LES study on traveling wave wall control for the wake of flow around a 3D circular cylinder at high Reynolds number

Abstract

Flow around a cylinder is an important research problem in fluid mechanics. However, studies on traveling wave wall (TWW) flow control for the wake generated by the flow around a 3D cylinder remain limited. This study performs large eddy simulation (LES) to investigate the wake characteristics of flow around a three-dimensional (3D) circular cylinder with the TWW. The influence of the main control parameters of TWW on the aerodynamic forces and wake of the 3D cylinder is analyzed. The wake characteristics of the 3D cylinder at Re = 4 × 10⁴ are obtained, and the traveling wave propagating downstream is achieved on the rear surface of the 3D cylinder using dynamic mesh. The control effects of wave amplitude, number of waves, and wave velocity of the traveling wave on the aerodynamic forces of the cylinder are analyzed, and the optimal control parameter combination is determined. The results demonstrate that the TWW control method completely eliminates the spanwise 3D flow characteristics of the cylinder when the amplitude ratio, number of waves, and velocity ratio are 0.02, 4, and 1.5, respectively. Moreover, it effectively suppresses the flow separation on the cylinder surface, eliminates the wake vortex, and suppresses vortex-induced vibration (VIV).

Keywords

traveling wave wall flow control three-dimensional circular cylinder wake large eddy simulation

Introduction

Flow around a cylinder remains a popular research problem in fluid mechanics because of its complex flow characteristics, including flow separation, vortex generation and shedding, and mutual interference between vortices. Circular cross-sections are commonly used in practical engineering, such as stay cables and slings in long-span bridges, chimneys, cooling towers, transmission lines, columns, and oil pipelines in offshore platform structures. When a fluid such as air or water flows over the cylinder surface, periodic vortex shedding typically occurs in its wake. The vortex, also known as the tendon of fluid motion, is one of the primary sources of unsteady aerodynamic forces acting on the cylinder surface. When the vortex shedding frequency approaches the natural frequency of the structure, a large vortex-induced resonance occurs, typically resulting in structural damage. In addition, vortex-induced vibration (VIV) is a limiting vibration and its long-term effects can reduce the fatigue life of structures. Therefore, controlling the alternating vortex shedding in the cylinder wake is crucial for eliminating the fluid-structure interaction vibration and improving the service life.

The flow control method controls the flow field and is divided into passive and active control according to whether external energy input is required. Passive flow control requires no auxiliary energy consumption and involves changing the flow environment using a passive flow control device. The most common passive control method controls the flow by adjusting the geometry of the surface and using surface attachments. For example, drag reduction approaches using wall ribbing and notching, additional splitter plates, and additional cylinders are representative passive flow control methods. Since the drag reduction research began in the 1930s, several studies have focused on the reduction of surface roughness, assuming that the smoother the surface, the smaller the drag. In 1967, the Kyiv Institute of Hydrodynamics in Ukraine proposed a mechanism for the reduction of hydrodynamic drag on a surface with striped grooves. Walsh (1982) and Walsh and Lindemann (1984), affiliated with the NASA Research Center, found that small V-shaped grooves placed along the flow direction on the surface of an object can effectively reduce the wall frictional resistance. This challenged the conventional belief that the smoother the surface, the smaller the drag. Subsequently, groove drag reduction technology received considerable attention in turbulent drag reduction research. Becher et al. (1997) evaluated the drag reduction performance of groove surfaces with various shapes and found that the V-shaped groove yields the best drag reduction effect, confirming the conclusion of Walsh's research on the optimal geometry; a maximum drag reduction effect of up to 10% could be achieved with this groove. Subsequent research has primarily focused on the optimization of groove dimensions and the improvement of testing methods (Park and Wallace, 1994; Viswanath, 2022; Lee and Jang, 2005). Hwang et al. (2003) performed numerical simulation with laminar flow and observed that the splitter plate could suppress vortex shedding and reduce drag and lift fluctuations. Kwon and Choi (1996) found that the vortex shedding would disappear when the length of the splitter plate was greater than a certain critical value. Setting small obstacles in the surrounding flow field can allow the control of the flow field structure to a certain extent. Through flow display experiments, Strykowski and Sreenivasan (1990) found that the vortex shedding of the main cylinder wake can be suppressed by setting small control cylinders at appropriate positions in the main cylinder wake. They revealed the drag reduction mechanism of the control small cylinders; that is, the concentrated vorticity in the boundary layer of the main cylinder is disturbed by the small control cylinders. Dipankar et al. (2007) used the overset mesh method to study the suppression of the main cylinder wake by small control cylinders for a low Reynolds number and compared their results with the experimental results of Strykowski and Sreenivasan (1990). Dalton and Xu (2001) used 2D LES and flow field visualization technology and found that placing control cylinders at suitable positions in the main cylinder boundary layer can reduce the drag.

Passive control does not require energy input and can achieve a good control effect with a low initial investment; therefore, it is extensively used in practical engineering applications. However, the passive control method is predetermined, and the control parameters cannot change with time. Therefore, this method cannot be adjusted according to the changes in the flow field, typically resulting in a suboptimal control effect. Therefore, more researchers focused on active flow control methods. By contrast, active flow control can be used when and where it is needed, and effective local or global flow changes can be obtained via local energy input. With the advancements in smart material technology, small-scale active flow control systems that are highly efficient, energy-saving, and integrated with controlled objects are replacing conventional large-scale flow control mechanical equipment. A variety of active flow control methods have been developed, including the bubble method, the oscillating wall drag reduction method, the angular momentum input method, the jet and suction method, and the bionic control method.

