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Experimental investigation of Reynolds number and spring stiffness effects on vortex induced vibrations of a rigid circular cylinder
European Journal of Mechanics / B Fluids 74 (2019) 34–40
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European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu
Experimental investigation of Reynolds number and spring stiffness effects on vortex induced vibrations of a rigid circular cylinder Alireza Modir a , Navid Goudarzi b , a b
∗
Department of Ocean and Mechanical Engineering, Florida International University, United States The William States Lee College of Engineering, University of North Carolina at Charlotte, Charlotte, NC 28223, United States
article
info
Article history: Received 17 January 2018 Received in revised form 5 June 2018 Accepted 17 October 2018 Available online 8 November 2018 Keywords: Flow induced motions Natural frequency Ocean energy Spring stiffness Towing tank Vortex induced vibration
a b s t r a c t The Vortex Induced Vibration for Aquatic Clean Energy (VIVACE) is a breakthrough technology, which harvests the hydrokinetic energy of ocean currents through enhancing Vortex Induced Vibration (VIV). In this paper, the VIV of an elastically mounted circular cylinder in a towing tank water channel is studied experimentally to investigate the effect of natural frequency, by using five different springs in a wide range of stiffness (125 N/m
1. Introduction
wake is known as the Strouhal number (St):
Vortex-Induced Vibration (VIV), which occurs in many fields of engineering, has been studied extensively, especially in the past fifty years, due to its significant implications in different engineering applications including offshore structures, cooling towers, bridges, heat exchangers and electrical transmission lines. VIV of an elastically mounted circular cylinder in fluid flow has been studied experimentally and numerically in the literature and have been covered in several overviews such as [1–4], and more recently [5]. Fluid flow passing a fixed circular cylinder generates vorticity due to the shear present in the boundary layer which results in the appearance of vortices on either side of the cylinder. By an increase in Reynolds number (Re > 40), the perturbations in the upstream flow cause one of the vortices to grow larger [6]. This vortex, with higher flow velocity and lower pressure, pulls the smaller vortex from the opposite side along the wake centerline and allow it to flow downstream. The repetition of this process leads to a periodic vortex shedding from either side of the body, altering the pressure distribution which results in acting periodic forces on the body [7]. The nondimensional frequency that the vortices are shed into the
St =
∗ Corresponding author. E-mail address: [email protected] (N. Goudarzi). https://doi.org/10.1016/j.euromechflu.2018.10.016 0997-7546/Published by Elsevier Masson SAS.
fv s D
(1) U where fv s is the vortex shedding frequency for a stationary cylinder, D is the cylinder diameter, and U is the relative velocity between the fluid and solid. Fig. 1 shows the relationship between Strouhal number and Reynolds number for stationary circular cylinders. In a large range of Reynolds numbers, the Strouhal number can be considered as constant. For the entire range of about 103 < Re < 105 , the Strouhal number is nearly 0.2 for smooth surfaces [8], which corresponds very well to the range of Reynolds number in our experiments. VIV phenomenon happens when the cylinder is elastically limited. Structural vibration, both inline and transverse to the flow, occurs due to the existence of a time-varying nonuniform pressure distribution around the body [9]. Lock in can be considered like linear resonance, as the vibration amplitudes increase significantly when the vortex shedding frequency becomes close to the natural frequency of the structure(fn ). In this situation, the nondimensional frequency (f ∗ = fosc /fn ) remain close to unity and the amplitude of oscillation increases dramatically [10]. Synchronization is defined as the matching of the vortex shedding frequency (fs ) with oscillation frequency (fosc ) [7]. In many papers ‘‘Lock-in’’ and ‘‘Synchronization’’ are used interchangeably.
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Fig. 1. St-Re relationship for circular cylinders [8].
It worth to mention that ‘‘Lock-in’’ and ‘‘Synchronization’’ have different meanings. ‘‘Lock-in’’ occurs when the vortex shedding frequency beaks from the Strouhal relationship and fits the natural frequency of the system and ‘‘Synchronization’’ happens when vortex shedding frequency beaks from Strouhal relationship but matches a frequency that is not essentially the natural frequency of the system. Consequently, lock-in is a special case of synchronization [11]. VIV is fundamentally a nonlinear phenomenon. In one hand, the structural vibration is induced by vortex shedding, on the other hand, the vibrating structure in turns affects the flow field and the resulted fluid forces applied on the structure. VIV is also known as a self-limiting phenomenon because as the cylinder displacement increases; the vortex shedding becomes weak, and hence tends to reduce further motion [12]. As described in [13], there are two distinct types of response for a VIV system, depending on the working fluid and mass-damping parameters (m∗ ζ ). The first one with a small amplitude which corresponds to high mass-damping parameter systems and mostly use air as its working fluid has two branch responses which are named ‘‘Initial’’ and ‘‘Lower’’ branches [10]. The second one which is known as a three branch response is often the result of low m∗ ζ . The working fluid in this type is commonly water. The third branch in this type has a higher amplitude that is called ‘‘Upper’’ branch of response [10]. The normalized oscillation amplitude can be defined as the ratio of amplitude A over the cylinder diameter D. Govardhan and Williamson suggested that A∗ = 0.6 criteria can be used to determine whether the amplitude has two or three branches [2]; if A∗ > 0.6, there will be three branches. The reduced velocity U ∗ in the literature is defined as U ∗ = U /fn D. Fig. 2 compares classical examples of two and three response branches at different reduced velocities [13]. Table 4 in the Results Section summarizes the Re number and mass ratios used in this experiment. In many engineering problems which involve fluid-flow passing a bluff body, VIV is classically treated as a ruinous phenomenon due to the resulting fatigue damages. Most previous researchers have mainly focused on reducing the destructive effects of Flow-induced motion. Unlike the former efforts, in suppressing the occurrence of VIV as a potentially damaging phenomenon for structures subjected to a fluid flow [14], in 2005, professor Bernitsas and his group at the University of Michigan invented a new method for extracting water current’s energy by utilizing VIV. The apparatus which is a hydrokinetic power-generating device is named VIVACE (Vortex Induced Vibration for Aquatic Clean Energy) [12]. The simplest form of the VIVACE converter comprises of an elastically mounted rigid circular cylinder which is connected to a power take-off system. The device can harness ocean and river’s current energy even in speeds as slow as 0.4 m/s [12]. They used different methods to increase the amplitude of oscillation for the purpose of increasing the efficiency of the extracted energy [15,16]. Although VIV of a cylinder with two degrees of freedom has been studied by
Fig. 2. Two and three branch response experiments, done by Feng and Williamson [13].
