u·m·i · 2014. 6. 13. · chapter 10: conclusions, discussions and recommendations 181 10.1...
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Design and performance evaluation of a wave-driven artificialupwelling device
Chen, Hsiao-Hua, Ph.D.
University of Hawaii, 1993
V·M·I300 N. Zeeb Rd.Ann Arbor, MI 48106
DESIGN AND PERFORMANCE EVALUAnON OF A WAVE-DRIVENARTIFICIAL UPWELLING DEVICE
A DISSERTATION SUBMITTED TO THE GRADUATEDIVISIONOFTHE UNIVERSITY OF HAWAI'I IN PARTIAL FULFILLMENT OF
THEREQUIREMENTS FORTIlE DEGREE OF
DOCTOR OFPHILOSOPHY
IN
OCEAN ENGINEERING
DECEMBER 1993
By
Hsiao-Hua Chen
Dissertation Committee:
Clark C. K. Liu, ChairpersonMichael J. Cruickshank
Yu-Si FokFrans Gerritsen
Hans-Jiirgen KrockHarold Loomis
Acknowledgments
I would like to thank all the members in my dissertation
committee, Dr. Cruikshank, Dr. Fok, Dr. Gerritsen, Dr. Krock, Dr. Liu
and Dr. Loomis for their valuable comments and careful
proofreadings. Special thanks to my committee chairman, Dr. Clark
C. K. Liu, for his creative guidance on this research and critical
review of this manuscript.
I want to thank Mr. Z. Qian, Ms. S. Wang, Dr. M. Tang, Professor Y.
Liu and Mr. F. Kuo who discussed with me and helped me to review
some of the important parts of my dissertation. I also want to thank
Miles, Andy and Mr. C. Wu who helped me to set up the
experimental instrumentation. Besides, I appreciate that Mr. T.
Culon and Mr. H. Donthi helped me to proofread the draft of my
dissertation.
I am grateful to the PICHTR, HNEI and NSF for providing the
financial assistance during my Ph.D. study.
Then, I want to thank my beloved wife for her unceasing
tolerance when I were not home days and nights, and the
encouraging support from my family back in Taiwan and the
fellows in the Christian fellowship here in Hawaii in past four years.
Finally, I want to thank God for giving me the chance to know
the world better and to know Him more.
III
•
Abstract
A wave-driven artificial upwelling device, consisting of a floating
buoy, an inner water chamber, a long tail pipe, and two flow
controlling valves, was developed for this research. Hydrodynamic
performance of the device to pump up nutrient rich deep ocean
water is evaluated by mathematical modeling analysis and
hydraulic laboratory experiments. The mathematical model of the
device is made up of four simultaneous differential equations. The
first three equations, which describe the motion of upwelled water
inside the device, were formulated based on momentum and mass
conservation principles. The fourth equation is the equation of
motion of the device in ambient waves. The model is solved
numericaIly by the fourth order Runge-Kuua method. The equation
of motion of the device in ambient waves contains several
parameters, including added mass ,damping coefficient, wave
exciting force and restoring coefficient. Values of these parameters
must be determined before the model equations can be solved. In
order to determine these variables, a hydrodynamic problem of
wave-device interactions must be solved. The boundary element
method is used to solve this hydrodynamic problem of radiation
and diffraction. Modeling results are verified by a series of
hydraulic experiments conducted in a wave basin in the James K. K.
Look Laboratory of Oceanographic Engineering at the University of
Hawaii. Comparing analytical and experimental results yields some
useful information concerning hydrodynamic coefficients under
IV
I
waves of large amplitude. The mathematical model developed in
this study was then used to evaluate the effects of five
configuration variables on the rate of upwelling flow at the design
wave conditions, and to establish design criteria.
v
Table of Contents
Page
Acknowledgments iii
Abstract iv
List of Tables x
List of Figures Xl
List of Photos xix
List of Symbols xx
Chapter 1.: Introduction 1
1.1 Wave Energy and Artificial Upwelling 31.2 Research Objectives and Procedures 4
Chapter 2: Literature Review........................................................... 8
2.1 Options of Artificial Upwelling...................................... 82.2 Wave and Floating Structure 122.3 Wave-Driven Artificial Upwelling Device 17
2.3.1 Motion of Floating Body 182.3.2 Motion of Internal Water Column 212.3.3 Operation of Valve 222.3.4 Pumping Effect of Artificial
Upwelling Device 23
Chapter 3: Wave and Device Interactions 28
3.1 Added Mass and Damping Coefficient 293.2 Wave ExcitingForce 383.3 Surrounding Water Level.. 483.4 Restoring Force 50
VI
Chapter 4: The Experiment of Wave and DeviceInteractions................................................................................... 52
4.1 Feasibility of Laboratory Work 524.1.1 Wave Forces on the Device 524.1.2 Model Feasibility Consideration....................... 544.1.3 Model Application 58
4.2 Experimental Setup and Measurement. 604.2.1 Radiation Test 63
4.2.1.1 Cylindrical Device 694.2.1.2 Cone-Shape Device 71
4.2.2 Diffraction Test. 724.3 Result Comparison 75
4.3.1 Added Mass and Damping Coefficient. 754.3.2 Wave Exciting Force 83
Chapter 5: Flow Motion in a Wave-Driven ArtificialUpwelling Device (I): Mathematical Formulation......... 85
5.1 Continuity Equation 875.2 Momentum Equation 885.3 Bernoulli Equation 925.4 Coordinate Conversion and the Governing
Equations............................................................................... 935.5 Equation of Motion of the Upwelling Device 955.6 Re-Arrangement of Governing Equations 98
Chapter 6: Flow Motion in a Wave-Driven ArtificialUpwelling Device (II): Mathematical Solution 100
6.1 Solution Procedure 1006.2 Modification of the Formulation on Different
Valve Operations 106
Chapter 7: Evaluation of the Performance of ArtificialUpwelling Devices with One and Two Valves 108
7.1 Momentum Equation for Water Movementin a One-Valve Device .l 08
Vll
7.2 Momentum Equation for the Movement ofOne-Valve Device 112
7.3 Simulation of the Pumping Effect of aOne-Valve Artificial Upwelling Device 1 137.3.1 Valve Operation, Pipe Length and
Pumping Effect 1 147.3.2 Resonance and Pumping Effect. 1207.3.3 Comparison of Different Approaches 120
7.4 Simulation of the Pumping Effect of aTwo-Valve Artificial Upwelling Device .1 26
7.5 Performance Comparison of One andTwo-Valve Devices 132
Chapter 8: Motions of Device and Inner Water Level: TheExperimental Approach .13 4
8.1 Verification Tests in the Laboratory 1348.1.1 Radiation Test.. .1 358.1.2 Diffraction Test .1 36
8.2 Model Setup and Feasibility Consideration .13 78.2.1 Radiation Test. .13 78.2.2 Diffraction Test .1 398.2.3 Free Oscillation Test .l 3 98.2.4 Model Feasibility Consideration .! 39
8.3 Result Comparison .1 458.3.1 Radiation Test 1458.3.2 Diffraction Test .1478.3.3 Free Oscillation Test. l 49
Chapter 9: Device Design and Performance Evaluation .1 53
9.1 The Design Variables 1549.1.1 Shape of the Float.. .1 559.1.2 Draft and Size of the Water
Chamber 1609.1.3 Length of Tail Pipe 164
9.2 Linear Consideration .1 719.3 Design Consideration 1769.4 Design Criteria 178
Vl11
I
Chapter 10: Conclusions, Discussions and Recommendations 181
10.1 Conclusions and Discussions 18110.1.1 Theory 18210.1.2 Experiment 18410.1.3 Application 186
10.2 Recommendations 190
References 192
lX
I
List of Tables
Table Page
9.1 The Base Conditions of Reference Pumping Rate .l 6 1
9.2 The Best Draft Ratio When L=300m 166
9.3 The Best Draft Ratio When L=800m .1 70
9.4 The Maximum Pumping Rate Under Unit WaveAmplitude 170
9.5 The Average Increasing Factors of Pumping Rate 177
x
List of Figures
Figure Page
1.1 The Proposed Wave-Driven Artificial UpwellingDevice 2
2.1 a Design of a One-Valve Artificial Upwelling Devicewith a Pipe of Constant Overhead 19
2.1b Design of a One-Valve Wave Power Generatorand Artificial Upwelling Device............ 19
2.2 The Graphic Solution of the Pumping Effect of aOne-Valve Artificial Upwelling Device byIsaacs et al. (1976) 24
2.3 The Capacity of Wave Pump Calculated byVershinskiy et al. (1987) 26
3.1 The Discretization of Boundary Element forRadiation Approach 35
3.2 Added Mass of a Cylindrical Artificial UpwellingDevice with Different Lengths of Tail Pipe 37
3.3 Damping Coefficient of a Cylindrical ArtificialUpwelling Device with Different Lengths ofTail Pipe 37
3.4 The Discretization of Boundary Element forDiffraction Approach 45
3.5 Wave Exciting Force of a Cylindrical ArtificialUpwelling Device with Different Lengths ofTail Pipe 47
3.6 Ratio of Wave Scattering Force and Total ExcitingForce for a Cylindrical Artificial UpwellingDevice 49
4.1 The Experimental Setup in the Laboratory 61
Xl
Figure Page
4.2 The Mechanical Heaving Oscillation on a CalmWater Surface: Radiation Test Setup 64
4.3 The Wave Exciting Force on Fixed Device:Diffraction Test Setup 73
4.4 Power Spectrum of the Mechanical Exciting Forceon a Cylindrical Artificial Upwelling Device forSmall Amplitude Oscillation 77
4.5 Power Spectrum of the Mechanical Exciting Forceon a Cylindrical Artificial Upwelling Device forLarge Amplitude Oscillation 77
4.6 Theoretical and Experimental Results Comparisonof Added Mass for a Cylindrical ArtificialUpwelling Device 78
4.7 Theoretical and Experimental Results Comparisonof Damping Coefficient for a CylindricalArtificial Upwelling Device 78
4.8 Power Spectrum of the Mechanical Exciting Forceon a Cone-Shape Artificial Upwelling Device forSmall Amplitude Oscillation ,. 79
4.9 Power Spectrum of the Mechanical Exciting Forceon a Cone-Shape Artificial Upwelling Device forLarge Amplitude Oscillation 79
4.10 Theoretical and Experimental Results Comparisonof Damping Coefficient for a Cone-ShapeArtificial Upwelling Device........ 81
4.11 Theoretical and Experimental Results Comparisonof Damping Coefficient for a Cone-ShapeArtificial Upwelling Device.... 81
Xll
Figure Page
4.12 Theoretical and Experimental Results Comparisonof Wave Exciting Force on a CylindricalArtificial Upwelling Device.. 84
4.13 Theoretical and Experimental Results Comparisonof Wave Exciting Force on a Cone-ShapeArtificial Upwelling Device.. 84
5.1 Schematic Diagram of a Wave-Driven ArtificialUpwelling Device 86
5.2 The Definition of Control Volume used to determinethe Flow Motion of an Artificial UpwellingDevice 86
7.1 Coordinates Definition of the Motion of a One-Valve Artificial Upwelling Device l09
7.2 Comparison of Wave, Motion of Device and Motionof Water Level of a One-Valve ArtificialUpwelling Device 115
7.3 Acceleration of Device and Water Column for aOne-Valve Artificial Upwelling Device WhenL/Dbuoy=5, T/Tn=1.5 and h/Dbuoy= 1/8 .116
7.4 Acceleration of Device and Water Column of aOne-Valve Artificial Upwelling Device WhenL/Dbuoy=20, Tffn=1.5 and h/DbuOy=I/8 .1 16
7.5 Velocity of Device and Water Column for aOne-Valve Artificial Upwelling Device WhenL/Dbuoy=5, TITn=l.5 and h/Dbuoy= 118 .1 18
7.6 Velocity of Device and Water Column for aOne-Valve Artificial Upwelling Device WhenL/D buoy=20, Trrn=I .5 and h/DbuOY=I/8 1 18
XIll
Figure Page
7.7 Pumping Volume Comparison for the Device withDifferent Lengths of Tail Pipe in One WaveCycle 119
7.8 Pumping Rate Comparison for a One-Valve ArtificialUpwelling Device with Different Tail Lengthsfor hs/Dbuoy= 1/8 119
7.9 Maximum Pumping Rate for a One-Valve ArtificialUpwelling Device with Different Tail Lengthsfor hs/DbuOY=I/8 121
7.10 Oscillation Amplitude of a One-Valve ArtificialUpwelling Device Under Different WavePeriods for L/Dbuoy=5 and h/Dbuoy=I/8 .122
7.11 Pumping Volume of One-Valve ArtificialUpwelling Device Under Different WavePeriods for L/Dbuoy=5 and h/DbuOy= 1/8 122
7. 12 Pumping Rate Comparison of Two Approaches onOne-Valve Artificial Upwelling Devicefor T/Tn=0.75 and h/Dbuoy= 1/8 124
7.1 3 Pumping Rate Comparison of Two Approaches onOne-Valve Artificial Upwelling Devicefor T/Tn=l.O and h/Dbuoy=I/8 124
7. 14 Pumping Rate Comparison of Two Approaches onOne-Valve Artificial Upwelling Devicefor T/Tn=1.6 and h/Dbuoy=1/8 .l25
7.15 Acceleration of Device and Water Column for aTwo-Valve Artificial Upwelling Device WhenT/Tn=I.2, DCh/DbUOY= 0.625, draft/Dbuoy= 1/3.2
L/Dbuoy= 6.25 and Dbuo/A= 0.18 127
XIV
Figure Page
7.16 Velocity of Device and Water Column for aTwo-Valve Artificial Upwelling Device WhenT/T =1.2, D h/Db = 0.625, draft/Db = 1/3.2n c uoy uoyL/DbuOY= 6.25 and Dbuo!A= 0.18 .127
7.17 Velocity of Device and Flow on the Outflow Openingfor a Two-Valve Artificial Upwelling DeviceWhen T/Tn=1.2, DCh/DbuOY= 0.625,
draft/Db = 1/3.2, L/Db = 6.25uoy uoyand Dbuoy/A= 0.18 129
7.18 Inner Water Level Oscillation of a Two-ValveArtificial Upwelling Device When T/Tn= 1.2,
D ch/Dbuoy= 0.625, draft/Dbuoy= 1/3.2
L/DbUOY= 6.25 and Dbuo!A= 0.18 129
7.19 Relative Outflow Velocity of a Two-Valve ArtificialUpwelling Device When T{fn=1.2,
DCh/Dbuoy= 0.625, draft/Dbuoy= 1/3.2
L/Dbuoy= 6.25 and Dbuo!A= 0.18 131
7.20 Pumping Volume of a Two-Valve Artificial UpwellingDevice When r/r =1.2, D h/Db = 0.625,n c uoydraft/DbuOY= 1/3.2, L/Dbuoy= 6.25 and
DbuOy/A= 0.18 131
7.21 Pumping Rate Comparison Between One-Valveand Two-Valve Artificial Upwelling Devicein When L/Dbuoy= 6.25 and Dbuo/A= 0.18 .133
8.1 Tests on Water Level Variation in the RadiationTest 138
8.2 Tests on Water Level Variation in the DiffractionTest 140
xv
Figure Page
8.3 Tests on Water Level Variation and OscillationAmplitude of the Device in Free OscillationTest 141
8.4 Neutral Position of Water Level in the Chamberin Radiation Test When Two Valves AreDeployed 146
8.5 Oscillation Amplitude of Water Level in the Chamberin Radiation Test When Two Valves AreDeployed 146
8.6 Neutral Position of Water Level in the Chamberin Radiation Test When There is no Valve 148
8.7 Oscillation Amplitude of Water Level in the Chamberin Radiation Test When There is no Valve l 48
8.8 Oscillation Amplitude of Water Level in the Chamberin Diffraction Test When There is no Valve 150
8.9 Oscillation Amplitude of the Device in FreeOscillation Test for a Cylindrical ArtificialUpwelling Device 1 50
8.10 Oscillation Amplitude of Water Level in the Chamberin Free Oscillation for a Cylindrical ArtificialUpwelling Device 152
9. 1 Selected Shapes of Float 156
9.2 Added Mass of an Artificial Upwelling Device WithSelected Shapes of Float .1 57
9.3 Damping Coefficient of an Artificial UpwellingDevice with Selected Shapes of Float.. l 57
9.4 Wave Exciting Force of an Artificial UpwellingDevice with Selected Shapes of Float.. l 59
XVI
Figure Page
9.5 Comparison of Pumping Rate of an ArtificialUpwelling Device with Different Shapes ofFloat, Same Draft and Same VolumeDisplacement 162
9.6 Comparison of Pumping Rate of an ArtificialUpwelling Device with Different Shapes ofFloat, Same Water Plane Area and SameVolume Displacement 162
9.7 Maximum Pumping Rate for Different Drafts andSizes of Water Chamber of an ArtificialUpwelling Device When Dbuoy= 16m
and L=300m 165
9.8 The Changing of the Frequency of MaximumPumping Rate for Different Draft of anArtificial Upwelling Device When 0 ch= 10m,
D buoy= 16m and L=300m 165
9.9 Maximum Pumping Rate for Different Tail Lengthsof an Artificial Upwelling Device WhenDch= 10m, DbuOY= 16m and draft=5m 16 8
9.10 The Changing of Frequency of MaximumPumping Rate for Different Tail Lengths of anArtificial Upwelling Device When Dch=10m,
D buoy=16m and draft=5m 168
9.1 1 The Changing of Frequency of MaximumPumping Rate for Different Draft of anArtificial Upwelling Device When Dch=10m,
D buoy=16m and L=800m 169
9.12 Relationships Between Oscillation Amplitudeand Pumping Rate in a Range of Frequencies: an Example 172
XVII
•
Figure Page
9.13 The Design Wave Frequency and the Moving ofthe Frequency of the Maximum Pumping Rate: an Example , 1 7 2
9.14 Increasing Factor of Oscillation Amplitude of anArtificial Upwelling Device When WaveAmplitude Increases for /n/ ·t=O.4m,
Un!
D ch/Dbuoy=0.625, draft/Dbuoy=0.95, Dbuoy=16mand L=800m 1 7 4
9.15 Increasing Factor of Pumping Rate of an ArtificialUpwelling Device When Wave AmplitudeIncreases for Inl 't=OAm, D h/Db =0.625,un) c uoydraft/Dbuoy=O.95, Dbuoy=I6m and L=800m I 74
9. I 6 Average Increasing Factor of Oscillation Amplitudeand Pumping Rate of an Artificial Upwelling
Device for In'unit=OAm, DCh/DbuOy=0.625,draft/Dbuoy=0.95, DbuOy=16m and L=800m .l 75
9. 17 Comparison of Maximum Pumping Rate of DifferentDb and Tail Length of an Artificial UpwellinguoyDevice Under Design Wave Conditions .l79
XV1l1
I
List of Photos
Photo Page
4.1 The Artificial Upwelling Device with DifferentShapes of Float.................... 62
4.2 The Data Acquisition System of the LaboratoryTest 62
4.3 The Mechanical Heaving Oscillation on CalmWater Surface: Radiation Test Setup 64
4.4 The Sinusoidal Motion Generator for MechanicalOscillation 65
4.5 The Stiff Strain-Gauge Dynamometer 66
4.6 The MINARIK® Speed Controllor for MechanicalOscillation.................................................................................... 67
4.7 The Adjustable Oscillation Arm of a MechanicalOscillation.................................................................................... 67
4.8 The Wave Exciting Force on Fixed Device:Diffraction Test Setup 73
4.9 Wave Maker and Wave Absorber in the WaveTank 74
8. 1 Tests on Water Level Variation in the RadiationTest 138
8.2 Tests on Water Level Variation in the DiffractionTest 140
8.3 Tests on Water Level Variation and OscillationAmplitude of the Device .1 4 1
XIX
Symbol
aaoa la'A'
1A' c
A.pipe
bb 'b I
b2boc
List of Symbols
Description
accelerationincident wave amplitude
constant value in incoming wave potential
exciting force coefficient for wave accelerationAl - A2 - A3water plane cross-sectional area of a cone-shape
artificial upwelling device when the device movesto the position of z
area of the inner wall of the tail pipe
cross-section area of the water chamber
cross-section area of the inflow pipe
cross-section area of the outflow opening
cross-sectional area of buoy
water plane cross-sectional area of cone-shape
artificial upwelling device at restwater plane cross-sectional area of cone-shape
artificial upwelling device when the device movesto the position of Z-Tl
cross-sectional area of the inflow pipe
water plane cross-sectional area of cylindrical
artificial upwelling devicearea of the outer wall of the tail pipe
damping coefficientexciting force coefficient for wave velocitydamping coefficient of first harmonic
damping coefficient of second harmonic
damping coefficient of zero harmonicrestoring force coefficient for cylindrical artificial
upwelling devicerestoring force coefficient for cone-shape artificial
upwelling device of z\t)
xx
•
Symbol
Dout
D.In
d rffoF I
F2F3Fh
FeFp _
KFpFRFdrFsFsh
F't
31F
Description
restoring force coefficient for cone-shape artificial
upwelling device of z\t)restoring force coefficient for cone-shape artificial
upwelling device of z(t)friction factor of a smooth plate
diameter of structuredisplacementdiameter of buoy of an artificial upwelling device
diameter of Acdiameter of water chamber of an artificial upwelling
devicediameter of outflow opening of an artificial upwelling
devicediameter of the long tail pipe of an artificial
upwelling devicedraft of the artificial upwelling device
friction factor of the pipe flowamplitude of the wave exciting force
first harmonic exciting forcesecond harmonic exciting forcethird harmonic exciting forcebody forcewave exciting forceFroude-Krylov forcepressure forcerestoring forcedriving force of water columnscattering forceshear forcedrag force on the inner or outer wall of the tail pipe
third order force which IS reflected on the first
harmonic
x 2 coordinate of the unit normal vector
XXI
Symbol
GgHhH
d.esign
Hm(x)
hoh sIm(x)Jm(x)kKm(x)
m'
m tm2
rnam
al
ma2
maoM ..
I,JnNl'N2,N3N ..
