FORCE COEFFICIENTS OF SPHERES
AND CUBES IN OSCILLATORY FLOW
WITH AND WITHOUT CURRENT
&
0
TECHNISCRE UNWERSITEIT
Scheepshydromerh caArebief
.Mekelweg2, 2628 cD Deift
Tél: 015-2-786873/Fax:2781836
DOKTOR INGENIØRAVHANDLING 1989:25 INSTITUU FOR MARIN HYDRODYNAMIKK TRONDHEIMLJNIVERSITETET I TRONDHEIM
in Oscillatory Flow with and without Current
An Experimental Study
by
Chang Li
Doktor Ingeniør Thesis
Division of Marine Hydrodynamics
Norwegian Institute of Technology
To those who gave their lives
for freedom and democracyThe present experimental study is concerned with the force coefficients of spheres
and cubes in periodical motions with and without current. These motions in-dude harmonical oscillations, circular and elliptical orbital motions. First, the experiments were carried out for the body oscillating in water at rest. Second,
the experiments were performed for the body oscillating in current flow.
A mechanical oscillating system, which consists of a hydraulic cylinder and an
oscillator, has been designed to obtain periodical motions. The body was moved
around a closed path without rotation. The in-line and lift forces were measured by a force transducer mounted inside the modeL The force coefficients were obtained through the Fourier analysis and the least square method.
In no-current conditions the drag and added mass coefficients, for both cube and
sphere in planar oscillation, were compared with the results of other authors. Comparisons between cube and sphere in planar oscillatory flow and between oscillatory and steady flow for smooth and rough spheres are presented. The measured lift force was small. The force coefficients for a smooth sphere and a cube in the orbital motion were different from those in the planar harmonical motion. In orbital motions the added mass coefficient sometimes had a negative
value.
The effects of currents on the drag and added mass coefficients, for a smooth sphere and a cube in various oscillatory motions, were discussed. The current
influences more significantly the mass coefficient than the drag coefficient.
The effect on the force coefficients due to the orbital motion is more important than that due to the current flow.
Acknowledgement
I wish to express my sincere thanks to my supervisor Professor Harald Aa. Walderhaug for his advie and support during my study. Thanks are also
ex-tended to Dr. Bjørn Sortland for many beneficial discussions and advice. I would also like to thank my colleagues and friends in the Norwegian Institute of Technology and Marintek A/S, and Mr. Ame Nervik and Mr. TerjèSotberg
for their help with the experiments.
The computer program used for the calculation of the force coecients is a mod-ified version of a program originally written by Dr. Bjørn Sortland.
This study has been financed by a scholarship from the Education Commission of People's Republic of China and The RKa1 Norwegian Ministry of Foreign Affairs. The support from the Norwegian Institute of Technology is acknowledged.
Abstract
j Acknowledge iiContents
jjjNomenclature
vi 1Introduction
1 1.1 General 1 1.2 Scope of investigation 22 Planar Oscillation-Part I
3 2.1 Introduction 3 2.1.1 General 32.1.2 Present Experimental study 5
2.2 Methods of Analysis 6
2.2.1 Mathematical Formulation 6
2.2.2 Dimensional Analysis 8
2.3 Description of the Experiments 9
2.3.1 Experimental Arrangement 9 2.3.2
Experimental Procedures ...12
2.4 Experimental Results . 18 2.4.1 Filter 18 2.4.2 Smooth Sphere 19 2.4.3 Rough Sphere 24 2.4.4 Cube 27 2.5 Summary . 313 Planar Oscillation-Part 11
323.1 New Arrangement of the Experiments 32
3.1.1 New Experimental Devices 32
3.1.2 Experimental Procedures 33
3.2 Results and Discussions 36
3.2.1 Sample Time Series 36
3.2.2 Transverse force 37
3.2.3 Added Mass and Drag Force Coefficients 37
3.2.4 Root-Mean-Square Force Coefficient and Relative Error 41 lii
3.2.5 Comparisons of Drag Coefficients between Steady and
Os-cillating Flow 45
3.2.6 Comparison between Cube and Sphere 47
3.3 Condusions 48
4 Orbital Motion
- 494.1 Introduction 49
4.1.1 General 49
4.1.2 Present Orbital Flow Study 50
4.2
Mathematical Formulation ...
524.2.1 Elliptical Orbital MotiOn 52
4.2.2 Circular Orbital Motion 53
4.2.3 Force COefficients 54
4.3 Description Of the Experiments 55
4.3.1 Experimental Arrangement 55
4.3.2 Experimental Procedures 55
4.4 Results and, Discussions 59
4.4.1 Circular Orbital Motion 59
4.4.2 Elliptical Orbital Motion 64
4.4.3 Further discussions 69
4.5 Conclusions 74
5 Effect of Currents
755.1
Introduction ...
755.1.1 General
...
755.1.2 Present Experimental Study 77
5.2 Mathematical Formulation 78
5.2.1 Motion Equation 78
5.2.2 Morison's Equation and, Force Coefficients 78 5.2.3 Root-Mean-Square Force Coefficients and Dimensional
Anal-ysis 81
5.3 Description of the Experiments 82
5.3.1 Experimental Arrangement 82
5.3.2 Experimental Procedures 82
5.4 Results and Discussions 85
5.4.1 Planar Oscillation Parallel to a Current 85
5.4.2 Planar Oscillation Perpendicular to a Current 89 5.4.3 Orbital Motion In-Line with a Current 93
5.5 Condusions 97
A Experimental Facilities
103A.l The Small Tank. 103
A.1.1 Tank.
- - 103A.1.2 Wavemalcer 103
A.1.3 Currentmaker 103
A.2 Instruments 103
A.2.1 Hyditulic Cylinder 103
A.2.2 Signal Generator No.1 104
A.2.3 Signal Generator No.2 104
A.2.4 DC Amplifier 104
A2.5 Frequency Filter 104
A.2.6 Tape Recorder 104
B Drag Force Study in Steady Flow
109B.I Low Reynolds Number. 109
B.2 Intermediate Reynolds Number 109
B.3 Large Reynolds Number 109
B.3.1 Subcritkal Reynolds Number 111
B.3.2 Critical Reynolds Number 111
B.3.3 Supercritical Reynolds Number 113
B.3.4 Transcritical Reynolds Number 113
B.4. Boundary Layer
...
113B.5 Vortex Shedding 114
B.6 Surface Roughness and Stream Turbulence Effect 114
B.7 Potential Flow 116
B.8 Sharp Edged Body 116
Nomenclature
A amplitude of oscillation
projected area of a body
Ca added mass coefficient
C added mass coefficient obtained by the least square method
Cd drag coefficient
C3
drag coefficient obtained by the least square methodC1 frictional force coefficient
CF force coefficient
CFrnaz maximum in-line force coefficient
CF,
root-mean-square in-line force coefficientCi,nna root-mean-square lift force coefficient
D sphere diameter
effective diameter of a rough sphere
I
frequency of oscillation ifiter cut-off frequencyF
total forceFd maximum drag force in a cycle
maximum inertia force in a cycle
Fm measured in-line force
F
Froude numberF
force component in x directionF
measured force in x directionF
measu±ed force in y directiong acceleration due to gravity
i unit vector parallel to x
unit vectOr parallel to y
roughness height
K
Keulegan Carpenter numberK
K=K(1+V/U)
K=K(1V/U)
M mass of a body radium of a sphere r, 9, a spherical coordinates Reynolds numbert
time in general T periodofacycle U velocity U = UrnCOS9 U velocity vector U = Ui + UjUrn amplitude of velocity
Uzm amplitude of velocity in x direction
U5
U5 = UsinO
U amplitude of velocity in y direction free-stream velocity
V volume of a body V current velocity
x,y Cartesian coordinates, the x-axis is horizontal and y-axis is vertical X
X = XmsinO
Xm amplitude of oscillation in x direction
Y
Y=Yco39
Y, amplitude of oscillation in y direction
Z
vectorZ=Xi+Yj
$
frequency parameter D2/(vT)8 2irt/T
relative error
v kinematic viscosity of water p density of water
pa body's density
(r, 9, c) velocity potential
1.1
General
The present work is devoted to an experimental investigation in order to
pro-vide data for calculating forces on subsea structures in waves and current. These forces are most conveniently described by the use of non-dimensional hydrody-naznic coefficients. The determination of the force coefficients is very important
with respect to the problem of the feasibility and regularity of the marine lifting
operations.
