Revisiting Hele-Shaw dynamics to better understand beach evolution

Revisiting Hele-Shaw Dynamics to Better Understand Beach

Evolution

O. Bokhove

, A.J. van der Horn

, D. van der Meer

, E. Gagarina

, W. Zweers

,

and A.R. Thornton

Wave action, particularly during storms, drives the evo-lution of beaches. Beach evolution by non-linear break-ing waves is poorly understood due to its three-dimensional character, the range of scales involved, and our limited un-derstanding of particle-wave interactions. We show how a novel, three-phase extension to the classic “Hele-Shaw” lab-oratory experiment can be designed that creates beach mor-phologies with breaking waves in a quasi-two-dimensional setting. Our thin Hele-Shaw cell simplifies the inherent complexity of three-phase dynamics: all dynamics become clearly visible and measurable. We show that beaches can be created in tens of minutes by several types of breaking waves, with about one-second periods. Quasi-steady beach morphologies emerge as function of initial water depth, at-rest bed level and wave-maker frequency. These are classi-fied mathematically and lead to beaches, berms and sand bars.

1

School of Mathematics, University of Leeds, Leeds, U.K.

2

Department of Applied Mathematics, University of Twente, Enschede, The Netherlands

3

Department of Physics, University of Twente, Enschede, The Netherlands

4

Department of Mechanical Engineering, University of Twente, Enschede, The Netherlands

5

FabLab, Saxion College, Enschede, The Netherlands

Copyright 2013 by the American Geophysical Union. 0094-8276/13/$5.00

Figure 1. Schematic of the Hele-Shaw cell. Wedge, waterline, particle bottom, and wave-maker are shown. Parameters are initial water depth W0 = H0 −B0 =

(10, 30, 50, 70) ± 3mm, bed level B0 = (20, 50, 80) ±

2mm at rest, and wave-maker frequency fwm =

(0.4, 0.7, 1.0, 1.3)Hz with angle θwm.

1. Introduction

Surf-zone dynamics concerns the formation and erosion of beaches by breaking waves. Beach evolution is strongest during storms, which leads to intermittent coastline changes Roelvink et al.[2009]. Prediction of beach dynamics requires an understanding of breaking waves, especially how waves pick up, transport and deposit sand particles. The combined interactions of water, air and sand are involved, making up a three-phase flow, with several spatial and temporal scales. These include wave action on shorter time scales, which is re-sponsible for the net/overall beach evolution on longer time scales. But beach evolution by non-linear breaking waves is poorly understood due to its three-dimensional character, the range of scales involved, and our limited understand-ing of particle-wave interactions. In principle, Navier-Stokes and continuum equations are available as models for water and air motion, and for interactions with and between sand grains. However, their direct numerical simulation remains literally incalculable because the degrees of freedom involved are simply too large. Models or closures of the unresolved processes such as particle-particle and particle-wave inter-actions are required. Despite great advances in coastal en-gineering [Soulsby, 1997; Calantoni et al., 2006; Garnier et al., 2010; McCall et al., 2010], our understanding of clo-sures of the averaged sand transport and wave breaking re-mains relatively weak. The question we posed was “How can we explore the fundamentals of beach evolution under breaking waves in a simplified laboratory and modelling en-vironment?” Our answer was to design a novel extension of Hele-Shaw’s classic 19th-century experiment with viscous fluid or Stokes’ flow around obstacles in thin cells [Lamb, 1993], as follows.

Imagine an isolated slice of beach that includes air, sand particles and water waves, and place it between two glass plates. The ’beach-slice’ is just over one-particle diame-ter wide, with all particle and wadiame-ter motion clearly visible and tractable. Based on analytical and numerical design calculations, we built a laboratory tank [Bokhove et al., 2010] to demonstrate beach evolution by breaking waves, see Figure 1. It is 1m long, 0.3m high, and 2mm wide, with Gamma Alumina particles 1.75 ± 0.1mm in diameter (density of 1750 ± 30kg/m3), separated from a wave-maker by a submerged wedge to avoid direct impact. We have used nearly spherical and uniform Gamma Alumina par-ticles. Two joined, metallic rods each with a diameter of 1.6mm act as wave-maker, driven by a PC-steered linear ac-tuator with sinusoidal output. We operate the wave-maker for monochromatic waves in the frequency range 0.4−1.3Hz, with a pendulum angle θwm. It is offset to produce a slightly

anti-symmetric wave, inducing less suction on the beach side. Fresh MilliQ-water was employed to ensure clean op-eration and limit variability in surface tension.

