Interaction between a tethered sphere and a free surface flow

Fluid Structure Interaction and Moving Boundary Problems 205 Interaction between a tethered sphere and a free surface flow M. Greco 1, S. Malavasi 2 & D. Mirauda 1 1 Department I.F.A., Basilicata University, Italy 2 Department I.I.A.R., Politecnico di Milano, Italy Abstract This paper analyses the vibrations and the flow field around a tethered sphere in a free surface flow in order to study the influence of the vortex shedding process on the dynamic response of the obstacle and on its oscillation frequency. The system is characterized by a low value of the mass ratio (ratio between the mass of the system and the added mass), a low value of damping and a low value of relative submergence (ratio between the depth of the current and the diameter of the sphere). The experiments are set up to measure the movements of the sphere in the main and transversal directions of the fluid flow with laser sensors and image analysis, whereas the description of the main wake structures of the flow downstream of the sphere are provided using the PIV technique. Keywords: dynamic response, sphere, flow field, vortex structures, resonance phenomenon. 1 Introduction Studies on the analysis of the two-dimensional and three-dimensional vibrating structures in bounded and free surface flows, have highlighted the existence of a strong dependence of the maximum transverse amplitude on some dimensionless groups, as shown below: ( U ) * * A max = A *; (1) where A* max =y 0 /D is the ratio between the maximum transverse amplitude (y 0 ) and the characteristic dimension of the body (D), U*=U 0 /f n D, called reduced velocity, is the ratio between the mean velocity of the flow (U 0 ) and the product S G

206 Fluid Structure Interaction and Moving Boundary Problems of the natural frequency of the body (f n ) with D, and S G is the Skop-Griffin parameter defined as follows: * ( ζ ) 3 2 S G = 2π Sh m (2) where Sh is the Strouhal number, m* is the ratio between the structural mass (m) and the added mass (m a ) and ζ is the ratio between the structural damping and the critical damping in water. More in detail, with reference to two-dimensional systems, experimentations conducted in the last decade (Khalak and Williamsom [3], [4], [5]; Govardhan and Williamsom [2]; Pesce and Fujarra [10]) have shown how the dynamic response of the obstacle changes depending on the values assumed by the parameter, m*, the damping coefficient, ζ, and the combined mass-damping parameter, m*ζ. In fact, in the case of low values of m*ζ, three different branches of response are observed: the initial, the upper and the lower, while for systems having high values of combined damping-mass parameter, m*ζ, there are only two branches of response: the initial and the lower. In both cases the transition from one branch to another are connected to resonance phenomena, lock-in and synchronization, and to different forms and configurations of the wake vortex downstream of the bodies itself. The vortex structures generated around a sphere are strongly 3D and the study of their evolution and interpretation under the condition herein considered has not yet been well developed. The scientific literature doesn t present studies concerning the analysis of flow field around three-dimensional tethered structures, but only related to the dynamic response of bodies, most of which have been carried out not long ago. Recently, Govardhan and Williamsom [1] and Jauvtis et al. [6] have proposed a study relating to the oscillations of a sphere in a uniform free surface flow, demonstrating 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 vortex shedding is close to the natural frequency of the body, and a synchronization regime is observed between the force and response. When the average velocity of the flow increases the system has showed the presence of periodic oscillations characterized by high values of displacement that represent the second mode of response, (Mode II). They have observed, also, a reduction in the synchronization regime as the mass of the vibrating structure increases, that is, as the parameter m* increases. The work deals with an experimental study of the interaction between a sphere and a steady free surface flow. It investigates the oscillation amplitude and frequency of the sphere and, under the same flow conditions, the vortex structures downstream of the obstacle. The analysis proposes the regime of vibration induced on a system characterized by low values of mass ratio, damping parameter and relative submergence.