Due to the difficulties in calculating viscous effects numerically, experimental methods based on force measurements were adopted to predict the force coef-ficients using Morison's equation and Fourier analysis. Small-scale experiments
provide an inexpensive method of interpreting phenomena 'which are usually
sim-ilar at small and large scale. Work with numerical models will also need more basic hydrodynamical data from experiments. Numerical methods should even-tually be checked against experimental data.
There are lots of possible body shapes for the subsea structures. Smooth and rough spheres and cubes were chosen. These are blunt shapes where pressure
resistance due to separation in viscous fluid is dominating. There is a significant difference between a sphere and a cube. For the sphere there is a scale effect due
to the moving separation points. -However, for the cube the scale effect can be neglected due to the fixed separation points by the sharp edges of the body.
Spheres and cubes in steady flow have been systematically reviewed in Appendbc B.. Few papers have been published concerning a body in planar oscillatory flow, a detailed description presented in the second chapter. The problem of separated
flow about a bluff body rern.ing theoretically unresolved in planar oscillatory flow due to the effect of the shed vortices. The force coefficients are dependent
on numerous parameters such as Reynolds number, Keulegan-Carpenter number,
relative roughness, etc. The experimental results in this thesis were compared
with experimental data obtained by Sarpkaya[27, Sortland[36] and øritsland and Lehn[41].
Little research has been done on a three-dimensional body in waves or in waves and current. A body in waves experiences orbital motion of fluid particles. The vertical component of the kinematics and dynamics must be considered. The stagnation point and the wake formed over the body will rotate around the body
as the waves pass. The presence of a vertical velocity will affect the hydrodynainic
force on the body. The added mass coefficient is reduced significantly in orbital
flow.
Ocean waves in general propagate on currents, not on still water. Hydrodynamic
forces on submerged bodies are directly related to the kinematics and dynamics of
the water particles and are significantly affected by currents The superposition approximation is still widely used Based on the superposition principle the
hydrodynamic forces in the co-existing flow field can be predicted using Morison's
equation. For a body in the co-existing flow field, there is a bias to the wake structure due to the mean flow. It affects the time-dependent forces, and then
the force coefficients Thus, the force coefficientsinthe co-existing flow field may be different from those obtained under planar oscillatory flow or orbital flow.
The oscillatory flow was achieved by moving a body around a closed path without
rotation. Both planar and orbital motions were obtained by a specially designed hydraulic-mechanical oscillating system. Simulating a wave-current system by oscillating a body linearly or in an orbital path in a. current, means an approxi-mation corresponding to a linear superposition system of waves and current.
1.2
Scope of investigation
The puxpoàe of the present investigation is to study the force coefficients of
spheres arid cubesin various loading situations and the effects ofdifferent
non-dimensional parameters This will provide the basic understanding of the hydro-dynamic loading on subsea structures in the sea.
Chapter 2 and 3 describes model experiments that were perfor.aed to
evalu-ate hydrodynamic coefficients in planar oscillatory flow considering the effects of
Reynolds number, Keulegan-Carpenter number and relative roughness. Exper-imental results were also compared with those from other authors and data for
steady flow.
In chapter 4, the scope of experimental investigations were extended from planar oscillatory flow to orbital flow Both circular and elliptical motions were obtained
The different results obtained from orbital flow and planar oscillatory flow were
discussed.
In chapter 5, experiments were carried out in the co-existing flow field. The body was oscillated parallel and perpendicular to a current and in orbital motion in-line
with a current The results were expressedin drag and added mass coefficients with respect to the relative current velocity.
2.1
Introduction
2.1.1
General
Iii order to study the hydrodynamic forces acting on a three dimensional body in the sea, the problem is considered at first for a body smusoidally oscillatmg in water at rest. The study of oscillatory flow goes back hundred years ago.
Early works were usually concentrated on the motion under very small Reynolds number with no separation in the boundary layer. Stokes in 1851 gave a formula
to calculate the force acting on a sphere oscillating in a liquid with the velocity U = Awcoswt. The drag and added mass coefficients in the Mórison's form for the Stokes' sphere problem can be expressed as
Cd =
(i+
(2.1)and
C0=+/
(2.2) where 24/R,, is the steady-flow drag coefficient for a sphere in the Stokes regime and the constant 0.5 is the ideal value of the added mass coefficient for a sphere.Both the drag coefficient and the added mass coefficient are above their corre-sponding steady-state values The Stokes formula is based on the unseparated
flow. Qdar and Hamilton[23] in 1964 studied forceS on a sphere accelerating in a viscous fluid with the Reynolds number from 0 to 62 They developed an equation expressing the fluid exerted force that depends on the velocity and ac-celeration of the sphere and the history of the motion The added mass and
history coefficients in the e uation were evaluated experimentally. The foregoing
is of little importance to the studies in marine operation, but they have given us a basic understanding about the nature of planar oscillatory flow.
In 1975 Sarpkaya measured the in-line force acting on spheres in a harmonirally
oscillating flow in a U-shaped vertical tunnel with vanous Keulegan Carpenter number and Reynolds number. Since 1975 a lot of laboratory and ocean ex-periments have been carried out and theories based on various vortex methods
have been studied for two dimensional bodies such as cylinders by many authors. There has not yet been a Sound application of such theoretical methods for a three dimensional body oscillating in a fluid With laminar or turbulent separation in the
boundary layer. Systematical paperreview about the study of separated flow has
been provided by Bearnian and Graham[7] in 1980, and by Sarpkaya in 198].[30]
and 1985[31], and by Stansby arid Isaacson[37] in 1987. Certain characteristic papers in the past development leading to the present experimental study are
siunmtrized in 'the following.
Three Dimensional Condition
In 1975 Sarpkaya[27] measured the in-line force acting on smooth spheres in a harmonically oscillating flow in a U-tube water tunnel. No ortelation was found between the Reynolds number and the added mass and drag force coefficients.
The range of the testing $ number was quite limited The effect due to the
different $ number was not considered.
In 1987 øritsland and Lehn[42] studied a sphere and some other idealized subsea
structures in free-decay tests for the evaluation of the added mass and the
4amp-ing force. They. suggested that the damp4amp-ing was most conveniently described by
a linear and a quadratic damping term. There was a relatively .lage scatter in their data
Two Dimensional Condition
In 1976 Sarpkaya[28J carried out an extensive series of' experiments to measure the hydrodynamic forces on both smooth and rough cylinders in a sinusoidally oscillating flow in a U-tube water tunnel at Reynolds number up to 7.0 x 1O,
Keulegan Carpenter number up 'to 150, and relative roughness from 0.002 to 0.02.
He found that the drag and inertia force coefficients depend on the Reynolds number, K number and the relative roughness This is also presented in the
book written by Sarpkaya and Isaacson[30].
In 1979 Pearcey[24] presented some observations on fundamental features of
wave-induced viscous flows past cylinders. The comparison between steady flow and
oscillatory flow was given Effects of current on the periodic drag force for
der with planar oscillatory incident flow have been considered Horizontal cylin-der orthogonal to orbital motions has been studied. The importance of vortex
shedding was discussed.
