Waves over variable bathymetry branched flow in the linear regime. - PDF document

Waves over variable bathymetry – branched flow in the linear regime. Adam Piotr Anglart 1,4 in collaboration with T. Humbert 2 , P. Petitjeans 1 , V. Pagneux 2 , A. Maurel 3 1 École Supérieure de Physique et de Chimie Industrielles de la ville de Paris, Laboratoire de Physqiue et Mécanique des Millieux Hétérogènes, Paris, France 2 Laboratoire d’Acoustique de l’Université du Maine, Le Mans, France 3 Institut Langevin LOA, Paris, France 4 Warsaw University of Technology, The Faculty of Power and Aeronautical Engineering, Warsaw, Poland Surface water waves High energy waves Intensity of a plane wave propagating from left to right in a random bathymetry Degueldre et al. Nature Physics 12 (2016) Branched flow seen in the wave energy map produced Shallow-water waves after the 2011 Sendai earthquake in Japan. High energy very sensitive to small fluctuations path heading for Crescent City in northern California. of the bottom topography National Oceanic and Atmospheric Administration (2011)

High energy paths Microwaves experiment antenna Randomly distributed conical scatterers. Höhman et al. 2010, Phys. Rev. Lett. 104 (2010) Microwave pattern at a frequency f = 30.95 Hz. Höhman et al. 2010, Phys. Rev. Lett. 104 (2010) No experimental results for surface water waves so far Outline 1 Numerical simulations 1.1 Shallow water equations 1.2 Numerical method 1.3 Periodic bathymetry. Bragg’s law 1.4 Disordered bathymetry. Branched flow 2 Experiment 2.1 Experimental setup 2.2 Dispersion relation validation 2.3 Measurement method 2.4 Results 3 Summary

Numerical simulations Shallow-water equations 1 Linearized shallow-water equation in z time domain ∂ 2 R ∂ η (x,y,t) ∂ t 2 η + e ∂ t η � r ( gh ( x, y ) r η ) = 0 h(x,y) 2 Linearized shallow-water equation in frequency domain (complex solution) y r ( gh ( x, y ) r η ) + ( ω 2 � i ω e R ) η = 0 x Final element method Shallow-water equations y perfect wall perfect amplitude of transmission the wave Neumann BC Dirichlet BC ∂ n η = ik η η (0 , y ) = A x perfect wall

Waves over periodic bathymetry Bragg’s law 1 Shallow-water equation r ( h r η ) + ω 2 g η = 0 2 Bragg’s law λ = 2 d sin θ n 3 Dispersion relation ω 2 = ( gk + γ k 3 ρ ) tanh( kh ) √ gh √ gh f 1 = = ≈ 2 . 7Hz λ d Waves over periodic bathymetry Form of the solution z incident wave transmitted reflected obstacle set 0 x x t x r x 1 x 2 x max ( ae − ikx + Rae ikx , if x ∈ [0 , x 1 ] R = − e − ikx r 1 − H r e − ikx r 2 T = − e − ikx r − Re ikx r η ( x ) = Tae − ikx , if x ∈ [ x 2 , x max ] e ikx r 1 − H r e ikx r 2 H t e − ikx t

Waves over periodic bathymetry Reflection and transmission coefficients for hemiellipsoid obstacles Waves over disordered bathymetry Branching patterns surface elevation, η surface elevation, η 1 Hz 3 Hz surface elevation, η surface elevation, η 11 Hz 27 Hz

Waves over disordered bathymetry Intensity maps 1 Hz 3 Hz Intensity, I Intensity, I Energy ∝ Intensity E ∝ I = | η | 2 + | ∇ η | 2 11 Hz 27 Hz Intensity, I Intensity, I Waves over disordered bathymetry Statistical analysis | Probability density function 1 st regime 2 nd regime 3 rd regime 1 Hz 3 Hz Gaussian distribution Rayleigh distribution branching patterns multiple scattering

Experimental setup video projector camera linear motor light 4 m 1.5 m wavemaker sloping bottom Experimental setup

Waves over a flat bottom Dispersion relation for water surface waes Wave propagation for a flat bottom and the Dispersion relation for water surface waves. frequency f = 2.8 Hz ω 2 = ( gk + γ k 3 ρ ) tanh( kh ) Experimental setup Disordered bathymetry camera scatterers

camera Measurment method Free-surface synthetic Schlieren H h dot pattern Mesure de la déformation d’une surface libre par analys du scatterers déplacement apparent d’un motif aléatoire de points resolution of ~ 10 -2 mm Moisy et al. 18 éme Congrés Français de Mécanique (2007) ∞ r η = � δ r h ∗ , where 1 h ∗ = 1 α h � 1 Z η ( x, y, t ) − i ω t d t η ( x, y, ω ) = ˆ H −∞ - optical displacement field δ r - free-surface elevation η - refraction coefficient (0.24 for air-water interface) α Parameters of the system h h 0 1 dimensionless wavelength a φ d 2 strength of the scatterer 3 density of scatterers 4 Ursell number

First regime | low frequencies wavelengths larger than the size of scatterer | ! *>1 Second regime | intermediate frequencies wavelengths comparable to the size of scatterer | ! * ≈ 1

Third regime | high frequencies wavelengths smaller than the size of scatterer | ! *<1 Comparison of numerical and experimental results 1 st regime 2 nd regime 3 rd regime 3 Hz 1 Hz Rayleigh distribution ! *<1 branching patterns Gaussian distribution ! *>1 ! * ≈ 1 multiple scattering

Summary simulations have been carried out to obtain suitable parametres for the experiment - num numerical al si - specified range of frequencies, where branched flow can be observed - experimental setup de designe gned and and ma manufact ctured ed - implentation of Free-Surface Synthetic Schlieren measurement method - construction of wa wavemaker that allowed to acquire needed regime of higher frequencies - thr hree re regimes of evolution of branched flow were found numerically and fo for the fir first ti time con confirmed med ex exper erimen mental ally fo for wa water-su surface wa waves - bra ranched flo flow patterns clearly visible for the wavelengths smaller than scatterers