Investigation of flow characteristics around a stationary circular cylinder with an undulatory plate

Accepted Manuscript Investigation of flow characteristics around a stationary circular cylinder with an undulatory plate J. Wu, C. Shu, N. Zhao PII: DOI: Reference:

S0997-7546(14)00060-0 http://dx.doi.org/10.1016/j.euromechflu.2014.04.007 EJMFLU 2775

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European Journal of Mechanics B/Fluids

Received date: 19 August 2013 Revised date: 31 March 2014 Accepted date: 7 April 2014 Please cite this article as: J. Wu, C. Shu, N. Zhao, Investigation of flow characteristics around a stationary circular cylinder with an undulatory plate, European Journal of Mechanics B/Fluids (2014), http://dx.doi.org/10.1016/j.euromechflu.2014.04.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Investigation of Flow Characteristics around a Stationary Circular Cylinder with an Undulatory Plate J. Wu1,*, C. Shu2 and N. Zhao1 1

Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street 29, Nanjing 210016, Jiangsu, China 2

Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260

Abstract The flow characteristics around a stationary circular cylinder with an undulatory plate have been numerically investigated in this work. This is extension of our previous study on flows over a cylinder with a rigid flapping plate [41]. In this study, the effect of plate flexibility on the flow characteristics is extensively examined by varying the frequency and amplitude of motion and the length of plate. The laminar flow at Reynolds number of Re = 100 is considered. Based on the numerical results obtained, some interesting flow patterns and drag reduction are observed. Meanwhile, the physical mechanisms elucidating good performance of the undulatory plate are also provided. As compared with the rigid plate, the flexible plate performs better in the flow control of bluff body.

*

Corresponding author; Email: [email protected].

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1. Introduction The problems of flows over a bluff body have been studied extensively through theoretical, experimental and numerical methods. Owing to its wide applications in engineering, the flow control is becoming an attractive subject [1]. Due to the unsteady flow separation, the vortex shedding behind the bluff body occurs, which is the cause of structural vibration and resultant fatigue damage. Hence, the control of vortex shedding is a major task in the flow control. Depending on whether energy inputs are required, the passive or active flow control can be defined respectively. In the field of the passive flow control, one choice is to modify the surface or geometry of bluff body. As a result, the drag can be reduced due to the separation delay or the wake change. Typical examples include the use of dimple [2, 3], roughness [4, 5], wavy surface [6, 7] and seam [8]. In addition, the use of sliding belt [9] or hydrophobic surface [10] is also a feasible way. On the other hand, the utilization of additive devices is an alternative method in the passive flow control. Strykowski and Sreenivasan [11] used a small circular cylinder to suppress the vortex shedding from main cylinder. Lee et al. [12] observed the drag reduction (maximum value is about 25%) by installing a small control rod in the upstream of circular cylinder. Kuo et al. [13] reported the reduction of form drag and lift fluctuation by symmetrically placing two small cylinders in the wake of primary cylinder. Galvao et al. [14] found the reduction of vortex-induced vibration and drag by arranging hydrofoils either at the sides of cylinder or in its wake. In addition, the splitter plate is frequently considered in the passive flow control. After the first work 2

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of Roshko [15], great efforts have been made to this topic [16-20], and some impressive conclusions have been achieved. For instance, complete suppression of vortex shedding can be obtained when the length of plate is five times of the diameter of cylinder. The frequency of vortex shedding can reach a minimum value when the length of plate equals the diameter of cylinder. Different from the passive flow control, external energy is consumed in the active flow control. One active control strategy is to lock the forcing frequency onto the frequency of vortex shedding. Koopmann [21] experimentally made a cylinder oscillate along the crosswise direction with various frequencies. Thereafter, flows over an oscillating cylinder with streamwise, crosswise and rotary motion have been intensively investigated [22-25]. Through these studies, some interesting results, such as Reynolds-number-dependent drag reduction and lock-on wake patterns, are obtained. Another technique in active flow control is to implement blowing/suction on the bluff body. Williams et al. [26] examined globally unstable flows over a cylinder with periodic disturbances. Some phenomena including the appearance of seahorse vortex patterns were observed. Using a pair of blowing/suction slots on a cylinder, Park et al. [27] numerically studied the feedback control of vortex shedding. Under some special conditions, the complete suppression of vortex shedding or the change of vortex shedding frequency was achieved. Since then, various optimal control models have been developed [28-30]. In the flow control approaches mentioned above, the control devices are generally rigid and cannot deform. On the other hand, it is known that aquatic animals

