Suppression of vortex-induced vibration of a circular cylinder using

Transcription

1 Suppression of vortex-induced vibration of a circular cylinder using thermal effects Hui Wan 1,2, a) 1 1, b) and Soumya S. Patnaik 1 Power and Control Division, Aerospace Systems Directorate, Air Force Research Laboratory WPAFB, OH 45433, USA 2 UES, Inc., 4401 Dayton-Xenia Road, Dayton, OH 45432, USA Abstract Transverse vortex-induced vibration (VIV) of a cylinder with various body-to-fluid density ratio and stiffness is studied. The cylinder is elastically mounted and heated, and the flow direction is aligned with the direction of the thermal induced buoyancy force. Amplitude of VIV can be reduced as the thermal control parameter Richardson number (Ri) increases, or even be fully suppressed when Ri is above a critical value. This critical Richardson number depends on both body-to-fluid density and structural stiffness. A higher critical Richardson is required to fully suppress the VIV of a structure with smaller density ratio. With the same density or mass, a structure with intermediate stiffness vibrating in lock-in regime needs higher critical Ri to suppress VIV than either rigid or flexible structures. Drag experienced by the body is also studied. It is found that for a flexible body, drag gradually increases with the Richardson number. For a body with intermediate stiffness, both drag and amplitude of VIV can be reduced until the Richardson number reaches the critical value, after which drag builds up if the Richardson number is further increased. A drag reduction of 30-40% can be obtained at the critical Richardson number. a) hui.wan.ctr.cn@us.af.mil b) soumya.patnaik.1@us.af.mil 1

2 I. INTRODUCTION Vortex-induced vibration (VIV) of structures is of practical interest to many engineering fields such as aerospace and mechanical systems. The suppression of VIV can be achieved passively or actively. In the category of passive control, people either directly change the surface shape or structure geometry by applying shrouds 1, helical strakes 2, splitter 3, 4, flow-parallel slit 5, hemispherical bumps 6, or change the nearby flow field by placing small control cylinders 7 or tripping wires 8. For example, Kwon et al. 3 used splitter plates to control the laminar vortex shedding behind a circular cylinder. They found that the vortex shedding completely disappeared when the length of the splitter plate is larger than a critical length. Assi et al. 4 measured the vibration response and drag of various splitter plate, pointed out low drag solutions for suppressing VIV of circular cylinders. Owen et al. 6 used hemispherical bumps inducing threedimensional effects to eliminate vortex shedding behind a cylinder. Strykowski et al. 7 found that vortex shedding behind circular cylinders can be suppressed by properly placed small control cylinder in the near wake of the main cylinder. Similarly, Hover et al. 8 used tripping wires modelling the effects of protrusions and attachments, and found that the wire introduces significant changes at certain reduced frequency. In the category of active control of VIV, methods of VIV suppression include acoustic feedback system 9, 10, blowing/suction-based flow control 11-13, oscillatory rotation 14, 15, synthetic jets 16, among others. It is reported 16 there exists a critical momentum coefficient (characterizing the strength of the jets), above which symmetric vortex structure and zero lift oscillation can be obtained. Although various methods mentioned above have been used to suppress the VIV, many of them actually share the same central objective of suppressing the alternate vortex shedding. In this sense, thermal effects can be considered as another factor to control the vortex shedding. 2

3 It has been realized that the importance of thermal effects on vortex shedding from a spatially fixed body. When flow passes a stationary cylinder with elevated temperature, changing of density of fluid near the cylinder induces buoyancy force, which is called thermal induced buoyancy force. Heat transfer in this scenario is so called mixed-convection, since both natural and forced convection are involved. The relative importance of natural convection and forced convection can be characterized by the Richardson number (Ri). In other words, the Richardson number characterizes the ratio between the thermal induced buoyancy force and the inertial force. Chang and Sa 17 numerically reported that periodic flow at Reynolds number (Re) 100 degenerated into steady twin-vortex pattern after certain Grashof number Gr (= Ri Re 2 ). Cheng et al. 18 numerically simulated lock-in effect on convective heat transfer from a circular cylinder under prescribed transverse oscillation. Hasan and Ali 19 reported vortex-shedding suppression in two-dimensional mixed convective flows passing circular and square cylinders. Using the Landau equation, Sengupta et al. 20, Hasan and Ali 19 also extracted the critical Ri for the onset of vortex shedding or suppression. With the flow direction against the thermal induced buoyancy force, Hu et al. 21 experimentally studied thermal effect on the wake of a heated cylinder in mixed convection regime, and found that small concurrent vortices are shed in the near wake when the Richardson number approaches unity. People have realized that the vortex shedding from a stationary body can be altered by certain thermal effects. However, very few studies have discussed the thermal effects on vortexinduced vibration, which is the interest of the current work. Structural properties of the cylinder such as mass and stiffness were translated into two non-dimensional parameters: reduced mass and reduced frequency, respectively. The problem is essentially fluid-structure-thermal coupled. In the present study, the body temperature is considered as another degree-of-freedom in the 3

