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

Waves over variable bathymetry – branched flow in the linear regime.

Adam Piotr Anglart1,4

1École Supérieure de Physique et de Chimie Industrielles de la ville de Paris,

Laboratoire de Physqiue et Mécanique des Millieux Hétérogènes, Paris, France

2Laboratoire d’Acoustique de l’Université du Maine, Le Mans, France 3Institut Langevin LOA, Paris, France 4Warsaw University of Technology,

The Faculty of Power and Aeronautical Engineering, Warsaw, Poland

in collaboration with T. Humbert2, P. Petitjeans1, V. Pagneux2, A. Maurel3

Branched flow seen in the wave energy map produced after the 2011 Sendai earthquake in Japan. High energy path heading for Crescent City in northern California. National Oceanic and Atmospheric Administration (2011)

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)

Shallow-water waves

very sensitive to small fluctuations

• f the bottom topography

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)

antenna

High energy paths

Microwaves experiment

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 time domain 2 Linearized shallow-water equation in frequency domain (complex solution)

y h(x,y) x z η(x,y,t)

r(gh(x, y)rη) + (ω2 iω e R)η = 0 ∂2 ∂t2 η + e R ∂ ∂tη r(gh(x, y)rη) = 0

Final element method

Shallow-water equations

perfect wall perfect wall perfect transmission Neumann BC amplitude of the wave Dirichlet BC x y ∂nη = ikη η(0, y) = A

Waves over periodic bathymetry

Bragg’s law

1 Shallow-water equation 2 Bragg’s law 3 Dispersion relation r(hrη) + ω2 g η = 0 λ = 2d sin θ n ω2 = (gk + γk3 ρ ) tanh(kh)

f1 = √gh λ = √gh d ≈ 2.7Hz

Waves over periodic bathymetry

Form of the solution

η(x) = ( ae−ikx + Raeikx, if x ∈ [0, x1] Tae−ikx, if x ∈ [x2, xmax] R = −e−ikxr1 − Hre−ikxr2 eikxr1 − Hreikxr2 T = −e−ikxr − Reikxr Hte−ikxt

x z xr x1 x2 xt xmax

incident wave reflected transmitted

• bstacle set

Waves over periodic bathymetry

Reflection and transmission coefficients for hemiellipsoid obstacles

Waves over disordered bathymetry

Branching patterns

1 Hz 3 Hz 11 Hz 27 Hz

surface elevation, η surface elevation, η surface elevation, η surface elevation, η

Waves over disordered bathymetry

Intensity maps

Energy ∝ Intensity E ∝ I = |η|2 + |∇η|2 1 Hz 3 Hz 11 Hz 27 Hz

Intensity, I Intensity, I Intensity, I Intensity, I

Waves over disordered bathymetry

Statistical analysis | Probability density function

1st regime 2nd regime 3rd regime

1 Hz 3 Hz Gaussian distribution Rayleigh distribution multiple scattering branching patterns

Experimental setup

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

Experimental setup

Waves over a flat bottom

Dispersion relation for water surface waes

Wave propagation for a flat bottom and the frequency f = 2.8 Hz Dispersion relation for water surface waves. ω2 = (gk + γk3 ρ ) tanh(kh)

camera

scatterers

Experimental setup

Disordered bathymetry

Measurment method

Free-surface synthetic Schlieren

Mesure de la déformation d’une surface libre par analys du déplacement apparent d’un motif aléatoire de points Moisy et al. 18éme Congrés Français de Mécanique (2007) camera

H

scatterers

h

dot pattern

• optical displacement field
• free-surface elevation
• refraction coefficient (0.24 for air-water interface)

rη = δr h∗ , where 1 h∗ = 1 αh 1 H

δr η α

resolution of ~ 10-2 mm

ˆ η(x, y, ω) =

∞

Z

−∞

η(x, y, t)−iωtdt

Parameters of the system

1 dimensionless wavelength 2 strength of the scatterer 3 density of scatterers 4 Ursell number

h0 h a

φd

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

1st regime 2nd regime 3rd regime

1 Hz 3 Hz Gaussian distribution Rayleigh distribution multiple scattering branching patterns

!*≈1 !*<1 !*>1

Comparison of numerical and experimental results

Summary

• num

numerical al si simulations have been carried out to obtain suitable parametres for the experiment

• 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