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could produce effective propulsion through the deformation of body. Among various types of aquatic locomotion, fish-like swimming is a representative one which has been investigated widely. Lighthill, who established a slender-body theory, is a pioneer in the study of fish-like swimming [31]. Based on numerous experimental results, it is found that a swimming fish could generate a wave propagating along its body [32]. Whereafter, continuously growing interest has been drawn to the study of undulatory body. Using a travelling wavy plate to model the swimming fish, Dong and Lu [33] systematically investigated the effects of phase speed, amplitude and relative wavelength on the flow structures and force behaviors. Zhu and Peskin [34] and Tian et al. [35] reported that the aquatic animals could benefit from the vortex interaction when they are in the side-by-side arrangement. Recently, Tian et al. [36] studied the propulsive performance of a foil with travelling wavy surface. They found that the use of wavy surface could produce higher propulsive efficiency than most aquatic animals. Due to the elegant hydrodynamic performance, the undulatory body has also been used to control the wake behind the bluff body. By generating the travelling transverse wave on the rear part of cylinder surface, Wu et al. [37] found that the vortex shedding of cylinder can be eliminated. Meanwhile, the drag reduction can reach 85% as compared to the case without travelling wave. Xiao et al. [38] carried out a numerical study on the hydrodynamic performance of undulatory foil in the near wake of D-section cylinder. They reported that the drag and lift fluctuation on the cylinder can be reduced at some specific undulation frequencies. For recent work on

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interaction between the flexible and rigid bodies, the reader can refer to [39, 40]. Due to its good performance, it is necessary to further explore the application of flexible body in the flow control. In this study, we numerically investigate the flow characteristics around a stationary circular cylinder with a flexible plate. The plate is attached to the base of cylinder and undergoes an undulatory motion. The current work is the extension of our previous study about flows over a cylinder with a rigid flapping plate [41]. By replacing the rigid flapping plate with the undulatory one, a more realistic motion of tadpole can be modeled. Hence, the flow characteristics may show some differences as compared with that of rigid plate. The motion of flexible plate is mainly governed by the Strouhal number of undulation motion (Stu), the amplitude of travelling wave (A) and the length of plate (l). To explore the flow characteristics in detail, systematical simulations by varying Stu, A and l are carried out. To accomplish this task, our developed boundary condition-enforced immersed boundary-lattice Boltzmann method (IB-LBM), which has been employed in the previous study [41], will also be adopted in this study. The paper is organized as follows. The problem and its governing equations are described in Section 2. The numerical validations are also performed in this part. Section 3 presents a complete numerical study of flows over the stationary cylinder with the flexible plate. At last, concluding remarks are presented in Section 4.

2. Problem Definition and Numerical Methods Figure 1 shows the sketch of two-dimensional viscous and incompressible flows 5

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over a stationary circular cylinder with a flexible plate. The plate can produce a travelling wave along the streamwise direction. Consequently, its motion equation is x p  x0  x

(1)

y p  y0  a  x  sin  kx  2 ft 

where

 x0 , y0 

is the leading point of plate which coincides with the rear point of

cylinder. k  2 /  is the wavenumber and  is the wavelength.

f

is the

frequency of oscillation. Since the leading point of plate is fixed, the amplitude a  x  is written as 2 a  x   A c1  x / l   c2  x / l   l  

(2)

where c1 and c2 are adjustable parameters, A is the amplitude of travelling wave, l is the length of plate. In this study, we do not investigate the effect of wavelength

and geometry of wavy plate on the flow patterns. Hence, we set   l for simplicity and c1  0.6 , c2  1.6 to ensure that the amplitude at plate tail is maximum. Based on the double amplitude, i.e. 2Al , the Strouhal number of undulation motion is defined as [42] Stu  2 fAl / u

(3)

where u is the free stream velocity. To numerically solve the flow problem defined above, a suitable choice is to adopt the developed immersed boundary-lattice Boltzmann method (IB-LBM), which has been successfully applied to simulate flows over a rigid flapping plate attached to a cylinder [41]. The governing equations are f  x  e  t , t   t   f  x, t   

 f  x, t   f  x, t   F  t  1



eq





(4)

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1   e  u e u   F  1   w   2   4 e   f cs  2   cs  1 2

 u   e f  f t 

(5)

