Control of the Swell by an Array of Helmholtz Resonators

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Resonators

Léo-Paul Euvé, Natalia Piesniewska, Agnès Maurel, Kim Pham, Philippe Petitjeans, Vincent Pagneux

To cite this version:

Léo-Paul Euvé, Natalia Piesniewska, Agnès Maurel, Kim Pham, Philippe Petitjeans, et al.. Con- trol of the Swell by an Array of Helmholtz Resonators. Crystals, MDPI, 2021, 11 (5), pp.520.

�10.3390/cryst11050520�. �hal-03244741�

Article

Control of the Swell by an Array of Helmholtz Resonators

Léo-Paul Euvé

*, Natalia Piesniewska

, Agnès Maurel

, Kim Pham

, Philippe Petitjeans

and Vincent Pagneux

Citation: Euvé, L.-P.; Piesniewska, N.; Maurel, A.; Pham, K.; Petitjeans, P.; Pagneux, V. Control of the Swell by an Array of Helmholtz Resonators.Crystals2021,11, 520.

https://doi.org/10.3390/

cryst11050520

Academic Editor: Mostafa Nouh

Received: 14 January 2021 Accepted: 27 April 2021 Published: 7 May 2021

Publisher’s Note:MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations.

Copyright: © 2021 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

1 PMMH, ESPCI Paris, PSL University, CNRS, 75005 Paris, France; natalia.piesniewska@espci.fr (N.P.);

phil@pmmh.espci.fr (P.P.)

2 Bluerium, av. L. Philibert, 13100 Aix-en-Provence, France

3 Institut Langevin, ESPCI Paris, PSL University, CNRS, 75005 Paris, France; agnes.maurel@espci.fr

4 IMSIA, ENSTA Paris, Institut Polytechnique de Paris, CNRS, 91732 Palaiseau, France;

kim.pham@ensta-paris.fr

5 LAUM, Université du Maine, CNRS, 72085 Le Mans, France; vincent.pagneux@univ-lemans.fr

* Correspondence: leo-paul.euve@espci.fr

Abstract:

We present a theoretical and experimental study of a resonator of the Helmholtz type for the control of the swell. An experimental demonstration of the shielding effect by a belt made of evenly distributed resonators is given. We then provide in-depth analysis of the Fano resonance resulting from the interference between the dock scattering (the background) and the resonant cavity scattering. This is done thanks to space–time resolved experiments which provides the complex- valued scattering coefficients and amplitude within the resonator. We provide a one-dimensional model derived in the shallow water regime owing to asymptotic analysis. The model contains the two ingredients of the Fano resonance and allows us to exhibit the damping due to leakage. When adding heuristically the damping due to losses, it reproduces the main features of the resonance observed experimentally.

Keywords:

Helmholtz resonator; water waves; protection belt

1. Introduction

Breakwaters are structures designed to create a sheltered area, protected from the wave attacks. Among the different strategies, floating breakwaters (FBW) are attractive when bottom-connected structures are not suitable, e.g., in sea area with deep waters or soft soil beds. Besides, they have less impact on the underwater fauna and flora [1,2]. The building block of such breakwaters is a rigid or flexible floating body, being either plain or partially empty. More generally, the modern design of devices able to control the energy flow of ocean waves has begun to benefit from the development of metamaterials. For instance, the propagation can be made anisotropic using varying bathymetry see e.g., [3–5], or it can be guided due to valley-locked transport [6]. In addition, a plethora of interesting phenomena has been proposed, such as the cancellation of the scattering by rigid obstacles and their cloaking [7–12], the perfect absorption of the wave energy in the nonlinear regime [13] and the trapping for energy harvesting using graded arrays of resonators [14,15].

