Numerical study on the suppression of the vortex-induced vibration of an elastically mounted cylinder by a traveling wave wall

Journal of Fluids and Structures 44 (2014) 145–165

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Numerical study on the suppression of the vortex-induced vibration of an elastically mounted cylinder by a traveling wave wall Feng Xu a,n, Wen-Li Chen b, Yi-Qing Xiao a, Hui Li b, Jin-Ping Ou b,c a b c

School of Civil and Environmental Engineering, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China School of Civil & Hydraulic Engineering, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o

abstract

Article history: Received 14 November 2012 Accepted 10 October 2013 Available online 12 November 2013

In the present paper, the commercial CFD code “Fluent” was employed to perform 2-D simulations of an entire process that included the flow around a fixed circular cylinder, the oscillating cylinder (vortex-induced vibration, VIV) and the oscillating cylinder subjected to shape control by a traveling wave wall (TWW) method. The study mainly focused on using the TWW control method to suppress the VIV of an elastically supported circular cylinder with two degrees of freedom at a low Reynolds number of 200. The cross flow (CF) and the inline flow (IL) displacements, the centroid motion trajectories and the lift and drag forces of the cylinder that changed with the frequency ratios were analyzed in detail. The results indicate that a series of small-scale vortices will be formed in the troughs of the traveling wave located on the rear part of the circular cylinder; these vortices can effectively control the flow separation from the cylinder surface, eliminate the oscillating wake and suppress the VIV of the cylinder. A TWW starting at the initial time or at some time halfway through the time interval can significantly suppress the CF and IL vibrations of the cylinder and can remarkably decrease the fluctuations of the lift coefficients and the average values of the drag coefficients; however, it will simultaneously dramatically increase the fluctuations of the drag coefficients. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Traveling wave wall Flow control Vortex-induced vibration Two degree of freedom CFD numerical simulation

1. Introduction The vortex shedding of a circular cylinder can induce dynamic loads or vibrations of structures, which are often reported in many actual engineering problems and become the main factors in structure instability and fatigue failure. Therefore, eliminating the alternating wake pattern of the cylinder and suppressing the VIV of the cylinder are of significance. Passive flow control does not consume external energy and achieves the purpose of the flow control, mainly by changing the flow conditions, such as the boundary conditions, the pressure gradient and so on. The most common approach to passive flow control is adjusting and optimizing the geometry of the structure surface to find the flow control purpose. Using riblets to reduce the drag is a typical passive flow control method (Choi, 1989; Bechert, 1997; Lee and Jang, 2005). Bearman and Owen (1998), Bearman and Brankovic (2004) found the reduction of the VIV of a square cylinder with a surface consisting of wavy separation lines and a circular cylinder with hemispherical bumps attached. Owen and Bearman

n

Corresponding author. Tel./fax: þ86 755 26033506. E-mail addresses: [email protected], [email protected] (F. Xu).

0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2013.10.005

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(2001) attached hemispherical bumps to the circular cylinder surface to control the VIV amplitudes because the regular vortex shedding near the wake of the cylinder is destroyed by the surface bumps. The passive flow control is determined in advance, although it can obtain certain control effectiveness when the flow condition changes and the control effectiveness of the passive flow control cannot reach the optimal status. An active flow control is inputting the external energy into the flow field and injecting the proper perturbation interacted with the flow inner mode to achieve the control purpose. So far, researchers have developed several active flow control methods to obtain low drag, high lift, vibration suppression and aerodynamic performance improvement, such as drag reduction, using wall vibration, the bubble method, injection and suction, momentum injection, traveling wave wall (TWW) bionic control methods and so on. Arturo and Maurizio (1996) employed the DNS numerical simulation to study wall vibrations to decrease turbulence and surface friction. Cui et al. (1991) conducted low-speed wind tunnel tests to study the influences of four means of excitations on the control effectiveness of flow over two-dimensional airfoils. Modi (1997), Modi and Deshpande (2001), Munshi et al. (1997, 1999), Patnaik and Wei (2002) investigated the momentum injections method to control the flow field around airfoils, flat plates, rectangular prisms and D-section prisms. The results showed that the momentum injected into the flow field by the rotating cylinders can delay flow separation and suppress the generation of a von Karman vortex street and flow-induced vibration. Korkischko and Meneghini (2012) placed each small circular cylinder near the top and bottom boundary layers of a circular cylinder, eliminated the oscillating wake and suppressed the vibration of the cylinder by injecting the angular momentum into the wake through these two small cylinders. When a viscous fluid flows through a rigid wall, the boundary layer cannot be eliminated; however, if the wall is designed to be flexible or movable, the boundary layer may be weakened. The experimental results (Choi, 1996) indicated the turbulent drag reduction rate could reach 7% and the corresponding reduction of the downstream surface friction of the flexible wall was 7%. Yang and Wu (2005) employed the “fluid roller bearing” (FRB) effects of the axisymmetric TWW to form a series of vortex rings separating the main stream from the near-wall flow and resulting in pressure drag and friction drag reductions. Wu et al. (2003) chose the proper wavelength, wave amplitude and ratio of wave velocity to the oncoming velocity to form FRB effects on the finite and infinite two-dimensional TWW by using numerical simulations. The TWW on airfoils has also been studied, and it was found the large flow separation was resisted. Vortex shedding from the streamline airfoils at large angles of attack was suppressed, and the lift was increased. Wu et al. (2007) developed a moving-wall, e.g., a TWW method, control strategy to manage the unsteady separated flow around a circular cylinder. This method introduced a fluid FRB and allowed the global flow to remain attached, resulting in vortex shedding elimination at Re ¼500. Ni et al. (2011) used the wavy wall to suppress the VIV of an elastically supported cylinder and studied the effects of the surface wave propagation directions and frequencies on the flow field. The results indicated that the proper propagation direction and frequency can dramatically decrease the cross-flow displacements. In the present paper, the entire process involving the flow around a fixed cylinder, the VIV of a free vibrating cylinder and the VIV controlled by the TWW was numerically simulated. The authors plan to study the control effectiveness of the TWW on the VIV of the elastically supported cylinder. The displacement time histories and their corresponding statistics, the centroid trajectory changing laws, the aerodynamic force time histories and their statistics at each stage versus the frequency ratios were studied to explain the power effectiveness of the TWW flow control method. Lastly, the mechanism of TWW flow control was revealed by analyzing the boundary vorticity flux and the flow field vorticity distributions. 2. Numerical simulation models and validation 2.1. Governing equations 2.1.1. Governing equations of fluid flow The governing equations of the fluid domain include the continuity equation in Eq. (1) and the Navier–Stokes equation in Eq. (2) for a two-dimensional incompressible flow. In a Cartesian coordinate system, the governing equations can be written as ∇ U u ¼ 0;

