Download Title of the Paper (18pt Times New Roman, Bold)

Flow-induced vibration of an elastically sphere at high combined mass-damping
parameter
D. MIRAUDA, M. GRECO
Department of Environmental Engineering and Physics
Basilicata University
Viale Ateneo Lucano 10
85100 POTENZA, ITALY
Abstract: - The paper reports the results of the experimental analysis on the transverse flow-induced vibration of an elastically
mounted rigid sphere in a free surface flow. Simultaneous displacement and velocity measurements have been used to
describe the free vibrations. Experiments have been performed referring to low mass ratio value (m*), high combined massdamping parameter (m*), small amplitude (A/D<1) and low values of relative submergence (ratio between water depth and
sphere diameter). Preliminary results compared to the literature case studies suggest a different behaviour between the test
case of the sphere and those referred to the cylinder mainly concerning the lock-in and hysteresis phenomena.
Key-Words: - sphere, transverse vibration, resonance, synchronization, lock-in, mass-damping parameter
1 Introduction
Flow-induced vibration of bodies, obstacles and structures,
invested by flow steady and not, represents a relevant topic
of research in both industrial ambits and large/small scale in
environmental contexts. The advanced research on
innovative industrial technologies and bio-technologies
requires the accurate understanding of the flow-body
interactions, addressed to the better definition of the
structural characteristics and flow field assessment,
mutually influenced by the presence and interference of the
obstacle.
Moreover, classical problems of scientific literature,
partially solved but that still remain, are represented by the
evaluation of resonance phenomena caused by the
interaction between obstacle and flow field (vortex
structures), generating critical conditions in terms of
stability and structural resistance (spans of bridges, piles in
river, off-shore structures, aerodynamic components, etc.).
Regarding such complex processes the research, carried on
through laboratory experiments on prototypes, represents
the unavoidable tool through which useful answers are
obtained and information for a robust approach in the
resolution of applicative problems is derived.
Main aspects in the study of resonance phenomenon and
connected effects are related to the relevant vibrations
induced in the structures as experimentally observed in
several test case [4,13].
The literature of flow-induced vibration presents a large
number of fundamental studies, which are summarized in
the comprehensive reviews of Sarpkaya (1979), Griffin and
Ramberg (1982), Bearman (1984), Parkinson (1989), and in
the books by Blevins (1990), Naudascher and Rockwell
(1994), and Sumer and Fredsøze (1997). Such works mainly
address the influence of the combined mass-damping
parameter, maximum amplitude, response modes and
synchronization of bodies emphasizing a different
behaviour between low and high values of mass-damping
ratio.
In literature several contributions address the twodimensional system, i.e. studies on flexible cylindrical
cantilevers [5,21,28], on elastically mounted rigid cylinders
[6,8,10,11,12,24] and on direct numerical simulations
(DNS) of long flexible cable and cylinders [3,18]. Only,
recently Govardhan et al. (1997) and Jauvtis et al. (2001)
have proposed a study relating to the oscillations of a sphere
in a uniform free surface flow, founding a behaviour for
three-dimensional structures, sensitively different from that
observed for two-dimensional structures.
The present work leads the study of simple threedimensional body vibrations following an experimental
approach and the body, represented by a sphere elastically
mounted in a flume, moves in a free surface flow with low
relative submergence.
Further, the presence of free surface has contributed to the
study of the transverse vibrations of structures. In fact, as
underscored by recent research [22], the free surface
influences the dynamic assessment of the system,
generating self-exciting phenomena, leading resonance and
lock-in phenomena once the value of relative submergence
remains close to the unity (h/D1).
Therefore, the present study is mainly addressed to observe
the obstacle response when low mass ratio (m*), high
combined mass-damping parameter (m*), small amplitude
ratio (A/D<1) and low values of h/D (see Table 1) take
place.
bottom. The flow field around the body was measured in
two different ways, using both a miniature current
flowmeter and laser Doppler anemometer in order to have
different representative velocity domains; in fact the
employment of two different meter systems allows one to
comprehend the flow features at different spatial and
kinematics scales.
