The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance

Abstract

This research demonstrates the physical feasibility of constructing a single degree of freedom ocean-wave energy capturing device of extremely high efficiency, which serves also as an almost-perfect floating breakwater. The device consists of an asymmetric heaving floater and a power-take-off (PTO) system. A special geometry was first designed to have minimal effects from viscosity. As such, potential-flow analysis proved adequate for predicting the behavior of the device even in a real-fluid environment. Per ideal-fluid theory, at the heave resonance frequency, when the absorption damping of the PTO is matched with the radiation damping of the floater, an energy extraction efficiency of 96.34% can be achieved. At this high-efficiency condition, the transmitted waves beyond the floater are 12.5% of the incident-wave amplitude (or a mere 1.56% of the incident-wave energy). A working two-dimensional physical unit of 0.227m beam and 0.8m draft was designed and fabricated to work with the permanent magnet linear generator developed in (Yeung et al., 2012). The floater-PTO system was tested at The Model-Testing Facility of the University of California (UC), Berkeley. Experiments showed that when the PTO operates at even just 48% of the optimal damping, the device yields already an energy-capturing efficiency of 82%, with only 2% of the incident-wave energy transmitted. These latter values are in agreement with the theoretical predictions. Restraining lateral force and pitching moment are shown, as well as how wave energy is directed to the PTO.

Full text

Abstract
This research demonstrates the physical feasibility of constructing a single degree of freedom ocean-wave energy capturing
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Keywords


1 Introduction
                
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 

The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of
high performance*
Farshad Madhi1, Meghan E. Sinclair2, and Ronald W. Yeung3





Manuscript submitted to MS&OT on August 15, 2013. Accepted on October 10, 2013. Editor: Marcelo A. S. Neves.
Article posted online on December 16, 2013: URL: www.sobena.org.br/msot/volume.htm
* US Provisional Patent No. 61/883,274 (UC No. BK-2014-037-1) URL: http://techtransfer.universityofcalifornia.edu/NCD/23530.html
Vol. 9 No. 1 pp. 05-16 June 2014
Marine Systems & Ocean Technology 5
USPTO Patent #9416766B2
(Aug,2016)

         
  
at a given site is available 
     


      

technical potential to serve as clean renewable energy sources


         
    
with the stipulation that when operating at resonance
   

        
        
       



device consists of foils rotating about a shaft perpendicular
   


      
      
                   
     

       

   

        


    

     
        
      


               

      
    
     
       
        
 

              
      
       
      
     
     





reducing complications arisen from wetness of the electrical

                  



       
        
      



2 Theoretical framework
We consider a two-dimensional body with motion restricted to

         
    


where A and k are the wave amplitude and wave number
      


to k in a water depth h

with g        
     k =

/g    
    (x,y,t) that
      

contour B and of the free surface y

time-harmonic factors





a is the
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung
6 Marine Systems & Ocean Technology
Vol. 9 No. 1 pp. 05-16 June 2014

 
       
potential is typically decomposed as

The terms





     
      


body surface B

B





where j
waves for unit-velocity of heave motion as depicted in
    f y     


The solution of

enables us to calculate the heave added
       


B
where





B



        
        
R and T be computed




j and k are the indices associated with any
 
problem
 
does not need to be solved to obtain R and T
    

2.1 Plausible asymmetric floater shape
         

       
     

       
         

by manipulating a shape function F   
and        
calm-water line being 

F 
The constants ABC can be found by applying the
F  F =
F _=F d* where = d* is the inflection-point
     A=+C
B= _ _C and C = d*d*+_ d* _ 
inflection point of d* = _ 
   

F 
       
variable is replaced by _ e where e was chosen to be
      
        
  

The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung
Vol. 9 No. 1 pp. 05-16 June 2014
Marine Systems & Ocean Technology 7
D
b
2(t)
h
y
x
A
(x,t) =A Re{-ie }
i(kx - t)