The bubble method generates a bubble coat on the object surface and controls the bottom flow structure through the low friction and lability of the bubbles. A representative example is the supercavitating underwater drag reduction technology. The results obtained by Arturo and Maurizio (1996) using direct numerical simulation showed that surface friction and turbulence could be suppressed by wall oscillations. Laadhari et al. (1994) conducted experiments and found that when the wall oscillated in the spanwise direction, the average velocity gradient in the turbulent boundary layer near the oscillating wall decreased, and the turbulence intensity weakened, thus verifying that the wall oscillations can reduce frictional resistance in the turbulent boundary layer. Patnaik and Wei (2002), Modi (1997), Modi and Deshpande (2001), and Munshi et al. (1997, 1999) employed the angular momentum input method to study the flow field around D-shaped columns with rotating cylinders, airfoils, and rectangular columns, respectively. The results showed that the momentum input to the flow field through small cylinders rotating at high speed could delay flow separation, reduce drag, and eliminate oscillating wakes, thereby reducing wind-induced vibration response. Bergmann et al. (2006) and Taneda (1978) found that the momentum input into the flow field by the rotation of the main cylinder itself could also eliminate the wake vortex street. Korkischko and Meneghini (2012) experimentally found that the momentum input to the wake field of the main cylinder by two small rotating cylinders enabled flow control and resulted in wake vortex shedding and drag reduction under different rotation speeds of the small cylinders. The steady jet and suction method can control the boundary layer of aircrafts, thereby improving the aerodynamic performance of aircraft and achieving the control effect of increasing the lift and reducing the drag. Through numerical simulation and experiments, Cui et al. (1991) determined that the flux and position of the jet and suction have an important influence on the control effect. Chen et al. (2013, 2014, 2015) used wind tunnel test and numerical simulation to study the control effect of the suction method on the aerodynamic forces and VIV of the cylinder. The research on jet and suction technology mainly focuses on the opening method, shape, position, strength of the jet or suction, changing method of the jet, and the active flow control mechanism of the jet and suction.

Among the active flow control methods, the TWW control method is closely integrated with bionics. The method was derived based on two photographs of dolphins swimming and jumping at high speed captured by Hertel (1963), and the corrugated wrinkles in the dolphin's lower abdomen skin are evident in the photographs. The researchers believed that dolphins use the undulations in the skin surface to reduce drag while swimming. The experimental results of Choi et al. (1996) using flexible walls indicated that the turbulent drag reduction rate could reach 7%, and the surface friction downstream of the corresponding flexible wall was reduced by 7%. Savchenko (1980) and Taneda (1978) generated transverse traveling wave walls with a certain amplitude and phase velocity and captured stable vortices at troughs. These vortices can replace the conventional boundary layer and form the “fluid roller bearing” (FRB) in the fluid layer between the solid wall and the main flow, which can achieve the effect of drag reduction. Based on the principle of traveling wave response from the skin surface of dolphins swimming at high speed and inspired by the experimental results, research on the TWW control method has been gradually carried out. Wu et al. (1990) found that the velocity between the wall and the captured vortex is in the same direction as that in the tangential motion upstream of the 2D transverse TWW; therefore, the shearing force is smaller than that in the conventional boundary layer. When the waveform and maximum amplitude of the traveling wave are known, there is a critical wave velocity at which the average shearing force in the fluid boundary layer near the wall disappears, and the total drag is zero at this time. Wu and Wu (1996) verified this predicted result using numerical simulation. Wu et al. (2003) found that the control effect of TWW was related to parameters such as wavelength, wave amplitude, and wave speed. The stable FRB effect was captured on the 2D TWW via numerical simulation, and the TWW was formed on the upper surface of the 2D airfoil. Moreover, the simulation results showed that TWW prevented the large flow separation on the airfoil surface, suppressed the vortex shedding on the airfoil surface with a large attack angle, significantly increased the lift. Yang and Wu (2005) used the vortex train formed by axisymmetric TWW to separate the main flow and the wall flow, which effectively reduced the frictional resistance and pressure difference resistance. The aforementioned studies focused on the flow control for plates or channels using the TWW method. Wu et al. (2007) studied TWW control of the flow field around a static 2D cylinder for the first time. They set TWW on the rear surface of the cylinder, analyzed the influence of control parameters on control effect, set reasonable parameter values, and eliminated the vortex shedding in the cylinder wake at a Reynolds number of 500. Xu et al. (2014, 2017) performed numerical simulation for TWW control of the flow around a 2D cylinder and VIV at a low Reynolds number of 200. They comprehensively analyzed the influence law of each parameter on the control effect for the cylinder wake. The results showed that appropriate propagation direction of the traveling wave and wave velocity could significantly reduce the downwind and crosswind vibrations of the cylinder. These observations suggest that the TWW flow control method has great potential in turbulence control for the flow field around a bluff body.

However, studies on TWW flow control for the wake of the flow around a 3D cylinder remain limited. In this study, LES is performed to examine the characteristics of the flow field around a 3D cylindrical section model. The 3D TWW is formed on the cylinder surface using dynamic meshing technology. The control effect of the traveling wave’s amplitude, number, and velocity on the aerodynamic forces on the 3D cylinder is analyzed. The mechanism of suppressing aerodynamic forces and VIV using the TWW method is revealed in terms of the instantaneous vortex structure, vorticity field, and time-averaged flow field of the 3D cylinder wake.

Numerical simulation model and validation

Geometric model and boundary conditions

In a rectangular Cartesian coordinate system, x is the downwind direction, y is the crosswind direction, and z is the spanwise direction of the cylinder. The center of the bottom surface of the cylinder is located at the origin, and the diameter D of the cylinder is 0.12 m. Karniadakis (1992) showed that the 3D flow characteristics could be effectively captured when the spanwise length of the cylinder is greater than πD. Therefore, the spanwise length of the cylinder L is set as 4D. Based on the study by Behr et al. (1995) on the size of the computational domain of the flow around a cylinder, the effective distance is obtained, and the influence of blockage ratio is ignored when the inlet boundary and the symmetry boundaries on both sides are not less than 8D from the cylinder center. The computational domain is a hexahedral region with a length, width, and height of 28D, 16D and 4D, respectively. The center of the cylinder is 8D from the upstream inlet, 20D from the downstream outlet, and 8D from the symmetry boundary on each side. The computational domain and boundary condition settings are shown in Figure 1. The flow direction is along the positive x-axis. The velocity-inlet condition is adopted at the inlet boundary, and a uniform inflow velocity U is assumed. The outflow boundary condition is used for the computational domain outlet. The remaining four surfaces of the computational domain are set as symmetry boundary conditions, and the cylinder surface is set as a no-slip wall boundary.

Figure 1.

Computational domain and boundary condition settings.

