# Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

## Abstract

Fundamentals of nonlinear wave-particle interactions are studied experimentally in a Hele-Shaw configuration with wave breaking and a dynamic bed. To design this configuration, we determine, mathematically, the gap width which allows inertial flows to survive the viscous damping due to the side walls. Damped wave sloshing experiments compared with simulations confirm that width-averaged potential-flow models with linear momentum damping are adequately capturing the large scale nonlinear wave motion. Subsequently, we show that the four types of wave breaking observed at real-world beaches also emerge on Hele-Shaw laboratory beaches, albeit in idealized forms. Finally, an experimental parameter study is undertaken to quantify the formation of quasi-steady beach morphologies due to nonlinear, breaking waves: berm or dune, beach and bar formation are all classified. Our research reveals that the Hele-Shaw beach configuration allows a wealth of experimental and modelling extensions, including benchmarking of forecast models used in the coastal engineering practice, especially for shingle beaches.

### Keywords

Hele-Shaw cell Laboratory experiments Mathematical design Shingle beaches Potential flow and shallow water simulations## 1 Introduction

### 1.1 A slice of beach

Natural beaches, including those partially enhanced or even completely man-made, are defensive zones that protect large tracts of the world’s coastlines against storm surges. This is especially the case in low-lying deltas such as the western and northern parts of The Netherlands, with its combination of a partially and totally reinforced coastline of beaches and dunes, and dikes. The dynamics of the surf zone, concerning the wet zone from the beach to the offshore line where the white capping of wave breaking starts, are very important for understanding how beaches, berms and sand or shingle banks form and erode. Good predictions of surf zone dynamics and the associated particle transport can reveal where suppletion of particulate material will be required to protect the coast, or where dredging is needed to keep navigational channels into harbours open.

Coastal engineers and fluid dynamicists have made great progress in formulating models for waves and currents, and particle transport in and around the surf zone. This progress has provided the fundamentals for operational forecast models such as Telemac, Delft-3D and Xbeach [22]. Current forecast models are founded on decades of research on wave dynamics and beach erosion, especially during storms [20, 27], on the formulation of the sediment and bedload flux transport laws [6, 9, 18, 29], and on direct mathematical analysis [14]. Despite great advances, the fundamental laws of how sand or gravel particles are picked up, transported and deposited by breaking waves remain relatively poorly understood at a fundamental level. This gap in knowledge motivates the goal of the presented work: demonstrating how a novel, compact Hele-Shaw beach configuration and laboratory experiment may shed new light on wave breaking induced particle transport.

### 1.2 Precursory results and goals

- (i)
to what extent is linear momentum damping the primary energy dissipation mechanism in the Hele-Shaw cell?

The primary advantage of the Hele-Shaw beach configuration is that everything becomes clearly visible: the dynamics of each particle and the free surface motion can be traced in time but not at the same time with the set-up used. Although the gap width can be adjusted to become a little over one to a few particle diameters wide, we presently limit ourselves to a gap width of circa 1.1 particle diameter. Particle beds investigated only involve on the order of ten thousand particles.

- (ii)
what breaker types do we observe in the Hele-Shaw cell?

- (iii)
what stable quasi-steady beach morphologies are observed in the Hele-Shaw beach as function of these three parameters?

## 2 Mathematics of experimental design

The main goal of this section is to determine the threshold gap width for which a broken wave can propagate from one end of the tank to the other. Waves generated at one end of the tank usually break when they reach the shallow region behind the wedge. The relevant dimensions concern a tank of a half to one metre in length and with a still water depth \(H_0\) of about 0–10 cm. We simplify the Navier–Stokes equations of motion for a fixed beach of gentle slope that remains dry onshore. First we scale the equations given the anisotropy of lateral, zonal and depth scales of space and velocity, and average them laterally assuming a Poiseuille-type flow profile across the gap. Subsequently, the resulting planar, incompressible Euler equations are depth-averaged, assuming hydrostatic balance to hold. The threshold gap width is found by using simulations of the resulting shallow water equations for suitable wave forcing.

