jump flow first. The essentially free turbulence shear flow is succesfully described by using vertically integrated conservation equations combined with chosen similarity functions. These equations form an extension of the nonlinear shallow water equations.

A description of a general bore of non-constant form may be based on the non-stationary versions of the vertically integrated conservation equations. This generalization of the steady flow model is presented in Svendsen and Madsen (1984), where it appears that it involves the solution to a hyperbolic system of four simultaneous partial differential equations. One important result is that the quasi-steady bore shape in the inner breaking region can be maintained due to the neutralization of the nonlinear steepening effect by the vertically nonuniform velocity distribution, which is accompanied by turbulence.

The apparently succesful description of several flow details of the single bore propagating on a beach (there are no measurements available for a verification) would yet need to be generalized for a system of periodic bores to be applicable in the present context.

2.2 DISCUSSION OF PRESENT CONTRIBUTION

The contribution discussed here is contained in Paper 2 (see Appendix A), the first in the series of papers that deal with these measurements and their analysis. The instantaneous values of surface elevations, velocities and pressures, and their simultaneous variations within a wave period are considered.

The preservation of wave shape

Analysis of the breaking wave surface elevations as presented in Paper 2 confirms the steadiness of the sawtooth bore-shape in the inner breaking region, at least as far as the wave's front face is concerned. Due to the virtual absence of turbulence at the back face there should be an anti-steepening effect, in accordance with the nonlinear shallow water wave theory for nondispersive waves. The measurements indeed indicate that the rearside of the bore slightly reduces its steepness as the bore propagates towards the shore.

Apart from this slight decrease of steepness of the bore's rearside, the characteristic form and also other flow features appear to be preserved; this indicates a locally controlled process which occurs in the present case because the water depth on the slope continues to decrease such that the bore remains turbulent and of approximately the same strength. Another external expression of this locally controlled process is the nearly constant ratio of wave height over mean water depth found to exist in the inner region for both the incident wave conditions.

The internal velocity field

In the analysis of the data the oscillatory velocity field was separated in a periodic part and a residue, designated as turbulence, by application of an ensemble-averaging method (see Appendix B).

-The asymmetry of the virtually constant surface shape, described above, is similarly found in the periodic, horizontal velocity field. The following characteristics of this velocity field in the inner breaking region for different relative surf zone positions are revealed in the paper. In the region Just in front of the bore and in the lower flow regions a vertically slowly varying flow pattern exists showing an asymmetry as can be expected from a non-breaking wave theory based on a similar asymmetric wave profile (this feature is actually used by Nadaoka, 1986, to determine the irrotational flow component). Just behind the bore front the shear between roller and undisturbed oscillatory flow disturbs the vertical slow variation, which then restores itself again to a vertically smooth variation until the next bore front arrives.

These flow characteristics indicate that the applicability of horizontal-bottom wave theories for symmetric waves are inadequate to predict the velocity field. When interested in the crest and trough values of the horizontal orbital velocity, it appears that when one chooses to use a wave theory for symmetric waves the linear theory prediction yields better results than the nonlinear, cnoidal prediction. The paper further gives detailed information on the simultaneous variation over time and depth of the periodic and turbulent velocity field. Based on this information Peregrine and Svendsen's model of the quasi-steady breaking wave could be interpreted and verified in a quantitative sense. The mixing layer resemblance of the turbulent wedge as it starts to spread out from the bore's toe and also the wake-flow

characteristics in the decay region "downstream" of the wave crest are j clearly confirmed.

I The internal pressure field

To the author's knowledge the present paper is sofar unique in its attempt to address the internal dynamic pressure field ir. breaking waves experimentally. The conclusions are as follows.

Under the wave troughs and at the rearside of the bore face the ensemble-mean dynamic pressure fluctuations are hydrostatic, but under the vreive fronts and crests they are non-hydrostatic. The deviation from the hydrostatic value Is strongest near the bottom and may amount up to 15%. As expected, the degree of non- hydrostaticlty can well be connected to the degree of streamline curvature, and in its turn again to the vertical periodic velocity fluctuations.

-CHAPTER 3

RADIATION STRESS AND SET-UP

3.1 GENERAL DISCUSSION

As the waves propagate through the surf zone both energy and momentum is lost through the turbulence generated by the breaking wave surface. These strongly related dynamic properties are treated from two points of view. This Chapter focusses on the gradient of the excess flux of momentum due to the waves (or radiation stress gradient) and the modelling of the wave set-up, while the next Chapter focusses on the energy loss and the modelling of the wave height decay.

