## Wave action on shallow water and applications to coastal hazard

##### Abstract

The mechanics of wave breaking in shallow water has been a major
research field for many years, and a very large number of published
results are available. No attempt is made here to review the whole
literature. Some interesting – if somewhat outdated - descriptions of
waves breaking on beaches are presented by Peregrine (1983), Battjes
(1988) or Liberatore-Petti (1992).
In fact, the most important process in the near coast zone of the
shoreline motion is wave breaking. Some waves break in shallow water,
some of them break at the water’s edge and in other circumstances
waves do not break at all (with steep beach slopes, incident waves with
low steepness - or long waves). In general, breaking in deep water is rarer
than breaking in shallow water. The latter is triggered by the bottom and
is more predictable, although the simple question ‘where breaking starts’
is far from having a unique answer, even in controlled physical
experiments. The breaker types are, generally, classified as spilling (where
the water spills down the front face), plunging (with a jet emanating from
the front crest), surging (characterized by a rise in water surface before
the breaking) and collapsing (between plunging and surging).
The fluid dynamics of non-breaking waves can be described using
potential theory in most of the flow field except near the bottom and
near the free surface, where vorticity develops and is confined to a
boundary layer. As long as the details near the free surface (e.g. necessary
for wind–wave interaction) and/or near the bottom (e.g. necessary for
sediment transport analysis) are not of interest, the potential theory
approach is sufficient. After breaking, ‘waves’ and ‘eddies’, essentially a
potential component and a rotational component of the flow field, are
intimately mixed.
The surf and swash zones are characterized by the complete
transformation of the organized motion of the incident, sea-swell, waves
into motions of different types and scales, including small-scale (less than
a wave period) turbulence, and large-scale (much greater than the wave
period) mean flows [Battjes, 1988].
It is obvious that [Stive and Wind, 1982; Lin and Liu, 1998a; Svendsen et
al., 2000; Svendsen, 2005] contributions from terms which have
traditionally been neglected in the traditional assumptions of hydrostatic
pressure, depth uniform velocity profile, and negligible turbulence, are
important and must be taken into full account in surf zone
hydrodynamics.
Non Linear Shallow Water equations (‘800) and Boussinesq models
[Peregrine, 1967] have intrinsic limitations and can only simulate wave
breaking and its evolution by assuming on semi-empirical ad hoc
assumptions and threshold values to represent wave dissipation.
Moreover, these models lack the capability to determine spatial
distribution of the turbulent kinetic energy, which is of great importance
for sediment transport studies [Lin and Liu, 1998b].
Given all this, it was only natural that the Navier-Stokes solvers now
widely tested and developed in other fields of fluid mechanics, with less
restricted assumptions involved, no wave theory assumed beforehand,
and the capability to simulate complex turbulent processes, should soon
become one of the main approaches to describe nearshore processes.
Numerical modeling of three-dimensional breaking waves is extremely
difficult. Several challenging tasks must be overcome. First of all, one
must be able to track accurately the free surface location during the wave
breaking process so that the near surface dynamics is captured. Secondly,
one must properly model the physics of turbulence production, transport
and dissipation throughout the entire wave breaking process. Thirdly,
one needs to overcome the huge demand in computational resources.
There have been some successful two-dimensional results. For instance,
more recent is the treatment of the free surface within such an Eulerian
framework with the marker and cell (MAC) method [e.g., Johnson et
al.1994] and the volume of fluid method (VOF) [e.g., Ng and Kot 1992,
Lin and Liu, 1998a].
The most common approach for simulating breaking waves is presently
the application of 2D-Reynolds Averaged Navier-Stokes (RANS)
equations with a Volume of Fluid (VOF) surface computation and a
turbulence closure model. Such an approach, while being often tested for
many years by many various Authors (see for instance Bovolin et al,
2004) only reached full maturity with a fundamental paper by Lin and
Liu (1998a). This line of research has been going on successfully for
many years to the point that reliable procedures now exist to simulate
wave breaking, run up and interaction with structures.
The next obvious step. i.e. the application of Large Eddy Simulation
(LES) models has so far not been equally successful [Watanabe and
Saeki, 1999; Christensen and Deigaard, 2001; Lubin et al, 2006;
Christensen, 2006].
LES models necessarily require a fully three-dimensional solution and
three-dimensional turbulence effects might be indeed important in the
prediction of velocity within the surf zone, especially in the case of
plunging breaker [Watanabe and Saeki, 1999]. Such models certainly are
a promising tool in the study of surf zone hydrodynamics; however, the
LES approach requires much finer grid resolution and a lager
computational domain than the RANS approach, resulting in the very
high demand on computational resource, at least for the time being.
They are however a definite perspective for the future.
Smoothed Particle Hydrodynamics (SPH) method, adapted from
astrophysics into a number of fields, is a relatively new method for
examining the propagation of highly nonlinear and breaking waves
[Monaghan et al, 1977; Dalrymple et al, 2005; Viccione et al, 2007-2008].
SPH offers a variety of advantages for fluid modeling, particularly those
with a free surface.
The Lagrangian method is meshfree; the equivalents of mesh points are
the fluid particles moving with the flow. The free surface requires no
special approaches, such as the volume-of-fluid method or a Lagrangian
surface tracking. Furthermore, the method can treat rotational flows with
vorticity and turbulence.
SPH is a technique based on computing the trajectories of particles of
fluid, which interact according to the Navier–Stokes equations. Each of
such particles carries scalar information, density, pressure, velocity
components, etc.
The work presented here is therefore mainly based on the application of
the Eulerian 2-dimensional RANS/VOF equations to the study of surf
zone processes on a beach. In particular the work is aimed at
demonstrating the capability of RANS/VOF to improve the current
modeling of surf zone hydrodynamics on sloping natural beach and in
front of shallow water coastal structures , comparing its performance
with laboratory observations and other theoretical and numerical results. [edited by author]