I,J
PPaPd,2
Pd,3
Descri ption
Green's functiongravitational accelerationwave heighttime step of Runge-Kutta methoddesign wave height
Hankel function of the second kind of order m
depth of water
overhead above sea water level
modified Bessel function of the first kind of order m
Bessel function of the first kind of order m
wave numbermodified Bessel function of the second kind of order
mdynamic pressure response factor
length of the tail pipethe length of the tube immersed in watermass of a two-valve artificial upwelling device at
resttotal mass of a two-valve artificial upwelling device
when oscillatingmass of a one-valve artificial upwelling device at rest
m 1 + mass of the water column in the pipe
added mass
added mass of first harmonic
added mass of second harmonic
added mass of zero harmonic
integration in radiation problem
unit normal vectornumber of the boundary sources
integration in radiation problem
reference point on the boundary surfaceair pressure inside the accumulator tank
dynamic pressure at the inflow opening
dynamic pressure at the outflow opening
XXll
Symbol
QR
r
Sf
SrTtT
d.
esrgn
Uo
u r I,U r,2
Description
pumping rate of an artificial upwelling devicevariable point on the integration surfacemaximum pumping rate for a certain configuration of
an artificial upwelling devicereference pumping rate of a two-valve artificial
upwelling device for base considerationsreference pumping rate of a two-valve artificial
upwelling device for different configurationsunit pumping rate of a one-valve artificial upwelling
device
average pumping ratedistance from x2 axis of the coordinate system
overhead of a one-valve artificial upwelling device at
restdistance between reference point and variable point
on the boundaries.radius of water plane cross-sectional area of the buoy
at restRenolds NumberRenolds Number of the pipe flow
radius of the radiation boundary
surface areawetted surface of the body
surface slope of cone-shape artificial upwel1ing
devicefree surface
radiation boundary surface
wave periodtimedesign wave period
maximum horizontal water particle velocity
relative velocity between water level and the device
relative velocity between water column in the tail
pipe and the device
xxiii
Symbol
u T,3
vvVcYd·ISVoutX
y
y, y,y
z
z*
ZstaticIzir I
Description
relative velocity between flow on the outflowopening and the device
pumping volumevelocitycontrol volumedisplacement volume of the devicepumping volumex I coordinate of the discretization planecoordinate with the direction of wave propagationcoordinate with the direction pointing upwardcoordinate with the direction parallel to the wave
frontx
2coordinate of the discretization plane
displacement, velocity, and acceleration of the waterlevel in the water chamber with respect to zerosea water level
displacement of device with respect to sea waterlevel
complex conjugate of zstatic displacement of a floating body = fol Creference oscillation amplitude of a two-valve
artificial upwelling device for differentconfigurations
dimensionless wave frequencyscale factor between prototype and modelangle of the body surface between point j and j+1
viscous damping coefficientfriction factor for the inner wall of the tail pipecoefficient of the expansion of <I> s
dimensionless damping coefficientphase lag wave and local pressure fieldphase lag of the first harmonic mechanic exciting
force
XXIV
I
Symbol
e
111 1
Illld.es.!gn<p, <p,<p
Ad .es i gn
/...0
Il
Description
phase lag of the second harmonic mechanic exciting
forcephase lag of the third harmonic mechanic exciting
forceratio of average to maximum velocity of a universal
velocity profilediffraction potential = <1>1+<1> s
time independent diffraction potential = <PJ+<l>sincoming wave potential
time independent incoming wave potential
radiation potential
time independent radiation potential
time independent radiation potential at the reference
point on the boundarytime independent radiation potential at the variable
point on the boundaryscattering wave potential
time independent scattering wave potential
total potential of the flow field
specific weight of the waterdisplacement of the wave with respect to sea water
levelwave amplitude, i.e., H/2design wave amplitude, i.e., Hd . /2esrgn
displacement, velocity, and acceleration of water inthe long tail pipe with respect to zero sea waterlevel
phase lag between wave and exciting forcevacuum gap between the valve and the water levelwave lengthdesign wave length
dimensionless wave frequency, w/ron
dimensionless added mass
xxv
I
Symbol
Jlovp
'tin
'tout
ueorod .esrgn
romax
ron
~
~o
'I'
~, ~,~
I I
( )neutralRe( )(R)
Im( )(I)
k
m
,m
,p~( )
Description
magnification factor
kinematics viscosity of waterdensity of the watershear stress on the inner wall of the tail pipe
shear stress on the outer wall of the tail pIpe
Y coordinates of the reference pointswave frequencydesign wave frequency
frequency that the maximum pumping rate happens
natural frequency of a floating structure
X coordinates of the reference pointsdimensionless damping factor
coefficient of the expansion of <1> S
displacement, velocity, and acceleration of water onthe outflow opening with respect to zero sea waterlevel
amplitude of the oscillationneutral level of the oscillation
real part of the complex parameter
real part of the complex parameterimaginary part of the complex parameter
imaginary part of the complex parameter
kth h .armoruc
mode of the velocity potential or order of the special
functionsvalue in the model test
value in the prototype
change of ( )
XXVI
Chapter 1Introduction
Coastal upwelling occurs off the west coasts of North America,
South America, West Africa and other regions where along-shore
wind stresses and surface temperature anomalies cause the surface
water to flow away from the coast and allow cold, nutrient-rich
deep ocean water to move upward (Sverdrup et aI., 1942). For
instance, in the California coast, the "period of upwelling" is from
the middle of February to the end of July when the prevailing
northwesterly occurs (Reid et al., 1958). Because coastal upwelling
provides a constant nutrient supply, high fish production has been
observed in areas of natural upwelling. Upwelling regions account
for only 0.1 % of the surface area of the world's ocean, but they yield
some 44% of the fish catch because the necessary nutrients are
supplied from deep ocean water (Crisp, 1975).
To increase the population of fish and other marine organisms by
the artificial upwelling of nutrients in deep ocean water has long
been considered as one of the possible means to enhance marine
fisheries, and, in the long run, to resolve the problem of future
world food supplies. Progress towards this goal, however, has been
very slow because of the high capital costs involved in delivering
deep ocean water to the surface. In this research, a wave-driven
artificial upwelling device is proposed and studied (shown as Fig.I.l
I
motionofthedevice
one-way tIowvalve dircc:lion
tailpipe
t ow direction
Fig.l.l The Proposed Wave-Driven
Artificial Upwelling Device
2
I). Through this device, deep ocean water can be pumped up
efficiently and economically.
1.1 Wave Energy and Artificial Upwelling
In this research, possible artificial upwelling techniques based on
thermal, chemical, mechanical, and pneumatic principles were
reviewed. Artificial upwelling by utilizing wave energy was found
to be the most viable alternative and most suited to the Hawaiian
ocean environment. Ocean waves have been long recognized as a
possible source of energy. Although they can be intercepted
anywhere, extreme variability creates a major problem in using
ocean waves as a power supply. However, in trade-wind ocean off
Hawaiian islands, the wave energy is reasonably consistent and
steady.
In order to capture energy from ocean waves, it is necessary to
intercept the wave with a structure which can respond In an
appropriate manner under the influence of the wave force. Two
kinds of wave energy devices have been utilized. One is a fixed
structure and the other, a floating structure. In this research, the
floating device was selected because it can be readily deployed in
the deep ocean.
Such a device riding on the heaving waves is a popular design
for extracting the energy from waves. As an example, the
pneumatic type of wave-power conversion buoy has been studied
3
by many researchers and has already been deployed as a self
powered navigation buoy. In this type of wave-power conversion
buoy, the wave energy is first used to compress the air in a
chamber located on the top of the buoy, then the compressed air
drives a turbine to generate electric power.
1.2 Research Objectives and Procedures
The prime objective of this research IS to design a wave-driven
artificial upweIling device which can pump up the largest amount of
nutrient-rich deep ocean water with given wave conditions. In
order to evaluate various design schemes, mathematical models and
laboratory physical models of the device are built and applied.
The basic concept of the two-valve wave pump proposed in this
study is that the deep ocean water will be pumped up through a
long tail pipe into a water chamber, and flow out through a
submerged opening. The performance of this artificial upwelling
device depends on the conjunctive movements of the ambient ocean
wave, the floating body, the water column inside the long tail pipe
and the water level in the water chamber.
The idea of wave-driven artificial upwelling was explored by
several researchers (Isaacs et aI., 1976; Vershinskiy et aI., 1987;
and Zhai, 1992). They adopted a very simple design of the device
which consists of a vertical pipe, a valve, and a float. These
researchers demonstrated successfully that the energy of ocean
4
waves can be used to pump up deep ocean water. However, these
studies did not provide the basis for engineering design of an
efficient device ready for real world application.
The performance of the upwelling devices was evaluated In
these earlier studies by using simple mathematical models based on
one or more of the following assumptions: (1) the device exactly
follows the wave motion, (2) added mass, damping coefficient, wave
exciting force and restoring force due to wave-floating body
interaction can be ignored, and (3) the friction of the water against
the walls of the tail pipe is negligible. The validity of these
assumptions, however, needs further investigation.
In this dissertation research, the simple versions of the device
adopted by past studies are revised in order to consider all possible
engineering design options and to identify the most efficient one.
Also, more rigorous mathematical analysis is introduced to examine
the limitation of applicability and simplified assumptions made in
past studies.
The wave-driven artificial upwelling device developed for this
research consists of a floating buoy, an inner water chamber, a long
tail pipe, and two flow-controlling valves (Figure 1.1). The rate of
upwelling flow out of the chamber as it heaves along with ambient
waves depends on the wave-device-flow interaction. A set of four
simultaneous differential equations describing these inter-related
movements are formulated based on momentum and mass
5
conservation principles. These governing equations, or the
mathematical model of the operation of an upwelling device, are
solved numerically by using the fourth order Runge-Kutta method.
The equation of motion of the upwelling device, which is one of
the four model governing equations, contains parameters of added
mass, damping coefficient, wave exciting force and restoring
coefficient. Values of these parameters must be determined before
the model equations can be solved. Determination of these
parameters, on the other hand, requires the sol ution of boundary
value problems describing the wave-floating body interactions.
These problems are solved in this research by using the boundary
element method.
Calculated parameter values and results of mathematical
modeling are verified by a series of hydraulic experiments
conducted in a wave basin in the James K. K. Look Laboratory of
Oceanographic Engineering at the University of Hawaii.
The mathematical model of upwelling is then applied to find the
best device configurations in relation to the rate of upwelling flow.
In each test run, different values of five configuration variables, i.e.,
the shape of the float, the size of the water chamber, the draft of
the device, the length of the long tail pipe and the size of the
floating buoy, are selected and the results are compared. Finally,
design criteria are established.
6
In test runs, ambient wave conditions are selected which have
wave periods ranging from 3.3 seconds to 40 seconds. A constant
wave amplitude of 0.40 m is used. The best device configuration is
finally determined for the design wave height and period of 1.5
meter (5 feet) and 10 seconds, respectively. The selection of this
design wave condition is based on a previous study of waves in the
deep ocean off the Hawaiian Islands in an area subjected to trade
winds (Homer, 1964).
7
Chapter 2Literature Review
2.1 Options of Artificial Upwelling
Methods of artificial upwelling have been proposed and tested in
many ways. Theoretically, artificial upwelling can be achieved by a
number of techniques based on established thermal, chemical,
mechanical, and pneumatic principles. The fundamentals of these
techniques and their past developments are briefly reviewed.
(1).OTEC
Facilities for artificial upwelling have been designed according to
the needs of OTEC plants, and the artificially upwelled nutrient-rich
deep ocean water has used for land-based aquaculture in past
years.
A 50KW Mini-OTEC system deployed in Hawaii at 1979 was the
world's first offshore moored OTEC system and operated for a short
period to demonstrate that power could be produced from the OTEC
system (Polino, 1983). In the Mini-OTEC system, the deep ocean
water was raised by using electric pumps and a long tail pipe which
is 656m long and O.6m in diameter with the pumping rate of
2m 3/sec (McHale, 1979). A field study for a combined Mini-OTEC
and discharge of deep ocean water for mariculture was conducted
on the platform HOYO anchored in the Toyama bay, Japan.
8
In the Mini-OTEC system of Hawaii, about 80% of the generated
power was used on pumping of deep ocean water and on operation
of the closed cycle system. Thus, the net power output was about
10KW.
(2). Wave pump
The wave pump was invented by Isaacs and Seymour (1973) at
Scripps Institute of Oceanography. It responds directly to wave
motion and extracts energy from the oscillatory motions to generate
power and to pump up deep ocean water.
In the beginning, Isaacs designed a small working model with a
30-meter pipe into the sea and its upper part was a floating buoy.
He concluded that very little hydroelectric energy could be derived
from this shallow water wave pump. Later in 1976, a modified
version of this model was tested off Kaneohe Bay, Hawaii (Wick and
Castel, 1978). The buoy used was about 6 feet in diameter and the
pipe was 300 feet long and 2 inches in diameter. In Hawaii's
experiment, deep ocean water was successfully brought to the
ocean surface by the wave pump.
Wick and Schmitt (1977) concluded that in the zones of trade
winds and westerlies, the waves are very energetic and consistent
and they are an inventor's delight.
9
(3). Current pump
The idea of using the energy of the ocean current for artificial
upwelling was first introduced by Liang et al. (1978) at the
Institute of Oceanography, National Taiwan University. Their
current pump used the suction force induced by an ocean current
when it flows through a submerged Venturi-type tube.
Liang et al. (1978, and 1979) conducted field and laboratory
experiments of artificial upwelling using the current pump. In the
field, the pipe attached to the Venturi tube was 45 meter long, 40
centimeter in diameter, and deployed at depths between just under
the sea water surface and 45m below. With a current of 18-27
centimeter per second (about 0.5 knot), the up-flow rate of deep
ocean water was around 12 liters per second. The maximum up
flow rate of the deep ocean water throughout this field experiment
was around 80 liters per second.
Based on their experimental results and analytical derivations,
Liang et aI. (1978) suggested that the performance of the current
pump could be enhanced by increasing the head difference between
the entrance and the exit of the Venturi tube.
(4). Perpetual salt fountain
In many tropical and subtropical regions of the ocean, warm
ocean water in the surface layers has a salinity exceeding that of
the colder water below. Stommel et al. (1955) tried to use this
I 0
natural density anomaly to pump up the deep ocean water through
a long pipe.
In the beginning, the deep ocean water was slowly upwelled by
a mechanical pump to the surface through a long pipe which was
lowered from the surface to the depth of the low salinity water.
When the pump was disconnected, the water would continue to
flow up. The reason for this phenomenon is because the low
salinity deep ocean water would first flows up by the mechanical
driving force and then gain heat from its surroundings when it
flows upward through the pipe. This heat-exchange would reduce
the density of water in the pipe and the difference In pressure
would cause the water to flow upward continuously.
They conducted a successful laboratory experiment USIng a glass
tu be of 3 inches in diameter with hot water flowing around the
tube. Based on their experimental results, they estimated that a
2000m long pipe might develop an extra 2m pressure head at the
sea water surface in the Central North Atlantic.
Later, Groves (1959) derived a water motion model of the
perpetual salt fountain by introducing heat conduction and flow
friction. He concluded that if a copper pipe with radius of IDem and
wall thickness 2mm, a pumping depth of 300m, temperature and
salinity differences between surface and 300m below as 6°C and
0.1 % respectively, then the flow rate of the salt fountain would be
about 5.5 liters per second. He also estimated that if the pipe's
1 I
radius and wall thickness were 100cm and Smm, all other
conditions remaining the same, then the upwelling flow rate would
be about 3000 liter per second.
(5). Surge energy converter
The surge energy converter was first invented by Wall (1911) to
capture the energy of a wave just as it enters the surf zone. The
surge energy converter is a deflector designed to absorb some of
the momentum of surge while turning the flow upward at an angle
to the horizontal. The momentum is absorbed to push a hydraulic
piston and generate power.
The use of surge energy to pump deep ocean water still presents
some practical problems. The "surf zone" is close to the beach, and
this means that the device should be deployed in the shallow water
area. If the power is generated close to the beach, and used
directly, the pipe for pumping deep ocean water will be very long
unless the depth profile is steep just outside of the surf zone like
some atolls at the South Pacific.
According to the mechanical analysis by McCormick (1981), the
conclusion is "In the view of the high capital costs of such a device,
the surge energy converter is not a cost-effective system".
2,2 Wa~e and Floating Structure
In order to extract energy from ocean waves using a floating
12
body, it is obvious that the study of the behavior of the floating
body under the influence of incident waves is of major importance.
Normally, the problems of the wave-structure interaction have
been solved mathematically by distributing singularities (sources or
dipoles) on the body boundary and introducing Green's third
identity to obtain the Fredholm integral equation of the second kind
for the unknown strengths of the distributed sources (Frank, 1967;
Brebbia, 1978). This is called the boundary element method.
The conventional way of setting up the Fredholm integral
equation of the second kind for all the unknown strengths is to use
a Green's function that has already satisfied all the boundary
conditions except the body boundary condition (Wehausen and
Laitone, 1960).
For a floating body with more complex geometry, the finite
element method is more popular for solving the problems of wave
structure interactions (Bai, 1975; Huang et al. 1985). When using
the finite element method, all the fluid domain must be discretized,
and thus, the number of equations is very large and time for
calculation is long.
The artificial upwelling device proposed in this research consists
of an axis-symmetric floating body with a long tail pipe and two
one-way valves. Here, only the heaving motion of the device is
allowed.
1 3
An efficient calculation of the diffraction wave forces on vertical
axis-symmetric smooth contoured bodies was done by Black (1975).
He developed a Green's function and an integral-equation
formulation that could reduce the surface integration into line
integral equations. The integration is carried out over the line
defined by the intersection of the body with the plane of 9=0 of the
incoming wave. Thus, the discretization of the boundary element is
only on one intersection line of the body boundary.
With regard to the work on finding radiation coefficients, Kritis
(1979) and Ferdinade & Kritis (1980) used the Yeung's method of
source distribution with the Bai's radiation condition (Bai, 1972) to
solve the radiation potential of a heaving axis-symmetric floating
body. They deployed the source points on all the boundaries of an
axi-symmetric plane and used the simplest Green's function, l/r.
Vantorre (1986) used third-order theory to determine the radiation
hydrodynamic coefficients of the axi-symmetric floating body when
the oscillation amplitude is no longer small.
In the past, some studies were focused on the motion of a
floating structure with a long tail pipe. PauIling (1979) and Hwang
(1979) studied the motion of a floating structure of ship-like form
with an OTEC cold water pipe oscillating In waves. In the study of
Hwang (1979), the dynamic response of a flexible long tail pipe
with constant internal flow velocity IS first formulated by
introducing the finite element method of structural analysis, and
I 4
•
then it is coupled with the motions of a ship-like structure. He
concluded that the change of the internal flow velocity had very
little influence on the motion of both the cold water pipe and the
whole system.
The one-degree-of-freedom heaving motion of the floating
cylindrical buoy is most commonly applied to the studies of the
wave energy extraction (Masuda, 1971; McCormick, 1974, 1976;
Isaacs et aI, 1976; Whittaker and McPeake, 1985). Most of the
heaving-type wave energy devices consist of a floating buoy, a
water column inside the buoy and an air chamber on top of the
inner water column. The motion of the buoy and the motion of the
water column are coupled by introducing the pressure in the air
chamber and the applied damping of the turbine.
A study of the optimal design of a power generator with aXI
symmetric tail tube buoys was done by Whittaker and McPeake
(1985). In their study, the peak capture factor is maximized when
the diameter of water column/diameter of buoy ratio equals 0.75,
and the frequency bandwidth of the capture factor is widened
when the length of the water column increases. The peak capture
factor of a floating system near resonance condition is about 20
times higher than the peak capture factor of a system with a fixed
device and an oscillating water column.
Comparisons of theory and experiment of the radiation and
diffraction problems of wave-body interactions have been done by
I 5
several researchers. Tasai (1960) compared wave amplitudes at
some distance from a vertical oscillating cylinder with calculated
asymptotic values. Portor (1960) and Paulling & Portor (1962)
measured pressure amplitudes, force amplitudes, and wave heights
generated by heaving motion for circular cylinder and several other
cylinders of ship-like section and compared them with experiment.
Vugts (1968a) made an series of experiments to measure added
mass, damping coefficient, wave exciting force and phase lag for
several columns: circular, triangular, rectangular and other modified
shapes. Recently, Ferdinande and Kritis (1980) conducted an forced
heaving oscillation experiment on a hemisphere and other circular
cylinders to verify their efficient method of determining added
mass and damping coefficient. Vantorre (1986) tested the added
mass and damping coefficient in large oscillation amplitude of a
floating conus and a submerged cylinder. For a freely floating body,
the experimental research is fewer. Vugts (1968b) investigated a
two-dimensional rectangular column with the motion of sway and
roll. McCormick et al. (1975) studied the performance of a floating
circular pneumatic wave-power generator by using forced
oscillation to measure the added mass and damping coefficient, and
using free oscillation test to measure the internal free surface,
internal chamber pressure and exhaust velocity of the au.
Whittaker & Wells (1978) tested the performance of a hydro
pneumatic buoy with a self rectifying rotor, and Whi ttaker &
McPeake (1985) tried different configurations of the device, i.e.,
1 6
tube diameter and tube length, for the purpose of optimal design of
the device.
2.3 Wave-Driven Artificial Upwelling Device
Isaacs and Seymour (1973) first suggested and deployed a model
test of a wave-power generator with a long round pipe and a slack
tethered float. The preliminary hydrodynamic study of a
exponential-shaped float with a long round pipe heaving at a design
wave frequency and significant wave height was investigated by
Isaacs et aI., (1976). In their research, the nonlinear restoring force
was considered. The added mass and damping coefficient were
considered to be constant while the wave exciting force had no
phase lag with the incoming waves.
A preliminary design and approach on the motion of a wave
pump as a long submerged pipe with a constant-head pipe above
water surface has been discussed by Isaacs et aI. (1976). They
assumed that the motion of the constant-head one-valve device
exactly followed a sinusoidal sea surface. Vershinskiy et al. (1987)
followed the analysis of Isaacs et al. (1976) and then deployed a
field test on one-valve device at the Black Sea in 1983 by a joint
expedition of VNIRO and the Institute of Oceanology near Cape B.
Utrish. In their research, the calculated and experimental results
showed good agreement.
Zhai (1992) evaluated the pumping effect of a' one-valve device
I 7
by calculating the motion of the floating body hydrodynamically.
He assumed that the added mass and damping coefficient are
constant for various oscillation frequencies and the outflow valve
would open when the acceleration of the pipe reduced to zero.
The details of the previous approaches are discussed as follow:
2.3.1 Motion of Floating Body
Fig.2.1 shows two designs of a wave pump designed by Isaacs et
aI. (1976). Fig.2.1 a is a one-valve artificial upwelling device while
Fig.2.1 b is a one-valve wave power generator and an artificial
upwelling device.
The one-valve artificial upwelling device consists of a long
submerged pipe (length=L), with a constant-head pipe (length=hs)
above water surface and a supporting float, while the wave
powered generator as a long pipe, an accumulator tank, a uni
directional check valve and a supporting float .
In Fig.2.1 a, Isaacs et aI. made an important assumption that the
device exactly follows a sinusoidal sea surface, the equation of
motion can be expressed as:
or
z=Tl
z = 11
1 8
(2.1 )
(2.2)
A. neutral position
ono>-WlIy valve
B. wave-excited condition
Fig.2.la Design of a One-Valve Artificial Upwelling Device
with a Pipe of Constant Overhead
A. neutral position B. wave-excited condition
Fig.2.tb Design of a One-Valve Wave Power Generator
and Artificial Upwelling Device
19
(2.3)
where Tl is the motion of the wave with respect to zero sea water
level.
In the above equations, the added mass and damping effect are
ignored, the restoring force equal the buoyant force and does not
show on the equation of motion because there is no relative motion
between wave and the device.
In Fig.2.1 b, Isaacs et a1. also investigated the motion of the
wave-powered generator with a long pipe, an accumulator tank, a
uni-directional check valve and a supporting float
hydrodynamically. The damping effect in this approach is assumed
to be linear, and the wave exciting force only consists of the second
derivative of the wave, i.e. Tl The equation of motion of the device
on open valve condition can be expressed as:
m 1i = m iii - z) - bi: - FR - Pa Ai n
On closed valve condition, it can be expressed as:
m 2z = m iii - z) - b z - F R(2.4)
where: m 1 = mass of the supporting structure and pipe
m 2 = m l + mass of the water column in the pipe
m a = added mass coefficient
z = displacement of the float with respect to sea water level
b = damping coefficient
F R = restoring force
20
I
P a = air pressure inside the accumulator tank
Ain = cross-sectional area of the inflow pipe
The consideration of Pa can be neglected if there is no air
accumulator chamber in the design of the wave pump.