### 2.1 Width and depth averaging

*Re*= 0.5–5. At leading order in \(\epsilon ^2\), (2) yields \(p=p(x,z,t)\) to be

*y*independent: only the viscous contribution in the

*y*direction remains of importance in the zonal and vertical momentum equations. Hence, the reduced equations consist of the underlined terms in (2). Next, we assume a balance between pressure gradients and the remaining viscous terms, such that dimensionally

*y*direction. The width-averaging step is followed by neglecting (Reynolds) stress terms. The resulting system in dimensional form reads

*y*independent pressure \(P=P(x,z,t)\) and \(\gamma =6/5\) for the quadratic flow profile used. The kinematic free surface and bottom boundary conditions at \(z=h(x,t)+b(x,t)\) and \(z=b(x,t)\) with water depth \(h=h(x,t)\) and bottom height \(b=b(x,t)\) are as follows

## 3 Experimental techniques

Particle properties. \({}^{*}\): Effective density with water-filled pores

Material | Diameter (mm) | Density (g/cm\(^{3}\)) | Porosity |
---|---|---|---|

Gamma Alumina | 1.75 \(\pm \) 0.1 | 2.08 \(\pm \) 0.2\({}^{*}\) | 0.53 \(\pm \) 0.05 |

Spherical Glass | 1.80 \(\pm \) 0.1 | 2.515 \(\pm \) 0.03 | 0 |

In both the wave sloshing validations and the wave breaking experiments, high speed cameras are used recording images from 500 to 1000 fps. The latter (beach morphology) experiments involve about \(10^{4}\) particles. In some cases, red dye is added to the water to increase the contrast between water and air. It has little discernible effect on the surface tension of water. The free-surface locations for arbitrary wave shapes are extracted from the video frames, using codes developed in MATLAB. At the start of a measurement series the set-up is flushed with clean MilliQ^{®} water to which in some cases surfactant was added. The surface tension is therefore not quite the same in all measurements.

For the experiments on beach morphology, photographs are taken every \(10\)s because the time scales involved are longer. The experiments are continued till a quasi-steady beach morphology emerges with little net variation over time. This is assessed visually. An automated analysis is conducted a posteriori to confirm or revoke this visual assessment. Snapshots from a typical measurement were shown in Fig. 2, in which the bed evolved from a flat state to a beach with a nearly constant slope. To analyse such beach formation, a particle tracking code has been developed in MATLAB, determining the location of each particle. The horizontal *x* direction is then divided into bins such that the highest (connected) particle in each bin defines the bed profile. To start each experiment a set procedure is followed. Particles are placed into the set-up filled with water a day in advance, such that particle pores saturate with water. The set-up is subsequently flushed with fresh MilliQ^{®} water, resulting in a nearly constant surface tension. The beach is levelled manually and the height \(B_0\) is measured. Subsequently, the set-up is drained and refilled with clean MilliQ^{®} . The range in this fixed room temperature was 23.5–\(28.7\,^\circ \text {C}\). The properties of water change only slightly with temperature in this range, and the influence on the results seems negligible. The beach morphology experiments are presented in Sect. 6. Further information can be found in Van der Horn [16].

## 4 Validating wave sloshing experiments

*a posteriori*with simple damped wave experiments in our Hele-Shaw cell without particles. We assume that the domain \(\varOmega \) consists of solid vertical walls at \(x=0\) and \(L_x\), and a fixed flat bottom at \(z=0\) together with a free surface at \(z=h(x,t)\). The next step is to impose the velocity field to satisfy \(\bar{{\varvec{u}}}=(\bar{u},\bar{w})={\varvec{\nabla }_{}}\phi \equiv (\partial _x\phi ,\partial _z\phi )\) with velocity potential \(\phi \) in (7). Consequently, the remaining (horizontal) vorticity component \(\partial _z\bar{u}-\partial _x\bar{w}=0\). Combined with the kinematic free surface equation (8a) with \(b=0\), the incompressibility condition yields Laplace’s equation with the dynamic and kinematic boundary conditions at \(z=h\), i.e.,

## 5 Breaking waves

Definition of the four wave breakers in the real world versus breakers in the Hele-Shaw set-up

Type | Description | Occurrence in cell |
---|---|---|

Spilling | Bubble-rich water appears at wave crest, spills down front face, sometimes proceeded by projected small jet | In our case bubbles collect at the face, no new bubbles are generated |

Plunging | Most of wave’s front face overturns and a prominent jet falls near the base of the wave, causing a large splash | Yes |

Collapsing | Lower portion of front face overturns and behaves like a truncated breaker | The bottom part of the breaker protrudes, but does not plunge |

Surging | No significant disturbance of the smooth wave profile occurs except near the moving shoreline | Yes |

Two series of experiments will be analysed, one in the first \(0.6\)m long Hele-Shaw tank with zeolite particles, and one in the \(0.96\)m tank with glass beads. In both experimental series, all four wave types are to a greater or lesser extent observed, with evidence for the collapsing breaker being the weakest.