Aj
-e-ikx Aj
+eikx
x1x2
E
E
1
2
O
B

 
 sec
 Dba
8 Marine Systems & Ocean Technology
Vol. 9 No. 1 pp. 05-16 June 2014
54321 0 1 2 3 4 5
t/T=4
CMML
UC BERKELEY
t/T=4
CMML
UC BERKELEY
5 0 5
2
1.5
1
0.5
0
0.5
1
1.5
2
54321 0 1 2 3 4 5
t/T=4.25
CMML
UC BERKELEY
t/T=4.25
CMML
UC BERKELEY
5 0 5 2
1.5
1
0.5
0
0.5
1
1.5
2
54321 0 1 2 3 4 5
t/T=4.5
CMML
UC BERKELEY
t/T=4.5
CMML
UC BERKELEY
5 0 5 2
1.5
1
0.5
0
0.5
1
1.5
2
54321 0 1 2 3 4 5
t/T=4.75
CMML
UC BERKELEY
t/T=4.75
CMML
UC BERKELEY
5 0 5 2
1.5
1
0.5
0
0.5
1
1.5
2
                     
                [sec−1]
          D  b  a2 
4
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung

dimensions of Db

resonance period of 
       
  


 



   

      
        
     
        
  

       
        
only minimal effects on the motion of our particular design
       
   
  
motion amplitude of am
2.2 Equation of motion, response
amplitude operator, and efficiency
         





where Kg and Bg     

         


K=

gb and
body mass M=

As where As is the submerged cross-sectional

in footnote        


where         Bg to the




e
 

 
where the group velocity is given by Vg = g

 
   


          
 
   
       
  




           
       






e          
radiation 
radiation
i.e.        


i.e. 
       

of

jj
  and ij


Vol. 9 No. 1 pp. 05-16 June 2014
Marine Systems & Ocean Technology 9
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung

2.3 Long-wave approximation of X2 and
corresponding RAO

        




  

of kb



gb
    
k =

g 



where 


These predictions are consistent with the results given in


3 Design and fabrication of the
floater

    
    

   

         




was designed to be attached to the carriage located above the
    
          
       

where wf


         

         
       
  

        



10 Marine Systems & Ocean Technology
Vol. 9 No. 1 pp. 05-16 June 2014
 and hydrodynamic shape factor 
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.1
0.2
0.3
0.4
0.5
0.6
|A±
j|
¯σ

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.5
1
γ
γ-RWYADMXA
γ-FSRVM
|A−
j|-RWYADMXA
|A+
j|-RWYADMXA
|A−
j|-FSRVM
|A+
j|-FSRVM
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
¯σ
¯µ22 ,¯
λ22

¯µ22 -RWYADMXA
¯
λ22 -RWYADMXA
¯µ22 -FSRVM
¯
λ22 -FSRVM

 and radiation damping
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung

The material for the mounting structure was chosen to be
         
 


      



3.1 Permanent magnet linear generator
(PMLG)
        
 was obtained to be
     
of  
   
        
         


       
          
 

The stator has two columns of metallic teeth with coils
winding around them and two sets of ball bearings which

    
between the coils and stator and by changing the resistance of
   
    

 
         
     
        
The actual values of the generator damping for the gap of
       Bg N _ s /
m      Bg
N _ s / m  = 

    
Bg

N _ s / m


       

 
 
 
 
Vol. 9 No. 1 pp. 05-16 June 2014
Marine Systems & Ocean Technology 11
 
 
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung

3.2 Restraining forces and moments
on mounting structure

      
moment analysis was conducted to obtain an estimation of the
moment acting on the bearings induced by hydrodynamic and




where Fext and Mext are the restraining sway force and pitching
       
          M
  whereas M= xg M     
and moment are exerted by the bearings that guide the heave
       
force and moment at the coordinate origin O




 
 



 
Fext
and pitch moment Mext

of


 
4 Experimental findings and
concept confirmation
The heave wave-exciting force was measured by restricting
          

xx


force X obtained from the present model tests for a wave
      

       

resonance 
the computed results for longer waves as one would
 
long-wave approximation results are in agreement with the




  
 
          
   

          
represents the case when 

      
       
 




 


D
>
6)
the experimental
       
consistent with the measurements of
X


            




where RT and TT are the complex wave amplitudes propagating

and

e

R2 T 2
    
D.

  
three values of constant damping Bg
The valley in the RT curves around resonance suggests that

TT
12 Marine Systems & Ocean Technology
Vol. 9 No. 1 pp. 05-16 June 2014
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
2
4
6
8
10
|Mext|
¯σ