some researchers, for example [17] that is out of the scope of this paper. The two most important performance metrics that can effectively characterize VIV are the frequency and amplitude of vibration which are influenced by some system parameters such as mass ratio, spring stiffness, structural damping, and surface roughness. The effect of mass ratio as an important parameter in the behavior of VIV has been studied by many researchers like [18] and [19,20]. Also, the effect of damping as an important parameter in extracting water energy in VIVACE converter is studied in [16]. To the best knowledge of authors, there are few studies on investigating the spring stiffness effect of a circular cylinder in a VIV system. Bao et al. have numerically studied the effect of natural frequency ratio (fx /fy ) in a two-degree-of-freedom single cylinder and tandem cylinders [21]. The Reynolds number in their experiments was low and fixed at Re=150. Zahari et al. [22] studied the effect of spring stiffness on VIV in five cylinders with different mass’ and diameters but constant length, spring stiffness, and fluid velocity. In a numerical investigation that is done to analyze the effect of natural frequency ratio in VIV of a cylindrical structure at low Reynolds number of 200, Huynh and his coworkers [23] experimentally studied the effect of nonlinear springs, called hardening springs, on VIV of a circular cylinder. They used cantilever beams and additional plates for modeling nonlinear spring stiffness and realized that using the nonlinear springs, broaden the resonance range. Lee and Bernitsas at [16] used a virtual damper–spring apparatus to investigate the effect of damping and spring stiffness on the performance of the VIVACE converter, by adjusting a software for controlling these parameters. Based on the limited range of stiffness’ that they used, a general conclusion could not be made. Although the natural frequency of the system is an essential parameter in analyzing the vortex-induced vibration, to the extent of our knowledge, still investigations on the effect of spring stiffness on the behavior of a cylinder in VIV has not been well documented. In the present paper, the effects of spring stiffness on the amplitude response and oscillation frequency of a rigid circular cylinder in one degree of freedom is experimentally studied. While there has been a number of literature on exploring the amplitude response at different Re ranges, there has been so few studies on the VIV response to a range of spring stiffness values. The experiments are done in the fully turbulent shear layer (TrSL3) Re number regime (1.5 × 104 < Re < 6 × 104 ) with a high mass-damping parameter (m∗ ζ = 0.11). This Re range starts from the high-end TrSL2 that was studied by Khalak and Williamson [18] and overlaps
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Table 1 Experimental parameters. Parameter
Symbol
Value
Cylinder diameter (mm) Cylinder Length (mm) Stiffness (N/m) Mass Ratio Velocity (m/s) Reynolds Number Damping ratio
D L K m∗ U Re
60 380 125-495 2.26 0.3-1.5 1.5 × 104 –6 × 104 0.05
ζ
with Raghavan and Bernitsas [15] results at TrSL3. Hence, this work fills the gap for experimental results between TrSL2 and TrSL3. Five different linear springs were used to have a better exploration of the effect of the natural frequency of the system on the behavior of the cylinder. The results from this work can be used by other researchers to design adjustable stiffness systems based on the flow velocity for extracting a maximum energy from currents. Fig. 3. Schematic of the VIV test model. Main components: 1: Cylinder, 2: Side strut, 3: Cylindrical bearings, 4: Springs, 5: Wheels, 6: Camera, 7: Rail, 8: Towing cable.
2. Experimental setup All experiments were conducted in a Towing Tank Water channel with 14 m length, 0.50 m width, and 0.60 m depth. This channel was designed and fabricated to study the flow in a wide range of velocities over a flexibly mounted circular cylinder. By towing the submerged cylinder along the channel, the situation of the circular cylinder subjected to fluid flow was modeled. The carriage set-up was running on top of two very smooth rails to eliminate the undesirable external vibrations. To simulate steadyand uniform-current conditions, the structure was towed through initially still water; hence, in comparison with free surface water channels, the turbulence intensity could be completely eliminated. By connecting an electrical inverter to a three-phase electric motor which had the maximum rotational speed of 3000 rpm, the speed of the carriage was controlled with a high accuracy. The range of towing carriage speed resulting in VIV was 0.3–1.5 m/s, so the corresponding reduced velocity and Reynolds number were 3 < U∗ < 13 and 1.5 × 104 < Re < 6 × 104 respectively. The cylinder oscillatory displacement was measured by an advanced digital camera, recording 30 frames per second, and an indicator which showed the position of the cylinder during the experiments. A smooth surface polyethylene cylinder with a density of 971 kg/m3 , the diameter of 0.060 m and length of 0.380 m was used in all experiments. The aspect ratio (Length/Diameter) of the cylinder was 6.3 and the blockage ratio inside the channel was 11.8%. The main properties of the experimental set-up are listed in Table 1. Figure three illustrates the schematic VIV test model. The cylinder flexible movement was achieved by connecting it to the coil springs through struts. It was constrained to oscillate transversely to the towing direction using two cylindrical bearings. One may refer to [19] for more details concerning the experimental set-up. Here we used five different springs with different stiffness’ to analyze the effect of natural frequency of the system on the behavior of a cylinder undergoing VIV. Natural frequency in water for different springs can be obtained from Eq. (2), where K = spring stiffness constant, mosc = oscillating mass and ma = ideal added mass. The added mass is ma = Ca md , where displaced fluid mass is md = ρw π D2 L/4 and Ca is the ideal added mass coefficient that is equal to 1 for a circular cylinder [2]. In this paper, the reduced velocity is defined by the natural frequency in still water (U∗ = U/fn,w D). fn,w =
1 2π
√
K mosc + ma
(2)
As it is shown in Eq. (3), the mass ratio is defined as the total mass divided by the displaced fluid mass (md ) where the total
Table 2 Spring stiffness and natural frequency of 5 different cases used here. Case
1
2
3
4
5
Spring stiffness (N/m) Natural frequency (1/s) Structural damping (N s/m)
125 0.95 2.3
200 1.2 2.9
250 1.3 3.2
365 1.6 3.9
495 1.9 4.6
mass includes the oscillating mass plus 1/3 of the spring mass (Components 1, 2, and 4 in Fig. 3) [15]. The weight of the overall oscillating mechanism that holds the cylinder and oscillates with it is 1.21 kg, the cylinder weigh is 1.19 kg; hence, the resulting mass ratio of the cylinder in the conducted experiments is 2.26. m∗ =
mosc md
=
mosc
ρw π4 D2 L
(3)
The source of the damping in our experimental setup is a combination of the fluid damping and structural (frictional) damping. By performing free-decay tests in water and using the logarithmic decrement method, damping ratio is calculated. By giving an initial displacement to the cylinder and measuring the two consecutive peaks (yn and yn+1 ) of the cylinder, the damping ratio can be calculated from Eq. (4). For each case, this test was conducted two times for four different initial displacements ranging from 20 mm to 50 mm and the calculated damping ratio stayed relatively constant for all 5 different cases. A sample free decay damping calculation test in water is shown in Fig. 4 for an initial displacement of 40 mm. By considering the metal–metal contact between each of the side struts and the bearings, the damping associated with the bearing system is moderately high, which results in a high mass-damping (m∗ ζ ) parameter in the experiments.