The energy dissipation rate is of first order importance for the radiation stress gradient, while the latter is only of second order importance for the former. This is due to the fact that the main part of the radiation stress gradient is caused by the energy loss, whereas the radiation stress gradient is only responsible for a generally marginal correction to the mean water depth. So, a logical order of treatment would be the other way around. However, the underlying thought which was followed during the consecutive stages of the analysis of the present measurements (and also here) is that a basic investigation of the radiation stress theory for the set-up should rely on the measured radiation stress gradients rather than on a theory relating the decay of energy flux and wave height to the gradients of radiation stress. So, we leave the modelling of energy dissipation and wave height decay for later treatment.

In the natural, horizontally two-dimensional case, nonuniformity of the wave field results in radiation stress gradients causing net forces of which the dissipation-induced rotational part can drive a depth-mean current and a tilting or set-up of the mean water level, and of which the irrotational part can cause a depression or set-down of the mean water level as depth decreases towards the surf zone. However, in our

stationary, one-dimensional case of normal incidence and alongshore uniformity the depth-mean current is essentially zero. Then the driving forces are balanced by a slope of the mean water level alone. The theory behind the role of the radiation stress is generally acknowledged to be

introduced by Longuet-Higgins and Stewart (1960).

The theoretical results for wave set-down and set-up are empirically supported by experiments. With respect to laboratory observations using periodic waves reference is made to Bowen et al. (19B8), to Stive and Wind (1982, see Paper 3) and to Svendsen (1984a). With respect to laboratory observations using random waves reference is made to Battjes (1972), Battjes and Janssen (1978) and Battjes and Stive (1985).

3.2 DISCUSSION OF PRESENT CONTRIBUTION

Paper 3 (Appendix A) firstly investigates the theoretical approximations behind the cross-shore momentum balance commonly made to derive the set-down/set-up results. Secondly, predictions of set-down/set-up on the basis of predicted radiation stress variations are investigated.

Common approximations

One important approximation behind the cross-shore momentum balance equation concerns the mean dynamic pressure terms. These are defined as the difference between the actual mean pressure and the mean hydrostatic pressure at a level within the water column. The contributions to the pressure may be estimated from the vertical distributions of the velocity. It appears that the mean dynamic pressures -even for nonlinear waves on a gently sloping beach- may be approximated according to the second order approximation for locally uniform waves. Hence the mean momentum balance can be written (see Paper 3, page 11 for notation):

-where

S « = ƒ (p u2 - p w2) dz + | p g (f - f )2 (3.2)

-d

This result indicates that the depth variation of the radiation stress is due to the convective momentum contributions. The measurements indicate that substantial contributions are found in the region above the level of the wave troughs, indicating that it is important to include these contributions in a theoretical estimate. Furthermore, the measurements indicate that the depth variation of the convective momentum contributions below wave trough level are relatively weak, and that the contribution of the turbulent flow to the convective momentum flux is negliglible.

These quantitative insights in the vertical distribution, of the time-mean momentum flux also appear to be important in the modelling of the wave-induced undertow and associated turbulence, as discussed in Chapter 5.

Another common approximation is that of the neglect of the mean bottom shear stress in the mean momentum balance. The validity of this could be checked with the present dataset using the measurements to estimate the terms in equation (3.1). It appears that away from the initial breaking region the measurements Justify the common neglect of this shear stress,

i.e. its magnitude falls within the measurement accuracy. In the later stages of the study during the modelling of the undertow (see Chapter 5) the relatively small magnitude of the bottom shear stress compared to the remaining force terms was confirmed.

Radiation stress decay and set-up

With neglect of the mean bottom shear stress, the mean water level variations may straightforwardly be derived from the radiation stress variations. In the case of predictions the problem is then reduced to the selection of a suitable wave theory from which the radiation stress variations may be derived. This is discussed in the paper. One of the

conclusions is that -although nonlinearity is important in the initial breaking region- it is not sufficient to apply a nonlinear theory which uses the assumption of a locally horizontal bottom, since the location of the breakpoint and the properties of the asymmetric near-breaking waves cannot be predicted with sufficient accuracy. This affects for instance the prediction of the total wave set-up, which is critically dependent on the correct prediction of the total radiation stress magnitude at the point of initial breaking.

