Vertical variation of the flow across the surf zone

Vertical variation of the f low across the surf zone

Erik Damgaard Christensen a,*, Dirk-Jan Walstra b, Narumon Emerat c

aDHI Water and Environment (DHI), Agern Alle 11, DK-2979 Hørsholm, DenmarkbDelft Hydraulics, PO Box 177, 2600 MH Delft, The Netherlands

cFluid Dynamics Group, Deparment of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, Scotland, UK

Abstract

This paper reviews recent advances that have been made in the numerical modelling and measurement techniques of the surf

zone. The review is restricted by the assumption of a long and uniform coastline case. Therefore, the frame of reference is the

2DV case, but including tree-dimensional processes important for this topic. During the last two decades, new measurement

techniques have become available (e.g. Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV)), which have

successfully been applied in numerous laboratory experiments. These methods have enabled detailed measurements of, for

instance, the production, transport and dissipation of turbulence and have made a valuable contribution to our understanding of

the processes in the surf zone. The first models that were developed were primarily based on assumptions directly derived from

such observations. Since the development of the first numerical models in the mid-eighties, much research effort has been put

into trying to improve these wave-averaged models because they can be applied at relatively low computational cost. The

improved understanding of the surf-zone processes has also led to the development of more advanced intrawave models such as

the Boussinesq-based models as well as the use of Navier–Stokes solvers. These new modelling techniques give a detailed

description of the processes in the surf zone. D 2002 Elsevier Science B.V. All rights reserved.

Keywords: Surf zone; Breaking waves; Laboratory experiments; Numerical models; Flow structures

1. Introduction

The surf zone has been the subject for the work of

many researchers as well as for coastal engineers, etc.

The study of the vertical flow structures in the surf

zone is an important and interesting subset of the

research area and there are plenty of reasons for that.

First of all, it is an area where the hydrodynamics is

very complex and, therefore, a natural challenge to any

researcher in hydrodynamics or fluid mechanics. Sec-

ondly, the violent transformation of smooth waves to

steep breaking waves and the generation of turbulent

flow structures beneath the waves is a scenic experi-

ence to anyone who visits a beach. However, one of

the most important reasons to do so is that the knowl-

edge of the mechanisms and the relation to, for in-

stance, sediment transport, is of outmost importance to

coastal engineering.

Earlier, Peregrine (1983) gave a review of the break-

ing waves on beaches. This was later followed by

another review by Battjes (1988). These reviews des-

cribed the dynamics of the breaking waves on beaches

in a rather broad manner. The paper by Peregrine

(1983) emphasises the approach to wave breaking,

the wave overturning and to a minor degree the post-

0378-3839/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.

PII: S0378 -3839 (02 )00033 -9

* Corresponding author. Tel.: +45-451-69168; fax: +45-451-

69292.

E-mail address: [email protected] (E.D. Christensen).

www.elsevier.com/locate/coastaleng

Coastal Engineering 45 (2002) 169–198

breaking mechanics. Battjes (1988) gave a more

detailed description of the post-breaking mechanics

and discussed this in relation to different parameterised

models of breaking waves.

This paper gives a review of the vertical variation

of the flow structures in the surf zone. It can be seen

as an extension to the earlier well-known reviews

since it is narrower in its scope and it includes the

latest research. Due to development of more advanced

measuring technology and more advanced numerical

models that make use of the faster computers now

available, the hydrodynamics can be studied in much

more detail than earlier. The study has been carried

out from 1997 to 2000 as a part of the Surf And

Swash zone MEchanics (SASME) project, supported

by the Commission of the European Communities,

Directorate General for Science Research and Devel-

opment. It will focus on the research within the

SASME group as well as the other work related to

the research topics dealt with by the SASME group.

The way a wave breaks on a slope has for many

years been categorised as spilling, plunging or surging.

From observations, the different breaker types can be

distinguished, see Galvin (1968) for a classification.

Peregrine (1983) suggested three possible splash-up

mechanisms at the plunge point: rebounding; some

rebounding and some penetration; and penetration and

pushing up of another jet. The spilling breaker might

start with a small jet as observed, for instance, by

Jansen (1986), but soon, the front develops into a

moving bore approaching the shoreline, while the

plunging breaker starts with a more violent jet.

Breaking waves can be characterised as the trans-

formation of well-ordered wave energy into turbulent

energy, which finally dissipates into heat. Wave break-

ing can be defined as the transformation of particle

motion from irrotational to rotational motion and tur-

bulence, Basco (1985). In the horizontal direction, the

surf zone can be divided into four zones as sketched in

Fig. 1. In the shoaling/deformation zone, the waves get

steeper and the wave profile deforms into asymmetric

shape, which is pitched forward. When the waves

reach the breaking point in the outer zone, the breaking

starts by an overturning jet, whose strength depends on

the type of breaker, i.e. spilling or plunging. In the

inner zone, the front of the wave can be characterised

by a moving bore until it reaches the swash zone,

where the toe of the wave moves forth and back.

As illustrated in Fig. 1, it is the outer and inner zones

that are of interest to this paper. In this area, the vertical

can by divided into the upper, middle, and lower

regions. In the upper region, a violent transformation

of the wave top into turbulent flow structures takes

place. These flow structures break down and interact

with the flow in the middle region to produce much

turbulence. The offshore-directed flow under the

trough of and the incoming wave motion interacts

together with the exchange of turbulence and momen-

tum to generate the undertow. The undertow interacts

with the bed that can be characterised as a wave–

current boundary layer, interacting with the externally

generated turbulence, cf. Fig. 2.

First, the experimentally based research is reviewed

followed by the Models of the Cross-Shore Processes

section. This section is divided into a wave-averaged

part and an intrawave part, where the intrawave part is

attached the greatest importance, since this area has

undergone a comprehensive progress within the last 5

Fig. 1. The surf zone dived into the shoaling/deformation zone, outer breaking zone, inner breaking zone, swash zone.

Fig. 2. A sketch of the production, diffusion and dissipation of

turbulent kinetic energy. The figure is based on Svendsen (1987).

E.D. Christensen et al. / Coastal Engineering 45 (2002) 169–198170

years. Finally, the conclusions and future perspectives

are summarised.

2. Experimental research

For decades, important features of surf-zone break-

ing waves have been studied qualitatively (Peregrine,

1983; Basco, 1985). Experimental studies providing

quantitative details have also been carried out. Earlier

experiments focused on the study of variation of free-

surface profiles, wave height, and wave celerity by

using a simple wave-gauging technique (see, e.g.

Svendsen and Hansen, 1976; Svendsen et al., 1978).

Although these measured values are of importance for

the determination of the wave-averaged quantities,

they do not provide information on the internal flow

of breaking waves. Most of the recent experimental

investigations have employed modern measurement

methods based on optical techniques, namely Laser

Doppler Velocimetry (LDV); sometimes referred to as

Laser Doppler Anemometry, LDA) and Particle Image

Velocimetry (PIV). Detailed discussion of these tech-

niques and their applications to wave kinematics can

be seen in the review paper by Greated and Emarat (in

press). Both methods are nonintrusive techniques and

can be used to measure the instantaneous Eulerian

velocity. However, the first technique provides a high-

frequency signal of the velocity at a fixed point in the

flow, whereas the latter gives a full-field map of the

velocity. Several other flow visualisations have also

been used when investigations of Lagrangian behav-

iour of the water particles were needed. This section

reviews experimental research, of the last decade or so,

whose results have supported or added to the knowl-

edge of physical dynamics and turbulence generation

in the three different vertical regions of surf-zone

breaking waves discussed earlier in the Introduction.

Most of the experiments were performed by using the

above optical measuring techniques. Finally, recent

experiments, compared with some of the numerical

models explained in Section 3, are discussed at the

end.

2.1. Initial breaking and upper region

It is known that breaking will occur when the

horizontal velocity of the water at the wave crest

exceeds the wave speed. The recent experimental work

done by Chang and Liu (1999) has supported this

condition. They made an attempt to measure kine-

matics in the breaking crest by using PIV technique.

Chang and Liu found that the fluid particle velocity at

the tip of the overturning jet reached 1.68 times the

phase velocity. They also found that the overturning jet

enters the horizontal front face with an acceleration of

1.1 times that of gravity, see Fig. 3. Moreover, Chang

and Liu (1999) also conducted another PIV measure-

ment and calculated the vertical vorticity (whose rota-

tional axis is perpendicular to the tank bottom). The

random vertical vortices were found after the wave had

broken, supporting the appearance of obliquely

descending eddies observed by Nadaoka et al. (1989)

as sketched in Fig. 4.

There have been experimental investigations per-

formed in the aerated region in the upper part of the

breaking waves. These include Jansen (1986) and,

more recently, Lin and Hwung (1992). Both studies

employed flow visualisation technique with the use of

ultraviolet light to illuminate fluorescent tracer par-

ticles, which were fed into the air-bubble region. Their

photographic and video images revealed a well-known

Fig. 3. An example of the initial breaking in a plunging breaker showing the strong horizontal eddy (the photograph was taken at the University

of Edinburgh).

E.D. Christensen et al. / Coastal Engineering 45 (2002) 169–198 171

sequence of jet-splash motions in both plunging and

spilling breakers. A number of jet-splash cycles and

vortex formations were found in both breaker types

and seemed to be reproduced in each individual wave

(in a regular wave train). In Jansen’s (1986) results,

smooth trajectories of the particles inside the jet-splash

motions suggested so-called coherent motions in the

flow. In Lin and Hwung’s (1992) results, the main

mechanism that drives the motion in the bubble zone

was found to be the vortex system that was generated

from the jet-splash cycles. Vortex stretching was also

found to occur due to the interaction between the jets,

the vortices, and the effect of the rising buoyant

bubbles. These effects are perhaps the main causes

of the development of the obliquely descending eddies

observed by Nadaoka et al. (1988a,b) and Nadaoka et

al. (1989). The vortices finally collapse under the

influence of the flow under the trough and the sub-

sequent vortex formations decrease rapidly in strength

due to dissipation and transfer of wave energy to the

turbulent motion in the air-bubble field. In the field

experiments, the eddies were found to involve large

amount of air bubbles which enhanced the upwelling

of sediment. Due to scale effects, the amount of

entrained air is relatively larger in large waves (field

experiments) compared to small waves (laboratory

experiments).