Table 1. Non dimensional groups used in the paper.
2 Experimental details
Simultaneous and direct measurements of the transverse
displacements and velocity of the obstacle submerged by
free surface flow have been collected through an
experimental apparatus built up in the Hydraulic Laboratory
of the Department of Environmental Engineering and
Physics of Basilicata University.
The main apparatus for displacements measurements is
located in the middle cross section of the water flume, 50
cm width.
The facility is made up by a plastic sphere of 9 cm in
diameter, solid to the bottom by an elastic cylindrical
support, (Fig. 1) with smooth surface to reduce defects
affecting the flow pattern.
An impermeable diaphragm has been inserted at the bottom
of channel. It is extremely elastic to permit the displacement
of the sphere in the main direction of the flow and
transversely, generating an elastic bottom constraint.
Several design criteria have been established for the
apparatus, as follows: 1) extremely linear system dynamic;
2) low mass ratio, m* (body mass)/(displaced fluid mass)
(m*<10); 3) high combined mass-damping parameter
( m* ζ ); 4) instantaneous and direct measurement of
displacements and velocity; and 5) possibility to change the
natural frequency of the system fixing different lengths of
support.
To keep the system dynamics linear and to avoid permanent
deformation during the runs when agitated by fluid, a derlin
rod, with an diameter of 10mm and sufficiently elastic in
response to the amplitude of the observed displacements,
has been used as support for the sphere.
The evaluation of the transverse displacements of the
obstacle was performed utilizing Laser Analog
Displacement Sensor (LAS) located under the channel
Fig. 1 - Schematic of experimental facility: frontal and side
view.
The possibility of changing the length of the rod and the
position of the joint, has been dictated by the necessity of
having a large range of natural frequencies in water of the
obstacle. These frequencies were calculated under the
hypothesis of perfect joint taking into consideration the
contribution of the added mass according to the following
relationship:
1
k  k'
(1)
fn 
2 m  m a
where k represents spring constant coefficient of system, k’
the added stiffness, which is usually included in the spring
constant of the system, m the mass of the sphere, ma the
related added mass calculated by the relation of Patton
(1965):
ma 
CA
6
D 3
(2)
with D diameter sphere, ρ fluid density and CA potential
added mass coefficient depending on the ratio between the
distance of the sphere centre from the free surface and the
diameter of the sphere (Fig. 2).
The used plastic sphere of 9cm and support rod of 50cm
lead to ranging the natural frequency between 3 Hz and 6
Hz. Further, such layout allows to obtain the velocity ratio
U* varying between 2 up to 8 due to the changes in water
discharge and natural frequency of the obstacle.
Moreover the relative submergence, defined through the
ratio between water depth, h, and the sphere diameter, D,
assumes value close to 1 (large scale roughness).
Fig. 2 - Added-mass coefficient for a sphere near a waveless
free surface.
3
Theory discussion and preliminary
results
As mentioned above, flow induced vibration on a bluff body
represents a relevant topic for study and research, as in
several fields of engineering, differentiating among low and
high mass-ratio. However, as recently discussed in literature
[12], typical answers and stimulus for ongoing research
mainly address the influence of the combined massdamping parameter, maximum amplitude, response modes
and synchronization.
These subjects are generally related to the relatively simple
case of an elastically mounted cylinder while few laboratory
investigations are focused on the submerged sphere in free
surface flow [7,14]. In this case, when low relative
submergence (large roughness) take place, the threedimensionality of the process occurs, drawing similar trends
for the typical parameters like those observed for a cylinder
with different threshold conditions.
While the study of cylinder behaviour becomes relevant
mainly for structural engineering, the test case of sphere
represents a strong theoretical base for environmental
engineering, likely for sediment transport processes in water
or air pollution for downwashing problems or local
morphology effects, etc.
In river engineering, in presence of large roughness
currents, flow induced vibration represents critical
conditions in which resonance effects and lock-in or
synchronization occurs generating uncontrolled effects,
sometimes even increasing the risk level for human
activities.