The computational domain is partitioned; the 4D × 4D range centered on the cylinder axis is considered the core refined region, and the rest is the exterior region. The entire computational domain is discretized using a hybrid grid. In the core refined region, structured hexahedral meshes are used inside boundary layers and unstructured tetrahedral meshes are used outside of boundary layers. Structured hexahedral meshes are employed to discretize the external area outside the core refined region. Mesh deformation only occurs within the core refined region, and mesh in the exterior region of the far field is not deformed. Specifically, the dynamic mesh technique is employed to facilitate the flexible deformation of the cylinder surface and enable Fluid-Structure Interaction (FSI). The traveling wave movement on the cylinder surface is achieved through the utilization of a User-Defined Function (UDF) and a grid node motion macro (Define_Grid_Motion). The structured grids within the boundary layers in the refined region deform synchronously with the TWW cylinder surface, while unstructured tetrahedral grids in the refined region undergo remeshing. Thus, the flexible traveling wave motion on the cylinder surface can be realized, and the computational efficiency can be improved. For the core refined region, thin hexahedral mesh elements are used in the boundary layer range of the cylinder surface, and the mesh height of each layer gradually increases from the inside to the outside, with a growth rate of 1.15. In addition, the tetrahedral mesh is used from the outside of the boundary layers to the boundary of the core refined region. During the traveling wave motion of the cylinder surface, the mesh in the boundary layers and the cylinder wall simultaneously undergo flexible deformation. Mesh reconstruction only occurs in the tetrahedral mesh, which ensures the orthogonality of the grid on the cylinder surface and improves the accuracy of calculated results. The meshing of the entire computational domain and the local region around the cylinder is shown in Figure 2.

Figure 2.

Meshing of the entire computational domain and the local region around the cylinder. (a) Global meshing (midspan section) (b) local meshing (3D view) (c) local meshing (2D view).

Solution setup and result validation

For the flow around a circular cylinder, vortex shedding occurs on the rear surface of the cylinder. Therefore, the TWW is designed to be set at the rear part of the cylinder. The governing equations for fluid motion and the wave equations of the TWW on the cylinder surface can be found in Xu et al. (2014). The movement of TWW is realized by governing equations of traveling wave and dynamic mesh technology. The wave equations of the TWW on the cylinder surface can be written as

\begin{array}{l} x = r \cos θ \\ y = r \sin θ, \begin{array}{l} - π / 2 \leq θ \leq \end{array} π / 2 \\ r = r_{0} + A (l) \cos [k (l - c t)] \end{array}

(1)

A (l) = {\begin{cases} \hat{A} \frac{l}{λ}, & 0 \leq l \leq λ \\ \hat{A}, & λ \leq l \leq (N - 1) λ \\ \hat{A} \frac{(N λ - l)}{λ}, & (N - 1) λ \leq l \leq N λ \end{cases}

(2)

λ = \frac{π D}{4 N} = \frac{π \times 0.12}{4 N} = {\begin{cases} 0.0471, & N = 2 \\ 0.0314 & N = 3 \\ 0.0236, & N = 4 \\ 0.0189, & N = 5 \\ 0.0157, & N = 6 \end{cases} (U n i t : m)

(3)

where

r_{0} = D / 2

, wave number

k = 2 π / λ

λ

is the wavelength, l is the arc length from point A to an arbitrary point P on the TWW cylinder surface as shown in Figure 2(b), c is the wave velocity,

A (l)

is the vibration amplitude,

\hat{A}

is the maximum vibration amplitude, and N is the number of waves. The

\hat{A}

of the first waveform increases linearly and the

\hat{A}

of the last waveform decreases linearly for each 1/4 arc surface.

Taking the number of waves N = 4 as an example, the movement pattern of traveling waves is illustrated in Figure 3. As depicted in the schematic diagram, the traveling wave propagates from points B and C to point A. The term ‘N = 4’ indicates the presence of four traveling waveforms along the 1/4 arc on the rear surface of the cylinder. The ratio c/U represents the traveling wave velocity relative to the incoming flow velocity, which is crucial parameter in the TWW control method. On both 1/4 arcs of the rear surface of the cylinder, the traveling wave simultaneously propagates downstream. Within the rear region of the cylinder, all grid node positions are calculated using equation (1), while equation (2) is exclusively employed to control the amplitude of each traveling wave. This ensures a smooth connection of boundaries at positions A, B, and C.

Figure 3.

Schematic diagram of traveling wave motion on a cylinder surface.

The flow field calculations are performed at a high Reynolds number of $R e = 4 \times 10^{4}$ . LES with the Smagorinsky subgrid-scale model is used to simulate the turbulent flow around the cylinder. Bounded central differencing is used for the discretization of convection terms, and pressure interpolation is selected as the standard. The coupling between pressure and velocity is solved using the SIMPLEC (semi-implicit method for pressure linked equations-consistent) algorithm. Bounded second order implicit discretization is used for the transient terms. The time step used in the calculations is $2.0 \times 10^{- 4}$ s, which is approximately 1/600 of the vortex shedding period.

The characteristics of the flow field around the cylinder are the basis for comparing the control effects of the TWW method. Therefore, LES is employed to study the 3D flow field around the cylinder at

R e = 4 \times 10^{4}

and verify grid independence. Parnaudeau et al. (2008) showed that the numerical simulation results were in good agreement with the experimental results when the spanwise grid length of the cylinder

Δ z

was less than 0.065D. Therefore, the spanwise grid length of the cylinder is set as 0.05D in this study. For grid independence verification, keeping the height of the first layer grid on the cylinder surface

h_{0}

and the spanwise grid length

Δ z

fixed, three types of grids with a total number

N_{t}

of grid elements are generated by varying the number of circumferential grid elements

N_{c}

, which are named GR1, GR2 and GR3, respectively. The parameters of each grid are listed in Table 1. The height of the first layer grid on the cylinder surface

h_{0}

is set as

4.17 \times 10^{- 4}

m. For the three grid schemes,

y_{\max}^{+} < 1

(see Table 1), which satisfies the accuracy requirement of LES for the wall grid. The instantaneous distributions of

y^{+}

values on the cylinder surface for the three grid schemes are shown in Figure 4.

Table 1.

Parameters for the three types of grid schemes.