In 1985 Bearman et al.[8] presented a comparison between theory and experiment
for the iñ..line forces on cylinders of genàal cross-section in planar oscillatory flows of small amplitude. The comparisons suggested that the theory is valid for K values below about 3 and moderately high values of the $ parameter.
In 1986 Sarpkaya[32] presented the in-line force coefficients for circular cylinders in planar oscillatory flows of small amplitude. The experimental data of the root-mean-square ralue of the in-line force coefficient fall onthe asymptotic theoretical line for K less than 9.
In 1988 Chaplin[1l] reported the loading on a smooth circular cylinder at high
harmonic motion through water initially at rest. Particular attention has been
paid to eliminate undesirable mechanical vibrations.
From the paper review above one can see that the problem has been more sys-tematically studied in the two dimensional condition. Few papers have been
found for three dimensional bodies. Some basic experimental studies in the three dimensional condition are needed.
2.1.2
Present Experimental study
This Chapter presents the in-line force coefficients for a sphere and a cube sinu-soidally oscillating in still water. A hydraulic cylinder system has been adopted to create the harmonically oscillatory motion. Morison's equation is used to analyse the hydrodynarnic force acting on the body. The drag and added mass
coefficients are obtained through the Fourier analysis or the least square method. Through simple dimensional analysis, the experiments have been systematically carried out concerning the influence of the following parameters:
f3 number, 13 = D2/(vT) where D is the diameter of the sphere, v the
kinematic viscosity of water and T the period of oscillation.
Keulegan Carpenter number, K = 2rA/D where A is the amplitude of
oscillation.
Reynolds number, R, =13. K.
Relative roughness parameter, kID where k is the diameter of the sands.
The experiments were performed at K number from 1 to 18 with R number
between 2 x i0 and 1.8 x105. The force coefficients of the smooth sphere were found to be dependent both on K number and /3 number(or R.,, number) in the present testing conditions. The force coefficients of a large scale sphere can be predicted. There is no significant correlation between the force coefficients and
K and R at K smaller than 18 and /3 larger than 9500 and the inertia f9rce
dominates. The roughness has significant effects on the drag force coefficient of the sphere. The force coefficients of the cube are independent of /3 number(or R number) due to the high turbulence level generated by the sharp edged separation of the body.
2.2
Methods of Analysis
2.2.1
Mathematical Formulation
The in-line force is assumed to be a linear combination of a velocity square dependent drag and an acceleration dependent inertia force. The drag and added mass cofficients can be obtainedthrough the Fourier analysis or the: least square
method.
The in-line force of a body osdllatiiig in water at rest can be then expressed in
the form based on Morison's equation[21]
F= CdpA,IUIU+(M+CaPV)
(2.3)in which C and C, represent respectively the drag and added mass coefficients
and U the instantaneous velocity of the oscillating body(U = U,,,cos9, with
0 = 2irt/T), V the volume of the body and M the mass of the body.
The Fourier averages of Ca and Ca can be obtained in theway given by Keulegan and Carpenter[17]. 3 ,2r FraCçSO th (2.4)
Ca_1j0
UmTAp ,2 FmT0
dO (2.5) Ca- 4a2V Jo
where Fm represents the measured force which has subtracted the mass force
component The force which is assumed to be an odd-harmonic function of 6,
can be expressed in trms of a Fourier series. caand Ca are given by the first two terms in the Fourier series.
The least square method minimizes the error between the measured force and the
calculated force from equatiàn 2.3. The drag, and added mass force coefficients are given by the least square method in theform
4 fFmCOs0Icos0I
C = --
dO (2.6)3,r Jo
Caja = Ca (2.7)
The Fourier analysis and the least square method yield identical C,, values and
the Cd values differ only slightly.
The measured rootmean-square force coefficient is defined as
Ii
2rfFm
doThe relative error between the measured maximum force and the calculated max-imum force according to Morison's equation is defined as
A Fmax(measured) - Fmcz(calculated) 2 16
-
Fm(calculated)The ratio of the maximum drag force to the maximum inertia force in a cycle can be written by Morison's equation for a sphere in the form
Fd3KCa
217FiSirCa
L )
and for a cube
F_KCd
(2.18)
CF.
(2.10)(2.11)
= +
and for a cube
CF,fl. +
The maximum force coefficient for a sphere is
cF=Ca+2g K>±
(2.12)K (2.13)
=
<and for a cube
CFma=Cd+j
K>Ea
(2.14)4irCa 2irCa
CFm
K<.
(2.15)and the measured maximum force coefficient given by
CFmaz = 2
maximum of the measured force in a cycle
}PUmAp (2.9)
The root-mean-square force coefficient and the maximum force coefficient can aiso be derived from the Morison's equation 2.3. The root-meansquare force
2.2.2
Dimensional Analysis
The hydrodynamic force, exerted on a body oscillating in water at
rest, is a
function of the body geometry, the fluid properties, the history of the flow and the instantaneous flow properties[30]. Through simple dimensional analrsis[38],the force coefficient of a sphere oscillating in water can be expresse4 as
F
i(UrnT.UrnD
UrnPk,!)
pUmAp D
v.s/DPbDT
where F is the hydrodynaxnic force, Urn the amplitude of the oscillating velocity,
A, the projected area of the body, p the water density, Pb the body's density, kID relative roughness parameter, i/T time parameter, K = UmT/D
Keulegan-Carpenter number, R, = UmD/P Reynolds number and F,, =
U//D Froude
number.
The ratio p/pb has no influence on the hydrodynamic forces since the body in-ertia force due to the sphere's absolute acceleration is subtracted from the total measured force. Foude number can be neglected in the deep water condition. To eliminate time as an independent variable, the Fourier average method is used
to get the mean value of the force coefficient. The force coefficient CF can be written as
CF=f(,UmD,,)
(2.20)CF=f(KRfl).
(2.21)If the body is a smooth sphere or a sharp edged body such as a cube, the relative roughness kID will not be takeninto account. The force coefficient CF can be rewritten in the form
CF=f(K,R.,,).
(2.22)Sarplya[28] introduced the new parameter
= Re/K = D2/(vT) called the
'frequency parameter' into the analysis of separated time-dependent flows. The
force coefficient can then be expressed a function ofKeulegan-Carpenter number
and number. That is
CF=f(K,fl).
(2.23)(2.19)
2.3
Description of the Experiments
2.3.1
Experimental Arrangement
This section describes briefly a Hydraulic Cylinder System on which the
experi-ments were carried out, together with the control system1 instrumentation, data collection and data processing arrangements. The hydraulic cylinder system is
shown in Figs. 2.1 and 2.2. The system gives the possibility to vary the fre-quency and amplitude of oscillation independently. The effects of the Reynolds number arid Keulegan-Carpenter number to the force coefficients can be then
studied separately.
The experiments were performed in a 25 meter long, 2.8 meter wide and 1 meter
deep towing tank in MTS, Trondheim. A steel frame was placed on the sides of the tank to support a hydraulic cylizder from which a test model was suspended
into the water. Achieving a high stiffness and lOw level of mechanical noise was
the dornhnt factor in the design of the experiments. The steel frame was built to have enough stiffness. The moving pipe attached to the hydraulic cylinder was guided by a hearing in order to minimi7.e the vibration of the model when the pipe oscillated during the tests. The arrangement of the bearing is shown in Figure 2.1. The amplitudes of vibration of the. system were kept iii very small
values during the experiments A detail description ofthe equipments is provided in Appendix A.