A brief recall needs to be address in the paper in order to
give useful basis for further discussion, therefore, referring
to the present test case, the system is thought like a simple
oscillator acted by a harmonic exciting force:
(3)
F t   FO cost   
where   2f with f the forcing frequency and  is the
phase angle extremely relevant in such kind of problems
[1,23] which leads to assume the motion equation in the
form:
(4)
mx  Bx  Cx  F t 
in which  Bx is the damping force. The solution of this
equation is:
(5)
x  e n t xo cosd t   
with  the damping factor or damping ratio.
Such coefficient, as mentioned by Naudascher and
Rockwell (1994), has been calculated reporting the
exponentially decaying response for the initial condition t =
0, x = xo and   0 (Fig. 3b).
For   1 , the displaced body simply returns to its
equilibrium position in an exponential fashion (Fig. 3c). The
damping for the limiting case (   1 ) is called critical
damping. In the underdamped case, 0    1 , as in the
present study, the coefficient has been obtained from
following equation:
x
2
(6)
  ln n 
xn 1
1 2
where  is called the logarithmic decrement.
Fig. 3 - (a) Simple body oscillator with linear damping (b,
c) Histograms of responses for an underdamped ( <1) and
an overdamped (≥case.
As well known at the resonance in the case of oscillating
cylinder when the oscillating frequency is relatively close to
the natural frequency, the values of the maximum amplitude
depends on the combined mass-damping parameter ( m *  )
[12]. Besides the effectiveness of the damping forces is
taking into account through the Skop-Griffin parameter, SG,
defined here as follows:
(7)
SG  2 3Str m*
In the present work this approach, generally developed for
regular cylinder, has been applied for the test case of sphere
in order to have some useful suggestion through the
comparison. In fact the observed values of the oscillating
frequency, derived by spectral analysis, are ranging around
the natural frequency in water of the sphere.
This allows one to assume the measurements as referred to a
 
2.00
Skop & Balasubramanian (1997)
Ramberg et Griffin in acqua (1981) m*= 34
Ramberg et Griffin in aria (1981) m*= 3.8
1.50
Mirauda et al. (2003) m*=6.95
A*
Present data m*=1.14
1.00
0.50
0.00
0.01
Feng (1968)
Skop & Balasubramanian (1997)
Khalak & Williamson (1999)
Govardhan & Williamson (2000)
Blackburn et Karniadakis (1993)
Newman et Karniadakis (1996)
Griffin (1980)
Mirauda et al. (2003)
Present data
1.0
0.10
SG
1.00
10.00
100.00
b)
Fig. 4 - Griffin plot showing maximum amplitude observed
in the experiment versus the combined mass-damping
parameter (a) and Skop-Griffin parameter (b).
Further information, concerning the resonance effects and
hysteresis can be revealed assuming the response amplitude
as a function of the normalized velocity U* (Fig. 5).
There exist two distinct types of response in such systems,
depending on high or low combined mass-damping
parameter (m*). In the case of low m* [10,11,12], the
dynamic response outline three different branches of
response: the initial, upper and lower one, which present a
"jump" in the magnitude of oscillating displacement. In fact,
moving from the initial branch to the upper one it has
been observed and demonstrated that the presence of a
phenomenon of hysteresis, while, passing from the upper
branch to the lower one, a sudden jump lacking in hysteretic
phenomenon is seen.
1.2
Feng (1968) m*=248
Khalak & Williamson (1999) m*=10.1
Mirauda et al. (2003) m*=6.95
Upper branch
Jauvtis et al. (2001) m*=2.8
Present data m*=1.14
Mode II
0.8
1.5
Upper
Khalak et al. (1996) m*= 2.4
A*
quasi-resonance state in which the maximum amplitude
observed can be assumed depending on SG or on the
parameter m*  C A  .
In the present study experiments were performed for high
Reynolds number around 105 and the measures have been
obtained fixing in each experiment the water discharge
across the channel and changing the length of support rod in
order to varying the natural frequency of the body. Each run
has been done for different water discharge and rod length.
