Gravity wave propagation in inhomogeneous media : wave

scattering and interference process

Vincent Rey

Mediterranean Institute of Oceanography (MIO)

University of Toulon (UTLN), France

rey@univ-tln.fr

Summer School and worshop« Wave in flows »

Prague, August 27-31, 2018

Environmental nearshore dynamics Ocean and coastal engineering

Vincent REY, hydrodynamics

RESEARCH ACTIVITIES

Applications

- Wave propagation and transformations in the coastal zone

- Sediment transport and morphodynamics

Some generalities on gravity waves

- Conservation equations and velocity potential

- Wave properties

Resonant interactions (2D cases) :

wave reflection and pressure oscillations

- Standing waves : basin oscillations or « seiching »

- Interference processes : application to nearshore structures

- Pressure oscillation : application to energy power device

Refraction – diffraction (3D cases) : energy focusing

and resonant interactions

- Wave celerity

- Wave focusing

- Resonant interactions : Bragg resonance

- Application to energy power device : Oscillating Water Column

CONTENTS

Swell Tp=10s

Wind wave Tp=3.3s

Water waves : sea states and wave models

Wind wave (+ swell) Swell (quasi-sinusoidal shape)

Sea states:

Waves characteristics : Hs, Tpeak, etc..

Wave models

- Monochromatic waves (phase-resolving models)

- Spectral waves (phase averaged models,

(based on the energy flux conservation)

Conditions for the potential

- Laplace’s equation (h>z>-h):

- Free surface conditions (z=h) :

- Kinematic :

- Dynamic (Bernoulli) :

- Bottom impermeability (z=-h) :

MONOCHROMATIC WAVES : STOKES THEORIES

Hypothesis

- Inviscid fluid : Euler equations

- Irrotational motion

Potential flow and

0y

x

z

z=-h

using

Total differentiation of the dynamic free surface equation and choice p=0 for the surface pressure

lead to the free surface condition (z=h) :

REMARKS

- The free surface condition is NON-LINEAR

- Expression is given at z=h, which depends on both time and space

Approximations:

- Linear : 1st order Stokes wave or Airy wave

- Perturbation methods : Stokes waves at higher orders

NON-LINEAR solutions : Asymptotic expansion of h and F

Free surface condition for F and Bernoulli expansion for h expressed at z=0

using taylor expansion:

O(e):

O(e):

O(e ): 2

O(e ): 2

Free surface (combined kinematic and dynamic (Bernoulli) conditions:

Bernoulli condition :

1st Order STOKES Wave :

Airy wave

Free surface boundary condition (z=0) :

For a free surface deformation of the form

The potential is given by

Elliptic

trajectories

Wave steepness ka<<1:

The wavenumber k verifies the dispersion relation

The wave celerity (phase celerity) is given by

The wave propagation is dispersive

Finite water depth

Shallow water approximation

Deep water condition

T=10s

shallow water deep water

(Non dispersive)

Deep water Intermediate water depth Shallow water

u,p

u,p

u,p

H

0<Dp< rgH

Dp=rgHO

Dp=0

h>l/2

h

l/2 >h>l/10 h<l/10

CHARACTERISTIC DATA and CLASSIFICATIONS

H

l 3 6T s - wind waves

Swell8 14T s -

In deep water conditions, no wave

impact near the bottom

In shallow water conditions, hydrostatic

conditions (in the linear approximation)

Wave energy :

Potential energy by unit volume:

Potential energy by unit wavelength λ and width dy for a monochromatic wave

Propagating along the x-axis :

Kinetic energy by unit wavelength λ and width dy :

Airy wave:

Total energy by unit length dx and width dy :

Mean energy flux across the y0z plane, normal to the direction of propagation:

Airy wave :

with

with

Wave energy is transported at the group velocity

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Profondeur d'eau relative h/ l

T=10s

H/H0

kH/k0H

0

Relative water depth h/l

0tE

x

Wave propagation on a gentle slope, the « shoaling »

Energy flux conservation for a progressive wave:

Wave steepness kH rapidly increases when h decreases

Non linear effects

Effect of a bed slope on the wave reflection

a

Rey, 1992

Gentle slopes weak reflection (< 1% in terms of energy)

Bottom step reflection cannot be neglected

Index discontinuity:

Partial reflection

12t

1 212r

1

121t

21r

Single step

Resonant interactions (2D cases)

Standing waves : basin oscillations or « seiching »

2

pL

l

Fundamental (p=1)

1st harmonic (p=2)

nodeantinode

an

tinod

ean

tinod

e

node

nodehalf-closed basin

closed basin

Harbour oscillations

Harbour

l/4l/2

node

antinode

antinode

antinode

Periods of seiching :