4 control of VIV. Depending on the body temperature and how flow direction is aligned with the thermal induced buoyancy force, the vibration can be either suppressed or agitated. In this paper, we suppress the vortex-induced vibration of a cylinder by elevating its temperature and aligning the flow in the same direction as the thermal induced buoyancy force to take full advantage of the thermal effects. The flow topology and vortex-induced vibration at different Richardson numbers are numerically studied for structures with various reduced mass and reduced velocity. Exploratory studies such as this can be later extended to provide basis for design consideration of aerospace heat transfer devices. II. MATHEMATICAL FORMULATION With Boussinesq approximation, the non-dimensional Navier-Stokes equation for incompressible flow and the thermal equation can be expressed as: 0 (1) 1 (2) 1 (3) in which,, and are the velocity, pressure, and temperature of the flow field. ( / ) is Reynolds number, the ratio of the inertial force to the viscous force. The diameter of the cylinder is taken as the characteristic length, and the free stream velocity is the characteristic velocity. The fluid properties such as density, thermal diffusivity, kinematic viscosity, and thermal expansion coefficient are denoted by,, ν, and, respectively. The pressure is nondimensionalized by, and the non-dimensional temperature is defined as, where and are the body temperature and the incoming flow temperature, respectively. Prandtl number ( / ) in equation (3) is the ratio of the kinematic diffusivity to the thermal 4

5 diffusivity, thus it is a property of the fluid medium only. In this study, water is considered and its Prandtl number is 7.1. Richardson number Ri ( / ) in equation (2) characterizes the ratio of buoyancy force to the inertial force. When heat transfer between the object and the fluid medium is dominated by forced convection, the Richardson number is far below unity, i.e., Ri 1. Whereas, when natural convection dominates the heat transfer, Ri 1. The term in equation (3) can also be written as the Peclet number Pe (= RePr). The motion of a solid body vibration in fluid can be described by a general dynamics equation of an elastically mounted body as follows: (4) in which the subscript i indicates the motion direction. is the displacement of vibration with respect to the mass center of the body, and is the corresponding force from fluid. The body displacement in one direction alters the flow field and surface pressure on the body, and therefore inherently affects the force in all directions. The mass of the object is denoted by, and, are the damping coefficient and spring stiffness, respectively. For a two-dimensional cylinder as studied in this paper, is the mass per unit length, i.e.,, in which is the density of the cylinder. The natural frequency of the cylinder is defined as /. Thus, of a stiff structure is higher than that of a flexible one, given the same mass. The equation of the body dynamics can be non-dimensionalized as: (5) in which is non-dimensionalized by the cylinder diameter in the current study, and is nondimensionalized by /. On the right hand side, is the fluid force coefficient in the unit 5

6 length, i.e.,. The reduced mass, denotes the mass ratio (equivalent to the density ratio) between the structure and displaced fluid. The reduced velocity on the left hand side is defined as, it hence scales the natural frequency of the structure. As the reduced velocity increases, the natural frequency and the stiffness of the structure decrease, thus the structure becomes more flexible. As damping increases, the vibration amplitude decreases, as shown in Newman et al. 22. Thus, suppression of lower damping or zero damping vortex-induced vibration is more challenging than cases with higher damping. In our study, the damping coefficient and (non-dimensional) are set to be zero to handle the worst scenario case, with a reduction of simulation cost simultaneously. III. NUMERICAL METHOD AND VALIDATION A. Numerical method The Navier-Stokes equations are discretized using a cell-centered, collocated arrangement of the primitive variables (, ), and are solved using Cartesian grid immersed boundary method 23. The equations are integrated in time using fractional step method Second-order central difference schemes in space are employed to both convection and diffusion terms in momentum and temperature equations. The problem setup of flow past an elastically mounted heated cylinder in shown in FIG. 1, in which is the lateral size of the computational domain, and are the domain size in the upstream and downstream of the cylinder. The dashed rectangle covering the cylinder is the zone with refined mesh, which is then stretched to outside boundaries. The flow is aligned with the buoyancy direction (FIG. 1b). The refined mesh around the cylinder is shown in FIG. 1c. 6

7 The velocity boundary condition on the cylinder surface in the current study is non-slip and impermeable. The temperature of the cylinder is set to unity. At the inflow boundary, the velocity is specified as (0,U), and the temperature is 0. At the outflow boundary, the Neumann boundary condition is specified, 0. At the lateral boundaries 0, 0. For pressure, the Neumann boundary condition 0 is imposed on all sides of the computational domain and the body surface, where is the normal direction of the boundaries. (a) (b) (c) FIG. 1. Problem setup of flow past an elastically mounted heated cylinder. The flow direction is aligned with the buoyancy. (a) Sketch of the computational domain; (b) Schematic of elastically mounted cylinder; (c) Grid near the cylinder. The Navier-Stokes equations and the body dynamics equation are implicitly coupled and solved iteratively. The implicit algorithm we have applied to fluid and rigid body coupling is summarized as follows. At each time step, the position of the body in fluid are updated based on the body velocities from the previous step. The Navier Stokes equations are solved to obtain the updated fluid velocity and pressure on the body surface. Surface pressure and shear stress are then integrated and the body velocities are calculated based on the body dynamic equations. This process of solving fluid dynamics and solid-body dynamics is iterated until the convergence 7

8 criteria of pressure and body velocities are obtained. More details on implicit algorithm, and validations on fluid-body interaction can be found in previous studies 25, 26. B. Computational domain and grid convergence The convergence study on the computational domain is shown in Table I, for flow past an elastically mounted cylinder at Re 150. The amplitude of lift coefficient shows good convergence. We thus selected D2 as the computational domain in simulations conducted in session IV. TABLE I. Convergence with computational domain size. Flow past an elastically mounted cylinder, Re = 150, = 2.0, = 4.0. Domain (unit: D) D D D The grid sensitivity study is carried out by varying the grid size in the refined mesh region, and the results of the amplitude of lift coefficient, time-averaged drag coefficient, and Strouhal number is listed in TABLE II. The 0.015D grid is used in our simulations in session IV. TABLE II. Grid convergence study using flow past a fixed cylinder at Re = 150. (unit: D) The time dependence was conducted using = 0.02, 0.01, and for flow past a freelyvibrating cylinder with 2.0, 4.0 at Re 150. The difference between the obtained amplitude of vibration using 0.01 and is 1.6%. Therefore, = 0.01 is used in simulations in session IV. 8