ð1Þ

∂u 1 þ ðu U ∇Þu ¼  ∇p þ ν∇2 u; ∂t ρ

ð2Þ

where ρ is the fluid density; ν is the kinematic viscosity coefficient; u is the velocity vector, where the components in the inline flow (IL, x direction) and the cross-flow (CF, y direction) directions are denoted as u and v, respectively; p is the pressure; and ∇2 is the Laplace operator. 2.1.2. Governing equations of cylinder vibration A cylinder undergoing a flow-induced vibration can be simplified as a mass–spring–damping system with the mass, stiffness and damping ratio denoted as m, k and c, respectively. The two DOF vibration equations in the IL and CF directions

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of the cylinder can be expressed as follows: mx€ þcx_ þkx ¼ F d ðtÞ;

ð3Þ

my€ þ cy_ þ ky ¼ F L ðtÞ;

ð4Þ

where m is the mass per unit length of the cylinder; t is the time; y, y_ and y€ are the displacement, velocity and acceleration of the cylinder in the CF direction, respectively; x, x_ and x€ are the displacement, velocity and acceleration of the cylinder in the IL direction, respectively; F d ðtÞ ¼ 1=2ρU 21 D C d ðtÞ and F L ðtÞ ¼ 1=2ρU 21 D C L ðtÞ are the drag and lift forces of the cylinder, respectively; C d ðtÞ and C L ðtÞ are the corresponding dimensionless drag and lift coefficients; U 1 is the uniform oncoming velocity; and D is the diameter of the cylinder. Eqs. (3) and (4) are then non-dimensionalized and can be written as 2

d X 4πζ dX 4π 2 2C d þ X¼ ; n þ n2 V r dt πmn dt Vr2

ð5Þ

2

d Y 4πζ dY 4π 2 2C L þ Y¼ ; ð6Þ n þ n2 V r dt πmn dt Vr2 pffiffiffiffiffiffiffi where ζ ¼ c=2 km is the damping ratio of the cylinder vibration system; X ¼ x=D and Y ¼ y=D are the dimensionless displacements in the IL and CF directions, respectively; mn ¼ 4m=ρπD2 is the mass ratio; V r ¼ U 1 =f n D is the reduced pffiffiffiffiffiffiffiffiffi velocity; f n ¼ ð1=2πÞ k=m is the natural frequency of the cylinder; and t n ¼ tU 1 =D is the dimensionless time, where 2 2 dx=dt ¼ U 1 ðdX=dt n Þ and d x=dt 2 ¼ ðU 21 =DÞðd X=dt n2 Þ. 2.1.3. Governing equations of the TWW Because vortex occurrence and vortex shedding are on the rear surface of the cylinder, the traveling wave wall was designed to be set at the cylinder rear part, i.e., the traveling waves propagated to the downstream direction are set on the upper and lower surfaces of the cylinder's rear part and eliminated vortex shedding from the cylinder. In a plane polar coordinate system ðr; θÞ with the origin O located on the center of the circle, the wave equations of the TWW on the cylinder surface can be written as 8 > < x ¼ r cos θ y ¼ r sin θ;  π=2 r θ rπ=2 ; ð7Þ > : r ¼ r þAðLÞ cos ½KðL CtÞ 0 8 L A^ ; > > < λ ^ AðLÞ ¼ A; > > : A^ ðNλ  LÞ ; λ

0 r L rλ λ r L rðN  1Þλ ;

ð8Þ

ðN 1Þλ rL r Nλ

where r 0 ¼ D=2 is the radius of a standard cylinder (no traveling wave on the surface); λ is the wavelength; K ¼ 2π=λ is the wave number; L is the arc length from point A to an arbitrary point P on the TWW as shown in Fig. 1; C is the wave velocity of the TWW; and AðLÞ is the vibration amplitude of each point on the TWW, where A^ is the maximum vibration amplitude and N is the number of wave. As shown in Eq. (8), the amplitude of the first wave increases linearly, the amplitude of the wave number N decreases linearly and the amplitudes of other middle waves are equal; this configuration can make the TWW connect sleekly with the other fixed surface of the cylinder.