Finally it is noted that in these experiments -as in many before and since- it appears that in the region just after the point of initial breaking there is virtually no gradient in the mean water level variation, in contrast to the predictions on the basis of the present approach. As was first remarked by Battjes and Janssen (1978) this is most probably due to the time needed to convert organized kinetic and potential energy into small-scale, dissipative turbulent motions.

-CHAPTER 4

ENERGY DISSIPATION AND WAVE HEIGHT DECAY

4.1 GENERAL DISCUSSION

The primary characteristic of the surf zone is the energy dissipation in the turbulent flow Induced by the breaking wave surface. For locally generated windwaves on a sandy beach, the energy loss loss Is virtually complete. Models describing this process are essential to an adequate understanding of nearshore dynamics and sediment transport. After focussing on the momentum loss and associated wave set-up in the foregoing Chapter, this Chapter focusses on the energy loss due to breaking and associated wave height decay. Again, though, it should be remarked that the phenomena are closely related.

The similarity model

An analytic model still frequently used by both coastal scientists and engineers to describe the wave height decay in surf zones on gently sloping beaches Is that of a constant ratio between breaking wave height and mean water depth. A commonly adopted value Is 0.8 (e.g. Longuet-Higgins, 1970), closely corresponding to the theoretical limit for the height of a solitary wave, originally 0.78 (McCowan, 1894) and most recently 0.833 (Williams, 1981). Furthermore, this approximation may be shown to lead to a similarity for the set-up as well, viz. the set-up gradient is proportional to the beach slope.

The physical assumptions behind this similarity model are (1) the bottom slope Is so small that locally the waves behave as If the water depth is constant, and (2) inside the surf zone the breaking wave motion is depth-controlled such that It can be sufficiently described by a'theory for nonbreaking waves of limiting height. However, already for two decades data have been presented (e.g. Horikawa and Kuo, 1966; Nakamura et al., 1966 and many others since) which indicate that the similarity

model is inappropriate on mild slopes, especially in the region of initial wave breaking. It appears that the criterion of theoretical limiting height provides a reasonable prediction of the initial breaking wave height, but that it does not hold inside the surf zone.

Energy dissipation models

More realistic results may be expected from models which introduce a dissipation source term for wave breaking in the averaged wave energy equation. For steady-state waves normally incident to the shore we have

in which 3 is the mean wave energy flux per unit span and e is the mean energy dissipation rate per unit area due to breaking. The most physically appealing approach, introduced by LeMéhauté (1962) and frequently used since, is the approximation of breaking waves as propagating hydraulic jumps (or bores). Extensions of this approach are by Divoky et al. (1970), Hwang and Divoky (1970), Svendsen et al. (1978), Stive (1984, see paper 4 in Appendix A) and Svendsen (1984a). Although each specific translation of the dissipation in a hydraulic jump to that in a periodic, breaking wave may have been performed differently, order of magnitude arguments (see e.g. Battjes and Janssen, 1978) show that in all models the dissipation source term is proportional to the wave height squared.

Aside from the jump model a number of alternative models for the dissipation source term have been presented. Horikawa and Kuo (1966) estimate the internal energy dissipation from the turbulent velocity fluctuations, which are assumed to decay exponentially with distance from the breakpoint. Sawaragi and Iwata (1974) refined this approach by

introducing the Prandtl mixing length model to describe the turbulent velocity fluctuations. Mizuguchi (1980) applies the analytical solution for internal energy dissipation due to viscosity with the molecular kinematic viscosity replaced by a turbulence viscosity. Finally, Dally et al. (1984) present a somewhat intuitive model which assumes that the

-wave energy dissipation is proportional to the -wave energy flux in excess of a threshold value for which the waves are apparently stable. Based on order of magnitude arguments it appears that in the first and third of these models the dissipation goes like the wave height squared as well, while for the last model it goes like the 3/2 power. The approach of the second model is not transparent enough to reveal such a power relation.

