Far-field maximal power absorption of a bulging ... · I FJM. Farley.\Far- eld theory of wave power...
Transcript of Far-field maximal power absorption of a bulging ... · I FJM. Farley.\Far- eld theory of wave power...
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Far-field maximal power absorptionof a bulging cylindrical wave energy converter
Matthieu AncellinUniversity College Dublin and MaREI Centre, Ireland
(Present address: ENS Paris-Saclay, France)
withA. Babarit (ECN), Ph. Jean (SBM Offshore) and F. Dias (UCD)
HyWEC 2, 18th June 2019
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Context
A. Babarit, J. Singh, et al. “A numerical model for analysing the hydroelastic response of a flexible
Electro Active Wave Energy Converter”. In: Journal of Fluid and Structures 74 (2017), pp. 356–384
I Goal: open lightweight tool for the hydrodynamical design of WEC withnon-trivial degrees of freedom.
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Summary
Theory of far-field maximal absorption width
Implementation
ApplicationsFloating buoyBulging cylindrical WEC
Conclusion and perspectives
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1 Theory of far-field maximal absorption width
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Strategy
I Linear potential flow in frequencydomain.
I Energy balance:
captured =
∮incoming and leaving waves
I For a given geometry, find the motiongiving the best captured width.
Scheme adapted from (Babarit and Delhommeau 2015)
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Preliminary definitions
Kochin function H
Φ(R, θ, z) = 4π
√1
λRH(θ)f0(z)e i(2π
Rλ+π
4 ) + O
(1
R32
)Complex-valued amplitudes of motions aj
Linear potential flow ⇒ linear combination of dofs
H =n∑
j=1
ajHj
Absorption width W
W =mean absorbed power [W]
mean wave power flux [W/m]
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Absorption width
For a given motion a = (aj)j∈{1,...,n}:
W (a)
8πk= =
n∑j=1
ajH∗j (π + β)
− k2∫ 2π
0
∣∣∣∣∣∣n∑
j=1
ajHj(θ)
∣∣∣∣∣∣2
dθ
Maximal absorption width
W optimal = maxa∈Cn [W (a)]
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Constraints on motion
Actually, aoptimal might be unphysical.
Maximal absorption width under constraint
W optimal = maxa∈{adimissible motions} [W (a)]
Example of constraints for a single degree of freedom:
|x | = max(|x(t)|)t∈[0,T ] < cst.
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Some previous works
I J.N. Newman. “Absorption of wave energy by elongated bodies”. In: Applied OceanResearch 1.4 (1979), pp. 189–196
I FJM. Farley. “Wave energy conversion by flexible resonant rafts”. In: Applied OceanResearch 4.1 (1982), pp. 57–63
I FJM. Farley. “Far-field theory of wave power capture by oscillating systems”. In:Philosophical Transactions of the Royal Society A: Mathematical, Physical andEngineering Sciences 370 (2012), pp. 278–287
I J. Falnes and A. Kurniawan. “Fundamental formulae for wave-energy conversion”. In:Royal Society open science 2.3 (2015), p. 140305
I A. Babarit. L’energie des vagues: Ressource, technologies et performance. ISTE editions,2018
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2 Implementation
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Structure of the code
Geometry
and dofsBEM
Inertia,
PTO, . . .
Eq. of
motionPower
A(ω), B(ω), Fe(ω)
Optimization
algorithm
Optimal
power
H(θ, ω)
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Capytaine: a Python-based distribution of Nemoh
I Full rewrite of Nemoh in Fortran 90 + Python.
I Improving user interface, documentation and testing.
I Integration with Python ecosystem (here Scipy optimization algorithm).
I Experimental optimizations.
I Openly available under GPL license:I conda install -c conda-forge capytaineI Documentation: https://ancell.in/capytaine/I Source code: https://github.com/mancellin/capytaine/
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3 Applications
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Application to a floating buoy
Vertical cylinder.
Dof: surge or heave or both.
free surface
1 m
1 m
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Absorption width as a function of wavelength
Dotted line: unlimited motion.
Plain line:√∑
j |aj |2 < 0.5 m.
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Absorption width as a function of wavelength
Dotted line: unlimited motion.