9 C. Validation Three validations related to the current study were presented here, the first is flow past a fixed cylinder, without considering heat transfer. The second validation is the mixed-convection problem of a fixed cylinder. The third validation is flow past an elastically mounted cylinder, to further validate the fluid-body interaction capability. 1. Flow past a stationary cylinder The first validation is flow passing a fixed cylinder. The lift coefficient listed in TABLE III is the maximum value of the non-dimensionalized transverse force. The drag coefficient is the mean value of the streamwise non-dimensional force. The Strouhal number in the table is defined as /, where and are the cylinder diameter and the characteristic velocity. Here is the vortex shedding frequency of the stationary cylinder, extracted from time history of the force coefficient. It can be seen that the force coefficients and Strouhal number match well with literature results. We also presented results of flow past a fixed cylinder at Re = 500 to show the capability of the current solver. The applicability of two-dimensional simulation on Re = 500 is discussed in Blackburn et al. 27 TABLE III. Force coefficients and Strouhal number in flow past a fixed cylinder. Fixed cylinder Re Shiels Qu Present Qu Present Liu Etienne Meneghini Present Blackburn Present

10 2. Mixed convection of stationary cylinders We then consider a mixed convection problem, in which cold flow passes a two-dimensional stationary heated cylinder. Aligning the flow in the same direction as buoyancy, i.e., pointing to the positive y-direction, we study the flow-assisting configuration as the second validation. The time-averaged drag coefficient obtained from numerical simulation on this mixed convection problem is listed in TABLE IV, in which most simulations are on flow past a circular cylinder, except the ones conducted on flow past a square cylinder to make comparison with results from Sharma et al. 33. The comparisons between the present simulation and literature results are reasonably well. For example, at Re = 40 and Pr = 10, the obtained from the current in-house solver matches well with Srinivas et al. s Fluent simulation results 34. For flow past a square cylinder, from the current study is also close to that from Sharma et al. 33 (zero blockage cases therein) at various Richardson numbers. 10

11 TABLE IV. Time-averaged drag coefficient in mixed convection of flow past cylinders. Re Pr Time-averaged Literature Present Patnaik 35 Circular Srinivas et al. 34 Circular , Patnaik 35 Circular Sharma et al. 33 Square 0 ~ ~ ~ Sengupta et al. 20 Circular Patnaik 35 Circular TABLE V. Time-average Nusselt number of flow past a heated stationary cylinder, Re = 100, Pr = Chang et al. 17 Sengupta et al. 20 Hasan et al. 19 Present For flow past a heated stationary cylinder at Re = 100, and Pr = 0.71, TABLE V presents the time-averaged Nusselt number. It is calculated by the integration of along the 11

12 cylinder, then averaged over 20 cycles. The comparisons between the present results and literature are reasonably well. We further examined the critical Richardson number by gradually increasing Ri. Vortex shedding is suppressed once Ri is above This critical Ri is close to the value 0.13 in Sengupata et al Flow past an elastically mounted cylinder Validations on the fluid-body interaction are also conducted against problems in Ahn et al. 36, in which flow passes a two-dimensional elastically mounted cylinder of reduced mass 2 under various reduced velocities (i.e., various structural stiffness). The flow stream is in horizontal direction and the cylinder can freely vibrate in vertical direction. The Reynolds number used in the simulation is 150. Figure 2 shows the phase changes between the lift coefficient ( ) and the transverse vibration displacement at selected structural stiffness. At a larger, e.g., 8.0, the force coefficient and displacement vary out-of-phase. When the lift coefficient is positive, the displacement of the cylinder is negative, and vice versa. At a smaller (e.g., 3.0), the lift coefficient and the cross-flow displacement vary in phase. At the intermediate reduced velocity ( 4.0), the amplitude of the vortex-induced vibration can be as high as 0.6. The phase plots of versus at various reduced velocity obtained in the present work show similar trend to both numerical results in Ahn et al. 36, and the experiments in Khalak et al

13 (a) = 3.0 (b) = 4.0 (c) = 8.0 FIG. 2. Phase plot of lift coefficient and transverse displacement of vibration, Re = 150, = 2.0. The flow structures of two selected reduced velocities are shown in FIG. 3. At 8.0, the structural stiffness is low. The oscillation magnitude of the cylinder is small, as shown in FIG. 2(c). The corresponding flow structure, presented in FIG. 3(a) for the cylinder at the uppermost (y= ) and lowermost (y= ) positions within a typical oscillation period, is therefore similar to the von Karman vortex street obtained from flow past a fixed cylinder. This typical Karman-type wake structure in FIG. 3(a) consists of single vortices (S) with opposite signs, and is identified as 2S mode 38. At 4.0, the oscillation of the cylinder is greater than its lower stiffness counterpart (FIG. 2). It can be seen from FIG. 3(b) that the generated vortices are shed in two rows at

14 (a) = 8.0 (b) = 4.0 FIG. 3. Vortex structure at the maximum and minimum transverse position, Re =150, = 2.0. (a) = 8.0. (b) = 4.0. Flow passes the cylinder from the left to the right. Solid and dash lines stand for positive and negative vorticity, respectively. TABLE VI. Comparison on vibration amplitude and force coefficient of a cylinder in vortex-induced vibration. Re Ahn Borazjani Present Ahn Present Borazjani Present TABLE VI shows comparison on vibration amplitude and lift coefficient amplitude between the present study and literature for VIV problem at Re = 150 ~ 200. The lift coefficients in the current simulation are close to those literature results. Note in TABLE VI that the vibration amplitudes obtained from our simulation changes between 0.58 and We will also see in the 14