In the present study, we envision a building block based on a resonance of the Helmholtz type. It is composed of a cavity open on its upper part to the air and con- nected to the surrounding sea water thanks to a thin neck (Figure 1b). Amusingly, this resonant cavity is the upside down version of the oscillatory water column devices (OWC) used to collect the sea wave energy [16], see Figure 1a. To begin with, we report in Figures 2 and 3 elements of the proof-of-concept of the shielding efficiency of a belt made of evenly distributed resonators. In a laboratory experiment, we have realised two rectan- gular belts surrounding a central target area that is to be protected. In the configuration (a) the belt is composed by cavities open to the sea by a neck (our Helmholtz resonators); in the configuration (b) it is composed by close cavities (a dock). In each case, two ping-pong

Crystals2021,11, 520. https://doi.org/10.3390/cryst11050520 https://www.mdpi.com/journal/crystals

balls on both sides of the belt are used to track the movements outside and inside the target area. For an incoming plane wave (blue arrow in Figure 2), video recordings evidence the efficiency of the configuration (a) to produce an unperturbed area within the belt (the videos are available in the Supplementary material; in these movies, top view corresponds to Figure 2a and bottom view to Figure 2b).

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oscillating water column

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oscillating water column

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water flow

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Figure 1.

(a) Oscillatory water column (OWC); the power of the rising and falling water column is transferred to the air through to a hole on the upper part of the cavity; the strong air flow is exploited by a power take off system [16]. (b) Helmholtz resonator with large oscillations of the water column due to strong water flow through a hole on the lower part of the cavity.

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Figure 2.

Proof-of-concept experiments: a belt aims to protect a rectangular target area from incoming waves (blue arrow). The belt is composed of Helmholtz cavities (a) or of close cavities (b).

(b)<latexit sha1_base64="IF465uYhfpKqxj6MMLudN9xFHo4=">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</latexit>

(a)<latexit sha1_base64="oSRzHv3hK0fb1qUcR8ea9SP6dhw=">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</latexit>

Figure 3.

Pictures at two times during the oscillatory motion of the ball outside the target area (at left) at its maximum (red plain lines) and minimum (blue plain lines) vertical positions. In the configuration (a) for the resonant belt, the ball inside the target area (dashed lines) has not moved; in contrast in (b) for the rigid dock belt, the two balls move with almost the same vertical amplitude.

In Figure 3, we report illustrations of our observations. The ball on the left with middle

line shown as plain lines is outside the target area. The ball on the right with middle line

shown as dashed lines is inside the target area. In (a) and (b) we have selected two times

during the oscillatory motion of the ball at left at its maximum and minimum vertical

positions. In (a), the ball on the right in the target area remains at the same heigh as the

area is efficiently sheltered by the resonant belt. In contrast for the rigid dock belt (b), the

ball on the right has an oscillation motion with nearly the same amplitude as outside the target area.

In this study, we characterize the resonance responsible for the shielding effect re- ported above. Firstly, we present in Section 2 experimental quantitative results obtained using the Fourier Transform Profilometry (FTP) which allows for space-time resolved measurements of the free surface elevation [17–19]. The set-up involves an incident-guided wave on a single resonator in a rectangular duct which is analog to an incident plane wave at normal incidence on a periodic array of resonators. The significant losses within the cavity weaken the sharp variations of the resonance around the perfect reflection of the Fano curve [20]. The combined effects of the resonance and of the losses produce a low transmission when compared to that produced by a dock. In Section 3, we use a one-dimensional model governing the surface elevation outside the resonant cavity. The model obtained in a recent study [21] reduces the effect of the resonator to jump conditions applying to the surface elevation and its horizontal derivative; next it encapsulates the resonant dynamics in an equation of a harmonic oscillator forced by the incident wave with damping due to the leakage by radiation. When accounting for the inherent viscous and nonlinear losses the experimental results are qualitatively well reproduced by the model.

Perspectives are drawn in concluding remarks Section 4.

Throughout the paper, in the numerics and the modelling, we consider an invis- cid, incompressible fluid, and an irrotational motion in the harmonic regime with time dependence e

. Under these assumptions, the velocity potential φ ( x, z ) satisfies



 



 



∆φ = 0,

∂

φ ( x, 0 ) = ^ω

2

g φ ( x, 0 ) , ∇ φ · ⁿ

= 0,

(1)

with x the horizontal coordinate, z the vertical one (z = 0 is the undisturbed free surface, z = − ^H the sea bottom) and Γ denotes the boundaries of the rigid parts of the resonator and of the sea bottom. The free surface elevation η ( x ) and the velocity u = ( u, v ) are then given by

u ( x ) = ∇ φ ( x ) , η ( x ) = ^iω

g φ ( x, 0 ) . (2)