In general, the effect of air entrainment is a research

area that has only gained little attention, perhaps due to

the complexity of the matter. However, Chanson and

Jaw-Fang (1997) investigated the entrainment of air in

plunging breakers. The breaking process was inves-

tigated with a high-speed video camera. The study

indicated that the rate of energy dissipation increases

with the bubble penetration depth and with the char-

acteristic length of the plunging jet shear flow. The

calculations of the energy dissipation were relatively

crude, which only emphases the difficulties dealing

with the region of mixed air and water in breaking

waves.

2.2. Middle region

More quantitative studies seem to have been carried

out in the middle region of surf-zone breaking waves

than in the upper region. This is due to the fact that both

of the main optical measuring techniques, LDV and

PIV, suffer from signal drop-out within the aerated

region which mostly appears in the upper surf zone

(although, in some cases, the air bubbles can spread

downward into the middle region). The measurements

could then only be performed under the trough level

(for LDV) and outside the air-bubble field (for PIV).

An important aspect of the results obtained from these

optical measuring techniques is the derivation of the

mean and turbulent components. For LDV experi-

ments, a time-averaging procedure would normally

be applied to the high-frequency records of the instan-

taneous velocity in order to obtain the mean velocity

and turbulent fluctuating velocity components. On the

other hand, if the experiments could be well repeated,

an ensemble-averaging procedure might be used for

Fig. 4. An illustration of horizontal and obliquely descending eddies as observed in Nadaoka et al. (1989).


both LDVand PIV data. This averaging technique was

mostly employed in the experiments discussed below.

An equivalent averaging technique to the ensemble

method, a so-called phase average, is also used to se-

parate the mean and turbulent components. In the

phase-average method, the measurement is made at

the same phase of each successive periodic wave and it

is based on the assumption that the wave period

remains constant from wave to wave.

Being an earlier optical measuring technique, LDV

has been used widely for the surf-zone breaking-wave

investigations. Stive (1980) was among the first, if not

the first, to apply the LDV technique to measure the

internal flow field under periodic breaking waves.

From the measurements, the velocities as well as the

turbulence intensities were found. Some of the results

are presented in Fig. 9 of the next section, in compar-

ison with model results. Nadaoka and Kondoh (1982)

presented LDV measurements for the internal velocity

field within the surf zone. They adopted the time-

averaging technique to obtain the ‘Eulerian’ mean

velocity and found a clear division of the velocity

field between the transition zone and the inner surf

zone. The results from these mean velocities also re-

vealed the onshore movement above the trough level,

while the offshore movement dominates the region

below the trough level. They concluded that the small-

scale turbulence and entrained air bubbles inside the

surf zone are associated with the large-scale eddies.

Nadaoka et al. (1989) used the LDV technique to

study the structures of turbulent flow field of spilling

breakers in the surf zone. Horizontal large-scale eddies

and obliquely descending eddies were observed around

the wave crest and behind the wave front, respectively.

The measured velocity was used to calculate the phase-

averaged velocity, Reynolds stress, vorticity, and tur-

bulent fluctuations. The generation of Reynolds stress

was found to be correlated with the large-scale eddies.

Pedersen et al. (1993) made an attempt to measure

the turbulent length scale in the surf zone, which is of

importance for the determination of eddy viscosity

and turbulent diffusion. The vertical fluctuations were

determined from the LDA data and were correlated to

calculate the length scale. They found that with the

vanishing of the bottom influence, the length scale

remains constant, around 0.2–0.3 times the water

depth. This range for the length scale is also suggested

by Svendsen (1987) and has been used to estimate

turbulent diffusion effectively (Ting and Kirby, 1995,

1996; Chang and Liu, 1999).

One of the recent comprehensive studies on turbu-

lence transport under surf-zone breaking waves, using

LDA technique, was that of Ting and Kirby (1994,

1995, 1996). The turbulence transport was studied in

detail by determining each term in the k-equation.

Interesting results were reported, especially on the

different mechanisms between different types of break-

ing waves. The correlation between the flow and the

turbulence level was used to qualitatively to explain

cross-shore sediment transport mechanisms. From sim-

ple reasoning, the sediment transport was found to be

offshore under spilling breakers, but onshore under

plunging breakers. Fig. 5 illustrates this phenomenon.

Under the spilling breaker, the turbulence level is

almost constant during a wave cycle. Consequently,

any sediment brought into suspension goes in the same

direction as the undertow, i.e. in the offshore direction.

Under a plunging breaker, a large plume of turbulence

is generated over the whole water column immediately

after the wave has broken. Outside this area, the

turbulence levels are generally low. Due to the high

turbulence level in the plume, large amounts of sedi-

ment are brought into suspension. Since the area of

high turbulence travels with the wave top towards the

shoreline, the sediment transport will be onshore. This

mechanism is well known to coastal engineers. How-

ever, important mechanisms, such as bed load and

phase lag effects, have to be considered in more de-

tailed studies of the sediment transport.

Although Laser Doppler Velocimetry technique

provides high-frequency records of the velocity and

has been used comprehensively as discussed above, its

disadvantages are that the simultaneous spatial distri-

butions of mean and turbulent quantities as well as

coherent structures could not be obtained. This can be

overcome by the new technique of particle image

velocimetry. However, due to the limited range in the

image resolutions and storage, PIV has never been used

to investigate the turbulence inside the surf zone until

recently. Haydon et al. (1996) and Chang and Liu

(1996) were among the researchers who initially

studied turbulence generated by breaking waves using

PIV. Haydon et al. (1996), however, used a rather

different method to separate mean and turbulent com-

ponents. They used a so-called ‘‘local-average’’ tech-

nique in which the velocity at each point is averaged,


using the nearby velocities, and the difference between

the instantaneous velocity and the averaged value is

determined as turbulence. Chang and Liu (1996), on

the other hand, used the ensemble-averaged approach

to calculate mean and turbulent velocities. Although

the breaking waves they studied were in deep water,

they proved that the PIV technique could be used as a

powerful tool for turbulent flow measurement. PIV has

been used effectively in their further study reported

later in Chang and Liu (1999).

A similar attempt was also made by Emarat and

Greated (1999), where PIV experiments on breaking

waves in the surf zone have been conducted as part of

the SASME project. Trains of periodic waves were

generated which then broke as spilling and plunging

breakers on a sloping beach. Phase-averaged techni-

que was used to determine the mean velocity and

turbulent intensity. Fig. 6 shows an example of the

PIV results obtained from these experiments. The

measurement in this figure was made just after break-

Fig. 6. Phase-averaged velocity vector map (a) and turbulent intensity distribution (b) of a plunging breaker, Edinburgh University.

Fig. 5. Illustration of the distribution of turbulence in the spilling and plunging breaker, respectively.


ing of the plunging breaker. The mean velocity vector

map is shown in Fig. 6a and the turbulent intensity

distributions in Fig. 6b. The aerated region in the

surface roller, where measurements could not be

made, is indicated. This shows that PIV results can

give an immediate picture of the mean flow and

turbulent structure under the plunging breaker. High

turbulent intensity can be seen around the surface

roller and the intensity gradually decreases towards

the bottom. This supports that the production of the

turbulence mainly occurs in the upper region.

2.3. Lower region

It is known that turbulence is also generated by the

shear stress, though with small intensity, in the oscil-

latory wave boundary layer along the bed (Fredsøe

and Deigaard, 1992). It has been argued that the

externally generated wave-breaking turbulence and

the shear-generated turbulence from the wave boun-

dary layer are statistically independent. However, little

information is available on the variation of the turbu-

lence and the effect of the interaction between the

turbulence generated from the two sources.

Early analysis on the interaction between the under-

tow and the boundary layer flow on a beach was made

by Svendsen et al. (1987). At that time, not many

measurements inside the boundary layer were avail-

able. Eddy viscosity was estimated theoretically and

compared to that calculated from the measured wave

parameters and velocity profiles above the boundary

layer. The accurate prediction of the undertow veloc-

ities close to the bottom was obtained under the

assumption that eddy viscosity in the boundary layer

is smaller than that produced by breaking waves.

Svendsen et al. (1987) suggested that the eddy viscosity

in the boundary layer varies continuously from zero, at

the bottom, to the same value as that in the undertow, at

a height of a boundary layer thickness.

Few experiments have been done to investigate the

effect of turbulence generated by surf-zone wave

breaking on the bottom boundary layer. These experi-

ments can be seen, for example, in the papers by

Deigaard et al. (1991b) and Cox et al. (1996), who

have investigated the bottom shear stress inside the

surf zone. Deigaard et al. (1991b) used a hot film

probe to directly measure the temporal variation of the

bottom shear stress under breaking waves on a smooth

bed of a constant slope. The measurements were also

made offshore from the breaking point for compar-

ison. The important results they obtained have shown

that although the temporal bed shear stress inside the

surf zone remains periodic, it is affected significantly

by the breaker-generated turbulence.

Instead of measuring the bottom shear stress

directly, as done by Deigaard et al. (1991b), Cox et

al. (1996) used the technique of LDV to measure

velocity profiles under spilling breakers on a rough

sloping fixed bed. The horizontal velocities, measured

well inside the boundary layer, were fitted to logarith-

mic profiles and were then used to estimate the shear

velocity, bottom roughness and, hence, the bottom

shear stress. The results indicate that, in the surf zone,

the temporal variations of the bottom shear stress can

be well estimated using the measured (phase-averaged)

horizontal velocity above the bottom boundary layer.

