Direct numerical simulations of breaking waves

Stéphane Popinet, Luc Deike, Ken Melville

Breaking waves at the water surface is a striking example of turbulent mixing across a fluid interface.

The impact of the jet generates turbulence, entrains air into the water and ejects droplets into the

air. A fundamental understanding of the general multi-scale properties of the resulting multiphase

turbulent flow is necessary to develop more accurate gas transfer or spray generation parameterizations.

In this talk, we will present direct numerical simulations of breaking waves at various scales, using the

open source solver Gerris including capillary effects. We will discuss the effects of surface tension on

the wave shape, the dissipation due to breaking and parasitic capillaries as well as air entrainment and

bubble statistics in breaking waves. The numerical results are carefully validated against laboratory

experiments and scaling models are proposed (Deike et al 2015, 2016).

References L. Deike, W. K. Melville and S. Popinet. Journal of Fluid Mechanics. (2016). Air entrainment and bubble statistics

in breaking waves. vol 801, pp 91- 129.

L. Deike, S. Popinet and W. K. Melville. Journal of Fluid Mechanics (2015). Capillary effects on wave breaking,

vol 769, p541-569.

Comparing hybrid vs eddy viscosity breaking models in weakly and fully

non-linear Boussinesq models

M. Kazolea and M. Ricchiuto

We consider the issue of wave breaking closure for Boussinesq type models, and attempt at providing

some more understanding of the sensitivity of some closure approaches to the numerical set-up, and in

particular to mesh size. For relatively classical choices of weakly dispersive propagation models, we

compare two closure strategies. The first is the hybrid method consisting in suppressing the dispersive

terms in breaking regions, as initially suggested by Tonelli and Petti in 2009. The second is an eddy

viscosity approach based on the solution of a turbulent kinetic energy. The formulation follows early

work by O. Nwogu in the 90’s, and some more recent developments by Zhang and co-workers (Ocean

Mod. 2014), adapting it to be consistent with the wave breaking detection used here. We perform a study

of the behavior of the two closures for different mesh sizes, with attention to the possibility of obtaining

grid independent results. Based on a classical shallow water theory, we also suggest some monitors to

quantify the different contributions to the dissipation mechanism, differentiating those associated to the

scheme from those of the partial differential equation. These quantities are used to analyze the dynamics

of dissipation in some classical benchmarks, and its dependence on the mesh size. Our main results show

that numerical dissipation contributes very little to the results obtained when using eddy viscosity

method.

This closure shows little sensitivity to the grid, and may lend itself to the development and use of non-

dissipative/energy conserving numerical methods. The opposite is observed for the hybrid approach, for

which numerical dissipation plays a key role, and unfortunately is sensitive to the size of the mesh. In

particular, when working, the two approaches investigated provide results which are in the same ball

range and which agree with what is usually reported in literature. With the hybrid method, however, the

inception of instabilities is observed at mesh sizes which vary.

Use of a phase-resolving model for wave breaking prediction

A. Varing, J.F. Filipot, V. Roeber, M.L. Yates

In the past 20 years, significant efforts have been made to simulate accurately wave conditions in shallow

water and the surf zone for a variety of nearshore activities and environmental issues. The motivation

for recent studies in coastal engineering are mostly centered on the need for accurate predictions of the

height and location of breaking waves in the surf zone and on the need for estimating detailed wave

characteristics at the breaking point [1, 2]. Breaking waves create significant dynamical loading on

ocean engineering structures. Optimal structural design requires accurate predictions of the maximum

nonlinear wave forces expected over the structure’s working lifetime. Most hydrodynamic models are

based on phase-averaged methods in which only integral wave parameters are calculated. The present

study focuses on phase-resolving models and more particularly Boussinesq-type models whose main

advantage is to describe individual wave characteristics and not wave parameters. In contrast to the

nonlinear shallow water equations, traditional Boussinesq models cannot simulate directly the effects of

wave breaking in shallow water. Many efforts have been made to extend these equations and include

the effects of wave breaking. This requires two steps: the introduction of a wave breaking criterion and

the introduction of breaking wave energy dissipation [3, 4]. Nowadays, Boussinesq models have been

shown to be effective even in situation with complex bathymetry and strong non-linear phenomenon [5,

6]. Extensive research on breaking waves is going on and a universal wave breaking criterion has not

been identified yet [7].

This study will focus on the deterministic wave-by-wave prediction of breaking waves with the BOSZ

model [8]. The wave breaking criteria implemented in BOSZ is based on the deactivation of dispersion

terms. From a literature review, different wave breaking criteria have been identified such as the

criterion of [3] based on the wave steepness of the front, the criterion of [4] based on the vertical speed

of the free surface, the Relative Trough Froude Number (RTFN) criterion [9], the Breaking Celerity

Index [1]. They will be implemented in the BOSZ model and compared to 1D and 2D experiments to

identify which criteria is most appropriate to take into account the effects of wave breaking with the

BOSZ model.

References [1] F. D’Alessandro and G. R. Tomasicchio, “The bci criterion for the initiation of breaking process in boussinesq-

type equations wave models,” Coastal Engineering, vol. 55, no. 12, pp. 1174–1184, 2008.

[2] R. Kurnia and E. Van Groesen, “High order hamiltonian water wave models with wave-breaking mechanism,”

Coastal engineering, vol. 93, pp. 55–70, 2014.

[3] H. A. Schäffer, P. A. Madsen, and R. Deigaard, “A boussinesq model for waves breaking in shallow water,”

Coastal Engineering, vol. 20, no. 3-4, pp. 185–202, 1993.

[4] A. B. Kennedy, Q. Chen, J. T. Kirby, and R. A. Dalrymple, “Boussinesq modeling of wave transformation,

breaking, and runup. i: 1d,” Journal of waterway, port, coastal, and ocean engineering, vol. 126, no. 1, pp. 39–47,

2000.

[5] P. Madsen and O. Sørensen, “A new form of the boussinesq equations with improved linear dispersion

characteristics. part 2. a slowly-varying bathymetry,” Coastal engineering, vol. 18, no. 3-4, pp. 183–204, 1992.

[6] S. Beji and J. A. Battjes, “Numerical simulation of nonlinear wave propagation over a bar,” Coastal Eng., vol.

23, pp. 1–16, 1994.

[7] Z. Tian, M. Perlin, and W. Choi, “Energy dissipation in two-dimensional unsteady plunging breakers and an

eddy viscosity model,” Journal of Fluid Mechanics, vol. 655, pp. 217–257, 2010.

[8] V. Roeber, “Boussinesq-type model for nearshore wave processes in fringing reef environment,” Ph.D.

dissertation, Honolulu, University of Hawaii at Manoa, 2010.

[9] T. Okamoto and D. R. Basco, “The relative trough froude number for initiation of wave breaking: Theory,

experiments and numerical model confirmation,” Coastal Engineering, vol. 53, no. 8, pp. 6

An attempt to unify wave breaking measurements formalism

JF Filipot (France Energies Marines), F. Leckler (Shom), R. Duarte (France Energies Marines)

This work is part of the DiMe project, a collaborative initiative lead by France Energies Marines and

aiming to provide a better characterization of the extreme sea state with a focus on their wave breaking

properties for the extreme design of Marine Renewable Energy converters (floating/fixed offshore wind

turbine, wave energy converters, floating tidal turbine, Thermal Energy plants).

