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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/D1). 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 cost where 2f 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) mx Bx Cx F t in which Bx is the damping force. The solution of this equation is: (5) x e n t xo cosd 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. 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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. 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