2.2. Geometrical model and boundary conditions Fig. 1 shows the cylinder with the TWW elastically supported by the spring and damper in a uniform flow. The cylinder can vibrate in the IL and CF directions, and the traveling wave propagates from point B and point C to point A. Fig. 2 shows the computational domain and the grid partition. The computational domain is a rectangle region of 40D  20D, and the cylinder center is located on the coordinate origin. The diameter of the cylinder D equals 0.01 m. The distances from the upstream inlet, downstream outlet and the two sides to the cylinder center are 10D, 30D and 10D, respectively. The computational domain is discretized by unstructured grids, and the grids near the cylinder surface and in the wake where the gradients of the parameters are large are locally refined. The smallest grid size of the cylinder surface is 0:018D, and the cell number is estimated to be approximately 57 000 for the present 2-D simulation. The flow direction is from left to right, and the boundary conditions are set as follows: the left side is a velocity-inlet with a uniform velocity U 1 ; the right side is a pressure-outlet where the relative pressure is set as 0; the upper and lower sides are set as symmetry walls; and the cylinder surface is set as a no-slip wall.

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Fig. 1. Vibrating model of an elastically mounted cylinder with the TWW.

Fig. 2. Computational domain and grid distribution.

2.3. Numerical solution procedure The present numerical simulation started from the flow around a fixed cylinder, when the alternating vortex shedding occurred. The cylinder is then set free to vibrate with two DOF. The vortex-induced vibration of the cylinder is caused by alternating vortex shedding, when the vibration tended to be steady, and the traveling wave wall is then activated to control the flow field. The entire numerical procedure includes the flow around a fixed cylinder (simplified as “FR”), the oscillating cylinder (vortex-induced vibration, VIV) and the oscillating cylinder controlled by using a traveling wave wall (“TWW”) method. The entire procedure can be simplified and written as “FR-VIV-TWW”. The numerical simulations are carried out at Re ¼ 200, and the laminar model is employed. The SIMPLE algorithm is used to calculate the coupling between the pressure and velocity fields. The format of the pressure interpolation is chosen as “Standard”. The second-order upwind scheme is employed for momentum discretization because of its stability and veracity. A first-order implicit forward discretization scheme is adopted for the time-derivative term. The time steps are changed according to the numerical simulation conditions. The dimensionless time step is set as Δt n ¼ 0:029 with the corresponding real time step of Δt ¼ 0:001 s for the “FR” and “VIV” stages. The smaller time step is used to simulate the “TWW” stage according to the wave period. The solution of the flow field is obtained using a CFD code (Fluent) based on the finite volume method (FVM) with a pressure-based algorithm. The aerodynamic coefficients C l and C d are first obtained by solving the pressure and velocity fields and then substituted into Eqs. (5) and (6) to calculate the cylinder responses in the IL and CF directions using the fourth-order Runge–Kutta method. According to Eq. (7), the moving speed of the TWW is obtained and combined with the cylinder velocities in the IL and CF directions. The composed speed of the TWW is then transferred to the cylinder surface, which is the flow field boundary. Creating the moving boundaries of the cylinder is accomplished through the dynamic mesh technique, which is controlled by a user-defined function (UDF) embedded into the Fluent code.

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2.4. Validity checking To validate the accuracy of the above-described grid and scheme, a uniform flow past a rigid circular cylinder at Re ¼ 200 is calculated first. Fig. 3 shows the dimensionless time histories and the corresponding spectra of the lift and drag coefficients, respectively. The lift and drag forces behave as regular periodic responses due to the alternating vortex shedding in the wake as shown in Fig. 4. The Strouhal number Stn is found to be 0.1959 and the dimensionless dominant frequency of the drag force is 0.3917 as shown in Fig. 3(b), which is about twice of Stn , as expected. The comparison results of the present calculations with some previous simulations of the flow around a fixed cylinder are tabulated in Table 1. Earlier study of Farrant et al. (2000) was carried out using the boundary element method, while

Fig. 3. Results of the flow around a rigid cylinder: (a) Lift and drag coefficient time histories and (b) spectral analysis of C l and C d .

Fig. 4. Vortex shedding in the wake of a rigid cylinder at t n ¼ 133:7234, while the C l reaches minimum values  0.6312.