### 5.1 Spilling

In the Hele-Shaw cell, the spilling breaker is characterised by pre-existing bubbles accumulating on the crest of the wave. Every cycle, the front face of the wave becomes almost vertical without full breaking. For real-world breakers, white water appears at the wave crest, indicating the presence of many small bubbles, and spills down the front face, sometimes preceded by the projection of a small jet. In contrast, neither bubble creation is observed at the interface when it is almost vertical, nor the creation of small jets.

### 5.2 Plunging

### 5.3 Collapsing

### 5.4 Surging

### 5.5 Iribarren number

Wave breaking in the first experimental series was filmed with a standard video camera recording at \(50\)fps during various stages of beach formation, using the horizontally moving wave-maker. In contrast to the second series, the bottom slope comprised of the zeolite or Gamma Alumina particles evolved naturally due to the action of the waves. A manual determination using an onscreen ruler and protractor of the wave height \(H\) (trough to crest) and a mean bottom slope \(\alpha \) near wave breaking reveals the following.^{1} A definite beach slope is difficult to define clearly in the zone where the wave breaks or starts to break. The corresponding estimated values \(\alpha =22,27,29,26^{o}\) of the beach slope from surging, collapsing, plunging to spilling breakers, respectively, are therefore prone to contain larger errors. The maximum wave heights \(H_b=4.2,4.0,3.8,6.0\) cm estimated are more accurate, as are the wave periods \(T=1.1,0.8,0.8\) and \(0.8\) s, respectively. Rough estimates of the resulting Iribarren numbers are \(I_b=1.9, I_b=1.7, I_b = 1.6\) and \(I_b=0.8\) for surging, collapsing, plunging and spilling breakers.

All four types of breakers are observed on the Hele-Shaw beach. They lead to a net particle transport and therefore evolution of the bathymetry. The time scale of this evolution is longer, on the order of minutes to an hour, than the time scale of the waves, which are on the order of one half to one-and-a half second. As expected, changes in the bathymetry have a strong effect on the wave dynamics, often leading to a change of breaker type. It is clear from the still photos shown in Fig. 7 that the bathymetry is different for each wave breaker. For the second series, it was not possible to extract the bottom profiles and extract the respective Iribarren numbers, due to the way the measurements were set up. We observe that: (i) the surging breaker occurs on a beach with two distinct angles; (ii) the collapsing breaker occurs in a beach with a submerged sand-bar (the small elevation under the vertical section of the wave profile); (iii) the plunging breaker is generated over a steep shallow section of the beach, whereas (iv) the spilling breaker rolls over a gently rising bathymetry. Note that these qualifications correspond roughly to to real-world situations, in which surging breakers are often long, low waves over steep bathymetry, while spilling breakers concern steep waves over mildly sloping bathymetry.

Broadly speaking, the plunging breaker transports a great deal of material onto the steep section of the beach; the collapsing breaker moves material from the apex of the submerged sand bar onshore; the spilling breaker is moving material all along the length of the bathymetry; and, finally, the surging breaker is moving material at the break in the bathymetric slope. The trend seems to be that most material is entrained by the wave when the depth is shallow and at locations where there is a change in shape of the bathymetry. The estimated Iribarren numbers are smaller than in nature but their ordering is appropriate.

## 6 Quasi-steady beach morphologies

Our next step is to demonstrate systematically how beaches and berms in the Hele-Shaw cell are formed by breaking waves. In total 80 measurements were performed to cover the parameter space spanned by wave-maker frequency \(f_{wm}\in [0.7,1.3]\) Hz, and the initially quiescent bed and water levels \(B_0\in [5,8]\) cm and \(W_0=H_0-B_0\in [1,8]\) cm. All measurements were performed in a semi-random order. This ensured that unforeseen variations of parameters not varied purposely, did not coincide with a gradual variation of the three parameters we did vary systematically. Since adding and removing particles to and from the tank is a slow process, mainly due to the porosity of the Gamma Alumina particles, measurements for each beach height \(B_0\) were performed successively. The results obtained are reproducible for two reasons: the phase diagram presented below shows a coherence that would otherwise be absent, and a total of five measurements was successfully repeated.