Two main approaches have been developed to quantify the wave breaking statistics. The first one is

based on the estimation of the breaking probability or rate (Q) measured at a single location in the ocean.

The main drawback of this method lies in the difficulty to assign the wave breaking events to a given

wave scale. To go around this difficulty, Phillips (1984) introduced the concept of spectral density of

breaking crest length per unit area of ocean, Λ. This quantity is in theory easily observable from image

of videos of the sea surface and allows for a simple spectral decomposition of the breaking event. Indeed,

it is assumed that the breaking their velocity can be related to the phase speed of the underlying wave.

Most of the theories and wave breaking statistics parameterizations fall in the first framework because

they are based on wave properties accessible from elevation signal collected at a single point. However,

because of the development of new detection method based on video (including stereo-video) most of

the wave breaking observations fall and will fall in Phillips's framework.

Here we will discuss how the two frameworks can be connected. Illustrations will then be presented of

validation from wave breaking statistics parameterizations with wave breaking observations coming

indifferently from the Q or Λ frameworks.

AN HP/SPECTRAL ELEMENT UNIFIED BOUSSINESQ FRAMEWORK

FOR WAVE-FLOATING BODY INTERACTION

Bosi Umberto, Centre de Recherche INRIA Bordeaux Sud-Ouest, umberto.bosi@inria.fr

Engsig-Karup Allan Peter, Technical University of Denmark, apek@dtu.dk

Eskilsson Claes, RISE Research Institutes of Sweden, claes.eskilsson@ri.se

Ricchiuto Mario, Centre de Recherche INRIA Bordeaux Sud-Ouest, mario.ricchiuto@inria.fr

This work is concerned with the development of a new efficient and accurate nonlinear tool for floating

Wave Energy Converter (WEC) analysis. Floating WECs are structures that can harvest energy from

the wave motion. Employed nearshore, they are gaining interest in the renewable energy community for

the low ecological and landscape impact and the predictability of wave state [1]. The interactions

between WECs and waves are traditionally modeled using linear radiation diffraction models [5]. Lately,

Reynolds averaged Navier-Stokes (RaNS) models are being employed for research purposes aimed at

maturing engineering practices through simulation tools that capture the nonlinear properties of waves

[6]. However, the first fails in capturing viscous

and higher orders nonlinear effects while the latter requires an impractical amount of computational

power and time to evaluate the solution. We present a new medium fidelity model for nonlinear wave-

structure interaction based on Boussinesq-type equations that have been a successful means to deliver

fast industrial wave propagation tools for decades. These are based on vertically integrated dimensions,

to obtain efficient models that take into account nonlinear effects and non-hydrostatic kinematics. We

have considered a wave-body coupling inspired by the work of Jiang [2]. The resulting model closely

resembles the coupling between two depth-averaged shallow water (or Boussinesq) models: a classical

one for the outer free surface region and a variant based on pressure-velocity coupling for the inner

region under the floating structure. We discuss the discretization of the problem: a high-order continuous

spectral/hp element method is employed to discretize the equations in space as it combines the generality

of the finite elements with the precision of the spectral technique described in [3]. The coupling fluxes

between the inner and outer domains are described by numerical fluxes as suggested in [4] for

discontinuous Galerkin spectral element method. The spectral method developed will present an

exponential speed of convergence and thereby provide a basis for trade-off between accuracy and

efficiency through the use of high-order numerical approximation. The work can be viewed as a concrete

case within a general unified theoretical framework that allows for coupling different shallow water and

Boussinesq-type models. A major advantage of this is that it is possible to demonstrate convergence to

meet practical accuracy requirements and enable numerical analysis that can help qualify the fidelity

needed from simulation tools. These results are employed to evaluate the benchmark for a one-

dimensional freely heaving box case, reproducing the results presented the work of Lannes [7] and

Rodriguez et al. [8] for nonlinear model and compared to VOF results for the Boussinesq model. The

wave-body interaction is explored to allow the introduction of mooring in the form of a spring and power

take-o_ unit in the form of a damper. Preliminary work on a proof-of-concept in a two-dimensional

setting are presented. The continuation of this work is aiming at bridging these fundamental

developments to applications of engineering relevance.

REFERENCES [1] Drew B., Plummer A.R. and Sahinkaya M.N. (2009): A review of wave energy converter technology, Proceedings of the

Institution of Mechanical Engineers, Part A: Journal of Power and Energy.

[2] Jiang T. (2001): Ship Waves in Shallow Water, VDI-Verlag.

[3] Karniadakis G. and Spencer S. (2013): Spectral/hp element methods for computational fluid dynamics, Oxford University

Press. 2

[4] Eskilsson C. and Sherwin S.J. (2002): A Discontinuous Spectral Element Model for Boussinesq-Type Equation, Journal of

Scientific Computing.

[5] Martinelli, L., and Ruol P. (2006):2D Model of floating breakwater dynamics under linear and nonlinear waves. COMSOL

users conference.

[6] Yu, Y.H., and Ye L. (2013): Reynolds-Averaged Navier-Stokes simulation of the heave performance of a two-body floating-

point absorber wave energy system. Computers & Fluids 73, 104-114.

[7] Lannes D. (2017): On the Dynamics of Floating Structures, Annals of PDE 3.1.

[8] Rodriguez M. and Spinneken J. (2016): A laboratory study on the loading and motion of a heaving box Journal of Fluids

and Structures 64.

Modeling and Forecasting of Wave-driven Coastal Hazards

Volker Roeber1, Assaf Azouri1, Martin Guiles1, Doug Luther1, Denis Morichon2, Florian Bellafont2 1Department of Oceanography, University of Hawaii at Manoa, USA

2Laboratoire des Sciences pour l'Ingénieur Appliquées à la Mécanique et au Génie Electrique,

Université de Pau & des Pays de l'Adour, FRANCE

volker@hawaii.edu

When large gravity wave swells, generated by distant storms, impinge on coastlines during high tides,

the resulting reach of the seawater is greater than by the tides alone, and the movement of the water

onshore is more violent. While these events don't have the devastating impact of large tsunamis, their

amplitudes are pushing into the range of moderate tsunamis and they occur more often, such that their

impacts on coastal erosion, freshwater aquifers, infrastructure and populations are already significant.

These events will grow more acute as their numbers and duration increase due to rising sea level and

wave energy expected from climate change. In order to strengthen the resilience of communities to

mitigate the impacts of episodic flooding events, quantitative forewarning is needed. Conventional wave

forecasts are based on computations from spectral models or measurements from wave buoys. These

techniques can provide sufficient accuracy for open beaches. Irregular reef-fringed shorelines and harbor

basins, however, are often subject to infra-gravity (IG) oscillations, which can drastically change the

wave environment nearshore; thus, requiring a more sophisticated approach and the use of phase-

resolving numerical models.

We will show the validation with field data and assess the applicability of three phase-resolving

numerical models (Xbeach, FUNWAVE, BOSZ) for characteristic swell events to wave processes at

beaches and harbors along the Northshore of Oahu (Hawaii) and the Atlantic coast near Biarritz. We

will also highlight critical features of the models and explain how they can be utilized for operational

nearshore wave and runup forecasting.

