26 July 2011 – WAVES 2011 - Vancouver
THE NUMERICAL COMPUTATION OF VIOLENT WAVES –
APPLICATION TO WAVE ENERGY CONVERTERS
Oyster Aquamarine Power 2009
Frédéric DIAS
School of Mathematical Sciences
University College Dublin
on leave from Ecole Normale Supérieure de Cachan
26 July 2011 - Vancouver
1
TOPICS COVERED IN TODAY’S TALK
LIQUID IMPACT
ON A WALL
TSUNAMIS
Thailand 2004
FREAK WAVES
South Carolina 1986
Sloshel
Spain 2010
LNG carrier
Japan 2011
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2
WAVE ENERGY CONVERSION
Aquamarine Power is a technology company that has
developed a product called Oyster which produces
electricity from ocean wave energy.
UCD and Aquamarine Power are
collaborating to deliver the nextgeneration Oyster 800.
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3
COLLABORATORS
Jean-Philippe Braeunig (INRIA & CEA)
Laurent Brosset (GTT)
Paul Christodoulides (Cyprus University of Technology)
Ken Doherty (Aquamarine Power Ltd.)
John Dudley (University of Franche-Comté)
Denys Dutykh (University of Savoie)
Christophe Fochesato (CEA)
Jean-Michel Ghidaglia (Ecole Normale Supérieure de Cachan)
Laura O‟Brien (University College Dublin)
Raphaël Poncet (CEA)
Themistoklis Stefanakis (University College Dublin & ENS-Cachan)
Research funded by
FP6 EU project TRANSFER (Tsunami Risk ANd Strategies For the European Region)
ANR (French Science Foundation)
SFI (Science Foundation Ireland)
GTT (Gaz Technigaz & Transport)
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PART 1
LIQUID IMPACT
ON A WALL
Sloshel
LNG carrier
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5
WAVE IMPACT AND PRESSURE LOADS
•Local phenomena involved during wave impacts are very sensitive to input conditions
•The density of bubbles, the local shape of the free surface, the local flow make the
impact pressure change dramatically even for the same experimental conditions
•How does one extrapolate wave impact from model (small scale) to prototype (full
scale)?
•In recent years, we have addressed the scaling issue by studying the various local
phenomena present in wave impact one after the other in order
 to better understand the physics behind
 to improve the experimental modelling
VIDEO 1 – Experiments in Marseille
(courtesy of O. Kimmoun)
6
VARIOUS SCENARIOS OF WAVE IMPACT
Slosh
« Flip-through »
condition with no air
bubbles. The impact
pressure has a single
peak.
Collision of a plunging
breaker with a thin air
pocket
Collision of a fully
developed plunging
breaker with a thick air
pocket
Source : Peregrine D.H. (2003), Water wave impact on walls, Annual Review of Fluid Mechanics
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THE FLIP-THROUGH PHENOMENON
•Run-up of wave trough
•Forward motion of almost vertical wave front
•Jet flow generated at the wall
8 m/s
55 bars
Bredmose et al. (2004), Water wave
impact on walls and the role of air,
Proceedings ICCE 2004
Brosset et al. (2011), A Mark III Panel Subjected
to a Flip-through Wave Impact: Results from the
Sloshel Project, Proceedings ISOPE 2011
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PHENOMENOLOGY OF A LIQUID IMPACT
 Global behaviour
 Global flow governed by Froude number
 Local behaviour
Escape of the gas between the liquid and the wall:
momentum transfer between liquid and gas
Compression of the partially entrapped gas during the last
stage of the impact
Rapid change of momentum of the liquid diverted by the
obstacle
Possible creation of shock waves: pressure wave within the
liquid and strain wave within the wall
Hydro-elasticity effects during the fluid-structure interaction
Braeunig et al. (2009), Phenomenological Study of Liquid Impacts through 2D Compressible
Two-fluid Numerical Simulations, Proceedings ISOPE 2009
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EXAMPLE OF BIAS INTRODUCED AT SMALL SCALE
Locally, the impact process is not similar for similar inflow conditions
 Gas compressibility bias: the equations of state should be scaled
 Liquid compressibility bias: speed of sound should be scaled
Liquid
Gas
Wall
Scale 1
Scale 1/40, with right
compressibility scaling
(Complete Froude Scaling)
 Biases are different for different impacts: no unique scaling law!
