The document presents a simplified model for understanding mechanisms in oscillating water columns (OWCs), particularly focusing on floating designs as a cost-effective option for energy extraction from waves. It discusses various modeling techniques and outlines the complexities involved in simulating these systems, highlighting the need for validation with experimental data. The findings suggest that losses in energy capture occur primarily at the column mouth and can inform future modeling efforts to prioritize areas requiring detailed analysis.

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A simplified model for oscillating water
column motion
Rebecca Sykes
Mechanical Engineer
Technical Directorate
May 23, 2012

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Oscillating water column
• Conventional OWC have been
shoreline devices
• LIMPET, Scotland
• Pico, Azores
• Sanze, Japan…
• Wave shoaling reduces energy to
shoreline
Power is extrac ted from the wave induced vertic al motion of the water free s urface c ompres s ing air
in a volume above. This can be us ed to drive an air turbine, s uch as the Wells turbine, whic h is
des igned for rec iprocating flows .

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Oscillating water column
• Conventional OWC have been
shoreline devices
J ac ket
• LIMPET, Scotland
• Pico, Azores
• Sanze, Japan…
G ravity bas ed
• Wave shoaling reduces energy to
shoreline
TLP
• Potential for greater energy
extraction offshore
• Options – fixed, semi-fixed or
floating
Floating OWC
Majority of propos ed/prototype offs hore OWC have been floating but there is potential to c ombine
with other tec hnologies and s o us e their s upport s tructure

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Objective
To present a model that furthers our understanding of the physical processes within
a floating Oscillating Water Column
Floating – cheaper CAPEX option (?)…
The diffraction and radiation problem which
existed for the fixed OWC must be
…but more complex to simulate
extended when floating to include radiation
from the device motion
THEREFORE: Increased complexity in predicting the energy capture

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OWC modelling
Three previous ly us ed modelling techniques
Numerical modelling Analytical modelling Physical modelling
Computational time/ Device/geometry specific High time and cost
accuracy trade-off
Specialist mathematical Scaling
Need for verification and skills
Increased potential for
validation
Need for verification and error at small scale
Application of OWC validation
boundary condition not
always easily available

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Simplified OWC model
• A simple geometrical model
was used to highlight the
fundamental physics
avoiding proprietary device
specific particularities
• An OWC is a highly resonant
device when undamped, and
is hydrodynamically narrow
banded in frequency
• Vertical oscillation -power
G eometry examined
producing oscillation

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Mathematical model
Initially c ons idering a fixed OWC to examine the diffraction pres s ure
Time domain piston model from [1]: ρS ( L + ηOWC )ηOWC + ρSgηOWC = SPOWC
 (1)
ηOWC and POWC can be expanded as series in powers of the small parameter ε
η
OWC
( t ) = η0 + εη1( t ) + ε 2η2 ( t ) + ε 3η3 ( t ) + ...
and P
OWC
( t ) = P0 + εP1( t ) + ε 2 P2 ( t ) + ε 3P3 ( t ) + ...
Substituting into (1) and taking those terms up to first order ηOWC
( 
0 1
)
ρS L + η η + ρSgη = P S
1 1 (2)
Assuming η1 and P1 are harmonic such that
η = Reη eiωt 
 ˆ1  P = Re p eiωt 
 1  POWC
1   1  
C ons idered internal
Which gives the frequency domain equation volume of water
− ω 2 ρ ( L + η 0 )η1 + ρgη1 = p1
ˆ ˆ

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Mathematical model prediction
3
45
0 40 0
2.5
-0.05 -0.05
35
-0.1 -0.1
2
30
-0.15 -0.15
z (m)
z (m)
-0.2 25 -0.2
1.5
-0.25 20 -0.25
-0.3 -0.3
15 1
-0.35 -0.35
0.1 0.1
10
0.05 0.1 0.05 0.1 0.5
0 0.05 5 0 0.05
0 0
-0.05 -0.05
-0.05 -0.05 0
y (m) -0.1 -0.1 y (m) -0.1 -0.1
x (m) x (m)
Pressure magnitude Pressure phase

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Structure for validation – fixed model to
validate diffraction solution
z
pz = -270mm • Vertical cylinder:
pz = -201mm
ηOWC o b = 50.5mm
pz = -144mm
x o a = 47.0mm
AI
o d = 300mm
o h = 1m
d • Tank: 2.65m x 23.27m
h • Regular waves
• Measurements:
a o Free surface elevation
Wave probe
b o Pressure at three depths
S c hematic of model us ed in wave flume experimental tes ting

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Validation
C omparis on of diffraction pres s ure (normalis ed by inc ident wave amplitude) in the frequenc y domain

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Mathematical model for floating OWC
When the water c olumn is defined by a floating s tructure, radiation effects mus t als o be c ons idered; for a s truc ture
that is axis ymmetric about a vertical axis in unidirec tional waves , the dominant lateral modes are s urge and pitc h.
Sloshing modes natural frequencies for fluid in a cylindrical tank:
gκ1n κ d  a
ω1n =
2
tanh 1n 
a  a  d
where κ1n = 1.8412, 5.3314, 8.5363, 11.706, 14.8636,…, κ1n = κ1(n –1) + π.
Pressure due to acceleration and Pressure due to acceleration and
sloshing in surge: sloshing in pitch:
Pξ1 ( t ) = ρ ξ1 ω 2 cos θ sin ( ωt )( r + A) Pξ5 ( t ) = ρ ξ5 ω 2 cos θ sin ( ωt )( rz + A)

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Mathematical model for floating OWC
Total dynamic pres s ure on the internal s urface of a floating OWC :
{
pT = p1 + pξ1 + pξ5 − ρ gz '+  g ( ξ3 + ξ 4 y '− ξ5 x ') 
  }
Pis ton model Due to pitc h Hydros tatic
Due to s urge
where pξ and pξ are the complex amplitudes of the acceleration and
1 5
sloshing pressures and (x', y', z') are the body fixed coordinates of a
general position on the wall.

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Floating structure for validation – floating model
Reflective marker Wave probe
z • Model dimensions:
• 2b =315mm
• 2 a = 104mm
• d = 300mm
AI • h = 1m
x
• Tank: 2.65m x 23.27m
PTO1 PTI1 • Regular waves
PTO2 PTI2
d • Measurements:
PTO3 PTI3
PTO4 PTI4
• Model displacement
h
PTO5 PTI5 • Free surface elevation
a • Pressure at three
Ballast b depths
Spacing
material
S c hematic of model us ed in wave flume experimental tes ting

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Validation
Wave
direc tion
C omparis on of dynamic pres s ure (normalis ed by incident wave amplitude) in the frequenc y domain

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Validation
Wave
direc tion
C omparis on of dynamic pres s ure (normalis ed by incident wave amplitude) in the frequenc y domain
Where model has been rotated with res pect to wave direction to as s es s lateral pres s ures

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What to take away from this…
• Simple model can be used to effectively relate the pressure and free surface
elevation for the piston mode of an OWC under certain conditions
• Majority of losses must occur around or outside the column mouth to explain
observed losses between Boundary Element Method model and physical
testing
• Model can be used to identify areas which can be modeled using simpler
inviscid theory such that computational resources can be focused on areas
with viscous phenomena

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For more information, please contact:
Rebecca Sykes
Mechanical Engineer – Renewable Energy,
Technology Directorate
Lloyd’s Register Group Services
Denburn House, 25 Union Terrace
Aberdeen, AB10 1NN
T +44 (0)1224 267694
E rebecca.sykes@lr.org
w www.lr.org/energy
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