(6)

where f is the distribution function and feq is its corresponding equilibrium state;

 is the single relaxation parameter;  t is the time step; e is the lattice velocity and w are the coefficients, which are related to the lattice velocity model employed. In the framework of lattice Boltzmann method, the relaxation parameter can be determined from the kinematic viscosity of fluid through     0.5 cs2 t , where

cs is the sound speed of the lattice model. In addition, f is the force density which is determined from the no-slip boundary condition. From Eq. (6), f can be written as

f  2 u  t

(7)

where  u is the fluid velocity correction. Besides the velocity, other macroscopic flow variables are calculated by

   f , 

P   cs2

(8)

For the detailed solution procedure and extensive validations of this method, the reader can refer to [41, 43-45]. As compared with the traditional IB-LBM, the current method can exactly satisfy the no-slip boundary condition and can accurately and efficiently deal with both the fixed and moving boundary problems. In this work, further validation of the adopted method is conducted. Two groups of parameters are chosen. One is A = 0.1, Stu = 0.2 and 0.6 and the other is A = 0.5, Stu = 0.4 and 1.0. Based on the diameter of cylinder d, the Reynolds number at Re = 100 is considered. Moreover, l = d. In current simulations, the size of computational

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domain is 50d×40d and the center of cylinder is fixed at (20d, 20d). It is noted that the same computational domain will also be used in the following study. Correspondingly, the non-uniform mesh is used, in which the mesh for the region (the region size is 2.2d×1.2d) around the cylinder and plate is uniform. Table I provides the calculated results, where Cd is the mean drag coefficient on the cylinder and plate, and Cl is the fluctuation of lift coefficient. Here, the drag and lift coefficients are defined by Cd 

2 FD ,  u2 d

Cl 

2 FL  u2 d

(9)

where FD and FL are the drag and lift forces, respectively. From the table, it is seen that accurate results can be obtained when the mesh spacing of uniform mesh is x  0.01 , which will also be adopted in following simulations. On the other hand,

for the lattice Boltzmann method used here, which is applied on the non-uniform mesh, the time step is not an independent variable, and needs to be chosen to equal the mesh spacing of uniform mesh in the vicinity of the cylinder and undulatory plate (i.e.,

 t  x ). The Lagrange interpolation modification to the standard LBM allows second-order accuracy to be maintained away from the uniformly meshed component of the computational domain [41, 43-45].

3. Study of Flows over a Circular Cylinder with an Undulatory Plate In this section, we systematically investigate the flow characteristics around a circular cylinder with an undulatory plate. The effects of parameters including the Strouhal number of undulation motion Stu, the amplitude of travelling wave A and the

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length of plate l are examined at Re = 100. Firstly, we fix the length of plate at l = d. Five different values of A, which range from 0.1 to 0.5, are used. At one fixed A, the Strouhal numbers vary in the range of 0.1 ≤ Stu ≤ 1. Then, the influence of the plate length varying from 0.5d to 2.5d is also studied.

3.1 Effect of frequency and amplitude When the length of flexible plate is fixed at l = d, its motion is controlled by the frequency and amplitude of oscillation. By changing Stu and A, the force and flow patterns would be significantly affected. Figure 2 plots the variation of mean drag coefficient Cd on the cylinder and plate as the function of Stu. As a reference, the result of the stationary plate case ( Cd  1.17 ) [41], denoted by the dashed line, is also presented in the figure. From this figure, it is obvious that the use of undulatory plate can reduce the drag. At low and medium Stu, the variation trend of Cd at different amplitude is similar. It changes from about 1.14 at Stu = 0.1 to about 0.9 at Stu = 0.4. As Stu further increases (Stu ≥ 0.5), the influence of amplitude on Cd becomes obvious. For A = 0.1 and 0.2, Cd decreases steeply and thrust force (i.e. negative drag) is generated at high frequency (minimum Cd is -0.192). This variation trend is similar to the case of flows over a travelling wavy plate or wall [33, 46]. For A = 0.3 and 0.4, Cd keeps decreasing and no thrust force is produced for all frequencies considered. For A = 0.5, however, Cd decreases mildly and finally converges to a constant value (around 0.8). Therefore, to achieve small mean drag force or even thrust force, one can drive the undulatory plate at high frequency and/or small 9