15 next session that the highest VIV amplitude is in the range of 0.6 ~ 0.65, depending on the reduced mass and the reduced velocity. These results are consistent with the comments made by Williamson et al. 37, 38 that VIV amplitude in laminar regime is around 0.6. The comparison of cylinder vibration frequency is also made. For example, as Re = 150, =2.0, and =4.0, the vibration frequency in Ahn et al. 18 is 0.232, while it is from our simulation. For Re = 200, =2.0, and =4.0, the vibration frequency calculated from Borazjani et al. 1 was 0.23, our result is IV. RESULTS AND DISCUSSIONS We consider an elastically mounted rigid cylinder that is restrained to only oscillate transverse to the free stream. Here we introduce temperature as another degree of freedom in the control of vortex-induced vibration. As shown in FIG. 1, flow passes through the cylinder from the bottom to the top, the cylinder is subject to vibrate in x-direction when generated vortices are shed from the body periodically. The cylinder is heated, and the thermal effects on the vortex-induced vibration are studied. Reynolds number 150 is used in the following study. The reduced mass varies from 0.8, 2.0 to 20, which can be transferred to the density ratio (specific gravity) as 1.0, 2.55 and 25.5, respectively, recalling the definition of ( ). The reduced mass 2.0 corresponds to aluminum and its alloy when water is selected as fluid medium, while = 20 stands for heavy metal. In the following, we first study the case = 2.0, then discuss the light and heavy reduced mass, respectively. Based on the setup in the current study as shown in FIG. 1, x-direction is interchangeable with the transverse direction. The lift coefficient ( ) and drag coefficient ( ) are the force in x- and y- direction, respectively. 15

16 A. Vortex-induced vibration of structures with various reduced mass Before studying thermal effects on a vortex-induced vibration of a cylinder, we consider a pure fluid dynamics problem of flow passing cylinders with various reduced mass (density) and reduced velocity (stiffness). Again, the Reynolds number Re = 150. The vibration responses such as amplitude (A) and frequency (f) variation with the increasing of the reduced velocity are shown in FIG. 4, with the selected reduced mass as the parameter. As increases, the natural frequency and structure stiffness decreases. In FIG. 4 (a), the amplitude response starts with an initial branch when is small, followed by a synchronization region in which the amplitude of vibration is high. The amplitude is then reduced to 0.1 or even lower when the reduced velocity further increases and enters into unsynchronized region. The amplitude response at this low Reynolds number is different from that at high Reynolds numbers in both amplitude magnitude and the response shape. At high Reynolds numbers, there typically exists an upper branch between the initial branch and the lower branch. The amplitude is close to or higher than unity in the upper branch 38, and is around 0.6 in the lower branch. In our simulations on low Reynolds number flow, there is no upper branch found. Therefore, it is inappropriate to further differentiate upper and lower level based on the amplitude response in low Reynolds number flow. It is also suggested by Khalak et al. 37 that the value of influences whether the upper branch appear or not. This could be the second reason of the absence of upper branch in our study, since the damping is assumed to be zero throughout this study. The maximum amplitude in our low Reynolds number simulations reaches 0.6~0.65, depends on the reduced mass. One similarity between low and high Reynolds number flow is that the reduced mass determines the width of the synchronization regime and maximum amplitude of VIV. More specifically, as shown in FIG. 4(a), increasing of leads to a narrower lock-in regime and a 16

17 smaller amplitude of motion. This is in consistent with the comments made by Jauvtis & Williamson (2004) 40 on effects of the mass ratio that lower mass ratio typically results in a larger amplitudes of motion and a widening response range of reduced velocity. Also note from FIG. 4(a) that as = 5.0~6.0, the vibration amplitude A is insensitive to the reduced mass. The body oscillation frequency after periodic vortex-induced vibration reached is shown in FIG. 4(b) for various reduced mass. Take = 2 as an example, the oscillation frequency at low, e.g., = 1.0, is basically identical to the vortex shedding frequency of a fixed body (see Table I). It rises as increases (the structure gets more flexible) and approaches maximum when = 4.0. It then decreases until reaches 7.0, after which it rises again and levels off as further increases. Consider the other two reduced mass, the lower reduced mass generates larger variation in the oscillation frequency over wider response range of. Also, for = 20, the structure oscillates at a similar frequency when is at the low and high extremes. FIG. 4(c) shows /, the ratio between the body oscillation frequency and the natural frequency, in which ~1 indicates the lock-in region where body oscillates near its natural frequency. We still take = 2 as an example. As the stiffness k reduces, the structure gets more flexible and its natural frequency decreases. When increases from 1.0 to 4.0, ascending of can be ascribed to two reasons: one is the increasing of body oscillation frequency as shown in FIG. 4(b); another is the decreasing of structural natural frequency. In this region of, it seems that the structure naturally adjust its oscillation frequency to match up with its natural frequency until ~1 is approached. As is in the range of 4.0~7.0, the natural frequency is further decreased. It can be seen from FIG. 4(b) that the oscillation frequency decedents in this region as well, so that is kept around unity. If is further 17