Long calculations over large numerical domains demonstrate that coastal flooding and nearshore

currents are dominated by both swell wave forcing and IG wave patterns. Wave breaking and transfer

of energy to low-frequencies are critical processes. While coastal reef features and man-made structures

can efficiently protect the nearby shoreline and harbors from energetic swell waves, freely propagating

IG waves can cause flooding and erosion far away from their location of origin. These IG waves can be

of over 20 min period - resembling the nature of tsunamis.

Our findings have important implications on flood risk assessment and future model development

strategies.

Critical jets in a plunging breaker.

Y.-M. Scolan

ENSTA-Bretagne (France), yves-marie.scolan@ensta-bretagne.fr

The twodimensional fully nonlinear free surface problem is solved in potential theory. By shaking a

rectangular basin with a simple cycle of oscillation, it is possible to introduce in the fluid enough energy

hence yielding to the appearance of a critical jet after the plunging crest well developed. That

phenomenon would never appear when dealing with a dam breaking case leading to a plunging breaker

as well. The fluid kinematics must be carefully analyzed for such cases as the one depicted below, since

it may lead to high loads when the fluid mass hits the wall of the tank.

The right figure is a closer view of the crest shown in the left figure. The green line shows the location

of the maximum velocity along each free surface profile.

Field Observations of Wave Breaking

J. Thomson, M. Schwendeman, S. Zippel, and A. Brown

Field observations of wave breaking provide essential information to make progress in understanding

the underlying processes. We present observations of wave breaking across deep, intermediate, and

shallow conditions, including in the presence of sheared currents and partial ice cover. Using shipboard

stereo measurements, we show that wave breaking is controlled by the local crest steepness and crest

speed, which are distinct from both the bulk parameters of the wave field and the parameters of the

carrier wave. The stereo results are superior to buoy results, because buoys must use dispersion to infer

steepness (rather than measuring wave geometry directly). Still, wave buoy measurements are useful

for the kinematics of individual breaking waves, as well as the relation of breaking statistics to bulk and

spectral wave parameters. The accelerations of wave buoys during breaking events far exceed the

motions predicted by weakly nonlinear wave theories, and these motions provide a proxy for wave

breaking severity. Wave breaking statistics are highly correlated with bulk wave steepness, with

important adjustments to the steepness associated with finite depth and sheared currents. In the presence

of partial sea ice cover, wave steepness is reduced and wave breaking is suppressed. These observations

of wave breaking are related to observations of the turbulent dissipation rate in the ocean surface layer.

Effect of vorticity on pre-breaking waves in shallow water" (M. Abid & C. Kharif).

On the absorption of breaking wave energy in fully nonlinear potential flow

model

By S.T. Grilli1, A. Mivehchi1, C.M. O’Reilly1, J.M. Dahl1, J.C. Harris2, and K. Kuznetsov2 1. Department of Ocean Engineering, University of Rhode Island, Narragansett, RI, USA

2. Laboratoire d’Hydraulique St-Venant, Université Paris-Est, EDF R&D, Chatou, France

In the past 30 years, increasingly accurate and efficient models have been developed to simulate

nonlinear wave propagation and transformations over a varying nearshore bathymetry as well as their

interactions with submerged and surface piercing fixed or floating structures. One successful approach

has been based on models solving Fully Nonlinear Potential Flow (FNPF) theory, by a higher-order

Boundary Element Method (BEM), in 2D [e.g., 2, 6, 7, 8, 9, 10, 11] or 3D [e.g., 1, 3, 4, 5, 14, 15]. Such

models can accurately simulate overturning waves and have been used to investigate their physical

properties just before breaking [e.g., 3, 5, 10]. However, in many naval hydrodynamics and

ocean/coastal engineering applications, it is desirable to prevent steep waves from overturning as this

eventually leads to instabilities and stops computations. A number of methods have been proposed to

do so, some based on specifying an “absorbing surface pressure” [12, 13], similar to the method used in

absorbing beaches [6, 7]. Here, we investigate such methods in the context of a hybrid model combining

a 3D-BEM FNPF Numerical Wave Tank [14, 15] and a Navier-Stokes (NS) solver, based on a Lattice

Boltzmann Method (LBM), for fluid-structure interactions [17]. Impending breaking is detected based

on a local maximum free surface slope/steepness criterion, and wave energy is absorbed using a local

“absorbing pressure” patch whose strength is calibrated with a physical criterion [e.g., 6, 7, 12, 13]. In

the hybrid model complex, the absorbed energy is reconciled with that actually dissipated in the NS-

LBM solver. Validation cases of this new “breaker model” are presented for strongly nonlinear waves

interacting with a vertical cylinder, for which filtering of free surface instabilities had otherwise been

necessary [16].

References

1. Corte, C. and Grilli S.T. 2006. Numerical Modeling of Extreme Wave Slamming on Cylindrical Offshore

Support Structures. In Proc. 16th Offshore Polar Engng. Conf. (ISOPE06), 3, 394-401.

2. Dombre E., Benoit M., Violeau D., Peyrard C. and Grilli S.T. 2015. Simulation of floating structure dynamics

in waves by implicit coupling of a fully nonlinear potential flow model and a rigid body motion approach. J.

Ocean Engng. and Marine Energy, 1, 55-76

3. Fochesato C., Grilli, S.T. and Dias F. 2007. Numerical modeling of extreme rogue waves generated by

directional energy focusing. Wave Motion, 44, 395-416.

4. Grilli, S.T., Guyenne, P. and Dias, F. 2001. A fully nonlinear model for three-dimensional overturning waves

over arbitrary bottom. Intl. J. Num. Methods in Fluids, 35(7), 829-867.

5. Guyenne, P. and Grilli, S.T. 2006. Numerical study of three-dimensional overturning waves in shallow water.

J. Fluid Mechanics, 547, 361-388.

6. Grilli, S.T. and Horrillo, J. 1997. Numerical Generation and Absorption of Fully Nonlinear Periodic

Waves. J. Engng. Mechanics, 123 (10), 1060-1069.

7. Grilli, S.T. and Horrillo, J. 1999. Shoaling of periodic waves over barred-beaches in a fully nonlinear

numerical wave tank. Intl. J. Offshore and Polar Engng., 9(4), 257-263.

8. Grilli, S.T., Skourup, J. and Svendsen, I.A. 1989. An Efficient Boundary Element Method for Nonlinear

Water Waves. Engng. Anal. Boundary Elems., 6(2), 97-107.

9. Grilli, S.T. and Subramanya, R. 1996. Numerical Modeling of Wave Breaking Induced by Fixed or

Moving Boundaries. Computational Mechanics, 17(6), 374-391.

10. Grilli, S.T., Svendsen, I.A. and Subramanya, R. 1997. Breaking Criterion and Characteristics for

Solitary Waves on Slopes. J. Waterway Port Coast. Oc. Engng., 123(3), 102-112.

11. Guerber, E., M. Benoit, S.T. Grilli and C. Buvat 2012. A fully nonlinear implicit model for wave

interactions with submerged structures in forced or free motion. Engng. Anal. Boundary Elemts., 36, 1,151-

1,163.

12. Guignard, S. and S.T., Grilli. 2001. Modeling of shoaling and breaking waves in a 2D-NWT by using a

spilling breaker model. Proc. 11th Offshore and Polar Engng. Conf. (ISOPE01), 3, 116-123.