ζ =
1 2π
ln
yn yn+1
.
(4)
3. Results and discussion In this section, experimental results for five different cases are presented and discussed to realize the impact of natural frequency on a one-degree-of-freedom rigid circular cylinder in VIV. To achieve this goal, observations are made on the following aspects of VIV: range of synchronization, the amplitude of oscillation and frequency ratio. Values of the spring stiffness (K) used in this paper are listed in Table 2.
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Table 3 Flow characteristic of the subcritical regime based on Zdravkovich’s classification [25].
Fig. 4. Time trace of a sample free decay test used for damping ratio calculation (K = 495 N/m).
Fig. 5. A sample time-trace plot for the first four seconds of the test, K = 495 N/m and U = 0.7 m/s.
By employing proper facilities in our experiments, a wide range of reduced velocities obtained (3 < U ∗ < 13), which is large enough to identify the lock-in phenomena. At lower U ∗ the cylinder’s movement is quasi-periodic, i.e. the maximum amplitude of oscillation might change on each cycle. As the U ∗ increases, the response suddenly becomes more periodic during synchronization region; i.e. the maximum amplitude in the majority of oscillations are constant. As it can be seen in Fig. 5, for the case of K = 495 N/m at U ∗ = 6.5, the cylinder reached its stable condition of vibration after only 3 s and oscillated in this situation for more than 15 s. Hence, there was enough time to analyze the cylinder’s behavior during each test. Several tests were repeated to inspect the repeatability of the tests and the result was promising. The amplitude of oscillation for each case is the average of maximum readings after the transition period. In our experiments performed for five different natural frequencies, maximum amplitude ratio about 0.98 was measured for K = 495 N/m at a reduced velocity around 7.
Reynolds range
Name
Characteristic features
150–200 < Re < 200–250
TrW1
200–250 < Re < 350–500
TrW2
350–500 < Re < 1 × 103 –2 × 103
TrSL1
1 × 103 –2×103 < Re < 2 × 104 –4 × 104
TrSL2
2 × 104 –4×104 < Re < 1 × 105 –2 × 105
TrSL3
Transition from laminar to turbulent vortices in wake Transition of irregular vortex during formation Development of transition vortices in free shear layer Formation of transition vortices in free shear layer Fully turbulent shear layer, laminar boundary layer separation
3.1. Reynolds number Reynolds number is one of the most important parameters in studying VIV of a circular cylinder. Achenbach and Heinecke [24] classified the flow around a circular cylinder into six general regimes as the following: Creeping flow (Re < 1) , steady separated regime ( 3–5 < Re < 30–40), periodic laminar wake (30– 40 < Re < 150-300), subcritical regime (150–300 < Re < 1.4 × 105 ), critical regime (1.4 × 105 < Re < 1 × 106 ), supercritical regime (1 × 106 < Re < 5 × 106 ) and transcritical regime (5 × 106 < Re). Based on the boundary layer characteristics of the flow over the cylinder surface, Zdravkovich [25] categorized the flow around a cylinder into 15 sub-regimes. The range of 1–2 × 104 < Re < 1– 2 × 105 is named TrSL3 and our experiments also lie in this range. The detail characteristic features of the subcritical regime are listed in Table 3. Also, Fig. 6 which is a reproduced from [15] and [25], shows the drag and fluctuation lift coefficient variation in each transition shear layer regime. It is clear that by increasing Reynolds number in the TrSL2, the lift coefficient increases and reaches its maximum in the TrSL3. By considering that most of the measurements in the literature corresponds to the TrSL2 regime, the different characteristics of TrSL2 and TrSL3 in the case of flow regime in the boundary layer, pressure coefficient and flow pattern can be found in [15]. Amplitude ratios (A/D) versus Reynolds number and reduced velocity for different values of K are compiled in Figs. 7 and 8 respectively. As shown in Fig. 7, the amplitude and range of synchronization in the upper branch increase with an increase in spring stiffness. The shift to higher operational Reynolds number increases the amplitude of oscillation and the range of synchronization which shows the strong dependence of VIV on natural frequency. As illustrated in Fig. 7, the onset of synchronization is more gradual for systems with higher K values. As an example, for the case with the lowest natural frequency (K = 125 N/m), the cylinder oscillates four times higher than the stiffest case (K = 495 N/m) at Re = 2 × 104 . This becomes important when a high amplitude ratio in a broad range of flow velocities is desired. 3.2. Reduced velocity Although the increase in the natural frequency shifts the upper branch to higher Reynolds numbers,but it can be seen in Fig. 8 that the Lock-In phenomena happens in 6 < U∗ < 9 for all the cases. By considering high Reynolds numbers in our experiments, and despite the high mass-damping parameter, it was expected to observe three branch responses in the amplitude response graphs.
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Fig. 6. Lift (CL ) and drag (CD ) coefficient as a function of Reynolds number for subcritical regime [15].
Fig. 7. Amplitude ratio versus Reynolds number for different spring stiffness cases.
It was interesting that we witnessed the disappearance of the lower branch in the amplitude response graphs. It can be seen in Fig. 8 that by increasing the U ∗ , the amplitude ratio jumps from the initial branch to the upper branch and it suddenly drops to the desynchronization region without any hump in the amplitude response which is not a common VIV character. This is in agreement with results from Raghavan and Bernitsas [15] where in the high mass-damping case, the lower branch vanished. They observed a rapid decrease in amplitude ratio from synchronization to desynchronization without experiencing the lower branch. Fig. 9 is a reform of Fig. 7 for illustrating that how the maximum amplitude ratio can be improved by using adjustable spring stiffness. For instance, it can be seen from Fig. 7 that at Re = 2 ∗ 104 the oscillation of the system with K = 125 N/m is four times higher than K = 495 N/m. Fig. 9 shows that by designing an adjustable spring stiffness system based on the flow velocity range, a higher energy extraction can be obtained; for instance, at a lower flow velocity, a reduced stiffness should be used. This method can be
Fig. 8. Amplitude ratio as a function of reduced velocity.
Fig. 9. Expectation of the amplitude ratio of the cylinder in an adjustable natural frequency system.
applied for systems such as VIVACE converter which need high amplitude ratio in a broader range of flow velocities. 3.3. Frequency response In Fig. 10, nondimensional frequency (f∗ = fosc /fn ) as a function of reduced velocity is plotted for five different values of spring stiffness. For a better comparison, the Strouhal frequency (inclined line) is also plotted. All five cylinders oscillate at a lower frequency than the natural frequency of their system in the initial branch (f∗ < 1). As the reduced velocity increases throughout the upper branch, f∗ deviates slowly from unity in the desynchronization section. This does not correspond to either the natural frequency of the system (f∗ = 1) or Strouhal frequency of a stationary cylinder (f∗ = 0.2U∗ ), but lies between them. As it can be seen in Fig. 10, the case with K = 495 N/m which experienced the highest amplitude of oscillation has the lowest frequency ratio in the upper branch and the desynchronization region. In other words, the oscillation frequency of the case with the highest amplitude is closer to its
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Table 4 The experimental characteristic of three well-known investigators and the present experiment.