Grid schemes	$h_{0} / D$	$N_{c}$	$y_{\max}^{+}$	${\bar{y}}^{+}$	$N_{t}$
GR1	3 × 10⁻⁵	120	0.782	0.359	1.04 × 10⁶
GR2	3 × 10⁻⁵	160	0.885	0.408	1.41 × 10⁶
GR3	3 × 10⁻⁵	200	0.803	0.402	1.80 × 10⁶

Figure 4.

Distribution of $y^{+}$ value on the cylinder surface for the three grid schemes. (a) GR1 (b) GR2 (c) GR3.

Figure 5 presents the results of time history and spectral analysis of the aerodynamic force for uncontrolled flow around a standard cylinder using the three grid schemes. $C_{l} = F_{l} / 0.5 ρ U^{2} D L$ and $C_{d} = F_{d} / 0.5 ρ U^{2} D L$ are the lift and drag coefficients of the cylinder, respectively; $F_{l}$ and $F_{d}$ are the corresponding lift and drag forces of the cylinder. The dimensionless main frequency obtained via spectrum analysis of the lift coefficient is the Strouhal number $S_{t}$ . Figure 5 shows that the $S_{t}$ values of the cylinders under the GR2 and GR3 grid schemes are almost identical, that is, 0.2054 and 0.2051, respectively. The dimensionless main frequency of the drag coefficient exhibits a two-times relationship with $S_{t}$ . The distributions of the mean pressure coefficient ${\bar{C}}_{p}$ at the cylinder midspan section and the time-averaged velocity ${\bar{U}}_{x}$ at the wake centerline under different grid schemes are depicted in Figure 6. Evidently, the results of the GR1 grid scheme differ significantly from those of the GR2 and GR3 grid schemes, whereas the corresponding results of the GR2 and GR3 grid schemes are consistent. In Figure 6(a), the mean pressure coefficient at the forward stagnation point of the cylinder midspan section ${\bar{C}}_{p d}$ under the GR1, GR2 and GR3 grid schemes is 1.019, 1.016 and 1.014, respectively, and the backpressure coefficient ${\bar{C}}_{p b}$ is −0.997, −1.185 and −1.206, respectively. The flow separation angle at cylinder midspan section for the GR1 grid scheme is 74.1°, and the corresponding maximum mean negative pressure coefficient is −1.589. For the GR2 and GR3 grid schemes, the flow separation angles are equal (72.0°), and the corresponding maximum mean negative pressure coefficients are −1.503 and −1.489, respectively. The length of recirculation in the cylinder wake region for the GR1 grid scheme is 1.432D, and those for the GR2 and GR3 grid schemes are close, that is, 1.289D and 1.278D, respectively.

Figure 5.

Time history and spectrum analysis of cylinder lift and drag coefficient under different grid schemes. (a) GR1 (b) GR2 (c) GR3.

Figure 6.

Mean pressure coefficient distribution at the cylinder midspan section and the time-averaged velocity distribution at the wake centerline for different grid schemes. (a) Mean pressure coefficient distribution at the cylinder midspan section (b) time-averaged velocity distribution at the wake centerline.

The statistical values of cylinder aerodynamic parameters, including drag coefficient mean value

{\bar{C}}_{d}

and fluctuation value

C_{d}^{'}

, lift coefficient fluctuation value

C_{l}^{'}

and Strouhal number

S_{t}

, are obtained according to the time history of the aerodynamic force over approximately 40 vortex shedding periods. The comparison of the statistical results of the aerodynamic parameters of a typical cylinder under the three grid schemes and the results of the flow around the cylinder at a similar Reynolds reported in the literature number are presented in Table 2. Evidently,

{\bar{C}}_{d}

C_{d}^{'}

and

C_{l}^{'}

calculated in this study gradually increase with the increase in the grid size, whereas

S_{t}

decreases as the grid size increases. According to the empirical formula of Norberg (2003), the lift coefficient fluctuation value

C_{l}^{'}

of the cylinder at

R e = 4.0 \times 10^{4}

is 0.517, which is close to the simulation result for the GR2 grid scheme. It indicates that the turbulence model, grid accuracy, and solution scheme used in the calculation accurately capture the characteristics of the flow field around the 3D cylinder. Based on this analysis, the results corresponding to the GR2 grid scheme satisfy the grid independence criterion, and therefore, this grid scheme is adopted for subsequent calculations. The 3D instantaneous vortex structure, time-averaged velocity contours, and streamlines at the midspan section of the uncontrolled standard cylinder wake corresponding to the GR2 grid scheme are shown in Figure 7.

Table 2.

Comparison of aerodynamic force statistical parameters and Strouhal number.

Study	Re	${\bar{C}}_{d}$	$C_{d}^{'}$	$C_{l}^{'}$	$S_{t}$
Surry (1972)	4.42 × 10⁴	1.200	—	—	0.210
Lu et al. (1997)	4.42 × 10⁴	1.180	—	—	0.200
Szepessy and Berman (1992)	4.3 × 10⁴	1.338	0.150	0.453	—
Szepessy and Berman (1992)	4.5 × 10⁴	—	—	—	0.190
Norberg (2003)	4.0 × 10⁴	—	—	0.517	0.185∼0.193
Present, numerical, GR1	4.0 × 10⁴	0.972	0.050	0.325	0.223
Present, numerical, GR2		1.184	0.078	0.516	0.205
Present, numerical, GR3		1.215	0.089	0.562	0.205

Figure 7.

3D instantaneous vortex structure and time-averaged velocity contours and streamlines at the midspan section of the uncontrolled standard cylinder wake. (a) Instantaneous vortex contours (b) time-averaged velocity contours and streamlines at the midspan section.

Results and discussion

The main parameters of TWW flow control include the propagation direction of the traveling wave, wave amplitude $\hat{A}$ , number of waves N, and wave velocity c. According to the research by Xu et al. (2014, 2017) on flow around 2D cylinders, the control effect of traveling waves propagating downstream is optimal. Therefore, the propagation direction of the traveling wave was fixed, and the control variable method was used in this study to analyze the influence of changes in the other three parameters on the aerodynamic forces, Strouhal number, and flow field vorticity characteristics of the 3D cylinder. The calculations were performed at a high Reynolds number $R e = 4.0 \times 10^{4}$ , and the motion of TWW was activated at the beginning of the calculations.