Table 1 Force coefficient and Strouhal number of flow around a rigid cylinder at Re ¼ 200. Investigation

Cd

C d0

CL0

Stn

Farrant et al. (2000) Meneghini et al. (2001) Lam et al. (2008) Current work

1.36 1.30 1.32 1.342

– 0.032 0.026 0.0273

0.51 0.50 0.426 0.444

0.196 0.196 0.196 0.1959

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recently more efforts have been made to solve the unsteady Navier–Stokes equations using FEM or FVM such as the studies of Meneghini et al. (2001) and Lam et al. (2008). In the table, the mean drag coefficient is given by C d , and the root-meansquare (RMS) values of lift and drag coefficients by CL0 and C d0 , respectively. All present calculations compare very well with the simulation results obtained by Lam et al. (2008) using the same FVM. The presently calculated Stn is very close to the published numerical results, showing that our result is reasonably good. It is indicated the grid resolution, time step and numerical scheme chosen in the present paper are proper for studying vortex shedding from a circular cylinder.

3. Results and discussion n

The frequency ratio f n =f s , mass ratio mn and reduced damping Sg ¼ 8π 2 Stn2 M n ζ are the important parameters of the VIV, n where f s and Stn are the vortex shedding frequency and the Strouhal number of the flow, respectively, over the standard n cylinder. In the present paper, the parameters are set as Sg ¼ 0:01, mn ¼ 1:0 and f n =f s in the range of 0.45–5.20. For flow control using TWW, the control parameters are carefully chosen to obtain an ideal control effectiveness, which includes the ^ Generally, C and N should wave velocity C, the number of waves on the upper surface N and maximum wave amplitude A. increase, and A^ should properly decrease with an increase in the Reynolds number. Among these parameters, the wave velocity served the most important role. According to the research of Wu et al. (2007), the ratio of the wave velocity to the oncoming velocity, the number of waves and the maximum wave amplitude are set as C=U 1 ¼ 4:0, N ¼ 4 and A^ ¼ 0:02D,

n

Fig. 5. Characteristic parameters of the CF and IL displacements of an elastically mounted cylinder versus f n =f s : (a) 2y0 =D and (b) x=D.

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respectively, in the numerical simulations. In the entire process of “FR-VIV-TWW”, the IL and CF displacement responses, the centroid motion trajectories, the aerodynamic coefficients, the boundary vorticity flux (BVF) and the vorticity contours in the wake of cylinder are used to study the control effectiveness using the TWW.

n

n

n

Fig. 6. CF and IL displacement time-histories of the cylinder with FR-VIV-TWW full process simulation: (a) f n =f s ¼ 0:55, (b) f n =f s ¼ 0:85, (c) f n =f s ¼ 0:90, n n n n n (d) f n =f s ¼ 0:95, (e) f n =f s ¼ 1:15, (f) f n =f s ¼ 1:30, (g) f n =f s ¼ 1:85 and (h) f n =f s ¼ 2:25.

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n

n

n

n

Fig. 7. Centroid trajectories of the cylinder with FR-VIV-TWW full process simulation: (a) f n =f s ¼ 0:55, (b) f n =f s ¼ 0:85, (c) f n =f s ¼ 0:90, (d) f n =f s ¼ 0:95, n n n n (e) f n =f s ¼ 1:15, (f) f n =f s ¼ 1:30, (g) f n =f s ¼ 1:85, and (h) f n =f s ¼ 2:25. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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153

n

n

n

Fig. 8. Lift and drag coefficients of the cylinder with FR-VIV-TWW full process simulation: (a) f n =f s ¼ 0:55, (b) f n =f s ¼ 0:85, (c) f n =f s ¼ 0:90, n n n n n (d) f n =f s ¼ 0:95, (e) f n =f s ¼ 1:15, (f) f n =f s ¼ 1:30, (g) f n =f s ¼ 1:85 and (h) f n =f s ¼ 2:25.

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3.1. IL and CF displacements Fig. 5 indicates that the fluctuation of the CF displacement 2y0 =D and the average of the IL displacement x=D change with the frequency ratios, as shown by the following four numerical simulation cases: Case 1. The VIV started at the “FR” stage, when the alternating vortex shedding was steady, and the cylinder was then free to vibrate (VIV). Case 2. The VIV was started at t n ¼ 0, which means the cylinder was set free to vibrate at t n ¼ 0. Case 3. The TWW started from the VIV stage, i.e., when the VIV tended to be steady, the TWW was started up. Case 4. The TWW was started at t n ¼ 0. n