Quasi-steady state beach morphologies emerge on a timescale of minutes to an hour. On this timescale, the type of wave motion and wave breaking adjusts to the changing bed forms. Initially the bed is flat, except near the wedge, and waves may not break. Once the water depth becomes shallower, wave breaking either sets in or becomes more pronounced. In most cases, particles of the bed keep moving during the monochromatic wave cycles, so the beach morphology remains quasi-steady. The experiments are terminated when the state is judged to be quasi-steady and an automated analysis is performed a posteriori.

This analysis, undertaken to establish whether the bed morphology is quasi-steady, is as follows. The difference between the initial state and an evolved state consists of a negative area of moved sediment and a positive area of deposited sediment. These areas differ because the deposited area is more compact by a few percent. An effective distance travelled by the sediment is represented by the distance between the centres of mass of these areas. This distance evolves over time. The bed state is now described by the cubical transport as the product of the (negative) sediment area times the effective distance the sediment travelled. The time derivative thereof yields a cubical transport rate, for which the wave-maker period \(T_\mathrm{per}\) is used as the relevant time unit. Finally, a bed is quasi-steady when this cubical transport rate falls below a threshold of \(100\mathrm {mm}^3/T_{per}\). That rate corresponds roughly with three bed particles transported over a distance of 1 cm per wave-maker period.

A classification is given of steady bed morphologies (SBMs) in the Hele-Shaw cell

SBM | Definition | Quantification |
---|---|---|

Dry beach | Beach on onshore side | Dry maximum at boundary/wall |

Immersed beach | Beach on onshore side | Wet maximum at boundary/wall |

Dune/berm | Island formation | Dry interior maximum, water on either side |

Dune-beach | Dry beach with dune | Dry interior maximum with dry land beyond |

Immersed bar | No dry bed parts | Wet interior maximum |

Quasi-static | Sediment transport small | Wet state |

Suction | Particles sucked to wave-maker | Interior/boundary minimum with largely quasi-static bed |

*berm or dune*formation starts with an initial heap of newly-transported sediment that is formed just behind the wedge (Fig. 13a). Subsequently, this particle mass starts moving towards the shore and grows, until at some point it breaks through the free surface. Because the water depth is shallow, it induces sufficiently heavy wave breaking, with corresponding dissipation of energy, that further onshore sediment transport is arrested. A berm or dune is formed with wave breaking on its active shore and a calm lake on the other side. Due to the porous structure of the bed, the water level of the lake moves slightly up and down.

*Beaches* form when the initial water level is deeper (Fig. 13d). The particle accumulation generated early in the evolution travels to the wall, and keeps growing until a maximum, stable beach angle is reached. Sometimes, the water layer is shallow enough for the beach to emerge from the water, constituting a (partially) dry beach. When the water layer is too deep, the beach stays submerged and wet, because there are insufficient particles available to form a stable dry beach.

A *dune-beach* appears as a transitional form between the beach and dune regions of the parameter space. The bed evolves like the dry beach case, but once it reaches the water surface it switches to a dune-like evolution, giving a rising bed on the onshore side, leading to the formation of a sharp cliff (Fig. 13b).

*Suction states* are clearly grouped in the part of parameter space where the frequency is high and the water depth is low. During suction, strong offshore sediment transport occurs. Hence, the suction part of parameter space is separated from beach and dune states by the quasi-static and bar morphologies (Fig. 13c).