Recently, Kozakiewicz et al. (1998) investigated

the effect of externally generated turbulence on the

boundary layer. However, the external turbulence was

not generated from breaking waves, but from a series

of grids placed at a certain distance above the bottom

of an oscillating water tunnel. It has to be noted also

that the boundary layer in this experiment was the

turbulent oscillatory boundary layer over a horizontal

bed. Nevertheless, their study provides more knowl-

edge of the vertical effect of turbulence generated

from different sources. Kozakiewicz et al. (1998)

presented the LDA measurements for the velocity

profiles from the grid to the bottom including the

boundary layer. In further equipment, a hot film probe

was also used to measure the bed shear stress. The

results were compared to those obtained from the

normal wave boundary layer without grids. The earlier

transition to turbulence and the increase in the bed

shear stress indicate the penetration of externally

generated turbulence down into the bed boundary

layer. Kozakiewicz et al. (1998) also investigated the

apparent roughness of the wave boundary layer and

found that, under externally generated turbulence, the

roughness increases with the increasing turbulent

intensity.

2.4. Summary of experimental research

The experimental research has to a large degree been

driven by the technological developments. Only within


the last two decades, the technology has evolved to a

level where extensive quantitative studies could be

performed. Earlier, the quantitative work was mainly

limitedtowaveheightandwaveceleritymea surements

by using simple wave-gauging technique. Even though

these measured values are of importance for the deter-

mination of the wave-averaged quantities, they do not

provide information on the internal flow of breaking

waves. Much research has used visualisation techni-

ques, giving good qualitative descriptions of the break-

ing process. Especially, the repeating of resplashing has

been reported in detail. Several researchers have also

found indications of coherent flow structures as, for

instance, the obliquely descending eddies. However,

these features are not reported as solid as the resplash-

ing mechanism. This might be due to the three dimen-

sionality of the phenomenon, which is more difficult to

study by means of visualisation techniques. The real

breakthrough for quantitative studies was the use of

optical techniques, i.e. LDA, LDV, and PIV. During the

last decade, this has led to comprehensive studies of the

flow region below trough level. Here, the undertow,

turbulence, and other statistical parameters can be

studied in great detail. However, the use of point-

measuring techniques does not give any direct infor-

mation about coherent flow structures in the surf zone.

PIV techniques have given some estimates of, for

instance, vorticity in a cross-shore section and, to some

extent, results of the flow field above trough level. The

present optical techniques fail to give reliable predic-

tions of the flow conditions in then aerated part of the

broken wave. Since fully three-dimensional techniques

are not available today, it has been very difficult to

obtain certainty about three-dimensional flow struc-

tures such as the obliquely descending eddies. There-

fore, some researches still question the existence of

these flow structures.

3. Models of cross-shore processes

3.1. Wave-averaged models

Wave-averagedmodels describe the hydrodynamics

in quantities that are averaged over the short-wave

period. Compared to other model types that are treated

in this paper, wave-averaged models are relatively

simple, robust, and computer-efficient. At present,

wave-averaged models are probably the most widely

applied model types in the mathematical modelling of

coastal hydrodynamics and morphology around the

world. In wave-averaged models, waves and currents

are usually modelled separately, the interaction of the

models is actually based on data coupling, where the

output of the one is used as input for the other (im-

plicitly assuming that there is no direct dynamic corre-

lation between the two). The wave–current interaction

may be modelled iteratively by executing both models

repetitively in a loop.

In this section, we focus on wave-averaged flow

models which describe the circulation current in the

vertical plane in the surf zone (1DV or 2DV). We

assume that the required wave parameters are available

as input. For a review of the available wave models, the

reader is referred to Hamm et al. (1993). Furthermore,

we assume a zero depth-averaged net flux so that the

wave-induced mass transport above wave trough level

is compensated by a seaward flow below this level.

The effects that (breaking) waves have on the wave-

averaged velocity profiles (undertow) can be summar-

ised as:

� The radiation stress gradient,� Wave breaking,� Wave-induced mass flux,� Increase in the turbulent production and the

eddy viscosity,� Wave-induced streaming in the boundary layer.

All these items are discussed below by giving a

global description of the underlying physical processes

and the scientific progress that has been made in trying

to understand and model them. Next, a chronological

description of the 2DV-undertow models that have

been developed the last 20 years. Finally, a brief

description is given of area models in which these

wave terms have also been included (in an often sim-

plified way).

3.1.1. The radiation stress gradient

Since the pioneering work done by Longuet-Hig-

gins and Stewart (1960, 1962, 1963, 1964), the depth-

integrated force balance in the surf zone of a uniform

coast has been well understood. This total force ba-

lance has been investigated by many researchers and

since become a basic approach in modelling surf-zone


dynamics. Essentially, the cross-shore gradient of the

horizontal momentum flux, associated with the wave

motion and the radiation stress, is balanced by a slope

in the mean water level (wave set-up). In a descriptive

paper by Dyhr-Nielsen and Sorensen (1970), it was

pointed out that the vertical distribution of the balanc-

ing forces is not similar and that a shear stress must be

introduced to obtain a force balance over the vertical.

Dally and Dean (1984) were the first to introduce this

concept in a numerical flow model. They derived a

second order mean flow model based on an expression

for the vertical distribution of the radiation stress. In

fact, all theories that have been developed since to

model the undertow (e.g. Svendsen, 1984b; Stive and

Wind, 1986; De Vriend and Stive, 1987) use this

imbalance as the principle driving force. Since the de-

velopment of these models, many researchers have

investigated the vertical shear stress distribution of

wave-induced currents (see, e.g. Fig. 7). It was found

that the correlation between the horizontal (u) and

vertical (w) components of the oscillatory wave mo-

tion can represent a significant contribution to vertical

shear stress distribution. Deigaard and Fredsøe (1989)

derived solutions for uw under wave decay for both

breaking and nonbreaking conditions. The overbar

indicate time averaging over the wave period A first

order approximation of bottom slope effects on the uw

term was derived in De Vriend and Kitou (1990).

Rivero and Arcilla (1993) showed, based on Peregrine

(1976), that there is a nonlinear vorticity transfer from

the current to the wave motion, which induces a

contribution to uw.

Fig. 7. Spatial variation of the local radiation stresses and set-up in the shoaling region (a) and breaking region (b), after Stive and Wind (1986).


3.1.2. Wave breaking

The breaking of waves is the most visible and

dramatic way in which waves dissipate energy. If we

consider the surface roller as a mass of water, which is

riding the underlying wave and stays stationary rela-

tively to the wave crest, it is straightforward to derive

the shear force between the wave and the surface roller

(Svendsen, 1984a). Roelvink and Stive (1989) used

the roller concept to explain the lag effects already

observed by Battjes and Janssen (1978). Roelvink and

Stive (1989) assumed that the organised wave energy

that is dissipated in the breaking process first is

converted into turbulent kinetic energy of organised

vortexes before being dissipated into small-scale tur-

bulent motion. Nairn et al. (1990) combined the roller

concept from Svendsen (1984a) and the result of

Deigaard and Fredsøe (1989) that the dissipation is

due to the work done by the shear stress, due to the

roller acting on the underlying water, to extend the

wave energy balance equation, as proposed by Battjes

and Janssen (1978), with a roller equation of similar

form. Note that Roelvink and Stive (1989) and Nairn et

al. (1990) follow a similar approach. Roelvink and

Stive (1989), however, assumed that this transfer from

organised wave energy to large-scale vortexes was

isotropic, whereas Nairn et al. (1990) adopted the

roller concept. In the appendix of Stive and De Vriend

(1994), Deigaard proposed a modification to the roller

equation by including the effects of momentum

exchange between the roller and the underlying wave.

It is now generally recognised that the energy, which is

dissipated in the breaking process, is first converted

into turbulent kinetic energy of organised vortexes

(defined as the roller) before being dissipated into

small-scale turbulent motion.

3.1.3. Wave-induced mass flux

In cross-shore direction, the wave-averaged total

flux is zero, if no other effects than the waves are

accounted for. However, from trough level to the

instantaneous water surface there is a non-zero flux

associated with the mass transport in the waves. This

wave-induced mass flux has to be compensated by a

net mean return flow below trough level. The mass

flux in the surface layer consists of two separate

contributions: one part due to the progressive character

of the waves (Phillips, 1977) and the other due to the

surface rollers in breaking waves (Svendsen, 1984a).

Srinivas and Dean (1996) use measurements of Na-

daoka (1986) to describe the total mass flux by fac-

torising the mass flux derived from linear wave theory.

Recently, Groeneweg and Klopman (1998) developed

a generalized Lagrangian mean (GLM) model in which

the mass flux is included in a natural way. This method

is a hybrid Eulerian–Lagrangian approach that also

enables the inclusion of a vertically nonuniform mass

flux distribution (Walstra et al., 2000), see also Fig. 8.

3.1.4. Wave-generated turbulence

Waves generate turbulence in the wave boundary

layer near the bottom due to the oscillating wave

motion and, if waves are breaking, in the surface layer.

The direct influence of these turbulent energy sources

is an increase of the kinetic turbulent energy, which

results in an increased eddy viscosity. In the wave-

averaged modelling approach, an energy cascade is

implicitly assumed in which there is a local equili-

brium between the production and dissipation of

energy. In this concept, the dissipated organised wave

energy in the top layer by the roller is taken as a source

of the turbulent kinetic energy below trough level.