10
26 July 2011 - Vancouver
Scale 1/40 with
Partial Froude
Scaling
INCOMPRESSIBLE EULER EQUATIONS WITH AN INTERFACE IN
PHYSICAL VARIABLES
(x,y,z) : spatial coordinates
u = (u,v,w) : velocity vector
p : pressure
r : density
g : acceleration due to gravity
h(x,y,t) : elevation of the interface
kinematic and dynamic
conditions on the interface
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INVARIANCE OF INCOMPRESSIBLE EULER EQUATIONS WITH AN
INTERFACE
Froude scaling
fs stands for full scale, ms for model scale
Dfs = l Dms
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TWO-FLUID COMPRESSIBLE EULER EQUATIONS WITH AN
INTERFACE IN PHYSICAL VARIABLES
Fluid equations
k = 1, 2
Boundary conditions
Equation of state
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INVARIANCE OF TWO-FLUID COMPRESSIBLE EULER EQUATIONS
WITH AN INTERFACE
Differences with the one-fluid incompressible case
pfs = λ(ρliqfs/ρliqms) pms
m = r ms / r fs
Scale the equations
of state
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Keep the same
density ratio
14
THE ROLE OF NUMERICAL STUDIES
• For sloshing inside the tank of a LNG carrier or for the motion of a wave energy converter,
numerical simulations can provide impressive results but the question remains of how
relevant these results are when it comes to determining impact pressures !
• The numerical models are too simplified to reproduce the high variability of the measured
pressures. NOT POSSIBLE FOR THE TIME BEING TO SIMULATE ACCURATELY BOTH
GLOBAL AND LOCAL EFFECTS ! (see ISOPE 2009 Numerical Benchmark)
• However, numerical studies can be quite useful to perform sensitivity analyses in idealized
problems (see ISOPE 2010 Numerical Benchmark)
15
COMPARATIVE NUMERICAL STUDY (2010)
Length (m)
Participants
ANSYS
Principia
ENS-Cachan
Hydrocean
BV
LR
Force
16
H
15
h
8
h1
2
h2
5
Case #
1
2
3
4
5
Scale
1:1
1:40
1:40
1:40
1:40
Organizers: GTT and UCD
(compressible bi-fluid software was required)
1D case
LNG = Liquefied Natural Gas
NG = Natural Gas
Liquid
LNG
LNG
Water
Water
1:40-scaled LNG
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Gas
NG
NG
Air
SF6+N2
1:40-scaled NG
1D SURROGATE MODEL OF AIR-POCKET IMPACT
 Piston model
 Perfect gas
H-h-z
 Adiabatic process
ρl
g
p
Gas
Liquid
Gas
p
h
Initial conditions: p=p0,
z( 0 )  0
z(0)=h1,
z
p0  
h2
  h1 
z( t )   g 

  
rl h   H  h  z   z 

17
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




THEORY : PRESSURES IN TIME DOMAIN
6.0E+04
Pressures at full scale: theoretical model
Pressure (Pa)
7.0E+04
Case 1
Case 2
Case 3
Case 4
5.0E+04
Warning:
4.0E+04
Case 5 = Case 1
 Compressibility matters,
when comparing at same scale
The 5 cases are very
smooth compression cases
3.0E+04