13. Guignard, S. and S.T., Grilli 2002. Implementation and validation of a breaker model in a fully

nonlinear wave propagation model. Proc. 4th Intl. Symp. on Ocean Wave Meas. Anal. (WAVES 2001),

1,012-1,021.

14. Harris, J.C., Dombre, E., Benoit, M. and S.T. Grilli 2014. Fast integral equation methods for fully

nonlinear water wave modeling. In Proc. 24th Offshore Polar Engng. Conf. (ISOPE14), 583-590.

15. Harris J.C., Dombre E., Mivehchi A., Benoit M., Grilli S.T. and C. Peyrard 2016. Progress in fully

nonlinear wave modeling for wave-structure interaction. Proc. JH2016, 12pps.

16. Harris J.C., K. Kuznetsov, C. Peyrard, A. Mivehchi, S.T. Grilli and M. Benoit 2017. Simulation of

wave forces on a gravity based foundation by a BEM based on fully nonlinear potential flow. Proc. 27th

Offsh. Polar Engng. Conf. (ISOPE17) 1,033-1,040

17. Mivehchi A., J.C. Harris, S.T. Grilli, J.M. Dahl, C.M. O'Reilly, K. Kuznetsov and C.F. Janssen 2017. A

hybrid solver based on efficient BEM-potential and LBM-NS models: recent BEM developments and

applications to naval hydrodynamics. Proc. 27th Offshore Polar Engng. Conf. (ISOPE17), 721-728.

Wave Breaking in Undular Bores. Henrik Kalish

University of Bergen, Norway

Henrik.Kalisch@uib.no

An undular bore is a laminar flow which transitions between two different flow depths. In rivers, bores

are generally due to tidal forcing. Bores can also be studied in the laboratory, and a number of studies

have been aimed at understanding the main features of bores. In particular, Favre conducted a dedicated

series of experiments with the aim of classifying different types of bores (Ondes de Translation, Dunod,

Paris, 1935).

In the relatively simple situation of a wavetank, one may without loss of generality assume that the

upstream flow depth is the undisturbed depth, and define the bore strength to be the incident wave

amplitude divided by the undisturbed depth.

It was found by Favre that there are three main bore types. If the bore strength is below 0.28, the flow

is laminar and oscillations of the free surface start to develop. Since in this case, none of the waves are

breaking, this case is termed the purely undular bore. If the bore strength exceeds 0.28, then the leading

wave behind the transition front starts to break. If the bore strength exceeds 0.75, a fully turbulent bore

appears. The main purpose of this lecture is to explore whether the ratio 0.28 can be found using some

fairly simple wave models such as Boussinesq systems.

Some recent results on the dynamics of a mechanically-forced spilling

breaker

Maurizio Brocchini, Alessia Lucarelli, Claudio Lugni

We propose an extension of the analyses illustrated at B’Waves 2016 on the dynamics that

characterize a mechanically-forced spilling breaker.

With the aim of better understanding the evolution of a strongly unsteady and curved spilling breaker

and collect information to validate the analytical model by Brocchini and co-workers [1,2,3], we focus

on specific dynamics that characterize: 1) the breaking onset, 2) the mean flow structure and 3) the

turbulence structures.

Mean and turbulent quantities of interest have been evaluated through a massive PIV experimental

investigation of an unsteady spilling breaker evolving in a sloshing tank [4]. Preliminarily the overall

repeatability of the phenomenon has been accurately checked.

Results will be proposed on the importance of extra rates of strain for an unsteady, curved spilling

breaker in opposition to what occurs in a hydraulic jump, a flow that is very often regarded as similar to

a spilling breaker and used as a proxy to investigate the turbulence of a breaker.

For the first time, the study assesses that for a spilling breaker: a) the extra rate of strain induced by

the streamline curvature (𝜕𝑉 𝜕𝑠⁄ ) is of the same order of magnitude of the mean simple shear (𝜕𝑈 𝜕𝑛⁄ );

2) the streamline curvature influences the turbulent structure of the flow. This significantly differs from

the observations by Misra and co-workers [3] for the hydraulic jump flow, where the importance of the

streamline curvature is ten times less than the mean simple shear.

References

[1] M. Brocchini, Flows with freely moving boundaries: the swash zone and turbulence at a free surface,

Ph.D. thesis, University of Bristol, School of Mathematics. (1996).

[2] S. K. Misra, J. T. Kirby, M. Brocchini, M. Thomas, F. Veron, C. Kambhamettu, Extra strain rates in

spilling breaking waves, in: Coastal Engineering Conference, Vol. 29, Asce American Society of Civil

Engineers, 2004, p. 370.

[3] S. K. Misra, M. Brocchini, J. T. Kirby, Turbulent interfacial boundary conditions for spilling

breakers, in: Coastal Engineering Conference, Vol. 30, Asce American Society of Civil Engineers, 2006,

p. 214.

[4] A. Lucarelli, C. Lugni, M. Falchi, M. Felli, M. Brocchini, On a layer model for spilling breakers: A

preliminary experimental analysis, In European Journal of Mechanics - B/Fluids, 2017, , ISSN 0997-

7546, https://doi.org/10.1016/j.euromechflu.2017.07.003.

Effects of 3D current structure on 2D horizontal circulation in surf zone

flows

James T Kirby

Center for Applied Coastal Research, Department of Civil and Environmental Engineering, University

of Delaware, Newark DE 19716 USA, kirby@udel.edu

Fengyan Shi

University of Delaware

Gangfeng Ma

Old Dominion University

Morteza Derakhti

Johns Hopkins University

Recent advances in numerical simulation of surf zone flows have provided a growing wealth of

information on the three dimensional (3D) structure of these flows. Simulations with wave-resolving

models have verified earlier predictions of the importance of the spatial structure of individual breaking

wave events in determining the scales of horizontal flow structures in the turbulent surf zone. Purely

wave-averaged models are known to under- predict the breakdown of coherent current patterns such as

shear waves in a number of observed cases, thereby suppressing the transition to a quasi-2D turbulence

field and it’s subsequent evolution through inverse cascade effects. The additional incorporation of

quasi-3D effects in 2D models, or the application of a full 3D wave-averaged model, improves

knowledge of the vertical structure of mean flows but further suppresses the transfer of energy to small

scale eddies. In contrast, wave-resolving, 2D Boussinesq models such as FUNWAVE are able to predict

the spatial structure and energetics of smaller scale flow structures, since the spatial pattern of breaking

and wave energy decay is resolved. These models have also been shown, however, to over-predict

mixing effects in comparison to results derived from dye or drifter studies. In this presentation, we use

the 2D Boussinesq model FUNWAVE and the 3D non- hydrostatic model NHWAVE to carry out

detailed simulations for several data runs from the Sandyduck experiment. These cases are chosen to

fall in the range of wave conditions where 2D wave-averaged models fail to predict the evolution of a

turbulent current field. We examine the effects of including 3D structure on apparent horizontal mixing,

and we examine the generation and persistence of the 3D vorticity field in order to examine the

importance of the two horizontal components (neglected in Boussinesq modeling) and their role in

organized coherent structures. We also examine the structure of the dissipation of organized wave

energy during wave breaking, in order to address the proper spatial distribution of dissipation-related

forcing in wave-averaged models.