18 increased to be greater than 7.0, the oscillation frequency fails to follow the reduction of, instead, the oscillation frequency rises up and levels off. The increasing of in this range is mainly due to the reduction of. The other two reduced mass = 0.8 and = 20 generate the similar trend of frequency variation as = 2. Again, it can be seen that the body with smaller reduced mass has a wider synchronization range, i.e. wider region of in which is close to the unity. We then study the thermal effects on vortex-induced vibration, mainly using = 2 as the example. Results of the other two reduced mass will also briefly covered. Based on the amplitude and frequency response of the structure with = 2 shown in FIG. 4, here for convenience we classify the structure with 8 as flexible, 3 as rigid, and 4 7, the lock-in region, as intermediate. (a) (b) (c) FIG. 4. Responses of cylinders versus reduced velocity at various reduced mass at Re = 150, (a) amplitude (b) oscillation frequency (c) ratio of oscillation frequency to natural frequency. B. Thermal control on structure with reduced mass = 2.0 For a cylinder with reduced mass 2.0, the reduced velocity = 8.0 and 4.0 were selected to study the flow field. Based on FIG. 4, these two reduced velocities correspond to the unsynchronized region and lock-in region, respectively. At = 4.0, the vibration obtains its maximum amplitude. Flow field generated by a vibrating rigid structure with small ( 3) is 18

19 not shown here, since its flow behavior is similar to that of a fixed cylinder. However, the amplitude response of thermal effects and experienced drag for = 3 are discussed and compared with those of structures with intermediate and high reduced velocities. 1. Reduced velocity = 8.0 At the reduced velocity 8.0, the cylinder vibrates in the unsynchronized region. We start with the uncontrolled case (Ri = 0), in which the natural convection effect is in absence. The flow field structure is presented in FIG. 5. At the initial stage, e.g., t = 30, the cylinder remains stationary and symmetric wake is developed (FIG. 5a). The cylinder therefore experiences zero net force in the transverse direction (x-direction in the current setup). At t = 50, instabilities develop in the wake, and the wake symmetry is broken. The wake away from the cylinder shows stronger asymmetry than the wake right after the cylinder. As time increases, the typical von Karman vortex street is obtained (FIG. 5c, d). FIG. 6 shows the time course of the cylinder x- displacement in the transient and the periodic states. The non-dimensional amplitude of the oscillation is about 0.08 (i.e., 8% of the cylinder diameter). The force in the transverse direction is also shown, and it can be seen that the force and displacement has a phase shift of 180⁰. This is consistent with the phase plot shown in FIG. 2c. 19

20 (a) t = 30 (b) t = 50 (c) t = 60 (d) t = 1950 FIG. 5. Vortices development generated by a cylinder in vortex-induced vibration, = 2.0, = 8.0, Ri = 0. (a) transient (b) steady-state oscillation FIG. 6. Time history of cross-stream displacement and lift, = 2.0, = 8.0, Ri = 0. Lift is not shown in the transient plot. Based on the configuration shown in Fig. 3, is the force coefficient in x- direction. The effects of natural convection increases gradually by increasing the Richardson number Ri. Vortex structures of periodic vibration is shown in FIG. 7 for various Ri. At small Ri (e.g., 20

21 0.1 ~ 0.2), the obtained flow structures are similar to those from pure forced convection (Ri = 0). As shown in FIG. 7, the width of the vortex street gets narrower when Ri increases. The x- displacement of the vibration is presented in FIG. 8, and it can be seen that the amplitude of the oscillation is gradually reduced when Ri increases. As the Richardson number increases to 0.25, the magnitude of VIV is about 3.7% of the cylinder diameter. When the Richardson number is increased to 0.3, there are no vortices formed and shed behind (above) the cylinder, whose typical flow structure is presented in FIG. 7(d) at t = 500. The wake behind the cylinder wiggles around the vertical centerline, but not large enough to induce strong vibration of the cylinder, with the vibration amplitude reduced to 0.8% of the cylinder diameter (FIG. 8). As Ri is increased to 0.5, the amplitude of VIV is around 0.3% of the cylinder diameter (not shown here). The time-averaged drag coefficient is shown in FIG. 17, in which the drag keeps increasing as Ri rises. (a) Ri = 0.1 (b) Ri = 0.2 (c) Ri = 0.25 (d) Ri = 0.3 FIG. 7. Vortex structure when the cylinder is located at the left-most position, = 2.0, =

22 FIG. 8. Time history of the x-displacement under various Ri. Structure with = 2.0, = Reduced velocity = 4.0 At the reduced velocity 4.0, the structure vibrates in the lock-in region with a high amplitude of oscillation (FIG. 4). Again, we first consider the case without heat transfer, i.e., Ri = 0. The oscillation displacement in the transient and the periodic states are shown FIG. 9, in which the oscillation is initiated at a non-dimensional time t~50. It then reaches a steady periodic oscillation after t~90. The force coefficient in transverse direction ( ) is also shown in FIG. 9. It can be seen that is in phase with x-displacement. The flow structure at = 4.0 is composed of two column vortices, as shown in FIG. 10, in which the width of the separation (S) between two fully developed vortices right after the cylinder is measured as 2.17D. The vortex structure in FIG. 10 is essentially similar to that in the validation case, except a 90º rotation. 22

23 (a) transient (b) steady-state oscillation FIG. 9. Time history of the displacement of cross-stream vibration 2.0, = 4.0, Ri = 0. (a) rightmost (b) middle (c) leftmost FIG. 10. Vortex structure of a VIV cylinder at various locations, = 2.0, = 4.0, Ri = 0. 23

24 We then investigate the thermal effects on the structure with = 2.0 and = 4.0, with Ri varies from 0.3 to 2. The displacement in the x-direction in transient and steady oscillation is shown in FIG. 11. It can be seen that, as the Richardson number increases, the initiation of the VIV is delayed, and it takes longer time to reach periodic vibration. The amplitude of the oscillation is also suppressed when Ri increases. At Ri = 0.3, the oscillation amplitude is 0.54, which is reduced by 11.5% compared to the case = 0 (FIG. 9). At Ri = 0.5, the amplitude is around 0.48, i.e. about one half of the cylinder diameter. At Ri = 0.7, the vibration magnitude is approximately reduced to The vortex structure for 0.3 is shown in FIG. 12, in which the separation S between two fully developed vortices right after the cylinder is about 0.82D, much smaller than that in the previous two-column vortex structure (FIG. 10). (a) transient (b) steady phase FIG. 11. Time history of the displacement of transverse vibration, Ri = 0.3, 0.5, 0.7 and 0.8. A structure with = 2.0, =