Influence of wave amplitude on control effects

The rear half of the 3D cylinder surface was set as the flexible TWW. The number of traveling waves on each 1/4-cylinder surface N was set as 4, and the ratio of wave velocity to incoming velocity $c / U$ was set as 1.5. The ratio of wave amplitude to cylinder diameter $\hat{A} / D$ was set as 0.01, 0.02, 0.03 and 0.04, respectively. The influence laws of wave amplitude on the control effect of the aerodynamic forces and wake of the 3D cylinder were analyzed.

Figure 8 shows the time history of cylinder lift and drag coefficients under different wave amplitudes. Compared the flow around the uncontrolled cylinder, the time history of the lift and drag coefficients for the TWW cylinder differ noticeably. When the wave amplitude is small ( $\hat{A} / D = 0.01$ ), the fluctuation frequency of the drag coefficient increases significantly. At this time, the amplitude of the lift coefficient is suppressed, but its periodic fluctuation characteristic is still observed. With the gradual increase in the wave amplitude, the drag coefficient curve exhibits zonal distribution with high-frequency fluctuation; the fluctuation amplitude gradually increases and the mean value gradually decreases, whereas the lift coefficient curve approaches a horizontal straight line with a mean value of 0. When the wave amplitude $\hat{A} / D \geq 0.02$ , TWW significantly reduces the lift coefficient fluctuation and drag coefficient mean value of the cylinder, and the degree of fluctuation of the time history of the lift coefficient almost disappears.

Figure 8.

Time history of lift and drag coefficients for the TWW cylinder with different wave amplitudes. (a) $\hat{A} / D = 0.01$ (b) $\hat{A} / D = 0.02$ (c) $\hat{A} / D = 0.03$ (d) $\hat{A} / D = 0.04$ .

The aerodynamic force time histories of the TWW cylinder under the four wave amplitudes are statistically analyzed. The mean value ${\bar{C}}_{l}$ and root mean square (RMS) value $C_{l}^{'}$ of the lift coefficient, and the mean value ${\bar{C}}_{d}$ and RMS value $C_{d}^{'}$ of the drag coefficient are plotted against the wave amplitudes, as shown in Figure 9. The dashed and dash-dot lines in the figure represent the statistical results of the aerodynamic parameters of the flow around the uncontrolled standard cylinder.

Figure 9.

Statistical value of lift and drag coefficients for the TWW cylinder with different wave amplitudes. (a) ${\bar{C}}_{l}$ and $C_{l}^{'}$ (b) ${\bar{C}}_{d}$ and $C_{d}^{'}$ .

As is evident from Figure 9(a), the lift coefficient mean value ${\bar{C}}_{l}$ of the TWW cylinder with different wave amplitudes is close to 0, which is consistent with the results of the uncontrolled standard cylinder. This suggests that the lift coefficient of the controlled TWW cylinder maintains the zero-mean characteristic owing to TWW motion and the symmetry of the surrounding grid. When $\hat{A} / D = 0.01$ , the RMS value of lift coefficient $C_{l}^{'}$ of the TWW cylinder decreases to 0.17, which is 67% lower than that of the uncontrolled standard cylinder. This indicates that the alternately shedding vortices in the cylinder wake cannot be completely eliminated due to the small amplitude, and the lift coefficient curve exhibits a certain degree of fluctuation, as shown in Figure 8(a). When $\hat{A} / D \geq 0.02$ , the RMS value of the cylinder lift coefficient $C_{l}^{'}$ is close to 0, and the results for each wave amplitude are reduced by more than 99% compared to that of the uncontrolled standard cylinder. This indicates that when the amplitude ratio of the TWW approaches 0.02, the TWW flow control method achieves a noticeable control effect, and the lift coefficient fluctuation is almost completely suppressed.

The statistical results of the mean and RMS values of the drag coefficient are presented in Figure 9(b). Evidently, the drag coefficient mean value ${\bar{C}}_{d}$ of the TWW cylinder with different wave amplitudes is smaller than that of the uncontrolled standard cylinder, and the RMS value of the drag coefficient $C_{d}^{'}$ exhibits a trend of initial decrease and subsequent increase. When $\hat{A} / D = 0.01$ , the ${\bar{C}}_{d}$ and $C_{d}^{'}$ values of the TWW cylinder decrease by 32% and 25%, respectively compared to the results of the uncontrolled standard cylinder. With an increase in the wave amplitude, the drag coefficient mean value of the TWW cylinder decreases significantly and gradually approaches 0. When $\hat{A} / D = 0.02 \sim 0.04$ , the ${\bar{C}}_{d}$ of the TWW cylinder decreases by 93%–99% compared to the results of the uncontrolled standard cylinder. However, when $\hat{A} / D = 0.04$ , the drag coefficient mean value of the TWW cylinder becomes negative. This indicates that the mean drag of the TWW cylinder is transformed into a thrust along the negative x-axis for a high wave amplitude. When $\hat{A} / D = 0.02 \sim 0.04$ , the $C_{d}^{'}$ of the TWW cylinder increases gradually with the increase in the wave amplitude, which is 45%, 108%, and 180% higher than that of the uncontrolled standard cylinder, respectively. Figure 8(b)–(d) show that the fluctuation degree of the drag coefficient increases; however, a significant difference exists compared to the drag coefficient time history of the uncontrolled standard cylinder shown in Figure 5, and the drag coefficient time histories under different wave amplitudes exhibit the characteristics of high frequency and equal amplitude.

The aerodynamic forces acting on the cylinder surface originate from the generation and evolution of the vorticity on the wall and in the wake field. When the wave amplitude affects the change in the cylinder aerodynamic coefficient, the vorticity around the cylinder also changes significantly. Figure 10 shows the instantaneous 3D vortex structure and vorticity contours at the midspan section of the TWW cylinder wake under different wave amplitudes. Here, red indicates positive vorticity with counterclockwise rotation, and blue indicates negative vorticity with clockwise rotation. Time-averaged velocity contours and streamlines at the midspan section of the TWW cylinder under different wave amplitudes are shown in Figure 11.