Long wave runup on plane and “non-reflected” beaches Efim Pelinovsky1-4), Ira Didenkulova2,5) and Artem Rodin2)

1) Institute of Applied Physics, Nizhny Novgorod, Russia

2) Nizhny Novgorod State Technical University n.a. R. Alekseev, Nizhny Novgorod, Russia

3) National Research University - Higher School of Economics, Moscow, Russia

4) Special Research Bureau for Automation of Marine Researchers, Yuzhno-Sakhalinsk, Russia

5) Department of Marine Systems, Tallinn University of Technology, Tallinn, Estonia

Runup of long sea waves on two kinds of coastal geometry is considered in the framework of the

nonlinear shallow-water theory taking into account the wave breaking effects. The first slope is a plane

beach widely used in laboratory and numerical experiments. The second one presents the so-called “non-

reflected” beach with the profile h(x)~x4/3 (where h is water depth and x is the offshore coordinate).

For the waves of very small amplitude the shallow-water equations are solved analytically, and it is

shown that the runup height on the non-reflected beach is bigger than on a plane beach. For large

amplitude waves the numerical solution is obtained with the use of the CLAWPACK code. The breaking

effects leading to the runup height decrease are significant on the non-reflected beach.

On the breaking onset of unsteady water wave packets evolving in the

presence of constant vorticity Michael Banner & Julien Touboul

The recent numerical study of Barthelemy et al. (2018) investigated the local properties of 2D and

3D nonlinear unsteady gravity wave packets in deep and uniform intermediate depth water. Their

study focused on the breaking onset transition zone separating maximum recurrence and marginal

breaking, and reported that a suitably normalized energy flux localized at the steepest crest in the

packet provides a robust breaking onset threshold parameter.

Our present study uses the fully-nonlinear BIEM solver developed by Touboul & Kharif (2016) to investigate breaking onset of 2D deep water nonlinear water wave packets propagating in the presence of a background current that varies linearly with depth. We seek to validate whether the proposed generic breaking onset threshold holds for the case of constant background vorticity. Results will be presented for different packet bandwidths and background vorticity levels.

The effect of wave breaking in numerical simulations of irregular sea waves

A. Slunyaev and A. Kokorina

Institute of Applied Physics, Nizhny Novgorod, Russia

and

Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia

The present state of the computer performance and efficiency of the numerical algorithms

makes possible direct numerical simulations of ensembles of irregular sea waves by phase-resolving

dynamical equations including the primitive hydrodynamic equations. This approach helps to obtain

reliable data on the surface wave statistics in controllable conditions. The rogue wave problem is a

particular task which has been tackled with the help of the direct numerical simulations of random

waves. The probability distribution functions for wave heights in given sea states are practically

important information which potentially may be obtained through these simulations. Effects of

nonlinearity alter significantly the probability distributions and may lead to the occurrence of extreme

sea states with enhanced likelihood of high waves. Rogue waves are still relatively rare events even in

rough sea states, though the effects of wave breaking start to play an essential role. Thus, in steeper

sea states the probability of high waves is expected to grow, while on the other hand the breaking

phenomenon limits the wave growth and hence decreases the probability of high waves.

Consequently, the appropriate accounting for wave breaking is essential for the accurate estimate of

the extreme wave probability.

It is clear that the fast codes used for the direct simulation of irregular waves cannot resolve

the wave breaking and require a parameterization of this effect. As emphasized above, an accurate

parameterization should be crucial for the adequate evaluation of the wave statistics. Physically wrong

parameterization would result in incorrect probability.

Our previous simulations of irregular sea waves with the JONSWAP spectrum in deep and

finite-depth basins were performed in the setting of strictly unidirectional waves [1-3]. The wave

breaking events of moderately steep waves were relatively seldom, so that rich wave statistics was

collected from purely non-breaking simulations. When the angle distribution is allowed (directional

waves), occasional breaking becomes much more frequent; then the regularization of wave breaking

is absolutely necessary.

In this paper two general approaches for the wave breaking regularization are discussed for

examples of strongly nonlinear 3D simulations of surface water waves: i) by virtue of the artificial hyper

viscosity, and ii) with the use of spectral filters at short scales. The focus is made on the effect of the

introduced regularization on the extreme wave statistical properties, and also on the robustness of the

simulated wave dynamics.

The research is supported in parts by RSF Grant No. 16-17-00041 and RFBR grant No. 16-55-52019.

[1]. A. Sergeeva (Kokorina), A. Slunyaev, Rogue waves, rogue events and extreme wave kinematics in

spatio-temporal fields of simulated sea states. Nat. Hazards Earth Syst. Sci. 13, 1759-1771 (2013).

[2]. A. Slunyaev, A. Sergeeva (Kokorina), I. Didenkulova, Rogue events in spatiotemporal numerical

simulations of unidirectional waves in basins of different depth. Natural Hazards 84(2), 549-565

(2016).

[3]. A.V. Slunyaev, A.V. Kokorina, Soliton groups as the reason for extreme statistics of unidirectional

sea waves. J. Ocean Eng. Marine Energy 3, 395-408 (2017).

Turbulent and Wave-Induced velocity fields over Wind-driven Surface

Waves Fabrice Veron, Kianoosh Yousefi, Marc Buckley, Nyla Hussain, and Tetsu Hara

In recent years, the exchange of momentum and scalars between the atmosphere and the ocean has been

the subject of several investigations. Although the role of surface waves on the air-sea momentum flux

is now well established, detailed quantitative measurements of the turbulent and wave-induced fields in

the airflow over surface waves remain scarce. The current incomplete physical understanding of the

airflow dynamics impedes further progress in developing physically based parameterizations for

improved weather and sea state predictions, particularly in high winds and extreme conditions.

Using a combined Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF) system, we

obtained laboratory measurements of the airflow velocity above surface waves for wind speeds ranging

from 0.86 to 16.63 m/s. The mean, turbulent, and wave-coherent velocity fields and then extracted from

instantaneous measurements.

In strongly forced cases in high wind speeds, Individual airflow separation events generate turbulence

and vorticity in the bulk flow. There, phase averaged turbulent Reynolds stress forms a negative-positive

pattern along the wave crest with a separation-induced maximum above the downwind side of the wave.

Concurrently, the wave-induced stress near the surface is a significant fraction of the total stress. At

lower wind speeds and larger wave ages, the wave-induced stress is positive very close to the surface,

below the critical height and decreases to a negative value further above the critical height. This indicates

a shift in the direction of the wave-coherent momentum flux across the critical layer. Our measurements

will be discussed in the context of available previous experimental, theoretical and numerical results.

Simulation of breaking waves in a High-Order Spectral model Seiffert, B.R. and Ducrozet, G.

We discuss the implementation of a wave-breaking mechanism into two non-linear potential flow

solvers HOS-NWT and HOS-ocean. These models are computationally efficient, open source codes

which solve for the free surface in a numerical wave tank and an open fluid domain, respectively, using

the High-Order Spectral (HOS) method. The goal of implementing a wave breaking mechanism into the

HOS solvers is to approximate the free surface as a single value. To do this, the solver identifies wave-

breaking onset using the criterion suggested by Barthelemy et al. (submitted), then removes energy by

adding a viscous dissipation term to the free surface boundary conditions, as introduced by Tian et al.

(2010, 2012).