Figs. 4a and 4b show the maximum amplitude plotted
versus a combined mass-damping parameter (m*+CA) and
the parameter SG, using data from several investigators.
From these figures, is noted that it is not possible to make a
“unique” curve of A*max versus SG for low values of
parameter m*. Sarpkaya (1979, 1995) himself states: one
should use the combined parameter SG only if SG>1 while
for SG < 1 the dynamic response of system is governed by
m* and  independently and non just by m*. In fact
analysing three pairs of low-amplitude response data with
each pair at similar SG but different m* values, he calculated
that there is a large (50-100%) influence of mass ratio on
A*max. This point is supported by Zdravkovich (1990), that
states: the mass ratio and damping should be treated as
separate parameters for values of m*<10.
In the figures the data of present work and those of previous
experiments of Mirauda et al. (2003) are also reported. The
first set, referring to a steel sphere in free surface flow, are
characterized by low oscillations and take place in the low
part of the graph. The second series, characterized by values
of m*lower of first, are spread in a different pattern from
the Skop’s curve, confirming the words of Sarpkaya and
others as well as the data reported in literature outlining the
different behaviour of the sphere regarding one of the
cylinders.
A preliminary explanation can be researched in the
complexity of the flow field which is affected by the
presence of the water surface and by the threedimensionality of the body as well as by high values of the
Reynolds number, the order of 100,000, while the
literature’s experiments refer to 1000-10,000.
Mode I
0.4
Lower branch
Initial branch
0.0
A*
0
.
0.5
0.0
0.00
a)
0.10
(m*+CA)
U*
10
15
Fig. 5 - Maximum amplitude versus velocity ratio.
Lower
Inferiore
0.01
5
1.00
10.00
Further, through the use of visualization techniques
(Digital Particle Image Velocimetry), it can be seen that
the change from the lower branch to the upper depends
on the jump in phase angle between the force induced by
the shedding of the main vortex and the displacement
related to a change in the form of the vortex wake
close to unity. In fact for the sphere, the frequency ratio
exhibits a different trend for low values of U* much steeper
at the beginning than afterwards, approaching slowly to the
value of f*=1. It is observed that the measured data, are not
distributed according to a straight line but, on the contrary,
the data set cause a relationship to occur among the
frequency f* and the velocity U* different from the linear
one.
Such behaviour can be also explained through the strong
three-dimensional aspect of the flow field, affected by the
shape of the obstacle and the free surface distortion due to
the low submergence governing the eddies structures
generated downstream and thus, the vibrating response of
the body [25].
4.0
Khalak & Williamson (1999) m*=2.4
Khalak & Williamson (1999) m*=10.3
Khalak & Williamson (1999) m*=20.6
3.0
Mirauda et al. (2003) m*=6.95
Present data m*=1.14
f*
downstream from the body. In this case the value of the
oscillating frequency of the body, f, passes across the natural
frequency in water generating a resonance phenomenon [8].
Instead, the passage from the upper branch to the lower one
is characterized by the presence of a phase-displacement
between the total force, the sum of the force induced by the
vortex and the potential force, and the displacement
which tends toward a periodic uniform trend. In such case
no change in the form of the wake is observed. For systems
having high values of combined damping-mass parameter
m* only two branches of response are observed: the
initial branch and the inferior one [4]. The passage
between the two branches occurs with the jump of a phase
for the force components, the total force and the force
induced by the vortex. This jump is related to a change in
the form of the wake. Even in this case the body reaches
conditions of resonance.
The behavior found for three-dimensional structures,
with elementary geometrical forms (ex. spherical), is
sensitively different from that observed for twodimensional structures. In fact, the data of Jauvtis et al.
(2001) relating to the oscillations of a sphere for values of
m*=2.8, demonstrate the presence of two distinct modes
of response. The first mode of response (Mode I) is
manifested in the presence of resonance conditions, when
the frequency of the shedding of the vortex is close to the
natural frequency of the body, and a synchronization
regime is observed between the force and the response.
When the average velocity of the flow increases, the
system shows the presence of periodic oscillations
characterized by high values of displacement that represent
the second mode of response (Mode II).