2. Experimental Result–Effect of the Losses

To characterize the resonator, we use a single resonator in a duct. The duct is 1.5 m long and p = 6 cm wide (along y); the water depth is set to H = 6 cm. The cavity has the same width p as the duct and it is fixed at the free surface with an immersion depth h = 2.8 cm; its dimensions are indicated in the Figure 4. This set-up is the equivalent of an array of resonators evenly distributed along y with spacing p for an incoming wave (along x) at normal incidence. In both cases, the solution far from the cavity region reads

η ( x < 0, ω ) = A

e

+ R ( ω ) e

, η ( x > L, ω ) = AT ( ω ) e

, (3)

with the complex valued amplitude of the incident wave A and scattering coefficients

( R, T ) . At the frequency ω is associated the wavenumber k.

H= 6.0 cm

<latexit sha1_base64="TDQfhFc4mkvkABgDMjPt53+Wgy0=">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</latexit>

L= 3.1 cm

<latexit sha1_base64="ajuUnoEuLlj13JnbHiOHp92WWQ8=">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</latexit>

h= 2.8 cm

<latexit sha1_base64="3iYuVG0RZ/VbI+aBY9INPsXLxeM=">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</latexit>

`= 0.5 cm

<latexit sha1_base64="/bbl1TEG/ucX4bLlTlCPJSCCZk8=">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</latexit>

e<latexit sha1_base64="pMULNZIzTSDYK7WYZcc0vLt8Fio=">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</latexit> = 0.2 cm x<latexit sha1_base64="IheOuiv69+R629/ejUoDhEpwiDo=">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</latexit>

z

<latexit sha1_base64="w9JmHXuxJb2WVk6ZfZV6FNv8uAI=">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</latexit>

Figure 4.

Experimental realization. Front view (in (

) coordinate) of the single resonator placed in a waveguide of width

along

with spacing

at normal incidence.

2.1. Experimental Characterization of Complex-Free Surface Elevation

Working in a duct has two advantages. Far from scattering regions, it gives rise to the propagation of a perfectly plane wave below the cut-off frequency ω

= 2π p

gH/

p ' ^{80 rad} · ^s

(only the mode 0 can propagate). Besides, it necessitates a relatively small volume of water hence allows for its coloration with anatase pigment rendering the free surface light diffusive (the coloration of water is visible in Figure 4). The FTP can, therefore, be implemented. It uses the field of a fringe pattern projected onto the moving free surface.

The phase map between the instantaneous deformed pattern and the reference undeformed one is used to reconstruct the field of free surface elevation [17–19]. The fringe patterns have been recorded using a camera Photron (1024 × 1024). With a recorded window x ∈ ( − ^{40, 40} ) cm and y ∈ ( − ^{1.5, 1.5} ) cm, the spatial resolution δx = δy = 0.84 mm is given by the pixel of the camera and the time resolution δt = 1/25 s by its acquisition frequency.

Our experiments have been performed in the harmonic regime with incident waves generated by a wavemaker at the left end of the duct imposing a sinusoidal motion at frequency ω (visible in the Figure 4); 11 values within the range (9,16) rad · ^s

^have been considered. To prevent the strong reflection at the right end of the duct we use a beach with an 8

angle. The experiments have been done using a resonant cavity and for comparison using a simple dock with the same dimensions. The instantaneous fields η ( x, y, t ) are stored which allows for pixel-by-pixel Fourier transforms resulting in the complex amplitude η ( x, y, ω ) . To reduce the noise-to-signal ratio, we use the mean profile η ( x, ω ) =

R

−1.5

η ( x, y, ω ) dy which provides the amplitude η

( ω ) in the cavity for

x ∈ ( 0, L ) and the scattering coefficients ( R, T ) by fitting the forms in (3) for x < 0 and

x > L. For the resonant open cavity, the Figure 5 shows an instantaneous free surface

measurement η ( x, y, t ) at an arbitrary time for the two frequencies realizing the minimum

and maximum transmissions (top panels) and the mean complex profiles η ( x, ω ) (bottom

panel, real part with blue lines and imaginary part with red lines); the result of the fits

using (3) are shown with dashed black lines.