Battjes (1975) was among the first to suggest such

approach. He related the depth-averaged turbulent

kinetic energy k to the energy dissipated by the waves

as: k =(D/q)1/3, where q is the density of water. Pere-

grine and Svendsen (1978) proposed a qualitative

model for the flow in turbulent bores. They related

the eddy viscosity in breaking waves using the sim-

ilarity between the flow field in breaking waves and

wake flow. Stive and Wind (1986) adopted this sim-

ilarity and estimated the (depth-averaged) eddy vis-

cosity as tt= 10� 2ch, where c is the wave celerity and

h the local water depth. Roelvink and Reniers (1994)

extended the quasi-3D modelling approach of De

Vriend and Stive (1987) by assuming a parabolic

distribution of the eddy viscosity and used Battjes’

(1975) expression to include the effects of wave

breaking on the eddy viscosity. A logical next step is

to apply k� l or k� e models to determine the vertical

distribution of the eddy viscosity. In Deigaard et al.

(1986), an intrawave flow model was developed,

which uses this concept; the vertical distribution of

the source term due to breaking waves is assumed to be

parabolic from the instantaneous water level to wave

trough. Walstra et al. (2000) include a source term in

the (wave-averaged) k- and e-equations, assuming a


linear distribution having its maximum at MWL and

reducing to zero at half a wave height below MWL. A

typical example from Deigaard et al. (1991a) of the

vertical distribution of the turbulent kinetic energy, k,

is given in Fig. 9.

Near the bed, the presence of waves also causes

additional turbulent kinetic energy production. In

breaking waves, however, the turbulence generated in

the bottom boundary layer is relatively limited and its

influence, in general, does not extend outside the

boundary layer. In nonbreaking waves, it is an impor-

tant mechanism which can significantly influence the

velocities over the entire water column. Often the

presence of waves is modelled by increasing the rough-

ness of the bed, often referred to as the apparent rough-

ness; an extensive review can be found in Soulsby

(1997). By introducing the wave energy dissipation in

the bottom boundary layer as a source term in the k- and

e-equations, an overall consistent approach is obtained(see, e.g. Walstra et al., 2000).

In Garcez Faria et al. (2000), the effects of various

eddy viscosity distributions (constant, linear and para-

bolic) on the velocity profile were investigated. Sur-

prisingly, the constant viscosity gives the best

agreement with the measurements. However, it has to

be noted that only two locations were considered for

three cases, where intense breaking occurred (in

regions where breaking is less intense, an accurate

description of turbulent quantities does become im-

portant). However, it does confirm earlier research

in which similar conclusions were drawn. This con-

clusion must, however, be seen in combination with the

used type of flow model, the measurements that

were taken and the wave model that was applied to

obtain the forcing. This is, however, illustrative for

the research efforts in the field. Due to the relative large

number of free parameters, most of which we cannot

predict accurately, it is difficult to point to a specific

part of the model without taking other inaccuracies

into account.

3.1.5. Wave-induced streaming in the boundary layer

The steady streaming generated in the bottom boun-

dary layer under nonbreaking waves in water of con-

stant depth was first derived by Longuet-Higgins

(1953). It can be regarded as a steady motion, which

also originates from the oscillatory wave motion uw

near the bed. These stresses can be modelled as an

additional shear stress at the top of the bottom boundary

layer based on the wave energy dissipation in the

bottom boundary layer, Fredsøe and Deigaard (1992).

3.1.6. Undertow models

Neglecting the advective terms and assuming the

wave-averaged eddy viscosity approximation for the

Reynolds’ stresses, the local momentum equation used

Fig. 8. Comparison of model (solid) with measurements of Klopman (1998) (symbols). Left: uniform mass flux no streaming. Middle: uniform

mass flux with streaming included. Right: nonuniform mass flux with streaming included (Walstra et al., 2000).


to solve the wave-averaged Eulerian flow becomes

(Stive, 1988):

@

@zqtt

@UðzÞ@z

¼ @

@xqðu2 � w2Þ þ @

@xqg1 þ @

@zquw

ð1Þ

where q is the fluid density, mt the turbulent viscosity, uand w are the horizontal and vertical components of the

wave orbital velocity, 1 is the mean surface elevation,

u2 and w2 the wave-averaged square of the wave and

turbulence–mean velocity components and, finally,

U(z) is the wave-averaged velocity over the depth.

Fig. 9. Measured and calculated vertical distribution of time-averaged turbulent kinetic energy (Deigaard et al., 1991a). The measurements are

from Stive (1980).


This equation is assumed to be valid throughout the

vertical domain.

To structure the discussion, three fluid layers are

distinguished following the approach of De Vriend

and Stive (1987), i.e. a surface layer above wave

trough level, a middle layer between the wave trough

level and the bottom layer, and the oscillatory bottom

layer.

Dally and Dean (1984) use linear wave theory to

derive the vertical momentum flux distribution in

which the region above wave trough is included. As

a first condition they adopt the constraint of total mass

flux in the two lower layers must compensate for the

flux above wave trough level. As a second condition,

a bottom velocity was taken: Ub = 1/8c(H/h)2, which

is 2/3 of long wave limit of Longuet-Higgins (1953)

conduction solution, Dally (1980) uses a no-slip

condition. Furthermore, Dally and Dean (1984) make

no distinction between the middle and bottom layer

and, also, the uw term in Eq. (1) is neglected. Due to

the adoption of a wave theory, which provides oscil-

latory wave velocities and free-surface variations in

the region above the wave trough, the Dally and Dean

(1984) approach actually implies the specification of a

shear stress (Dally and Dean, 1986).

Both Svendsen (1984b) and Hansen and Svendsen

(1984) use measurements of the driving terms in Eq.

(1) to describe the vertical distribution of vertical flux

of horizontal momentum. The boundary condition

types are similar to Dally and Dean (1984). The

bottom velocity condition is taken as Ub = 3/16c(H/

h)2, which is the long wave limit of Longuet-Higgins

(1953) conduction solution. By including the wave

shape and roller effect for the determination of the

mass flux, a more accurate prediction is obtained.

Similar to Svendsen (1984b), Stive and Wind

(1986) also use measurements to describe the vertical

distribution of momentum flux over the middle layer.

As boundary conditions, a compensating mass flux in

the middle layer is prescribed and, in contrast to above

discussed approaches, a shear stress at the wave trough

level is applied.

Based on comparison with experiments, in which

negative (i.e. offshore) mean flow velocities were

found, Svendsen (1985) concludes that the viscosity

in the bottom layer should be relatively small com-

pared to the viscosity in the middle layer. He used this

result, in combination with a coupling of the interface

of the bottom and middle layer, to obtain realistic

velocities near the bottom. This technique was inves-

tigated further and compared with measurements in

Svendsen et al. (1987). In Svendsen et al. (1987), still,

measurements were used to obtain the driving forces

of Eq. (1), which implies that the solution of the flow

in the upper layer is disregarded. Essentially, there is

no boundary condition at the interface between the top

and middle layer.

In the models discussed above, it was assumed that

the contribution of uw in the middle layer could be

neglected. However, it was found that this contribu-

tion to the shear stress can be significant in cases of

sloping bottom and dissipative waves. Deigaard and

Fredsøe (1989) found that the uw term varies linearly

with the depth when waves dissipate energy; they

derived expressions for breaking waves and for non-

breaking waves with energy dissipation in the boun-

dary layer. De Vriend and Kitou (1990) derived a first

order estimate of uw in case of a sloping bottom.

Nielsen and You (1996) found that the term uw grows

very strongly in the presence of currents. The term is,

in general, not well understood; however, it can be

modelled by numerical 2DV models.

In Garcez Faria et al. (2000), the Svendsen et al.

(1987) and Stive and Wind (1986) models are com-

pared. It was found that overall agreement with meas-

urements was comparable for both models. The

Svendsen (1984a,b) model gives better agreement in

the lower half of the water column, but its applicability

is hampered by the fact that it essentially requires

velocity measurements near the bed to derive boun-

dary conditions.

The findings of the cross-shore flow has been in-

corporated in full three-dimensional mean flow models

that now are routinely applied in coastal areas. The ex-

tension of the existing (tidal) flow models is to include

wave-forcing terms which are primarily based on the

same terms as used by the 2DV models. De Vriend and

Stive (1987) were among the first to present a (quasi)

3D model for these more general cases. Another

approach, which is quasi-3D, is based on Deigaard

(1993). This model has been applied for calculation of

the long-shore as well as the cross-shore sediment

transport in the surf zone, (Elfrink et al., 1996, 2000).

Walstra et al. (1994) described the wave-driven cross-

shore currents in a 2DV version of a 3D hydrostatic

flow model.


In Pechon and Teisson (1994), a fully 3D flow

model is used to model the vertical circulation; how-

ever, it has to be mentioned that they model the effects

of wave-induced return flow separately by a Boussi-

nesq-type model. Walstra et al. (1994) add an extra

term to the continuity equation, which implies it was

allowed to have vertical velocities through the surface

layer. However, this approach will not work in 3D as

the mass flux is treated as a scalar and not a vector.

Based on the GLM model developed by Groeneweg

and Klopman (1998), Walstra et al. (2000) describe the

flow in wave-averaged Lagrangian velocities (total

flux) in which the mass flux is included in a natural

way. In Elias et al. (2000), the model is validated using

comprehensive field data from the Egmond site.

3.2. Boussinesq models

The use of Boussinesq models to model the prop-

agation of waves in shallow water has been extensively

used in the past. For instance, the wave disturbances in

harbours can be modelled by the Boussinesq equations.

To be able to simulate the set-up and wave height in the

surf zone, additional terms have to be added. The in-

clusion of these terms will be discussed in this section.