Zhai (1992) expressed the equation of motion of the cylindrical
float device by using the simplified Froude-Kryloff force as the
exciting force.
(m 1+ m a)Z + bz + F R =pgAbuoy "
where Abuoy is bottom area of a cylindrical buoy.
2.3.2 Motion of Internal Water Column
(2.5)
In Fig.2.1 a, a constant water column acceleration was obtained
simply by introducing the force balance on the water column.
(2.6)
where the upward direction is set to be positive on the expression
of acceleration, velocity and position.
Here, the friction loss is neglected and the relative motion
between the wave and the device is neglected as well.
In Fig.2.1 b, Isaacs et aI. expressed the governing equation of the
water column within the pipe with an air chamber as:
2 1
pA. L (p = pA. (L + rU g - pA. (L + r) g - p a A.In In In In
- frictional loss (2.7)
If Pa is neglected and r is equal to hs ' then the water column
acceleration inside the pipe (<P) then can be obtained:
- ,,-h s fri II<P = L g - rrctrona oss(2.8)
Zhai (1992) expressed the equation of motion of the inner water
column by considering the motion of the device (z) and the vacuum
gap (~<p) between the valve and the water level on the top of water
column and neglecting the overhead, hs '
- T)-z+~<p fri I<P = L g - rrctrona loss
2.3.3 Operation of Valve
(2.9)
The motion of the inner water column is controlled by the
operation of the valve. Isaacs et aI. (1976) suggested that the valve
would open when the acceleration of the water column is equal to
or larger than that of the device. When the valve opens, the motion
of the water column follows Eq.(2.4). The valve will be closed when
the velocity of the water column is equal to or smaller than that of
the device. When the valve is closed, the water column IS
motionless and moves along with the device.
Thus, when the displacement of a device or sea surface at a
22
certain position can be can be expressed as:
z = Izl cos cot (2.10)
the occasion for valve opening (t 1) and valve closing (t2 ) can be
obtained by comparing the acceleration and velocity of the water
column and the device:
obtain) t1 (2. II)
t
- col z] si n rot ~ <j>( t ) + I ip( t) d t ---!t!lli~ t2 i 2
t I (2.12)
Zhai (1992) suggested that the valve would open when the
acceleration of the device reduces to zero, and the valve would be
closed when the velocity of the water column is equal to or smaller
than that of the device.
2.3.4 Pumping Effect of Artificial Upwelling Device
By using Eq.(2.2), Eq.(2.8), Eq.(2.11) and Eq.(2.l2), Isaacs et al.
graphically showed that the pumping effect would occur which is
shown as Fig.2.2.
Vershinskiy et al. (1987) used the criteria of Isaacs et al. (1976)
and obtained the pumping volume by the integration of the velocity
difference when the valve is open.
23
• device's.. device's
-0-- device's
accelerationvelocitydisplacement
valve closes
-----']I>-~ time
Fig.2.2 The Graphic Solution of Pumping Effectof a One-Valve Artificial Upwelling Deviceby Isaacs et at (1976)
24
V 1 = r2 [<p( t) - z(t)] A. d tou 1
11 n
where:
(2.13)
(2.14)
The average pumping rate (Q) for one wave cycle (period T)
then can be obtained.
v- outQ=
T (2.15)
Fig.2.3 shows the pumping capacity for the Vershinskiy et al.'s
analysis (1987). The pumping rate of the constant-head one-valve
device is related to the dimensionless water column acceleration,
i.e., the length of the tail pipe, the cross-sectional area of the tail
pipe, the overhead, and the wave frequency. Fig.2.3 also shows that
there will be no pumping effect when Iq> I/ I11 I> 1.0.
Zhai (1992) used Eq.(2.5), Eq.(2.9) and the criteria of valve
operation to calculate the pumping effect of a wave pump with
specific dimension of:
diameter of the buoy (Db ) = 1.5 muoy
diameter of the tail pipe (D. ) = 0.2 mIn
length of the pipe (L) = 200 m
wave amplitude (1111) = 1 m
25
Fig.2.3 The Capacity of Wave Pump Calculatedby Vershinskiy et al. (1987).
26
wave period (T) = 10 sec.
and obtained a discharge rate of 0.06 m3/sec.
27
Chapter 3Wave and Device Interactions
The fluid-body interactions involve the oscillation of the artificial
upwelling device which induces added mass and damping effect,
while the surrounding wave exerts an exciting force which leads to
the device's oscillation. Once the device oscillates in the wave
environment, the restoring force of the device should be considered.
In order to set up the equation of motion of the artificial upwelling
device, the added mass, damping coefficient, wave exciting force,
and the restoring force must be determined.
The boundary element method is used in this chapter. When
solving the radiation problem, an efficient method developed by
Kritis (1979) is used. In this method for an axi-symmetric floating
body, the unknown strengths of sources are distributed on all the
boundaries of an axi-symmetric plane. Thus, all the boundary
conditions are directly plugged into the Fredholm integral equation
of the second kind.
When solving the diffraction problem, another efficient method
developed by Black (1975) is used and modified to find the wave
exciting force. In this method for axi-symmetric body, a Green's
function which satisfies all the boundary conditions except the body
boundary condition is used by modifying from Black's research.
The surface integration in the Fredholm integral equation of the
28
second kind is then reduced to a line integration on the intersection
of the body surface with the plane 9=0.
In the diffraction approach, the dynamic pressures on the
openings of the device are obtained and used as variables to solve
the flow motion of a artificial upwelling device.
Even though the surrounding water level of the device will be
influenced by the radiation oscillation and the wave diffraction
effects, a simplification is made. The surrounding water level is
assumed to be equal to the incoming wave and it is used ira later
theoretical investigations.
Finally, the restoring force IS investigated by considering the
change of the shapes of the device, the influence of the wave, and
the motion of the device.
3.1 Added Mass and Damping Coefficient
The radiation velocity potential of a non VISCOUS fluid (4) r) is
introduced to represent the flow field, and the velocity potential is
expressed in the cylindrical coordinates. By obtaining the radiation
potential of the flow field, the added mass and damping coefficient
can be obtained.
The Kirchhoff decomposition is first introduced to define a
velocity potential which is space dependent and time independent:
(3.1)
29
where x2 is pointing upward, i. (t) describes the harmonic heaving
velocity vector of the device which is not a function of space, i.e.,
z(t) = ~I e-irot
i.( t) =- iro z(t)
(3.2)
(3.3)
and cl>r (R, 6, x2 ) IS the complex form of the space dependent and
time independent velocity potential in cylindrical coordinate which
can again be defined as sum of real and imaginary parts:
'" = ""(R) • '"(I)'fir 't"r + 1 'fir (3.4)
The velocity potential <I> r should satisfy the Laplace equation and
all the linearized boundary conditions. Thus, the governing
equation and boundary conditions for the time independent
velocity potential q>r at deep water condition can be expressed as:
(at x2=0)
(3.5)
(3.6)
(3.7)
(3.8)
where n IS the x2 component of the unit normal vector on theX2
body surface.
30
By introducing Euler's Integral, the radiation pressure (Pr) and
force (Fr ) acting on the body which makes the prescribed motion
can be obtained.
(3.9)
Thus, the radiation forces on the x2 direction can be obtained.
(3.10)
By introducing Eq(3.l), Eq.(3.3), and Eq.(3.4), Eq.(3.10) can be
rearranged as:
F JJ( (R) • (I».. d Sr=-P CPr +lcj>r z*n x
S 2b
JJ( (R) - (l) .) d S= - P cj> r Z + cP r roz * n x
S 2b
(3.11)
The added mass (rna) and linear damping coefficient (b) then can
be obtained by setting:
F =-m z-bir a
Thus,
3 1
(3.12)
(3.13 )
(3.14)
For an x2 axis axis-symmetric floating body, the radiation
boundary condition Eq.(3.8) can be replaced by the Bai '5 radiation
condition (Bai, 1972) and the Yeung's method of source distribution
(Yeung, 1973) can be introduced.
The Bai's radiation condition is:
( valid when R --7 oc )(3.15)
In Yeung's method of source distribution, all the boundary
conditions are directly used on the Fredholm integral equation of
the second kind. The unknown source strengths are distributed on
all the boundaries of an axis-symmetric plane. Thus, the
relationship between the velocity potential at the reference point
on the boundary epr(p) and the velocity potential at the variable
points on any other boundary surfaces epr(q) can be expressed by
the Fredholm integral equation of the second kind, i.e.,
where: p = the reference point on the boundary surface
32
•
q = the variable point on the integration surface
r = the distance between p and q.
By introducing body boundary condition, free surface boundary
condition, and radiation condition into Eq.(3.I 6), Eq.(3.16) becomes:
21t <pr(p) + JJ <pr(q)[ ~ (~) ] dSsb
= JJ [ n (q) rl ] dSx 2(3.17)
where Sb IS body surface, Sf is the free surface, Sr is radiation
boundary surface, and Ro is about 5 to 6 times the body's radius.
The boundary surfaces can be discretized into plane co-axial
rings, vertical co-axial elementary cylinders, and elementary co
axial conical surfaces. Hence, the complete integral equation can be
approached by the expression:
N1
N2
21t <pr(T.) + L <pr(T )M.. + L <pr(T.)( M .. - wg2
N.. )1 . 1 r 1J . N 1 J 1J 1J
J= J= 1+
<PrCT,) [M .. + CZRI - ik)N.. ]
J 1J 0 1J
33
N]
=I n x (T.)N... 1 2 J IJJ=
where
M .. = JJ jd (~) dSIJ san
N.. = JJ ~ dSIJ S
(3.18)
(3.19)
(3.20)
Thus, a set of N3 linear equations with N3 unknowns (jlr(T j , i=1 to
N 3) is then established.
Here, the subscript i refers to the reference point, j refers to the
variable surfaces (rings, cylinders, or conic surfaces), and r IS the
distance from a reference point Tj to any point Tj of any ring. Thus,
there are N=N3 reference points. The locations of the reference and
variable points are chosen in the middle of each linear segment.
The coordinates of each reference and variable point and each
segment of the boundary element are shown as Fig.3.!.
In Fig.3.1, the coordinates of the beginning and ending points of
each boundary element are expressed as (X., Y.) in the X-V plane.J J
The coordinates of the reference points T j are chosen in the middle
of the linear segment which can be expressed as (~ , 'U) in the X-Y
plane:
X1+X~. = 1+ 1
1 2
34
(3.21)
YSf
XN2+1,Y NHl)
X
Fig.3.1 The Discretization of Boundary Elementfor Radiation Approach
(X NHl,Y NHl)
35
y.1+y.1+ I
'\). = ---=--I 2 (3.22)
The x2 component of the normal vector on the body n can be"2
expressed as:
nx(T)=cosa.2 r' J
where
y. 1- Y.-1 J+ J
a.=tan (X -X)J j+l j
(3.23 )
(3.24)
The solution of Eq.(3.l8) then gives N3 values of <l>s(T) and <l>r(T)
in complex forms.
Thus, the added mass and damping coefficient then can be
obtained by introducing Eq.(3.13) and Eq.(3.14).
Fig.3.2 shows the result comparison of the added mass for the
cylindrical artificial upwelling device with different lengths of tail
prpe. This device has the ratio of the draft and diameter of buoy
(draft/Db UOY
) = 1/3.2. Fig.3.3 shows the result comparison of the
damping coefficient with different lengths of tail pipe. The results
of a 3-D model are also introduced in both figures to verify the
results of this model.
Both Fig.3.2 and Fig.3.3 show that the influence of the tail pipe
on added mass and damping coefficient is very small.
36
1.8
1.6-•;;1.4
~<,•a 1.2-flO.,ClI 1a
"dQJ 0.8"d"dClI
0.6
0.4
0.5
-•;;~ 0.4~.c-~ 0.3c:lQJ-c.Ji::....QJ 0.20c.J
tlCc:l- 0.1PoeeClI"d
0
3-D model• LID -0"-or• LID -1.5
~"-G LID -2.5"-or
6 LID -3.5"-or-,r-; .-- • .......
o 0.5 1 1.5 2wave period, (CJ)2 Rb /g)
Iloy
Fig.3.2 Added Mass of a Cylindrical ArtificialUpwelling Device with Different
Lengths of Tail Pipe
--3-D model• LID -0
"-•• • LID -1.5"IIOYji' 0 LID -2.5
~"IIOY
6 LID -3.5"uy
1 <,~~
~
o 0.5 1 1.5 2wave period, (CI) 2R / g)
buoy
Fig.3.3 Damping Coefficient of a CylindricalArtificial Upwelling Device withDifferent Lengths of Tail Pipe
37
I
In Fig.3.2 and Fig.3.3, this efficient model can get a result very
close to the 3-D model.
The added mass and damping coefficient of other draft/Dbu o y
ratio can then be obtained and used as known variables in the
equation of motion of the artificial upwelling device.
3.2 Wave Exciting Force
Wave exciting forces on offshore structures can be characterized
by three categories and they are solved in different ways:
I. When the structure is small (D is the diameter of the structure)
and the wave height (H) is large compared to the wave length
(A) and, i.e. D/').. < 0.15 and HID > 1, both inertia and drag force
are important. The Morison equation is introduced.
II. When the structure is small and the wave height is small
compared to the wave length, i.e. D/A < 0.15 and HID < 1, the
inertia force is still important but the drag force can be
neglected. The Froude-Krylov potential flow theory IS
introduced by considering the incident wave pressure only.
III. When the structure is not small and the wave height is small
ccmpared to the wave length, i.e. D/A > 0.15 and H/D < 1, the
existence of the structure will influence the flow field and the
diffraction effect should be considered. In this situation, the
potential flow Diffraction theory is introduced by considering
38
(3.25)
both the incident wave pressure and the scattered wave
pressure.
Here, the third category IS used because of the diameter of the
designed artificial upwelling device is big.
In the potential-theory approach, the Euler's Integral IS also
introduced to express the relationship between the exciting
pressure (Pe) and the diffraction potential (<1>e)'
a<1>ePe=-P-al
where: <1> e = <1>} + <1> s' <1>1 is the velocity potential of the incoming wave
and <1> s is the velocity potential resulting from the disturbance of
the incoming wave by the presence of the structure.
The wave exciting force on the device (Fe) IS the surface
integration of Eq.(3.25). It is the summation of the forces which are
induced by the incoming wave and the scattering wave, i.e., Froude
Krylov force (FF_K) and the scattering force (Fs) '
(3.26)
where:
S b = the wetted surface of the body
nX2
= the x2 coordinate of the unit normal vector on Sb
(x l' x2, x3) = the unit vector on the Cartesian coordinate system
39
which sits on the zero water level where xl is the direction of
wave propagation, x2 is the direction pointing upward, and x3
is the direction parallel to the wave front.
The Froude-Krylov force can be obtained by introducing the
incoming wave potential which satisfy the Laplace equation and the
free surface boundary condition in the deep water. The incoming
wave can be expressed as the complex form:
(3.27)
where k is the wave number and CJ) is the wave frequency.
The incoming wave potential for deep water can be expressed as:
(3.28)
Different from the incoming wave potential, the scattering
potential (<1> s) must satisfy the Laplace equation, the free surface
boundary condition, body boundary condition and the radiation
boundary condition.
By setting that the diffraction potential can be expressed as one
space dependent term and one time dependent term,
(3.29)
then. the space dependent scattering potential must satisfy the
following equations:
40
aep s ol---41 =0ax g s2
(a t x2
= 0)
(3.30)
(3.31)
;l.i,. acj>IV\l"s
an = an (3.32)
Iim vIR(a~ - i k) cj> s = 0R--+o<
where:
(3.33)
n = the unit normal vector point outward from the fluid into the
body
R = the distance from the x2 axis of the coordinate system
For an x2 axis axis-symmetric floating body, the line-integration
method developed by Black (1975) is used to solve the wave
exciting force more efficiently.
The space dependent incoming wave potential (cj>I) In cylindrical
coordinate (R, 8, x2) is first introduced.
cj>I=a/1(x2
)f £m(-i)m Jm(kR)cos(m8)m=O
where:
a = - iro a f (0)o gIl
(3.34)
ao = incident wave amplitude
x 2 = positive on upward direction from the zero water level
41
;~,
k = wave number
R = radius from the center of the coordinate
J (kR) = Bessel function of the first kind (of order m) with anrn
argument kR.
The Green's function which satisfies the Laplace Equation, free
surface, and radiation boundary conditions is used. This Green's
function is modified from Black (1975) for finite water depth and it
is now applicable for deep water condition. It can be expressed as a
a independent term grn times a a dependent term.
DC
G(R, e,x 2'Sl'S3,S2) = I Ern g rn cosme cosms 3rn=O
where
(3.35)
(3.36)
here, (R, a, x2) is the cylindrical coordinate of a point in the fluid
domain, and (SI' ~3' S2) is the coordinate of a source point. The
upper terms in the brackets are used for R>S1 and the lower terms
are used when R< l; i: The terms appear in Eq.(3.34), (3.35) and
(3.36) are defined as:
E = 2 wh en m ~ 2rn
42
2k = CO /g.
It is noted that this Green's Function is non-singular. The point
source is represented by a discontinuity in the gradient of the
function rather than a singularity.
The Hankel function Hm(x) is the second kind which satisfies the
radiation condition. Jm is the standard Bessel function, Km is the
modified Bessel function of the second kind (or modified Hankel
function), and 1m
is the modified Bessel function of the first kind.
The Fredholm integral equation of the second kind for the wave
scattering potential is then introduced.
(3.37)
where p is the reference point on the body surface and q is the
variable point on the body surface. It is also noted that the
integration is only performed over the surface of the body.
By defining:
oc
<l>s = a 1 L lm cosmsm=O
oc
<I>} =a1L 'l'm cosrn O
m=O
43
(3.38)
(3.39)
the line-integral formula can be obtained by introducing Eq.(3.38),
Eq.(3.39) and Eq.(3.35) into Eq.(3.37) and integrating over /;3'
J dgm dgmXm = 41t [ Xm( -:1- n r +~ n;lt ) r d I
at aA.2
2
J ti\jrm d\vm ) ]+ g (~n +-:l-n rdl, m=0,1,2, ......m al r aX x 2
2 (3.40)
The integration is carried out over the line defined by the
intersection of the body with the plane 8=0.
The boundary line can be discretized into line segments. The
discretization of the boundary element is shown as Fig.3.4. The
location of the reference and variable points of each segment are
chosen in the middle of the linear segments. Thus, one set of N
linear equations with N unknowns X· (j=1 to N) is established forj.rn
one certain m.
N 0..
~ (a i.jrn - 4 J~) Xj,m = b i,m'J=l
where:
0.. = the Kronecker deltaJ,J
m =0,1,2, ...
(3.41)
ago .I,J,m
a .. =[ ::Jr n r ·+I,J,m ~ .. ,J
1 I (3.42)
44
(3.43)
I
Y
(X N+l,Y N+l)
X
(X N,YN)
(Xl,Yl)
Fig.3.4 The Discretization orBoundary Elementfor Diffraction Approach
45
The only unknowns in Eq.(3.41) are X· . The subscripts on g andj.rn
'" indicate that i is the reference point, j is the variable point, and m
is the mode number of the velocity potential. The subscripts on the
normal vector n indicate the components and the variable point.
The length of the segment at jlh variable point is L\lj'
By introducing the Euler's Integral, the pressure at the point (R,
9, x2) on the body then can be obtained.
oc
P .=-i ropa] eiro t L (X .+'1' .r cos mee .r m,1 rn.r'
m=O (3.44 )
The vertical force depends only on the m=O angular mode
because of the symmetry. It can be obtained by integrating
Eq.(3.44) with the nX2 component of the body surface and can be
expressed as:
21t ioi pal e iro t NFe = - L (X· + 'V. ) R. n x L\1.
Eo i=l 1,0 1,0 I 2 I(3.45)
Fig.3.5 shows the companson of the wave exciting force for the
cylindrical artificial upwelling device with different lengths of tail
pipe. The device has the draft and diameter of buoy ratio
(draft/DbuOY
) = 1/3.2. The result of a three dimensional model is
also included and verified. Besides, the pressure on the openings of
the device can be obtained from Eq.(3.44). The inflow and outflow
openings are set at the coordinates of (0, 0, dr+L) and (Rbuo / 2, 0,
dr).
46
0.16
-!:! 0.14't:l
>bO0. 0.12<,
p.,C1/-... 0.1CIJu1-4
cB 0.08
coc:: 0.06-~-u><G' 0.04.,>CId 0.02~
0
3-D model<l LID -0buoyT LID -1.5
buD)'0 LID -2.5
buoyA LID -3.5
buD)'
~~ 8
~ 0 •I--
o O.S 1 1.S 2
wave frequency, (0)2Rb / g)uoy
Fig.3.5 Wave Exciting Force of a CylindricalArtificial Upwelling Devicewith Different Lengths of Tail Pipe
47
The results of the pressure at the openmgs will be used on
solving the motion of the internal flow of the artificial upwelling
device.
Fig.3.5 shows that the influence of the tail pipe on the wave
exciting force is very small. Besides, this efficient model can get
results very close to the 3-D model.
For the device with cylindrical float, Fig.3.6 shows the ratio of
scattering force and total exciting force (F/Fe) for various ratio of
buoy diameter and wave length, Db /1... The result shows that theuoy
influence of scattering wave force is relatively small compared to
the incoming wave force .
3.3 Surrounding Water Level
The water level surrounding the artificial upwelling device can
be obtained by introducing the dynamic free surface boundary
condition.
(3.46)
where
(3.47)
Here, it is assumed that the incoming wave level is the same as
the surrounding water level of the device, i.e., 11(t).
48
0.2
0.15
0.05
o
p
V/
VV
/'
VV
o 0.2 0.4 0.6 0.8 1 1.2Db fA.uoy
Fig.3.6 Ratio of Wave Scattering Forceand Total Exciting Force for aCylindrical Artificial Upwelling Device
49
By making this assumption, the influences of the wave
diffraction and radiation on the surrounding water level of the
artificial upwelling device are neglected.
3.4 Restoring Force
The waves acting upon the device will cause the device to
oscillate. The relative motion between waves and the oscillation of
the device will then generate the restoring force of the device.
On formulating the motion of the artificial upwelling device, the
restoring force is a dominant force among all forces, in addition, it
plays a important role in determining the natural frequency of the
artificial upwelling device.
The restoring force is the excess buoyant or gravitational force
when the floating body IS not in its neutral rest position. It
depends on the shapes of the floater and the relative motion
between the floating body and the surrounding wave, i.e., z-11.
When the device has a cylindrical floater, the restoring force IS
linear.
(3.48)
where Ao is the water level cross-sectional area of the cylindrical
device, and c is the restoring coefficient of the cylindrical device.
When the device has a cone-shape float, the restoring force is no
50
longer linear. The nonlinear restoring force can be expressed as:
(3.49)
where Ac is the water level cross-section area at rest, and Act is the
cross-section area when the device moves to the position of Z-TJ.
Thus, the restoring force of the cone-shape float can be obtained
by defining the slope of the float as S .c
(3.50)
where Dc = the diameter of Ac '
5 1
Chapter 4The Experiment of Wave and Device Interactions
In this chapter, the dynamic, kinematic, and geometric
similitudes are first considered in the laboratory work in order to
apply the experimental results with minimum errors.