(a) minimum of transmission<latexit sha1_base64="f8ly1NyQo0i7Kx1vJW23/DFxbl4=">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</latexit>

(b) maximum of transmission

<latexit sha1_base64="CkgEr2hP2bWlCfBRFfE8a93Bt3A=">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</latexit>

Im(⌘)

<latexit sha1_base64="fqgrFk0rD9RQU+u4JRWsc304GQQ=">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</latexit>

Re(⌘)<latexit sha1_base64="+rM70zucf7a64tphwWZQ8PGxgNs=">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</latexit>

-0.4

<latexit sha1_base64="GS2fv2RujiDgX/15vmmV3WBUpTI=">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</latexit>

0

<latexit sha1_base64="69NSq0xGmfxZGTvS/VfDWUUgDVk=">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</latexit>

0.4

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0.8<latexit sha1_base64="VQJmNytz28umqn5vn1qHERtGjP0=">AAACxnicjVHLSgMxFD2Or1pfVZduBovgakjHhV0W3HRZ0T6gFpmZpjV0XmQySimCP+BWP038A/0Lb+IUFCmaYZKbc885yc3101BkirG3JWt5ZXVtvbRR3tza3tmt7O13siSXAW8HSZjInu9lPBQxbyuhQt5LJfciP+Rdf3Ku8907LjORxFdqmvJB5I1jMRKBpwi6ZE79plJljsv0sH8HNcesrIpitJLKK64xRIIAOSJwxFAUh/CQ0ddHDQwpYQPMCJMUCZPneECZtDmxODE8Qic0j2nXL9CY9tozM+qATgnpl6S0cUyahHiSYn2abfK5cdboIu+Z8dR3m9LqF14RoQq3hP6lmzP/q9O1KIxQNzUIqik1iK4uKFxy8yr65va3qhQ5pITpeEh5SXFglPN3to0mM7Xrt/VM/t0wNar3QcHN8aFvSQ2ed9FeHHRcp3bquBdutcGKVpdwiCOcUD/P0EATLbTJe4wnPOPFalqxlVv3X1RrqdAc4MewHj8BB6WPpQ==</latexit>

0

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-40

<latexit sha1_base64="RiyYUpH2aMy7m+Yhgpv4e1wGlrU=">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</latexit> 40<latexit sha1_base64="Aki9wl85bRKCdOCqsAm2ughXI7M=">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</latexit>

-0.4

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0

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0.4

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0

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-40

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x (cm)

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x (cm)

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x

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y

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-1.5

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1.5

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x<latexit sha1_base64="qsMkCrB1GEGP2PR+zSFVt8z4fwA=">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</latexit>

y

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-1.5

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1.5

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Figure 5.

FTP measurements for a resonant open cavity (a) at the minimum transmission for

= 12.5 rad · ^s

and (b) at the maximum transmission for

= 9.3 rad · ^s

. In (a,b), top panels show the instantaneous free surface measurement

(

) at an arbitrary time. Bottom panels show the real (blue) and imaginary (red) parts of

(

) obtained by pixel-by-pixel Fourier transforms of the recorded instantaneous patterns and after

using (3) providing the complex valued coefficients (

) ; for

∈ ( 0,

) the relative complex valued amplitude

, normalized to

2.2. Experimental Evidence of Fano Resonance

Figure 6 shows ( R, T ) and η

against the wavelength λ = 2π/k obtained experimen- tally (circles) along with the same quantities calculated numerically (plain and dashed lines). The dashed lines in the lossless case are characteristic of a Fano resonance with striking variations of the scattering curves around the smooth curve of the dock on its own.

The measurements of ( R

, T

) for the dock on its own (green circles) coincide with the numerics without a loss (plain green line). In contrast to the resonant, cavity-significant shifts between measurements and lossless numerics are observed. Accordingly in the nu- merics, we have accounted for the losses in the cavity only by adding a small complex part to the real wavenumber satisfying k tanh kh =

, namely we have considered k → k + ik

(the plain lines show the best agreement for k

= 1.5 m

). Although the losses weaken

the resonance, the cavity efficiently reduces the transmission (compared to a dock) for

wavelengths smaller than about 35 cm.