In summary, virtually all models adopt a dissipation function which is approximately proportional to the wave height squared. In their further elaborations all the models introduce coefficients which are fitted empirically. An important feature, however, that is not represented in most models is that of disappearance of breaking as the depth stops to continue its decrease. For instance the Horikawa and Kuo (1966) data indicate that the wave height stabilizes when the depth becomes constant. Only the models of Mizuguchi (1980) and Dally et al. (1984) include this effect. The latter, more general approach intuitively proposes the energy dissipation to be proportional to the difference between the actual energy flux and the "stable" energy flux for which there is no dissipation. This approach is physically attractive and is in fact closely corresponding to the model for random, breaking waves proposed by Battjes and Janssen (1978). For practical applications these types of model are preferable over other models, because they are also applicable on non-monotonic beach profiles.

4.2 DISCUSSION OF PRESENT CONTRIBUTION

The present contribution (Paper 4 in Appendix A) pursues the verification of the idea that a quasi-steady breaking wave resembles not only superficially but also dynamically the hydraulic jump. To this end, firstly a detailed comparison is made between the flow fields in a hydraulic jump and a periodic bore of approximately the same strength. Secondly, the energy dissipation quantities in both cases are compared. The resemblance between quasi-steady breakers and weak hydraulic jumps is confirmed. With respect to the dissipation it is found that the

periodic version of the classical hydraulic jump formulation underestimates the dissipation rates by 30% to 50%. This discrepancy appears to be mainly associated with the turbulent flux of flow momentum at the crest section.

Because of the close physical resemblance the hydraulic jump dissipation model should be considered the physically most appealing approach to describe quasi-steady breaking wave dissipation. For the simple case of monotonically decreasing depth the present article proposes to use simplified forms of the time- and depth-averaged momentum and energy equations -with empirical corrections to account for the differences found- to solve for both the set-up and wave height decay. The generality of the model is larger than that of the earlier models in that it takes account of both set-up and the initial wave steepness.

-CHAPTER 5

UNDERTOW

5.1 GENERAL DISCUSSION

For the case of a steady, vertically two-dimensional motion of normally incident waves, it was discussed In Chapter 3 that the depth-integrated total mass and momentum equations yield a depth-mean zero flow. In essence a balance then exists between the depth-integrated wave-induced momentum flux (i.e. the radiation stress) and the wave set-up. However, locally (in the vertical) the mass and momentum fluxes need not be in balance. This can result in so-called secondary flows that are a correction on the depth-averaged flow.

The potential role of the local imbalance of momentum flux in the surf zone was first pointed out by Dyhr-Nielsen and Sorensen (1970). They hypothesized that the imbalance between the depth-varying momentum flux and the depth-uniform hydrostatic pressure due to the set-up grows so strong in the surf zone that it provides a driving force for a vertical circulation.

One of the causes in the local imbalance of the mass flux is well-known, i.e. the non-zero mean mass flux of the irrotational fluctuating wave motion above the wave trough level due to the in-phase relation between horizontal oscillatory velocity and surface elevation. A second cause is the presence of the surface roller in the quasi-steady bores. The first effect is theoretically known for periodic waves of constant form relative to an irrotational wave motion below trough level. This theoretical estimate combined with the mass flux in the roller as estimated by Svendsen (1984b) may be considered to give an acceptably accurate estimate of the total mass flux above wave trough level. For the vertically two-dimensional case we have the obvious condition that the mass flux over total depth is zero, so that the mass flux integrated over the layer below trough level is seaward to compensate for the

positive mass flux above this level due to the described effects. This relatively strong flow is called undertow.

Since 1980 several quantitative evaluations of the above concept have been presented in the literature. In contrast to the approach in the other chapters, the following section discusses the present contribution as an integral part of a discussion of the several contributions on undertow modelling which have appeared. The reason is that these papers including the present one play a part in a fast process of convergence towards a more general picture on the problem of undertow modelling. Discussion of the present paper in this context seems to be more relevant than an isolated discussion. Here it seems appropriate to describe the general outline of the approach , not specified in the elaboration of several details, that may be considered to yield the most consistent results based on the present knowledge.

To structure the discussion three fluid layers are distinguished following the approach of De Vriend and Stive (1987), i.e. (1) a surface layer above trough level, (2) the middle layer between trough level and bottom layer and (3) the oscillatory bottom boundary layer. The reasons for this distinction are twofold. Firstly, the relevant forces in the flow equations of these layers appear to be significantly different. Secondly, we are primarily interested in the vertical mean flow structure near the bottom and not necessarily interested in that of the upper region. In fact, we are satisfied with a theoretical estimate of the integral properties of the upper layer alone, providing boundary conditions for the bordering middle layer. So, we adopt the proposition by Stive and Wind (1986) to take account of the surface layer effects via an effective shear stress at the trough level, compensating for the momentum decay above it, and via the condition that the net undertow must compensate for the mass flux in the upper layer.