Several to tens minutes

Partially standing waves : Interference processes

Interferences : physical approach

Inhomogeneous media (sinusoidal bed)

Incident wave

x

Ref

lect

ed w

ave

d

Phase lag after one return path:2

2 x

l

D

Phase matching Reflection « allowed »

Opposition of phases Reflection «forbidden »

Interference process

Transmitted wave

Maximum of reflection for l=2d (Bragg Resonance)

d

Reflection forced by the bottom boundary condition

Bottom boundary condition for

Taylor expansion (z=-h): ...z x x

F F

z x x

F F -

If K=2k : cos( )t kxx x

F

reflected wave

z

iKx1= De + cos( )

2D Kx

x

-h( )z x h -

( , )x th

x

( )A x

Slow variation with respect to x( )A x

Rapid variation with respect to x

Multi-scale expansion method:

Mei, 1983

Reflection forced by the bottom boundary condition

Belzons et al, 1991; Rey, 1992

Sinusoidal bed Doubly sinusoidal bed

Bragg resonance for K=2k :

2k/K

Strong reflection at the Bragg conditions

Oscillations on both parts due to the

finite length of the bed

1st order and subharmonic (2nd order)

Bragg resonance

High interaction at low frequency

even if « 2nd order » Bragg

Reflection tends to 1 for increasing

bed length

Reflection forced by the bottom boundary condition

Doubly sinusoidal bed

Interference process :

Analogy with the PF interferometer

Guazzelli et al, 1992; Rey, 1992

Patch i Patch j

interference process

Section

of the

La Jolla

canyon

Reflection coefficient for a

normally incident wave

Reflection in the presence of canyon : the La Jolla canyon (califormia, USA)

Magne et al, 2007

Application to nearshore structures:

Rectangular bar : Influence of the length on the reflection H1=4mH2=1.5m

L

L=2m L=4m

L=8m L=16m

Reflection R

Transmission T

Frequency (Hz)Frequency (Hz)shallow water deep water

h1=4m

h2=1.5m

L=4m

0 evanescent mode1 evanescent mode2 evanescent modes

Application to nearshore structures:

Expression of the velocity potential : presence of evanescent modes

Integral matching method, at a step:

with

Pressure:

Velocity:

Frequency (Hz)

Weight of evanescent modes

Rey, et al 1992

Model taking into account

the evanescent modes

Frequency (Hz)

1st harmonic

Bar location

Fundamental

Ref

ecti

on

co

effi

cien

t Arb

itra

ry a

mp

litu

de

Significant weight at the vicinity of the structure

Evanescent modes involved in the interference process

Application to nearshore structures: Porous structures: reflection and dissipation

Arnaud et al, 2017David de Drézigué et al, 2013

3.00 m

0.15 m0.30 m

1.20 m

0.23 m

D=0.050 m

Porosity g=0.7

Porosity g=0.3

Porous

locationWave

generatorAbsorbing

beach

Experimental set-up, SeaTech wave tank, UTLN

Frequency (Hz)

Dispersion relation

with

and

Added mass

Application to nearshore structures:

Submerged plate : Interference process and Influence of the beach on the reflection

Ocean basin FIRST, La Seyne/Mer, France

H1=3m

H2s=0.5m

L=1.53m

H2f=2.4m

Interference process above the plate

Oscillating behaviour due to reflection from the beach (absorber)

Reflection due

to the beach

Reflection

Upwave the plate

Rey and Touboul, 2011

O

h>l/2

Dp=0 Dp=?

h

Application to nearshore structures:

Submerged plate : bottom induced pressure

Symbols :measurements

Dashed line and full line :

calculations (linear model)

Location of the plate

0<X<1.53m

Partially

standing wave

Free surface deformation and bottom pressure amplitudes,

T=1.4s, a=54mm

X(m) X(m)

)cosh(kh

gaP i

N

r

Harmonic frequency 2f

Fundamental

frequency f=1/T

- 1st order bottom pressure amplitude (frequency f) 30 times

the bottom pressure due to the incoming wave

- « Longuet-Higgins effect » at 2nd order (frequency 2f) Touboul and Rey, 2012

Application to wave energy device:

bottom induced pressure at twice the wave frequency for « deep water » conditions

Non linear free surface condition at z=h:

Perturbation method (for 2nd order Stokes waves):

for z=-h (in fact for any z)

Jarry, 2009

Spectre des pressions pour le capteur C6

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.5 1 1.5 2 2.5 3

Frequence (Hz)

Pre

ssio

n (

cm

)

C6

Wave maker

Reflective

wall

1.3m

19.12m

C6

Pressure

sensor

C6

Wave frequency f=1Hz

Frequency (Hz)

0 0.5 1 1.5 2 2.5

Pre

ssu

re (

cm)