2.0E+04
1.0E+04
0.0E+00
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Pressures and times are Froude-scaled for cases 2, 3, 4, 5
with λ = 40, fs = full scale, ms = model scale
18
1.0
Time (s)
1.1
1.2
pfs = λ.(ρliqfs/ρliqms).pms
tfs = √λ.tms
0.7
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0.8
0.9
CONCLUSIONS FOR THE 1D CASE
 The different numerical methods are able to simulate adequately a
simple smooth compression of a gas pocket without escape of gas
 Very good agreement on the maximum pressure
 For all methods :
 Complete Froude Scaling (CFS) works (same result for cases 1
& 5)
 Partial Froude Scaling (PFS) generates a bias
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COMPARATIVE NUMERICAL STUDY (2010)
VIDEO 2 – ENS-Cachan code
2D case
Length
H
h
h1
h2
L
l
l1
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(m)
15
8
2
5
20
10
5
20
COMPARATIVE NUMERICAL STUDY (2010) – 2D CASE
Case # Scale
6
7
8
9
10
1:1
1:40
1:40
1:40
1:40
Liquid
Gas
LNG
LNG
Water
Water
1:40-scaled LNG
NG
NG
Air
SF6+N2
1:40-scaled NG
Participants
Software
Method
ANSYS
Fluent
Finite Volume/VOF
Principia
LS-DYNA
FEM Euler/lagrange
ENS-Cachan
Flux-IC
Finite Volume/NIP
Lloyds Register Open-Foam Finite Volume/VOF
Force
Comflow
Finite Volume/VOF
UoSFSI
in House Finite Differences/VOF
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6
X
X
X
X
X
X
7
X
X
X
X
X
X
8
X
X
X
X
X
9
X
X
X
X
X
X
10
X
X
X
X
X
21
6.0E+05
5.0E+05
4.0E+05
Pressure (Pa)
ENS-CACHAN : MESH SENSITIVITY FOR CASE 6 (SCALE 1:1)
Pressure at P1 for three different meshes
•Warning:
P-P0 cas6 dx=0.05m
P-P0 cas6 dx=0.1m
•High refinement
is required for
P-P0 cas6
dx=0.0333m
capturing the pressure
peak
3.0E+05
2.0E+05
1.0E+05
0.0E+00
-1.0E+05
0.5
0.6
0.7
Time (s)
0.8
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0.9
1
22
HORIZONTAL TIME SERIES AT P5 : ALL PARTICIPANTS
130
120
110
Case 6
Case 7
Case 8
Case 9
55
45
100
120
110
Case 6
Case 7
Case 8
Case 9
Case 10
90
80
70
35
130
90
80
70
60
50
50
40
15
40
30
30
5
20
20
10
0.90
Time (s)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-15
10
1
0
Velocity (m/s)
55
50
45
Horizontal velocity in P5 at full scale: Lloyds Register
0.1
26
24
22
20
Case 6
Case 7
Case 8
Case 9
Case 10
40
35
30
0
Time (s)
Velocity (m/s)
-5 0
Case 6
Case 7
Case 8
Case 9
Case 10
100
60
25
Horizontal velocity in P5 at full scale: ENS-Cachan
0.2
10
0.6
0.7
Time (s)
0
0.1
40
30
Case 6
Case 7
Case 8
Case 9
Case 10
14
12
0.5
35
16
20
0.4
Horizontal velocity in P5 at full scale: Force Technology
18
25
0.3
Velocity (m/s)
65
140
Horizontal velocity in P5 at full scale: PRINCIPIA
Velocity (m/s)
75
Velocity (m/s)
horizontal velocity in P5 at full scale: ANSYS
Vx (m/s)
85
0.2
0.3
0.4
0.5
0.6
0.