Unidirectional wave breaking has been validated in HOS-NWT for single and multiple breaking waves,

and for irregular waves in two-dimensions, by a series of experiments conducted at the Hydrodynamics,

Energetics & Atmospheric Environment Lab (LHEEA) at École Centrale de Nantes (ECN). The model

has demonstrated success in calculating surface elevation and corresponding frequency/amplitude

spectrum before and after breaking events, as well as wave-breaking onset time, location and energy

dissipation (Seiffert et al., 2017, Seiffert & Ducrozet 2017).

The success of the wave-breaking model in a unidirectional wave field provides the basis for application

of the model in a multidirectional wave field. Once a multidirectional wave breaking mechanism is

validated in HOS-NWT, it can be incorporated into HOS-ocean as well as other nonlinear potential flow

models. This will result in a powerful and efficient computational tool which can solve for the large

scale evolution of nonlinear sea states, including breaking waves. These models provide a useful tool in

the study of extreme sea states, nonlinear wave phenomena, the development of rogue waves, dynamic

response of offshore vessels and marine renewable energy devices, and predicting loads on marine

structures, for example.

References

Barthelemy, X., Banner, M. L., Peirson, W. L., Fedele, F. Allis, M., Dias, F. (submitted). On a unified

breaking onset threshold for gravity waves in deep and intermediate depth water. Journal of Fluid

Mechanics, arXiv:1508.06002v2 [physics.ao-ph].

Seiffert, B. R., Ducrozet, G. & Bonnefoy, F., (2017). Simulation of breaking waves using the High-

Order Spectral method with laboratory experiments: wave-breaking onset. Ocean Modelling, doi:

10.1016/j.ocemod.2017.09.006.

Seiffert, B. R. & Ducrozet, G., (2017). Simulation of breaking waves using the High-Order Spectral

method with laboratory experiments: wave-breaking energy dissipation. Ocean Dynamics, doi:

10.1007/s10236-017-1119-3.

Tian, Z., Perlin, M., & Choi, W. (2010). Energy dissipation in two-dimensional unsteady plunging

breakers and an eddy viscosity model. Journal of Fluid Mechanics, 655, 217-257.

Tian, Z., Perlin, M., & Choi, W. (2012). An eddy viscosity model for two-dimensional breaking waves

and its validation with laboratory experiments. Physics of Fluids, 24(3), 036601.

Vorticity effect on the onset of breaking waves in shallow water Malek Abid and Christian Kharif

IRPHE

Two-dimensional nonlinear gravity waves travelling in shallow water on a vertically sheared current of

constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyperbolic

equations for the surface elevation and the horizontal velocity are derived. Using Riemann invariants of

these equations that are obtained analytically, a closed-form nonlinear evolution equation for the surface

elevation is derived. A dispersive term is added to this equation using the exact linear dispersion relation.

Within the framework of this new model and Whitham equations a study of the effect of vorticity on the

onset of breaking wave is carried out.

Study on Acceleration Features of Long Waves over Sloping Beaches

- A comparison between Solitary Wave and Undular Bore

Chang Lin*, Wei-Ying Wong, Ming-Jer Kao

Department of Civil Engineering, National Chung Hsing University, Taichung, Taiwan

*Corresponding Author: chenglin@nchu.edu.tw

One of the burning issues in civil engineering is the evolution of free-surface flows near shoreline

when the incident oscillations of considerable water masses caused by long waves, like tsunami,

bores and storm surf, can eventually result in tremendous destructions and casualties. Most of such

damages are related to run-up and run-down motions of long waves around the shoreline. Therefore,

understanding the features of related hydrodynamics during the whole process is very important for

the evaluation of any kind of mitigation effort.

Solitary wave and dam-break generated bore experiments were conducted in a 14.0 m long glass-

walled wave flume. A piston-type wave generator driven by a precision servo-motor was used to

generate the solitary waves with the incident wave-height to water-depth ratio varying from 0.01 to

0.04. Furthermore, undular bores were generated right downstream of a suddenly lifted gate (over

which its supporting frame-structure was mounted upon the wave flume). The generated bores travel

first over a horizontal bottom and then propagate over a 1:20 sloping beach. The ratio of the upstream

water depth to the downstream one ranges from 1.5 to 1.8 to allow the generation of undular bores. Flow

visualization technique and high-speed PIV were employed to observe the flow patterns and velocity

fields of both solitary waves and undular bores.

Based on visualized images captured and velocity fields measured near the still-water shoreline, the

following flow features at different stages of solitary wave and undular bores will be explored: (1) The

temporal and spatial variations in the velocity fields during run-up and run-down stages; (2) Flow

reversal zone and evolution of vortex structure close to the sloping-beach surface; (3) Variation

characteristics of acceleration and pressure gradient; (4) Comparison of acceleration and deceleration

features for solitary wave and undular bore for addressing prominent distinction in between.

Furthermore, the effect of breaking leading waves on the velocity fields of undular bore will be also

discussed.

On the connection between whitecap foam signatures and breaking wave

energy dissipation.

A. H. Callaghan

Air entraining breaking waves, also known as whitecaps, are thought to be dominant mechanism

dissipating surface wave energy in deep water. Due to the broadband scattering of light associated

with the sub-surface bubble plume and the associated surface foam, whitecaps are readily identifiable

features on the ocean surface. As such, detection of whitecaps present an unambiguous way to detect

surface wave breaking in digital images of the sea surface. Consequently, sea surface imaging for

whitecaps has long been used to generate statistics of average quantities such as total whitecap

coverage of the sea surface, or more detailed phase-averaged descriptions of breaking crest lengths

following the Phillips “lambda” distribution framework. Such measurements have provided a wealth

of knowledge relating to various aspects of oceanic breaking covering breaking wave kinematics,

dynamics and as well as providing useful data for scaling exchange processes such as marine aerosol

production and gas exchange. However, there is a paucity of data related to the severity of, or total

energy dissipated by, individual oceanic whitecaps, and the exact relationship between oceanic

whitecap coverage, breaking wave energy dissipation and wind energy input rate remains unclear.

In this talk I will present results from a laboratory experiment aimed at calibrating the whitecap foam

signature measured using digital cameras to infer the total energy dissipated by individual breaking

wave whitecaps [Callaghan et al., 2016]. The experimental data show that the volume of the two-phase

flow integrated in time during active wave breaking is almost linearly proportional to the total energy

dissipated by breaking. The value of a turbulence strength parameter ( ) is determined relating the

breaking wave “volume-time-integral” to total energy dissipation. When data from 3 previous

experimental campaigns are included, the value of is found to be relatively constant over almost 3

orders of magnitude in measured breaking wave energy dissipation. Using a whitecap decay time as a

proxy for bubble plume penetration depth allows estimates of individual whitecap energy dissipation

to be inferred from measurements of surface whitecap foam. The experimental results are used to

formulate an energy-balance model for oceanic whitecap coverage, with model estimates compared to

in-situ whitecap coverage measurements made in the North

Atlantic at wind speeds up to 23 m/s. Best agreement is found when bubble plume penetration depth

increases with increasing forcing and energy dissipation due to other processes is also accounted for.

Reference:

Callaghan, A. H., G. B. Deane, and M. D. Stokes (2016), Laboratory air entraining breaking waves:

Imaging visible foam signatures to estimate energy dissipation, Geophys. Res. Lett., 43,

doi:10.1002/2016GL071226.