Fig. 5(a) shows that the variation trend of 2y′=D versus f n =f s is coincident with those of Zhou et al. (1999) for Case 2, n although the present results were slightly larger than those of Zhou et al. (1999). Besides f n =f s ¼ 0:45, the value of 2y′=D in n Case 1 is larger than that in Case 2. When f n =f s ¼ 0:55 and 0:65, 2y′=D in Case 1 was 50% and 65% larger, respectively, than n those in Case 2; when f n =f s ¼ 0:85–1:30, 2y′=D in Case 1 was 10–15% larger than that in Case 2, and at other frequency ratios, the results of the two cases are close to each other. Comparing the results between Case 1 and Case 3, we could find great differences between the CF displacement fluctuations of the cylinder before and after starting up the TWW. When n f n =f s ¼ 0:45 and 0:55, 2y′=D decreased by 61% and 87%, respectively; moreover, 93–99% reduction of the 2y0 =D was n achieved at the other frequency ratios, after starting the TWW. When f n =f s ¼ 1:15, the amplitude and fluctuation of the CF 0 displacement reached their maximum values, ymax ¼ 0:698D and 2y =D ¼ 0:983, and then decreased to ymax ¼ 0:02D and 2y′=D ¼ 0:0142, i.e., reduced 97.1% and 98.6% after starting up the TWW for Case 1. Fig. 5(b) indicates that the average of the IL displacement x=D in Case 1 is 6–16% smaller than that in Case 2 in the entire frequency ratio range. The variation of the IL displacement at the frequency ratios is the opposite of that of the CF displacement, and the difference was mainly induced by the different initial conditions of the VIV of these two cases, which resulted in different aerodynamic forces. When the TWW is activated, the IL flow displacement dramatically decreased. x=D

(a)

(b)

Fig. 9. Spectral analysis of drag coefficients of the TWW cylinder: (a) Case 3, the wave starts at t n ¼ 290 and (b) Case 4, the wave starts at t n ¼ 0.

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in Case 3 reduced 74–88% and 78–89% compared to that in Case 1 and Case 2, respectively. The averages of the IL n displacements reached the maximum values x ¼ 2:04D (Case 1) and x ¼ 2:27D (Case 2) at f n =f s ¼ 0:45. When the TWW started up, the averages of the IL displacements of the two cases decreased to 0:483D and resulted in a 76.3% and 78.7% reduction, respectively. Therefore, combining the results of IL and CF displacement, the TWW flow control method can effectively suppress the IL and CF vibrations of the cylinder. Case 4 mainly studied the control effectiveness of the TWW on the VIV cylinder starting at t n ¼ 0, i.e., the TWW is activated without waiting for the occurrence and stabilization of the VIV. The results indicate the operation in Case 4 could also suppress the IL and CF displacements of the cylinder. However, 2y0 =D and x=D in Case 4 were slightly larger than those in Case 3, when the wave started at t n ¼ 290, in the entire frequency ratio range. Fig. 6 shows the dimensionless displacement time histories at different frequency ratios in the entire simulation process from the “FR” stage to the VIV stage and then to the TWW stage, which started at t n ¼ 290. At each frequency ratio, there are 1.5

Present, VIV from RL, two-degree Present, VIV from t=0, two-degree C.Y. Zhou, VIV from t=0, two-degree Present, TWW from VIV, two-degree Present, TWW from t=0, two-degree

1.2 0.9 0.6

C' (Standard Cylinder)

0.3 0.0 1

2

2.5

3

4

5

Present, VIV from RL, two-degree Present, VIV from t=0, two-degree C.Y. Zhou, VIV from t=0, two-degree Present, TWW from VIV, two-degree Present, TWW from t=0, two-degree

2.0 1.5

C (Standard Cylinder)

1.0 0.5 0.0 1

2

3

4

5

Present, VIV from RL, two-degree Present, VIV from t=0, two-degree Present, TWW from VIV, two-degree Present, TWW from t=0, two-degree

0.6

0.4

0.2

0.0

C' (Standard Cylinder)

1

2

3

4

5

Fig. 10. Characteristic parameters of the lift and drag coefficients of the cylinder at different conditions: (a) C L0 , (b) C d and (c) C d0 .

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three periods, which were divided into 0 rt n o 145, which was the result of the “FR” stage; 145 rt n o290, which was the VIV without control that behaved like a harmonic vibration feature; and t n Z 290, which was the stage of the TWW that resulted in the dramatic reduction of the average of the IL displacement and the fluctuation of the CF displacement, i.e., the suppression of the IL and CF vibrations. Fig. 7 indicates the centroid trajectories of the cylinder before and after starting up the TWW. Before activating the TWW, the cylinder irregularly vibrated at low frequency ratios, and the centroid trajectories look like figure 8 shapes with a dominant vibration in the CF direction when the frequency ratios increase. The dashed blue curves in the figures indicate the centroid trajectories of the cylinder after the TWW started up. The results show when the TWW started up, Y′ and X were immediately suppressed, and the centroid trajectories gradually approached the initial positions. 3.2. Aerodynamic forces In the entire simulation process of “FR-VIV-TWW”, the aerodynamic forces at each stage are key parameters. Fig. 8 shows the lift and drag coefficients at each stage at 8 typical frequency ratios. In each figure, the interval 0 r t n o 145 for each time history indicates the lift and drag coefficients of the stage of “FR”. The interval 145 r t n o 290 shows the lift and drag n coefficients of the VIV cylinder (Case 1). When f n =f s ¼ 0:85–1:30, the averages and fluctuations of the drag coefficients are larger than those of the fixed cylinder, and the fluctuation of the lift coefficient is first less than that of the fixed cylinder, n then gradually increases and finally exceeds that of the fixed cylinder; when f n =f s 4 1:85, the averages and fluctuations of the drag coefficients and the fluctuations of the lift coefficients gradually decrease and then approach those of the fixed cylinder. In each figure, the interval t n Z 290 for each time history indicated the lift and drag coefficients of the VIV cylinder with TWW flow control (Case 3). The results show the averages of the drag coefficients and the fluctuations of the lift coefficients dramatically decreased, and the fluctuations of the drag coefficients increased in the entire frequency ratio range. The vibrating amplitudes in the IL direction are small, as shown in Figs. 6 and 7, despite the occurrence of VIV resonance. The reason should be that the natural frequencies in the IL and CF directions are equal, but the frequency of the IL force is twice that of the CF force frequency. Fig. 8 shows the mean value of the drag coefficient will decrease after starting up the