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amplitude. Besides the mean drag force, instantaneous drag and lift forces also depend on the frequency and amplitude. Figure 3 shows the fluctuation of drag and lift coefficients ( Cd , Cl ) varying with Stu at different A. Roughly, both Cd and Cl increase with Stu. As shown in Fig. 3(a), Cd at different amplitude increases smoothly when Stu ≤ 0.4. The larger the amplitude is, the lower the rate of increase is. As Stu further increases (Stu ≥ 0.5), for A = 0.1, Cd first shows a little change and then increases steeply when Stu > 0.6. It finally comes to the maximum value (= 1.709) at Stu = 1. On the other hand, for other amplitudes (A = 0.2-0.5), Cd increases gradually with Stu and the maximum values of Cd are always smaller than that at A = 0.1. It is found that higher amplitude would induce smaller maximum fluctuation. The similar variation tendency of Cl can be found in Fig. 3(b). From the figure, it is found that the maximum Cl is 10.698, which appears at Stu = 0.1 and A = 0.1. Again, for other amplitudes, the maximum Cl (around 4.5) is much smaller than that at A = 0.1. To further investigate the effect of plate flexibility on the force behaviors, some typical time histories of Cd and Cl are given in Fig. 4. Meanwhile, the corresponding phase diagram about Cd and Cl is also plotted. As shown in Fig. 4(a1), the variation of Cd and Cl is irregular at low frequency and small amplitude. As a result, the phase diagram about Cd and Cl in different periods cannot overlap with each other. When the frequency becomes higher and higher, the variation of Cd and Cl is dominated by the undulatory plate. When Stu = 1 as plotted in Fig. 4(a2),

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the variation of Cd and Cl becomes regular and periodic. As a consequence, a smooth "butterfly" shape with one cross point can be observed. This situation is similar to that in the rigid flapping plate case [41]. As A increases to 0.3, the variation of Cd demonstrates slight irregularity at Stu = 0.1 (Fig. 4(b1)) and the variation of

Cl is still periodic. Hence, the misalignment around the peak value of Cd can be observed. As the frequency increases (Fig. 4(b2)), a sub-maximum value of Cd appears, which causes the generation of two concave portions in the phase diagram. When the amplitude is further increased (Fig. 4(c)), the force performance is similar to that at A = 0.3. The only difference is that the variation of Cd around the maximum value and the sub-maximum one becomes obvious. Consequently, another two cross points are formed in the phase diagram. It is known that to generate the wave along the flexible plate, the input of external energy is required. Since the plate only executes the vertical oscillation, following the definition in [33, 46], the power consumed due to the undulation motion is defined as Pu   FL l

dy p dt

dx . Using  , u and d , it can be non-dimensionalized

as C pu 

2 Pu  u3 d

(10)

On the other hand, the power needed to overcome the drag force is Pd  FDu . After non-dimensionalization, it is the same as the drag coefficient Cd . Therefore, the total non-dimensional power is C pt  C pu  Cd . Figure 5 presents the time-averaged non-dimensional power for undulation and total power varying with Stu. Three

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amplitudes of A = 0.1, 0.3 and 0.5 are selected. With the increase of Stu, both of the powers increase for all amplitudes. For A = 0.1, the mean undulation power C pu is smaller than the mean total power C pt at low and medium frequencies. The reason is that Cd is positive under this condition. Nevertheless, C pu is approaching C pt as Stu increases and finally exceeds it at Stu = 1 due to negative Cd . This behavior is similar to that of a single travelling wavy plate [33]. In contrast, C pu is always less than C pt at medium and large amplitudes. Based on the above results, it is found that the use of a flexible plate can efficiently reduce the drag force. At low amplitude of undulation motion, the thrust force together with high force fluctuations could be generated. As the amplitude increases, the fluctuations decrease gradually and the minimum mean drag coefficient increases. On the other hand, high drag reduction is achieved with the cost of large input of external energy. Accompanying the change of force behaviors, the flow patterns are also altered by the undulatory plate. Figures 7 and 8 illustrate some representative results, which are displayed as instantaneous vorticity contours and time-averaged streamlines. To make a comparison, the results of the stationary plate case are also drawn in Fig. 6. In these figures, the solid line means the vorticity with counterclockwise direction and the dashed line for clockwise direction. Because of the undulation motion, the secondary vortex shed from the flexible plate continuously interacts with the main vortex from the cylinder. The resultant flow field is then dependent on the strength of secondary vortex. At low frequency, as shown in Figs. 7(a) and 8(a), the separation