Implementation and test of a modeling strategy for depth-induced breaking

in fully nonlinear potential flow models

Christos E. Papoutsellis (a) (b), Bruno Simon (a), Michel Benoit (a) (b) and Marissa L. Yates (c)(d)

(a) Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), Marseille, France

(b) École Centrale Marseille, Marseille, France

(c) Saint-Venant Hydraulics Laboratory, Université Paris-Est, Chatou, France

(d) Cerema, Water, Sea and Rivers, Margny-les-Compiègne, France

A modeling approach is tested for the treatment of breaking waves in the coastal zone, within the limits

of numerical wave models based on fully nonlinear potential flow theory. Assuming irrotational flow

and a non-overturning free surface, fully nonlinear models such as the Hamiltonian Coupled-Mode

System (HCMS) [1] and models using a spectral approach in the vertical direction [2], are not able to

predict the overturning and breaking of waves and their post-breaking evolution.

In order to overcome this limitation, a strategy is adopted involving a breaking wave identification

algorithm, based on certain local criteria such as the slope or vertical velocity of the free surface

elevation. The identified breaking waves are forced to dissipate energy through the action of an

appropriate external surface pressure applied locally, which is computed by assuming an analogy

between a breaking wave and an hydraulic jump [3]. In order to assess the performance of the present

approach, the calculations are compared with experimental measurements of plunging or spilling

breakers over barred and sloping bathymetries (e.g. [4], [5]). Numerical aspects and the sensitivity of

the method to free parameters (e.g. thresholds for breaking initiation and termination, duration and

spatial extent of the pressure term) will be discussed.

References:

[1] Ch. Papoutsellis, G. Athanassoulis, A new efficient Hamiltonian approach to the nonlinear

water-wave problem over arbitrary bathymetry, (Submited)). (2017).

http://arxiv.org/abs/1704.03276.

[2] C. Raoult, M. Benoit, M.L. Yates, Validation of a fully nonlinear and dispersive wave model

with laboratory non-breaking experiments, Coast. Eng. 114 (2016) 194–207.

doi:10.1016/j.coastaleng.2016.04.003.

[3] S. Guignard, S. Grilli, Modeling of wave shoaling in a 2D-NWT using a spilling breaker

model, in: ISOPE 2001, Stavanger, Norway, 2001.

[4] S. Beji, A. Battjes, Experimental investigation of wave propagation over a bar, Coast. Eng. 19

(1993) 151–162. doi:10.1016/0378-3839(93)90022-Z.

[5] F.C.K. Ting, J.T. Kirby, Observation of undertow and turbulence in a laboratory surf zone,

Coast. Eng. 24 (1994) 51–80. doi:10.1016/0378-3839(94)90026-4.

Review of wave breaking characteristics

Florian DESMONS1*, Maria KAZOLEA2, Pierre LUBIN1

and Mario RICCHIUTO2

1 Laboratoire I2M, Université de Bordeaux, France

2 Centre de Recherche INRIA Bordeaux Sud-Ouest, France

* florian.desmons@u-bordeaux.fr

Surface wave breaking, occurring from the ocean to the coastal zone, is a complex and

challenging two-phase ow phenomenon which plays an important role in numerous processes,

including air-sea transfer of gas, momentum and energy. Recent modeling attempts are

struggling with the lack of physical knowledge of the _nest details of the breaking processes.

Furthermore, no universal scaling laws for physical variables have been found so far. Hence,

parameterizing breaking effects becomes very difficult. The pre- and post-breaking events can

be quantified, detected, qualified (breaking detection, breaker classification, breaker intensity

estimation, energy dissipation evaluation, etc.). We aim at presenting and discussing the

common practices, and highlight the gaps and limitations.

References [1] B. Robertson, K. Hall, R. Zytner, and I. Nistor, Breaking waves: review of characteristic

relationships, 55 (2013).

[2] M. Kazolea, A. Delis, and C. Synolakis, Numerical treatment of wave breaking on

unstructured finite volume approximations for extended boussinesq-type equations, J. Comput.

Phys. 271, 281-305 (2014).

[3] P. Lubin and H. Chanson, Are breaking waves, bores, surges and jumps the same flow?

Environmental Fluid Mechanics 17, 47-77 (2017).

Wave propagation and wave breaking using adaptive mesh refinement

method and coupled models

Frédéric Golay1, Kevin Pons1,2, Mehmet Ersoy1

1 IMATH, Université de Toulon, CS 60584, 83041 Toulon cedex 9, France

2 Principia S.A.S., Zone Athélia 1, 215 voie Ariane, 13705 La Ciotat cedex, France

In the waves propagation and waves breaking context, efficient numerical methods are necessary to

simulate multi scale events. Such numerical modelization is always a compromise between numerical

accuracy, physical model’s relevance and computational cost. One way to reduce the computational cost

is to use an adaptive mesh refinement method on unstructured meshes. The adaptive mesh refinement

method follows a block-based decomposition [2,3,5], which allows quick meshing and easy

parallelization. The mesh refinement parameter, based for example on the numerical entropy production,

benefits of an automatic thresholding which allows to determine appropriated mesh refinement

parameters. This approach is used here with a finite volume scheme solving the multi-dimensional Saint-

Venant system [4,6] and isothermal bi-fluid Euler system [1]. We propose also a method to couple in a

two way nesting approach those two models.

References (1) F. Golay, P. Helluy, “Numerical schemes for low Mach wave breaking”, International Journal of

Computational Fluid Dynamics, vol.21 n°2, pp 69-86, Février 2007.

(2) Ersoy M., Golay F., Yushchenko L., “Adaptive multi-scale scheme based on numerical entropy

production for conservation laws”, Central European Journal of Mathematics, 11(8), 1392-1415, 2013.

(3) Golay F., Ersoy M., Yushchenko L., D. Sous, “Block-Based Adaptive Mesh Refinement scheme

using numerical density of entropy production for three-dimensional two-fluid flows”, International

Journal of Computational Fluid Dynamics, volume 29, issue 1, pp67-81, 2015.

(4) Marcer R., Pons K., Journeau C., Golay F., “Validation of CFD models for tsunami simulation.

TANDEM Project”, Revue Paralia, Vol. 8, pp n04.1–n04.6, 2015.

(5) Altazin T., Ersoy M., Golay F., Sous D., Yushchenko L., “Numerical investigation of BB-AMR

scheme using entropy production as refinement criterion”, International Journal of Computational Fluid

Dynamics, 2016.

(6) Pons K., Golay F., Marcer R., “Adaptive mesh refinement method applied to shallow water model:

a mass conservative projection”, 17ème conférence “Topical problems of fluid mechanics”, Prague,

Février 2017.