Hydraulic Cylinder System
The hydraulic cylinder system[1J which consists of a hydraulic cylinder, a
servo-control system and a potentiometer, gives the harmonically oscillatory motion. The hydraulic cylinder has a maximum stroke of 0.6 meter, a maximum force about 3500 N and a maximum velocity about 2.5 meter per second. The move-ment of the hydraulic cylinder is controlled by a servo system to produce the desired motion. The position of the hydraulic cylinder is measured by a poten-tiometer. The hydraulic cylinder is linked to the valve and pump arrangement of the wavemaker of the tank with long hoses. Thi arrangement of the hydraulic
cylinder system has problems in the following
Vibration of the hydraulic cylinder in relatively small amplitudes of
oscil-lation at low frequencies.
Irregular motion and shockwise phenomenon at the position of the maxi-mum displacement of the hydraulic cylinder in relatively large amplitudes
of oscillation at high frequencies.
Maximum available amplitude decreased with increasing the oscillating fre-quency in the range of very high frequencies.
Figure 2.1:. Arrangement of the hydraulic cylinder.
It
Wave Maker
Figure 2.2: Connection between the hydraulic system of the wavémakerand the
Figure 2.3: Arrangement of the force transducer and the model. The vibration at low frequencies is probably caused by the unsteady oil transfer in the hydraulic system. The irregular motion and shockwise phenomenonare
caused by the long hoses and some uncertainty of the system. The maximum
amplitude is reduced at high frequencies when the velocity of oscillation exceeds
its maximum value 25 rn/s. To avoid the problems mentioned above the most of the experiments were carried out in a medium range of oscillating frequencies from 0.2 Hz to 1.5 Hz. For high frequencies up to 2.5 Hz the experimentswere
only performed in relatively small amplitudes. The motions of the hydraulic cylinder, were relatively smooth arid harmonical in the range of the frequencies covered by the present experiments. Further improvement could be made by decreasing the length of the hoses.
Force Transducer
The force was measured by a strain gauge force transducer. The strain gauge was mounted on a membrane in each of the two holes of a force transducer shown
in Figure 2.3. The force transducer was then mounted on the upper part of the
steel supporting rod. The other end of the force transducer was fitted with a
pipe which was attached to the moving part of the hydraulic cylinder shown in
Figure 2.3. To protect the strain gauge from water the force transducer was
coated with Hottinger sealing compound. This sealing compound gave some kind of drifting-in after the force transducer had been loaded or unloaded, but
only a negligible effect was observed. The force transducer hac a maximum
load capacity of 100 N. The one-component force transducer measured only the vertical force , that is the in-line force acting. on the testing body. Tansverse forces are not considered in the present chapter. They will be discussed later.
Instruments
The signal which was taken from force transducer, were amplified by a DC am-plifier made by HBM. The output from the. amam-plifier is ±10 Volt(Appendix A). Because of the disturbances of high frequency, composed by theelectrical noise in the amplifier and cables and mechinical vibration of thehydraulic cylinder,
the signals from both the DC amplifier and the potentiometer were thenfiltered
through 4 pole lowpass butterworth filters. When the signals were passedthrough the filter, higher frequency components would be cut off and have phase shift as discussed by Sortland[35].
The signals from the filter were recorded in analog form on tape by a Tandberg tape recorder with output ±5 volt.
Data Acquisition
The analog, signals from the tape recorder were digitized by a A/D converter and stored on a disc-file by running the computer programs on a VAX 730 computer The sampling frequency varied as a function of the frequency of oscillation, always gave more than 100 samples per cycle The data were registered for more than 10 cycles of oscillation for each specific experiment The data were then analysed by
the computer programs using the Fourier average and the least square methods to obtain the force coefficients The final results were presented by the averaged values of the force coefficients for at least 10 cycles with the standard deviation of all coefficients provided More information about the computer programs
and detail description of the method to obtain the force coefficients refer t
Sortlánd[35].
2.3.2
Experimental Procedures
Description of Models
Two typical three-dimensional models have been chosen in the present
exper-iments Those are a sphere(D=0 1 in) with moving separation points and a
cube(a=0 1 in) with fixed separation pomts Both models are made of PVC The original PVC surface is assumed to act as a hydraulic smooth surface. The
sphere and the cube were made h011Ow to reduce the mass component of the inertia force and 'the disturbance due to the vibration of the system. A more detailed description about the models is given in Table 2.1.
Table2.1: Description of the models under Planar Oscillation-I.
A steel supporting rod with diameter 0.005 m and length 0.15 in connected the
model with the force transducer. The effects of the rod on the in-line force
measured in the experiments can be listed as follows:
Increasing the inertia force due to the mass of the rod.
Increasing the friction force due to the increasing area of the body surface by the rod.
Pressure loss due to the attached area between the rod and the sphere. Altered flow pattern around the sphere due to the presence of the rod. Altered flow pattern around the rod due to the presence of the sphere. The mass force has been subtracted from the force measured before the
calcula-tionof the force coefficients. There isno contribution from the mass of the model
to the force coefficients. The first item do not need to be considered. The effect of
the second item can be neglected for two reasons. One is that the dynamic force
is much larger than the friction force for an oscillating bluff body. The second is
that the surface area of the rod is much less than the surface area of the model. The rod is placed in the oscillating direction. The only contribution from the flow is the friction force. The effect due to the fifth item can be neglected as we have discussed the effect from the second item. The supporting rod couldhave a
significant stabilizing effect on the sphere wake. However, this does not affect the drag force signiflcantly[26j. The diameter of the rod is 0.005 m and the diameter
of the sphere is 0.1 m. As the ratio of two diameters is very small d/D =0.05,
the effects due to the third and forth items to the drag force in steady flow can be neglected as discussed by Hoerner[15]. The effects to the added mass can be neglected as presented in the book written by Sarpl.ya and Isaacson[30]. It has
Model Type Surface D k k/D Weight
m m kg sphere smooth 0.100 0.222 sphere rough 0.102 0.0010 1/100 0.268 sphere rough 0.1014 0.0007 1/140 0.258 sphere rough 0.101 0.0005 1/200 0.247 cube smooth 0.100 0.280
been assumed that the effects on the force coefficients due to the small rod would not increase significantly from steady flow tooscillating, flow.
Roughiess was made by gluing sands on the surface of the sphere. Different sizes
of sand have been chosen to create different roughness. Roughness tests were only performed with the sphere Three relativeroughnesses have been chosen, that is the relative roughness k/Dk equal to 1/100, 1/140 and 1/200 where k is the diameter of the sands grains and Dk equal to D + 2k. The sphere diameter is increased when roughness partides are glued to the sphere.
Calibration
Calibration tests were carried out to find out calibration factors both for the potentiometer andforce transducer. The potentiometer wascalibrated by reading the. voltage output from the amplifier when the hydraulic cylinder was placed at
different positions A straight hne through these pomtscalculated by least square method gave the calibration factor. The test isiilts have shown a good estimate of the calibration fattor and, the linearity of the system.
Calibration for the force transducer was done by loading weights on the model. The steel rod which connecte the model and force transducer was kept in the vertical direction during the calibration. The output voltage from theforce
trans-ducer was read when different weights were loaded and removed The maximum weight was 1 kg with 0 1 kg load increment This sequence was repeated three times. The calibration of the fOrce transducer was carried out both before and after a set of experiments. The method of least square was used to get the cal-ibration factor The calcal-ibration factors from different series of calcal-ibration tests
Table 2.2: Calibration tests on the effect of the cutoff frequency.
differed onlr slightly.
Test No.
f/Ic
Cl 5 0.5 0.4-20 0.025-L25
C2 5 1.0 0.8-20 0.050-1.25
C3 10 0.5 0.4-20 0.025-1.25
Effects of the Filter
A number of tests were performed to investigate .the effect of the filter on the
force coefficients. Two oscillatingfrequencies 0.5 Hz arid 1.0 Hz were chosen with
K numbers 5 and 10 in the tests. The cut-off frequency of the filter was changed during the tests. The arrangement of the tests is listed in Table 2.2.