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bubble behind the plate is elongated and shrunk at A = 0.2 and 0.4 respectively as compared with the stationary plate case (Fig. 6). At the same time, the regular vortex shedding can still be observed. It means that the vortex shedding of cylinder is partially affected by the flexible plate. As the frequency increases up to medium value (Figs. 7(b), 8(b) and 8(c)), the separation bubble is shrunk clearly. Meanwhile, the secondary vortex pushes the main vortices away from the centerline. As a result, no vortex shedding happens in the wake and the flow becomes steady. The above variation of flow pattern has already been observed in the case of rigid flapping plate [41], and similar flow pattern was also presented in the work of Tian et al. [39]. When the frequency reaches the maximum value considered (Figs. 7(c) and 8(d)), the separation bubble is further shrunk. At the same time, the steady flow pattern is still observed, which is different from the rigid flapping plate case where the vortex shedding reappears at high frequency [41]. Moreover, the strength of secondary vortex is significantly increased. Consequently, a jet-like vorticity profile, which is caused by the reverse von Karman vortex sheet, can be found. This phenomenon is similar to that reported by Bergmann et al. [25] who investigated the laminar flow over partially rotating cylinder. It explains why small mean drag force or even thrust force is produced at high frequency. From the results in Figs. 7 and 8, we know that the flow field is obviously changed by the undulatory plate. The effect of oscillating frequency and amplitude on the flow characteristics is significant. Some flow patterns similar to those in the rigid flapping plate case are reproduced. In addition, different flow patterns due to the

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flexible plate are also generated.

3.2 Effect of plate length In the sub-section above, the effects of plate flexibility have been studied at l = d. Nevertheless, the length of flexible plate also influences the flow behaviors. In current sub-section, we carry out numerical simulations by fixing the amplitude at A = 0.2 and varying the plate length in the range of 0.5d ≤ l ≤ 2.5d. Figure 9 exhibits the mean drag coefficient at different plate length and frequency. The results of the stationary plate case are also plotted in the figure. Same as the situation at l = d, Cd decreases with Stu at different plate length. Except for the case at l = 1.5d and Stu = 0.1, Cd is always smaller than that of the stationary plate case. When l increases from 0.5d to d, there is a clear drop of Cd . With increase of Stu, the variation of Cd becomes more and more steep. The underlying reason might be that the plate at l = 0.5d cannot effectively affect the main vortex for all frequencies. However, Cd becomes sensitive to the frequency at l = d. As l increases up to 1.5d, Cd shows a little increase at Stu = 0.1. For other frequencies, Cd continues to decrease and its variation becomes smooth. If l keeps increasing, Cd would change slightly at low and medium frequencies (Stu ≤ 0.5) and decrease smoothly at high frequency (Stu ≥ 0.7). Therefore, longer flexible plate can obtain larger drag reduction and is more possible to generate the thrust force. It has been proven that by attaching the stationary plate to cylinder can reduce the drag. To measure the effectiveness of drag reduction of undulatory plate, the ratio 14

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of total power in the undulatory plate case over that required to overcome the drag in the stationary plate case can be defined as



Pu  Pd  flexible Pd  stationary



C pt Cd  stationary

(11)

where C pt and Cd  stationary are the mean non-dimensional total power of undulatory plate case and the mean drag coefficient of the stationary plate case, respectively. Clearly, the condition to both reduce the drag and save the energy is   1 , which is known as the net saving situation. Figure 10 plots the variation of  in the range of 0.1 ≤ Stu ≤ 0.5 at A = 0.2 with different plate length. From the figure, it can be seen that  is always larger than 1 when Stu ≥ 0.4. Therefore, the net saving situation only occurs at low frequency for all plate lengths. When l = 0.5d and d,  increases with Stu monotonously and the minimum values are 96.26% and 96.4% respectively. For other plate lengths,  first decreases and then turns to increase rapidly. The minimum values appear at Stu = 0.2. The longer the plate is, the smaller the minimum

 is. The minimum  is 91.86%. However,  of longer plate increases much faster than that of shorter plate when Stu ≥ 0.3. Thus a short plate could achieve drag reduction with less consumption of energy and a long plate might reduce drag more effectively. In addition to the force behaviors and power consumption, the length of flexible plate also influences the flow characteristics. Figure 11 illustrates the instantaneous vorticity contours at different plate length with A = 0.2, Stu = 0.2 and Stu = 0.8. When the frequency is low (Stu = 0.2), the size of vortices in the wake firstly increases with the plate length and then starts to decrease as l > 1.5d. This phenomenon could be 15