The Boussinesq equations include nonlinearity as

well as frequency dispersion. Basically, frequency

dispersion is introduced in the flow equations by taking

into account the effect that vertical accelerations (or the

curvature of the streamlines) have on the pressure

distribution. The set-up and use of the Boussinesq

equations for nonbreaking waves have gained much

interest during the last three decades. The papers by

Peregrine (1967), Svendsen (1974), McCowan (1985),

and Madsen et al. (1991) are just a few of the research

works made within this field. Since the main topic of

this review is the flow structures in the surf zone, the

Boussinesq theories outside the surf zone will not be

discussed in any detail.

3.2.1. The Boussinesq equations

A description of the Boussinesq equations in the

surf zone was given in Schaffer et al. (1993). It consists

of the depth-integrated equations of continuity and

horizontal momentum expressed as:

@S

@tþ @P

@x¼ 0 ð2Þ

@P

@tþ @M

@xþgd

@S

@xþ w ¼ 0

z z

impulse slope of the surface

ð3Þ

where S is the surface elevation, P is the depth-in-

tegrated velocity, M is the momentum flux, and w is

the dispersive pressure term.

Schaffer et al. (1993) assumed a uniform velocity

throughout the depth, giving

M ¼ p2

dand w ¼ h2

6Uxxt �

h

2ðnhÞxxt: ð4Þ

In these equations, the effect of wave breaking has

not yet been accounted for. The different ways are

described in the following:

3.2.2. Modelling wave breaking by the viscosity and

roller concepts

In, for instance, Karambas and Koutitas (1992), the

energy dissipation due to wave breaking was repre-

sented by including an eddy viscosity term in the

depth-averaged velocity components, disregarding the

vertical variation of the velocity profile that is far from

uniform in the breaking zone. Generally, the local

energy dissipation will depend on the vertical gra-

dients in the velocity profile rather than horizontal

gradients (see, e.g. Madsen, 1981).

Schaffer et al. (1993) included the concept of sur-

face rollers described by Deigaard (1989). In the

surface roller concept, the wave breaking is assumed

to start if the slope of the local water surface exceeds a

certain value. The roller was considered a passive bulk

of water isolated from the rest of the wave motion. The

geometry of the surface roller was defined as illus-

trated in Fig. 10. The geometry was found by Deigaard

(1989) by considering the energy loss in a hydraulic

jump following a model by Engelund (1981).

The influence of the roller is taken into account

through an additional convective momentum term, see

Fig. 11. The profile is in principle the one suggested

by Svendsen (1984a).

The nonuniform velocity profile changes the depth-

integrated velocity and momentum to be: P= u0d+

(c� u0)dM = u02d+(c2� u0

2)d. The excess momentum

due to the nonuniform velocity distribution is defined


as: R =M� (P2/d ), which was found to be equivalent

to R = d(c�P/d )2 (1� d/d ) � 1, in Schaffer et al.

(1993). The wave celerity, c, given the velocity of the

passive bulk of water, was set to 1.3 times the linear

shallow water wave celerity. The thickness, d, of thepassive bulk of water travelling with the front of the

wave has to be determined. This was done by incorpo-

rating the heuristic geometrical approach of Deigaard

(1989). Assuming that, for a nonbreaking wave, the

local gradient of the wave front has a maximum tanu,they simply took the wave to be breaking when this

gradient was exceeded. The use of a hydraulic jump

analogy showed that: uc 10j. With a few modifica-

tions, which include spatial temporal variation of the

angle, u, this approach was used to estimate the thick-

ness, d, and that closed the system of equations.

It was found that the model was able to represent

many surf-zone phenomena such as initiation and

cessation of wave breaking over a bar; the fluctuating

break point for irregular waves; and, especially, the

shoreward shift of the set-up relative to the point of

wave breaking for regular waves.

Rahka et al. (1997) applied a wave model similar to

the one presented in Schaffer et al. (1993) with minor

modifications. The aim of their work was to inves-

tigate sediment transport over a cross-shore profile.

Therefore, a hydrodynamic model was included for the

wave boundary layer and the undertow. The velocity

profile in the hydrodynamic model was not in any way

coupled to the wave model. From the wave calcula-

tions, an estimate of the instantaneous production of

turbulence was used, based on the bore analogy of

Deigaard (1989). Using the shear stress contribution as

explained in Deigaard (1993), the vertical variation of

the undertow was estimated. The model was found to

predict the undertow and the sediment transport rates

reasonably well.

The thesis research by Quinn (1995) has reported

experiments on laboratory surf-zone breaking waves.

He used PIV techniques to study the hydrodynamics of

wave breaking on mildly sloping beaches as well as on

profiled beaches. Boussinesq and Serre models were

used to predict the depth-averaged horizontal velocity

and compared with PIV data. Additionally, the PIV

velocity data was also used to calculate the radiation

stress, mean energy flux and mean momentum flux.

The predictions agreed quite well with the measured

velocities, especially near the bed, but there were

Fig. 10. The geometry of the surface roller, two values of / are defined. /B, /0 is the angle at breaking and a smaller terminal value,

respectively. From Schaffer et al. (1993).

Fig. 11. Illustrating of the assumed velocity profile of a breaking wave with a surface roller, From Schaffer et al. (1993).


fewer comparisons in the crest region. This discrep-

ancy is due to the known problem of the Boussinesq

models in which they overestimate the surface curva-

ture and, hence, they cannot well predict the velocity

near the crest of a breaker, see Svendsen and Putrevu

(1996).

In Bayram and Larson (2000), a comparative study

between the model proposed by Watanabe and Dibaj-

nia (1988) (W–D model) and the roller model by

Schaffer et al. (1993) is presented. Sato et al. (1992)

rewrote the model by Watanabe and Dibajnia (1988) to

an eddy-viscosity model. Data on water surface eleva-

tion, collected during two extensive field experiments

on surf-zone hydrodynamics and sediment transport,

carried out at the U.S. Army Field Research Facility

FRF (Duck, NC), were employed to evaluate the wave

transformation model. The wave measurements during

the two experiments known as DUCK85 and SUPER-

DUCK were discussed in Ebersole and Hughes (1987)

and Ebersole and Hughes (1988), respectively. The

models were used with the default parameters as pre-

sented in the original papers.

Themaximumwater surface elevations found by the

models were somewhat underestimated prior to break-

ing. The arrival of the individual bores was predicted

reasonably well. It was found that the W–D model

gave a smoother water surface elevation compared to

the roller model, implying that wave disintegration is

less pronounced.

In summary, the agreement between the simulation

results and the measurements was acceptable, both for

the roller model and W–D model, considering that the

model parameters were not adjusted in order to

objectively assess the predictive capability of the

models. Nonlinear shoaling was underestimated in

the calculations due to the inherent limitations of the

Boussinesq equations. The steepening of the wave

profile was reasonably well simulated by both models.

Considering that the two models of wave breaking

were developed for spilling breakers and that plunging

breakers prevailed in the experiments, the simulation

results must be regarded as satisfactory. Overall, the

roller model predicted the statistical parameters

slightly better than the W–D model.

3.2.3. The vorticity model

The velocity profile used in the roller concept gave

a better prediction of the set-up and wave height

distribution over the surf zone compared to earlier

attempts though the description of the velocity profile

in itself was a little crude. In order to improve the

description of the velocity profile, a third approach of

the breaking description was introduced by Svendsen

et al. (1996). In this approach, the vertical distribution

of the vorticity was obtained by solving the transport

equation for vorticity. This method was further devel-

oped in Veeramony and Svendsen (2000).

Applying the stream function in the surf zone, it was

possible to split up the velocity in a potential compo-

nent of the velocity (up) and a rotational component of

the velocity (ur).The Boussinesq equations finally

contain terms that depend on the vorticity in the surf

zone. Therefore, the vorticity has to be estimated. A

vorticity transport equation can be derived from the

Reynolds-averagedNavier–Stokes equations (RANS),

where the convective terms are neglected. Hereby, the

transport equation for the vorticity looks as:

@x@t

¼ j@2x@r2

þ Oðd,l2Þ ð5Þ

where j is assumed to be constant and r is a trans-

formed ordinate.

The boundary conditions for the vorticity at the

surface is found by a curve fit to experiments in three

hydraulic jumps, see Bakunin (1995) and Svendsen et

al. (2000). At the bottom, the vorticity is set equal to

zero, which is consistent with the assumption of a

free-slip condition. This actually closes the system of

equations.

The point where the breaking is initiated for each

wave has to be specified explicitly. Wave breaking was

assumed to start when the maximum slope in front of

the wave crest exceeds 20j (see Madsen et al., 1997).

The transition period, during which the roller devel-

ops, lasts for 1/10th of the wave period. Drawing on

the results from the measurements in hydraulic jumps,

the fully developed roller extends from the wave crest

to the location in front of the wave where the gradient

is zero.

Comparison with the experimental data of wave

heights and set-up by Hansen and Svendsen (1979)

was as good as for the roller model. The comparison

with horizontal velocities and the undertow showed

also good agreement with the experimental results in

Cox et al. (1995) as illustrated in Fig. 12. Furthermore,

the radiation stress calculated from the model is in


qualitative agreement with the results obtained by

previous researchers.

3.2.4. Summary of Boussinesq modelling

As it has been shown in this section, the inclusion

of the vertical variation of the velocity profile has

improved the Boussinesq modelling in the surf zone.

The eddy viscosity approach was found to give a

reasonable prediction of the set-up and the wave

heights in the surf zone, but could not capture the

onshore shift in the set-up compared to the breaking

point. The use of the roller model incorporated the

vertical variation in a rather crude manner, but gave a

more precise prediction of, for instance, the set-up and

the wave heights in the surf zone.

When the sediment transport is simulated, this

assumption could not stand alone, but two models

were added in Rahka et al. (1997), one for the wave

boundary layer and one for the undertow in the surf

zone. This combination of models gave very realistic

profiles of the undertow as well as realistic sediment

transport rates. The formulation in Rahka et al. (1997)

is not limited to the two-dimensional case, which is

the strength of this model.