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
2
4
6
8
10
|Fext|
|Fext
|¯
f=0
|Fext
|¯
f=1
|Mext
|¯
f=0
|Mext
|¯
f=1
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung


D b 
 
Bg


       
 
achieve

e . 

e 
this experiment with 
 
 
the TT value over T      




Wedge mounted on the bottom of the moving outer cylinder


         
 
        

 



 
5 Conclusions
       
   
      
        
     


Bg of the generator is matched with the radiation damping

             

  

       

   
For the case of Bg  



res  

 
      
Bg =
Vol. 9 No. 1 pp. 05-16 June 2014
Marine Systems & Ocean Technology 13
0.4 0.5 0.6 0.7 0.8 0.9 1
0.2
0.4
0.6
0.8
1
1.2
1.4

¯σ
!
!¯
X2!
!
|¯
X2|RWYADM XA
|¯
X2|Long Wave Aprx
|¯
X2|Expt.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5

¯σ
!
!
!
a2
A
!
!
!
!
!
!
a2
A
!
!
!¯
f=0
!
!
!
a2
A
!
!
!¯
f=0.2652
!
!
!
a2
A
!
!
!¯
f=0.4818
!
!
!
a2
A
!
!
!¯
f=1
Expt.!
!
!
a2
A
!
!
!¯
f=0
Expt.!
!
!
a2
A
!
!
!¯
f=0.2652
Expt.!
!
!
a2
A
!
!
!¯
f=0.4818
2.2
3.6
λ
2.8
4.9
7.1
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
0.2
0.4
0.6
0.8
1
ηe
¯σ

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
0.25
0.5
0.75
1
|T|,|R|,|TT|,|RT|
|TT|¯
f=1
|RT|¯
f=1
|TT|¯
f=0.4818
|RT|¯
f=0.4818
|T|
|R|
Expt. |T|¯
f=0.4818
ηe|¯
f=1
ηe|¯
f=0.4818
Expt. ηe|¯
f=0.4818

Bg values
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung




resBg 



res
 
    
        

     

      


 
the restraining force and moment do get transferred to the


6 Acknowledgements

        
        
      
      
       

  
    
     
     

References
AmAr
   

AubAult
 
st


bedArd


bessho        
     

brito-melo
of wells turbine design parameters by numerical simulation
      

evAns 

FAgley          
“Computational investigation of irregular wave cancellation
      
st

he

     

JiAng
of rolling cams for wave-energy capture in a viscous
     st
     

KAnner

     nd


KeArny

 
KoutAndos    
     

liAo        
     

mAdhi                 
      


mAlArA       
an u-oscillating water column and performance in random

14 Marine Systems & Ocean Technology
Vol. 9 No. 1 pp. 05-16 June 2014
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung
Video Clip
https://www.youtube.com/watch?v=D_wPAhw-cu0

mørK     
     
th
     

shugAn
    

siegel

      
st


sinclAir
   

son
and validation of a two-coaxial cylinders system as a wave-
    
rd

teh
      
     

tom
      
   st  
     

uzAKi
        

       

WehAusen

WehAusen



yeung       
      
st

yeung         
from  to        
  th   

yeung      
      
  

yeung         

notes for course  

Appendix A. Derivation of heave
wave-exciting force using the long-
wave approximation
The heave wave-exciting force can be calculated using the


B
       X = X2
F +
X2
D X2
F includes the incident wave potential

 and
X2
D includes the diffraction potential



 corresponds to the



where 

as follows

The term f ( y)
ekix+yikxkykixy
ik2xy to simplify the




 
B



Vol. 9 No. 1 pp. 05-16 June 2014
Marine Systems & Ocean Technology 15
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung

For obtaining the X2
D  
conditions for

 and



B B B


B
B and
hdenotes the
bottom boundary and F  
The contributions due to and F can be discarded
because

and

satisfy the same boundary conditions on
h can also be discarded because


B B


B
We can once again use the

 
evaluate 

B
where the terms of and   
of the long wave approximation kb     

to evaluate X2
D and obtain

 X2
F+X2
Dk =

g and non-dimensional
        
wave-exciting force

16 Marine Systems & Ocean Technology
Vol. 9 No. 1 pp. 05-16 June 2014
The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance
Farshad Madhi, Meghan E. Sinclair, and Ronald W. Yeung