The mean flow velocity, U(z), in the middle layer may now be calculated from a local mean horizontal momentum balance equation -with the Reynolds stress modelled using the eddy viscosity concept- as given by Stive and Wind (1986), which does not principally differ from that given

-by Dally (1980) or Svendsen (1984b) (for notations see Paper 5, Appendix A):

d au(z) a t ~ 7 w

„ ; ,n ,,

aï

"t ~ET~

aï

' . '

S '

-

The assumptions basic to this equation are that the only deviation of the pressure distribution from hydrostatlcity is caused by the wave motion and that the mean advection of momentum is negligible. Further it is assumed that the vertical distribution of viscosity v is known. To solve for the mean flow U(z) we need the magnitude of the terms on the right hand side, and two appropriate boundary conditions, e.g. the above estimated shear stress condition on the layer's upper edge and a mass flux constraint describing the total mass flux through the layer. Then, by integrating the balance equation once we may solve for the other shear stress condition. Integrating once again yields the mean flow distribution. Let us first discuss the driving terms in equation (5.1).

When we adopt a wave theory, the magnitude of the right hand side terms of (5.1) may be derived theoretically if the variation of wave height and mean water level in wave propagation direction is known. The accepted procedure is to solve the mean depth-integrated momentum balance equation for the set-up and the mean depth-integrated energy balance equation for the wave heights adopting a theoretical wave description including breaking dissipation (e.g. as done in Paper 4 ) . However, if we create a mutually consistent set of mean momentum balance equations and appropriate boundary conditions for each of the layers, the mean momentum balance equation integrated over the total depth does not add new information to our system. This implies that the solution of the mean water level variation should follow from the set of the three, layer-integrated mean momentum balance equations mutually coupled by continuity conditions at the interfaces. So, at this stage the mean water level variation is still an unknown.

In the bottom boundary layer the mean horizontal momentum balance equation has an additional term compared to the one for the middle

region, i.e. the vertical gradient of the wave-induced crossproduct (uw) yielding:

k

' »«> f ' - k "

- '

' * k >s < + k " » »

-

A further relevant difference between the balance equations of bottom layer and middle layer is that the turbulent viscosity in the bottom boundary layer is significantly less than that in the middle layer (see Svendsen, 1985).

Analogously to the middle layer, solution of the mean flow distribution in this bottom layer requires double integration of the above equation and two boundary conditions. A logical choice for reasons of continuity at the interface would be to require continuity in velocity and mean shear stresses, as suggested by Svendsen (1985), and evaluated by Svendsen et al. (1987). Also, we have available the no-slip condition at the bed. Whatever choice we decide to make we always have an extra boundary condition known, since we only need two conditions. This may be used to solve for an unknown in the system, viz. the variation of mean water level. This is the logical consequence of the fact that a consistent set of equations was created which provides as much information (and more) as e.g. the depth-integrated momentum balance equation.

5.2 DISCUSSION OF PRESENT AND OTHER CONTRIBUTIONS

As described the several contributions on undertow modelling which have appeared in the literature since 1980 indicate a fast process of convergence towards a more general picture on the problem of undertow modelling. Discussion of the present paper in this context seems to be more relevant than an isolated discussion.

-The approach of Dally (1980). Börecki (1982) and Daily & Dean (1984)

Both Dally (1980) and Börecki (1982) use the linear, theoretical wave description to derive the momentum flux distribution over depth,

including the region above wave trough level. As a first condition they adopt the constraint of total mass flux in the lower layer compensating for the mass flux above trough level. As the second condition a bottom velocity condition was taken, for which Dally and Börecki use the no-slip condition and Dally & Dean (1984) the condition Ub = i/8 c (H/h)2, which is 2/3 of the long-wave limit of

Longuet-Higgins' (1953) conduction solution result. Further, Dally and Dally & Dean make no distinction between the bottom and the middle layer and use the middle layer equation (5.1) down to the bottom. Börecki does use the boundary layer equation (5.2), but assumes that the turbulent viscosity in both the middle and the bottom layer are of the same order of magnitude.