Fig. 10. Nondimensional frequency of response (f∗ ) versus nondimensional velocity (U∗ )
m∗ ζ
Investigator
Reynolds range
Reynolds regime
m∗
Feng Khalak & Williamson Present study Raghavan & Bernitsas
6.0 × 103 –2.3 × 104 5.0 × 103 –1.6 × 104
TrSL2 TrSL2
250 0.36 7.6 2.40 0.013 5.1
0.55 0.94
1.5 × 104 –6.0 × 104 2.5 × 104 –7.5 × 104
TrSL3 TrSL3
2.26 0.11 3.14 0.26
0.98 1.31
D (cm) A/D
6.0 6.4
common characteristic of VIV systems, the maximum amplitude ratio obtained in 6 < U∗ < 8 for all of the cases. Unlike other cases, Feng [27] conducted his experiments in a wind tunnel and used air as the working fluid; due to very high mass ratio, he did not observe the upper branch in the results. As it is illustrated in Fig. 11, although the mass-damping parameter in the current study was about nine times higher than that from Khalak and Williamson [18], the maximum amplitude of oscillation and the range of synchronization of two experiments are very close to each other. This can be understood from the higher Re number and consequently higher lift coefficient in the present study. 4. Conclusion
Fig. 11. The comparison of the experiments done by Feng [27], Khalak and Williamson [18], Raghavan and Bernitsas [15] and the present study.
natural frequency than other cases. Wang et al. [26] studied the flow patterns of a cylinder undergoing VIV during initial and upper branches which explains the behavior of the cylinder 3.4. Comparison To compare the present study results with three of the most well-known published works in the field of VIV of a flexibility mounted circular cylinder, the results for the case 5 (K = 495 N/m) are plotted against those from Feng [27], Khalak and Williamson [18] and Raghavan and Bernitsas [15] in Fig. 11. The characteristics of these four experiments including Re number, mass and amplitude ratios are listed in Table 4. The Re number in the first two works [27,18] was in the TrSL2 regime. The last work [15] Re number was in the TrSL3. Different characteristics of TrSL2 and TrSL3 regimes are explained in [15]. The Re number range in the present study lies between what [18] and [15] investigated in their experiments: changing from TrSL2 to TrSL3. As a
Vortex-Induced Vibration of a rigid circular cylinder on elastic springs with a smooth surface condition was investigated experimentally in a towing tank water channel at high Reynolds and mass-damping parameter. In this study, the focus was on identifying the effect of natural frequency on the behavior of a cylinder undergoing VIV, by employing five different spring stiffness. The results including amplitude response, frequency response and the range of synchronization of the cylinder in VIV are analyzed as a function of Reynolds number and reduced velocity. The major conclusions of this study are the following: The studied Re range showed the gradual transition between two distinct turbulence shear layers from previously published works at Re ≈ 4000 to Re ≈ 100,000. The present study upper branch keeps expanding overtaking the previously published works at Re ≈ 4000 which is completely overtaken when Re reaches TrSL3, Re ≈ 100,000. An amplitude ratio of 0.98 was achieved for a smooth cylinder in VIV even with high mass and damping. By considering high value of m∗ ζ , we observed that the amplitude of oscillation in upper branch fell sharply to the desynchronization without experiencing a detectable lower branch. From our experiments, it can be concluded that the maximum amplitude of oscillation and the range of synchronization strongly depend on spring stiffness. The increase in natural frequency led to the increase in the amplitude of oscillation and extension of the range of synchronization. According to the amplitude ratio versus Reynolds number graphs, it is suggested to use a nonlinear spring stiffness system when it is desired to achieve maximum amplitudes in different flow velocities. Adjusting the system for having a lower natural frequency in lower current velocities can help VIVACE converter to have a higher efficiency in a wider range of ocean and river current velocities. References [1] C.H.K. Williamson, R. Govardhan, Vortex-induced vibrations, Annu. Rev. Fluid Mech. 36 (2004) 413–455. [2] C.H.K. Williamson, R. Govardhan, A brief review of recent results in vortexinduced vibrations, J. Wind Eng. Ind. Aerodyn. 96 (6) (2008) 713–735.
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[3] T. Sarpkaya, A critical review of the intrinsic nature of vortex-induced vibrations, J. Fluids Struct. 19 (4) (2004) 389–447. [4] P.W. Bearman, Circular cylinder wakes and vortex-induced vibrations, J. Fluids Struct. 27 (5) (2011) 648–658. [5] A.C. Fernandes, S. Mirzaeisefat, L.V. Cascão, Fundamental behavior of Vortex Self Induced Vibration (VSIV), Appl. Ocean Res. 47 (2014) 183–191. [6] M.M. Zdravkovich, Flow around Circular Cylinders; I Fundamentals, J. Fluid Mech. 350 (1) (1997) 377–378. [7] B.M. Sumer, J. Fredsœ, Hydrodynamics around cylindrical structures (Vol. 12), World Scientific, 1997. [8] R.D. Blevins, Flow-Induced Vibration, second ed., Van Nostrand Reinhold, New York, 1990. [9] S.A. Hatton, Update on the Design of Steel Catenary Riser Systems, Society of Underwater Technology, Deep and Ultra Deep Water Offshore Technology, 2H Offshore Engineering, 1999. [10] J.T. Klamo, Effects of Damping and Reynolds Number on Vortex-Induced Vibrations (Doctoral dissertation), California Institute of Technology, 2007. [11] J.T. Klamo, A. Leonard, A. Roshko, The effects of damping on the amplitude and frequency response of a freely vibrating cylinder in cross-flow, J. Fluids Struct. 22 (6) (2006) 845–856. [12] M.M. Bernitsas, K. Raghavan, Y. Ben-Simon, E.M. Garcia, VIVACE ss(Vortex Induced Vibration Aquatic Clean Energy): A new concept in generation of clean and renewable energy from fluid flow, Journal of Offshore Mechanics and Arctic Engineering 130 (4) (2008) 041101. [13] A. Khalak, C.H.K. Williamson, Motions, forces and mode transitions in vortexinduced vibrations at low mass-damping, J. Fluids Struct. 13 (7) (1999) 813– 851. [14] S. Muddada, B.S.V. Patnaik, An active flow control strategy for the suppression of vortex structures behind a circular cylinder, Eur. J. Mech.-B/Fluid 29 (2) (2010) 93–104. [15] K. Raghavan, M.M. Bernitsas, Experimental investigation of reynolds number effect on vortex induced vibration of rigid circular cylinder on elastic supports, Ocean Eng. 38 (5) (2011) 719–731. [16] J.H. Lee, M.M. Bernitsas, High-damping, high-Reynolds VIV tests for energy harnessing using the VIVACE converter, Ocean Eng. 38 (16) (2011) 1697– 1712.