Figure 10.

Instantaneous 3D vortex structure and vorticity contours at the midspan section in the TWW cylinder wake for different wave amplitudes. (a) $\hat{A} / D = 0.01$ (b) $\hat{A} / D = 0.02$ (c) $\hat{A} / D = 0.03$ (d) $\hat{A} / D = 0.04$ .

Figure 11.

Time-averaged velocity contours and streamlines at the midspan section of the TWW cylinder for different wave amplitudes. (a) $\hat{A} / D = 0.01$ (b) $\hat{A} / D = 0.02$ (c) $\hat{A} / D = 0.03$ (d) $\hat{A} / D = 0.04$ .

As shown in Figure 10(a), when $\hat{A} / D = 0.01$ , the motion of the wall elongates the shear layer separated from the cylinder surface, and curls to form alternatingly shedding vortices further downstream of the cylinder wake. Compared with the time-averaged flow field results of the uncontrolled standard cylinder shown in Figure 7(b), when $\hat{A} / D = 0.01$ , the width of the wake region of the TWW cylinder is narrower, the time-averaged main vortex in the wake is elongated to an elliptical shape, and the vortex center is further away from the cylinder, as shown in Figure 11(a). Moreover, alternately shedding vortices in the cylinder wake are not completely eliminated when the wave amplitude is small, but TWW delays the vortex shedding of the cylinder wake, thereby reducing the lift coefficient fluctuation of the cylinder to a certain extent, as shown in Figure 9(a). As shown in Figure 10(b)–(d), when $\hat{A} / D = 0.02 \sim 0.04$ , no alternating vorticity shedding occurs in the cylinder wake region; at this time, the vorticity is strap-shaped and distributed stably and symmetrically on both sides of the cylinder’s wake centerline. Consequently, the lift coefficient fluctuation value of the TWW cylinder is completely suppressed and close to zero. Notably, the control effect of TWW is sensitive to wave amplitude. When the amplitude ratio $\hat{A} / D > 0.02$ , the vortex shedding in the cylinder wake is completely eliminated, the 3D flow characteristics of the cylinder wake disappear, the correlation of the spanwise flow field is significantly enhanced, and the flow fields at different locations along the span exhibit similarity and apparent 2D characteristics. The vorticity contours at the cylinder midspan section shown in Figure 10 show that large positive and negative vorticities occur at the wave trough of the TWW cylinder. This indicates that a series of near-wall vortices are captured by the TWW, and the captured vortices are located between the main flow region and the near-wall shear layer, resembling the generation of a virtual slip fluid boundary and generating a FRB effect, which weakens or eliminates the adverse effects caused by conventional boundary layers. As shown in Figure 11(b)–(d), as the wave amplitude gradually increases, the width of the cylinder wake region reduces and the streamlines become straighter and do not curl. When $\hat{A} / D = 0.04$ , the streamlines flowing through the cylinder are no longer separated and remain attached to the cylinder surface; moreover, they are symmetrically distributed on both sides of the wake centerline, which is similar to streamlines flowing through a streamline body. Thus, the flow characteristics such as flow separation, vortex generation, and shedding, which are prominent in the flow around a bluff body, are completely eliminated.

Influence of number of waves on control effect

The amplitude ratio $\hat{A} / D$ of the traveling wave was set as 0.02, and the ratio of the wave velocity to the incoming velocity was set as 1.5. The influence of the number of waves on the control effect of the cylinder wake was investigated. The number of waves N was set as 2, 3, 4, 5, and 6, and the variation laws of the cylinder aerodynamic force and the wake vortex shedding mode for different number of waves were analyzed. The time-history curves of the lift and drag coefficients are shown in Figure 12.

Figure 12.

Time history of the lift and drag coefficients of the TWW cylinder for different number of waves. (a) $N = 2$ (b) $N = 3$ (c) $N = 4$ (d) $N = 5$ (e) $N = 6$ .

As shown in Figure 12(a), when the number of waves $N = 2$ , periodic fluctuations in the lift and drag coefficients of the cylinder are observed, which indicates that the TWW does not suppress the vortex shedding in the cylinder wake. This may be because an amplitude reduction function is set at the starting and ending wavelengths of the traveling wave (Xu et al., 2014) to ensure the smooth connection between the TWW and the static boundary; consequently, the wave amplitude gradually decreases toward both ends. Therefore, when $N = 2$ , no complete traveling wave is generated on the cylinder surface, and few near-wall vortices are captured by the TWW. Therefore, the FRB effect is not generated. As shown in Figure 12(b), when $N = 3$ , the fluctuation degree of the lift coefficient is significantly reduced compared to the results of flow around the uncontrolled cylinder, and the drag coefficient mean value is also reduced. By contrast, the fluctuation degree and frequency of the drag coefficient increase because of the high frequency traveling wave motion. When $N = 4$ , the control effect of the TWW on the cylinder wake begins to appear, the lift coefficient fluctuation disappears completely, and the drag coefficient mean value is close to zero, which indicates that the TWW completely eliminates the vortex shedding in the cylinder wake.