5

Upper surface

p

4

t 5=335.0124

3

t 4=328.8842

2

t 3=322.7559

1

t 2=316.6247

0

t 1=310.4965

0

0.004

0.008

0.012

0.016

0.02

Upper surface

5

p

4

t 5=61.5978

3

t 4=55.4692

2

t 3=49.3407

1

t 2=43.2121

0

t 1=37.0836

0

0.004

0.008 n

0.012

0.016

0.02

Fig. 11. Static pressure on the upper surface of the TWW cylinder (f n =f s ¼ 1:30): (a) Case 3, the wave starts at t n ¼ 290 and (b) Case 4, the wave starts at t n ¼ 0.

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157

TWW, but the fluctuating component is much larger than that of the fixed cylinder and the VIV cylinder. However, the response amplitudes in the IL direction, as shown in Fig. 6, are small. The main reason should be that the frequency of drag force is determined by the traveling wave frequency [f ¼ C=λ ¼ nU 1 =λ ¼ 16n=π  U 1 =Dðn ¼ 4:0Þ ¼ 595:06 Hz], which is much n larger than the IL natural frequency of the cylinder [f n ¼ f s ¼ Stn U 1 =D ¼ 5:72 Hz]. Fig. 9 shows the spectral analysis of the drag coefficients of the TWW cylinder at the frequency ratios ranging from 0.45 to 2.60. The main frequencies of the drag n coefficients are kept as 594.6 Hz (Case 3) and 594.4 Hz (Case 4) in the entire f n =f s range, respectively, which are very close to the frequency of the traveling wave. Fig. 10 indicates the statistical analysis of the lift and drag coefficients at different simulation stages, i.e., the fluctuation of the lift coefficient C L0 and the average and fluctuation of the drag coefficients C d and C d0 versus the frequency ratios. In Fig. 10(a), n C L0 of Case 2 is close to the results of Zhou et al. (1999) at f n =f s ¼ 1:3–1:5 and less than the results obtained by Zhou et al. (1999) n at other frequency ratios. C L0 reached its maximum value at f n =f s ¼ 1:5, and the maximum value of Case 1 is 26% larger than that

5

Upper surface

σa

4

t 5=335.0124

3

t 4=328.8842

2

t 3=322.7559

1

t 2=316.6247

0

t 1=310.4965

0

5

0.004

0.008

0.012

0.016

0.02

Upper surface

σp

4

t 5=335.0124

3

t 4=328.8842

2

t 3=322.7559

1

t 2=316.6247

0

t 1=310.4965

0

5

0.004

0.008

0.012

0.016

0.02

Upper surface

σ

4

t 5=335.0124

3

t 4=328.8842

2

t 3=322.7559

1

t 2=316.6247

0

t 1=310.4965

-1

0

0.004

0.008

0.012

0.016

0.02

Fig. 12. sa , sp and s on the upper surface of the TWW cylinder, when the wave starts at t n ¼ 290: (a) sa , (b) sp and (c) s.

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of Case 2. The value of C L0 should be the reason why the fluctuation of the CF displacement of Case 1 was larger than that of Case 2. When the TWW started up, C L0 is suppressed to approximately 0.03 in the entire frequency ratio range for Case 3. For Case 4, the TWW started at t n ¼ 0; C L0 is also dramatically decreased to the level close to that in Case 3 and is less than the result of the flow around a fixed cylinder, which is equal to 0.45. Fig. 10(b) shows that the value of C d calculated in the present paper was slightly larger than the results of Zhou et al. (1999), and the changing trends are very close to each other under the same conditions (Case 2). C d reached its peak at n f n =f s ¼ 1:3 for the different cases. The value of C d in Case 1 is slightly larger than that in Case 2 in the entire frequency ratio range, but the difference is small and less than 10%. When the TWW started up, C d is decreased approximately to 0.29 in Case 3. In Case 4, C d is 20% larger than that in Case 3 and is found to be 0.35, which is less than the value of 1.34 of the FR stage in the entire frequency ratio. In Fig. 10(c), it can be seen that the C d0 in Case 1 is larger than that in Case 2 at n n f n =f s ¼ 0:85–1:5, and the results are close to each other at other frequency ratios. As the f n =f s further increases, C d0 tends to

5

Upper surface

σa

4

t 5=61.5978

3

t 4=55.4692

2

t 3=49.3407

1

t 2=43.2121

0

t 1=37.0836

0

5

0.004

0.008

0.012

0.016

0.02

Upper surface

σp

4

t 5=61.5978

3

t 4=55.4692

2

t 3=49.3407

1

t 2=43.2121

0

t 1=37.0836

0

5

0.004

0.008

0.012

0.016

0.02

Upper surface

σ

4

t 5=61.5978

3

t 4=55.4692

2

t 3=49.3407

1

t 2=43.2121

0

t 1=37.0836

-1

0

0.004

0.008

0.012

0.016

0.02

Fig. 13. sa , sp and s on the upper surface of the TWW cylinder, when the wave starts at t n ¼ 0: (a) sa , (b) sp and (c) s.

