Tapping wave energy through Longuet Higgins microseism effect B Molin1 D Lajoie2 N Jarry2 G Rousseaux3

David Lajoie

Description

Tapping wave energy through Longuet-Higgins microseism effect B. Molin1, D. Lajoie2, N. Jarry2, G. Rousseaux3 1 Ecole Centrale Marseille & IRPHE, 13 451 Marseille Cedex 20 () 2 ACRI, 260 route du pin Montard, BP 234, 06904 Sophia-Antipolis Cedex () 3 Laboratoire Dieudonne, UMR 6621, Parc Valrose, 06 108 Nice Cedex 02 () This paper is dedicated to the memory of Pierre Guevel. Introduction It is well-known, since the works of Miche (1944) and Longuet-Higgins (1950), that, under a standing wave system, second-order pressures at twice the wave frequency penetrate the water column down to the sea-floor, whatever the waterdepth. Recently Guevel proposed that energy could be extracted from the waves with a heaving horizontal plate at the sea bottom, located next to a reflective cliff or sea-wall, and tuned to oscillate at twice the wave frequency. Encouraging preliminary experiments were conducted in ACRI's wavetank (Lajoie et al. 2007). In this paper we address the theoretical modeling of wave energy extraction with such a device, in the asymptotic case when the waterdepth is very large compared to the wavelength. In section I we assume that the first-order wave system is little modified, i.

• ?i ?


• ?2 ?


• reflected wave


• longuet-higgins


• surface d'equation


• tion de l'oscillateur de longuet-higgins en presence


• velocity potential

Sujets

• Rousseau
• Lajoie
• Velocity potential

Informations

• Publié par: pefav
• Nombre de lectures: 15
• Langue: English

Extrait

Tapping wave energy through Longuet-Higgins microseism effect

1 22 3 B. Molin , D. Lajoie , N. Jarry , G. Rousseaux 1 ´ Ecole Centrale Marseille & IRPHE, 13 451 Marseille Cedex 20 ([e-mail]) 2 ACRI, 260 route du pin Montard, BP 234, 06904 Sophia-Antipolis Cedex ([e-mail]) 3 LaboratoireDieudonn´e,UMR6621,ParcValrose,06108NiceCedex02([e-mail])

ThispaperisdedicatedtothememoryofPierreGue´vel.

Introduction It is well-known, since the works of Miche (1944) and Longuet-Higgins (1950), that, under a standing wave system, second-order pressures at twice the wave frequency penetrate the water column down to the sea-floor, whateverthewaterdepth.RecentlyGu´evelproposedthatenergycouldbeextractedfromthewaveswitha heaving horizontal plate at the sea bottom, located next to a reflective cliff or sea-wall, and tuned to oscillate at twice the wave frequency.Encouraging preliminary experiments were conducted in ACRI’s wavetank (Lajoie et al.2007). In this paper we address the theoretical modeling of wave energy extraction with such a device, in the asymptotic case when the waterdepth is very large compared to the wavelength.In section I we assume that the first-order wave system is little modified, i.e.the power taken from the waves is a small portion of the power carried by the incoming wave.In section II we relieve this assumption and we show that one hundred percent of the wave power can be extracted, notwithstanding how large the waterdepth. I. Classical perturbation theory We assume a regular incoming wave system, with amplitudeAand frequencyω, that is fully reflected from a vertical wall inx= 0.We use a coordinate systemOxzwith−∞< x≤0;z= 0 the unperturbed free 2 surface. Thewaterdepth ishand we assumekhÀ1 wherek=ω /gis the wave number. We make use of potential flow theory and we look for the velocity potential Φ(x, z, t) under the form (1) 2(2) 3(3) Φ(x, z, t) =²Φ +²Φ +²Φ +. . .(1) with the small parameter²identified with the wave steepnesskA. At first order the free surface elevation is (1) η(x, t) =Acos(kx−ωt) +Acos(−kx−ωt) = 2Acoskxcosωt(2) and the velocity potential ½ ¾ n o 2A g2 iA g (1)kz kz−iωt(1)−iωt Φ (x, z, t) =−e coskxsinωt=< −e coskxe =<ϕ(x, z(3)) e ω ω (2) The second-order velocity potential Φsatisfies the free surface equation, inz= 0 (2) (2) 1) ∂ (1) (1) ( (1)(1) 23 Φ +gΦ =−η(Φ +gΦ )−2rΦ∙ rΦ =−4A ωsin 2ωt(4) tt ztt zt ∂z (2) As a result, Φis simply (2) 2 Φ =A ωsin 2ωt(5) (2) 2 2 The associated pressure−ρΦ =−2ρ Aωcos 2ωtis independent of the space coordinates. t At the foot of the reflective wall, a heaving plate with lengthl¿his thus subjected to the load (2) 22 F= 2ρ Aω lcos 2ωt(6) The vertical velocity of the plate being ½ ¾ n o (2) q (2) (2)−2iωt−2iωt V=<ve =<e (7) l