Horizontal velocity in P5 at full scale: U. of Southampto
Case 6
Case 7
Case 8
Case 9
Case 10
25
20
15
10
8
15
10
5
6
5
4
0
2
0
Time (s)
0
0.1
0.2
0.3
0.4
0.5
0.6
-5
0
0.7 0
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.1
0.2
0.3
0.4
0.5
0.6
Tim
0.7
-10
Velocities and times are Froude-scaled for cases 7, 8, 9, 10
Vfs = √λ.Vms , tfs = √λ.tms
with λ = 40, fs = full scale, ms = model scale
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0.6
23
MAXIMUM PRESSURE AT P1 : ALL PARTICIPANTS
Pmax (bar) - all participants
35
Case 6: scale 1 - LNG/NG
Case 10: scale 1:40 - scaled LNG/NG
Case 7: scale 1:40 - LNG/NG
Case 8: scale 1:40 - water/air
Case 9: scale 1:40 - water/(SF6+N2)
30
25
20
15
10
5
0
ANSYS
Principia
ENS-Cachan
LR
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ForceT
UoS
24
COMPARATIVE NUMERICAL STUDY (2010) – 2D CASE
 Absolute values for Vmax and Pmax are very scattered
 The meshes are not refined enough to capture sharp peak pressures
 After some work on the models, results should be much less scattered
(work in progress)
 Such a simple test should be passed adequately before
attempting to calculate more complex impacts
 For all methods, whether relevant or not :
 Complete Froude Scaling (CFS) is satisfied
 Partial Froude Scaling generates a bias
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BACK TO WAVE IMPACT
• Strategy used to compute
wave impact : couple potential
flow solver with two-fluid
compressible flow solver
• Potential flow solver
computes the wave all the way
to overturning (Fochesato &
Dias 2006)
• Two-fluid (gas + liquid)
compressible flow solver
computes the liquid impact on
the wall
work in progress
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PART 2
TSUNAMIS
Thailand 2004
Japan 2011
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MATHEMATICAL GENERATION OF A TSUNAMI (CLASSICAL)
Sea bottom deformation (green) due to
the earthquake (a few metres)
The water surface (blue) which was flat
before the earthquake is suddenly
deformed. It takes the same shape as
the deformed sea bottom (water is
incompressible) – Kajiura 1963
Sea bottom deformation obtained by
Okada‟s solution
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Before
After
28
SEA BOTTOM DEFORMATION
Okada‟s solution (or Mansinha &
Smylie) based on dislocation theory
and fundamental solution for an elastic
half-space
Fault consisting of one or a few
segments (Grilli et al. (2007) for 2004
megatsunami, Yalciner (2008) for Java
2006 event, …)
Passive tsunami generation
Illustration – July 17, 2006 Java event
– single-fault based initial condition
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FINITE-FAULT SOLUTION
Based on the multi-fault representation of the rupture (Ji et al.)
Complexity of the rupture reconstructed using a joint inversion of the static
and seismic data
Fault‟s surface parametrized by multiple segments with variable local slip,
rake angle, rise time and rupture velocity
Can recover rupture slip details
Use this seismic information to compute the sea bottom displacements
produced by an underwater earthquake with higher geophysical resolution
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FINITE FAULT SOLUTION – 17 JULY 2006 JAVA EVENT
BATHYMETRY
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FINITE FAULT SOLUTION – 17 JULY 2006 JAVA EVENT
Vertical displacement
Nx = 21
Ny = 7
VIDEO 3
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SOLUTION AT VARIOUS GAUGES – 17 JULY 2006 JAVA EVENT
Three methods used to integrate the water wave equations
Weakly nonlinear method (Formulation involving the Dirichlet-to-Neumann
operator, numerical evaluation of the operator using Fourier transforms, time
integration using the standard 4th order Runge-Kutta scheme, approach in the
spirit of Craig, Sulem, Guyenne, Nicholls, etc)
Cauchy-Poisson problem (Linearized water wave equations)
Boussinesq-type method (Formulation derived by Mitsotakis (2009), integration
with the standard Galerkin/finite element method, time integration using the 2nd
order Runge-Kutta scheme)
Virtual gauges
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SOLUTION AT VARIOUS GAUGES – 17 JULY 2006 JAVA EVENT
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OTHER TOPIC IN TSUNAMI RESEARCH – EFFECT OF SEDIMENTS
Science 17 June 2011
As the rupture neared the seafloor, the movement of the fault grew rapidly, violently
deforming the seafloor sediments sitting on top of the fault plane, punching the overlying
water upward and triggering the tsunami.
This amplification of slip near the surface was predicted in computer simulations of
earthquake rupture, but this is the first time we have clearly seen it occur in a real
earthquake.
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OTHER TOPIC IN TSUNAMI RESEARCH – EFFECT OF SEDIMENTS
80%
amplification
0
0.25
sediment thickness / fault depth
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OTHER TOPIC IN TSUNAMI RESEARCH – RUN-UP AMPLIFICATION
It has been observed that sometimes the most dangerous tsunami wave is not the leading wave.