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that of the FR stage, which is equal to 0.0273. For Case 3 or Case 4, C d0 dramatically increased and did not change with the variation of the frequency ratios, which are equal to 0.31 and 0.46, respectively. C d0 at various frequency ratios are significantly larger than the results of the flow around a fixed cylinder.

3.3. Boundary vorticity flux and flow vorticity The boundary vorticity flux (BVF) and the vorticity distribution of the flow field are key in revealing the control mechanism of the TWW. The BVF denotes that the vorticity flux passes through a unit area of the boundary per unit time and can be expressed as s ¼ νn U∇ω;

ð9Þ

where ω is the wall vorticity flux and n is the outer normal unit vector. For a two-dimensional incompressible flow, the BVF equals sa , which is induced by the tangent acceleration of the moving boundary of the cylinder surface, plus sp , which is induced by the tangent pressure gradient of the boundary. The sum can be written as    ∂ p s ¼ sa þ sp ¼  as þ ; ð10Þ ∂s ρ where the subscript s denotes the tangential direction of the flexible wall, where the counter-clockwise direction is taken as the positive direction; sa is the controllable part in the BVF and the source of the flow control by the TWW; and sp is the part not controllable and the result under the control of sa . n f n =f s ¼ 1:30 with a higher level of CF displacement was chosen to show the corresponding BVF and flow vorticity distribution. Fig. 11 shows the normalized static pressure distribution of the upper surface of the TWW cylinder at five moments for Case 3 and Case 4. We define φ ¼ π  θ A ½0; π so that φ ¼ 0 at the front stagnation point of the standard cylinder and φ increases in the clockwise direction as shown in Fig. 1. The results indicate that the pressure distributions propagated with the upper surface propagation of the TWW cylinder and that the pressure peak is located on the trough of the traveling wave.

Fig. 14. Vorticity magnitude on the upper surface of the TWW cylinder: (a) Case 3, the wave starts at t n ¼ 290 and (b) Case 4, the wave starts at t n ¼ 0.

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According to the pressure distribution on the cylinder upper surface and Eq. (10), the distributions of sa , sp and s can be obtained. Figs. 12 and 13 show the normalized BVF for Case 3 (t n ¼ 290) and Case 4 (t n ¼ 0), which are plotted with the TWW of the cylinder's upper surface. Each subfigure indicates the results of the five moments after starting up the TWW, and the time bucket (t 1 –t 5 ) is approximately equivalent to the 500 wave cycle. Although the starting moment of the TWW is different, fine control effectiveness is achieved, and the distribution changing laws of BVF are similar for the two cases. The period of sa is two times that of the traveling wave of the cylinder's upper surface as shown in Figs. 12(a) and 13(a). sp for the trailing edge of the cylinder upper surface is close to 0, i.e., flow separation would not occur at this position. The distribution of sp changed with the variation of TWW propagation. The positive peak of sp existed on the leeward of the TWW peak, which means the boundary layer separated, changed into the free shear layer and rolled into vortices; the negative peak of sp existed on the windward of the next TWW peak, which means the free shear layer separating from

Fig. 15. Vorticity contours and streamline diagrams close to the TWW cylinder surface at a given time: (a) at t n ¼ 335:0124, when the wave starts at t n ¼ 290 and (b) at t n ¼ 61:5978, when the wave starts at t n ¼ 0.

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n

n

n

Fig. 16. Comparison of vortex shedding pattern in the wake of a vibrating cylinder with Zhou et al. (1999): (a) f n =f s ¼ 2:60, (b) f n =f s ¼ 1:75, (c) f n =f s ¼ 1:50, n n n (d) f n =f s ¼ 1:30, (e) f n =f s ¼ 0:95 and (f) f n =f s ¼ 0:85.

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Fig. 17. Vorticity contours of TWW cylinder, when the TWW starts up at t n ¼ 290: (a) t n ¼ 310:4965, (b) t n ¼ 316:6247, (c) t n ¼ 322:7559, (d) t n ¼ 328:8842 and (e) t n ¼ 335:0124.

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Fig. 18. Vorticity contours of TWW cylinder, when the TWW starts up at t n ¼ 0: (a) t n ¼ 37:0836, (b) t n ¼ 43:2121, (c) t n ¼ 49:3407, (d) t n ¼ 55:4692 and (e) t n ¼ 61:5978.