25 (a) rightmost (b) middle (c) leftmost FIG. 12. Vortex structure when cylinder located at two extreme positions (a) and (c), and the middle (b). = 2.0, = 4.0, Ri = 0.3. The development of flow structure and the temperature contour for Ri = 0.7 are shown in FIG. 13. It can be seen that at initial stage t = 5, typical symmetric vortices were developed above the cylinder. At t = 15, the two typical vortices right behind the cylinder were stretched. Further behind these two typical vortices, the vorticity near the centerline of the cylinder changed the sign. The fully developed 2S vortex structure at t = 500 is presented in FIG. 13c. The temperature field basically resembles the vorticity field once the flow is fully developed. 25

26 (a) t = 5 (b) t = 15 (c) t = 500 (d) t = 5 (e) t = 15 (f) t = 500 FIG. 13. Development of vortex structure (a-c) and temperature field (d-f) at = 2.0, = 4.0, and Ri = 0.7. As Richardson number is further increased to 0.8, the vibration is fully suppressed during the simulated time range. The flow structure in transient stage is similar to the case of Ri = 0.7. The 26

27 vortex structure for Ri = 0.8 at t = 1000 is shown in FIG. 14, in which the flow is symmetric and there is no alternate vortex shedding behind the cylinder. The two-column vortex structure in the uncontrolled case has been changed into two stationary symmetric vortices right behind the cylinder. A stable thermal plume is formed in the wake of the cylinder (FIG. 14b). The pressure field is symmetric about the cylinder centerline, with high and low pressure near the frontmost and the rearmost points, respectively (FIG. 14c). Thus, the body experiences zero lift, in this case, in the x-direction. Note that the flow structure experiences an abrupt change when Ri increases from 0.7 to 0.8. The VIV amplitude is reduced from 0.42 to the order of 10-4, and the vibration is literally fully suppressed. Hence, there exists a critical Richardson number between 0.7 and 0.8, above which the vortex shedding is fully suppressed. (a) vortex structure (b) temperature (c) pressure field FIG. 14. Flow past an elastically mounted cylinder with = 2.0, = 4.0 at t = 1000, Ri =

28 3. Time-averaged flow field, = 2.0 and = 4.0 The time-averaged vorticity field is shown in FIG. 15(a) for flow past fixed cylinder, and in FIG. 15(b-f) for elastically mounted cylinders with = 2.0 and = 4.0. The average is conducted over 20 steady-state oscillations for cases with vortex shedding. For cases without alternative vortex shedding, the time average is taken over 100 non-dimensional time, since the cylinder may still experience tiny oscillation as mentioned in session 2. The time-averaged velocity vector at various streamwise positions is also presented, as indicated in FIG. 15 (a). For example, y = 1D specifies the position one diameter after the body center. For the fixed cylinder, there is a recirculation region in which the flow is reversed, as shown by the velocity vector at y = 1D in FIG. 15 (a). The dashed circle in Ri = 0 and Ri = 0.7 outline the averaged-location of the vibration body. When flow past elastically mounted cylinders, there is no obvious recirculation region founded, especially when Ri increases. The velocity deficit region is formed in the wake of fixed cylinder, and of cases Ri = 0 and Ri = 0.7 even at y = 5D. As Ri is higher than 0.8, a velocity deficit region exists only right after the cylinder. Since the vortex-induced vibration is fully suppressed and a jet is formed above the cylinder, a velocity excess region therefore exist near the centerline. Here we define the apparent half-width ( / ) of velocity deficit/excess region at certain streamwise position as the distance to the centerline from the point where the averaged velocity component is 99% of the upcoming flow. The measured half-width is listed in TABLE VII, in which a number followed by e in parenthesis indicates it is velocity excess region, otherwise it is velocity deficit region. For the flow past fixed cylinder, the apparent halfwidth is 0.89D at the cutline y = 2D, and 1.68D at y = 5D. The apparent half-width in the wake of the case Ri = 0 is 1.48D at y = 2D, and 1.74D at y = 5D. Apparently, the wake of case Ri = 0 is wider than that of fixed cylinder, especially in the near field. For Ri = 0.7, the half-width velocity deficit region is reduced compared to that in Ri = 0. As Ri further increases to 0.8, velocity 28

29 deficit region in the near field is gradually changed to velocity excess at y = 5D. When Ri is even higher, such as 1 or 2, the velocity excess region extends into the near field due to the generated jet. For instance, y = 2D is changed to the velocity excess region when Ri is 2. TABLE VII. Apparent half-width ( / ) of velocity deficit/excess region in wakes. A number followed by e in a parenthesis indicates the velocity excess region, otherwise it is velocity deficit region. The value is non-dimensionalized by cylinder diameter D. / Fixed Ri = 0 Ri = 0.7 Ri = 0.8 Ri = 1 Ri = 2 y = 2D (e) y = 5D (e) 0.40 (e) 0.46 (e) (a) Fixed (b) Ri = 0 (c) Ri = 0.7 (d) Ri = 0.8 (e) Ri = 1.0 (f) Ri = 2.0 FIG. 15. Time-averaged vorticity field and velocity vectors at various streamwise locations. = 2.0, =