Due to the adoption of a wave theory which provides oscillatory wave velocities and free surface variations in the region above wave trough level the approach of Dally & Dean actually implies (see the discussion of Svendsen, 1984b, by Dally & Dean, 1986) that a shear stress boundary condition is specified at the mean water level, similar to the condition at trough level described earlier. Because of this Implicit introduction of a third condition the mean gradient in water level can be solved for explicitly in terms of the mass flux below wave trough level, the bottom boundary velocity and the gradient in energy density.

The strength of these basically similar approaches is that given the wave height variation an Internally consistent theoretical formulation is given to derive the undertow. A physically weak point in each approach however is the treatment of the bottom boundary layer. Dally neglects the important force term due to the time mean cross-product of (üw) in the boundary layer. Börecki includes this term but assumes that the turbulent viscosities in both layers are of the same order, which yields a physically'unrealistic result (see below). Dally & Dean take the boundary velocity from Longuet-Higgins' (1953) conduction solution,

which implies that the important pressure gradient term is neglected. The approach of Svendsen (1984b) and Hansen & Svendsen (1984)

Both Svendsen (1984b) and Hansen & Svendsen (1984) use measurements of the driving terms in equation (5.1) to describe the vertical distribution of horizontal momentum flux over the middle layer. As conditions they use the constraint of compensating mass flux in the middle layer and a bottom velocity condition given by Ub = 3/i6 c (H/h)2, which is the long-wave limit of Longuet-Higgins'

(1953) conduction solution result. The mass flux above wave trough level is evaluated more accurately than with linear theory, by including the actual wave shape and the breaking wave roller effect.

The further analysis in Hansen & Svendsen already showed (see also Svendsen, 1985) that the above bottom velocity (which is in the direction of wave propagation) disagrees with the observed large seaward directed velocity close to the bottom, which is caused by the set-up gradient.

The present approach (Stive & Wind. 1986. Paper 5 in Appendix A)

Similar to Svendsen (1984b) in Paper 5 also measurements of the force terms in equation (5.1) are used to describe the vertical distribution of momentum flux over the middle layer. As boundary conditions the constraint of compensating mass flux in the middle layer and a shear stress at wave trough level are applied. The mass flux above wave trough level is evaluated with linear theory, including the breaking wave roller effect.

The strength of this approach is that the effect of the flow in the region above wave trough level on the undertow is expressed through a shear stress at wave trough level. This stress appears to be quite significant and may well be estimated to first order by setting up an upper-layer-integrated momentum balance in which the dominant driving terms are rather easily evaluated on the basis of the present knowledge

-of wave decay due to breaking. The result for the vertical pr-ofile -of the undertow velocity including the values near the bottom is realistic. However, to arrive at an accurate result near the bottom, so that for instance the bottom shear stress is predicted, requires in essence the inclusion of the .solution of the flow in the bottom boundary layer.

The approach of Svendsen et al. (1987) and Stive & De Vriend (1987)

From the analysis of the magnitudes of the terms in the momentum balance equation for the boundary layer (5.2), Svendsen (1985) concludes that the viscosity in the bottom layer should be far smaller than that in the middle layer to result in negative mean flow velocities just above the bottom boundary layer, as observed. This result used in combination with a patching technique requiring continuity at the layer's interfaces

(Svendsen, 1985) should yield a realistic variation near the bottom of the mean flow and accompanying bottom shear stress. This idea is evaluated in Svendsen et al. (1987).

The strength of the latter approach is that specifically the flow near the bottom is well analyzed, and this seems after all the region of most

interest for sediment transport applications. A weak point in the evaluation is that measurements of the force terms in equation (5.1) were used. In other words, the solution of the flow in the upper layer and therewith the upper boundary condition for the middle layer Is disregarded. This was resolved by Stive & De Vriend (1987); they combined the patching technique between bottom and middle layer with the upper layer representation by the shear stress at trough level, compensating for the momentum decay above it. The result Is a consistent set of momentum balance equations for each layer, such that the depth-integrated equation adds no new information to the system; the mean water level variation is part of the solution.

CHAPTER 6

SUMMARY AND CONCLUSIONS

A limited, but detailed set of measurements of the fluid flow properties of periodic, breaking waves was conducted in a laboratory flume. The programme comprised the breaking of periodic wave trains on a plane beach of a 1 in 40 slope for two different incident conditions. The subsequent analysis has been directed towards the understanding of the internal hydrodynamics of these waves in the quasi-steady breaking mode. This mode is attained soon after the strongly unsteady process of

initial breaking is passed.