### 6.1 Bed activity and beach angles

The free water layer depth, \(W_0\), proves to be the most dominant parameter to determine the type of steady bed morphology. Dunes are observed at shallow water depths \(W_0=1\) cm, beaches at larger depths of about \(W_0=5\) cm, and hybrid dune-beaches at intermediate water layer depths of \(W_0=3\) cm. Bed dynamics is further determined by \(f_\text {wm}\) and \(B_0\). When the initial bed height \(B_0=5\) cm, enough sediment is transported to form beaches in only a few cases, while no dunes or dune-beaches are observed. Suction only occurs in one case. For larger values of \(B_0\), dunes and dune-beaches are formed and more beaches are created, but more instances of suction also occur. Hence, bed activity increases with increasing \(B_0\). We note that the height \(5\) cm of the fixed wedge between wave-maker and bed correlates with the most pronounced jump in bed activity. Concerning the wave-maker frequency, especially the measurements with \(B_0=5\) and \(7\) cm suggest a slight optimum in bed transport for \(f_{wm} \approx 0.9\) Hz. We note that the observed bed morphologies are quite reproducible: the phase diagram in Fig. 12 is coherent. It is clear that beaches, both immersed and dry, are formed when the waves can reach the end of the Hele-Shaw cell without dissipating their wave energy beforehand.

## 7 Conclusions and discussion

We presented the mathematical design of a Hele-Shaw cell for the study of bed dynamics by breaking waves. The design shows that damping can be controlled by calculating the gap width of the cell such that driven nonlinear wave motion survives across the tank, while greatly reducing the effects of turbulence and yielding very tractable dynamics.

We showed that all types of real-world wave breakers were also observed on the Hele-Shaw beaches, albeit in idealized forms due to the effects of surface tension. Iribarren numbers were roughly estimated from one of the measurement series, showing the right ordering but other (smaller) values than found on real-world beaches. Finally, a comprehensive parameter study of quasi-steady bed morphologies revealed definite trends in the parameters varied: the levels of the initially flat bed and water at-rest, together with the wave-maker frequency. We could thus identify distinct states at longer times in the bed evolution, such as berms/dunes and beach-dunes, dry and immersed/wet beaches, and bars.

More work is required to relate wave breaking to the bed shape underneath, and to the Iribarren number. Further investigation to assess the role of the wedge (used for technical reasons in the present study), the role of the length and width of the tank, and the role of the wave-maker is also desirable. A more elaborate video capturing system, such that the fast wave motion can (intermittently) be recorded alongside the recording of the natural long-time bed evolution, would be of value.

There are numerous and sensible variations to be made on the laboratory work. These include study of Hele-Shaw beach dynamics under a systematic increase of the gap width to a few particle diameters, and varying the material properties, such as particle properties (size, shape, and density), liquid properties (alcohol-water mixtures), and the effects of glass coatings (to reduce contact line effects).

We presented two preliminary models to enable and assess the mathematical design. Further research is required to extend these models to include the multiphase dynamics observed in the Hele-Shaw cell, in a more or less detailed or averaged manner. The advantage of the Hele-Shaw configuration remains that the quasi-two-dimensional nature of the set-up in principle allows the formulation, study and experimental validation of a hierarchy of models. These can range from the Navier–Stokes equations with explicit particle dynamics for brute-force calculations, to multiphase continuum models and their wave-, width- and depth-averaged versions. Finally, our Hele-Shaw methodology appears useful for benchmarking current wave and sediment forecast models used in coastal engineering [22].

Our beach profiles are remarkably similar to those found in Powell’s report [25] for shingle beaches in a \(42\times 1.5\times 1.4\mathrm{m}^3\) wave tank. The difference is that our mean slope is with circa 1:3 about two times steeper than the one in Powell [25]. This could possibly be attributed to the greatly diminished long wave reflections caused by the side wall friction. Akin to shingle beaches, beach porosity and porous flow play a visible role in our set-up. We studied the building of berms and beaches from nearly flat ground states. This seems to contrast with (most) larger scale laboratory and numerical studies in which the bed dynamics and total bed transport from the onset lie closer to the equilibrium profiles [32]. A comparison between models and data of Hele-Shaw beach experiments could therefore potentially cover a broader range of bed evolution by breaking waves.

## Footnotes

- 1.
Source file

*golfbakonno2.avi*contains surging, collapsing, plunging and spilling breakers at \(8,133,115\) and \(63\) s, respectively.

## Notes

### Acknowledgments

We acknowledge Gert-Wim Bruggert for technical assistance and dr. Boudewijn de Smeth from the International Institute for Geo-Information Science and Earth Observation in Twente for lending technical equipment. Financial support came from the MultiScale Mechanics and Physics of Fluids groups of Stefan Luding and Detlef Lohse (University of Twente), respectively; the Stichting Free Flow Foundation; and, the Geophysical Fluid Dynamics Program at the Woods Hole Oceanographic Institution, where O.B. did some of the work.

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