Fluid Structure Interaction and Moving Boundary Problems 207 The experimental activities have been concentrated on direct measurements of longitudinal and transversal displacements as well as on kinematical characters of flow field downstream of the sphere in order to begin the study of the correlation between the flow structure and the dynamic behaviour of the body. The displacement measurements have been obtained through laser sensors and CCD camera, while the reconstruction of the flow field has been obtained through Particle Image Velocimetry technique. 2 Experimental apparatus The experiments have been performed in a non-tilted Plexiglas open water channel with rectangular cross section, 0.6 m height, 0.5 m width and 5.0 m length. The obstacle used has been a water filled sphere with a diameter D=0.087 m. The sphere surface is made of PVC and covered with an episodic paint to reduce the surface roughness. A rod has been used to connect the sphere to a fixed structure. The rod is made of stainless steel and Derlin. The stainless steel part, which is connected to the sphere, is 0.2 m long and 3 mm in diameter. The Derlin part completes the rod with a diameter of 9 mm. The distance between sphere and channel floor has been set at 3 mm. Figure 1 shows the sketch of the obstacle rest. Figure 1: Sketch of the obstacle rest mounted on the channel (cross-section). The experiments have been characterized by a range of Reynolds number Re 10 4 3 10 4 and a relative submergence h/d 1. The movements of the sphere have been measured using two methods: an analog laser displacement sensor able to provide displacements in one direction and an image analysis of CCD acquisitions, able to provide all 2D displacements of the sphere but with less accuracy. The flow field reconstruction downstream of the sphere has been provided by a PIV (Particle Image Velocimetry) technique specifically developed for hydraulic applications (Malavasi et al. [7]). The 2D velocity fields have been obtained measuring the two-dimensional trajectories of seeding particles in a

208 Fluid Structure Interaction and Moving Boundary Problems defined time interval. This is possible by filming the seeding flow on the measurement plane defined by means of a light sheet with a proper exposure time; it is set on the base of the mean velocity. The image acquisition apparatus is composed by an illumination system, a particle dispenser, a progressive CCD camera with a proper frame grabber board and a personal computer. In this arrangement, it is possible to take monochromatic images with a resolution of 763x576 pixels at a frame rate of 50 Hz. The light sheet is created by a linear light-converter connected to a light source of 150 W, which provides a light sheet of 5 mm with a diverging angle of about 0.4 degrees. The tracer particles are white spherical polystyrene particles with an average diameter of φ = 400 µm and density of ρ p =1050 kg/m 3. The polystyrene density, being very close to that of the water, and the particle dimension assure the minimization of gravitational and inertial effects on the particle movements. The numerical analysis of the image captured differs from the correlation image method usually applied in PIV analysis. Using a blob-analysis algorithm on every frame, it is possible to measure the geometrical dimensions of each recognized trajectory obtaining the location, the module and the orientation of the displacement, and thus, through the exposure of the images, the velocity. The direction ambiguity is solved using the velocity information on two consecutive frames. The main wake structures of the flow downstream of the sphere have been then obtained by the analysis of the velocity fields. 3 Results 3.1 Displacement analysis As mentioned above, the maximum transverse amplitude of a structure depends on the reduced velocity, U*, and on the Skop-Griffin parameter, S G. As regarding the influence of parameter S G (figure 2), the results obtained for m*=1.33 are compared with several experiments related to two-dimensional structures [5, 11, 12] and results of previous works on three-dimensional structures [8, 9] at different values of m*. Present data, characterized by values of m*ζ, are very close to and sometimes lie along the curve proposed by Skop and Balasubramanian [12] to interpret the behaviour of two-dimensional bodies in water. This represents an important result because the dynamic behaviour of both two-dimensional and threedimensional structures could be interpreted by a unique curve. Moreover, they confirm and extend the results of Mirauda et al. [8, 9], which refer to a steel sphere in free surface flow with high values of m*ζ. By plotting the maximum amplitude versus the reduced velocity, U*, (figure 3), it is possible to observe how the system for the investigated range of U* is close to the first mode of response found by Govardhan and Williamsom [1] and Jauvtis et al. [6]. This shows that the system tends to reach resonance conditions, where vortex-shedding frequency is equal to the natural frequency of body, but it is still not close to the synchronization regime (second mode of response).