0

0.08

high peak pressure at twice the wave frequency (Longuet-Higgins, 1950))

Sig

nif

ican

t p

ress

ure

(m

)0

0.012

Period (s)

Incident wave angle (degree)0 500.8 1 1.2

0.035

0

pre

ssu

re (

m)

Application to wave energy device:

Real sea states

Sea bed

Energy converter device

Sea surface

Sea wall

Incident wave reflected wave

Experimental set-up,

ACRI-in, France

« Longuet-Higgins » effect significantly decreases for wave frequency and direction spreadingJarry, 2009

HS=5cm

H=5cm

H=10cm

HS=10cm

Refraction – diffraction

Réfraction

Diffraction

Slow changes of wave properties :

Plane waves, ray theory

Rapid change of amplitude and/or direction :

No more plane waves, diffraction parameter

Refraction – diffraction

Canyon of Capbreton, France

caustics

Wave celerity C

raycrest

ray crest

Wave celerity C=/k depends on

- bathymetry :

- Currents :

- Porous media :

H Energy spreading

through scattering

Bonnefille, 1992

Refraction – Descartes – Snell’s Law

(WKB approximation, mild slope)

Free surface

h1h2

Incident wave

Reflected

wave Transmitted

wave

Descartes-Snell’s Law

q2

q1-q1

l1

l2

Side view

1 21 2

1 2 1 2

sin sin avec = , =C C

C C k k

q q with

1D step effectSlowly varying depth

since :

where

in the plane wave approximation

and

Eikonal equation

Diffraction due to rapid bathymetric

changes

Diffraction from abrupt structures

(dikes, jetties)

Diffraction

Refraction-diffraction due to « rapid » bathymetric changes

Berkhoff equation

Velocity potential :

Conservation equation and boundary conditions :

0y

x

z

z=-h(x,y)

Hypothesis (mild slope) :

with the Green function

Weak formulation of the Laplace’s equation :

Berkhoff equationMei, 1983

Rigid breakwater

cos( )

cos( )

4( , ) sin

2

4sin

2

ikr

ikr

krF r I e

krI e

a q

a q

a qq

a q

-

-

- -

-

qa

r M(x,y)

Rigid breakwater

F verifies the Helmotz equation :

Velocity potential :

Solution (polar coordinates) :

Diffraction due to vertical walls : Diffraction behind a breakwater

Wave of incidence a : Normal incidence (a=/2) :

with

Penney and Price, 1952, Horikawa, 1988

amplitude

Wave diffraction behind the breakwater

crest

Rey et al, 2018

Propagation in the presence of vertical walls : Expression of the velocity potentials

d2M

h = constant water depth

Integral matching method, at a width discontinuity:

with

Pressure:

Velocity:

where

x

y

0

d2m

d1M

d1m

Top view

n < nprop Propagating modes of direction

Evanescent modes along the x-axis

q

Wave direction

x

y

n > nprop

Propagation in the presence of vertical walls : examples

Sub-wavelength hole Rectangular emerging structure

dimensionless amplitude with respect to the incoming wave dimensionless amplitude with respect to the incoming wave

d=0.2m

Top views

L=0.2m

l=0.39m

d=0.3m

L=1.2m

l=1.56m

either spreading or focusing effects

Incoming wave

direction

Jarry et al, 2009

Propagation in inhomogeneous media : wave focusing above a shoal

PeriodMaximum of

amplification

Distance from the

end of the mound

(m)

T=0.3 1 -

T=0.4 1.12 -0.20

T=0.5 1.31 -0.10

T=0.6 1.53 -0.10wave focusing especially for longer waves

Experimental set-up, SeaTech wave tank, UTLN

Holtuijsen et al., 2003, Jarry et al, 2009

Wave focusing above a shoal : influence of the diffraction term

Propagation in inhomogeneous media : wave focusing in the presence of currents

Wave gages

Rey et al, 2014

Ocean basin FIRST, La Seyne/Mer, France

X(m)

Wav

e am

pli

fica

tio

n

Wave amplification

current lield forced by the underwater mounds

Amplification up two twice the incident wave amplitude

in deep water conditions for wave-opposing current conditions

Mean curent field

Propagation in inhomogeneous media : wave scattering in the presence porous media

Porous rectangular structure

made of vertical cylinders

L x l = 0.30m x 1.20m

wave scattering:

Reflection upwave, refraction/diffraction

propagation across the porous medium

Arnaud, 2016

Porous, 3d

Impervious, 3D

Porous, 2D

X(m)

Y(m

)