The aerodynamic force time histories of the TWW cylinder are statistically analyzed at five different number of waves, and the curves of the mean value ${\bar{C}}_{l}$ and RMS value $C_{l}^{'}$ of the lift coefficient and the mean value ${\bar{C}}_{d}$ and RMS value $C_{d}^{'}$ of the drag coefficient as a function of number of waves are obtained (Figure 13). As shown in Figure 13(a), the lift coefficient of the TWW cylinder with different number of waves exhibits the characteristic of zero mean value, which is consistent with the results of the uncontrolled standard cylinder. When $N = 2$ , the lift coefficient RMS value $C_{l}^{'}$ of the TWW cylinder is 0.68, which is 32% higher than that of the uncontrolled standard cylinder. This indicates that the disturbance introduced by the TWW enhances the vortex shedding in the cylinder wake, resulting in a large fluctuation in the lift coefficient, as shown in Figure 11(a). In Figure 13(b), the drag coefficient mean value ${\bar{C}}_{d}$ of the TWW cylinder is 1.16, which is only 2% lower than that of the uncontrolled standard cylinder. The drag coefficient RMS value $C_{d}^{'}$ is 0.088, which is slightly higher than that of the uncontrolled standard cylinder. When $N = 3$ , the lift coefficient RMS value $C_{l}^{'}$ of the TWW cylinder is reduced to 0.06, which is 88% lower than that of the uncontrolled standard cylinder, indicating that the TWW has already introduced the control effect in the cylinder wake. In addition, ${\bar{C}}_{d}$ and $C_{d}^{'}$ of the TWW cylinder are 0.69 and 0.079, respectively, and ${\bar{C}}_{d}$ is 41% lower than that of the uncontrolled standard cylinder. Figure 13(a) clearly indicates that when $N = 4$ , $C_{l}^{'}$ of the TWW cylinder is close to zero, which is reduced by 99.8% compared to the result of the uncontrolled standard cylinder. At this time, the vortex shedding in the cylinder wake is completely eliminated by the TWW such that the lift coefficient fluctuation is completely suppressed. As the number of waves continues to increase, the vortex shedding in the cylinder wake is also completely eliminated. When $N \geq 4$ , the drag coefficient RMS value $C_{d}^{'}$ of the TWW cylinder exceeds that of the uncontrolled standard cylinder, and $C_{d}^{'}$ reaches a maximum value of 0.168 at $N = 5$ . When $N = 4$ , ${\bar{C}}_{d}$ of the TWW cylinder is reduced further to 0.029. When $N = 5, 6$ , ${\bar{C}}_{d}$ of the TWW cylinder changes from 0.009 to −0.04, which indicates that the TWW motion has completely changed the direction of the drag. It can be seen that $N = 4$ is the optimal number of waves for the TWW control method to suppress the cylinder wake.

Figure 13.

Statistical value of the lift and drag coefficient for the TWW cylinder with different number of waves. (a) ${\bar{C}}_{l}, C_{l}^{'}$ (b) ${\bar{C}}_{d}, C_{d}^{'}$ .

Figure 14 shows the instantaneous 3D vortex structure and vorticity contours at midspan section in the TWW cylinder wake with different number of waves. As shown in Figure 14(a), when $N = 2$ , there are alternately shedding vortices in the cylinder wake, and the unstable shear layer in the near-wake region immediately curls to form vortices. The vortices in the near-wake region mainly appear as spanwise vortex tubes, and they are transformed into streamwise strap vortices in the slightly far downstream wake region. This also confirms that the lift coefficient fluctuation of the cylinder has not been effectively controlled, and vortex-induced vibration may still occur. As shown in Figure 14(b), when $N = 3$ , there are still 3D vortices shedding irregularly in the cylinder wake, and the spanwise vortex tubes in the near-wake region are gradually broken and transformed into irregular streamwise vortices. It can be clearly observed from the vorticity contours at the midspan section that the shear layers separated from the cylinder surface do not curl rapidly to form vortices but are elongated and distributed nearly symmetrically in the near-wake region, and vortex shedding occurs further downstream in the wake region. The TWW has the effect of delaying the vortex shedding in the wake, such that the lift coefficient fluctuation value and the drag coefficient mean value of the cylinder are significantly reduced. When $N \geq 4$ , as the number of waves increases, the control effect of the TWW on the cylinder wake becomes increasingly obvious. Owing to the FRB effect, the shear layers on both sides of the cylinder do not curl, and the vortices in the wake completely disappear. From the vorticity contours, the positive and negative vorticity are closely attached to the cylinder wall and are distributed symmetrically on both sides of the wake centerline in the strap shape behind the cylinder rear edge point. Consequently, the lift coefficient fluctuation of the cylinder is completely suppressed, approaching a straight line with a mean value of zero, such that the possibility of VIV occurring is completely eliminated.

Figure 14.

Instantaneous 3D vortex structure and vorticity contours at the midspan section in the TWW cylinder wake with different number of waves. (a) $N = 2$ (b) $N = 3$ (c) $N = 4$ (d) $N = 5$ (e) $N = 6$ .

Figure 15 shows the time-averaged velocity contours and streamlines at the midspan section of the TWW cylinder with different number of waves. When $N = 2$ , the main vortex in the TWW cylinder wake is close to the cylinder rear edge. Compared to the results of the flow around the uncontrolled standard cylinder shown in Figure 7(b), the distance between the center of the time-averaged main vortex and the cylinder is lower, so the lift coefficient fluctuation value of the cylinder is larger than that of the uncontrolled standard cylinder. At this time, the TWW enhances the cylinder wake. When $N = 3$ , the time-averaged main vortex in the cylinder wake is elongated, the center of the main vortex is farther away from the cylinder, and the interaction between the vortex and the cylinder is weakened, resulting in a weakening of the cylinder aerodynamic force. When $N \geq 4$ , the backflow in the near-wake region of the TWW cylinder completely disappears, and the streamlines no longer curl, gradually become straight, and are symmetrically distributed on both sides of the cylinder’s wake centerline. The resistance of the cylinder surface is mainly composed of two parts: one is the friction resistance, and the other is the pressure difference resistance. In turbulent flow, the resistance of the cylinder wall mainly comes from the pressure difference resistance, whereas the viscous resistance can be ignored. As the number of waves continues to increase, it can be observed from Figure 15 that the low-velocity and low-pressure regions in the cylinder wake gradually decrease, indicating that the TWW flow control method makes the drag coefficient mean value of the cylinder gradually decrease with the increase in the number of waves.

Figure 15.

Time-averaged velocity contours and streamlines at the midspan section of the TWW cylinder with different number of waves. (a) $N = 2$ (b) $N = 3$ (c) $N = 4$ (d) $N = 5$ (e) $N = 6$ .