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explained as follows. The secondary vortex from the flexible plate is weak when l ≤ 1.5d. Its impact on the main vortex from the cylinder is not significant. The elongation of vortices is mainly caused by the increase of plate length which is similar to the stationary plate case [15]. As l reaches 2d, the interaction of secondary vortex and main vortex cannot be neglected. Hence, the growth of vortices is prevented by the strengthened secondary vortex. When Stu = 0.8 however, with the increase of l, the vortices are first shrunk and then elongated. At this frequency, the secondary vortex is strong and it tries to dominate the wake. For the short plate (l = 0.5d), the secondary vortex cannot compete with the main vortex. Consequently, the long vortices are observed. Since the strength of secondary vortex increases with the plate length, the vortex sheding is completely suppressed when l ≥ 2d. As a result, the vortex street behind the plate is almost the same as that observed in a single travelling wavy plate [33]. From the discussion above, it is clear that the length of flexible plate can evidently change the flow behaviors. A short plate can partially influence the flow field and the drag reduction is small. A long plate may completely suppress the vortex shedding and the formed wake is similar to that in the individual undulatory plate case. Accordingly, large propulsive force could be generated.

3.3 Comparison between flexible and rigid plates It is noted that the current study is the extension of our previous work [41]. To demonstrate the difference of performance between the flexible plate and the rigid one, 16

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we make a comparison about the force peformance and the power consumption. The plate length is fixed at l = d. For the flapping motion, the amplitude of flapping angle is denoted by θ. Then the excursion of plate tail is A  l sin  . In present simulations, two angles with θ = 15o and 25o are considered. The corresponding amplitudes of undulation are A = 0.2588 and 0.4226, respectively. Moreover, the frequency of motion changes from 0.1 to 0.5. Figure 12 shows the variation of mean drag coefficient with frequency. The drag fluctuation represented by the error bar is also presented in the figure. It is noted that only the positive half bar is shown. For θ = 15o and A = 0.2588 as plotted in Fig. 12(a), Cd of the flapping plate case (denoted as Cd  f ) is smaller than that of the undulating

plate case (denoted as Cd u ) when Stu ≤ 0.4. After that, Cd  f exceeds Cd u since

Cd  f starts to increase with Stu when Stu ≥ 0.3 while Cd u decreases monotonously. For θ = 25o and A = 0.4226 (Fig. 12(b)), the variation tendency of Cd is similar to that in Fig 12(a). At this time, Cd u is always smaller than Cd  f . Additionally, at the same frequency and amplitude, the drag fluctuation caused by the flapping motion is higher than that by the undulation motion. Figure 13 displays the comparison about the effectiveness of drag reduction. At θ = 15o, the net saving situation (  = 96.6%) appears at Stu = 0.1 for the flapping plate case (denoted as  f ). When Stu > 0.1,  f is larger than 1 and increases steeply. Correspondingly, at A = 0.2588,  of the flexible plate case (denoted as  u ) decreases roughly from the balance situation (  ≈ 1) to the net saving situation (  = 97.4%) when Stu increases from 0.1 to 0.2. After that,  u stops decreasing and

17

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increases with Stu gradually. When Stu > 0.2,  u is much smaller than  f . On the other hand, no net saving situation can be found for the flapping plate at θ = 25o. The increment of  f is less than that at θ = 15o. Meanwhile, the minimum  u at A = 0.4226 can be obtained at Stu = 0.3 (  = 94.7%). Moreover,  u at A = 0.4226 is always smaller than that at A = 0.2588 when Stu > 0.1. Based on the comparison above, it can be concluded that the undulatory plate performs better than the flapping one and it can produce the effective drag reduction. It seems that the flexible plate is a more suitable choice for the active flow control. To better understand the good performance of flexible plate, the physical mechanisms are required to be explored. It is known that the pressure drag is dominant in the drag force for the Reynolds number considered in this work [47]. Due to the motion of plate, the pressure distribution in the near wake is consequently influenced. Figure 14 diagrams the mean pressure coefficient profiles at x/d = 21.55 (It is located at the position, which is 0.05d away from the tail of plate) as a function of y position. The pressure coefficient C p is defined as

Cp 

2  Pw  P0   u2

(12)

where P0 is the free stream pressure, Pw is the local pressure in the near wake. In the figure, the amplitudes are A = 0.4226 for the flexible plate and θ = 25o for the rigid one. Two frequencies with Stu = 0.2 and 0.3 are considered. It is shown from the figure that the obvious change of C p only happens in the region spanning about 2d symmetric to the centerline. At Stu = 0.2, C p of the flexible plate case (denoted as

C p u ) is larger than that of the rigid one (denoted as C p  f ). Since the pressure in 18

