Bragg scattering by a line array of small cylinders in a waveguide. Part 1. Linear aspects

Abstract

Motivated by potential applications for offshore airports supported on vertical piles, we report a theory of wave diffraction by a periodic array of circular cylinders. The simple case of normal incidence on a rectangular array is studied here, which is equivalent to a line array along the centre of a long channel. An asymptotic theory is developed for cylinders much smaller than the incident wavelength, which is comparable to the cylinder spacing. Focus is on Bragg resonance near which scattering is strong. A combination of the method of multiple scales and the Bloch theorem leads to simple evolution equations coupling the wave envelopes. Dispersion of transient wave envelopes is investigated. Scattering of detuned waves by a large but finite number of cylinders is investigated for frequencies in and outside the band gap. Quantitative accuracy is assessed by comparisons with numerical computations via finite elements. The analytical theory prepares the ground for nonlinear studies and may facilitate future inclusion of real-fluid effects such as vortex shedding.

Water wave scattering by dense or sparse arrays of surface-piercing bodies by integral matching method

“…We can notice that since the channel width W = L, only the first mode (n = 0) is propagative for kL < π. As in [23], we can observe for mode 1 a strong reflection near kL = π and kL = 2π. These two peaks correspond to respectively 1st and 2nd order Bragg resonances which correspond to resonant wavemedium interactions when the spatial periodicity of the medium is half the wave wavelength at first order (here kL = π), or a multiple of half the wave wavelength for higher orders (here kL = pπ, p > 1).…”

“…These two peaks correspond to respectively 1st and 2nd order Bragg resonances which correspond to resonant wavemedium interactions when the spatial periodicity of the medium is half the wave wavelength at first order (here kL = π), or a multiple of half the wave wavelength for higher orders (here kL = pπ, p > 1). Let us note that these peaks are slightly shifted towards low wavenumbers, due to the finite size of the cylinders (see also [23]). High values of the reflection coefficient above one is observed for mode 2 and for mode 3, for kL = 2π and kL = 4π respectively.…”

“…In this first section, we consider the case of sparse cylinders as studied by Li and Mei [23]. In their study, only the first mode is propagative in the range of frequency considered.…”

“…We will consider here the same configuration as Fig. 9 of [23], where they compare results from their asymptotic solution with those from a finite element method, for a/L = 0.10, L/W = 1, N = 10, a, L, W and (2N + 1) being the cylinder radius, the distance between two successive cylinders along the x-axis, the channel width and the number of cylinders along x-axis, respectively. A sketch of the configuration is shown in Fig.…”

Advances and Challenges in Wave Energy Park Optimization—A Review

“…Both the iterative and non-iterative versions of the multiple scattering theory have been further developed and used to model the hydrodynamics of wave energy parks (Ji et al, 2015;Konispoliatis and Mavrakos, 2016;Göteman, 2017;Ruiz et al, 2017;Fang et al, 2018;Giassi and Göteman, 2018;Zheng et al, 2018Zheng et al, , 2019Liu et al, 2019). Several other analytical methods have been developed and used for wave energy park applications, including matched asymptotic expansions (McIver and Evans, 1988), multipole expansions (Linton and Evans, 1993), and Bragg scattering (Li and Mei, 2007).…”