Oscillation in Air
The tests of the model oscillated in air were performed in the beginning. The mass component of the inertia force was found through these tests. The tests were carried out at several different oscillating frequencies and amplitudes. The
inertia force coefficients obtained from those tests showed a good agreement with the values from the mass calculation.
Main Experiments
The test programme is described in Table 2.3 with the description of the test conditions and model types. The range of parameters covered by the current
experiments is given in Table 2.4.
The oscillations were executed more than 10 cycles from rest, and then at least 20 oscillations were recorded. The experiments were carried out in a continuous
sequence. Tests conddcted after a long period of setting were tried. Between runs
sufficient time was left for the water surface in the tank to become completely
calm. The results showed no thgniflcant departure from those obtained in a
Table 2.3: Test programme for Planar Oscillation-L
Test No. Model Type k/D
I
Icf/Ic
Hz Hz 1 sphere 0.2 1.0 0.2 2 sphere 0.35 2.0 0.175 3 sphere 0.6 4.0 0.15 4 sphere 0.8 4.0 0.2 5 . sphere 1.0 4.0 0.25 6 sphere 0.2-2.5 2,4,8,10 0.1-0.25 7 sphere 0.2-2.0 2,4,8. 0.1-0.25 8 sphere . 1/100 0.2-2.2 2,4,8 0.10.28 9 sphere 1/100 0.2-2.0 2,4,8 0.1-0.25 10 sphere 1/140 0.2-2.5 4,8,10 0.1-0.25 11 sphere 1/140 0.2-2.0 2,4,10 0.1-0.20 12 sphere 1/200 0.2-2.5 4,8,10 0.1-0.25 13 sphere 1/200 0.2-2.0 2,4,8 0.1-0.25 14 cube 0.35 2.0 0.175 15 cube 0.8 8.0 01 16 cube 1.0 8.0 0.125
Table 2.4: Range of parameters covered by the experiments under Planar Oscil-lation-I. TestNo. K $ 10 iO 1 1.2-16.3 0.2-3.1 1.9 2 1.0-15.6 0.3-5.2
34
3 1.0-16.8 0.6-9.6 5.7 4 1.2-17.8 0.9-13.5 7.6 5 1.0-17.4 1.0-16.4 9.5 6 5 0.9-10.7 1.9-23.4 7 10 1.9-17.6 1.9-19.2 8 5 0.9-9.3 1.9-20.1 9 10 1.9-17.6 1.9-19.1 10 5 0.9-10.6 1.9-23.3 11 10 1.9-17.7 1.9-19.2 12 5 0.9-92 1.9-19.2 13 10 1.9-17.7 1.9-19.2 14 1.3-17.1 0.4-5.6 3.4 15 1.2-16.6 0.9-12.6 7.6 16 1.2-16.1 1.1-15.2 9.52.4
Experimental Results
The experimental results in planar oscillation are plotted in figures presented in this chapter. The variation of the force coefficients with K at constant values
of /3 was provided by a series of experiments conducted at constant values of osdilatory frequency. For the smooth and rough spheres the force,coefficients were also presented with R,,, number at constant values of K by keeping, constant
oscillating amplitudes during the tests.
The force coefficients presented here are: mean values with standard deviations over at least 10 cycles of data. The,force coefficients and some. other important
parameters are also given in table form by running the computer program by
SOrtland[35]. An example of the results at Test No.3 is given in Appendix C
Tables C.1 and C.2. In general the values of the standard deviations f Ca, Cd, CF, and CF are smaller than 0.05 at K larger than 2, and up to 0.20 for
C, Cd and CF, and 0.50 for CFm at K smaller than 2. The. large values of
the standard deviations of the force coefficients at low K are m'rn1y due to the
rise of the noise-to-signal ratio at small amplitudes oscillation. The scatter of the
data from cycle to cycle becomes smaller at larger K number.
2.4.1
Filter.
The effect of the cut-off frequency of the ifiter. to various coefficients has been shown in Appendix A from Figure A.3 to Figure A.7, wherethe' coefficients are
made non-dimensional by dividing the value of the coefficients at f/fe = 0 25
where f is the frequency of the oscillation and f the cut-offfrequency of' the
filter All of the present experiments were carried out at f/fe < 03 The filter has no significant effects on any of the coefficients at different cut-off frequencies
for f/fe <0.3 except to the maximum force coefficient. Figure A.6 shows large scatter Of the maximum force coefficient for various cut-off frequencies. It means
that the maximum force coefficient is correlated to the high frequency compo-nents of the force measured. The high frequency compocompo-nents of the measured
force consist of mi%inly two parts. One is the high frequency componentsof the
hydrodynainic force of the surrounding flow acting on the modelwhich should
be counted in the maximum force coefficient calculation. The other is the high frequency noise due to the mechanical vibration and electric disturbance which should not be included in the maximum force coefficient calculation. The use
of the filter has cut off the amplitudes of all the highfrequency components and mtroduced a phase shift of the high frequency components which may in-clude useful parts of the hydrodynainic force component. The mai4mumforce coefficient calculated from the measured force might therefore be inaccurate.
2.4.2
Smooth Sphere
Drag and Added Mass Coefficients
Drag and added mass coefficients for the five values of are plotted against K in Figs 2.4 and 2.5. Figures 2.6 and 2.7 show the data obtained as a function of R,. at three constant values of K. The experimental results show that the scale effect for a sphere is quite large. The force coefficients are dependent on both, parameters K and &(or ) Small correlation between the force coefficients and
K number is found for larger number. For large scale spheres in general sea
condition Ca could be expected to take the values around 0.5 and Ca in the order
of 0.1, the effect of K and R numbers is small.
The added mass coefficient in Figure 2.4 shows that Ca decreases with increasing
K number. The experimental data in the figure give a very little scatter for K lower than 8 except the data at / equal to 1900. It means that the correlation with 3 number is not remarkable and the scale effect is small at K lower than 8. The scatter of data for /9 equal to 1900 could be initiated by the vibration of the hydraulic cylinder at relatively small amplitude and low frequency oscillations. For small values of K number Ca is a little bit higher than its non-viscous value
0.5. For K larger than 8 Ca decreases first, and reaches a minimum value, and then rises slightly. For larger /9 number Ca becpmes increasingly more independent on, K.
There is a significant correlation between the drag coefficient, K number and /9 number shown in Figure 2.5. At K number between 2 and 18 C increases with
increasing K for /9 number lower than 3400. For /9 larger than 3400 C increases
first, and reaches a maximum value, and then decreases. The variation of the
drag coefficient with K becomes increasingly more insignificant at larger values of /9 number. For K less than 2 the friction component in the drag force becomes
more and more dominant. The drag. cOefficient could become infinite at very small values, of K number. Separation of the boundary layer eaist already at R,,
number around 102 in steady fiow[15]. In the present experimental, condition the
minimum R number is in the order of i0. It means that separation occurs in
the boundary layer of the testing sphere in all the experiments.
From the discussion above we can conclude that the variation of the force coeffi-cients with K becomes more and, more insignificant at larger values of /9 number.
In the tests at /9 equal to 9500 Ca has values around 0.5 and values of C be-tween 0.1 to 02. A full scale diving bell, for example, with diameter 2 meters in sea condition with significant wave height 8 meters and corresponding
pe-riod 10 seconds[18], has /9 number /9
D2/(vT) = 3.5 x iO and K number
K = 2vA/D = 12.5.