The latest model presented in Veeramony and

Svendsen (2000) modifies the velocity profile in the

roller from being a passive bulk of water that travels

with the wave front to a profile taking into account the

velocity variation in the roller. To do this, an equation

for the transport of vorticity was included as well as

boundary conditions estimated from experimental

results. The wave boundary layer at the bottom was

not accounted for in this model and the eddy viscosity

was set to a constant; consequently, the model has to

be further developed before it can be used to simulate,

for instance, the sediment transport. Outside the

boundary layer, the velocity profiles and undertow

were estimated with good accuracy.

In all the above Boussinesq models, an explicit

breaking criterion has to be added. In the viscosity

models, the breaking criterion is often related to the

depth and slope of the bottom, while in the roller and

vorticity approach, it is related to the slope of the

surface only.

Fig. 12. A comparison of the undertow profiles from the model results by Veeramony and Svendsen (2000) (—) and the data (.) of Cox et al.,

1995. The ordinate zV= z+ h is zero at the bottom. From Veeramony and Svendsen (2000).


3.3. Navier–Stokes solvers

The most direct way to investigate the flow in the

surf zone numerically is to solve the basic equations

for Newtonian fluids called the Navier–Stokes equa-

tions. In many other areas than coastal hydrodynam-

ics, such as aerodynamics and fluid mechanics, the

method has gained much attention during the last few

decades evolving into a whole discipline called Com-

putational Fluid Dynamics (CFD). The method is

capable of calculating the flow in complex geometry

to give very refined information about the velocities,

turbulence, transport properties, etc. The advantage is

the very detailed information that can be obtained

from the model, while the large CPU time requirement

is a disadvantage.

3.3.1. The Navier–Stokes equations

The Navier–Stokes equations consist of a continu-

ity equation and a momentum equation as sketched

below:

Continuity equation:

@ui@xi

¼ 0 ð6Þ

Momentum equations:

q@ui@t

þ quj@ui@xj

¼ � @p

@xiþ @

@xjl

@ui@xj

þ @uj@xi

� �ð7Þ

where q is the density of the fluid, ui is the velocity

components, p the pressure, l the dynamical viscosity,

and t and x the time and spatial independent variables.

For small Reynolds numbers, the Navier–Stokes

equations can be solved directly without a turbulence

model. This approach is called Direct Numerical

Simulation (DNS) since it does not include a turbu-

lence model. It is only small-scale flows that can be

solved by DNS such as a wave boundary layer for

small Reynolds numbers. Another approach is Large

Eddy Simulation (LES), which is quite similar to DNS,

the difference being that only the larger eddies are

directly simulated, while the smaller-scale eddies, i.e.

smaller than the grid scale, are accounted for through

a subgrid-scale model. In general, DNS and LES

require a fine resolution in three spatial dimensions;

thus, the CPU time for such calculations is rather

excessive and, therefore, the practical engineering ap-

plications are limited.

Another, and extensively used, approach is the one

based on the time-averaged Navier–Stokes equations,

yielding the Reynolds-averaged Navier–Stokes equa-

tions (RANS), in which the velocity components are

assumed as consisting of a fluctuating and time-

averaged component:

u ¼ uþ uV ð8Þ

where u is the time-averaged velocity, uV is the

fluctuating part of the velocity, and u is the total

velocity. When inserting the expression for u in Eq.

(7), the Reynolds-averaged Navier–Stokes equations

emerge:

q@ui@t

þ quj@ui@xj

¼ � @p

@xiþ @

@xjl

@ui@xj

þ @uj@xi

� �

� @

@xjqujVuiV ð9Þ

The only unknown term in Eq. (9) is �qujVuiV ,which is called the Reynolds stress tensor:

sij ¼ �qujVuiV ð10Þ

The Reynolds stresses have to be modelled, this is

the so-called ‘‘closure problem’’, since the model

closes the system of equations. This is often done

by the introduction of an eddy viscosity model linked

to a turbulence model like, for instance, a k� e model

or a k�x model, see Wilcox (1988). The Reynolds-

averaged Navier–Stokes equation and the k� e trans-port equations with the appropriate boundary condi-

tions have been used to successfully predict many

complex turbulent flows (e.g. Rodi, 1984).

3.3.2. The free surface

In all the work presented below, the Navier–Stokes

equations are solved on a rigid grid, where the free

surface cuts through the cells. For overturning surfa-

ces, a number of different methods have been applied

in the past.

Miyata (1986) used a line segment method, where

the line segments endpoints were situated on the grid-

lines. A Lagrangian movement of the endpoints to a


new position found the new position of the surface.

The positions of the endpoints of the free-surface

segment were interpolated to be located on the grid

again. Miyata (1986) used this method to simulate

breaking waves in front of an advancing floating body

and over submerged bodies.

Another method is the marker-and-cell (MAC)

method that was developed by Harlow and Welch

(1965). It is based on markers that are distributed all

over the fluid domain. Each marker follows the veloc-

ity field in a Lagrangian way. At the new time step, a

cell that earlier contained markers might now be empty

and is, therefore, not a part of the fluid domain. Cells

that contain markers, but have empty neighbours, are

defined as surface cells, etc. An example of the MAC

method used for breaking waves is given in Sakai et al.

(1986).

A similar method to the MACmethod is the Surface

markers method presented by Chen et al. (1991) and

used for breaking waves in Christensen (1998) and

Christensen and Deigaard (2001). Here, the markers

are only situated at the surface, which reduces the

computational costs and improves the accuracy.

All the above methods find the position of the

surface in a Lagrangian manner.

An approach that has been widely used during the

recent years is based on a continuity equation for a

conservative quantity, F, as sketched by the following

equation:

@F

@tþ ui

@F

@xi¼ 0 ð11Þ

The fluid is located where F is equal to 1.0 and the air/

void region is equal to 0. F = 0.5 determines the

position of the free surface.

Eq. (11) is solved in a Eulerian way. The first and

straightforward way to solve the problem is to use a

very accurate higher order convection scheme such as

the QUICK. Kawamura and Miyata (1994) used this

approach. In their cases, both the air and fluid flows

were simulated around ships and submerged bodies. In

Hirt and Nichols (1981), a special advection scheme

was used to avoid the smearing of the surface, which

they called the ‘‘Volume of Fluid’’ method, also known

as VOF. This method has been extensively used and

modified by several researchers. Here, Lemos (1992),

Lin and Liu (1998a,b) should be mentioned.

3.3.3. RANS modelling of the surf zone

The first attempt to model the flow in the surf zone

and turbulence was given in Lemos (1992). He applied

the original VOF method developed by Hirt and

Nichols (1981) together with a k� e model to repre-

sent the turbulent scales in the simulations. The

numerical scheme was rather simple due to the Carte-

sian coordinate system without any stretching and a

rather coarse mesh (300� 40) for a computational

domain of 9 m long and 0.4 m high. The periodic

waves broke on a slope simulated by a staircase-type

out-blocking of computational cells. The results

showed that the approach could be used to simulate

the surf-zone turbulence though the turbulence levels

were found to be overpredicted.

Breaking-wave transformation processes on a slope

were studied through an experiment and numerical

analysis by Takikawa et al. (1997). Velocities in the

surf zone were measured, however, not by a nonin-

trusive optical technique, but directly by a two-dimen-

sional electromagnetic current meter. An uncertainty

was then introduced to the velocity measurement due to

the effect of turbulence generated around the sensor

and the influence of air bubbles. Vorticity, skewness,

and Reynolds stress were calculated from the meas-

ured velocity. The results indicated that, in the upper

region of the surf zone, the Reynolds stress is highly

related to the vorticity and the skewness. Additionally,

Takikawa et al. (1997) also recorded video images of

vortices generated at the time of water mass inrush.

They then developed a numerical system to solve the

Reynolds equations, using the Simplified Marker and

Cell (SMAC) method, and compared with the experi-

ments. Reasonable agreement between experimental

and numerical results was reported, especially for the

scale of vorticities.

Lin and Liu (1998a,b) used a similar approach to

Lemos (1992), but with a code of Hirt and Nichols

(1981) further developed by Kothe et al. (1991).

Again, the k� e model was used to represent the

turbulent scales. The numerical resolution was fine,

but still on a Cartesian mesh. The slope was accounted

for by letting the bed cut through the cells at the bottom

boundary, with a special treatment of the neighbour

cells opposite of the fluid boundary. Hereby, the stair-

case treatment as in Lemos (1992) was avoided.

As in Lemos (1992), Lin and Liu (1998a,b) found

that the turbulence levels at breaking were overesti-


mated. The error was of the order of two to three times

the measured quantity. In the inner zone, the turbu-

lence is, in general, 25% to 50% higher than measured

in Ting and Kirby (1994). They argued that the

primary reason for the overestimated turbulence was

that the current turbulence closure model cannot

accurately predict the initiation of turbulence in a

rapidly distorted shear flow region such as the initial

stage of wave breaking.

The numerical results of the averaged flow below

trough level were examined in the paper by Lin and

Liu (1998a,b). The flow field averaged over a wave

period showed that the undertow was significantly

smaller or even directed towards the shore instead of

offshore as found from measurements.

The results were extracted from 8.3 wave periods to

9.3 wave periods after the beginning of the simulation.

They found that the set-up and set-down had not fully

developed and, thus, the calculated mean water depth

could be inaccurate, which might explain the difference

in the undertow predicted by the numerical model.

The roller region exists in the upper level of the

breaking wave front with the mean particle velocity

being about 80% of the local phase velocity, C. In the

lower part, beneath the trough level, the mean velocity

field is little affected by the breaking process, which

leads to a rather uniform velocity distribution. This

value is actually quite different compared to the value

of 1.3Clinear theory used in the roller model, the differ-

ence might be due to different ways of defining the

roller velocity, whether it is in the top or at the bottom

of the roller.