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front of the cylinder is nearly not affected by the motion of plate, it almost keeps constant. Therefore, higher C p in the near wake would generate lower pressure drag. Moreover, the variation of C p  f is more dramatical than that of C p u . This explains why Cd of the flexible plate case and its fluctuation are smaller than those of the rigid one in Fig. 12(b). As Stu increases up to 0.3, C p u is enhanced. So, the pressure drag is reduced as compared with that at Stu = 0.2. In contrast, C p  f increases slightly and sharp oscillation can be seen. As a result, smaller pressure drag with high fluctuation would be produced. This is also consistent with the force behaviors in Fig. 12(b). Another mechanism to explicate the good performance of flexible plate might be the wake width. As reported by Niu and Hu [48] who experimentally studied the viscous flow over a hairy disk very recently, smaller wake width can produce lower drag. Figure 15 compares the instantaneous vorticity contours of the flexible and rigid plates at Stu = 0.1 and 0.4. The amplitudes are the same as those in Fig 14. At Stu = 0.1 as shown in Fig. 15(a), the cylinder with the flexible plate sheds narrower vortices than the rigid one. It means that the wake width behind the flexible plate is smaller. When Stu = 0.4 (Fig. 15(b)), the flexible plate generates a stable wake with limited width. But the vortex induced by the rigid plate is complicated and it keeps shedding with broad width. This behavior corresponds well to the drag difference between the flexible and rigid plates in Fig. 12(b). To further dig into the flow characteristics of the flexible and rigid plates, the vortex interaction in the near wake could be examined. It has been indicated in our

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previous work [41] that the mode of vortex interaction can be classified into the constructive interaction and the destructive interaction. When the secondary vortex from the plate constructively interacts with the main vortex from the cylinder, the strength of the combined vortex would be enhanced. As a consequence, the mean drag force on the cylinder and plate is increased. In contrast, the destructive vortex interaction could decrease the mean drag force. Figure 16 provides the evolution of vortex interaction in one cycle for the flexible and rigid plates. The sketch maps of the vortex positions are also presented. The amplitude and frequency of motion are the same as those in Fig. 15(b). The time interval is T/4, where T is the period of motion. When the tail of plate comes to the top of stroke at T/4 (Fig. 16(a)), both the flexible plate and the rigid plate are generating a counterclockwise vortex (labeled as p1). After one time interval (Fig. 16(b)), the vortex from the flexible plate is detached by the plate and tries to interact with the vortex with clockwise direction on the upper surface of the cylinder (labeled as c1). This phenomenon may be caused by the travelling wave along the plate. It enables the vortex to move along the plate smoothly. In contrast, the vortex of the rigid plate is pulled by the plate and tries to interact with the vortex with counterclockwise direction on the lower surface of the cylinder. When the tail of plate reaches the bottom of stoke (Fig. 16(c)), the vortex from flexible plate has destructively interacted with the vortex from cylinder (forming a combined vortex, labeled as s1) and a clockwise vortex is being produced on the tail (labeled as p2). Accordingly, the constructive vortex interaction happened for the rigid plate. At the end of one period (Fig. 16(d)), the vortex from the plate is going to interact with the

20

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newly generated vortex from cylinder (labeled as c2). In addition, it can also be found from the figure that as compared with the rigid plate, the flexible plate almost does not change the position where the vortex from cylinder sheds. Consequently, the force fluctuations for the flexible plate are smaller than those for the rigid plate. Based on the analysis above, it is known that owing to the undulation motion, the flexible plate could weaken the vortex from the cylinder, which may increase the pressure in the near wake consequently. At the same time, the width of wake could be well controlled in a limited range due to the existence of travelling wave along the plate, which generally contributes to the formation of stable flow.

4. Conclusions In this study, the numerical investigation of the laminar flow over a stationary circular cylinder with an attached undulatory plate has been performed. The current work is the direct extension of our previous study on the flow characteristics behind a cylinder with a rigid flapping plate. By fixing the Reynolds number at Re = 100, we systematically examine the effect of parameters including the oscillating frequency Stu, amplitude A and length of plate l on the flow characteristics. When l = d, the flow behaviors are highly dependent on the oscillating frequency and amplitude. The mean drag coefficient is always smaller than that in the stationary plate case. It monotonously decreases with Stu for all amplitudes considered. The influence of amplitude becomes obvious at medium and high frequencies. Higher Stu produces larger drag reduction. On the other hand, the fluctuations of drag and lift 21