The simplified situation of normally incident, periodic waves breaking on a plane slope was chosen for the following reasons. Firstly, the emphasis on the cross-shore quasi-steady breaking motions more or less justifies the choice to consider the more easily studied situation of alongshore uniformity of the beach and of a normally incident wave field, as encountered in a laboratory wave flume. Secondly, the periodicity of the waves allows the application of ensemble-averaging and other phase-coupling techniques to reveal the structure in space and time of the internal properties. Thirdly, on the plane slope the breaking strength of the bore-like waves remains nearly constant, so that the quasi-steady character can be well studied.

The relevance of these small scale, laboratory measurements to large scale, natural conditions was investigated by a scale comparison of the present measurement set with a similar set in a large scale wave flume.

It appeared that in the scale range investigated (wave heights of 0.1 m to 1.5 m) there are no significant deviations from the Froude scaling. The quantities considered are wave heights, set-up and vertical profiles of maximum seaward, maximum shoreward and time-mean horizontal velocities. This result implies that for instance the visually observed difference in air entrainment has no significant influence dynamically.

-Most of the previous studies on wave breaking are restricted to water surface characteristics. This study includes the internal flow characteristics. Specifically, an analysis is made of the turbulent, the oscillatory and the wave-mean flow field as such and of their relation with integral properties as radiation stress, energy flux and energy dissipation. Knowledge of the instantaneous flow properties In breaking waves is of importance for investigations focussing on phenomena at timescales smaller than the wave period, such as the problem of sediment transport modelling in the surfzone or that of wave forces on structures in the surfzone. An understanding of dynamic properties integrated over the timescale of a wave period and/or integrated over depth, like radiation stresses and energy dissipation, is important because these properties play an important role in the understanding and modelling of surfzone phenomena like set-up, wave height decay and mean currents. The analysis of the instantaneous, internal oscillatory and turbulent flow field has yielded quantitative confirmation of earlier hypotheses on the wake flow character of the quasi-steady, borelike breaking wave motion. Also, for practical applications some suggestions are given with respect to the -limited- applicability óf horizontal-bottom wave theories and with respect to the -moderate- deviation of the dynamic pressure field from its hydrostatic value. The understanding of the instantaneous properties forms a basis for the analysis of the integral properties.

The first integral property investigated is the radiation stress or rather its decay as the waves propagate towards the beach. It is found that the common approximations made to derive the mean water level set-up from the loss of radiation stress lead to satisfactory results. Specifically, confirmation was found of the common mean dynamic pressure approximations and, with some reserve for the outer breaking region, of the negligibility of the time mean bottom shear stress. However, common theories to predict the radiation stress decay are shown to have important shortcomings. Nonlinear, horizontal-bottom theories predict the radiation stress magnitude at the breakpoint reasonably well, but

fail to predict the breakpoint position, which is important for accurate predictions of for instance the mean water level set-up. In addition to the latter shortcoming, linear theory also overestimates the radiation stress magnitude substantially, which makes its predictions even less accurate.

The second integral property investigated is the energy flux gradient and the related energy dissipation. It is shown that the flow field of quasi-steady breaking waves closely resembles that of hydraulic jumps. This supports the use of a -slightly corrected- hydraulic jump formulation for the breaking wave energy dissipation. The analysis of the loss of energy flux and associated wave height decay due to the breaking-induced dissipation confirms the weakness of the still common similarity approach and the strength of the energy balance method; for the latter, modelling suggestions are made.

Finally, an analysis of the undertow confirms that it is driven by the imbalance between the vertically nonuniform convective momentum flux and the vertically uniform pressure gradient due to the set-up, as had been proposed earlier. In addition to the considerations on the internal force balance, the boundary conditions are discussed which are needed to find a theoretical formulation for the undertow. It is argued that contrary to the assumptions in earlier studies it is not primarily the influence of the near-bottom oscillatory boundary layer that imposes a condition on the mean flow in the fluid interior. Instead it appears that the conditions imposed by the strong, spatial decay of the wave motion -expressed by a shear stress at the trough level- dominates the flow. These findings are used to suggest a predictive model for the undertow.

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