Plain line:√∑
j |aj |2 < 0.5 m.
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Absorption width as a function of wavelength
Dotted line: unlimited motion. Plain line:√∑
j |aj |2 < 0.5 m.Matthieu Ancellin – HyWEC2 – June 2019 11 / 19 ,
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Absorption width as a function of wavelength
Dotted line: unlimited motion. Plain line:√∑
j |aj |2 < 0.5 m.Matthieu Ancellin – HyWEC2 – June 2019 11 / 19 ,
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Absorption width as a function of wavelength
Dotted line: unlimited motion. Plain line:√∑
j |aj |2 < 0.5 m.Matthieu Ancellin – HyWEC2 – June 2019 11 / 19 ,
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Absorption width as a function of wave direction
Dotted line: unlimited motion.
Plain line:√∑
j |aj |2 < 0.5 m.
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Absorption width as a function of wave direction
Dotted line: unlimited motion.
Plain line:√∑
j |aj |2 < 0.5 m.
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Absorption width as a function of wave direction
Dotted line: unlimited motion.
Plain line:√∑
j |aj |2 < 0.5 m.
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
j |aj |2 < 0.5 m.Matthieu Ancellin – HyWEC2 – June 2019 12 / 19 ,
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
j |aj |2 < 0.5 m.Matthieu Ancellin – HyWEC2 – June 2019 12 / 19 ,
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
j |aj |2 < 0.5 m.Matthieu Ancellin – HyWEC2 – June 2019 12 / 19 ,
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Application to a simplified S3
Small scale model of bulging WEC:
10 m
0.4 m 0.3 m
with only 4 simplified degrees of freedom:
sin (1× 2πx/L)
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Application to a simplified S3
Small scale model of bulging WEC:
10 m
0.4 m 0.3 m
with only 4 simplified degrees of freedom:
sin (1× 2πx/L)
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Application to a simplified S3
Small scale model of bulging WEC:
10 m
0.4 m 0.3 m
with only 4 simplified degrees of freedom:
sin (2× 2πx/L)
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Application to a simplified S3
Small scale model of bulging WEC:
10 m
0.4 m 0.3 m
with only 4 simplified degrees of freedom:
sin (3× 2πx/L)
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Application to a simplified S3
Small scale model of bulging WEC:
10 m
0.4 m 0.3 m
with only 4 simplified degrees of freedom:
sin (4× 2πx/L)
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Absorption width as a function of wavelength
Dotted line: unlimited motion. Plain line:√∑
j |δr j |2 < r0/2.Matthieu Ancellin – HyWEC2 – June 2019 14 / 19 ,
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Absorption width as a function of wavelength
Dotted line: unlimited motion. Plain line:√∑
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Absorption width as a function of wavelength
Dotted line: unlimited motion. Plain line:√∑
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Absorption width as a function of wavelength
Dotted line: unlimited motion. Plain line:√∑
j |δr j |2 < r0/2.Matthieu Ancellin – HyWEC2 – June 2019 14 / 19 ,
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Absorption width as a function of wavelength
Dotted line: unlimited motion. Plain line:√∑
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Absorption width as a function of wavelength
. . .
Dotted line: unlimited motion. Plain line:√∑
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
j |δr j |2 < r0/2.Matthieu Ancellin – HyWEC2 – June 2019 15 / 19 ,
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
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λ λ λ λ λ λ
λ
λ
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
j |δr j |2 < r0/2.Matthieu Ancellin – HyWEC2 – June 2019 17 / 19 ,
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
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Absorption width as a function of wave direction
Dotted line: unlimited motion. Plain line:√∑
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4 Conclusion and perspectives
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Perspectives
I Continuing work at UCD:I Comparison with equations of motion from (Babarit et al., 2017).I Use of the actual modes of deformation of the S3 and parametric studies;
I To-do:I More physical constraints, including incoming wave height;I Dedicated optimization algorithm?I Analytical resolution of the optimization problem?
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Conclusion
I Proof-of-concept of a lightweight model for WECs design:
I Based on far-field radiated waves;
I Purely hydrodynamical;
I Few inputs: the geometry, the active dofs and their maximal amplitude;
I Including any non-trivial dofs.
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