The full scale /9 number is much larger than the largest /9 number of the present
d
N
d
a
d0
Figure 2.4: Ca versus K number with various values of number for a smooth
Sphere. á,fl = 1900; v,fl = 3400; ±, = 5700; x, = 7600; D,fl = 9500. a e A A
=
V A A V A A A A 77 * A A A w V + + I- -'a 'C AA AA a AA A + A + 0*Figure 2.5: Ca versus K number with various values of
fi
number for a smoothsphere.
J3 = 1900; V, = 3400; +, = 5700; x, = 7600;
D, = 9500. a 'C AA V + V V V 0 d a-A a0 £ A A V 1 + V+ + + + c V + N * + + 0
-d,..-,---
0 30000 100000 . 150000 '(ftFigure 2.6: C0 versus R, number with various values of K number for a smooth
sphere. ,K 5; V,K = 10; +,K = 15.
A A
+
V
50000 100000 150000 500000
Figure 2.7: Cd versus R number with various values of K number fora smooth
0 In La x 9 'OXAI. A AL a + q A 20 K
Figure 2.8: CF3 versus K number with various values of number for a smooth
sphere. = 1900;
v,fl
= 3400;+,fi
= 5700; xj3 = 7600;D,fl= 9500. A 4. A 0 aI
A X + a + 5XAA a 10 15 KFigure 2.9: CFma versus K number with various values of number for a smooth
4 4 V V
4?
.4 V 4 + * + a + x + x x a a a + x a ia . -KFigure 2.10: Ratio of the maximum drag to the maximum inertia force in a
cycle versus K number with various values of $ n3imber for a smooth sphere. = 1900; v,fl = 3400; +,fl = 5700; x,fl = 7600; D,$= 9500.
around 0.5 and Ca values in the order of 0.1.
Figs. 2.6 and 2.7 provide the variation of the added mass anddrag coefficients of the smooth sphere with R, number at various constant values of K number. C0
increases and C decreases with increasing R, number. Both coefficients reacha
relatively constant value at large R number; C0 is close to its theoretical value of 0.5 at larger R, number. Ca. is dose to its super-critical value in steady flow
in the order of 0.1 at larger R, number. The transition in the boundary layer
occurs much earlier in oscillating flow than in steady flow, specially in the cases of lower K number. For example, the boundary layer is already turbulent and C
takes the value of 0.11 at R, number around io with K about 5 in oscillating
flow as shown in Figure 2.7. In steady flow C reaches the value of 0.1 .at 1?,, number larger than 3 x 10[15J in the supercritical region.
Root-Mean-Square and Maximum Force Coefficients
Figs. 2.8 and 2.9 show that the root-mean-square and the maximum force co-efficients decrease significantly with increasing K number at K lower than 8, and
then gradually reach relatively constant values. Both coefficients are quite inde-pendent of $ number at K lower than 8. This is because the inertia components of CFrma and CFrn, are predominant and independent of $ at K lower than 8 as discussed before(refer to Equations 2.10 and 2.13 and Figure 2.4). Due to
the high frequency noise and the effect of using the ifiter as we have discussed earlier the measured maximum force coefficient could be maccurate The data of the maximum force coefficient show much larger scatter than the data of the na force coefficient. At K larger than 8 the drag component becomes more and more important and the experimental data in Figs. 2.8 and 2.9 start to show
relatively larger scatter.
Ratio of Maximum Drag to Maximum Inertia Force
Figure 2.10 shows the ratio of the maximum drag force to the maximum inertia force in a cycle. The inertia force component iS dominant at K number lower than 8 for number larger than 1900 and at K up to 18 for fi larger than 7600. The drag force component becomes more and more predominant with increasing K number for smaller than 7600. The drag force component will be dominant at larger K number and smaller j number.
2.4.3 - Rough Sphere
Dragand added mass coefficients of rough spheres are plotted against R number in Figs. 2.11 to 2.14 and the smooth sphere data are plotted for comparison.
The force coefficients of the rough Sphere presented here show small dependence
on R,number at relatively larger R, number. The scale effect of the sphere has been reduced due to the roughness effect The roughness has no significant con-tribution to the added mass coefficient at low K number as shown in Figure 2 11 At low K, as we have discussed that the drag force component is unimportant the roughness will not affect the rms and maomum force coefficients remarkably The roughness effect to Ca is larger at small R,1 than at large R,. The
rough-ness effect to Cd is smaller at small R than at large R. Further increases in roughness have a smaller effect than the initiSi change from a smooth to the first rough sphere.
Figs. 2.11 and 2.13 show that the added mass coefficients of the rough spheres increase sharply at low R, number to a nearly constant value with increasingR,,
number The nearly constant value of C at large R, number are larger at low
K(K=5) thaii at high K(K=10). Further increases in roughness seem to have an
insignificant effect on Ca values.
Figure 2.12 shows that the data for the rough spheres at low K do not differ from
those corresponding to the smooth sphere at R number smaller than 2 x
where the flow is supposed to be attached The R,1 number must be sufficiently high for the roughness to play a role on the drag and flow characteristics of the sphere. The smaller the relative roughness the higher is the R, number needed
d
a a
a a a a N d aI....
-I 1 COOCOFigure 2.11: Ca versus R,1 number with various values of relative roughness
kID
at K equal to 5.
O,smooth3phere; V,k/D = 1/200; +,k/D = 1/140; x,k/D =1/100.Figure 2.12: C versus number with various values of relative roughness
kID
at K equal to 5.
O,smocthsphere; ,k/D = 1/200; +,k/D = 1/140;x,k/D = 1/100.
a
N
a - .
Figure 2.13: Ca versus R number with. various values of relative roughness
kID at K equal to 10.
O,srnooth.sphere; V,k/D = 1/200; +,k/D = 1/140;x,k/D= 1/100.
B x x + a a5 sasas 155055 ascoaoFigure 2.14: C versus R number with various values of
relative roughnessk/D at- K equal to 10. O,smoothsphere;
,k/D = 1/200; +,k/D = 1/140;
x,k/D = 1/100. 0 * * a. a a ;,4 . , * V B V V V vq a a a B
at which the roughness effect begins to occur. Due to the disturbances brought about by the roughness elements the transition to turbulence of the shear layers occurs at relatively low R,. number for a rough sphere. Figure 2.14 shows that
the drag coefficient is smaller than its smooth sphere value at low R,, number due
to the transition of the shear layers from the laxninai flow to turbulent flow. Cd
decreases to a minimum value and then rises slightly to a nearly constant
super-critical value. The smaller the relative, roughness the smaller is the magnitude of
the minimum C and the larger is R, number at whichm that minimum occurs. Due to the retardation of the boundary layer flow by roughness the separation in the boundary layer occurs early. The separation angle for the rough sphere remains smaller than that for the smooth sphere. The drag coefficient for the rough sphere has a larger value than that for the smooth sphere and increases with increasing relative roughness in the supercritical range.
2.4.4
Cube
Figs. 2.15 to 2.19 present the force coefficients of a cube oscillating harmonically
in water at rest. The force coefficients are plotted with K number for three
different values of /9 number There is no correlation between the force coefficients
and /9 number(or R number). That is due to the fixed separation points by the sharp edges of the cube. The flow pattern around the cube can be assumed to vary similarly for different values of R,, number at a fixed K number. The scale
effect can be neglected. For K larger than 10 the change of the added mass
and drag coefficients with K number is not significant. The force coefficients
are independent on both K and R. The value around 1.0 for the added mass
coefficient and 1.05 for the drag coefficient can be assumed for a cube at K values larger than 10 in planar oscillating flow.
Added Mass and Drag Force Coefficients
The added mass coefficient C shown in Figure 2.15 approaches its potential flow
value(0 7 for a cube by Sarpkaya[30] ) at low K Ca increases with increasing K
number at K values smaller than 7. For K larger than 7 C0 reaches a nearly
constant value around 1.0.