The data collected, for the investigate range of U*, are close
to the first mode of response outlining how the system tends
to reach the resonance conditions where vortex-shedding
frequency is equal to the natural frequency. Previous
experiments show only the initial branch without “jumps” in
the amplitudes and, therefore, do not exhibit the hysteresis
phenomena. This can be roughly explained considering the
high value of the damping coefficient that should inhibit
instability phenomena.
One more point of the discussion concerns the
“synchronization” or “lock-in” that occurs when the value
of f* remains close to the unity as the velocity U* increase.
Better the condition of lock-in requires also that the vortexshedding frequency remain close to the oscillation
frequency f and thus to fN. In this case f*=1.
Further increasing of U*,over a range of synchronization,
pulls the shedding frequency away from its non oscillating
value and such transition is characterized by the presence of
hysteresis [12]. Fig. 6 reports the data observed by Khalak
et al. (1999) for a vibrating cylinder with mass ratio equal to
2.4, 10.3, and 20.6 and the data obtained through the present
experiments referred to m*=1.14.
The case of the sphere seems to show different behaviour
from the cylinder for low values of U* and for values of f*
2.0
1.0
0.0
0
2
4
6
U*
8
10
12
14
Fig. 6 - Frequency ratio versus velocity ratio.
4 Conclusion
An experimental apparatus has been designed and built to
study transverse flow-induced vibration of an elastically
mounted rigid sphere.
Simultaneous and direct measurements of the displacement
and velocity of the obstacle have been obtained for high
combined mass-damping parameter (m*) and amplitude
A/D<1.
From the relationship between the maximum amplitude and
the parameter U*, a first synchronization regime of
oscillating structure is observed similar to that of Jauvtis et
al. 2001. Finally, all figures have showed, at the moment,
different behaviour of the three-dimensional system with
respect to that observed for cylindrical structures. At the
present such behaviour can be also justified by the presence
of free surface flow on the body, the high range of Reynolds
number (105) and the three-dimensionality of the process.
Such conditions can generate complex fluid-dynamic
phenomena not easily explainable through the available
data, therefore, it seems to be important to continue the
experimental activity, utilizing new data, in order to extend
the range of velocity ratio U*, the maximum amplitude A*
and the frequency ratio f* investigated.
References:
[1] Bearman P. W., Vortex shedding from oscillating bluff
bodies, Annual Review of Fluid Mechanics, Vol. 16,
1984, pp. 195-222.
[2] Blevins R. D., Flow-induced vibrations, New York:
Van Nostrand Reinhold, 1990.
[3] Evangelinos C., Karniadakis G. E., Dynamics and flow
structures in the turbulent wake of rigid and flexible
cylinders subject to vortex-induced vibration, Journal of
Fluid Mechanics, Vol. 400, 1999, pp. 91-124.
[4] Feng C. C., The measurement of vortex-induced effects
in flow past a stationary and oscillating circular and Dsection cylinders, Master’s Thesis, University of British
Columbia, Vancouver, B.C. Canada, 1968.
[5] Fujarra A. L. C., Meneghini J. R., Pesce C. P., Parra P.
H. C. C., An investigation of vortex-induce vibration of
a circular cylinder in water, Conference on Bluff Body
Wake and Vortex Induced Vibration, Washington, 1998.
[6] Gharib M. R., Leonard A., Gharib M., Roshko A., The
absence of lock-in and the role of mass ratio,
Conference on Bluff Body Wake and Vortex Induced
Vibration, Washington, 1998.
[7] Govardhan R., Williamson C. H. K., Vortex induced
motions of a tethered sphere, Journal of Wind
Engineering and Industrial Aerodynamics, Vol. 69-71,
1997, pp. 375-385.
[8] Govardhan R., Williamson C. H. K., Modes of vortex
formation and frequency response of a freely vibrating
cylinder, Journal of Fluid Mechanics, Vol. 420, 2000,
pp. 85-130.