This Hele-Shaw beach configuration allows precise track-ing of the few thousands particles involved, in both time and in approximately two spatial dimensions, because only one layer of particles fits the tank laterally. Similarly, using high-speed cameras the air-water interface is easily tractable 1

as a water line. By design, the dynamics in the Hele-Shaw cell is quasi-two-dimensional [Rosenhead , 1963; Wilson and Duffy, 1998]. Classically, Hele-Shaw hydrodynamics is dominated by side-wall boundary layers leading to a nearly parabolic flow profile [Batchelor , 1967]. The Navier-Stokes equations can be width-averaged, while ignoring Reynolds-stress and the small in-plane viscous terms. This leads to linear damping of momentum in the vertical plane, inversely proportional to the square of the gap width between the glass plates. The crucial insight is that for a gap width 2l > 1.5mm a damped breaking wave propagates to the end of a fixed beach of about 0.5m, which we determined numer-ically. Otherwise, the wave dissipates too quickly, and then the desired phenomenology of wave breaking and particle transport is too weak or absent.

In Section 2, we confirm that the observed decay of wave modes in the Hele-Shaw laboratory tank, filled with water but without particles, is captured reasonably well by nu-merical simulations using a potential flow model [Lighthill , 1978] with only this linear damping. This comparison sug-gests that other flow profiles or damping of the contact line [Vella, 2005] at the water-air-glass interface may be of sec-ondary importance. Lee et al. [Lee et al., 2007] consid-ered one particle settling in a Hele-Shaw cell, and observed quasi-two-dimensional behaviour for gap widths 2l < 1.05d, with particle diameter d, and three-dimensional behaviour for 2l > 1.1d. We hence chose a gap width of 2mm, given that our particles are 1.75 ± 0.1mm in diameter. While the stronger damping mechanisms due to glass walls and con-tact lines are absent at natural beaches, these simplify the dynamics in the Hele-Shaw cell, and may be readily included in mathematical models. Anyhow, the Hele-Shaw beach captures the complex problem of wave breaking within a quasi-two-dimensional realm, which lends itself very well for fundamental research on wave-particle interactions. Both realistic wave breaking and beach dynamics are observed, as we show in Sections 3 and 4. Finally, we conclude and draw inferences for future research in Section 5.

2. Validity of Linear Momentum Damping

The design of the Hele-Shaw beach experiment was based on analysis of approximate models for the hydrodynamics. A narrow gap width between the two glass plates, just over one particle, is desirable to enhance the visualization and limit the dynamics to be nearly two-dimensional. At the same time, this gap still needs to be wide enough to allow nonlinear breaking waves and beach formation. In the clas-sic Hele-Shaw case, the parabolic flow profile in the vertical plane is

(u, w) = 3 2(¯u, ¯w)(l

2

− y2)/l2 (1) with u = u(x, y, z, t) and w = w(x, y, z, t) the velocity com-ponents in the horizontal, x, and vertical, z, directions, lat-eral direction y, gap width 2l and the latlat-erally averaged mean velocities ¯u = ¯u(x, z, t) and ¯w = ¯w(x, z, t), and time t. Under the approximation that the local flow profile is parabolic in the lateral direction, we can substitute (1) into the three-dimensional Navier-Stokes equations for the hy-drodynamics of water and average these across the gap. Given the anisotropy in the length and velocity scales along and normal to the vertical plane, the viscous balance across the gap is assumed to be dominant. An approximation is then made in which slow variations in the vertical plane on scales larger than the gap width are still permitted. After averaging and neglecting Reynolds-stress terms, the result-ing two-dimensional Euler equations with linear momentum

damping become ∂ ¯u ∂t + β ¯u ∂ ¯u ∂x+ β ¯w ∂ ¯u ∂z = − 1 ρ ∂p ∂x− 3ν ¯u l2 , (2) ∂ ¯w ∂t + β ¯u ∂ ¯w ∂x + β ¯w ∂ ¯w ∂z = − 1 ρ ∂p ∂z− 3ν ¯w l2 , (3) ∂ ¯u ∂x+ ∂ ¯w ∂z = 0, (4)

in which ν is the viscosity of water, g the acceleration of gravity, p the pressure, ρ the constant density of water, and β = 6/5 a scaling factor due to the width-averaging. The first two equations are the momentum equations in the x– and z–directions, and the last equation expresses that this width-averaged velocity in the vertical plane is incompress-ible. Partial derivatives are denoted as usual. Higher-order contributions to the lateral flow profile (1) are ignored in (4), which especially concerns omissions in the three-dimensional boundary layers near the free surface, side walls and near particles. Nevertheless, for the flow in the bulk away from

Figure 2. Initial conditions. Initial conditions are shown for a) the simulations that start at rest flow with a tilted free surface, and b) for the experiments that start at rest with a tilted tank.