Fluid Structure Interaction and Moving Boundary Problems 209 2,0 1,5 Skop & Balasubramanian (1997) Ramberg et Griffin in acqua (1981) m*= 34 Ramberg et Griffin in aria (1981) m*= 3.8 Khalak et al. (1996) m*= 2.4 Mirauda et al. (2004) m*=7.90 Present data m*=1.33 A* 1,0 0,5 0,0 0,01 0,10 S G 1,00 10,00 100,00 Figure 2: Maximum transverse amplitude (A*) versus the Skop-Griffin parameter (S G ). 1,2 0,8 Jauvtis et al. (2001) m*=2.8 Mirauda et al. (2004) m*=7.90 Jauvatis et al. (2001) m*=0.8 Jauvatis et al. (2001) m*28 Present data m*=1.33 Mode II A* 0,4 Mode I 0,0 0 5 U* 10 15 Figure 3: Maximum amplitude versus reduced velocity for three-dimensional bodies. Finally, figure 4 shows the trajectory of the movement on the horizontal plane of the sphere reproduced by the image analysis obtained for the case of U*= 2.44, m*=1.33 and h/d=1.38. As for this example, oscillation amplitudes

210 Fluid Structure Interaction and Moving Boundary Problems transverse to the fluid flow are much larger than the stream wise motions presenting typical trajectories of crescent topologies. These results agree with those of Govardhan and Williamsom [1] and Jauvtis et al. [6] obtained for values of m* 1, when the sphere is defined light. An asymmetry in the system can also be noted when the location of the zero point of the sphere is considered. 0.1 0-0.1 X/D -0.2 zero point -0.3-0.4-0.5-0.6-0.4-0.2 0 0.2 0.4 0.6 Y/D Figure 4: Typical trajectory of sphere motion for light sphere (U*=2.44; m*=1.33; h/d=1.38). 3.2 Flow field analysis In the following figures, some examples of flow field reconstruction downstream of the tethered sphere are reported. Even if PIV technique provides 2D flow field, the analysis of the vortex shedding on the symmetrical planes of the sphere could give significant contributions to understand the phenomenon. Figure 5 describes the flow field structure downstream of the sphere on a vertical plane which is the symmetrical plane of the sphere when it is in static position. In the case of low mean velocities of the flow (0.2 m/s) and, consequently, of low transverse and streamwise oscillations, it is possible to observe the development and evolution of coherent vortex structures similar to ones downstream of static spheres in steady flow. The significant difference between the two wake structures in figure 5 are due to the distortion on the evolution of the wake structures caused by the sphere movement. Figure 5 shows the presence of contra-rotation and asymmetrical vortex shedding especially in the lower part of the obstacle near the bottom of the channel, where the presence of the boundary layer of the surface helps the vortex formation. Although the acquisitions are limited to low values of mean velocity, the measurements show a different system response for different boundary conditions. When the mean velocity increases (0.4 m/s), and thus the transverse

Fluid Structure Interaction and Moving Boundary Problems 211 oscillations achieve values close to the diameter of the sphere, it is possible to observe two different conditions corresponding to the position of the sphere (figure 6). The first condition occurs when the sphere is moving near the vertical symmetry plane of the sphere when static; the second occurs when the sphere is reaching the maximum amplitude in oscillation. As depicted in figure 6, in the first case the wake structure is similar to one observed for low velocities, in the second case the vortex structures disappear and the flow field is quite regular. Increasing the flow velocity, it observes a growing dominance of the near wake configuration between the free surface and the obstacle. In this area, a jet dampens the development and evolution of coherent vortex structures downstream to the sphere. Figure 5: Sideways average flow field on a time interval of 0.4 s (U*=1.63; m*=1.33; h/d=1.38). The movement of the sphere on the time interval is depicted by the dashed line. Figure 6: Sideways average flow field on a time interval of 0.4 s (U*=3.25; m*=1.33; h/d=1.38). The movement of the sphere on the time interval is depicted by the dashed line. Finally, the images obtained from the bottom and sideways views show the simultaneous presence of macro-vortex with vertical and horizontal axes influencing each other. Such feature strongly underlines the three-dimensional nature of the flow field. Figure 7 depicts the average flow field structure on the