[9] Griffin O. M., Ramberg S. E., Some recent studies of
vortex shedding with application to marine tubulars and
risers, SME Journal of Energy Resources Technology,
Vol. 104, 1982, pp. 2-13.
[10] Khalak A., Williamson C. H. K., Dynamics of a
hydroelastic cylinder with very low mass and damping,
Journal of Fluids and Structures, Vol. 10, 1996, pp.455472.
[11] Khalak A., Williamson C. H. K., Fluid forces and
dynamics of a hydroelastic structures with very low
mass and damping, Journal of Fluids and Structures
Vol. 11, 1997, pp. 973-982.
[12] Khalak A., Williamson C. H. K., Motion, forces and
mode transitions in vortex-induced vibrations at low
mass-damping, Journal of Fluids and Structures Vol.
13, 1999, pp. 813-851.
[13] Koopmann G. H., The vortex wakes of vibrating
cylinder at low Reynolds number, Journal of Fluid
Mechanics, Vol. 28, 1967, pp. 501-512.
[14] Jauvtis N., Govardhan R., Williamson C. H. K.,
Multiple modes of vortex-induced vibration of a sphere,
Journal of Fluids and Structures, Vol. 15, 2001, pp.
555-563.
[15]Windhorst A., Flatterschwingungen von zylindern im
gleichmassigen flussigkeits-strom, Mitteilungen des
Hydraulischen Institut der Technischen Hochschule,
Munchen, Vol. 9, 1939, pp.3-39.
[16] Mirauda D., Greco M., Transverse vibrations of an
sphere at high combined mass-damping parameter,
Proceedings of International Symposium on Shallow
Flows, Delft, Olanda, 2003.
[17] Naudascher E., Rockwell D., Flow induced vibration:
an engineering guide, Rotterdam, Balkema, 1994.
[18] Newman D. J., Karniadakis G. E., Simulation of flow
past vibrating cable, Journal of Fluid Mechanics. Vol.
344, 1997, pp. 95-136.
[19] Parkinson G., Phenomena and modelling of flowinduced vibration of bluff bodies, Progress in Aerospace
Sciences, Vol. 26, 1989, pp.169-224.
[20] Patton K. T., Tables of hydrodinamic mass factors for
translational motion, Winter Ann. Meeting ASME,
Chicago, Illinois, Paper 65-WA/UNT-2, 1965.
[21] Pesce C. P., Fujarra A. L. C. Vortex-induced vibrations
and jump phenomenon: experiments with a clamped
flexible cylinder in water, International Journal of
Offshore and Polar Engineering, Vol. 10, 2000, pp.2633.
[22] Sakamoto H., Haniu H., The formation mechanism and
shedding frequency of vortices from a sphere in uniform
shear flow, Journal of Fluid Mechanics, Vol. 287, 1995,
pp.151-171.
[23] Sarpkaya T., Vortex-induced oscillations, Journal of
Applied Mechanics, Vol. 46, 1979, pp.241-258.
[24] Sarpkaya T., Hydrodynamics damping, flow-induced
oscillations, and biharmonic response, ASME Journal of
Offshore Mechanics and Artic Engineering, Vol. 117,
1995, pp. 232-238.
[25] Sheridan J., Lin J.C., Rockwell D., Flow past a
cylinder close to a free surface, J. Fluid Mech., Vol.330,
1997, pp.1-30.
[26] Skop R. A., Balasubramanian S., A new twist on an
old model for vortex-excited vibrations, Journal of
Fluids and Structures, Vol. 11, 1997, pp. 395-412.
[27] Sumer B. M., Fredsøze J., Hydrodynamics around
cylindrical structures, Singapore, World Scientific,
1997.
[28] Vickery B. J., Watkins R. D., Flow-induced vibrations
of cylindrical structures, Proceedings of the First
Australian Conference on Hydraulics and Fluid
Mechanics, New York, Pergamon Press, 1964.
[29] Zdravkovich M. M., On origins of hysteretic responses
of a circular cylinder induced by vortex shedding.
Zeitschrift fur Flugwissenschaften Weltraumforschung
Vol. 14, 1990, pp.47-58.