[17] C.H.K. Williamson, N. Jauvtis, A high-amplitude 2T mode of vortex-induced vibration for a light body in XY motion, Eur. J. Mech. B/Fluids 23 (1) (2004) 107–114. [18] A. Khalak, C.H. Williamson, Investigation of relative effects of mass and damping in vortex-induced vibration of a circular cylinder, J. Wind Eng. Ind. Aerodyn. 69 (1997) 341–350. [19] A. Modir, M. Kahrom, A. Farshidianfar, Mass ratio effect on vortex induced vibration of a flexibly mounted circular cylinder, an experimental study, Int. J. Marine Energy 16 (2016) 1–11. [20] R. Govardhan, C.H.K. Williamson, Critical mass in vortex-induced vibration of a cylinder, Eur. J. Mech. Fluids 23 (2004) 17–27. [21] Y. Bao, C. Huang, D. Zhou, J. Tu, Z. Han, Two-degree-of-freedom flow-induced vibrations on isolated and tandem cylinders with varying natural frequency ratios, J. Fluids Struct. 35 (2012) 50–75. [22] M. Zahari, H.B. Chan, T.H. Yong, S.S. Dol, The effects of spring stiffness on vortex-induced vibration for energy generation, in: IOP Conference Series: Materials Science and Engineering (Vol. 78, No. 1, p. 012041), IOP Publishing, 2015. [23] B.H. Huynh, T. Tjahjowidodo, Z.W. Zhong, Y. Wang, N. Srikanth, Nonlinearly enhanced vortex induced vibrations for energy harvesting, in: 2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM), IEEE, 2015, pp. 91–96. [24] E. Achenbach, E. Heinecke, On vortex shedding from smooth and rough cylinders in the range of reynolds numbers 6 × 103 to 5 × 106 , J. Fluid Mech. 109 (1981) 239–251. [25] M.M. Zdravkovich, Flow Around Circular Cylinders Volume 1: Fundamentals, Oxford Science Publications, 1997. [26] X.K. Wang, C. Wang, Y.L. Li, S.K. Tan, Flow patterns of a low mass-damping cylinder undergoing vortex-induced vibration: Transition from initial branch and upper branch, Appl. Ocean Res. 62 (2017). [27] C.C. Feng, The Measurements of Vortex-Induced Effects in Flow Past a Stationary and Oscillating Circular and D-Section Cylinders (M. Sc. thesis), University of British Columbia, 1968.
Contents lists available at ScienceDirect
European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu
Experimental investigation of Reynolds number and spring stiffness effects on vortex induced vibrations of a rigid circular cylinder Alireza Modir a , Navid Goudarzi b , a b
∗
Department of Ocean and Mechanical Engineering, Florida International University, United States The William States Lee College of Engineering, University of North Carolina at Charlotte, Charlotte, NC 28223, United States
article
info
Article history: Received 17 January 2018 Received in revised form 5 June 2018 Accepted 17 October 2018 Available online 8 November 2018 Keywords: Flow induced motions Natural frequency Ocean energy Spring stiffness Towing tank Vortex induced vibration
a b s t r a c t The Vortex Induced Vibration for Aquatic Clean Energy (VIVACE) is a breakthrough technology, which harvests the hydrokinetic energy of ocean currents through enhancing Vortex Induced Vibration (VIV). In this paper, the VIV of an elastically mounted circular cylinder in a towing tank water channel is studied experimentally to investigate the effect of natural frequency, by using five different springs in a wide range of stiffness (125 N/m
1. Introduction
wake is known as the Strouhal number (St):
Vortex-Induced Vibration (VIV), which occurs in many fields of engineering, has been studied extensively, especially in the past fifty years, due to its significant implications in different engineering applications including offshore structures, cooling towers, bridges, heat exchangers and electrical transmission lines. VIV of an elastically mounted circular cylinder in fluid flow has been studied experimentally and numerically in the literature and have been covered in several overviews such as [1–4], and more recently [5]. Fluid flow passing a fixed circular cylinder generates vorticity due to the shear present in the boundary layer which results in the appearance of vortices on either side of the cylinder. By an increase in Reynolds number (Re > 40), the perturbations in the upstream flow cause one of the vortices to grow larger [6]. This vortex, with higher flow velocity and lower pressure, pulls the smaller vortex from the opposite side along the wake centerline and allow it to flow downstream. The repetition of this process leads to a periodic vortex shedding from either side of the body, altering the pressure distribution which results in acting periodic forces on the body [7]. The nondimensional frequency that the vortices are shed into the
St =
∗ Corresponding author. E-mail address: [email protected] (N. Goudarzi). https://doi.org/10.1016/j.euromechflu.2018.10.016 0997-7546/Published by Elsevier Masson SAS.
fv s D
(1) U where fv s is the vortex shedding frequency for a stationary cylinder, D is the cylinder diameter, and U is the relative velocity between the fluid and solid. Fig. 1 shows the relationship between Strouhal number and Reynolds number for stationary circular cylinders. In a large range of Reynolds numbers, the Strouhal number can be considered as constant. For the entire range of about 103 < Re < 105 , the Strouhal number is nearly 0.2 for smooth surfaces [8], which corresponds very well to the range of Reynolds number in our experiments. VIV phenomenon happens when the cylinder is elastically limited. Structural vibration, both inline and transverse to the flow, occurs due to the existence of a time-varying nonuniform pressure distribution around the body [9]. Lock in can be considered like linear resonance, as the vibration amplitudes increase significantly when the vortex shedding frequency becomes close to the natural frequency of the structure(fn ). In this situation, the nondimensional frequency (f ∗ = fosc /fn ) remain close to unity and the amplitude of oscillation increases dramatically [10]. Synchronization is defined as the matching of the vortex shedding frequency (fs ) with oscillation frequency (fosc ) [7]. In many papers ‘‘Lock-in’’ and ‘‘Synchronization’’ are used interchangeably.
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Fig. 1. St-Re relationship for circular cylinders [8].