Experimental study on the influence of directional spread on wave

breaking

Yuxiang Ma1, Dianyong Liu1, Marc Perlin2, Guohai Dong1

1State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian,

116024, China

2Department of Ocean Engineering, Texas A&M University, Galveston, TX, 77554, USA

There are many experiments focusing on 2D wave breaking, and all previous studies

expanded the knowledge of wave breaking. However, in the open ocean, waves usually

propagate in multi-directions, which will cause a large difference in wave breaking. Until

now, the dynamics of fully 3D wave breaking has far from being understood due to its

complexity. Liu et al. (2015) designed an experiment with an ‘X’ configuration (Figure 1) to

study wave breaking induced by two identical wave groups which were propagating with an

intersecting angle of 16o (2). It is shown that, compared with unidirectional results, the

directional spreading plays an important role in wave breaking. Very recently, a new

experiment has been conducted in a configuration with an intersecting angle of 24o (2) to

examine the influence of directionality further. Therefore, the data obtained in the two

experiments is used in the present study to reveal the propagation directionality on the wave-

wave interactions and the wave breaking. The nonlinear energy transfer during the wave-wave

interactions is particularly sensitive to the directional spread. Furthermore, it is found that the

severity of breaking is depending on propagating directionality. Greater breaking occurs when

the wave packets propagate with a relatively larger angle, and hence the energy loss increases

with the increase of the approach angle generally (Figure 2).

Figure. 1. Experimental set-up for the surface elevation measurements. (Not to scale.)

Figure 2. Energy dissipation as a function of wave steepness. The squares represent the case with θ = 8°; the

circles represent the case with θ = 12°.

Modelling shoaling and breaking waves on a mild sloping beach

Gaël Richard, Maria Kazakova, Arnaud Duran

The mathematical modelling of coastal wave is a quite challenging issue since it is difficult to describe

in the same model the dispersive effects in the shoaling zone and the energy dissipation of breakers in

the surf zone. In this study we propose a new model for waves propagation over a mild slopping

topography. Shearing and turbulence effects in breaking waves are taken into account by a third variable

called enstrophy. With Teshukov’s hypothesis of weakly sheared flows, the system is closed and features

depth-averaged balance equations for mass, momentum and energy. In the absence of enstrophy, the

system reduces to the equations of Green-Naghdi. Enstrophy production is handled with a turbulent

viscosity hypothesis and enstrophy dissipation is governed by an empirical law. Since the model is

dispersive and not hyperbolic, the enstrophy equation can replace conveniently the energy equation for

the numerical resolution. The equations were numerically solved with the strategy of Le Métayer et al.

(2010). The scheme is rewritten for the new variables and allows us to use a hybrid method which consist

in the resolution of a hyperbolic system by a Godunov-type method and an elliptic equation. The non-

dissipative part of the model possesses a solitary wave solution which is confirmed numerically. The

numerical simulations were successfully compared to the experimental data of Hsiao et al. (2008).

Multi-dimensional shear shallow water flows

S. Gavrilyuk1

1Aix{Marseille Université, CNRS, IUSTI, UMR 7343, 5 rue E. Fermi, 13453 Marseille Cedex 13, France,

sergey.gavrilyuk@univ-amu.fr

Abstract

The multi-dimensional equations of shear shallow water flows represent a 2D hyperbolic non-

conservative system of equations which is reminiscent of generic Reynolds averaged equations for

barotropic turbulent fluids. The model has three families of characteristics corresponding to the

propagation of surface waves, shear waves and average flow.

I present a new splitting technique to define the weak solutions to this non-conservative system of

equations. The full system is split into several subsystems for which the notion of the weak solution is

almost classical. Each split subsystem contains only one family of waves (either surface or shear waves)

and contact characteristics. The accuracy of such an approach is tested on the exact 2D solutions

describing the flow where the velocity is linear with respect to the space variables, and on the solutions

describing 1D roll waves. The capacity of the model to describe multi-D experimentally observed

phenomena was shown: ‘fingering’ of plane one-dimensional wave fronts, and the formation of

singularities in radially convergent flows.

I will also discuss the perspectives of such a `splitting approach' for the description of 3D compressible

turbulent flows.

This is joint work with K. Ivanova and N. Favrie.

References

[1] 2017 S. Gavrilyuk, K. Ivanova, N. Favrie, Multi-dimensional shear shallow water flows:

problems and solutions (submitted).

On breaking wave modelling with pseudo-compressible and mixture theory

models

Onno Bokhove

School of Mathematics, University of Leeds, UK

Abstract

The modelling of wave breaking will be explored using two models: a pseudo-compressible model based

on a Van-der-Waals-type equation of state and a mixture theory model, both with potential-flow theory

as limiting systems.

Lagrangian transport by breaking deep-water surface waves

Nicholas Pizzo, Luc Deike, and Ken Melville

Abstract

The Lagrangian transport due to breaking deep water surface gravity waves is examined using

theoretical, numerical, and observational studies. First, a theoretical criterion for particles to surf an

underlying surface gravity wave is presented. It is found that particles traveling near the phase speed of

an underlying wave, in a geometrically confined region on the forward face of the crest, increase in

speed. Next, this theory is employed to study the Lagrangian transport due to non-breaking and breaking

waves. Direct numerical simulations of focusing wave packets are presented, and it is found that

breaking may enhance drift by up to an order of magnitude, compared to the classical Stokes drift

prediction. A simple scaling argument implies that the drift due to breaking is proportional to a measure

of the slope of the waves as compared to the unbroken Stokes drift which scales with the slope squared,

where this measure of the slope is always less than 1. Finally, this model is combined with ocean field

data of breaking wave statistics to estimate the Lagrangian drift due to breaking. This is compared with

the classical prediction of Stokes drift, and it is found that breaking may provide a large contribution to

the wave induced mass transport. Implications for a better description of upper ocean processes are

discussed.

EXPERIMENTAL INVESTIGATION MICROSCALE BREAKINGWAVES IN TWO-PHASE

PIPE FLOW

Vollestad Petter1, Jensen Atle1, Ayati A. Anis1

1Department of Mathematics, University of Oslo, N-0316 Oslo, Norway

Abstract

We present an experimental study of stratified gas-liquid pipe flow conducted at the Hydrodynamics

laboratory, University of Oslo. The experimental setup consists of a 31 meter long, 10 cm diameter pipe,

the test fluids are air and water at atmospheric conditions. Simultaneous two-phase particle image

velocimetry (PIV) is used to evaluate two-dimensional velocity fields along the center plane of the pipe.

In addition, wave statistics are acquired using conductance wave probes. For further details on the

experimental setup see [2, 1]

(a) (b)

Figure 1. a) Wavy air-water flow inside a horizontal pipe. b) Overview of experimental setup.

Previous analysis [2] has shown that at a given superficial liquid velocity increasing the gas flow rate

will eventually lead to a regime of amplitude saturation, where the rms amplitude of the wave field is

independent of gas velocity. Based on visual observation in the pipe, microscale breaking of the waves

is assessed to be a likely cause of the observed amplitude saturation. In the present work we analyse the

possible breaking waves using PIV in both the gas and liquid phase. The goal of this study is to establish

where in the Usg/Usl flow map microscale breaking occurs, and at what frequency. We also want to

investigate whether airflow separation previously observed above waves in the amplitude saturation

regime is related to the onset of microscale breaking waves.

References

[1] A.A. Ayati, J. Kolaas, A. Jensen, and G.W. Johnson. A PIV investigation of stratified gas-liquid flow in a

horizontal pipe. International Journal

of Multiphase Flow, 61:129 – 143, 2014.

[2] A.A. Ayati, J. Kolaas, A. Jensen, and G.W. Johnson. Combined simultaneous two-phase {PIV} and interface

elevation measurements in stratified

gas/liquid pipe flow. International Journal of Multiphase Flow, 74:45 – 58, 2015.