Lin and Liu (1998a) found that the vorticity gen-

erated on the bottom is almost one order of magnitude

smaller than that generated by the turbulent bore and it

can be either positive or negative, varying in both time

and space. Further, they found that the turbulence

transport mechanisms are quite different at different

elevations in the surf zone. In the transition region,

between the roller region and the bottom region, the

dominant mechanisms are turbulence production, ver-

tical diffusion, and vertical convection. The first two

contribute to the increase of turbulence, but the last

reduces the turbulence. In the bottom region, all turbu-

lence transport mechanisms become very weak (two

orders of magnitude smaller than those in the transition

region) and, thus, the turbulence level is nearly constant

during one wave period.

A similar approach as the one sketched above was

used in Lin and Liu (1998b) to investigate the turbu-

lence transport and vorticity dynamics in the surf zone

under plunging breakers. The results of the plunging

breaker case compare better with measurements of the

undertow than similar results for a spilling breaker. The

turbulence levels are too high just after the breaking

point, but closer to the shoreline the turbulence levels

seem to be of the same order of magnitude as in the

experiments by Ting and Kirby (1995). Still the turbu-

lence levels are 25% to 100% larger than the measured

levels. The model does capture the pulse of turbulent

production at the breaking point and the turbulence

almost dies out between the breakers. This might

explain the better agreement with measurements. Since

the turbulence dies out between each breaking wave,

the average level of turbulence is reached already after

a very few numbers of waves, which might not be the

case for the spilling breaker presented above.

Bradford (2000) made a comparative study of three

turbulence models. All three turbulence models used

the turbulent viscosity concept combined with differ-

ent formulations of k� e like models. One model is

based on an algebraic estimation of the dissipation

based on a length-scale approach, another is the

commonly used k� e model, and, finally, the Renor-

malized Group (RNG) extension of the k� e model. In

general, the model like k� e model and k-model gave

an average turbulence level that was twice as large at

the experimental levels reported by Ting and Kirby

(1994) for the spilling breaker, while the RNG model

gave slightly smaller overestimations. The turbulence

levels were found to be very close to the measured

ones in the case of a plunging breaker, which agreed

well with the results shown in Lin and Liu (1998a).

The undertow, which balances a strong onshore

flux between the wave top and the wave trough and is a

strong offshore-directed flow below the trough, is

actually found by as subtracting a large figure from

another large figure. This implies that the undertow, in

general, is hard to model completely correct. Still, the

undertow found by both Bradford (2000) and Lin and

Liu (1998a) was, in general, too low or directed

towards the shore instead of offshore in the spilling

breaker and, therefore, the model could not be used to

model, for instance, sediment transport. In Christensen

et al. (2000), it was found that the model had to run for

at least 15 wave periods before stationary conditions


from wave to wave were achieved. This gave a better

prediction of the undertow, but the turbulence levels

were still too high to allow anything else than qual-

itative description of the sediment transport.

Lin and Liu (1998a,b) used a more advanced

description of the Reynolds stresses than Bradford

(2000), their formulation did not show substantial

improvements over the isotropic models. The choice

of the boundary conditions, grid resolution, and the

model coefficients all seem to have more impact on the

solution. Mayer and Madsen (2000) found that the

traditional turbulence models never find a stationary

level of turbulence and eddy viscosity. The problem

arises due to stability problems in the k�x model in

wave-induced orbital motion. A preliminary work

around was presented where the turbulent production

term was related to the vorticity instead of the strain

rate. The strain rate is not equal to zero in waves, cf.

Fig. 13, the water elements are stretched, while the

vorticity is zero, i.e. no rotation of the water elements.

Using this approach, a much better agreement between

modelled and measured water level elevations was

achieved. In Zhao et al. (2000), a multiscale turbulence

model is set-up based on a k� l model. Since the

production term still is related to the strain rate, the

waves produce turbulence before they actually break.

The instability reported by Mayer and Madsen (2000)

was avoided and, therefore, the simulated water ele-

vations agreed well with measurements.

The turbulence level within the surf zone, i.e. after

wave breaking, was overpredicted in all the reported

studies using the RANS approach. The intrusion of air

in the roller and the upper area of the surf zone might be

the most important reason for the deviations between

numerical and experimental results in this region. One

could argue that since the mixture of air and water in

the roller, on average, has a smaller density than water,

the turbulence produced in the roller would have

difficulties in penetrating the underlaying fluid as

illustrated in Fig. 14. Therefore, a larger part of the

production and dissipation takes place in the roller

before it is diffused downward, which explains the

overestimated turbulence levels in the surf zone by the

numerical models. This was actually already accounted

Fig. 13. The deformation of water elements under different phases of a wave cycle.

Fig. 14. The distribution of the density of the air/water mixture through the roller region.


for in Deigaard et al. (1986), where a 1DV model was

set up for the turbulence. Seventy percent of the

turbulent kinetic energy were assumed dissipated

before it reaches the fluid beneath the aerated part of

the water column. A simple approach like this would be

difficult to apply for a full 2DV solution of the flow

field. Instead, the influence of the intrusion of air on the

hydrodynamics and the turbulence has to be modelled

perhaps by adding another model that accounts for the

dynamics in the mixed air/water region.

Very recently, Emarat et al. (2000) studied the

mechanics of a surf-zone plunging breaker. Results

from 2D PIV measurements were compared against

those from a numerical model based on the Navier–

Stokes equations and the VOF method. Fig. 15 shows

an example of a velocity vector field of a plunging

breaker at the time of breaking. A result from the

numerical model is shown in Fig. 15a, while the PIV

measurements are shown in Fig. 15b. Good agreement

between both results was found for the comparison of

the flow field and velocity magnitude distribution. This

suggests that further attempts at investigating the

vorticity and turbulence structures could also be made

by the model. This study actually shows that the main

problem in the modelling of the surf zone with Navier–

Stokes solvers is the turbulence model and perhaps also

the boundary conditions for the turbulence model.

If a Navier–Stokes solver is to be used to simulate,

for instance, sediment transport, it is very important to

model the wave boundary with a reasonable accuracy,

since the sediment concentrations even in breaking

waves are much larger near the bed. Duy et al. (1998)

made an attempt to model the wave boundary layer in

the surf zone under breaking waves. Apparently, they

assumed that the effect of the turbulence in the break-

ing wave did not influence the turbulence generated in

the wave boundary layer in a spilling breaker. The

turbulence levels in the boundary layer were found to

be of the same order of magnitude as in the measure-

ments by Cox et al. (1996).

3.3.4. LES of the surf zone

As explained, LES is another way of simulating

turbulence in wave breaking, with turbulence model

for the subgrid turbulence only. Since it is a smaller

part of the turbulent regime the model has to take into

account, the model can be much simpler than the mo-

dels used in RANS.

Two-dimensional modelling cannot be said to be

true LES since the simulation of eddies is only two-

dimensional and, therefore, the stretching of eddies

that is characteristic for true turbulence is not simu-

lated at all. However, the work of Zhao and Tanimoto

(1998) shows a surprisingly good comparison with

measurements of the vertical distribution of, for ins-

tance, the maximum and minimum orbital velocities.

The wave height and mean water level distribution also

compared well with measurements. Despite the defects

of only two dimensions and that the slope of the

submerged reef was 1:2, the model surely gave some

good indications of the strength of a LES model.

Christensen (1998) and Christensen and Deigaard

(2001) used a full three-dimensional Navier–Stokes

solver, combined with a two-dimensional free-surface

model based on the surface-markers method, to study

the three-dimensional turbulent flow structures in the

breaking zone. A Smagorinsky subgrid model, Sma-

gorinsky (1963), was used for the simulations.

The turbulent structures were investigated under

different breaker types: spilling, weak plungers, and

strong plungers. The turbulent kinetic energy was

found by averaging over the transverse direction. In

Fig. 15. Velocity vector map of a plunging breaker at the breaking point. (a) the Navier–Stokes solution and (b) the PIV measurement.


spilling breakers, the turbulence is generated in a series

of eddies in the shear layer under the surface roller.

After the passage of the roller, the turbulence spreads

downwards. In the strong plunging breaker, the turbu-

lence originates to a large degree from the topologi-

cally generated vorticity. The turbulence generated at

the plunge point is almost immediately distributed

over the entire water depth by large organised vortices.

Away from the bed, the length-scale of the turbulence

(the characteristic size of eddies resolved by the mo-

del) is similar in the horizontal and the vertical direc-

tion; it is found to be in the order of one-half of the

water depth.

In some cases, an eddy structure was identified

from one cross-section to the next and, in that case, the

position of the centres was marked. A line through the

centre shows the orientation of an eddy and the pattern

of obliquely descending eddies, which was identified

by Nadaoka et al. (1988a,b) and Nadaoka et al. (1989),

can clearly be recognised, as sketched in Fig. 16.

The mechanism of the generation of the obliquely

descending eddies and longitudinal eddies beneath

breaking waves might be in the same manner as for

the Langmuir circulation cells under wind-driven cur-

rent, cf. Nepf et al. (1995), see the section on exper-

imental investigations. Langmuir circulation is longi-

tudinal vortices aligned in the wind direction. If this is

the case, two different mechanisms produce rather

coherent turbulence flow structures that are either long-

itudinal eddies or obliquely descending eddies. The

Fig. 16. An example of detecting obliquely descending eddies, from Christensen and Deigaard (2001).


longitudinal eddies might be a further development of

the very coherent horizontal turbulent structure as

sketched in Fig. 17. Due to a perturbation of the

onshore-directed flow, the original horizontal eddy is

stretched giving a longitudinal eddy. Due to the per-

turbation, vorticity around a vertical axis is generated

as shown in Fig. 18. The velocity gradient giving co-

herent obliquely descending eddies stretches this eddy.