It worth to mention that ‘‘Lock-in’’ and ‘‘Synchronization’’ have different meanings. ‘‘Lock-in’’ occurs when the vortex shedding frequency beaks from the Strouhal relationship and fits the natural frequency of the system and ‘‘Synchronization’’ happens when vortex shedding frequency beaks from Strouhal relationship but matches a frequency that is not essentially the natural frequency of the system. Consequently, lock-in is a special case of synchronization [11]. VIV is fundamentally a nonlinear phenomenon. In one hand, the structural vibration is induced by vortex shedding, on the other hand, the vibrating structure in turns affects the flow field and the resulted fluid forces applied on the structure. VIV is also known as a self-limiting phenomenon because as the cylinder displacement increases; the vortex shedding becomes weak, and hence tends to reduce further motion [12]. As described in [13], there are two distinct types of response for a VIV system, depending on the working fluid and mass-damping parameters (m∗ ζ ). The first one with a small amplitude which corresponds to high mass-damping parameter systems and mostly use air as its working fluid has two branch responses which are named ‘‘Initial’’ and ‘‘Lower’’ branches [10]. The second one which is known as a three branch response is often the result of low m∗ ζ . The working fluid in this type is commonly water. The third branch in this type has a higher amplitude that is called ‘‘Upper’’ branch of response [10]. The normalized oscillation amplitude can be defined as the ratio of amplitude A over the cylinder diameter D. Govardhan and Williamson suggested that A∗ = 0.6 criteria can be used to determine whether the amplitude has two or three branches [2]; if A∗ > 0.6, there will be three branches. The reduced velocity U ∗ in the literature is defined as U ∗ = U /fn D. Fig. 2 compares classical examples of two and three response branches at different reduced velocities [13]. Table 4 in the Results Section summarizes the Re number and mass ratios used in this experiment. In many engineering problems which involve fluid-flow passing a bluff body, VIV is classically treated as a ruinous phenomenon due to the resulting fatigue damages. Most previous researchers have mainly focused on reducing the destructive effects of Flow-induced motion. Unlike the former efforts, in suppressing the occurrence of VIV as a potentially damaging phenomenon for structures subjected to a fluid flow [14], in 2005, professor Bernitsas and his group at the University of Michigan invented a new method for extracting water current’s energy by utilizing VIV. The apparatus which is a hydrokinetic power-generating device is named VIVACE (Vortex Induced Vibration for Aquatic Clean Energy) [12]. The simplest form of the VIVACE converter comprises of an elastically mounted rigid circular cylinder which is connected to a power take-off system. The device can harness ocean and river’s current energy even in speeds as slow as 0.4 m/s [12]. They used different methods to increase the amplitude of oscillation for the purpose of increasing the efficiency of the extracted energy [15,16]. Although VIV of a cylinder with two degrees of freedom has been studied by
Fig. 2. Two and three branch response experiments, done by Feng and Williamson [13].
some researchers, for example [17] that is out of the scope of this paper. The two most important performance metrics that can effectively characterize VIV are the frequency and amplitude of vibration which are influenced by some system parameters such as mass ratio, spring stiffness, structural damping, and surface roughness. The effect of mass ratio as an important parameter in the behavior of VIV has been studied by many researchers like [18] and [19,20]. Also, the effect of damping as an important parameter in extracting water energy in VIVACE converter is studied in [16]. To the best knowledge of authors, there are few studies on investigating the spring stiffness effect of a circular cylinder in a VIV system. Bao et al. have numerically studied the effect of natural frequency ratio (fx /fy ) in a two-degree-of-freedom single cylinder and tandem cylinders [21]. The Reynolds number in their experiments was low and fixed at Re=150. Zahari et al. [22] studied the effect of spring stiffness on VIV in five cylinders with different mass’ and diameters but constant length, spring stiffness, and fluid velocity. In a numerical investigation that is done to analyze the effect of natural frequency ratio in VIV of a cylindrical structure at low Reynolds number of 200, Huynh and his coworkers [23] experimentally studied the effect of nonlinear springs, called hardening springs, on VIV of a circular cylinder. They used cantilever beams and additional plates for modeling nonlinear spring stiffness and realized that using the nonlinear springs, broaden the resonance range. Lee and Bernitsas at [16] used a virtual damper–spring apparatus to investigate the effect of damping and spring stiffness on the performance of the VIVACE converter, by adjusting a software for controlling these parameters. Based on the limited range of stiffness’ that they used, a general conclusion could not be made. Although the natural frequency of the system is an essential parameter in analyzing the vortex-induced vibration, to the extent of our knowledge, still investigations on the effect of spring stiffness on the behavior of a cylinder in VIV has not been well documented. In the present paper, the effects of spring stiffness on the amplitude response and oscillation frequency of a rigid circular cylinder in one degree of freedom is experimentally studied. While there has been a number of literature on exploring the amplitude response at different Re ranges, there has been so few studies on the VIV response to a range of spring stiffness values. The experiments are done in the fully turbulent shear layer (TrSL3) Re number regime (1.5 × 104 < Re < 6 × 104 ) with a high mass-damping parameter (m∗ ζ = 0.11). This Re range starts from the high-end TrSL2 that was studied by Khalak and Williamson [18] and overlaps
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Table 1 Experimental parameters. Parameter
Symbol
Value
Cylinder diameter (mm) Cylinder Length (mm) Stiffness (N/m) Mass Ratio Velocity (m/s) Reynolds Number Damping ratio
D L K m∗ U Re
60 380 125-495 2.26 0.3-1.5 1.5 × 104 –6 × 104 0.05
ζ
with Raghavan and Bernitsas [15] results at TrSL3. Hence, this work fills the gap for experimental results between TrSL2 and TrSL3. Five different linear springs were used to have a better exploration of the effect of the natural frequency of the system on the behavior of the cylinder. The results from this work can be used by other researchers to design adjustable stiffness systems based on the flow velocity for extracting a maximum energy from currents. Fig. 3. Schematic of the VIV test model. Main components: 1: Cylinder, 2: Side strut, 3: Cylindrical bearings, 4: Springs, 5: Wheels, 6: Camera, 7: Rail, 8: Towing cable.