1D numerical simulations in the framework of the Nonlinear Shallow Water Equations are used to
investigate the Boundary Value Problem (BVP) for plane and non-trivial beaches.
Monochromatic waves, as well as virtual wave-gage recordings from real tsunami simulations, are
used as forcing conditions to the BVP.
Resonant phenomena between the incident wavelength and the beach slope are found to occur,
which result in enhanced runup of non-leading waves.
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THE 25 OCTOBER 2010 EVENT – RUN-UP AMPLIFICATION
Magnitude 7.7 earthquake on 25 October west of South
Pagai, a small island off the west coast of Sumatra
Death toll > 700
Bathymetry
Runup amplification
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VOLNA CODE FOR THE NUMERICAL MODELING OF TSUNAMI WAVES:
Generation, propagation and inundation
 Unstructured triangular meshes – can be run in arbitrary
complex domains
 Finite volume scheme implemented in the code
 Robust treatment of the wet/dry transition
 Similarities with GeoClaw (LeVeque) – see yesterday
Benchmark – Catalina 2
Reproduces at 1/400 scale the
Monai valley tsunami, which struck
the Island of Okushiri (Hokkaido,
Japan) in 1993. The computational
domain represents the last 5 m of
the wave tank. Initial incident wave
offshore given by experimental data,
and fed as a time-dependent
boundary condition.
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VIDEO 4
39
PART 3
FREAK WAVES
South Carolina 1986
Spain 2010
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EXTREME OCEAN WAVES
Rogue Waves are large oceanic surface waves that represent statistically-rare
wave height outliers
Foremast
1934
February 1986 - It was a nice day with light breezes and
no significant sea. Only the long swell, about 5 m high
and 200 to 300 m long. We were on the wing of the
bridge, with a height of eye of 17 m, and this wave broke
over our heads. Shot taken while diving down off the
face of the second of a set of three waves. Foremast
was bent back about 20 degrees (Captain A. Chase)
Damage and fatalities when wave height is outside design criteria
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EXTREME OCEAN WAVES
Definition of a Rogue Wave: H / Hs > 2 where H is the wave height (trough to
crest) and Hs the significant wave height (average wave height of the one-third
largest waves)
Rogue waves are localised in space (order of 1 km) and in time (order of 1 minute)
Existence confirmed in 1990‟s through long term wave height measurements and
specific events (oil platform measurements)
Draupner 1995
Elevation of the sea
surface in metres
Time in seconds
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HOW ARE EXTREME OCEAN WAVES GENERATED ?
The physics of rogue wave formation is a subject of active debate.
Very few observations!
•
•
•
•
•
•
Linear Effects
Focusing due to continental shelf topography
Directional focusing of wave trains
Dispersive focusing of wave trains
Waves + opposite current
Nonlinear Effects
Exponential amplification of surface noise (instabilities)
Formation of quasi-localised surface states
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EXTREME OCEAN WAVES
The physics of rogue wave formation is a subject of active debate and studies are
not helped by practical observational difficulties
VIDEO 5 –
Experiments in
Nantes
Observation of swell propagation from satellite
data (IFREMER, BOOst Technologies)
Evidence of directional wave focusing in
a « numerical » wave tank (Fochesato,
Grilli, Dias, Wave Motion, 2007 – based
on BEM)
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NONLINEAR EFFECTS – NONLINEAR SCHRÖDINGER EQUATION
What does the Nonlinear Schrödinger equation contain? (Osborne‟s book)
(1) The superposition of weakly nonlinear Fourier components.
(2) The Stokes wave nonlinearity.
(3) The modulational instability.
Two populations of nonlinear waves:
Population 1: Weakly nonlinear superposition of sine waves (1) together with the Stokeswave correction (2).
Population 2: Item (3) predicts the existence of unstable wave packets, which can rise up
to more than twice the significant wave height.
Unstable packets are a type of nonlinear Fourier component in NLS spectral theory. This
spectral contribution arises from the Benjamin-Feir instability and obeys a threshold effect:
Only large enough wave fields have unstable modes in the spectrum.