212 Fluid Structure Interaction and Moving Boundary Problems horizontal symmetrical plane of the sphere. The time interval considered is of 0.4 s and the movement (right to left) of the sphere in this time is evidenced using the dashed line. Figure 7 also shows the distortion of the vortex due to the movement of the sphere. The analysis of the flow field evolution highlights the synchronism of the vortex shedding with the sphere oscillation. Figure 7: Horizontal average flow field on a time interval of 0.4 s (U*=2.44; m*=1.33; h/d=1.38). The movement of the sphere on the time interval is depicted by the dashed line. 4 Conclusions An experimental apparatus has been designed and built to study transverse flowinduced vibration and flow field of an elastically mounted rigid sphere, characterized by low combined mass-damping parameter and low values of relative submergence. The maximum transverse amplitude of the sphere oscillation has been investigated depending on the Skop-Griffin parameter, S G, and the reduced velocity, U*. Under the fixed boundary conditions, the experimental results have highlighted that the dynamic behaviour, both for two-dimensional and threedimensional structures, could be interpreted with a unique curve in the plane A*- S G. Moreover, they have confirmed and extended the results of Mirauda et al. [8, 9]. From the relationship between the maximum transverse amplitude and the reduced velocity (figure 3), a first resonance regime of oscillating structure is observed similar to one of Govardhan and Williamson [1] and Jauvtis et al. [6]. Moreover the typical trajectories of the sphere oscillation are agreed with those suggested by the same Authors for values of m* 1. The analysis of flow field images has shown the nature of the flow field around the obstacle, underlining the typology and the organization of the vortex structures controlling the process. Although the acquisitions are limited to low values of mean velocity, the measurements have allowed to recognize two main processes: firstly, the vortex shedding due to both the obstacle and body oscillations and, secondly, the

Fluid Structure Interaction and Moving Boundary Problems 213 occurrence of jets which slow down the development and evolution of coherent vortex structures, especially in conditions of high velocity. Image acquisition has shown the strongly three-dimensional nature of the flow field, with the simultaneous formation of macro-vortex with vertical and horizontal axes influencing each other. References [1] Govardhan, R. & Williamson, C. H. K., Vortex induced motions of a tethered sphere. Journal of Wind Engineering and Industrial Aerodynamics, (69-71), pp. 375-385, 1997. [2] Govardhan, R. & Williamson, C. H. K., Modes of vortex formation and frequency response of a freely vibrating cylinder. Journal of Fluid Mechanics, (420), pp. 85-130, 2000. [3] Khalak, A. & Williamson, C. H. K., Dynamics of a hydroelastic cylinder with very low mass and damping. Journal of Fluids and Structures, (10), pp. 455-472, 1996. [4] 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, (11), pp. 973-982, 1997. [5] Khalak, A. & Williamson, C. H. K., Motion, forces and mode transitions in vortex-induced vibrations at low mass-damping. Journal of Fluids and Structures, (13), pp. 813-851, 1999. [6] Jauvtis, N., Govardhan, R. & Williamson, C. H. K., Multiple modes of vortex-induced vibration of a sphere. Journal of Fluids and Structures, (15), pp. 555-563, 2001. [7] Malavasi, S., Franzetti, S. & Blois, G., PIV Investigation of Flow Around Submerged River Bridge. River Flow, Napoli (Italy), June 23-25, pp. 601-608, 2004. [8] Mirauda, D. & Greco, M., Transverse vibrations of an sphere at high combined mass-damping parameter, Shallow Flows Jirka & Uijttewaal (eds) Balkema Publisher, Taylor & Francis Group, London, ISBN 90 5809 700 5, pp. 111-116, 2004. [9] Mirauda, D. & Greco, M., Flow induced vibration of an elastically mounted sphere at high combined mass-damping parameter. IASME Transactions, 3(1), pp. 486-491, 2004. [10] 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, (10), pp.26-33, 2000. [11] Ramberg, S. E. & Griffin, O. M., Hydroelastic response of marine cables and risers. In Hydrodynamics in Ocean Engineering, Norwegian Institute of Technology, Trondheim, Norway, pp. 1223-1245, 1981. [12] Skop, R. A. & Balasubramanian, S., A new twist on an old model for vortex-excited vibrations. Journal of Fluids and Structures, (11), pp. 395-412, 1997.