2. Experimental setup All experiments were conducted in a Towing Tank Water channel with 14 m length, 0.50 m width, and 0.60 m depth. This channel was designed and fabricated to study the flow in a wide range of velocities over a flexibly mounted circular cylinder. By towing the submerged cylinder along the channel, the situation of the circular cylinder subjected to fluid flow was modeled. The carriage set-up was running on top of two very smooth rails to eliminate the undesirable external vibrations. To simulate steadyand uniform-current conditions, the structure was towed through initially still water; hence, in comparison with free surface water channels, the turbulence intensity could be completely eliminated. By connecting an electrical inverter to a three-phase electric motor which had the maximum rotational speed of 3000 rpm, the speed of the carriage was controlled with a high accuracy. The range of towing carriage speed resulting in VIV was 0.3–1.5 m/s, so the corresponding reduced velocity and Reynolds number were 3 < U∗ < 13 and 1.5 × 104 < Re < 6 × 104 respectively. The cylinder oscillatory displacement was measured by an advanced digital camera, recording 30 frames per second, and an indicator which showed the position of the cylinder during the experiments. A smooth surface polyethylene cylinder with a density of 971 kg/m3 , the diameter of 0.060 m and length of 0.380 m was used in all experiments. The aspect ratio (Length/Diameter) of the cylinder was 6.3 and the blockage ratio inside the channel was 11.8%. The main properties of the experimental set-up are listed in Table 1. Figure three illustrates the schematic VIV test model. The cylinder flexible movement was achieved by connecting it to the coil springs through struts. It was constrained to oscillate transversely to the towing direction using two cylindrical bearings. One may refer to [19] for more details concerning the experimental set-up. Here we used five different springs with different stiffness’ to analyze the effect of natural frequency of the system on the behavior of a cylinder undergoing VIV. Natural frequency in water for different springs can be obtained from Eq. (2), where K = spring stiffness constant, mosc = oscillating mass and ma = ideal added mass. The added mass is ma = Ca md , where displaced fluid mass is md = ρw π D2 L/4 and Ca is the ideal added mass coefficient that is equal to 1 for a circular cylinder [2]. In this paper, the reduced velocity is defined by the natural frequency in still water (U∗ = U/fn,w D). fn,w =
1 2π
√
K mosc + ma
(2)
As it is shown in Eq. (3), the mass ratio is defined as the total mass divided by the displaced fluid mass (md ) where the total
Table 2 Spring stiffness and natural frequency of 5 different cases used here. Case
1
2
3
4
5
Spring stiffness (N/m) Natural frequency (1/s) Structural damping (N s/m)
125 0.95 2.3
200 1.2 2.9
250 1.3 3.2
365 1.6 3.9
495 1.9 4.6
mass includes the oscillating mass plus 1/3 of the spring mass (Components 1, 2, and 4 in Fig. 3) [15]. The weight of the overall oscillating mechanism that holds the cylinder and oscillates with it is 1.21 kg, the cylinder weigh is 1.19 kg; hence, the resulting mass ratio of the cylinder in the conducted experiments is 2.26. m∗ =
mosc md
=
mosc
ρw π4 D2 L
(3)
The source of the damping in our experimental setup is a combination of the fluid damping and structural (frictional) damping. By performing free-decay tests in water and using the logarithmic decrement method, damping ratio is calculated. By giving an initial displacement to the cylinder and measuring the two consecutive peaks (yn and yn+1 ) of the cylinder, the damping ratio can be calculated from Eq. (4). For each case, this test was conducted two times for four different initial displacements ranging from 20 mm to 50 mm and the calculated damping ratio stayed relatively constant for all 5 different cases. A sample free decay damping calculation test in water is shown in Fig. 4 for an initial displacement of 40 mm. By considering the metal–metal contact between each of the side struts and the bearings, the damping associated with the bearing system is moderately high, which results in a high mass-damping (m∗ ζ ) parameter in the experiments.
ζ =
1 2π
ln
yn yn+1
.
(4)
3. Results and discussion In this section, experimental results for five different cases are presented and discussed to realize the impact of natural frequency on a one-degree-of-freedom rigid circular cylinder in VIV. To achieve this goal, observations are made on the following aspects of VIV: range of synchronization, the amplitude of oscillation and frequency ratio. Values of the spring stiffness (K) used in this paper are listed in Table 2.
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Table 3 Flow characteristic of the subcritical regime based on Zdravkovich’s classification [25].
Fig. 4. Time trace of a sample free decay test used for damping ratio calculation (K = 495 N/m).
Fig. 5. A sample time-trace plot for the first four seconds of the test, K = 495 N/m and U = 0.7 m/s.
By employing proper facilities in our experiments, a wide range of reduced velocities obtained (3 < U ∗ < 13), which is large enough to identify the lock-in phenomena. At lower U ∗ the cylinder’s movement is quasi-periodic, i.e. the maximum amplitude of oscillation might change on each cycle. As the U ∗ increases, the response suddenly becomes more periodic during synchronization region; i.e. the maximum amplitude in the majority of oscillations are constant. As it can be seen in Fig. 5, for the case of K = 495 N/m at U ∗ = 6.5, the cylinder reached its stable condition of vibration after only 3 s and oscillated in this situation for more than 15 s. Hence, there was enough time to analyze the cylinder’s behavior during each test. Several tests were repeated to inspect the repeatability of the tests and the result was promising. The amplitude of oscillation for each case is the average of maximum readings after the transition period. In our experiments performed for five different natural frequencies, maximum amplitude ratio about 0.98 was measured for K = 495 N/m at a reduced velocity around 7.
Reynolds range
Name
Characteristic features
150–200 < Re < 200–250
TrW1
200–250 < Re < 350–500
TrW2
350–500 < Re < 1 × 103 –2 × 103
TrSL1
1 × 103 –2×103 < Re < 2 × 104 –4 × 104
TrSL2
2 × 104 –4×104 < Re < 1 × 105 –2 × 105
TrSL3
Transition from laminar to turbulent vortices in wake Transition of irregular vortex during formation Development of transition vortices in free shear layer Formation of transition vortices in free shear layer Fully turbulent shear layer, laminar boundary layer separation
3.1. Reynolds number Reynolds number is one of the most important parameters in studying VIV of a circular cylinder. Achenbach and Heinecke [24] classified the flow around a circular cylinder into six general regimes as the following: Creeping flow (Re < 1) , steady separated regime ( 3–5 < Re < 30–40), periodic laminar wake (30– 40 < Re < 150-300), subcritical regime (150–300 < Re < 1.4 × 105 ), critical regime (1.4 × 105 < Re < 1 × 106 ), supercritical regime (1 × 106 < Re < 5 × 106 ) and transcritical regime (5 × 106 < Re). Based on the boundary layer characteristics of the flow over the cylinder surface, Zdravkovich [25] categorized the flow around a cylinder into 15 sub-regimes. The range of 1–2 × 104 < Re < 1– 2 × 105 is named TrSL3 and our experiments also lie in this range. The detail characteristic features of the subcritical regime are listed in Table 3. Also, Fig. 6 which is a reproduced from [15] and [25], shows the drag and fluctuation lift coefficient variation in each transition shear layer regime. It is clear that by increasing Reynolds number in the TrSL2, the lift coefficient increases and reaches its maximum in the TrSL3. By considering that most of the measurements in the literature corresponds to the TrSL2 regime, the different characteristics of TrSL2 and TrSL3 in the case of flow regime in the boundary layer, pressure coefficient and flow pattern can be found in [15]. Amplitude ratios (A/D) versus Reynolds number and reduced velocity for different values of K are compiled in Figs. 7 and 8 respectively. As shown in Fig. 7, the amplitude and range of synchronization in the upper branch increase with an increase in spring stiffness. The shift to higher operational Reynolds number increases the amplitude of oscillation and the range of synchronization which shows the strong dependence of VIV on natural frequency. As illustrated in Fig. 7, the onset of synchronization is more gradual for systems with higher K values. As an example, for the case with the lowest natural frequency (K = 125 N/m), the cylinder oscillates four times higher than the stiffest case (K = 495 N/m) at Re = 2 × 104 . This becomes important when a high amplitude ratio in a broad range of flow velocities is desired. 3.2. Reduced velocity Although the increase in the natural frequency shifts the upper branch to higher Reynolds numbers,but it can be seen in Fig. 8 that the Lock-In phenomena happens in 6 < U∗ < 9 for all the cases. By considering high Reynolds numbers in our experiments, and despite the high mass-damping parameter, it was expected to observe three branch responses in the amplitude response graphs.