The experimental equipment is set up for both radiation and
diffraction tests. Data are recorded by a data acquisition system
comprismg an analog/digital board, a Compaq 386 PC, a NoteBook®
software, and a Laser Printer.
The experiments are undertaken for both small and large
amplitude modes for both mechanical oscillation and wave
generation. The experimental results of added mass, damping
coefficient and wave exciting force are compared with the
theoretical results. The influences of the large amplitude oscillation
and large amplitude wave on added mass, damping coefficient, and
wave exciting force are then evaluated.
4.1 Feasibility of Laboratory Work
4.1.1 Wave Forces on the Device
For a model testing of the wave effect on an offshore floating
body with a long tail pipe in the deep water condition, the main
factors involved in the analysis of wave force, F, are: uO' A, p, g, T, L,
H, D, and v (maximum horizontal water particle velocity, wave
52
length, water density, gravitational acceleration, wave period,
length of long tail pipe, wave height, structure characteristic
dimension, and kinematic viscosity of water).
F = \V(uo' A, p, g, T, L, H, D, v) (4.1)
According to 1t- theorem, set uo ' p , D as three basic physical
quantities, the dimensionless wave force then can be obtained as a
function of dimensionless factors.
(4.2)
There are six dimensionless factors that influence the wave
are:factorsdimensionlessthese
forces on the floating offshore structure. By replacing D, A and L,
Froude number (u 1-liiJ),o
Keulegan-Carpenter number (KC number = UOTID), wave steepness
(H/A), diffraction parameter (D/A), tail Length ratio (LID), and
Reynolds number (uoL/v). The Froude number and Reynolds
number are dynamic considerations, KC number is a kinematic
consideration, while wave steepness, diffraction parameter, and tail
length ratio are geometric considerations. The KC number is
equivalent to the ratio of fluid particle orbit diameter to structure
diameter whereas the diffraction parameter is the ratio of the
structure diameter to wave length.
It should be noted that the equivalences of Froude number, KC
number, diffraction parameter, and tail length ratio can be satisfied
53
(4.3)
simultaneously while only the equivalence of Reynolds number is
not satisfied.
The Froude number equivalence between model and prototype is
required since the gravitational acceleration plays a key role In
wave action. In order to avoid the effect of Reynolds number miss
scaling for a model test, large diffraction parameter (Db lA, Dbuoy uoy
is the diameter of the buoy) equivalence should be used to
minimize the drag effect of wave motion. Thus, the design Db II.uoy
of the artificial upwelling device is large enough and the Reynolds
number equivalence can be neglected.
If A is the wave length in the field and A is the wave length in,p ,m
the laboratory, the linear scale factor (usually takes values between
30 and 200) is set as:
A,p(X=-
A,m
The scale factor a is selected from 40 to 80 in the laboratory
model study, so that the wave conditions in the laboratory can
cover the significant wave conditions in the real world.
4.1.2 Model Feasibility Consideration
Is Froude number similitude suitable for our model study of the
motion of the floating body?
There are two parts in this experiment that viscous effect may
54
be important. The first one is the wave drag force on the device.
The second part is the drag force both on the inner wall and outer
wall of long tail pipe.
Concerning the wave drag force, large diameter of the floating
buoy will reduce the wave drag force. The buoy diameter of the
prototype is 16 meter to 32 meter, and the wave length of a 10sec
period deep water wave is 156 meter, therefore, the Dbuo/A. ratio of
the prototype is always greater than 0.1. Under this Dbuo/A. ratio,
the wave drag force is no longer dominant according to Fig.3.6.
Therefore, wave drag force can be neglected without any
impropriety if the Dbuo/A. ratio can be preserved.
The drag effect on the pipe In this experiment includes (1) the
drag force on the outer wall of the pipe which is induced by the
oscillation of the device, and (2) the drag force on the inner wall of
the long tail pipe which is induced by the inner water flow.
The drag force on the outer wall of the long tail pipe can be
considered conservatively as the drag force on a flat plate without
the influence of curvature. It is investigated and compared with
the restoring force of the device. The restoring force is the
dominant inertial force which influences the motion of device.
The oscillation motion and velocity of the device can be
expressed as:
55
z(t)=lzl sin(rot)
Z(t)= Izi ro cos(rot)
(4.4)
(4.5)
The significant wave amplitude and period are selected as 0.75
m and 10 sec in the field. The scale factors of a= 40 is used.
According to Froude number similitude, the wave amplitude and
period in the laboratory are 0.01875 m and 1.58 sec. If the
oscillation amplitude is the same as the wave amplitude, the
velocities of both model and prototype oscillation can be obtained
as:
IZ,m' = IZ,rn/w,rn = 0.01875 * (21t/1.58) = 0.075 (m/s)
Ii. ,pi = IZ,pl ro,p = 0.75 * (21t/10) = 0.47 (m/s)
The Reynolds numbers (R e) are taken with respect to the length
of the tail pipe (L) in this investigation. The length and diameter of
the tail pipe in the laboratory are 0.6m and 0.05m while 24m and
2m are considered in the field. The kinematic viscosity is set as
1*10-6 m2/s
and density of water is 1000kg/m3
•
Jirnl L rn 4Re L = v =4.5 * 10,m
When ReL < 5* 10 5, the surface resistance of a laminar boundary
on a smooth plate can be introduced. When ReL > 107, Schlichting
56
(4.6)
5for ReL,m < 5 * 10
formula of surface resistance has better agreement with the
experimental results for the turbulent boundary layer on a smooth
plate. These two formulas can be written as follow:
2
F = 1.3 3 A pliml't,m Rel/2 pipe 2
L,m
2
F = [ 0.4 5 5 _ 1700 ] A PIi pi't,p 2.58 Re pipe 2
(Log Re) L,p10 L,p'
7for Re
L> 10,p (4.7)
The restoring force of both model and prototype are:
F R =pgA b /zml = 2 3.1 (N).m uoy,m
6F R =pgA b /zpl=I.48*10 (N),p uoy,p
The ratio of the drag force and the restoring force (F./FR) then
can be obtained for both model and prototype by introducing
Eq.(4.6) and (4.7).
-3F, m = 1.66 * 10 (N),, F't,m/FR,m = 1/13900, for model (4.8)
F't,p = 46.5 (N), for prototype (4.9)
Thus, the drag force on the outer wall of the long tail pipe is very
small compare to the restoring force both in the laboratory and in
57
the field, and thus, it is proper to neglect the factor of Reynolds
number similitude by considering the drag force on the outer wall
of the long tail pipe.
The effect of the drag force on the inner wall of the tail pipe is
considered later in chapter 8.
4.1.3 Model Application
Since the drag effects of (l) the wave on the structure and (2)
the existing of the lone tail pipe, can be neglected, the absence of
Reynolds number equivalence is not improper.
From the equivalence of Froude number, KC number and
diffraction parameter, the scale factors of various quantities for the
model work of the floating offshore structure in waves are listed as
follow:
Length a
Time a°.5
Velocity aO.s
Acceleration 1
Volume a 3
Mass a 3
Force a 3
Discharge rate a 2.5
There are two important factors that should be considered when
58
making waves In the laboratory. First one IS to generate a deep
water wave. Second one is to keep the wave steepness, H/A.,
smaller than 1/7 so that the wave would not break, or wave
steepness smaller than 0.01 so that linear wave theory can be
applied with less mistake.
The wave maker at Look Laboratory can generate the waves of:
T (wave period): O.3sec - 5.0sec.
A. (wave length): 0.2m - 4.0m
H (wave height): 1.0cm - 16cm
For deep water wave, the dispersion relationship is:
2A. = 1.56 *T (4.10)
Thus, the deep water condition is satisfied when the depth of the
tank is larger than the limiting depth of the deep water wave (i.e. h
> L/2).
The wave maker in the Look Laboratory can generate a wave
that satisfies the deep water condition when T < 1.2sec at a water
depth of 1.1 meter. Besides, under the condition of T = 1.2sec, the
wave steepness is smaller than 0.01 in the laboratory when the
wave height H < 2.25 ern. In the laboratory test, the wave period
Tc-O.Ssec is selected because the wave is more stable and it is more
closed to the wave conditions in the real world when the scale
factor is between 40 and 80.
59
For different scale factors, the corresponding wave period, wave
length and wave height of the laboratory and field for deep water
condition and linear wave steepness are:
T H
Laboratory 0.8sec - 1.2sec 1.0m - 2.25m 1.0 cm - 2.25cm
a=40
a=80
5.1sec - 7.6sec 40m - 90m
7.2sec - 10.7sec 80m - 180m
DAm - 0.9m
0.8m - 1.8m
Thus, when a ranges from 40 to 80, the wave conditions In the
laboratory can cover the design wave conditions in the real world.
In the ranges state above, when the diameter of the buoy is 40cm
In the laboratory, the ratio of Dbuo/A. is from 0.18 to 004 both in the
laboratory and in the field.
4.2 Experimental Setup and Measurement
The experiments of both radiation and diffraction tests were
carried out in a 80 x 12 x 4 ft wave tank in the Oceanographic
Engineering Laboratory of the University of Hawaii at Manoa
(shown as FigA.I). Devices with a cylindrical float and a cone-shape
float were introduced in both radiation and diffraction tests (shown
as PhotoA. I). Data were recorded by a data acquisition system with
an analog/digital board, a Compaq 386 PC, a NoteBook® software,
and a Laser printer (shown as Photo.4.2).
60
l
I~ 12 feet ~I
120 fee
" """ ,,,"" '" """" "'" """""" ,Wave Maker
Direction orWave Making
Wave Tank
Wave-,
Direction
t
ArtificialUpwelling Device./
Signal ~V/ Cables ') :,,/
./ Working Platform(
Wave Energy Dlsslpator(Wave Absorbor) .......
\..~ AID Board IControl Room
Fig.4.1 The Eeperimental Setup in the Laboratory
61
I
Photo.4.1 The Artificial Upwelling Devicewith Different Shapes of Float
PhotoA.2 The Data Acquisition Systemof the Laboratory Test
62
•
4.2.1 Radiation Test
The experiment of obtaining added mass and damping coefficient
of an artificial upwelling device is to operate the heaving
oscillations on the free surface of an calm fluid (shown as FigA.2
and PhotoA.3).
The exciting force is exerted on the device by a sinusoidal motion
generator, adequately driven to give a nearly pure harmonic motion
(shown as PhotoAA). The mechanical exciting force is measured by
a stiff strain-gauge dynamometer (PhotoA.5). The oscillation
frequency and amplitude of this sinusoidal motion generator are
adjustable. A MINARIK® speed controller (PhotoA.6) is set to
control the oscillation frequency while the oscillation amplitude can
be adjusted by moving the oscillation arm (PhotoA.7). The
oscillation amplitude is measured by oscillation arm, while the
travel peak of the device is recorded by two touch-response chips
which send out the signals when they touch each other on top of the
guide.
Thus, the motion of the device, z(t), and the mechanical exciting
force, F(t), can be obtained and they are recorded simultaneously.
The frequency (re) as well as the phase lag (0) between the motion
of the device and the mechanical exciting force then can be
obtained based on the observation of z(t) and F(t).
63
Photo.4.3 The Mechanical Heaving Oscillation on aCalm Water Surface: Radiation Test Setup
fixedworkingplatform
head 8l!!="d-,,-c_..
pressur~::::::t:-'''''''':r1gauge r::I
30cm
float which shapecan be changed
water tank
Fig.4.2 The Mechanical Heaving Oscillation On aCalm Water Surface: Radiation Test Setup
64
Photo.4.4 The Sinusoidal Motion Generator forMechanical Oscillation
65
Photo.4.5 The Stiff Strain-Gauge Dynamometerand Force Meter
66
0\-.....J
"
-~"Tb-, •.....r.-,~~
I ',,' ..•··0.,Photo.4.6 The MINARIK® Speed Controllor
for Mechanical OscillationPhoto.4.7 The Adjustable Oscillation Arm of
Mechanical Oscillation
--------------------
The motion of the model can be described by:
z(t) = Izi eiro t (4.11)
The mechanical exciting force, F(t), can be expressed as the
summation of several harmonics.
(4.12)
The kt h harmonic exciting force component (IFkl) and phase lag
«\) can be obtained by the integration of the mechanical exciting
force signal:
2 ITrklcOS Ok = T 0 F(t) sin koit d t(4.13 )
(4.14)
The added mass, damping coefficient, and oscillation frequency
of the artificial upwelling device can be expressed by non
dimensional forms as following:
mJ.1 = a
pVd
,IS
cS = bproV d ·
IS
(4.15)
(4.16)
[5=
68
(4.17)
I
where Vdis is the displaced water volume of the device, and Rbuoy is
the radius of water plane cross-sectional area of the buoy at rest.
The heaving oscillation tests have done in two phases for
different purposes.
1. Oscillating in different amplitudes: This is to find the
influence of nonlinearity caused by the oscillation of large
amplitude.
2. Oscillating by controlling the operation of inflow and outflow
valves: This is to find the influence of the inside flow and
valve opening on the radiation hydrodynamic coefficients.
Here, only the first phase is considered. The second phase will
be discussed in Chapter 8 after obtaining the inflow and outflow
velocities on the openings.
The added mass and damping coefficient of the cylindrical and
cone-shape device can be derived by using the recorded data in the
laboratory. They are discussed as follow:
4.2.1.1 Cylindrical Device
By neglecting the effect of VISCOUS damping, the equation of
motion of the cylindrical device is:
(m + rna) 2(t) + b i(t) + c z(t) =F(t) (4.18)
By assuming that only first 3 harmonic exciting forces exist, the
69
exciting force can be expressed as:
(4.19)
Besides, the added mass and damping coefficient on the left
hand side of Eq.(4.18) can be expressed by the summation of
harmonics 0, 1, and 2, i.e.:
m a = m a + m eioo t + m e i 200to at a 2 (4.20)
(4.21 )
Thus, by introducing Eq.(4.1l), (4.19), (4.20), (4.21) into
Eq.(4.18), one can derive the added mass and damping coefficient of
harmonic 0, 1, and 2 by:
- ~..£m+m a -- I 2 + 2o IZIw 0> ~
s i nOlb =o zro
(4.22)
~m a =- 2t Izlo>
where:
(4.23)
(4.24)
c = restoring coefficient = pgA bu oy
A buoy = water-plane area of the buoy.
70
4.2.1.2 Cone-Shape Device
The equation of motion of cone-shape device is different from
that of cylindrical device because the restoring force term is no
longer as simple as Eq.(4.18). For a cone-shape device with a
surface slope Sc' draft dr' bottom diameter Db' and water plane area
Ac• the restoring force (FR ) can be expressed as:
where:
(4.25)
Thus, the equation of motion of cone-shape device can be
expressed as:
where:
7t I 2d rc2 = - 2 pgs (Db + -5-)
c c
7t 2d r 2C3 = 4 Pg (Db + -5-)
c
7 I
(4.26)
By introducing Eq.(4.11), (4.19), (4.20), (4.21) into Eq.(4.26), the
added mass and damping coefficient of harmonic 0, I, and 2 can be
obtained as:
~ c3m+m a =- 2 +2
o Izlro 00
IFz/COSO z czlzlrna =- 2 +-2-
1 Izloo 00
~s i nO l
b =o z ro(4.27)
(4.28)
rna2
IF3/coS03 c 1/z1
2
=- +--Izloo2
002
b _IF3/sino32 - Izloo
(4.29)
4.2.2 Diffraction Test
The wave exciting force on the fixed device can be obtained by
generating progressive waves and recording the data of exciting
forces and the waves simultaneously (shown as FigA.3 and
PhotoA.8) while the pressure gauge and wave gauge are deployed
at the same wave front.
In order to generate the wave with wider frequency range, two
gears with different diameters are used. The wave absorber at the
end of the tank is deployed to absorb the wave energy and to
maintain the progressive wave in the wave tank (PhotoA.9).
The progressive waves pass a certain position can be obtained
and expressed as:
(4.30)
72
Photo.4.8 The Wave Exciting Force on Fixed Device:Diffraction Test Setup
fixedworkingplatfonn
FigA.3 The Wave Exciting Force on Fixed Device:Diffraction Test Setup
73
I
Photo.4.9 Wave Maker and Wave Absorber in the Wave Tank
74
I
The exciting force on the fixed device can be obtained and
expressed as:
Fe == IFel e i(rot +ip)
where , is the phase lag between wave and exciting force.
Eq(4.31) can also be expressed as:
Fe == a'11 + b'it
(4.31)
(4.32)
By comparing with Eq.(4.3l) and Eq.(4.32) and introducing
Eq.(4.30), the relationship between a', b l, 1,,1, IFel, ro and cP then can
be obtained.
4.3 Result Comparison
4.3.1 Added Mass and Damping Coefficient
In the first testing phase, the oscillation amplitude of Izl==7.0cm
(large amplitude) and Izl==2.Scm (small amplitude) are introduced In
this test. Here, small oscillation amplitude means the ratio of
oscillation amplitude and draft is 1IS (Izl/draft= 1IS), and big
oscillation amplitude means the ratio of oscilJation amplitude and
draft is 1/1.8 (lzl/draft=1/1.8).
The small amplitude represents I.Om and 2.0m oscillation
amplitudes in the field when the scale factor is ex =40 and ex = 8 0
while the large amplitude represents 2.8m and 5.6m oscillation
amplitudes in the field when the scale factor is ex=40 and ex=80.
75
For cylindrical device, the diameter of 40 em is used. The data of
z(t) and F(t) are recorded simultaneously. Through the Fourier
Transform and power spectrum analysis of F(t), the higher
harmonic mechanical exciting force can be obtained.
For the oscillation of small amplitude, the higher harmonic forces
are very small compare to the first harmonic force (see FigAA as
example). This energy spectrum of mechanical exciting force IS
generated by taking Fast Fourier Transform of the records of
mechanical exciting force. The energy spectrum is defined as the
sum of the squares of the real and imaginary parts of the Fourier
transform. This indicates that only first harmonic exciting force
should be considered. The added mass and damping coefficient can
be determined by using Eq.(4.22). For the oscillation of large
amplitude, the higher harmonic forces can be neglected as well (see
FigA.5 as example). The added mass and damping coefficient of
small amplitude oscillation is determined theoretically and
compared with the experimental results which include both small
and large amplitude oscillations. They are shown as FigA.6 and
FigA.7.
For cone-shape device, the dimension of S=2.5, L= 12.5cm and
D=35cm are used. Same as the test on cylindrical float, the test
results show that only first harmonic exciting force should be
considered when the oscillation amplitude is small (see Fig.4.8 as
example).
76
~ power of firs harmonic fora
o 2 4 S 8 10frequency, liT (l/sec)
Fig.4.4 Power Spectrum or Mechanical Exciting Forceon a Cylindrical Artificial Upwelling Devicefor Small Amplitude Oscillation
-NIIIU...
.E-
-' power of firs hannonic force~
'~
10o 2 4 6 8frequency, liT (l/sec)
Fig.4.5 Power Spectrum or the Mechanical Exciting Forceon a Cylindrical Artificial Upwelling Device forLarge Amplitude Oscillation
77
1.4
-.!l'0
~ 1.2.......
Et-rn 1II)
asa'tJQ.I 0.8'tJ'tJas
0.6
0.5-•;;>a 0.4~,0-.;
0.3dQ.I-ui:::....Q.I 0.20u
llGd- 0.1~eas'tJ
0
-- theoreticalQ small amplitude+ big amplitude
n
.~
~Q
++ <,
++ + Q G~+ r-.-++ G
~++ +
o 0.2 0.4 0.6 0.8 1 1.2 1.4
oscillation frequency, (co 2Rb
/g)aoy
Fig.4.6 Theoretical and Experimental ResultsComparison of Added Mass for aCylindrical Artificial Upwelling Device
-- theoreticalG small amplitude+ large amplitude
+ + +
+~
v
~+
+1 ~
I ~r-----o 0.2 0.4 0.6 0.8 1 1.2 1.4
oscillation frequency, (co2Rb
/g)uoy
Fig.4.7 Theoretical and Experimental ResultsComparison of Damping Coefficient for aCylindrical Artificial Upwelling Device
78
I
N~Col...~-
.J powerof firstIhannonic forcer-;-
~ .\.
o 2 4 • 8 10frequency, liT (l/sec)
Fig.4.8 Power Spectrum of Mechanical Exciting Forceon a Cone-Shape Artificial Upwelling Devicefor Small Amplitude Oscillation
-N~Col...tE-....CII~o~
-- powerof first harmonicforce....
~
o
Fig.4.9
2 4 6 8 10frequency, liT (llsec)
Power Spectrum of the Mechanical Exciting Forceon a Cone-Shape Artificial Upwelling Device forLarge Amplitude Oscillation
79
The added mass and damping coefficient can be determined by
introducing Eq.(4.22). For the oscillation of large amplitude, the
higher harmonic forces can be neglected as well (see FigA.9 as
example). Thus, the added mass and damping coefficient of the
cylindrical device can be obtained by using Eq.(4.22). The added
mass and damping coefficient of small amplitude oscillation is
determined theoretically and compared with the experimental
results which include both small and large amplitude oscillations.
They are shown as FigA.lO and FigA.Il.
The added mass and damping coefficient of the small amplitude
motion in Fig.4.6, 4.7, 4.10, and 4.11 show that the experimental
results coincide with the theoretical results of linear approach.
Thus, the result of added mass and damping coefficient can be
applied to the field for the oscillation amplitude of 1.0m of a=40
and 2.0m of «=80 without impropriety.
When the oscillation has large amplitude, Fig.4.6, 4.7, 4.10, and
4.11 show that the experimental results of the added mass are
smaller than the theoretical result, and the experimental results of
the damping coefficient are larger than the theoretical result.
For cone-shape device, the additional damping coefficient is
quite obvious (FigA.11).
Comparing with the research of Vantorre (1986) on heaving
con us of 45°, the change of added mass (~Il) and damping
80
2
1.8-.!~
~ 1.6
~.1.4-
",.
'"nf 1.2a'tJQI
1'tJ'tJ~
0.8
0.6
1.4-•:;1.2
~~,C 1-.s= 0.8QI-Uc:::.... 0.6QI0u
tlG 0.4I:l-e::l.e 0.2as'0
0
theoretical0 small amplitude+ big amplitude
0
~ 0
++ OJ"
A+ +Q
+ +~r-n ~
+~~
r---
o 0.2 0.4 0.6 0.8 1 1.2oscillation frequency. (Ci)2Rb Ig)
lIoy
Fig.4.10 Theoretical and Experimental ResultsComparison of Added Mass for aCone-Shape Artificial Upwelling Device
theoretical
" .mall amplitude+ bl, amplitude
++ +
++ ++
vor ----+ Q r--
O:Yv
o 0.2 0.4 0.6 0.8 1 1.2
oscillation frequency. (Ci)2 R Ig)buoy
Fig.4.11 Theoretical and Experimental ResultsComparison of Damping Coefficient for aCone-Shape Artificial Upwelling Device
81
I
(L\O) due to large amplitude oscillation do exist and they come from
the influence of third-order force on the first harmonic (31F).
L\m a 1 (31F
) 2L\J.L = V = 2 Re 2-; Izip dis pro V
d' Z ZIS (4.33 )
L\o= L\bpc.oV
d,IS
=(4.34)
where z" is the complex conjugate of z.