Figure 3. Potential energies. Potential energies a) P (t), and b) P (t)exp(3νt/l2

), versus time t. Profiles for differ-ent initial angles α = 4.0o

, 4.2o

5.4o

, 6.1o

, 6.5o

are shown for the model simulations (thin lines) and the laboratory experiments (thick, jagged lines).

boundaries, these linear damping terms in the momentum equations of (4) are reasonable, as we will see.

Furthermore, when we assume that the velocity has the form (¯u, ¯w) = (∂φ/∂x, ∂φ/∂z) with velocity potential φ, then the potential flow equations emerge with a linear damp-ing term. To simplify the calculations, we set β = 1 here-after, but this does not change the argument below regard-ing the potential energy. The resultregard-ing system of equations can in principle be derived from the following variational principle 0 = δ ZT 0 „Z L 0 φs∂h ∂tdx − (P (t) + K(t)) « e3νt/l2 dt ≡ δ ZT 0 „Z L 0 φs∂h ∂t − 1 2g(h − H) 2 dx − Z h 0 1 2|∇φ| 2 dx dz « e3νt/l2 dt (5)

with water depth h = h(x, t), the potential at the free sur-face φs, mean still water depth H, suitable boundary and

end-point conditions for a tank of length L and with a fi-nal time T . Principle (5) is an extension of Miles’s clas-sic variational principle [Miles, 1977], with the exception of the additional factor exp 3νt/l2

. That factor is the in-verse of the integrating factor due to the linear momentum damping. The consequence of formulation (5) is that it is better to plot the potential energy, P (t), times this factor exp 3νt/l2

. When, therefore, we start with the release of water from a rest state with available potential energy, this corrected potential energy, P (t) exp 3νt/l2

, is expected to oscillate around a constant mean value.

A straightforward comparison between the potential flow model arising from (5) and an experiment can now be used to assess the validity of linear momentum damping. In the nu-merical discontinuous Galerkin finite element approximation to (5) (cf. Gagarina et al. [2012]), we start at rest with a lin-early slanted free surface with angle α, see Fig. 2a. Instead, in the experiment we start with a tank at rest lifted upward with an angle α, see Fig. 2b, and lower it down quickly to a horizontal level such as to obtain, approximately, the still profile in Fig. 2.a. The small phase difference is removed by shifting the time profiles of P (t) and P (t)exp(3νt/l2

) such that the first zero crossings are lined up. For five values of α, these results are shown in Fig. 3. For the larger angles α = 6.1o

and α = 6.5o

, our experimental approximation at the start is less satisfactory, but for the other angles the match between the simulations (thin lines) and experiments (thick noisy lines) is good for about one second. Within two seconds, most energy has dissipated, but after one second, the comparison with the model is less good, see Fig. 3b. This discrepancy is presumably due to the neglected three-dimensionality of the profile near the free-surface boundary layer in combination with effects of surface tension. The latter effects have been analysed in the context of a related yet different, two-dimensional Faraday experiment by Vega [2001]. We conclude that for driven flows on scales much larger than the gap width, linear momentum damping is a good leading-order model approximation, but that mod-elling the fine-structure in breaking waves may require res-olution of three-dimensional free-surface boundary layers.

3. Wave breaking

We found all types of wave breaking [Peregrine, 1983] at natural beaches, including spilling, plunging, collapsing and surging breakers. To trace waves, we dyed the water red

to enhance the contrast and facilitate analysis of the mea-surements, obtained by a high speed Photron SA2 camera at 1000fps. Space-time renderings of the four wave types are shown in Fig. 4, and discussed in turn. In a plung-ing breaker, the wave front overturns and a prominent jet falls at the base of the wave causing a large jet, Fig. 4a. We observe two bubbles caught during overturning. For a surging breaker, a significant disturbance and vertical face in an otherwise smooth profile occurs only near the mov-ing shoreline, Fig. 4b. In a spillmov-ing breaker, whitish water at the wave crest spills down the front face sometimes with the projection of a small jet, Fig. 4c. The grooves indicate that the presence of (pre-existing) bubbles. In a collapsing breaker, the lower portion of the wave’s front face overturns and then behaves like a plunging breaker, Fig. 4d, where the lower portion of the wave is seen to shoot forward after circa

Figure 4. Space-time plots of breaking waves. The wa-ter surface (blue) and bed (red) of measured a) plunging, b) surging, c) spilling and d) collapsing breakers.