30 The velocity component in y-direction along the body vertical centerline (x = 0) is drawn in FIG. 16, in which -0.5D 0.5D is the position of the cylinder. As y -0.5D, flow approaches the cylinder with reducing speed. For the fixed cylinder, the velocity is reversed in the region of -0.5D 1.55D, and gets its minimum of around y=1.06d. After the recirculation region, starts to increase and reaches at y = 5.45D, it then decreases and approaches to after y >10D. For the case Ri = 0, the cylinder subjects to strong vibration, along the centerline approaches to 0.21 at y = 12D, indicating a larger velocity deficit and a higher drag compared to the fixed cylinder. For Ri = 0.3, the velocity component is smaller than unity in the region shown in FIG. 16. At Ri = 0.7, recovers to unity when is around 7. As Ri further increases, velocity takes shorter length to recover to unity, indicating shortening of velocity deficit region. FIG. 16. Time-averaged velocity component along the centerline. = 2.0, =

31 4. Drag coefficient The vibration amplitude A and the time-averaged drag coefficient at various Richardson numbers and reduced speeds are shown in FIG. 17 for structures with reduced mass 2.0. Time average is conducted in the same way as that in the previous section. The reduced speed = 3.0 stands for a rigid body with high stiffness, while = 8.0 means a flexible body with low stiffness. It can be seen that the amplitude of VIV monotonically decreases as Ri increases, until fully suppression is obtained. At = 3.0, VIV is fully suppressed when Ri is around For 4.0, a critical Ri exists near At = 7.0, VIV is suppressed once Ri reaches 0.4. For the flexible structure with 8.0, VIV is fully suppressed when Ri is around 0.3. Obviously, higher Ri is needed to suppress the vibration of structures in the lock-in region. The time-averaged drag coefficient is presented in FIG. 17 (b), in which the solid dot on y-axis specifies of a fixed cylinder at Reynolds number 150. Note for the structure with = 8.0, at Ri = 0 is around 1.23, which is even smaller than the drag coefficient of a fixed cylinder. This indicates that a flexible structure may experience smaller drag compared to its fixed-body counterpart, under the same Reynolds number. For the case of = 8.0, almost monotonically rises as Ri increases. At Ri = 0.3, reaches about the same drag value of fixed cylinder. Further increase of Ri unnecessarily generates higher drag, since the VIV is already suppressed when Ri is around 0.3. For the rigid structure with 3.0, the drag almost keeps at a constant until Ri is increased to 0.25, at which a sharp force reduction occurs. For = 7.0 structure, gradually decreases as Ri rises to 0.25, after which starts to increase. For the more interesting case 4.0, the time-averaged drag coefficient is as high as 2.37 at Ri = 0, which is way greater than that of the fixed cylinder. The high results from 31

32 high amplitude of vibration in the lock-in region. The corresponding surface pressure distribution around the cylinder is presented in the next session. We take at Ri = 0 as the reference. When Ri rises, is reduced until Ri reaches 0.8, at which is lowered to 1.53, a reduction of 35.4% compared to the reference. As Ri further increases, starts to increase. When Ri reaches 2.0, is about 1.87, which is a reduction of 21.2% with respect to the reference. Thus, for the structure with 4.0, both VIV amplitude and drag experienced by the body can be reduced, at an optimized Richardson number such as 0.8. Note once VIV is fully suppressed, the structure stiffness ceased to play a role in the flow, the drag coefficients of structures with various stiffness collapse to each other, as shown in FIG. 17. (a) vibration amplitude (b) time-averaged drag coefficient FIG. 17. Vibration amplitude (a) and time-averaged drag coefficient (b) at various Richardson number. 2.0, Re = 150. The solid dot in (b) indicates the drag coefficient of the fixed cylinder at same Reynolds number. 5. Surface pressure distribution 32

33 Time-averaged surface pressure distribution of flow past elastically mounted cylinder under various Richardson numbers is presented in FIG. 18. The angle α was labeled in FIG. 1(b) in which = 0 and = 180 correspond to the frontmost and the rearmost points on the cylinder, respectively. The surface pressure of a fixed cylinder at Re = 150 is also shown, with a maximum around 1.09 at the frontmost point and a minimum around near 81 ~82, after which the surface pressure recovers and levels off with around at the rearmost point. This result is quite close to that of Qu et al. 29 For an elastically mounted cylinder of = 2.0, = 4.0 (FIG. 18a), the body subjects to strong vortex-induced vibration at Ri = 0. The pressure near the frontmost point is 0.46, and it monotonically drops to at the rearmost point. There is no pressure recovery region on surface of the cylinder. Compared to the fixed body case, the pressure at the frontmost point is lower, due to the dramatic cylinder vibration. The pressure at the rearmost point is much more negative than that of the fixed cylinder. This explains high drag coefficient experienced by the VIV cylinder at Ri = 0. As Ri increases, the amplitude of VIV decreases, the pressure coefficient at = 0 falls between the cases of fixed cylinder and the VIV cylinder without thermal control (Ri = 0). When Ri is above 0.8 and the vibration is fully suppressed, the pressure at the frontmost point is 0.98 ~ 1.0, which is slightly lower than that obtained in the fixed cylinder. This may be ascribed to the thermal boundary layer created by heated cylinder. The pressure distribution in the backside of the cylinder is more interesting. At Ri = 0.3 and Ri = 0.7, the minimum pressure coefficient occurs respectively around 117 and 106, after which the surface pressure increases and levels off. At Ri 0.8 and 1.0, descents to the minimum value of and -2.36, respectively, near the region ~ The surface pressure coefficient then rises up to