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NONLINEAR OCEAN WAVES & INVERSE SCATTERING TRANSFORM
Space like Benjamin-Feir parameter
Time like Benjamin-Feir parameter
Tracy 1984
If one wants more rogue waves in an experiment, one needs to increase the significant
wave height and increase the number of carrier oscillations under the modulation (that is
make the spectrum narrower).
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EXTREME WAVES GENERATED BY MODULATIONAL INSTABILITY
The initial wave condition is a monochromatic deep water wave with a small disturbance added
(modulation with a wavelength much larger than the wavelength of the carrier).
After several wave periods instabilities develop and energy becomes concentrated at a single
peak in the wave group.
Surface elevation in a timespace representation (during
10 wave periods)
1929
-1995
Snapshots of the surface elevation
during 10 wave periods
T.B. Benjamin
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MODULATION INSTABILITY IN THE TIME NLSE
Exponential growth of a periodic perturbation
Linear stability analysis yields gain, etc
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BREATHER SOLUTIONS SOLVE THE SAME PROBLEM
An ideal growth-return cycle of MI is described as a breather
“Akhmediev Breather” (AB)
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WHAT ABOUT THE CASE a = 0.5 ?
Emergence “from nowhere” of a steep wave spike
Soliton on finite background
Maximum contrast between peak and background
Polynomial form (H. Peregrine)
But zero gain !
1938
-2007
1938
-2007
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OBSERVING AN UNOBSERVABLE SOLITON
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UNDER INDUCED CONDITIONS OUR TEAM FOUND THIS ROGUE WAVE SOLUTION
VIDEO 6
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THIS ROGUE WAVE SOLUTION IN A WAVE TANK ?
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READING
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OCEAN WAVE ENERGY : AN ASSET
WAVE ENERGY
CONVERSION
Oyster Aquamarine Power 2009
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OYSTER TECHNOLOGY
•
•
•
•
•
Large mechanical „flap‟ moves
back and forth with motion of
waves
Two hydraulic pistons pump high
pressure water via pipeline to
shore
Conventional hydroelectric
generator located onshore
Secured to seabed at depths of
8 – 16m
Located near shore, typically 500
– 800m from shoreline
VIDEO 7
56
OYSTER
PROJECT MILESTONES
• Oyster 1 Project – 315kW demonstrator successfully installed and gridconnected at European Marine Energy Centre (EMEC) in Orkney, October
2009
• Oyster 2 Project – 2.4MW project – on schedule for 2011
• Oyster 3 Project – 10MW development on track – commissioning 2013 or
2014
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WHAT DOES MATHEMATICS BRING ?
High end computational modeling for wave energy systems
1. Wave impact and pressure loads on a single Wave Energy Converter
2. Optimal device spacing for an array of Wave Energy Converters
3. Preferred geographical locations for near shore Wave Energy
Converter sites in Ireland
4. Biofouling (biological growth on surfaces in contact with water)
58
SURVIVABILITY
Scale effects in experiments at
small scale
•
One of the most commonly acknowledged
difficulties of conducting experiments with
Wave Energy Converters : presence of
scale effects (Reynolds much larger at full
scale than at small scale – for example, at
scale 1/40, viscous forces on the model
are multiplied by a factor 253 if only
Froude scaling is satisfied)
•
This makes mathematical and numerical
modelling a particularly valuable tool in the
development of Wave Energy Converters
Experiments performed at Queen‟s University Belfast
59
CONCLUDING REMARKS
Flaps might be simple devices from the engineering point of view. But they are really
challenging from the fluid mechanics and numerical simulation points of view
•Intermediate water depth (kh of order unity)
•3D problem
•Neither laminar flow nor potential flow (vortex shedding)
•Moving solid boundary (large motion) + free surface
•Intermediate size structure (no simplification)
•Nonlinear waves
•Coupling with biological growth
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THANK YOU FOR YOUR ATTENTION
MERCI POUR VOTRE ATTENTION
Howth, Ireland
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