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Fig. 6. Lift (CL ) and drag (CD ) coefficient as a function of Reynolds number for subcritical regime [15].
Fig. 7. Amplitude ratio versus Reynolds number for different spring stiffness cases.
It was interesting that we witnessed the disappearance of the lower branch in the amplitude response graphs. It can be seen in Fig. 8 that by increasing the U ∗ , the amplitude ratio jumps from the initial branch to the upper branch and it suddenly drops to the desynchronization region without any hump in the amplitude response which is not a common VIV character. This is in agreement with results from Raghavan and Bernitsas [15] where in the high mass-damping case, the lower branch vanished. They observed a rapid decrease in amplitude ratio from synchronization to desynchronization without experiencing the lower branch. Fig. 9 is a reform of Fig. 7 for illustrating that how the maximum amplitude ratio can be improved by using adjustable spring stiffness. For instance, it can be seen from Fig. 7 that at Re = 2 ∗ 104 the oscillation of the system with K = 125 N/m is four times higher than K = 495 N/m. Fig. 9 shows that by designing an adjustable spring stiffness system based on the flow velocity range, a higher energy extraction can be obtained; for instance, at a lower flow velocity, a reduced stiffness should be used. This method can be
Fig. 8. Amplitude ratio as a function of reduced velocity.
Fig. 9. Expectation of the amplitude ratio of the cylinder in an adjustable natural frequency system.
applied for systems such as VIVACE converter which need high amplitude ratio in a broader range of flow velocities. 3.3. Frequency response In Fig. 10, nondimensional frequency (f∗ = fosc /fn ) as a function of reduced velocity is plotted for five different values of spring stiffness. For a better comparison, the Strouhal frequency (inclined line) is also plotted. All five cylinders oscillate at a lower frequency than the natural frequency of their system in the initial branch (f∗ < 1). As the reduced velocity increases throughout the upper branch, f∗ deviates slowly from unity in the desynchronization section. This does not correspond to either the natural frequency of the system (f∗ = 1) or Strouhal frequency of a stationary cylinder (f∗ = 0.2U∗ ), but lies between them. As it can be seen in Fig. 10, the case with K = 495 N/m which experienced the highest amplitude of oscillation has the lowest frequency ratio in the upper branch and the desynchronization region. In other words, the oscillation frequency of the case with the highest amplitude is closer to its
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Table 4 The experimental characteristic of three well-known investigators and the present experiment.
Fig. 10. Nondimensional frequency of response (f∗ ) versus nondimensional velocity (U∗ )
m∗ ζ
Investigator
Reynolds range
Reynolds regime
m∗
Feng Khalak & Williamson Present study Raghavan & Bernitsas
6.0 × 103 –2.3 × 104 5.0 × 103 –1.6 × 104
TrSL2 TrSL2
250 0.36 7.6 2.40 0.013 5.1
0.55 0.94
1.5 × 104 –6.0 × 104 2.5 × 104 –7.5 × 104
TrSL3 TrSL3
2.26 0.11 3.14 0.26
0.98 1.31
D (cm) A/D
6.0 6.4
common characteristic of VIV systems, the maximum amplitude ratio obtained in 6 < U∗ < 8 for all of the cases. Unlike other cases, Feng [27] conducted his experiments in a wind tunnel and used air as the working fluid; due to very high mass ratio, he did not observe the upper branch in the results. As it is illustrated in Fig. 11, although the mass-damping parameter in the current study was about nine times higher than that from Khalak and Williamson [18], the maximum amplitude of oscillation and the range of synchronization of two experiments are very close to each other. This can be understood from the higher Re number and consequently higher lift coefficient in the present study. 4. Conclusion
Fig. 11. The comparison of the experiments done by Feng [27], Khalak and Williamson [18], Raghavan and Bernitsas [15] and the present study.
natural frequency than other cases. Wang et al. [26] studied the flow patterns of a cylinder undergoing VIV during initial and upper branches which explains the behavior of the cylinder 3.4. Comparison To compare the present study results with three of the most well-known published works in the field of VIV of a flexibility mounted circular cylinder, the results for the case 5 (K = 495 N/m) are plotted against those from Feng [27], Khalak and Williamson [18] and Raghavan and Bernitsas [15] in Fig. 11. The characteristics of these four experiments including Re number, mass and amplitude ratios are listed in Table 4. The Re number in the first two works [27,18] was in the TrSL2 regime. The last work [15] Re number was in the TrSL3. Different characteristics of TrSL2 and TrSL3 regimes are explained in [15]. The Re number range in the present study lies between what [18] and [15] investigated in their experiments: changing from TrSL2 to TrSL3. As a
Vortex-Induced Vibration of a rigid circular cylinder on elastic springs with a smooth surface condition was investigated experimentally in a towing tank water channel at high Reynolds and mass-damping parameter. In this study, the focus was on identifying the effect of natural frequency on the behavior of a cylinder undergoing VIV, by employing five different spring stiffness. The results including amplitude response, frequency response and the range of synchronization of the cylinder in VIV are analyzed as a function of Reynolds number and reduced velocity. The major conclusions of this study are the following: The studied Re range showed the gradual transition between two distinct turbulence shear layers from previously published works at Re ≈ 4000 to Re ≈ 100,000. The present study upper branch keeps expanding overtaking the previously published works at Re ≈ 4000 which is completely overtaken when Re reaches TrSL3, Re ≈ 100,000. An amplitude ratio of 0.98 was achieved for a smooth cylinder in VIV even with high mass and damping. By considering high value of m∗ ζ , we observed that the amplitude of oscillation in upper branch fell sharply to the desynchronization without experiencing a detectable lower branch. From our experiments, it can be concluded that the maximum amplitude of oscillation and the range of synchronization strongly depend on spring stiffness. The increase in natural frequency led to the increase in the amplitude of oscillation and extension of the range of synchronization. According to the amplitude ratio versus Reynolds number graphs, it is suggested to use a nonlinear spring stiffness system when it is desired to achieve maximum amplitudes in different flow velocities. Adjusting the system for having a lower natural frequency in lower current velocities can help VIVACE converter to have a higher efficiency in a wider range of ocean and river current velocities. References [1] C.H.K. Williamson, R. Govardhan, Vortex-induced vibrations, Annu. Rev. Fluid Mech. 36 (2004) 413–455. [2] C.H.K. Williamson, R. Govardhan, A brief review of recent results in vortexinduced vibrations, J. Wind Eng. Ind. Aerodyn. 96 (6) (2008) 713–735.
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