34 and at ~ 180, respectively, which are very close to at the rearmost point of a fixed cylinder. Comparing the cylinder with = 4.0 at various Ri with the fixed cylinder, surface pressure coefficient near the front portion of the cylinder (e..g, = 0 ~30 ) is positive but lower than that of the fixed cylinder; in the back portion of the cylinder ( = ) is negative and lower than that of fixed cylinder (except the very short region near = 180 when Ri = 2.0). Therefore, all VIV cylinders with = 2.0, = 4.0 experience higher drag compared to the fixed cylinder. Considering the VIV cylinder under thermal control (Ri > 0) and the one without thermal control (Ri = 0), we can see that the surface pressure recover region appears when Ri > 0.3. As Ri increases to 0.8, the recover region starts earlier (with smaller ), both the minimum pressure coefficient and at = 180 get less negative compared to the uncontrolled case. Thus, the minimum drag is obtained at Ri around 0.8. As Ri further increases from 0.8 to 2.0, there is no dramatic change in the starting angle of the recover region, as well as value near ~ 180. However, the surface pressure gets more negative over a wide region (e.g., 60 < <150 ) compared to obtained in Ri = 0.8. Hence, further increasing of Ri after 0.8 causes drag coefficient rising up, as shown in FIG. 17 (b). For an elastically mounted cylinder of = 2.0, = 8.0 (FIG. 18b), the cylinder performs mild vibration with amplitude only 8% of its diameter when Ri = 0. The surface pressure coefficient near the frontmost point is lower than that of the fixed cylinder, and on the rest part of the surface is fairly close in these two cases. Thus the cylinder reduces the front surface pressure and therefore the drag coefficient by mild vortex-induced vibration. As Ri increases, near the regions of = 0 and = 180 varies little, but the minimum surface 34

35 pressure is lowered consistently and the cylinder surface subjects to more negative pressure. This explains the rising of drag coefficient in FIG. 17 (b) when Ri increases. (a) = 4.0 (b) = 8.0 FIG. 18. Surface pressure distribution of flow past a cylinder at various Ri, = 2.0, Re = 150. C. Thermal control on structure with reduced mass = 0.8 When the reduced mass is 0.8, the density ratio between the solid body and the fluid is around 1. As shown in FIG. 4, for the case of 0.8, the maximum oscillation is obtained when the reduced velocity is around 4.0. We thus select = 4.0 as our study case. The displacement and force coefficient in the cross-stream direction are shown in FIG. 19 for the uncontrolled case. The cylinder vibrates in lock-in regime, and the transverse force and body displacement oscillate in phase. The Richardson number is then increased to see its effects on vibration amplitude A and the experienced time-averaged drag, as shown in FIG. 20. The time averaged drag coefficient is 2.32 when Ri = 0. When Ri increases from 0 to 0.9, the vibration amplitude and the drag decrease almost linearly. Near Ri = 1, the VIV is fully suppressed, both A and drop sharply. The minimum drag is obtained at Ri = 1.02, with a value of 1.60, accounting for 69.2% of the drag 35

36 compared to the case of Ri = 0. As Ri further increases, the cylinder remains stationary and the drag starts to increase. At Ri = 2.0, the drag coefficient rises to 1.87, a reduction of 19.4% compared to the uncontrolled case. (a) Transient (b) Steady-state oscillation (c) Phase plot FIG. 19. Time history of the displacement, force coefficient in cross-stream direction, and phase plot, 0.8, = 4.0, Ri = 0. FIG. 20. Vibration amplitude and time-averaged drag coefficient versus Ri. The structure with = 0.8, = 4.0, Re =

37 D. Thermal control on structure with reduced mass = 20 When the reduced mass is 20, the density ratio between the solid body and the fluid is high. Referring to FIG. 4, the maximum oscillation is obtained when the reduced velocity is around 5.0 for the reduced mass 20. Thus, = 5.0 is selected to study the thermal effect. The vibration displacement and lift coefficient are shown in FIG. 21 for the uncontrolled case, in forms of time history and phase plot. The transient stage for high reduced mass is much longer than that for low reduced mass (e.g. 0.8 in FIG. 19), since it takes longer time to overcome the inertia of heavy bodies. The lift coefficient and transverse displacement oscillate in phase, but the lift coefficient contains more frequency components. The lift coefficient increases up to 0.85 in the transient stage then gradually decreases to 0.25 in steady oscillation, while the vibration amplitude monotonically increases during the transient. This combined effect results in the phase plot as shown in FIG. 21 (c). The vibration amplitude A and the time-averaged drag coefficient are shown in FIG. 22 as Ri varies from 0 to 1.0. At Ri = 0, the oscillation amplitude and the time-averaged drag coefficient are 0.59 and 2.10, respectively. For this high reduced mass, the critical Ri is around 0.30, at which the vibration of the cylinder is considered as fully suppressed. The drag coefficient experienced by the cylinder is around 1.27, a reduction of 39.5% compared to the case of Ri = 0. As Ri further increases above the critical value, drag increases gradually and reaches 1.60 when Ri is 1.0. The critical Ri and time-averaged drag coefficient are summarized in TABLE VIII, in which cylinders in all the listed cases vibrates in the lock-in region if without thermal control. It can be seen that the critical Ri increases as the reduced mass decreases, i.e., a cylinder with smaller body-to-fluid density ratio needs stronger thermal control to fully suppress VIV. At the critical Ri, 30%-40% reduction in can be achieved, depending on the reduced mass. 37

38 (a) transient (b) steady-state oscillation (c) phase plot FIG. 21. Time history of the displacement of cross-stream vibration 20, = 5.0, Ri = 0. FIG. 22. Vibration amplitude and time averaged drag coefficient versus Ri, M red = 20, U red = 5.0. TABLE VIII. Summary on critical Richardson number and time-averaged drag coefficient. Critical Ri at Ri = 0 at critical Ri Reduction % % % 38