Y=0.10m

X

Y

Propagation in the presence of periodic structures :

wave diffraction and Bragg resonance

Rey et al, 2018

Periodic breakwaters of rectangular shape

Periodic cylinder arrays

Incident wave

scattered waves

Incident wave

scattered waves

shore

shore

L

D=2a

W

Periodic breakwaters of rectangular shape

wave scattering

k, wave wavenumber,

Periodicity d of the breakwaters along y-axis

Number of propagating modes:

Reflection coefficient versus kb:

(-), 1st mode (n=0); (-) second mode (n=1);

(-) third mode (n=2); (-) fourth mode (n=3);

Relative reflected energy flux versus kb :

(-.-),1st mode (n=0), (- -), total energy. Vertical

dashed lines correspond to the locations

of the frequency cases presented in the following

Energy flux conservation:

d-l

kd

d

1 mode 2 modes 3 modes 4 modes

resonance along y-axis for kd=n

2 3

(a) (b)

(c) (d)

n =1prop

n =2prop n =2prop

n =1prop

Propagation in the presence of periodic cylinders :

Convergence with respect to the numbers of modes and steps

Propagation in the presence of periodic cylinders :

Convergence with respect to the number of steps ( for 10 modes)

Reflection coefficient in the case of sparse array of 11 cylinders (a/L = 0.10,

L/W = 1) versus kL/: (- - ), 1st mode, P = 2 (3 modes); (-), 1st mode, P = 3 (4

modes); (- - ), second mode, P = 2 (3 modes), (-), second mode, P = 3 (4 modes);

(- -), third mode, P = 2 (3 modes); (-), third mode, P = 3 (4 modes).

1st order

Bragg resonance

for the 1st mode

(direction Ox)

higher orders

Bragg resonance

for the 1st mode,

kL=n

Propagation in the presence of periodic cylinders : Bragg resonances

higher orders

Bragg resonance

for 2nd and 3rd modes

Amplification due to

resonance along y-axis

Propagation in the presence of periodic cylinders :

Bragg resonances for scattered waves

(a) Reflection versus kx1L/:

(-), reflection coefficient for the second mode;

(-), total reflected energy;

(-), reflected energy for the 1st mode;

(-.-), reflected energy for the second mode;

(- -), reflected energy for the third mode;

(b)Reflection versus kx2L/:

(-), reflection coefficient for the third mode;

(-), total reflected energy;

(-), reflected energy for the 1st mode;

(-.-), reflected energy for the second mode;

(- -), reflected energyfor the third mode.

Resonance conditions:

Application to wave energy device : Oscillating water column (OWC)

Research project at ISITV, UTLN (years 2000):

How to provide energy, for punctual needs

in the natural park of Scandola in Corsica,

France for public lighting?

Bay of Girolata, Corsica,

France

Prototype, ISITV (now SeaTech), UTLN

Oscillating water column

(from Delauré and Lewis, 2003)

Wells turbine

How to increase incoming wave energy?

Wave focusing above a shoal

Gouaud et al, 2010

Application to wave energy device : Oscillating water column (OWC)

A B

0

0,5

1

1,5

2

2,5

0 2 4 6 8 10 12

X longitudinal postion (m)

Am

pli

ficati

on

co

eff

icie

nt

H/

Hi

experimental results

numeric

A B

Numerical (REF-DIF)

A

B

Experiments

T=1.7s

Ocean basin FIRST

Gouaud et al, 2010

Numerical (stepwise

discretization, integral

matching method)

Application to wave energy device :

Wave amplification

Wave amplification depends on the geometry

of the OWC

Wave amplification enhanced by the shoal

Channel length

sloshing

Piston-type

resonance

Gouaud et al, 2010

Wave amplification in the water column

Linear wave behaviour

channel 0.9 m

0

1

2

3

4

5

6

7

0 1 2 3 4Period (s)

H /

Hi

irregular waves : Tp=2.1s

irregular waves : Tp= 2.6 sHs=60mm

0

0,00005

0,0001

0,00015

0,0002

0,00025

0,0003

0 0,5 1 1,5

Frequency (1/s)

Sp

ectr

al

den

sit

y

Tp=2.6s

Tp=2.1s

• Use of two Jonswap spectra

• Both transfer functions are almost identical

Non-linear effects are rather negligible in that case

Wave spectraSloshing

resonancePiston-type

resonance

Conclusions

Irregular waves : interference process less important due to

wave energy spreading on both frequencies and direction of propagation

In the 3d Case of constant water depth : Helmotz equation, analogy to other types of waves as acoustic waves

Inhomogeneous medium : water wave celerity changes (bathymetry, current, porous medium, presence of surface-piercing structures)

Scattering : 2D, reflection, transmission

3D, + refraction, diffraction, focusing or dispersion

Leads toInterference process due to wave partial reflection at domain boundaries

and Bragg resonance for periodic inhomogeneous media

References