Figure 5. Phase diagram of beach morphologies. a) States in the parameter plane of initial water depth W0 = H0− B0 and wave frequency f_wm, for an initial

bed level of B0 = 8cm. (For B0 = 2 and 5cm the state

is quasi-static, except for fwm = 1Hz, B0 = 5cm and

W0 = 5cm, for which a sand bar developed.) b) Initial

water levels and bed heights (blue and red lines, respec-tively) and final bed height for a: b1) wet beach, b2) dry beach, b3) sand bar, and b4) dune.

100ms, and to separate from the top part of the wave. In the Hele-Shaw set-up these wave types, see the supplemen-tary videos Supplemensupplemen-tary Videos [2013]. are small-scale versions smoothed by surface tension, when compared to vi-olent three-dimensional breakers at beaches. These types of wave breaking emerge during various stages of the bed evolution into sandbars, beaches or dunes, observable in the Hele-Shaw cell.

4. Quasi-steady beach morphologies

Time-dependent beach morphologies form on a time scale of minutes to hours, forced by wave action that occurs on a typical time scale of one second. We undertook a pa-rameter study [Horn, 2012] of beach formation by break-ing waves in which we varied the initial water depth W0 =

(10, 30, 50, 70)mm and bed level B0= (20, 50, 80)mm at rest,

before turning on the wave-maker in a monochromatic fre-quency range of (0.4, 0.7, 1.0, 1.3)Hz. The state of the bed was captured every 10s with a Nikon D5100 camera. De-pending on the parameters, several quasi-steady beach mor-phologies were observed to develop after 10 to 30 minutes. They are classified mathematically. When the bed profile has an interior maximum, its maximum is either submerged (“sandbar”), or dry on top or its onshore side (“berm-dune or beach-dune”). When the bed profile has a boundary max-imum, it is either wet or dry (‘wet or dry beach’). A dune emerges when water is found at both sides of a berm, see the movie on berm formation ([Movie5berm, 2012]). A beach-dune is a beach-dune near the right wall, also with an interior max-imum bed height away from the boundary. A submerged

Figure 6. Bed heights in space-time. Time-space dia-grams of bed height versus spatial coordinate x for a) dry and b) wet beaches (boundary maxima), and c) a sub-merged sand bar and d) a dune (interior maxima). When dry land emerges the initial water level (black line) is indicated.

sand bar arises when a significant amount of sediment area transport is taking place, more than 10cm2

, with an interior, wet maximum of the bed height, and none of the above def-initions hold. Finally, in the quasi-static state hardly any sediment area transport is taking place (less than 10cm2

and no parts fall dry). Four of these states are displayed in Fig. 5, displaying both initial and final states, together with a phase diagram for the case B0= 80mm. Space-time plots

of the bed height evolution from a nearly flat submerged bed to the dune, dry beach, wet beach or sandbar are given in Fig. 6, and range from 35 to 60 minutes. It is clear in the phase diagram in Fig. 5 that the states evolve smoothly from one to the other. Furthermore, wave activity is less efficacious when: the wave has already lost its energy due to its breaking over the wedge; the water is very shallow; the frequency is too high such that the viscous damping be-comes too high; or, when the water is too deep such that particles do not get picked up. The wet beach state emerges when there are insufficient particles available to create a dry beach.

5. Conclusions

All Hele-Shaw beach dynamics presented are robust and reproducible. Details depend on the precise wave-maker motion, its placement and asymmetry, and compaction of the initial bed. Bed compaction in the quasi-static state is about 1%, and about 2% for displaced particles. These small changes are, however, known and can be included precisely as initial conditions and forcing into mathematical predic-tions.

The innovation of our Hele-Shaw beach experiment is that particle, wave and water motions can be measured and analysed thoroughly. This will trigger the development and precision validation of new forecasting models [McCall et al., 2010; Vega, 2001; Thornton et al., 2006; Cotter and Bokhove, 2010; Dumbser , 2011; George and Iverson, 2011]. The measurements will form a demanding benchmark for existing coastal engineering forecast models of sand trans-port ([Operational wave and sediment forecast models, 2013]: Delft3D, Telemac and XBeach). Finally, the scope of our Hele-Shaw research is broad: researching its dynamics for different initial bed slopes, particles of varying size and den-sity, wider gap widths, and more complex wave-maker mo-tion will facilitate breakthroughs in mathematical modelling of near-shore three-phase flows.

Acknowledgments. We thank the Physics of Fluids and Multi-Scale Mechanics groups at the University of Twente and the Stichting Free Flow Foundation for financial support. We gratefully acknowledge the careful proofreading by Valerie Zwart and Professor Menno Prins.

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O. Bokhove, School of Mathematics, University of Leeds, LS2 9JT, Leeds, U.K. (O.Bokhove@leeds.ac.uk)