In Langmuir circulation, these mechanisms are contin-

uously fed by the wind shear stress, which introduces

the surface shear. In breaking waves, the surface shear

is only present in the breaking process. This is why

these coherent flow structures soon break up into more

chaotic three-dimensional turbulence when they are

outside the influence from the roller region.

These observations agree well with the conclusions

of Christensen (1998) and Christensen and Deigaard

(2001), where horizontal, obliquely descending, and

longitudinal eddies were observed. Watanabe and

Saeki (1999) found similar results from three-dimen-

sional simulations of breaking waves. The coherent

flow structures were generated almost instantly at the

breaker point and enhanced by the breaking process.

Finally, the flow structures broke down outside the

region affected by the surface roller. The processes

of growth of vorticity around vertical and trans-

verse axis were further discussed in Watanabe et al.

(2000).

Mutsuda and Yasuda (2000) used a similar model

mainly to study the entrainment of air in a solitary

Fig. 17. Generation of the obliquely descending eddies from an original horizontal eddy.

Fig. 18. Generation of the obliquely descending eddies from an undulation in the shear layer in the upper region.


wave breaking over a reef. Three-dimensional turbu-

lence appeared later in this study than in the other

referred studies. This might be due to the study of a

solitary wave instead of periodic waves where a small

amount of turbulence is left over from the previous

breaking wave. Generally, the effect of air has not yet

been studied in detail with three-dimensional models.

The model by Mutsuda and Yasuda (2000) used a

relatively coarse resolution (402� 37� 62) points

for a computational domain that was 6.5 m long, 0.6

m width and 0.5 m high, when simulating the entrain-

ment of air bubbles. The smallest resolvable air bubble

uses at least a few grid points. Therefore, the resolution

has to be much finer or a special model for the

influence of the smaller air bubbles has to be added.

However, the entrainment of air was found to enhance

the bursts of turbulence.

3.4. Other mathematical models

Apart from the models already discussed, which

have been used extensively and gained much interest

during the last 5 years, there are other approaches as

well. One of these is based on the discrete vortices to

describe the dynamics of plunging breakers and

another model that describes the interaction between

the air and water.

3.4.1. Discrete Vortex Model

In Pedersen et al. (1995), the Discrete Vortex Model

(DVM) was used to simulate surf-zone dynamics. The

DVM method directly simulates the distribution of

rotation in the fluid by representing the rotation as

vorticity particles, which for each time step are as-

signed convective and diffusive translations in space.

This should fulfil the requirement of spatial resolution

of vorticity in the interior of the computational do-

main. A simple technique based on a nonbreaking po-

tential theory wave was implemented to simulate the

free surface.

The work presented in Pedersen et al. (1995) deals

with the breaking wave as it plunges, i.e. from the

moment when the description by the potential theory

breaks down and before the quasi-steady bore ap-

proach becomes valid. The model is set up with a

rigid lid. Therefore, the free surface is not simulated

directly. The effect of the free surface is found by

superimposing the boundary conditions for the stream

function obtained for a nonbreaking wave with the one

of a model of the jet penetrating the surface due to the

plunging breaker.

One of the major ideas is to set up models for the

generation of vorticity and turbulence. The vorticity is

generated at the surface and at the bed in the wave

boundary layer. The two mechanisms that are domi-

nant for the production of turbulence and vortices at

the surface are the topologically generated vorticity

and the flow pattern generated by the jet impingement

on the surface in front of the wave crest, cf. Fig. 19. At

the bottom, the approach by Asp Hansen et al. (1994)

was implemented, which was based on the model by

Fredsøe (1984).

Fig. 19. Generation of topologically induced vorticity. From Pedersen et al. (1995).


The sediment transport was modelled and the com-

parison with measured concentrations showed a rea-

sonable agreement, i.e. within a factor two. In general,

the concept depends to a high degree on the way the jet

is modelled. As it was discussed in the earlier sections,

this is actual a crucial point since the strength and shape

of the jet to a high degree depends on the type of

breaker. This relationship between the shape and

strength of the jet is, in general, not well known.

3.4.2. The upper zones mixture of air and water

Only few researchers have investigated the impor-

tance of the air–water mixture in the upper layer of a

breaking wave. Recently, Brocchini and Peregrine

(2000a,b) studied the dynamics of turbulence near a

free surface, where preliminary results were presented

in Brocchini and Peregrine (1998). A detailed descrip-

tion of the model can be seen in Brocchini (1996).

Brocchini and Peregrine (2000a) set up a diagram

that depends on the length scale of the turbulence and

its strength. For very small length scales, capillary

forces suppress the turbulence near the surface, while

it is damped by gravity for large scales. Whether the

turbulence results in water bubbles, or any other water

configuration, leaving the surface depends on the

strength of the turbulence as well as the length scale.

If all the fluid is in continuous contact with the main

water body, smaller undulations may be present such

as a wavy surface or scars in the region with little

influence by capillary forces. Or the surface might be

rippled or knobbly in a case where the capillary forces

are dominating.

In the case of breaking waves, gravity is often taken

to be the dominating damping force. This might also

be true for real breaking waves on a beach since the

length scale is relatively large compared to a laboratory

wave flume. In a wave flume, the scales in some

experiments become so small that capillary forces

might influence the result. Waves can also be said to

be undulations of the surface that is in balance with

gravity. When the waves approach the shore, this

balance is violated and parts of the water body might

be thrown in front of the wave as in a plunging breaker.

Based on the qualitative description of the surface

layer, i.e. the mixture of air and water at the surface in,

for instance, a breaking wave, a system of equations is

set up in Brocchini and Peregrine (2000b). These

equations are integral equations of mass, momentum,

and turbulence. One of the results is boundary con-

ditions, for instance, for the turbulence at the mean free

surface. Still, the equations have to be closed when

applied in other models, but the approach is certainly

one of the first attempts to quantify the influence of the

air/water mixture on the flow below the surface layer.

4. Concluding remarks

The research related to the vertical variation of flow

structures has been reviewed. A qualitative description

of the processes is followed by a description of the

experimental research. Until the last decade, much of

the experiments of flow structures in the surf zone was

based on qualitative analysis. Recently, the use of la-

ser–optical technologies, i.e. LDA and PIV, has meant

that research also gives good qualitative results. For

instance, the breaker mechanics in both spilling and

plunging breakers has been investigated thoroughly

and the production, transport, and decay of turbulence

and vorticity have been analysed.

Phase-averaged models of cross-shore processes

have become the state-of-the-art in many different

kinds of engineering tools. The methods are very ro-

bust and, due to their low computational cost, wide-

spread in many types of engineering software, where,

for instance, the cross-shore sediment transport is

calculated. Lately, this type of models has been ex-

tended to the three-dimensional or quasi three-dimen-

sional case. It has been found that the roller has to be

taken into account to give reasonable estimation of the

set-up and cross-shore flow in the surf zone.

Intrawave models, such as the Boussinesq equa-

tions and Navier–Stokes solver, have been applied

recently. The roller model in the Boussinesq approach

has showed to give a good description of the wave

height decay and set-up in the surf zone. To model, for

instance, the sediment transport, extra models for the

undertow and wave boundary layers are needed since

the applied velocity profile is too simple to be used

without modification. Still, such a composite model

gives a very realistic result with respect to wave height,

set-up, undertow, and sediment transport. Recently, the

velocity profile was improved by adding an equation

for the vorticity. This certainly improved the modelling

of the undertow, but the accuracy on the wave set-up

was the same as earlier. In some ways, the model has to


be improved since it does not take into account the

wave boundary layer and the eddy viscosity is

assumed constant.

Up till breaking and in the initial breaking process,

the Navier–Stokes solvers have shown to be very

accurate. Traditional turbulence models generally pro-

duce turbulence before the actual breaking. This

problem has been linked to stability problems for

the turbulence models in wave-driven orbital motion.

After breaking, the turbulence in models based on the

RANS method is, in general, too high. The reason for

this has not yet been found, but it might be related to

either an insufficient turbulence model or the fact that

the mixture of air and water has not been accounted

for. Most turbulence models are optimised for fully

developed boundary layer flows like, for instance,

channel flow, and only little is known about their

application to the wave case. The use of LES models

gives very detailed information about the flow struc-

tures. A number of characteristic coherent turbulent

flow structures, such as horizontal eddies, obliquely

descending eddies, and longitudinal eddies, have been

recognised. The comparisons with experiments have

mainly been qualitative.

Taking the mixture of air and water into account

might be one of the challenges that has to be addressed

as well as better turbulence models if the Navier–

Stokes or Boussinesq-type of models are to be used to

study the flow in the surf zone in detail.

Acknowledgements

The authors would like to acknowledge the support

of the Commission of the European Communities

Directorate General for Science Research and Devel-

opment on the Surf and Swash Zone Mechanics

(SASME) project under contract number MAST3-

CT97-0081. This review has been undertaken during

the final period of the project. Participants of the

SASME project, who have contributed their work and

made valuably comments and suggestions to this

paper, are gratefully acknowledged. Narumon Emarat

was also supported by the Ministry of Education, Thai

Government for her PhD studies. D.J.R. Walstra was

also party funded by the Delft Cluster Project Coasts

03.01.03. E.D. Christensen was also partly funded by

the Danish Technical Research Council (STVF grant

no. 9801635) under the frame programme Computa-

tional Hydrodynamics and partly by the Commission

of the European Communities Directorate General for

Science Research and Development on the Environ-

mental Design of Low-Crested Coastal Defence

Structures (DELOS) project under contract number

EVK3-CT-2000-00041.

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