Her e, the third-order force on the first harmonic (31 F) has
already included in the experimental result of first harmonic
exciting force (F1) . According to the research of Vantorre (1986),
the third order theory for the oscillation of large amplitude can be
applied to find radiation hydrodynamic coefficients when m<1.2.
In this experimental study, tiJ<1.2 means T>O.82sec for cylindrical
device, and T>O.77sec for cone-shape device. Nevertheless, in
Vantorre's third-order research, the theoretical result did not have
good agreement with the experimental result when the oscillation
amplitude is large.
It is noted that the experimental results show an additional
damping effect exists when the device has the movement of large
amplitude oscillation. In the resonance condition, the oscillation
amplitude of the device is relatively large, thus, the additional
damping effect will decrease the oscillation amplitude so that the
pumping effect of the artificial upwelling device will be reduced.
82
4.3.2 Wave Exciting Force
The theoretical result of IFe' for both cylindrical and cone-shape
device are calculated and compared with those of the experimental
results. They are shown as Fig.4.12 and Fig.4.13.
In Fig.4.l2, the calculated wave exciting forces show good
agreement with the experimental results for both small and large
amplitude waves. Even when the wave period is T=2sec, the wave
is no longer under deep water condition, the agreement still exists.
Here, the "small amplitude" means that the wave height is smaller
than 2.25cm while "large amplitude" means that the wave height is
larger than 2.25cm which is in the range non-linear wave condition.
For the cone-shape device, Fig.4.13 shows that the experimental
result coincides with the theoretical result when the wave
amplitude is small and CJ) > 4 (T < 1.6sec). When the wave
amplitude is large, the wave exciting force tested in the laboratory
is smaller than the theoretical results. The reason for this may be
because of the stronger non-linear effect for cone-shape device
when the wave amplitude is large. This argument needs further
investigation.
83
0.08-!l'tI 0.07>
tilla.<, 0.06~CI-~ 0.05
Q)u~
~ 0.04
till= 0.03...........u>< 0.02Q)
Q)
> 0.01t1S~
0
0.1-..:a
>tlO 0.08a.<,~ ..-Q) 0.06u~
~
tlO0.04=...........u
><Q)
Q)0.02
>t1S~
0
~
\ --- theoretical0 small amplitude+ big amplitude
~\
1\+~ 0++ .......
+ + -------- ---r--
o 0.5 1 1.5 2 2.5 3 3.52wave frequency, (£0 R
b/ g)
uoy
Fig.4.l2 Theoretical and Experimental ResultsComparison of Wave Exciting Force on aCylindrical Artificial Upwelling Device
~\I
theoretical0 small amplitude
~\+ big amplitude
+ o 0\+
~0"""r---
+ + + ~+ ~ .......
~'<,
o 0.5 1.5 2 2.5 3 3.5
wave frequency, (ro2 R /g)buoy
Fig.4.13 Theoretical and Experimental ResultsComparison of Wave Exciting Force on aCone-Shape Artificial Upwellin g Device
84
Flow Motion in(I):
Chapter 5a Wave-driven Artificial Upwelling Device
Mathematical Formulation
A schematic diagram of a wave-driven artificial upwelling device
developed by this study is shown as Fig.5.1. This device consists of
a float, a long tailpipe, and two control valves. The float has a much
larger diameter than the tailpipe such that the water level inside
the float would move slower than the flow in the tailpipe. The
device oscillates by abstracting energy from ambient ocean waves,
which, in tum, cause the water column inside the device to oscillate
and produce upwelling flow.
In this study, upwelling flow is analyzed by taking the water
column inside the device as a control volume. Furthermore, the
control volume is defined in such a way that the inflow and outflow
take place only at openings which are controlled by two one-way
valves (shown as Fig.5.2). Because the control volume is attached to
the device which moves with an acceleration in a wave field, the
control volume would also move with an acceleration and a non
inertial moving coordinate must be introduced in the analysis.
The neutral position of the device is used In this study as a non
inertial moving coordinate. If the ocean is completely at rest, the
neutral position would coincide with the zero ocean water level, and
is at a distance of d above the bottom of the float. Here, d, is alsor
known as the draft of the device.
85
A. neutral position B. wave-excited condition
water level inside the device
Fig. 5.1 Schematic Diagram of a Wave-DrivenArtificial Upwelling Device
2(t) 1
L
Ifriction00 theouter wall
Fig.S.2 The Definition of Control VolumeUsed to Determine the Flow Motionof an Artificial Upwelling Device
86
(5.1 )
In general, the displacement of neutral position of the device is
z(t) which is measured from the zero ocean water level (Fig.5.2).
And thus, the acceleration of the neutral position is z(t).
5.1 Continuity Equation
Continuity equation of a non-inertial system in the artificial
upwelling device takes the following general form:
~fc.v.pdVc+(.s.PUr·dA=0
where: Vc = the controi volume
....u r = the relative velocity vector on the control surfaces
with respect to the non-inertial control volume
A = the normal area vector pointing out of the control
surface.
The first integral in Eq.(5.1) is the rate of increase of mass inside
the control surface at any time. If the water density is a constant, it
becomes:
(5.2)
where:
L = length of the tail pipeu 1 = velocity of the water level inside the floatr.
87
(5.3)
Al = cross-sectional area of the water chamber
A 2 = cross-sectional area of the inflow pipe
A 3 = cross-sectional area of the outflow opening
Assuming that the velocity distribution over the cross-sectional
areas of the inflow and outflow openings are uniform, thus, the
second integral in Eq.(5.1) becomes:
f p iir . dA = - pA 2 u 2 - pA 3 u 3c.s, r , r ,
It is noted that ,during the device operation, upper control
surface is always above the water level inside the chamber
(Fig.5.2). Therefore, no flow takes place at its upper control surface.
With these simplifications, the continuity equation takes the
form of:
(5.4)
where:
u = relative velocity of the water flow at inflow opemngr,2
u = relative velocity of the water flow at outflow openingr.3
5.2 Momentum Equation
The momentum equation for the motion of water inside the
control volume, which is attached to a non-inertial coordinate, can
be expressed as:
88
where: Fp = pressure force
F sh = shear force
Fb = body force
Assuming that the pressure distribution over the cross-sectional
area of the inflow and outflow openings are uniform, the pressure
force can be expressed as the summation of static and dynamic
pressure forces as follows:
where: Pd.2 = dynamic pressure at the inflow opening
Pd,3 = dynamic pressure at the outflow opening.
(5.6)
The shear force is proportional to the velocity square of the flow
10 the tail pipe and can be estimated by:
pu lu I, f r;2 r,21' ,F = 't * A = - * * A = P ~ * u lu Ir.in in pipe 4 2 pipe r.21 r;l
,andfA' .
/3' = pIpe8 (5.7)
where: A'. = area of the inner wall of the tail pipepipe
89
Friction factor f in Eq.(5.7) is a function of Reynolds number of
the pipe flow (Reo)' For laminar pipe flow, friction factor is related
with Reynolds number by a simple linear function. For turbulent
pipe flow, Blasius and Prandtle empirical expressions can be
introduced.
f - 0.316- 1/4
(ReJ
for Reo s 3000
5for 3000< ReD < 10
(5.8)
(5.9)
(5.10)
The body force is the gravitational force acting on entire water
body inside the control volume and can be expressed as:
(5.11)
The force F(t) is the summation of the gravitational force acting
on the water body inside the device and the force due to the
acceleration of the device:
2
F(t)=pA'l(y-z+d r)(g+ d :)dt (5.12)
The last term on the left hand side of the momentum equation is
the extra body force due to the acceleration of the control volume.
90
If the water is continuous at the connection of the float and the tail
pipe, and the entire water body inside the control volume moves
with the same acceleration z, it can be expressed as:
(5.13 )
Assuming uniform velocities at the water level 1 inside the float
and at two openings, the first integral on the right hand side of the
momentum equation can be expressed as:
;t fe.v.ur pdV=Pddt[ur,2AzL+urJAl(dr-z+ y)]
(5.14)
Similarly, the second integral on the right hand side of the
momentum equation can be expressed as:
(5.15)
With these simplifications, the momentum equation becomes:
[pg(d -z+L)+p ,2JAz+[pg(d -z)+p 3]A _p~/u 1u,2/r d r d, 3 r,2f r
2
- p[A L + A (d - z + y)] d 2Z
2 1 r d t
du ;2 du 1 2 2 2= p[A2 L d ~ + AId r ---crT- + A 1U r ,I] - pA 3 u r,3 - pA2 U r.2
9 I
(5.16)
By rearrangmg, the static force terms can be deleted. Thus, the
momentum equation can be reduced as:
P A + P A - pp'u /u /- pg y (A + A )d, 2 2 d .3 3 r ;2 r ;2 2 3
+A u2
-A3u
2,3 -Azu
Z .,J1 r ,I r r ... (5.17)
where -pgy(Az+A 3) is the additional body force and pA 'I (d r -z+y):i
IS the dynamic part of the force F(t).
The dynamic pressure on the opening, i.e. Pd,Z and Pd,3' can be
obtained from Chapter 3, and thus, they are not unknowns.
5.3 Bernoulli Equation
The Bernoulli equation between the water level in the float and
the outflow openings constitutes the third governing equation for
the movement of water inside the device. It is assumed that the
inner water surface and the outflow particle are on the same
streamline. This assumption is made because the driving force on
the outflow opening is mainly from the head difference between
inner water level and the outflow opening. Thus, it takes the form
of,
92
P U2
U2
3 r , 3 _ r J + ( + d _ )-+----- Y z'Y 2 g 2g r (5.18)
Again, it is assumed that the pressure distribution over the
cross-sectional area of the outflow opening is uniform and can be
expressed as::
P3 = pg(d , - z) + P d,3
Then, the Bernoulli equation becomes:
2 2P d, 3 u r.3 U r J--+--=-+Y
'Y 2g 2g
5.4. Coordinate Conversion and the Governing Equations
(5.19)
(5.20)
Velocities introduced above are relative to the non-inertial
moving reference coordinate z(t). They can be expressed as
absolute velocities with respect to the fixed zero water level. The
conversion of these velocities from a non-inertial moving coordinate
to a fixed coordinate can be accomplished as follows:
dy dzu =---r,l d t d t
dq> dzu =---
r,2 dt dt
d~ dzu =---r,3 dt dt
(5.21 )
(5.22)
(5.23)
where: y = the displacement of water in the water chamber with
respect to zero sea water level
93
q> = the displacement of the water in the tailpipe with
respect to zero sea water level
~ = the displacement of the water on the outflow opening
with respect to zero sea water level
By taking first derivative of u I' u 2 and u 3 in above equationsr, r.z> r,
with respect to time, similar conversion for corresponding
accelerations can be obtained.
(5.24)
(5.25)
(5.26)
Thus, the governing equations of the flow motion in a artificial
upwelling device can be written as Eq.(5.27), (5.28), and (5.29) with
respect to the zero sea water level.
A d y _ A dq> _ A d~ _ A' dz =01 dt 2dt 3dt i d t
2 2 2d y d q>, d z
A I(dr-z+Y)-2 +A2L-2 -AI(d r -z+Y)-2dt dt dt
94
(5.27)
d Y d z 2 d ~ d z 2 P d, 3(---) -(---) +2gy -2 -=0
dt dt dt dt P
5.5 Equation of Motion of the Upwelling Device
(5.28)
(5.29)
Thus far, mathematical formulation of fJow in a two-valve
artificial upwelling device has produced following three
independent equations, i.e. Eq.(5.27), (5.28), and (5.29). Because
there are four unknowns, or yet), <p (t), ~ (t) and z(t), one more
independent equation is needed. However, as the upwelling device
is heaving in a wave field, the motion of the device and the motion
of the water inside it are inter-related.
In addition to added mass, damping effect, wave exciting force,
and restoring force as discussed in Chapter 3, two extra forces are
considered. They are (1) viscous damping force due to the
movement of the tail pipe relative to surrounding ocean water, and
(2) friction force on the inner wall of the long tail pipe due to
upwelling flow in the pipe.
Ignoring the effect of curvature, the viscous damping force on
the outer wall of the tail pipe can be expressed as:
F = t *A. =C * pili/ *A. = ~ * i Iiir.out out pipe f 2 pipe
95
,andCf P A .
~ = pipe2 (5.30)
where A. is the area of the outer wall of the pipe.pipe
When ReL < 5* 10 5 • the surface resistance of a laminar boundary
on a smooth plate can be introduced. When Re L > 105, the boundary
layer becomes turbulent and the 1/7 power-law is applied. When
R eL > 107 , Schlichting formula of surface resistance has better
agreement with the experimental results for the turbulent
boundary layer on a smooth plate. These three formulas can be
written as follow:
C = 1.33f ReI/2
L
C = 0.074f Rel~ 5
5for Re < 5 * 10
L
5 7for 10 <Re <10
L
(5.31)
(5.32)
C = 0.455 _ 1700f ~58 Re
(L R LoglO eJ7
for ReL>10
(5.33)
The friction force on the inner wall of the pipe has already been
discussed earlier.
The force exerted by the water in the chamber on the device can
be analyzed as both the reaction force of F(t) and the body force F'b
as shown in Fig.S .2.
96
The force F(t) has been introduced as Eq.(5. I2), while the body
force F'b is the gravitational force on the area A'l'
(5.34)
By considering the existence of the water chamber inside the
floating, and the deep ocean water flow in the long tail pipe, the
equation of motion of the device is considered to be influenced by
the incoming waves, the viscous effect, and the water mass in the
water chamber.
The Newton's second law is introduced to set up the equation of
motion of the floating body. The motion of the floating body IS
induced by the summation of all the external forces.
m z = F - m z- b i - ~ilil + p ~'u2 - c f(z- TVe a r~
- F(t)- Fb
where m is the mass of the floating body.
(5.35)
It is noted that the summation of F(t) and F'b is the force exerts
on the device by the water in the chamber, i.e., the dynamic part of
F(t).
Rearranging Eq.(5.35), we can obtain:
2
m' U + b d z + ~ d zId z 1- p ~' U2 + C f (z - TV
dt 2 dt dt dt r~
97
(5.36)
where m'= m + rna + additional mass of water inside the device
when device is oscillation.
By introducing Eq.(5.22) into Eq.(5.36), Eq.(5.36) can be
expressed with respect to zero sea water level.
, d2z
dz dZ/dzl ' dcp dz 2m -+b-+ ~-- -p ~(---) +c f(z-T})dt 2 dt dt dt dt dt
(5.37)
5.6 Re-Arrangement of Governing Equations
Upwelling hydrodynamics of a wave-driven artificial device has
been formulated mathematically in previous sections as a set of
simultaneous differential equations, or Eq.(5.27), (5.28), (5.29), and
(5.37). In order to facilitate a mathematical solution, these
governing equations are re-arranged as a system of four ordinary
differential equations, as shown below. In these equations, four
unknowns at their highest order, or ( Cp, Y, and z are put at the
left hand side. Unknowns at lower orders and other coefficients are
put at the right hand side.
(5.38)
98
+ kIf 5 ii + k" 6 1] = f i z, cP, Z, 11 )
where:
(5.39)
(5.40)
(5.41)
A(d -z+y)k ' _ --,-I-,-r-::-_
- - A L1 2
, 2Ak 3 •
4=AL Z,2
kIf1=- 2 z , k"2 = 2g , kIf =- ~ .3 P ,
k" =-~1 m' ,
k" = ~6 m"
kIf =_~2 m' ,
kIf =£13 m"
99
k" =- .s,4 m' ,
Chapter 6Flow Motion in a Wave-driven Artificial Upwelling Device
(II): Mathematical Solution
6.1 Solution Procedure
Because three higher order unknowns stay In two ODEs, the
rearrangement of Eq.(5.38) to Eq.(5.41) are needed in order to solve
the system of ODEs. First, we can introduce Eq.(5.38) and Eq.(5.41)
into Eq.(5.39), and thus, Eq.(5.39) becomes:
(6.1)
Then, we can introduce Eq.(5.38) into Eq.(5.40), thus, the
equation of ~ can be separated from the other three equations in
the system of ODEs. Eq.(5.40) becomes:
(6.2)
By taking first derivative of Eq.(6.2) with respect to time, the
new equation with three second order unknowns <P, Y, and z , can
be 0 btai ned.
2f ·(k Y+ k cp + k i) + kIf (k Y+ k cp + k i) - 2 f . Z11 2 3 11 2 3 )
100
=> (2 fl' k 1 + kill k 1 - 2 Y- kIf1):Y + (2 fl' k 2 + kIf1k 2) ij>
+ (2 fl' k 3 + k"1k 3 - 2 f 1 + 2 y) z=k" 2 Y+ k"3 Pd ,3
By introducing Eq.(5.41) into Eq.(6.3), Eq.(6.3) becomes:
f 6 ( y, <p, i ) ij> + f 7( Y, <p, i )Y
= - (2 fl' k 3 + k"1k 3 - 2 f 1 + 2 y) . f 4 + k"2 Y+ k"3Pd, 3
= f 8 ( t, y, <p, i, z)
(6.3)
(6.4)
Now, the governing equations become one first order ODE for ~,
i.e. Eq.(5.38), and three simultaneous second order ODEs for CP, y,
and z, i.e., Eq.(6.1), (6.4), and Eq.(5.41). Therefore, the system of
three second order ODEs is solved first to find the unknowns of cp(t),
yet), and z(t). The solution can then be introduced to the single
first order ODE to solve for ~(t).
In solving this system of ODEs, the water level outside the device
IS first assumed to be the same as the incoming waves without
considering the influence of wave diffraction and radiation. Thus,
the water level outside the device can be expressed as a function of
time:
11(t)=/11I sin(rot), for a fixed ro (6.5)
The local dynamic pressure has a phase lag (<P) with the wave
and can be expressed as:
10 I
p /t) = /P dl sin(rot + 0), for a fixed ro
and P d can be expressed as:
(6.6)
(6.7)
Thus, the governing equations of the second order ODEs can be
rewritten as:
z= fit, z, <p, z)
where:
, , 2 ,f =-k f -k f -k f +f
5 24 31 412
f = 2f -k + k" k - 2 . - k"7 1 1 11 Y 1
(6.8)
(6.9)
(6.10)
The system of ODEs can then be solved by introducing the
fourth-order Runge-Kutta multi-step numerical method.
In the multi-step numerical calculation, the variables In f1 to f8
in the above equations are different in all the sub-steps of the
multi-step in order to accomplish the simultaneous relationship.
102
When solving the system of ODEs, the initial conditions are first
assumed.
yet =O)=y 0' yet =O)=Y o ;
z(t= 0)= zo' z(t =0) =i o •
(6.11 )
It is important to choose a proper time step !J. t and the proper
number of time step for all the motion of the system to reach
harmonic state.
The unknowns of <P(t), y (t), i(t), and ~(t) at the time step
(n + 1)~ t can be obtained by knowing the parameters in the time
steps of n!J.t, (n+O.5)!J.t, and (n+l)~t of a multi-step Runge-Kutta
numerical scheme. These parameters in the time steps of n!J. t,
(n+O.5)!J.t, and (n+ l)~t are Ejn , Fjn, Gjn , and Hjn•
By setting that h=!J. t, the following unknown terms can be
obtained by knowing E. , F. , G. , and H. :In In JD In
y = y + -31 (E + 2F + 2 G
2+ H
2)
n+l n 2n 2n n n
i =zn + -31
( E + 2F + 2°3
+ H3
)n s-l 3 n 3 n n n
103
~ =k Y +k cj> +k in+l 1 n+l 2 n+l 3 n-sl
Then, the other unknowns can be derived.
z =n+l
where:
z -zn+\ n
h
solving the simultaneous linear equations => ii> and y
E h M
=-'q>In 2
E h··=-. y2n 2
and
104
f 6 ( Yn + E2n , <Pn + E1n' in + E3 n ) <P
+f (y +E ,<P +E ,i +E )y7 n 2n n In n 3 n
solving the simultaneous linear equations ~ (p and y
F h=-'c:pIn 2
F h··=-. y2n 2
and
G h=-'c:pIn 2
G _h."2n - 2 Y
G3 n = ~ . f 4 ( t n + ~' in + F3 n' <Pn + FIn' Zn + Z n )
and
105
Y" n = h ( Yn + G2 n )
H _1!. ...In-2 <p
H h"=-. Y2n 2
It is noted that above derivations applied to the situation when
both inflow and outflow valves are open.
modifications are required.
Otherwise, some
6.2 Modification of the Formulation on Different Valve Operations
(l). When inflow valve is open and outflow valve is closed, the
following conditions must be introduced.
(a) u 3 = 0r,
du
(b). ~ =0
106
(2). When inflow valve is closed and outflow valve is open, the
following conditions must be introduced.
(a) u 2 =0r,
du(b). d:,2 = 0
(3) When both inflow and outflow valves are closed, the
following conditions must be introduced.
(a) ur ,3 = ur ,2 = 0
du du(b). d:,3 = d:,2 =0
The opening and closing of the inflow and outflow valves will be
discussed in the next chapter along with their influence on the
motion of the device and water flow.
107
Chapter 7Evaluation of the Performance of Artificial Upwelling
Devices with One and Two Valves
Wave-driven artificial upwelling devices developed in past
studies (Isaacs et a1., 1976; Vershinskiy et a1., 1987; and Zhai, 1992)
have rather similar design features. In general, these devices
consist of a long vertical pipe, a one-way flow valve located on the
top of the pipe, and a float to keep the entire device afloat (Figure
7.1). In this chapter, governing equations for a one-valve device
are developed by simplifying the equations for a two-valve device.
The performance of these one-valve artificial upwelling devices,
and the two-valve device developed by this study are then
evaluated and compared.
7.1 Momentum Equation for Water Movement in a One-Valve Device
Based on mathematical formulation conducted in Chapter 5,
hydrodynamic performance of a two-valve artificial upwelling
device can be analyzed in terms of four independent ordinary
differential equations with four independent unknowns. It will be
shown below that, for a one-valve device only two independent
ordinary differential equations, describing momentum conservation
principles for the movement of the water in the device and the
movement of the device itself, would be required to solve for two
unknowns. These two unknowns, which are defined relative to the
zero water level, are the vertical displacement of the water column
108
I
A. neutral position
L
B. wave-excited condition
L
Fig.7.1 Coordinate Definition for a One-ValveArtificial Upwelling Device
109
in the pipe, q>(t) and the displacement of the device, z(t), which are
shown in Fig.7.!.
In a one-valve artificial upwelling device, there is no water
chamber inside the float, and both inflow and outflow are controlled
by a single valve located at the top the vertical pipe. Therefore, the
momentum equation formulated as Eq.(5.39) in Chapter 5 for the
flow in a two-valve deice can be simplified as,
(7. I)
(7.2)
Note that q>(t) in Eq.(7.l) is the displacement of water In the long
pipe of a one-valve device and is equal to yet) in Eq.(5.39).
With a few additional simplification assumptions as given below,
Eq.(7. I) can be further reduced to the forms used by past studies:
(1). If one assume that q> does not vary with time, or q> = hs =
constant, L+d r is the length of the tail pipe, and the dynamic
pressure is pg kp(x 2)*Tl, Eq.(7.I) then becomes:
_ k p(x2)Tl-(z+h s) v~- _f_J _n
q> = L + h s g 2 D.J n
where:
f = the friction factor = f(v in)
Din = the diameter of the long pipe.
110
k p(x 2) = the dynamic pressure response factor which is
always smaller than unity.