The drag coefficient Cj shown in Figure 2.16 has a constant value about 2 at low K between 1.8 to 3.2. This constant value ofCdcan be explained by the matched,
isolated-edge theory[8J. The vortices generated by the sharp edges of the cube are small compared to the distance between the separation points.. There is no
interaction between the vortices shed from different edges of the cube. The flow pattern is Similar. If K is larger than 3.2, the size of the vortices reaches the order
of the dimension of the cube(a=0.1 m at the present èondlition). There may be strong interaction between the vortices shed from different edges. That causes
continuously with increasing K until a constant value(1.05 corresponding to the
drag coefficient in steady flow for a. cube[15] ) is approached at larger K number.
Rins and Maximum Force Coefficients
Figs 2.17 and 2.18 show the root-mean-square force coefficient and themaximum
force coefficient versus K number with three va1ue of $ number Bothcoefficients
drops significantly with increasing K number at K below 4. Afterwards the coefficients decrease slowly with increasing K number. There is small scatterof the data shown in both figures except for the maximum force coefficient shown
in Figure 2.18 at low K number as we have discussed before.
Ratio of Maximum Drag to Maximum Inertia Force
Figure 2.19 shows the ratio of the maximum drag to the maximum inertiaforce
in a cycle.. The Ratio.increases continuously with increasing K number. The in-line force is predominant by the inertia component at low K numberand the drag force becomes more and mare important at larger K number. The abrupt diange around K equal to3.2 on the curve confirms further the conclusionwhich
we have made on the discussion about Cdvalues.
Figure 2.15: Ca versus K number. with various values of $ number for a cube.
0 VI
a-0do
10 15 50 KFigure 2.16: C versus K number with various values Of 9 number for a cube.
v,13 = 3400; +j3 = 7600; 0,13 = 9500. V
-
a 0+ 30 15 50 KFigure 2.17: C,, versus K number with various values of /3 number for a cube. vJ3 = 3400; +,$ =. 7600; 0,13 = 9500.
t
V
F
K
Figure 2-18: CF,, versus K number with various values of number for a cube.
= 3400; +, 7600; 0j3= 9500. V + TV a 10 10 < 30
Figure 2.19: Ratio of the maximum drag to the maximum inertia force in a cycle
versus K number with various valuesof number for a cube. ',fl = 3400;
2.5
Summary
In the experiments described in this chapter, the hydrodynainic loading on a smooth sphere, rough spheres and a cube has been systematically studied under
laboratory conditions by driving a hydraulic cylinder oscillating harmonically in
water at rest. The Reynolds number range reached is 2 x i0 to 1.8 x iO with
the Keulegan Carpenter number covered from 1 to 18.
The force coefficients of a smooth sphere are dependent on both K and R num-bers in general. For large 9 number, for example, a large scale sphere in general sea conditions, the force coefficients can be expected to be independent of both K and R,. Ca has values around 0.5 and Cd around 0.1 and the inertia compo-nent of the in-line force is predominant. The scale effect can be neglected. The
drag component of the in-line force can only be important at small 13(or R) -and
large K number. The critical R, number is much lower in oscillatory flow than
in steady flow, aspecially at low K number.
The roughness has increased the stability of the flow around the sphere. The force
-coefficients are less dependent of R and K. The roughness has less influenceon
the added mass coefficient than on the drag coefficient. The critical R number is reduced further due to the roughness. The value of the drag coefficient in the supercritical range is increased due to the roughness. Further increases in
roughness have a smaller effect than the initial change froma smooth to the first
rough sphere.
There is no correlation between the force- coefficients and fl(or R) number due to the fixed separation points by the sharp edges of the cube. The scale effect can be neglected. The value around 1.0 for the added mass- coefficient and 1.05 for the drag coefficient can be assumed for a cube at K values larger than 10 in
hi this chapter comparisons have been made between the present experimental data and the existing: data in both steady flow and oscillating flow by other
au-thors. Past development of the drag force study in steady flow has been reviewed
in Appendix B to have a better understanding about oscillating flow. The ex-periments concerning transverse force measurement are carried out under the
new experimental àxtangement. The transverse force hasbeen found to be much smaller than the in-line force.
31 New Arrangement of the Experiments
The new arrangement of the experiments are based on the arrangement described
in the previous chapter Some of theexperimental devices have been changed in
order to satisfy the new purose in the present experiments. These changes are
described in the following.
A two-component forcetrarisducer was used to measure both the in-lineforce and
the transverse force. The force transducer was mounted inside the model. The hydraulic cylinder was replaced. by an oscillator driven by a DC motor. There
are two reasons for using the oscillator instead of the hydraulic cylinder. First it is expected that we will have smaller mechanical vibration at the positionof
the maamum displacement The other reason is to test the oscillator system
which will be used later to create the orbital motion together with the hyd±aulic
cylinder.
3dd New Experimental Devices
Oscillator
An oscillat6r drivenby a DC motor replaced the hydraulic cylihder. The position of oscillation was measured by a potentiometer The oscillator was expected to create less vibration during the experiments, especially at the position of the maximum displacement of oscillation. An steel framework which connected the model with the oscillator was built to have enough stiffness to minimize the amplitude of vibration. Due to the limited capacity of the motor a spring system was adopt to connect the moving part of the oscillator with the roof to balance
the weight.
The oscillator system consists of a DC motor and a set of wheels and bands. The mOtor is operated by a servo-control system which obtains signals from a sinuagenerator. The oscillator is driven by the motor through wheels and rubber
bands. A detail description of the oscillator system is provided by Aakenes [2]. A view of the oscillator is shown in Figure 3.1. The oscillator has a maximum
stroke of 0.6 meter.
Force Transducer
A two-component force transducer has been designed to measure both the in-line
and transverse forces by Marintek A.S.. The force transducer is the strain gauge type with a maximum load capacity of 100 N in both directions.
The force transducer was placed inside the model as shown in Figure 3.2. The supporting steel rod has a diameter of 5 mm The gaps between the supporting rod and the model are covered by rubber bands. The elasticity of the rubber bands is large enough to give no contribution to the force measurement. The gaps are about 1 mm which is large enough to allow the deflection of the model
to take place during the tests. The flow pattern around the model would be
changed due to existence of the supporting rod However, The dimension of the model is 12 times larger than the diameter of the rod, hence, the effect on the force measurement due to the rod can be neglected.
The supporting rod is placed horizontally by two, vertical pipes with diameter 0.015 m of the steel supporting framework. The distance between the two pipes
is 0.4 meter. The effect on the force measurement due to these twO pipes can be neglected.
Instruments
The signal generator used in the present experiments is type Sinusgenerator 8038 developed by Marintek A.S.. The sinusgenerator has a period range from 0.25 to 2.1 second.
The signals from the force transducer were amplified by a DC amplifier. The signals from the DC amplifier together with the signal from the potentiometer were then filtered through 4 pole lowpass butterwortli ifiters. All signals were
recorded in analog form on tape. The analog data were later digitized by a
A/D converter and analysed by running the computer programs on a VAX 730 computer as described in the previous chapter.
3.1.2
Experimental Procedures
Calibration Tests
The same calibration tests were carried out as described in the previous chapter for the potentiometer and the two-component force transducer. The test results showed very little dispersion and gave good estimates of the calibration factor and the linearity of the system.
1].
Figure 3.1: Arrangement: of the oscillator system.
Table 3.1: Description of the experiment under Planar Oscillation-Il.
The tests of the cube oscillated in air were performed to get the mass component
of the inertia force. The tests were carried out at several different oscillating frequencies and amplitudes. The inertia coefficients from different tests differed
only slightly.