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the previous leeward reattached and fixed the vortices into the trough of the traveling wave; see Figs. 12(b) and 13(b). A series of clockwise rotating small-scale vortices are formed along the TWW propagation direction, and the separating vortices did not occur on the trailing edge of the TWW. Therefore, the purposes of suppressing the flow separation and eliminating the oscillating wake of the cylinder are achieved. Figs. 12(c) and 13(c) indicate the distributions of s, which is the superposition of sa and sp . The value of s for the trailing edge of the cylinder is close to 0, which is similar to sp . Fig. 14 indicates the flow vorticity distributions of the cylinder's upper surface for Case 3 and Case 4. The results show the flow vorticities propagated along the flexible surface, and each trough always corresponded to a flow vorticity peak. The vorticity contours and streamline diagrams close to the TWW cylinder surface at t n ¼ 335:0124 (Case 3) and t n ¼ 61:5978 (Case 4) are shown in Fig. 15. The troughs of the TWW cylinder can stably capture a series of clockwise rotating vortices on the upper surface and anticlockwise rotating vortices on the lower surface. The stable small-scale vortices in the troughs of the TWW cylinder can separate the near-wall shear flow from the main stream, effectively eliminate the vortex street in the wake and suppress the VIV of the cylinder. The instantaneous vortex shedding patterns in the wake of a vibrating cylinder are compared with the results of Zhou et al. (1999) under the same simulation conditions, which include the Reynolds number, reduced damping, mass ratio and n frequency ratio, as shown in Fig. 16. When f n =f s ¼ 2:60, the natural frequency of the cylinder is far from the vortex shedding frequency and the vortex pattern (Fig. 16(a)) is very similar to the result of the rigid cylinder in Fig. 4. As the frequency ratio n decreases, the wake vortex shedding modes are influenced by cylinder vibration. When f n =f s ¼ 1:50, the vortex spacing in n the near wake in both the CF and IL directions becomes much smaller, as shown in Fig. 16(c). When f n =f s ¼ 1:30, two parallel rows of oppositely rotating vortices are formed in the near wake. The vortex spacing seems to be smaller in the IL direction and wider in the CF direction. Two parallel rows of vortices become unstable at some downstream position and attempt to restore the Karman-vortex-street-like pattern of the “2S” mode (Fig. 16(d)). As the frequency ratio decreases to 0.95 and 0.85, the IL vortex spacing is enlarged because the vortex shedding frequency decreases as shown in Fig. 16(e, f). The abovecompared results indicate the vortex patterns at different frequency ratios close to the results of Zhou et al. (1999). It is shown that the flow pattern in the wake of an elastically mounted cylinder undergoing VIV is accurately obtained in the present simulation. Figs. 17 and 18 indicate the evolutions of the flow over the TWW cylinder at five different moments for Case 3 and Case 4, respectively. As the TWW started up halfway for Case 3, the cylinder was in the VIV status before activating the TWW, resulting in an alternating vortex shedding in the wake; after starting up the TWW, the free shear layer separating from the cylinder surface would not roll and form into new vortex streets in the wake, and the existing vortex streets in the flow field were gradually eliminated. The final vorticity was concentrated in the region near the symmetric line of the downstream flow field behind the cylinder. Fig. 18 shows the flow vorticity contours at five different moments for Case 4, which activated the TWW at t n ¼ 0. From the initial moment when the TWW was activated, the free shear layer separating from the TWW surface did not develop into vortices and distributed near the symmetric line of the downstream flow field behind the cylinder. That is, from beginning to the end, there is no vortex street formation in the wake of the TWW cylinder, and the purpose of eliminating the wake vortex street and suppressing the VIV of the cylinder was accomplished. 4. Conclusion The present paper studied the TWW flow control method and its control on the VIV of a circular cylinder. A numerical simulation of the entire process of “FR-VIV-TWW” was performed, and the conclusions obtained from the study are as follows. The troughs of the TWW cylinder can produce a series of stable small-scale vortices moving with the TWW propagation. The stable small-scale vortices can separate the main stream from the near-wall shear flow, effectively suppressing the occurrence of the detached vortices from the cylinder surface, eliminating the wake vortex street and suppressing the VIV of the cylinder. The TWW activated initially (Case 4) or halfway (Case 3) can suppress the IL and CF vibrations of the VIV. For Case 4 and Case 3, the control effectiveness is 98.6% and 98.9% for the maximum fluctuation of the CF vibrations, and 71% and 76% for the maximum average of the IL vibrations, respectively. The TWW flow control can decrease the fluctuation of the lift coefficient and the average of the drag coefficient; however, the fluctuation of the drag coefficient will significantly increase in the entire frequency ratio.

Acknowledgments This research was funded by the National Natural Sciences Foundation of China (NSFC) (51008103, 51008093, 51178086, 51378153) and the Fundamental Research Funds for the Central Universities (HIT. NSRIF. 2014127). References Arturo, B., Maurizio, Q., 1996. Turbulent drag reduction by spanwise wall oscillations. Applied Scientific Research 55, 311–326. Bearman, P.W., Owen, J.C., 1998. Reduction of bluff-body drag and suppression of vortex shedding by the introduction of wavy separation lines. Journal of Fluids and Structures 12, 123–130.

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