(2). When kp(x 2)=1 and neglecting the motion of z, Eq.(7.2) will
become Eq.(2.8), or
M n- h s fricti IIcP = L g - ncnona oss
(3). When kp(x 2)=1 and considering the vacuum gap when the
valve is closed, Eq.(7.2) will take the form of Eq.(2.9) as
proposed by Zhai (1992), or
M n-z+~<p fricti IIcP = L g - rrctrona oss
(4). When the motion of the device follows the wave exactly,
k p(x 2)=O, and no frictional effect, Eq.(7.2) will take the form
of Eq.(2.6) as proposed by (Vershinskiy,1987), or
In Eq.(7.1), when the device is motionless, i.e. z=O, the equation
of motion of the water column becomes:
(7.3)
By setting L'=L+dr = the length of the pipe immersed in water,
and assuming that the damping effect in the tube is linear, Eq.(7.3)
becomes,
111
(7.4)
where:
b = linear damping coefficient.
L'= L+dr = the length of the tube immersed in water.
Eq.(7.4) is the same as the equation formulated by Isaacs (1949)
to describe the motion of the water column in a stationary pipe
under the influence of the surrounding waves.
7.2 Momentum Equation for the Movement of One-Valve Device
By following the definition in Fig.7.1, the equation of motion of
the a one-valve artificial upwelling can be expressed in the same
manner as a two-valve device, except that the term of additional
mass is different.
= a' ft + b' n , when valve is open
(m 2 + m a) Z + b z + ~zlzl + c fez - tV
= a' ft + b' 'Ii when valve is closed
(7.5)
(7.6)
In Eq.(7.5) and Eq.(7.6), additional mass is considered when the
valve is closed. Drag force due to the up-flow of the water column
exists only when the valve is open.
1 12
When the valve is closed, the VISCOUS effect does not exist and
the equation of the movement of water in the device, or Eq.(7.2)
becomes:
(7.7)
The Froude-Krylov force is used, and the dynamic pressure
response factor on the bottom of the long pipe can be expressed as:
for deep water(7.8)
The Runge-Kutta numerical method is introduced to solve the
simultaneous ordinary differential equations Eq.(7 .2), Eq.(7.5),
Eq.(7.6) and Eq.(7.7).
The valve operates with the following criteria based on the
research of Isaacs et al. (1976):
(1). the valve will open when the acceleration of the water
column is equal to or larger than that of the device.
(2). The valve will be closed when the velocity of the water
column is equal to or smaller than that of the device.
7.3. Simulation of the Pumping Effect of a One-Valve Artificial
Upwelling Device
The following three examples are used to discuss the key
characteristics of waves and the motion of floating body that
1 I 3
influence the pumping effect of a one-valve artificial upweJJing
device.
7.3.1 Valve Operation, Pipe Length and Pumping Effect
The first example is to compare the pumping effect of a
cylindrical device with two different length of tail pipe.
The selected conditions of this example are: (1) device has a
natural frequency about 1.5 times the wave frequency (T/Tn=I.5),
and (2) overhead hs is 1/8 of the float diameter (h/D buoy=I/S) (3)
draft is 1/5.2 of the float diameter (draft/Db =1/5.2).uoy
When the tail length is 5 times the float diameter (L/Dbuoy=5),
Fig.7.2 shows that the osciJIation of the device will reach steady
harmonic state after 32 cycles. After 32 cycles, the device has an
displacement about two times the wave amplitude. This bigger
oscillation amplitude will induce bigger amplitudes of acceleration
and velocity of the device as well.
Fig.7.3 and Fig.7.4 show the influence of the length of tail pipe
and the additional mass on the acceleration of the device. The
longer L is, the closer the acceleration of the water column is to zero
when the valve is open. The longer L is, the closer the <P drop is
when the valve is closed or open.
In Fig.7.3 and Fig.7.4, at the time after 32.8 cycles, i.e. t}, the
valve will open when the acceleration of the water column is equal
I 14
I
15
...~10
..E"<,
"-......c:: 5eueeuuas-Q,
~ 0
"
• wave" device, z
--0- water level, z+hJ----------1H • 1--------1
-5
32 32.5 33cycles, (tIT)
33.5 34
Fig.7.2 Comparison of Wave, Motion of Deviceand Motion of Water Level of a One-ValveArtificial Upwelling Device
115
32.5 33 33.5 34cycles, (tIT)
Fig.7.3 Acceleration or Device and Water Columnor a One-Valve Artificial Upwelling Device
When LlDbuo
=5, Ttr =1.5 and b ID ...~=118f II • ~~
--device, z"~----+-f .· ..o.... water column, ,"
4
3-.....Cl
2•r="...-~ 1~-c:s" 00-..~
-1""QI-QIUu -2eu
-3
32
4
--device, z·3 ••••0. •. - water column. ,.
-~Cl
2:I
r="...a
1<,s-c:s 00-..s
-1""QI-QIUu -2s
-3
32 32.5 33 33.5 34
cycles, (tiT)
Fig.7.4 Acceleration or Device and Water Columnof a One-Valve Artificial Upwelling Device
When UDbuo
=20, TIT =1.5 and b IDbuo
=1/8'1 II • f
116
to or larger than that of the device. This t} can be seen in Fig.7.5
and Fig.7.6 as well.
When the valve is open, the velocity of the water column and the
device is calculated and compared. Fig.7.5 and Fig.7.6 show the
influence of the length of tail pipe on the velocity of the device.
The longer L is, the longer is the opening period of the valve.
In Fig.7.5, at t2=33.4 cycles the valve is closed when the velocity
of the water column is equal to or smaller than that of the device.
In Fig.7 .6, t2=33.6 cycles which is larger than the case of the shorter
pipe. These occasions can be seen in Fig.7.3 and Fig.7.4 as well.
Thus, the device with a longer pipe has an earlier t} and a later
t2 • Therefore, increasing the length of the pipe will enhance the
pumping effect of the device. This effect is shown in Fig.7.7.
Fig. 7.8 shows the pumping rates for different tail lengths under
different wave frequencies. Here, on the x-axis, ron is the natural
frequency of the device, and on the y-axis, Qunit is defined as:
1111 . A.Q = unIt In T . =1 sec in the laboratory,
unit T ' Unitunit
and T . = va sec in the fieldunit
Fig.7.8 indicates that the device will have a pumping effect only
in a certain range of wave frequencies and longer pipe lengths. If
the tail pipe is too short, the pumping effect will be greatly reduced.
1 I 7
32.5 33 33.5 34cycles, (tIT)
Velocity of Device and Water Columnof a One-Valve Artificial Upwelling Device
When UD..._=S, Tff =1.5 and hID L •• _ =1/8_~ D 1"""
1------+----1 -- device, z'•••.g ••• , water column, "
i
j.,.e
5
4
- 3..i• 2~
~ 1>-r; 0-u0 ·1-~>
·2
·3
32
Fig.7.S
5
4
- 3..C:l
2F=" 0-.a '0."0.<,
1"0.
>->; 0w-U0 ·1-~>
·2
• device, z'• -- -0"" water column, "
·3
32 32.5 33 33.5 34cycles, (tIT)
Fig.7.6 Velocity of Device and Water Columnof a One-Valve Artificial Upwelling Device
When UDbuoy=20, TIT =1.5 and h IDL ••_ =113D I """1
118
I
• V (LlD~aGl'"'S)--o-y (LlD~aGl'-20)
J~ "fiT
I // /
J/ J/~ N::.IFo
32 32.5 33 33.5 34cycles, (tiT)
Fig.7.7 Pumping Volume Comparison for the Devicewith Different Lengths of Tail Pipein One Wave Cycle
12
-.!l 10<......aJI
E 8<,>-.
6CI.I
S=s-0 4>till
=-~ 2S~~
60
--LID -150
...,.• LID -1.5... ...,.
C --LID -2.5
if...,.
--0-- LID -5
a 40 .aGl'---+- LID -20
.aGl'--6-- LI D -40
CI.I...,.
'-JtU 30~
till.s 20PoS=s
s:a..10
00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
wave frequency, o>/roD
Fig.7.8 Pumping Rate Comparison for a One-ValveArtificial Upwelling Device with DifferentTail Lengths for b /D
buo=1/8
I 1
119
Fig.7.9 shows the maximum pumping rate for the device with
different tail lengths. It indicates that after the tail length of
L/Db
=15, the pumping rate will not increase but decrease underuoy
the selected conditions stated above.
7.3.2 Resonance and Pumping Effect
The second example is for a one-valve device oscillating under
the surrounding waves which are close to the natural frequency of
the device.
Two surrounding wave periods are selected, i.e. T/Tn=1.1 and T/
Tn =1.5, to evaluate the pumping effect. The selected conditions of
this example are: (1) length of tail pipe is L/D b u Oy=5, and (2)
overhead hs is 1/8 of the float diameter (h/Dbuo y=1/8).
Fig.7.l0 show the increase of the oscillation amplitude when the
wave period approaches the natural period of the device. Thus, the
pumping effect is greatly enhanced as seen in Fig.7.11.
7.3.3 Comparison of Different Approaches
Fig.2.3 shows the result of the research of Vershinskiy et al.
(1987). The relationship between two dimensionless factors, i.e.
pumping rate and water column acceleration, is shown as a curve.
The results of this pumping curve come from the assumptions that
the acceleration of the water column is a constant and the device
moves with the wave exactly.
120
~r-r: t-
/ ~~
p
~.
45
40...'ii
drJ
35<,
M•if 30or...al...tIO
25~-Dca 20='Dc
~ 15e
10o 10 20 30
LIDbuoy
40 50
Fig.7.9 Maximum Pumping Rate for a One-ValveArtificial Upwelling Device with DifferentTail Lengths for h lOb =118
I uoy
121
10..i:iII 8
£<,N 6
074'0a
oool- 2e,SIf
0d0"t: -2~---Col -4VI0
-6
40
Fig.7.10
• TIT -1.5• I---t----t-----i
~T/T -1.1•
40.5 41 41.5 42 42.5 43
time, (sec)
Oscillation Amplitude of One-Valve ArtificialUpwelling Device Under Different Wave Periodsfor L/Dbuoy=5 and hlD
booJ=118
Fig.7.11
40.5 41 41.5 42 42.5 43
time, (sec)
Pumping Volume of One-Valve ArtificialUpwelling Device Under Different Wave Periods
for L/Dbuoy=5 and b./Dbuoy=118
I--TIT -1.5
•--TIT -1.1•
/ / / /
1 I I II I I/ / !/
l V_ n/It Avo
40
30
.....I:l 25<:'ii:5
s: 20<,
>-. 15cue::s-0 10>tilld
oool5Q,
S::sQ,
122
In this section, the result of a hydrodynamic approach is
compared with the result of the approach of Vershinskiy et aI.
(1987). In the hydrodynamic approach, the length of tail pipe is
changed and hs is fixed (h/D b u o y=1/8) in order to simulate
different conditions of <P which is used on the horizontal axis of the
research of Vershinskiy et aI.
Fig.7.12, Fig.7.13 and Fig.7.14 show the result comparison for two
approaches at wave period of T/Tn=O.75, T/Tn=l.O and Tffn=1.6.
When the wave period is T/Tn=l, Fig.7.13 shows that the
resonance occurs, and thus, the pumping rate of the hydrodynamic
approach reaches a maximum much higher than the result of the
approach of Vershinskiy et aI.
When the wave period IS T/Tn=O.75, Fig.7.12 shows that the
pumping rates for the hydrodynamic approach are smaller than the
result of Vershinskiy et aI.'s analysis. This is mainly because the
oscillation amplitude of the device is smaller than that of the waves
under the hydrodynamic calculation.
When the wave period is T/Tn=1.6, Fig.7.14 shows good
agreement between both calculations. This is mainly because the
oscillation amplitude of the device is similar to that of the
surrounding waves. Thus, the assumptions of the approach of
Vershinskiy et al. (1987) can be accepted.
Thus, when the surrounding waves period IS much bigger than
123
Fig.7.12
\ i-- hydrodynamic model~
\ -0-- Vershlnskly et al.
I\.
\ \\ <,
I'\. r-,
~~ " r-.
Fig.7.13
~"r-. --hydrodynamic model
--Vershlnskly et al.
<, -.<,
~<,
r-,~
'w
~
1......s<a 0.8:E-d-. 0.6lU...~...coc:l 0."-~El='~
0.2lUbeCf...lU> 0~
7
.....Cl
<- 6a
:E-d 5-.lU 4...~...co 3c:l-~El 2='~
lU 1co~...lU> 0~
o
o
0.2 0.4 0.6 0.8 1 1.2 1.4
water column acceleration, (cp" /lllloo2)
Pumping Rate Comparison of Two Approacheson One-Valve Artificial Upwelling Device
for Trr =0.75 and h IDbuo
=118D .,
0.5 1 1.5 2 2.5
water column acceleration, (cp" fllli 002)
Pumping Rate Comparison of Two Approacheson One-Valve Artificial Upwelling Devicefor Trr =1.0 and b /D
buo=118
D I'
124
1
-c:l
<-0.8a
E
a-or 0.6..,cd...tills:: 0.4-Q.S~Q.
QJ 0.2tillcd
'"'GI>cd 0
\• hydrodynamic model
\ ~ Vershlnskly et al.
"~~
<,~
o 0.2 0.4 0.6 0.8 1 1.2 1.4
water column acceleration, (cp" /ITllol)
Fig.7.14 Pumping Rate Comparison of Two Approacheson One-Valve Artificial Upwelling Devicefor TIT =1.6 and h ID
bu=1/8
• • 01
125
the natural period of the device, the amplitude of the device is
similar to the waves, and the estimation of both approaches are
similar.
7.4. Simulation of the Pumping Effect of a Two-Valve Artificial
Upwelling Device
An example is used to explain some key characters of the
operation of the valves and their influences on the pumping effect.
The selected conditions of this example are: (l) the device has a
natural frequency about 2.5 times the wave frequency (T/Tn=2.5),
(2) Dch/Dbuoy= 0.625, (3) draft/Dbuoy= 1/3.2, (4) L/Dbuoy= 1.5, and
(5) Dbuo/A.= 0.18.
For a two-valve device, the operation of the inflow valve
depends on the following criteria:
(I). The valve will be open when the acceleration of the long
water column is greater than that of the device, i.e., ip> Z. The
occasion of the inflow valve opening can be defined as t=t1
•
In Fig.7.15, at t=t1 , the inflow valve is open because cP> z. Thus,
cP and cP are different from z and z after this moment. The
comparison of the velocity can be seen 10 Fig.7.16.
In Fig.7 .15, the acceleration of the device will be changed when
the inflow valve is operating because the mass of the device is
changed. The acceleration of the water column will also be changed
126
I
20
15
ode -5'tl~
~ -10~~ -15~
-20
32 32.5 33 33.5 34cycles, (tIT)
Fig.7.15 Acceleration of Device and Water Columnfor a Two-Valve Artificial Upwelling DeviceWbeII Trr.",2.5, D..ID...,,,,O.625, drartJD...,1:1I3.2,LID...,..l.5 altd D"'/)...,II.lI
10
bl.l.£ -5CI.I>
...ii:l
~
~>-
5
o
--z'••••G- •• - .'
-1032 32.5 33 33.5 34
cycles, (tIT)
Fig.7.16 Velocity of Device and Water Columnfor a Two-Valve Artificial Upwelling DeviceWhen TfT.=2.5, DdllD...,=0.625, drartlD...,'"1/3.2,LID...,=l.5 and D...,IA.=O.18
127
the velocity of the fluid particle on the outflow
The occasion of the outflow valve closing can be
because of the operation of the outflow valve when the inflow valve
IS open.
(2). The valve will be closed when the velocity of the device is
greater than that of the long water column Z > q>. This occasion of
inflow valve closing can be defined as t=t2•
In Fig.7.16, at t=t2 , the inflow valve is closed because 2><i>.
Thus, q> and q> are the same as z and i after this moment.
For a two-valve device, the operation of the outflow valve
depends on the following criteria:
(1). The outflow valve will open when the velocity of the device
IS larger than the velocity of the flow on the outflow opening, i.e.,
i > ~ . The occasion of the outflow valve opening can be defined as
t=t3·
in Fig.7 .17, at t=t3, the outflow valve is open because 2 > (
Thus, the velocity ~ is different from i after this time.
(2). The outflow valve will be closed when the velocity of the
device is less than
. . i < ~openmg, I.e., .
defined as t=t4•
In Fig.?I?, at t=t4
, the outflow valve is open because i<~
Thus, the velocity ~ is the same as i after this time.
Fig.? .18 shows the effect of the valve operation on the motion of
128
10
cv.s -6QI>
...-l:l•j:-
~>-
5
o
--z'····0···· ~.
.,'Q
.d
-1032 32.5 33 33.5 34
cycles, (tIT)
Fig.7.l7 Velocity of Device and Flow on the Outflow Openingfor a Two-Valve Artificial Upwelling DeviceWben Trr.s:2.5. D•.'D-,..0.625. drart/D-," 1/3.2.UD-,zl.5 and D-/A.=O.18
8
... 6Ci:I
E-,- 4NI~-....." 2>"-...GJ
0wlU~
-232 32.5 33 33.5 34
cycles, (tIT)
Fig.7.18 Inner Water Level Oscillation of a Two-ValveArtificial Upwelling DeviceWhen Trr.=2.5. D,.ID-,=0.625, drartlD-,=1/3.2.UD-,=l.5 and D-,/A.=O.18
129
water level in the chamber while Fig.7.19 and Fig.7.20 show the
influence of valve operation on the pumping effect.
The water level of the water chamber will remain constant when
both of the valves are closed. In Fig.7.18, at t > ~, the water level
will remain constant when both of the valves are closed.
At t > t3 in Fig.7.18, the water level is decreasing because the
inflow valve is closed and the outflow valve is open.
The water level of the water chamber will be raised again when
the outflow valve IS closed and the inflow valve is open. In
Fig.7.18, at t > t4, the outflow valve is closed while the inflow valve
IS open.
The pumping effect occurs when the outflow valve IS open.
When t3<
t< t4 , the relative outflow velocity is shown as Fig.7.19.
The pumping volume is the time integration of the outflow velocity
and the area of the outflow opening.
(7.9)
The pumping volume between t3< t< t4 is the time integration of
Fig.7.19 and area A3 which is shown on Fig.7.20.
If the time intervals t} < t< t2 and t3< t< t4 do not overlap, this
means that the two valves will not be open at the same time. Then,
the pumping volume can be obtained directly by introducing the
130
3432.5 33 33.5cycles, (tiT)
Relative Outflow Velocity or a Two- ValveArtificial Upwelling DeviceWhen Trr.=2.5, D..ID...,,,O.625, dran/D...,=1/3.2,1JD...,=1.! and D...,tM:O ...
t-~?'~., '\
~ lt-t
A
\, \,\ \
I \ / ,I • ~'-z'
V V
Fig.7.19
-12
32
-2
o
-~ -4•~> -&-b -8volU> -10
3433.533cycles, (tiT)
Pumping Volume or a Two-ValveArtificial Upwelling DeviceWbm Tff.-=2.5, D..ID...,:O.625, drartlD...,=1/3.2,1JD...,=1.5 and D...,IM:O.U
32.5
Fig.7.20
o32
20
.ILl 10a::I-0>tie
5d-~a::I
Po.
c<:C;=' 15
E
>-
131
I
product of water level difference, i.e., y-z, and the area of water
chamber.
7.5. Performance Comparison of One and Two-Valve Devices
Fig.7.21 shows the pumping rate comparison between a one
valve and two-valve artificial upwelling device under the same
length of tail pipe, the same size of the inflow pipe and the same
initial mass of the system. Here, on x-axis, rodesign is defined as:
eo . = eo (10 sec) = 121tO (radl sec) ,design
in the field
rod . = va * to (1 Osec)e srgn
_ r::- 21t=V cxTQ (radl sec),
where: cx = scale factor
in the laboratory
The overhead (h/Dbuoy) in the one-valve device is selected as 0
and 118 of the Dbuo y ' From Fig.7.21, the preference of the two-
valve device IS obvious.
Main reasons for the preference of a two-valve device are: (1)
the effect of "lift-up" in the water chamber of a two-valve device
increases the outflow velocity, and thus, the inflow velocity IS
increased, and (2) the natural frequency of a two-valve device IS
smaller than that of a one-valve device, and is closer to the
frequency of design wave.
132
I
150
• one-valve,
Y one-valve,
--<>-- two-valve
h /D -118• bUoy
h /D -0• buoy
i 100ifa
oo 0.5 1 1.5 2 2.5
wave frequency, 00/00design
3 3.5
Fig.7.21 Pumping Rate Comparison Between One-Valveand Two-Valve Artificial Upwelling Device
When LlDb
=6.25 and Db /A=O.18uoy uoy
133
•
Chapter 8Motions of Device and Inner Water Level: The
Experimental Approach
After the mathematical model is developed, a series of
laboratory model tests are conducted. In a mechanical oscillation
test and a wave exciting test, water levels in the chamber are tested
and verified, while in a free oscillation test, both the motions of the
device and the water level are tested and verified.
In addition, the influence of the flow-induced drag force In the
tail pipe is investigated as well.
8.1 Verification Tests in the Laboratory
The governing equations in Chapter 5 can be verified by a series
of laboratory model tests which will be discussed in section 8.2. In
the radiation (mechanical oscillation) test, the water flow in the
device will be studied by using different arrangement of valves. In
the diffraction (wave exciting) test, the water motion in the device
will be studied by closing the outflow opening and removing the
inflow valve. In the free oscillation test, the motion of the device
and water level in the chamber are both studied.
The governing equations for the radiation and diffraction tests
can be obtained by modifying the governing equations of Chapter 5,
and they are discussed as follows:
134
8.1.1 Radiation Test
When the floating device is subject to an assigned harmonic
motion z(t),
z(t) =Izi e-icot
the governing equations of motion of the water body could merely
use Eq.(5.38), Eq.(5.39), and Eq.(5.40). Thus, the motion of the
water column in the long tail pipe, the water level inside the water
chamber, and the outflow velocity on the outflow opening can be
solved.
The governing equations are:
(8. I)
(8.2)
(8.3)
where kO and f = the constants when z or z expresses as a
harmonic function of t.
By following the same step as Eq.(6.2) and Eq.(6.3), the system
with two second order ODEs and one first order ODE can be
obtained. Then by introducing Runge-Kutta numerical method, the
unknowns can be solved.
Two phases of the experiment are tested In the laboratory. The
135
mam purpose of these tests are to verify the feasibility of the
theoretical model and to verify the operation of the valves.
The first phase IS to test the water level variation in the water
chamber with both valves deployed. The second phase is to test the
water level variation in the water chamber without deploying the
inflow and outflow valves.
8.1.2. Diffraction Test
When the device is fixed in the wave tank without the outflow
opening and without the inflow valve, the motion of the water level
inside the water chamber can be obtained by solving Eq.(5.38), and
Eq.(5.39) with the conditions of:
z=O, z e D , and z=O.
Thus, the governing equations become:
(8.4)
(8.5)
(8.6)
By taking first derivative of Eq.(8.5) with respect to time, two
simultaneous second order equation can be obtained to solve two
unknowns, i.e., q> and y.
(8.7)
136
(8.8)
where:
A ~ p-f 1 • 2 1. 2 tJ 'I~ g d.2=--y +-<p ---<p --Y+-
2 A 2 L L A2 L L pL
By introducing the fourth-order Runge-Kutta numerical method,
the unknowns can be solved.