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forces increase with frequency and amplitude, which causes high power consumption and large drag reduction. Due to the existence of secondary vortex, the flow patterns become stable at some specific frequencies and amplitudes. Moreover, the jet-like vorticity profiles are also formed under some conditions, which is associated with the appearance of thrust force. Additionally, by varying the plate length at some fixed amplitude, it is found that the maximum drag reduction increases with the plate length. Due to the weak secondary vortex, the flow pattern is partially changed for a short plate. However, a long plate can successfully suppress the vortex shedding. As compared with the rigid flapping plate, the undulatory plate produces more drag reduction associated with smaller force fluctuations at the expense of lower energy consumption. This may be due to the relatively high pressure distribution in the near wake, the narrow width of wake and the presence of destructive vortex interaction. The current study reveals that a flexible plate may be more effective than the rigid one to actively control the vortex shedding of bluff body.

Acknowledgments This work was supported by the National Natural Science Foundation of China (11302104 and 11272153) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

References

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2011.

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Table I Parametric study of flows over a cylinder with flexible plate at Re = 100 l

A

Stu

0.2

x

Cd

Cl

0.01

1.077

1.914

0.0125

1.089

1.926

0.00625

1.074

1.911

0.01

0.641

4.949

0.0125

0.652

4.971

0.00625

0.643

4.945

0.01

0.91

1.345

0.0125

0.923

1.358

0.00625

0.911

1.344

0.01

0.789

4.11

0.0125

0.806

4.264

0.00625

0.785

4.115

0.1

0.6

d

0.4

0.5

1.0

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l

d

Al

 x0 , y0  y

x

u Fig. 1

x  x, t   x0  x y  x, t   y0  a  x  sin  kx  2 ft 

Flows over a circular cylinder and a flexible plate with undulation motion

Fig. 2

Variation of mean drag coefficient Cd with Stu at l = d

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(a)

(b)

(a) Cd , (b) Cl Fig. 3

Variation of drag and lift coefficient fluctuations with Stu

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(a1)

(a2)

(b1)

(b2)

32

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(c1)

(c2)

(a) A = 0.1, (b) A = 0.3, (c) A = 0.5; (1) Stu = 0.1, (2) Stu = 1; In the time history plots (left row), solid line means drag coefficient and dashed-dotted line means lift coefficient; Fig. 4

Time histories of drag and lift coefficients and their phase diagram

33

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Fig. 5

Fig. 6

Variation of mean non-dimensional power with Stu

Instantaneous vorticity contours and time-averaged streamlines of flows over a cylinder with a stationary rigid plate at Re = 100

(a)

(b)

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(c)

(a) Stu = 0.1, (b) Stu = 0.4, (c) Stu = 1 Fig. 7

Flow patterns vary with Stu at A = 0.2

(a)

(b)

(c)

(d)

(a) Stu = 0.1, (b) Stu = 0.3, (c) Stu = 0.6, (d) Stu = 1 Fig. 8

Flow patterns vary with Stu at A = 0.4

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Fig. 9

Variation of mean drag coefficient with plate length at A = 0.2

Fig. 10 Effectiveness of drag reduction varies with frequency at A = 0.2

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(a)

(b)

(c)

(d)

(e)

Stu = 0.2

Stu = 0.8

(a) l = 0.5d, (b) l = d, (c) l = 1.5d, (d) l = 2d, (e) l = 2.5d Fig. 11 Flow patterns vary with plate length at A = 0.2

37

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(a)

(b)

(a) flexible plate: A = 0.2588, rigid plate θ = 15o (b) flexible plate: A = 0.4226, rigid plate θ = 25o Fig. 12 Comparison of drag coefficients between flexible and rigid plates

38

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Fig. 13 Effectiveness of drag reduction for flexible and rigid plates

Fig. 14 Mean pressure coefficient profiles in the near wake of plate (x/d = 21.55) at A = 0.4226 for flexible plate and θ = 25o for rigid plate

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(a)

(b)

(a) Instantaneous vorticity contours at Stu = 0.1 (b) Instantaneous vorticity contours at Stu = 0.4 Red line: flexible plate, Blue line: rigid plate. Fig. 15 Comparison of wake width at A = 0.4226 for flexible plate and θ = 25o for rigid plate

(a)

c1 p1

p1

c1

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(b)

c1 p1

p1 c1

(c)

c2 s1

p2

s1

p2 c2

(d)

41

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c2 p2

s1

s1

p2 c2

flexible plate

rigid plate

Fig. 16 Evolution of vortex interaction in a motion period for flexible and rigid plates

42