Main Experiments
The transverse forces were found to have much smaller values than the in-line forces for both the sphere and the cube by the experiments. A set of systematic tests were performed on the cube to test the validity of the oscillator system and the effect due to the new arrangement of the experiments. The cube has the same dimension(a=0.1 m) as that cube described in the previous chapter. A description of the model, test condition and range of parameters covered by the experiment is given in Table 3.1.
m kg
17 cube 0.10 0.49 0.67,0.83 4 .0 2.5-12.5
3.2
Results and Discussions
The results of the present tests(Test No.17) are shown in a table form in
Ap-pendix C Table C.3. Examples of the time series of the motion and force traces included the transverse force traces are also presented in Appendix C. For the sphere the time series of the motion and in-line force traces of Test No.2 in the previous chapter are selected as an example for fifrther discussion. Comparisons of the drag force coefficient of spheres between oscillatory flow and steady flow
are presented.
For both cube and smooth sphere the drag and added mass coefficients are com-pared with the results of Other authors. The measured root-mean-squareforce
coefficients together with their corresponding calculated values according to
Moii-son's equation and potential theory are presented. The relative error between the
measured maximum force and the calculated maximum force is given.
3.2.1
Sample Time Series
Examples of the motion and force traces obtained in the experiments are pte-sented in Appendix C. Both of the motion and in-line force signals for the. cube and the sphere repeat themselves clearlyin the tests. The in-line force traces are not the same in all the figures. Tbisphenomenon indicates the dependence of the
in-line force on K number. The mOtion traces show the rise of the noiseto-signal
ratio at small amplitude oscillations.
The motion traces of both the oscillator and the hydraulic cylinder follow the sinusoidal curve quite well. Vibration occurs at the maximum displacement of the oscillator and the hydraulic cylinder. The hydraulic cylinder seems to create a smoother motion than that by the oscillator.
Examples of the motion and force traes obtained in the present tests for the cube are shown from Figure C.1 to Figure C.6. The maximum in-line force peaks lag behind the motion peaks. The in-line force traces are much smoother at large K number than at small K number. The effect on the force traces due to
the vibration noise in the motion is not significant, especially at large K number
The transverse force traces show that .the transverse force is small and irregular.
Examples of the motion and the force traces obtained in Test No.2 for the sphere
are shown from Figure C.7 to Figure C.14. The maximum in-line force peaks stay around the motion peaks as shown in Figs C.7 and C.8 atsmall K number
due to the dominating inertia force. The effect due to the vibration noise is more significant' at small K number than at large K number, the measured maximum fOrce coefficient at small K number can be inaccurate due to this reason. For large K number the noise-to-signal ratio in the motion signals decreases and the
The force traces become two peaks and more irregular at large K number as shown
in Figs C.12 and C.14. The highest peak of the force traces moves to the peak of the velocity of oscillation with increasing K number due to the drag component being more and more important. Small amplitude mechanical vibrations were observed at large amplitude. oscillation during the tests. The irregularities in the force traces at large K number might be caused by the vibrations and the
irregularities of the vortex shedding.
3.2.2
Transverse force
The root-mean-square transverse force coefficients are given in Appendit C Table C.3. The time series of transverse force traces are shown in Figs. C.2, C.4 and
C.6. From both values in the table and the time series of force traces in
the figures the transverse forces have been found in vexy small values which are quite insignificant compared with the in-line force values.
3.2.3
Added Mass and Drag Force Coefficients
Cube
The present experimental results together with the results obtained in the pre-vious chapter and by Sortland[36] in his U-tube tests are presented in Figs. 3.3
and 3.4.
Figure 3.3 shows that the present data of added mass coefficient have the lowest
values with the data of Test No.15 from the previous chapter in the middle and
the data from Sortland's U-tube on the top. The data from the U-tube stay
constantly around 15% higher than the present data. The data of Test No.15 have about 10% higher values than the present data at K smaller than 10 and have the same values as the present data at K larger than 10.
Figure 3.4 shows that the present data of drag coefficient agree well with the data
from Sortland's U-tube tests and the previous data of Test No.15 at K number larger than 3.5. A relative large scatter occurs at small K number as expected
in the inertia force dominated region. The data of Test No.15 have slightly lower
values than the present data and the U-tube data in large K number.
Sphere
The experimental results of Test No.1, Test No.3 and Test No.5 in the previous chapter are presented in Figs. 3.5 and 3.6 together with the results obtained by
Sortland[36] in his U-tube tests, Sarpkaya[27J in his U-tunnel tests and øritsland and Lebn[42] in their free-decay tests.
Figure 3.5 shows that Sortland's U-tube data for j3 = 1200 have larger
va.l-ues than that of the present results. Sarpkayadata have smaller values and a
relatively large scatter Sarpkaya camed out the tests in the U-tunnel without
considering the influence of/3number in his early work in1975 The diameter of his testing spheres varies from 2 86 cm to 10 10 cm with the naturalfrequency
I
= 0.3495 of the Utunnel. It meansthat the/3number in his experiments vansfrom 250 to 3127 He seems to use smSll spheres which have small values ofj3 to
get large K values, and to use large spheres which have large values of/3 to get snall K alues. It makes his data close to the present results oflarge /3numbçr at small K number range, and close to the present resultsof small/3 number at large K number range.
Figure 3.6 shows that the measured data of drag coefficients show relatively
good agreement with the data from Sortland's and Sarpkaya's U-tube tests and with the data from ønitslaid's and Lebn's free-decay tests. Sortland's data are slightly smaller than the present results. As we have discussed aboveSarpkaya's data are obtained at different/3 numbers. His data agree well with the present
rults of large/3 number atK less than 4 and small /3number at K larger than
ønitslad and .Lehn[42] carried out their free-decay tests at /3,number about
4870. The free-decay method is much simpler to use for estimating the drag force
coefficients than the U tube method and the forced oscillation method However, the free-decay method assumes that the added mass coefficient is constant at different amplitudes of oscillation. This might not be true at large K andsmall /3 numbers Obviously the force coefficients will also depend on the wakeestablished by the previous oscillation; A large scatter has been found in their data[40] in the different initial amplitudes of oscillation for the same /3number
Further discussions
The disagreement in the results presented above in the same testing condition is mainly due to the different experimental arrangement The following parameters
*111 influence both the added mass and drag coefficients. '1. Blockage effect.
Effect due t the supporting rod.
Free surface effect.
The main reason for large values of the added mass coefficient in theU-tube tests is blockage effect by the limited cross section area. of U-tube. TheU-tube added
mass coefficient stays higher than the present data for both cube and sphere
as discussed abOve. Blockage effect of the U-tube on the drag coefficient is not significant. The blockage effect is neglected. in. the present test condition.
X 0
á0x0
xo a 0 o 10 18 1(20Figure 3.4: Ca versus K number for cubes. x,Sortland; D,Test No.15; Q,present
results. X XXX % X 'C 0 U a C 0 10 18 20 K
Figure 3.3: C versus K number for cubes. x,Sortlaiid D,Test No.15; Q,present
results.
0
S
is
KFigure 3.5: C1 verSus K number for spheres x,Sortland /3 = 1200; +,Sarpkaya
1975; ,Test No.1 /3 = 1900; D,Test No.3 ,8 = 3400; O,Test Nb.5 /3 = 9500.
+ a + xo 0 0
iL
+ 00 A + +L * l.a 0xxdcy13 0 8 + + 88 0 + A 0 S 10 20 '5 KFigure 3.6:: C versus K number for spheres. x,Sortland 3 = 1200; +,Sarpkaya 1975; *,øritsland and Lehn; A,Test No.1 /3 -= 1900; D,Test No.3 /3 3400;
Q,Test No.5 /3 9500.
00 0 0
*