8.2 Model Setup and Feasibility Consideration
8.2.1 Radiation Test
Similar to the radiation test discussed in Chapter 4. the harmonic
motion z(t) is adequately given by a sinusoidal motion generator.
The relative change of water level variation. i.e., y-z, is recorded
because the wave gauge 2 moves along with the device. The
experimental setup is shown as Fig.8.1 and Photo.8.1. Here, wave
gauge I is set to check the disturbance of wave field by the
oscillation of the device.
There are two phases In this test; the first one is to deploy both
valves while the second one is to remove both valves. The changes
of water level would be different for these two test phases.
137
Photo.8.1 Tests on Water Level Variation in the Radiation Test
fixedworkingplatform
105ft
wa egauge 1
outflowcheckvalve
--Sc:nJ4-
water tank
Fig.8.l Tests on \Vater Level Variation in the Radiation Test
138
8.2.2 Diffraction Test
Similar to the diffraction test discussed in Chapter 4, the device
is fixed in relation to the waves. Wave gauge 2 is to test the inner
water variation while wave gauge 1 is set in front of the device to
record the wave data. The experimental setup is shown as Fig.8.2
and Photo.8.2. In this test, the outflow opening is absent and the
inflow valve is removed.
8.2.3 Free Oscillation Test
In this test, the device is released in the waves, and it oscillates
by following the head guide. Thus, only heaving motion is allowed.
The motion of the device is recorded by a displacement gauge on
top of the device. Wave gauge 2 is fixed on the device to record the
relative water level variation, i.e., y-z, while wave gauge 1 is set in
front of the device to record the wave data. The experimental
setup is shown as Fig.8.3 and Photo.8.3. In this test, both inflow
and outflow valves are deployed to simulate the real operation of a
wave-driven artificial upwelling device.
8.2.4 Model Feasibility Consideration
In this section, the same question as 4.1.2 is asked. Is Froude
number similitude suitable for our model study of the motion of the
water column and the device?
The drag force on the inner wall of the long tail pipe caused by
139
[r
Photo.S.2 Tests on Water Level Variation in the Diffraction Test
fixedworkingplatform
fied
head gt!ide
wave direction
wagauge I
weight
p3' water tank
Fig.S.2 Tests on Water Level Variation in the Diffraction Test
140
Photo.8.3 Tests on Water Level Variation and Oscillation Amplitudeof the Device in the Free Oscillation Test
fixedworkingplatform
wave direction
wa egauge I
displacementgauge
ot water tank
Fig.S.3 Tests on Water Level Variation and Oscillation Amplitudeof the Device in the Free Oscillation Test
141
the upwelling flow has been formulated as Eq.(5.7). Eq.(5.7) is also
the drag force on the boundary of the water column.
In Fig. 7.16, the relative velocity of the pipe flow varies from 0
to 6.0 coill/unit when Trrn=2.5, DCh/DbuOY= 0.625, draft/Dbuoy= 1/3.2,
L/D buoy =1.5 and Dbuo/A= 0.18. In this example, the drag force on
both model and prototype can be obtained when the scale factor a=
40 is used. When the relative velocity of the pipe flow is 6.0
0> 1111 't' T is I sec and Db =OAm for model test, the friction factorUnJ n uoy
can be obtained by introducing Eq.(5.9) and Eq.(5.10). The relative
velocity and the Reynolds number for model and prototype are:
u =6.0*0>1111, =6.0*221t5*0.01=0.15(m/sec)r,2,m umt.m .
u = 6.0 *co1ll1 , = 6.0 * ~ *OA = 0.9 5 (m1 sec)r,2,p u n i t.p 2.5 40
U 'l D,Re = r"",m m,m=7500
D,m v
u 2, D. 6Re = r, ~ m,p = 1.9 * 10
D,p
From Eq.(5.7), Eq.(5.9) and Eq.(5.l0), the shear force on the inner
wall of the tail pipe can be determined.
-3F't,m = 7.0 * 10 (N),
F't,p = 143 (N),
for f =0.034,m
for f =0.0105,p
Thus, the ratio of drag force and restoring force are:
142
"~,
F't,m/FR,m = 1/3300,
F't,p/FR,p = 1/10000,
for model
for prototype
(8.9)
(8.10)
Thus, the drag force on the inner wall of the long tail pipe is very
small compared to the restoring force both in the laboratory and in
the field.
Concerning the motion of the water column in the tail pipe, two
effects should be considered when the drag force is involved. The
first one is the ratio of the drag force and the driving force, and the
second one is the change of velocity profile due to the influence of
the drag force.
In the first effect, the driving force of the water column can be
estimated by assuming that the surrounding water level and water
column are motionless while the water in the chamber has the level
of Izl. By using Izl=3.1111Iunit in this example, the driving force can be
determined.
Fd
= pg A. [z] = 1.08 (N)r,m m.m ,m
Fd
= p gA, Izl = 69,340(N)r,p m.p .p
Thus, the ratio of drag force and driving force are:
F 't,p/Fdr,p = 1/488,
for model
for prototype
143
(8.11)
(8.12)
Thus, the drag force on the water column ]S still small compare
to the driving force both in the laboratory and in the field.
In the second effect, the ratio of the average to maximum
velocity, E, is introduced by setting that ur , 2 is the maximum
velocity of the velocity profile in the tail pipe. Thus,
u 2=EU "r , r... (8.13)
The Reynolds number, ReD' can be obtained.
(8.14)
The universal velocity profile in a smooth pipe can be expressed
by a power-law which is related to Reynolds number for the
turbulent pipe flow (Nikuradse, 1932).
By introducing Nikuradse empirical data for the power-law
equation, the ratio of the average to maximum velocity can be
estimated as:
C.m = 0.791 (power n=6.0), for model
£.P = 0.859 (power n=9), for prototype
(8.15)
(8.16)
Here, £.m and e.p are similar. Thus, it is proper to apply the
Froude number similitude for the test of an artificial upwelling
device.
It is noted that the velocity distribution In the tail pipe is
144
assumed to be uniform when deriving the governing equations.
8.3 Result Comparison
8.3.1 Radiation Test
There are two testing phase in the radiation test. The first
testing phase is to test the water level variation in the water
chamber with both inflow and outflow valves are deployed.
Fig.8A shows the companson of theoretical and experimental
results of the neutral position of the inner water level when both
valves deployed.
Fig.8A indicates that there is a water level pile-up in the water
chamber when the oscillation frequency is high. This pile-up
phenomenon comes from the initial imbalance between the inflow
and outflow flow rate. After several oscillation cycles, the water
level will reach its steady state and fluctuates on its pile-up neutral
position. The water level pile-up also shows that the outflow
discharge rate is smaller than the inflow pumping rate before the
steady state occurs.
Fig.8.5 shows the results of the amplitude of the inner water
level. The amplitude of the inner water level is smaller when the
oscillation frequency is high.
It is noted that the ampli tude of the water level is not directly
related to the pumping rate of the device. The pumping rate may
145
3I="
......-! 2.5•"1:1
2-NI~- 1.5~1IJ> 1ell-500 0.5ell-tC~
0-~...OJ -0.5::sell~
·1
.
I ---- theoretical IQ test data
»>.....
/ '"
s>~
~%
~
o 0.5 1 1.5 2 2.5 3 3.5
2
E......N
I
~ 1.5.cu
"::sOJ--~ 1S~
I:l0-- 0.5cf---u.,0
0
(fJ/(fJdesllD.
Fig.8.4 Neutral Position or Water Level in the Chamberin Radiation Test When Two Valves are Deployed
---- theoreticaliii test datA
\~-0 0 G
o 0.5 1 1.5 2 2.5 3 3.5(fJ/(fJ
desllD
Fig.8.S Oscillation Amplitude or Water Level in the Chamberin Radiation Test When Two Valves are Deployed
146
be large when the amplitude of the water level IS small, because
both valves may be open at the same time.
The second phase is to test the water level variation in the water
chamber when both valves are removed.
Fig.8.6 shows the comparison of theoretical and experimental
results of the neutral position of the inner water level when both
valves are removed.
The neutral position is basically zero without a pile-up
phenomenon when there is no valve. Fig.8.7 shows the water level
oscillation amplitude inside the water chamber when there is no
valve. It is noted that the water level amplitude of no-valve
oscillation is higher than the case of oscillation with valves. The
operation of the valves does influence the flow condition of the
system.
8.3.2 Diffraction Test
Isaacs (1949) set up a test to verify a theory of water level
oscillation inside a cylindrical semi-submerged pipe under waves.
Isaacs concluded that the water level inside a semi-submerged pipe
would have a magnification factor of 12 for a 2-feet pipe, and 10.8
for a I-feet pipe under the influence of the dynamic pressure of the
waves. The magnification factor is defined as the ratio of the water
column oscillation amplitude and wave height.
147
~2
'-e... 1.5":l-NI~ 1-.-"> 0.5"-...".... 0tC~-tC -0.5.......::t
"c:l-1
-- theoreticalQ test data
- ~ 0 - ~~ .~... '/ ...... -... - --
o 0.5 1 1.5 2 2.5 3oscillation frequency, m/o>
de.IID
3.5
Fig.8.6 Neutral Position of Water Level in the Chamberin Radiation Test When There is no Valve
::£ 1."......N
I
~
. 1.2
""=....--Q, 1El(If
~0 0.8-....m---v.,0 0.6
1-- theoretical 1Q test data
t»(
I~'" ---. ......
GG
G G
o 0.5 1 1.5 2 2.5 3oscillation frequency, 0>/0>
de.I,D
3.5
Fig.8.7 Oscillation Amplitude of Water Level in the Chamberin Radiation Test When There is no Valve
148
For a two-valve device in this study, the water chamber above
the long tail pipe will decrease the water level amplitude when the
resonance occurs, i.e., only approximately the ratio of AziA},
Fig.8.8 shows the comparison of the water level oscillation
amplitude of the theoretical data and the test data when there is no
outflow opening and no inflow valve. From Fig.8.8, the water level
oscillation is very small. Thus, the water level variation influenced
by the wave dynamic pressure on the bottom of the tail pipe can be
neglected in this research.
8.3.3 Free Oscillation Test
The water level variation in the water chamber and the
oscillation amplitude of the device on the wave-driven free
oscillation test is finally verified.
The theoretical result is obtained from the solution of all the
governing equations and compared with the laboratory test data.
Fig.8.9 shows the comparison of the oscillation amplitude of a
cylindrical device. The experimental data are smaller than the
theoretical data especially when the wave frequency approaches
the natural frequency of the device. This condition has two possible
reasons. The first one is the additional damping effect when the
oscillation amplitude is large, and the second one is the viscous
force of the guide which holds the device so that it moves vertically.
Neither of these effects are considered in the theoretical equations.
149
I • theoretical ICil test data
\
~I \
) <,
.....0.25(&I
>(&I-...(&I.. 0.2~
~
~
0
(&1- 0.15'0£:3'.. ---;p"--~ 0.1El~
d0- 0.05..~---v'"0
0
o 0.5 1 1.5 2 2.5TIT
D
3 3.5 ..
Fig.8.8 Oscillation Amplitude or Water Level in the Chamberin Diffraction Test When There is no Valve
10
E,N 8
.(&I
'tI:s 6..--~El~
d 40-..~.......- 2v'"0
0
~ I-- theoretical~'\ I • le""to
I \) G \
(I\.
~0
~..,.........
o 0.5 1 1.5 2 2.5 3 3.5ro/co
destiD
Fig.8.9 Oscillation Amplitude of the Cylindrical Devicein Free Oscillation Test
150
I
Fig.8.10 shows the comparison of the inner water level variation
of a cylindrical device. The water level amplitude is also influenced
by the oscillation amplitude of the device. The experimental result
IS always smaller than the theoretical result because of the
additional damping effect when the wave frequency approaches the
natural frequency of the device.
151
•
6
.£<,
5NI~
..Q.l 4
'tS:3.........-0. 3S~
-Q.l
> 2Q.l-a...Q.l.... 1t'd~
0
• theorettcal '-0 test data
r-,I \
10
[J c~-----,~~0
0 1--.
o 0.5 1 1.5 2 2.5wave frequency, ro/ro
design
3 3.5
Fig.S.IO Oscillation Amplitude of Water Level in the Chamberin Free Oscillation Test for A Cylindrical Device
152
Chapter 9Device Design and Performance Evaluation
Design variables are evaluated relative to the pumping efficiency
of an artificial upwelling device. These five design variables are:
(1) shape of the float, (2) draft of the device, (3) size of the water
chamber, (4) length of the tail pipe, and (5) buoy diameter. The
length of the tail pipe selected is within a range of 300m to 800m
which corresponds to the minimum ocean depth where nutrient
rich water occurs.
Evaluation is made under selected wave periods ranging from
3.3 seconds to 40 seconds, which represent possible wave periods
one would encounter in real ocean conditions. A constant wave
amplitude of 0.40 m is used. Under these selected wave conditions,
linear wave theory is still applicable.
By considering the nonlinear pumping increment when the wave
amplitude is large, the best combination of device variables are
eventually determined relative to a design wave condition which
has a wave period of Tdesign=lO second, and wave height of
Hdesign=1.5 meter, or, amplitude Tl design = 0.75 meters.
The nonlinear pumping increment is that the pumping rate will
increase in a nonlinear way when the wave amplitude increases.
Thus, the pumping rate under the design wave height IS
investigated by considering this nonlinear increasing factor.
153
•
9.1 The Design Variables
Before going into the discussion of the influence of the
configuration of the device on the pumping rate, the factors that
influence the oscillation amplitude of the device is first considered.
The resonance of a device in an ambient wave, which produces
the maximum oscillation amplitude, will occur if the natural
frequencies of the device and the wave are identical.
frequency of a floating structure (ron) can be defined as:
ron = Jm +cm a
where c is the restoring coefficient.
The natural
(9.1 )
The oscillation amplitude is influenced by natural frequency,
wave frequency, and damping coefficient. The magnification factor
of the oscillation amplitude is defined as the oscillation amplitude
divided by Zstatic' It can be expressed as:
where:
A =~o ron
~o = 4(m+~ )roa
[0Z =-static C
fo = the amplitude of the wave exciting force
154
(9.2)
(9.3)
(9.4 )
(9.5)
•
I-I';
Thus, the factors that influence the oscillation amplitude are: (l)
the mass and added mass of the device, (2) the restoring coefficient,
(3) the damping coefficient, and (4) wave exciting force. All these
factors will be used on the discussion of the shape of the float.
9.1.1 Shape of the Float
The device with different shapes of float will have different
added mass, damping coefficient, wave exciting force, and restoring
coefficient. Thus, the natural frequency and magnification factor
are different for the device with different shapes of float.
How to increase the magnification factor and Zstatic of the
oscillation amplitude are the first concern of choosing the shape of
an artificial upwelling device.
Fig.9.1 shows the cylindrical device and two groups of cone
shape device. All the devices have the same displacement. The
first group of cone-shape devices has the same draft while the
second group has the same water surface cross-sectional area as the
cylindrical device.
Fig.9.2 and Fig.9.3 are the comparisons of the added mass and
damping coefficient between the cylindrical device and two groups
of cor.~-shape device.
Generally speaking, the oscillation amplitude will be greatly
enhanced when the wave frequency approaches the natural
155
Dn.tc.!Dbuo,...1/3.2
Shapeof Cylindrical Float
~\ "r j::-Orlll, S-2.S
I 0.115 DbuOl --I
~\ "Eo f-S-2.0o..t.
I . o.m Dboo, • I
~\ "f j-SoLSo..t,
I• 0.775 Obuo, • I
Shapeof Cone-Shape Float: Group1
,. 0.... Db.'" '1
Shapeof Cone-Shape Float:Group2
Fig.9.1 Selected Shapes of Float
156
5
• cylindrical.. cone-I: S-2.5• cone-I: S-2.0...... 4 cone-I: S-1.5~.(
'tl • cone-2: S-2.5~ 6 cone-2: S-2.0- ---0--- cone-2: S-1.5<,Sal 3-lillillU
S 2"'0Q)
"'0"'0lU 1
oo 0.5 1 1.5 2 2.5 3
oscillation frequency, ro/rodesign
3.5
Fig.9.2 Added Mass of an Artificial Upwelling Devicewith Selected Shapes of Float
1.6..:a> 1.4
Co:9
" 1.2.0-
2.5
group
0.5 1 1.5 2oscillation frequency, ro/ (j)
dedlgn
--+-- cylindrical-----A- cone-It S-2.5 I----j-----+------I---.- cone-It S-2.0--- cone-I: S-1.5--- cone-2: S-2.5------6- cone-2: S-2.0 1----j-----::=::-=-iT------I---0-- cone-2: S-1.5
o
1
o
0.4
0.8
0.6
0.2
ws:lQ)u5Q)
ou
Fig.9.3 Damping Coefficient of an Artificial Upwelling Devicewith Selected Shapes of Float
157
I frequency if the damping coefficient is small. From Fig.9.3, the
device with a cylindrical float has the smallest damping coefficient
and thus it has the chance of obtaining a greater magnification
factor.
Fig.9.4 shows the comparison of the exciting force for the device
of two different shapes. The group 1 of the cone-shape devices has
a greater exciting force because of the greater water surface cross
section area. Nevertheless, the restoring force coefficient of the
group 1 of the cone-shape device is also greater, thus, the effect of
enhancing Zstatic is reduced.
How to decrease the natural frequency of the device so that it
will be closer to the design wave frequency is another concern in
choosing the shape of device.
In order to decrease the natural frequency of the device, a
greater mass and smaller restoring force coefficient (smaller water
surface cross sectional area) is needed. In Fig.9.2, the added mass
of the first group of the cone-shape device is greater than that of
the cylindrical device. Nevertheless, the restoring coefficient of the
first group of the cone-shape devices is greater than that of the
cylindrical device as well. For the second group of the cone-shape
device, the added mass is smaller than that of the cylindrical device
and the restoring coefficient is the same as that of the cylindrical
device. Thus, the device with a cylindrical float has the chance of
reaching smaller natural frequency.
158
• cylIndrical• cone-It S-2.S" cone-I: S-2.0 1----+----1)( cone-I: S-l.S• cone-2: S-2.Sis cone-2: S-2.0
1--_0;;:-+----+---1 ---<>- cone-2: S-l.S 1----+----1
o
0.3
eLlU"" 0.15cEbOc::-= 0.1-u>CeLl
ell 0.05>;
-III
>:; 0.25bOQ.-<,~" 0.2-
o 0.5 1 1.5 2 2.5
wave frequency, ro/rodeslgn
3 3.5
Fig.9.4 Wave Exciting Force of an Artificial Upwelling Devicewith Selected Shapes of Float
159
Before going into the comparison of the pumping effect of the
artificial upwelling device. a reference pumping rate (Q) is firstr
obtained and defined under the given configurations of the device
and the unit wave conditions, i.e., T=10 sec, and Ill'unit=OA m. They
are listed as Table.9.1.
Thus, the reference pumping rate «» is calculated as 7.2 m3/sec.
Fig.9.5 shows the pumping rate comparison between group 1 of
the cone-shape devices and the cylindrical device when the sizes of
the device are the same as the- base conditions except the length of
the tail pipe is 100m. Fig.9.6 shows the pumping rate comparison
between group 2 of the cone-shape devices and the cylindrical
device when the size of the water chamber is Dch=3 .17m, and
outflow opening Dout= 1. 17m.
Both Fig.9.5 and Fig.9.6 show that the cylindrical device has a
better pumping performance. Besides, the maximum pumping rate
of the cylindrical device occurs in the lower frequency zone which
is closer to the design wave frequency. Thus, the cylindrical device
is chosen for the artificial upwelling design.
9.1.2 Draft and Size of the Water Chamber
In this section, a device with a buoy diameter of 16m ~ Db 'uoy
and a fixed tail length L=300m are selected to investigate some
basic parameters of the design of the artificial upwelling device.
According to the selection of the buoy diameter 16m ~ Db ,so thatuoy
160
Table 9.1 The Base Conditions of Reference Pumping Rate
confi gurati ons given
shape cylindrical
Dpipe 2m
Dout 2m
draft 5m
Dbuoy 16m
Dch 10m
L 300m
161
3.532.521.5
f----+----+----f--I -- cylindrical-0-- cone-I: S-2.5----.- cone-I: S-2.0
~---+------+---~---t--.r-- cone-I: S-1.5 J-----i
lI
I 4
3.5
- 3eta 2.5-QJ~ 2tIS~
tlO 1.5s:l....Pce 1::'
I=l..
0.5
00 0.5 1
wave frequency, rol rodesign
Fig.9.S Comparison of Pumping Rate of an Artificial UpwellingDevice with Different Shapes of Float, Same Draft
and Same Volume Displacement
-c!'a
--cylindrical--- cone-2: S-2.5----.- cone-2: S-2.0
1.5 f----f-----+---f---f -0-- cone-2: S-1.5
1
0.5
oo 0.5 1 1.5 2 2.5 3 3.5
wave frequency, ro/rodesign
Fig.9.6 Comparison of Pumping Rate of an Artificial UpwellingDevice with Different Shapes of Float, Same Water PlaneArea and Same Volume Displacement
162
0.1 S Dbuo/"'design' the wave drag effect can be neglected. The tail
length L=300m is selected in order to reach the nutrient rich zone
In the deep ocean.
The draft of the device is directly proportional to the weight or
displacement of the system. When the size of the water chamber is
fixed, the initial draft of the device is proportional to the initial
mass when the device is at rest.
Another parameter is the size of the inner water chamber.
When the water chamber is small, the effect of "up-lift" of the
water in the chamber is small when the system is oscillating. When
the water chamber IS large, the additional mass in the water
chamber is big when the device is oscillating. The increasing of the
size of the water chamber results in the obvious effect of "up-lift".
The big amount of additional mass will decrease the natural
frequency of the device.
When two devices have the same draft but different size of the
water chamber, the initial mass of two devices are different. Once
they begin to oscillate, the mass of two systems are then similar,
except for the mass in the long tail pipe which is controlled by the
operation of the inflow valve.
When two devices have the same size of water chamber but
different draft, the mass of two devices are different and the
natural frequencies of two devices are different too.
163
Fig.9.7 shows the maximum pumping rate for various draft for
four different sizes of water chamber when the diameter of the
buoy is 16m. In Fig.9.7, the maximum pumping rate are similar for
all different size of water chamber. The case of D h/Db =0.625c uoy
seems better than the others. Besides, the maximum pumping rates
are also similar for devices with the same size of water chamber
but different draft.
Fig.9.8 shows the change of the frequency of the maximum
pumping rate (oomax) because of the increasing of the draft (weight)
under constant size of water chamber DchiDbuoy = 0.625, diameter of
the buoy Db =I6m, and length of the tail pipe L=300m. Fromuoy
Fig.9.8, the best draft ratio is obtained when the frequency of the
maximum pumping rate equal to the design wave frequency.
Similarly, the best draft ratios for the buoy with other diameters
under the same conditions (Dch/Dbuoy=0.625 and L=300m) can be
obtained as well. They are listed as Table.9.2.
9.1.3 Length of Tail Pipe
In this section, the influence of the length of the tail pipe is first
considered. Then, the length of the tail pipe is changed to L=800m.
In Chapter 7, Fig.7.9 shows that the length of the tail pipe is the
key control factor to increase the pumping rate when L/Db isuoy
smaller than 5 for a one-valve artificial upwelling device. For the
case of a two-valve device, this effect exists as well. Thus, the tail
164
----+-- D ID -0.5ell buoy
---+-- Dell/D....oy -0.625
--<r- D ID -0.75ell buoy
~D ID -0.875ell buoy
-40.. ..........~ -----..
- ~~~~.;:;... ....---.- 0"""
~../ -..-.... <,~ .>v -w-
\\\
<,r-,-,
-.-~
\ "-'--
1.5
0.8
5
-ef....... 4.5
~
cI-~ 4~
C'lS~
tlO
= 3.5-~S;:s~
~3
e2.5
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
draft/Dbnoy
Fig.9.7 Maximum Pumping Rate for Different Drafts and Sizesof Water Chamber of an Artificial Upwelling Device
When Dbuoy=16m
and L=300m
1.6
o 0.5 1 1.5 2 2.5draftiD
buoy
Fig.9.8 The Changing of Frequency of Maximum Pumping RateFor Different Drafts of an Artificial Upwelling Device
When Dbuoy=16m
and L=300m
165
Table 9.2 The Best Draft Ratio When L=300m
draftD
buoy
1.07
0.82
0.65
0.42
0.35
166
Dbuoy
16m
20m
24m
28m
32m
I
length at which the maximum pumping rate occurs IS first
investigated before going into the case of L=800m.
Fig.9.9 shows the maximum pumping rate of a cylindrical device
for various tail length while Dch/Dbuoy=0.625, DbuOy=16m, and
draft=5m. From Fig.9.9, the maximum pumping rate for various tail
length will reach its maximum at L/Dbuoy=8.75, and almost
approaches constant number when L/Dbuoy is greater than 10.
Fig.9.10 indicates that ro max decreases when the tail pipe is long.
Compared with the case of a one-valve device in Fig.7.8, the
influence of the length of the tail pipe on co is more obvious formax
the two-valve device.
Fig.9.1l is another plotting of Fig.9.8. Fig.9.11 shows the co formax
L=800m. In Fig.9.8 and Fig.9.11, the influence of tail length on romax
is considered. Thus, the best design draft for the tail length
L=800m is listed as Table.9.3.
Thus, the maximum pumping rate of the device under the unit
wave amplitude (llllunit=OAm) can be listed as Table.9.4 when
D ch/Dbuoy is 0.625.
It is noted that the maximum pumping rates In the above list are
under the condition of unit wave amplitude. Under the design
wave amplitude, the increasing factors of the pumping rate are
different for different draft/Db ratios.uoy
167
3.5
-cI'<,
:let 3
-ci'...CIS~
ee 2.5c:l....~
13::s~ 2
~13
1.5
~.~
}~
Ii
o 10 20 30 40 50 60LlD
b uoy
Fig.9.9 Maximum Pumping Rate for Different Tail Lengthsof an Artificial Upwelling DeviceWhen D
cb=10m,Dbuoy=16m and drafteSm
1.8eu...CIS~
ee 1.7c:l....~
Ei 1.6::s cIlO
Po iiII. "
~ S 1.5Ei .......=
'C) sa
1.4>.us=eu~ 1.3coeu~
1.2
:\\\~-,
I---- -------.
o 10 20 30LID
buoy
40 50 60
Fig.9.10 The Changing of Frequency of Maximum Pumping RateFor Different Tail Lengths of an Artificial UpwellingDevice When D
cb=10m,D
buoy=16mand draft=5m
168
1.5
1.4
1.3
1.2
1.1->.C.J 1s:lCI)
::scrCI) 0.9et:
0.8
,\\.~
~~, "<.
o 0.5 1 1.5 2 2.5draft/Dbuoy
Fig.9.11 The Changing of Frequency of Maximum Pumping RateFor Different Drafts of an Artificial Upwelling Device
When Db =16m and L=800muoy
169
Table 9.3 The Best Draft Ratio When L=800m
draft DD buoy
buoy
0.95 16m0.73 20m0.55 24m0.39 28m0.30 32m
Table 9.4 The Maximum Pumping Rate Under Unit WaveAmplitude
Dbuoy
16m16m20m20m24m24m28m28m32m32m
draftD
buoy
1.070.950.820.730.650.550.420.390.350.30
170
Maximum Pumping Rat~
(m 3/sec)
L=300m L=800m
20.6821.21
20.2620.70
19.8920.46
19.2020.09
18.8519.54
9.2 Linear Consideration
In this section, the oscillation amplitude of the device, and the
pumping rate under different frequencies are first investigated.
The increase of oscillation amplitude and pumping rate when the
wave amplitude increases is then considered.
The pumping rate is not linearly increased by increasing the
wave amplitude and increasing the oscillation amplitude of the
device.
Fig.9.12 shows an example of the relationship between the
pumping rate and the oscillation amplitude of the artificial
upwelling device.
The correlation between the oscillation amplitude and the
pumping rate is obvious. In Fig.9.12, the maximum pumping rate
occurs at the natural frequency of the device, but the pumpmg rate
is not linearly proportional to the oscillation amplitude. The
frequency bandwidth of pumping rate is wider than that of the
oscillation amplitude.
Fig.9.13 shows that the maximum pumping rate may occur at the
frequency that is smaller than the natural frequency of the device.
It is noted that the maximum pumping rate never occurs at the
frequency that is larger than the natural frequency of the device,
and the design wave frequency is always smaller than the natural
frequency of the device. And thus, the frequency of the maximum
171
~S
tool<,
~tool
d S~ a 1.0CI.I
't);:3 CI.I~~.... ~- ~~ ~
S be~ c:l....c:l ~
0 e.... ;:3~
~ ~--....vCIl0
--oscillation amplltude I~ pumping rate
J1\;;/ ~J/ \~
.»r/ ~'<,--~ ""-- --,-- --1.0
wave frequency, 001 CJ)D
Fig.9.12 Relationship Between Oscillation Amplitude andPumping Rate in a range of Frequencies: an Example
Ie•atool<, Ie
tool •~1.0
CI.I't);:3 ~
~ CI.I.... ~- ~~ a.cs be~ c:l-c:l ~0 e.... ;:3...~ ~--....vCIl0
9 oscllla tion amplitude I.., .""... - pumping rate
.' I\....:' '6
l. \: 'j '.
\.4
"" / ····l"6- ...-e ---' 00 1100 ~-- V· n
I00 /00
design n
1.0
wave frequency, 00/ 00n
Fig.9.13 The Design Wave Frequency and the Moving ofthe Frequency of the Maximum Pumping Rate:an Example
172
pumping rate (ro max) sometimes is closer to the design wave
frequency.
The increasing factor is defined as the times of increase of
oscillation amplitude and the pumping rate when the wave
amplitude increases.
Fig.9.l4 and Fig.9.15 show the increasing factor of the oscillation
amplitude and the pumping rate for various wave frequencies at
the case of ITj'unit=OAm, Dch/Dbuoy=O.625, draft/Dbuoy=O.95,
D buoy=16m and L=800m. The variable Qr l is the reference pumping
rate for unit wave amplitude but different configurations of draft,
Dbuoy and L.
From Fig.9.14, the increasing factor of the oscillation amplitude is
bigger when the wave frequency is close to the zone of natural
frequency. From Fig.9.15, the increasing factor of the pumping rate
does not seem to be regulated by different wave frequencies.
Thus, if one wants to predict the performance of the device, the
average increasing factor of the oscillation amplitude and the
pumping rate for various wave height must be investigated.
By taking an average about the selected frequency range,
Fig.9.16 shows the average increasing factor for oscillation
amplitude and pumping rate at the same conditions as Fig.9.14 and
Fig.9.15. Basically, the oscillation amplitude increases in a quasi
linear way, while the pumping rate increases in a non-linear way
173
14
12
10
...8..
N
<,
N 6
4
2
I I
• hlll ITt' ...u·2-
---0-- ITtI/ITt'''I,.3.. ITtI/ITtI....u.4
I-- II 11l1 / ITt I..1,· S
1\ ;
1\ /A\~.
~l)/ ~- -.r--
-/\/"'"-----.. .....
o 0 0.5 1 1.5wave frequency, rol ro
n
2 2.5
Fig.9.14 Increasing Factor of Oscillation Amplitude of anArtificial Upwelling Device When Wave Amplitude
Increases for 11'\1 ~=O.4m, D ./D._ =0.625, draftld =0.95,••• c ~f11 DII01
Dbtlcy=16m and L=800m
0.5 1 1.5 2 2.5
wave frequency" ro/ron
Increasing Factor of Pumping Rate of an ArtificialUpwelling Device When Wave Amplitude increasesfor Illl••i&=0.4m, D•.'Dbtocy=0.625, draft/dbU"l'=O. 9S,
Dbtlcy=16m and L=800m
1-----+----+-1 -- ITtJ/ITlI..u. 2 1-+-----1
---0-- ITlI/ITlluau-3f------+----+-i ---'I'- hTIIITll
uau-4 1-+------1
II ITlI/ITlI"al'-S
8
7
6
S...
if 4d3
2
1
00
Fig.9.15
174
12
10
8
....... ....N if'<, CY
6N
4
2
o
• oscillation ampli tude /c--- --0- pumping rate
~VV
V ./'
~.r.r/--..>: ...
/' y
o 2 4 6 8 10 121111/1111
unit
Fig.9.16 Average Increasing Factor of Oscillation Amplitudeand Pumping Rate of an Artificial Upwelling Device
for ItlIUDlt
=O.4m, D../D .....,=O.62S, drartld....,=O.9S,
D....,=16m and L=800m
175
which is only has about half of the increasing rate of wave
amplitude when the wave amplitude is large.
Similar results for the increasing factor can be obtained for
different draft/Dbuoy ratio and Dbuoy. Generally speaking, the
smaller is the draft/DbuOY ratio, the larger is the increasing factor of
the pumping rate.
9.3 Design Consideration
According to the discussion above, the draft, size of the water
chamber (Dch)' length of the tail pipe (L), and size of the buoy
(D b ) of the device are the main factors that influence theuoy
maximum pumping rate (Qmax) and the frequency of the maximum
pumping rate (ro max ) . Among these factors, the draft of the buoy
and the length of the tail pipe are the dominant factors to influence
CJ) , while the length of the tail pipe are the dominant factors tomax
infl uence Qmax.
It is noted that all these factors will influence the pumping rate
Q, but only the length of the tail pipe is the dominant factor to
influence the maximum pumping rate Qrnax.
The design wave amplitude, 111'design=Hdesign/2, is 0.75 m. Thus,
the ratio of 111ldesig/ITllunit is 1.875.
From the linear analysis, the increasing factors of the pumpmg
rate for ITJ 'design/Ill lunit are listed as Table.9.5
176
I
ITable 9.5 The Average Increasing Factors of Pumping Rate
Increasing Factor
(Q/Qrl)
D draft L=300m L=800mDbuoy buoy
16m 1.07 1.5616m 0.95 1.57
20m 0.82 1.62
20m 0.73 1.6324m 0.65 1.6424m 0.55 1.6628m 0.42 1.6728m 0.39 1.6832m 0.35 1.6832m 0.30 1.69
177
Therefore, the maximum pumping rate of the device with a
different diameter of buoy and a different length of tail pipe under
the design wave condition can be obtained and shown as Fig.9.17.
From Fig.9.17, the best diameter of the buoy of the two-valve
artificial upwelling device is about 20m for L=300m or 24m for
L=800m under the design wave conditions. Thus, from Table.9.2
and Table.9.3, the best draft of the device IS 16.4m
(draft/Dbuoy=0.82) for L=300m or 13.2m (draft/DbuOy=0.55) for
L=800m under the design wave condition.
The maximum pumping rate of the artificial upwelling device is3 332.8 m /sec for L=300m or 34 m /sec for L=800m.
It is noted that the maximum pumping rate will be smaller when
the additional damping effect is considered.
9.4 Design Criteria
From the above discussion, the best design criteria for L=300m
and L=800m can be listed as follow:
(1). The shape of the float of the artificial upwelling device IS
cylindrical.
(2). The size of the water chamber is about D h/Db =0.625.c uoy
(3). The diameter of the buoy of the artificial upwelling device is
about 20m for L=300m or 24m for L=800m under the design
wave conditions.
178
35-(J4) 34.5fI.)
<,..,a 34-
><CIS
if 33.5
4)"
33...~s..bD 32.5d....Poee 32::sPoe
>< 31.5~e
---0--- L=800m• L=300m
~~ ~
»> <,r-,/ ~
-: <...............
~
3115 20 25 30 35
D (meter)buoy
Fig.9.17 Comparison of Maximum Pumping Rate of DifferentDb and Tail Lengths of Artificial Upwelling Device
uoyUnder Design Wave Conditions
179
•
(4). The draft of the device is 16.4m for L=300m or 13.2m for
L=800m under the design wave condition.
180
Chapter 10Conclusions, Discussions and Recommendations
10.1 Conclusions and Discussions
A wave-driven artificial upwelling device is proposed and
investigated. The added mass, damping coefficient, and wave
exciting force for an artificial upwelling device with a long tail pipe
are theoretically determined by introducing the boundary element
method to solve the velocity potential on the body boundary.
A laboratory test is then conducted to verify the hydrodynamic
coefficients for both small and large amplitudes of wave and
oscillation of the device.
The motion of the internal flow is then investigated by using the
mass and momentum conservation principles, and then coupled
with the motion of the device. A numerical model is developed to
solve these simultaneous ordinary differential equations. A
laboratory test is conducted to verify the motion of the device and
the motion of the inner water level.
By defining design wave conditions, the design of an artificial
upwelling device is achieved. The configurations of the device are
carefully selected so that the maximum pumping rate would occur
under the design wave conditions.
18 I
10.1.1 Theory
1. An efficient model is developed to find the added mass and
damping coefficient for an axi-symmetric floating device with
a long tail pipe by using the boundary element method. The
calculated added mass and damping coefficient shows good
agreement with a previously developed 3-D radiation
diffraction model. The tail pipe has very little influence on
added mass and damping coefficient of the device. Besides,
another efficient model is developed to calculate the wave
exciting force on the axi-symmetric floating device by usmg
the boundary element method as well. The calculated wave
exciting force shows good agreement with the 3-D model as
well. The tail pipe has very little influence on the wave
exci ting force as well.
2. The motion of the water flow in the device can be formulated
by considering a control volume which moves in a non
inertial acceleration field. The motion of the floating body is
then coupled with the motion of the water flow. The Runge
Kutta numerical method is used to solve these simultaneous
ODEs. The governing equation is simplified and can verify the
governing equation of the one-valve device. The solution is
also verified in the laboratory with the experimental data and
good agreements are shown.
3. The influence of the valves' operation and the additional mass
182
are also considered when solving the governing equations.
The additional mass of water in the water chamber is
considered for the two-valve device once the device is excited
by the surrounding waves. The water in the long tail pipe is
counted as part of the device when the inflow valve is closed.
These additional mass of deep ocean water enable the two
valve artificial upwelling device to enhance its natural period,
and the oscillation of the two-valve device also enables the
additional mass of deep ocean water to be lifted up and to
flow out easily.
4. The factors that control the outflow velocity of both a one
valve and a two-valve device are different. For the two-valve
device, the discharge is underwater. The gravitational force
pushes out the water in the chamber when the water level in
the chamber is higher than the zero water level plus the wave
dynamic head. For the one-valve device, the discharge is
above the water surface. The discharge velocity of a one
valve device is almost equal to the velocity of the device
when the tail pipe is long. It is easy to estimate that the
outflow velocity of a two-valve device IS about 5 to 10 times
that of a one-valve device. And thus, the performance of the
two-valve device is always better.
5. The scattering wave exciting force IS larger than 17% of the
total exciting force when the ratio of Dbuo/A is larger than 0.1.
183
Thus, the influence of the wave drag force can be neglected in
this analysis. The large DbuOY is introduced for two reasons:
one is to extract the wave energy more efficiently; the other is
to apply the potential theory approach which can be verified
in the laboratory.
10.1.2 Experiment
1. The Froude number similitude is suitable for the model study
of the motion of artificial upwelling device. The diameter of
the floating buoy of the device is larger than 0.1 A so that the
wave drag effect is not dominant. The drag force on the outer
and inner wall of the pipe is only about 1/104
and 1/103 of
the restoring force, the drag force on the water column IS
about 1/102
of the driving force of the water column and they
can be neglected.
Besides, velocity profiles of the water column in the tail
pipe subject to the drag force on the boundary layer are
similar both in the model test and in the prototype. Thus, the
Froude number similitude is still applicable.
2. In the forced oscillation (radiation) test for added mass and
damping coefficient, the higher harmonic exciting forces are
very small compared with the first harmonic force for both
cylindrical and cone-shape device. The experimental result
shows good agreement with the theoretical result when the
184
Ioscillation amplitude is small. When the oscillation amplitude
is large, the added mass of the experimental result is smaller
than the theoretical result while the damping coefficient of
the experimental result is larger than the theoretical result.
The change of added mass and damping coefficient mainly
comes from the non-linear third order force on the first
harmonic when the oscillation amplitude is large.
3. In the diffraction test for the wave exciting force, for the
cylindrical device, the experimental results show good
agreement with the theoretical result even when the wave is
no longer under deep water conditions. For the cone-shape
device. the experimental result coincides with the theoretical
result when the wave amplitude is small. When the wave
amplitude is large, the wave exciting force tested in the
laboratory is smaller than the theoretical results. The reason
for this may be because of the stronger non-linear effect for
the cone-shape device when the wave amplitude is large.
This argument needs further investigation.
4. The radiation test on the water level variation shows that the
neutral position of the water level in the water chamber will
change when the valves are operating. A final balanced
neutral water level will be reached. The neutral water level
is higher when the oscillation frequency is higher. The
experimental and theoretical results show good agreement.
185
5. The diffraction test on water level variation indicates that the
resonance effect of the inner water column is greatly reduced
by the existence of the water chamber. The experimental
result matches the theoretical result on this test.
6. The oscillation amplitude of the device is recorded smaller
than the theoretical result on the free oscillation test
especially when the wave frequency approaches the natural
frequency of the device. Two possible reasons are considered.
The first one is because of the additional damping effect when
the oscillation amplitude is large, and the second one is
because of the viscous force of the guide which holds the
device to move vertically. Both of these effects are not
considered in the theoretical equations.
10.1.3 Application
1. For the two-valve device, the resonance of the water column
in the long tail pipe has a very small effect on increasing the
pumping rate.
The existing of the water chamber above the long tail pipe
will reduce the amplitude of the water column when the
resonance of the water column occurs. Nevertheless, this
design of a two-valve device can make the best use of the
ocean surface wave in lifting its floating body and water
chamber, and thus, the wave energy is directly extracted by
186
the device instead of the oscilJation of the water column In the
long tail piope.
2. The water level amplitude in the water chamber and the
pumping rate are closely related, but their relationship is
influenced by some other factors.
From the governing equations, the amplitude of water
level, i.e. y, in the chamber is influenced by:
(1). the oscillation amplitude of the device,
(2). the oscillation frequency of the device,
(3). the operation of inflow and outflow valves, i.e., the
opening occasion and flow velocities on the openings.
and the pumping rate of the device can be considered as the
ol'(~ow velocity, i.e. ( which is influenced by:
(1). the oscillation velocity of the water level in the chamber,
(2). the operation of inflow and outflow valves, i.e., the
opening occasion and flow velocities on the openings.
Thus, a larger oscillation amplitude for the device does not
result in a larger water level amplitude in the chamber. On
the other hand, a small water level amplitude in the chamber
does not mean that the pumping rate of the device is small. If
both the inflow and outflow openings are open, the oscillation
187
velocity of the water level in the chamber will be small, and
thus, the amplitude of the water level will be small too. But,
in this situation, the pumping rate is not necessarily small.
3. The condition of a small water chamber in the two-valve
device is different from the case of no water chamber in a
one-valve device.
The main difference is that the outflow condition for the
two-valve device is controlled by the water level difference
between the inner and outer water level, while for one-valve
device, it is controlled by the velocity and acceleration of the
device and water column. Thus, even when the two-valve
device is operating with a small water chamber, the pumping
rate is still larger than that of the one-valve device.
4. Pumping rate IS not linearly increased by increasing wave
amplitude. The application of linear superposition of the
pumping effect should be used with caution. Theoretically,
oscillation amplitude increases in a linear way, while the
pumping rate only has about half of the increasing rate when
the wave amplitude increases. It is noted that the oscillation
amplitude does not increase in a linear way in the laboratory
model test because of the existing of additional damping and
drag on the head guide.
5. Five factors that influence the pumping rate of a two-valve
188
device are considered. They are: draft of the device, size of
the water chamber (Deh) , length of the tail pipe (L), and size of
the buoy (Dbuoy). All these factors will influence the pumping
rate Q. For the best design consideration, the draft of the
buoy and the length of the tail pipe are the dominant factors
which influence the frequency of the maximum pumping rate
ro max' while the length of the tail pipe is the dominant factor
which influences the maximum pumping rate Qrnax.
6. The best design criteria for the device with a tail length of
L=300m and L=800m can be listed as follow:
(1). The shape of the float of the artificial upwelling device is
cylindrical.
(2). The size of the water chamber is about Deh/Dbuoy =0.625.
(3). The diameter of the buoy of the artificial upwelling device
is about 20m for L=300m or 24m for L=800m under the
design wave conditions.
(4). The draft of the device is 16.4m for L=300m or 13.2m for
L=800m under the design wave condition.
7. The experimental results show an additional damping effect
and the drag force on the guide exist when the device has the
movement of large amplitude oscillation. It is noted that this
additional damping effect is not considered in this research.
189
These effects will decrease the oscillation amplitude so that
the pumping effect of the artificial upwelling device will be
reduced. The decreasing of the pumping effect may need
further investigations.
10.2 Recommendations
1. The experiment in the laboratory was limited by the water
depth. A real deep-water wave can be made in this tank only
when the wave period is smaller than 1.2 second because the
depth of the water is only l.lm. Beyond T=1.2 second, the
wave is no longer a deep water wave and subject to bottom
friction, thus, the wave length will be smaller. A deeper wave
tank is recommended for the further laboratory approach. An
experiment in the field is another option. A field location
with a steady wave condition is preferred and may be chosen
as the testing site.
2. There is an outflow velocity on the outflow opemng which is
on the body boundary. The influence of the outflow velocity
on the hydrodynamic coefficients and the wave exciting force
should be considered when the outflow velocity is large. To
formulate the additional unsteady outflow velocity on the
body boundary need further investigations.
3. When the device has a smaller diameter of buoy (Db /A<O.l),uoy
the wave drag effect need to be considered. The effect of the
190
wave drag force on the motion of the device may reduce the
oscillation amplitude of the device. The performance of an
artificial upwelling device with a smaller buoy may be
investigated.
4. The limitation of the laboratory equipment prevent us from
observing the pumping rate directly. Thus, other options are
used in this research. Through the verification of the water
level variation and the oscillation amplitude, the pumping
rate is then predicted. Better instrumentation for the testing
of the outflow velocity IS recommended. Through verification
of the outflow velocity, the performance of the artificial
upwelling device is fully controlled.
5. The investigation of the non-linear effect on the performance
of an artificial upwelling device both on large wave amplitude
and large oscillation amplitude are needed further to more
accurately estimate the pumping rate.
191
•
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