Influence of wave velocity on control effect

To study the influence of wave velocity on the control effect of the cylinder wake, the number of waves N on the quarter cylinder surface is set to 4, and the ratio of the maximum wave amplitude to the cylinder diameter $\hat{A} / D$ is 0.02. The ratio of the wave velocity to the incoming wind velocity $c / U$ is selected as 0.5, 1.0, 1.5, and 2.0, respectively, and the variation laws of the cylinder aerodynamic force and the wake vortex shedding mode under different wave velocities are analyzed. The time histories of the lift and drag coefficient of the TWW cylinder under different wave velocities are shown in Figure 16. Figure 16(a) indicates that when the wave amplitude and number of waves are taken as appropriate values, even if the wave velocity is small, the lift coefficient fluctuation of the TWW cylinder has been significantly reduced, and the drag coefficient mean value has also decreased. As the wave velocity increases gradually, the effect of the TWW in controlling the cylinder wake becomes increasingly significant. When the velocity ratio $c / U \geq 1.5$ , the lift coefficient tends to be a horizontal straight line, and the TWW has completely suppressed the large fluctuation of the cylinder lift coefficient, but it is accompanied by an increase in the frequency and amplitude of the drag coefficient fluctuation. The time history of the drag coefficient of the TWW cylinder exhibits a strap-shaped distribution with a stable amplitude, as shown in Figure 16(c) and (d).

Figure 16.

Time histories of the lift and drag coefficient for the TWW cylinder with different wave velocities. (a) $c / U = 0.5$ (b) $c / U = 1.0$ (c) $c / U = 1.5$ (d) $c / U = 2.0$ .

When the number of waves N is 4, the corresponding wavelength $λ$ is $π D / 4 N$ . According to the Reynolds number and the velocity ratio, the wave velocity c can be obtained. Further, the traveling wave motion frequencies at different velocity ratios can be calculated from $f_{t w w} = c / λ$ , which are 103.33, 206.66, 310.00, and 413.34 Hz, respectively. The spectrum analysis results of the drag coefficient for the TWW cylinder with different wave velocities are shown in Figure 17, and the main frequencies of the drag coefficients are 110.91, 208.87, 310.61, and 412.30 Hz, respectively, which are very close to the fluctuation frequencies of traveling waves. The high-frequency fluctuation of the drag coefficient of the TWW cylinder is clearly derived from the disturbance frequency caused by the traveling wave motion, which also reflects the active control characteristic of the TWW method. Although the magnitude of the drag coefficient is large, its high frequency is usually far from the natural frequency of the structure, so it does not cause adverse structural vibrations. As shown in Figure 17(a) and (b), when $c / U \leq 1.0$ , there is a low-order dominant frequency in the spectral analysis result of the drag coefficient. This indicates that the drag coefficient mean value has low-frequency fluctuation, which can be also observed from the drag coefficient time history in Figure 16(a) and (b). When $c / U \geq 1.5$ , there is no low-order main frequency in the spectrum analysis result of the drag coefficient for the TWW cylinder, the traveling wave motion frequency is obviously dominant, and the second-harmonic frequency exists, as shown in Figure 17(c) and (d).

Figure 17.

Spectral analysis of the drag coefficient for the TWW cylinder with different wave velocities. (a) $c / U = 0.5$ (b) $c / U = 1.0$ (c) $c / U = 1.5$ (d) $c / U = 2.0$ .

To further analyze the variation law of cylinder aerodynamic force with different velocity ratios, the lift and drag coefficient mean values ( ${\bar{C}}_{l}$ and ${\bar{C}}_{d}$ ) that characterize the time-averaged characteristics of cylinder aerodynamic force and the lift and drag coefficients RMS values ( $C_{l}^{'}$ and $C_{d}^{'}$ ) that characterize the fluctuation characteristics of cylinder aerodynamic force are plotted in Figure 18. As shown in Figure 18(a), when $c / U = 0.5$ , the RMS value of the lift coefficient $C_{l}^{'}$ is 0.324, which is 37.2% lower than that of the uncontrolled standard cylinder. Furthermore, it can also be found from Figure 16(a) that the fluctuation amplitude of the lift coefficient time history is significantly reduced. As the velocity ratio continues to increase, $C_{l}^{'}$ continues to decrease. When $c / U \geq 1.5$ , the decrease in $C_{l}^{'}$ only increases from 93.8% to 98.3% relative to the uncontrolled standard cylinder. At this time, the drag coefficient fluctuation value $C_{d}^{'}$ increases slowly as the velocity ratio gradually increases. As shown in Figure 18(b), the drag coefficient mean value ${\bar{C}}_{d}$ decreases rapidly with the increase in the velocity ratio and reaches a positive minimum value of 0.029 at $c / U = 1.5$ , which is only 2.45% of the uncontrolled standard cylinder result. Therefore, $c / U = 1.5$ is considered to be the optimal velocity ratio to achieve the aerodynamic control effect. When $c / U = 2.0$ , the drag coefficient mean value ${\bar{C}}_{d}$ is −0.094, which means that drag changes to the $- x$ direction due to the action of the high-velocity traveling wave, and its absolute value is 7.94% of ${\bar{C}}_{d}$ of the uncontrolled standard cylinder.

Figure 18.

Statistical values of the lift and drag coefficient for the TWW cylinder with different wave velocities. (a) ${\bar{C}}_{l}, C_{l}^{'}$ (b) ${\bar{C}}_{d}, C_{d}^{'}$ .

The change in the cylinder aerodynamic characteristics originates from the change in the vortex structure and vortex shedding mode in the flow field. The instantaneous 3D vortex structure and vorticity contours at the midspan section in the TWW cylinder wake with different velocity ratios are shown in Figure 19. When $c / U = 0.5$ , the Karman vortex street with alternating shedding can still be observed in the wake field of the TWW cylinder. When the velocity ratio increases to $c / U = 1.0$ , the TWW motion plays a role in delaying the vortex shedding, the separated shear layer presents a nearly symmetrical stable state at one diameter behind the cylinder, and the alternating shedding vortices only appear further downstream. This phenomenon also explains why the lift coefficient RMS value decreases greatly at $c / U = 1.0$ . When $c / U \geq 1.5$ , the complex 3D vortex structure in the wake region of the TWW cylinder disappears, the instantaneous vorticity at the cylinder midspan section is strap-shaped and distributed symmetrically and stably on both sides of the wake centerline, and the positive and negative vortices shedding alternately have completely disappeared. Consequently, the fluctuation of the lift coefficient of the TWW cylinder disappears completely, approaching a straight line with a mean value of zero. The TWW flow control method delays or suppresses the flow separation and vortex generation such that the flow field returns to a stable state, which fundamentally eliminates the fluctuation characteristic in the lift coefficient.

Figure 19.

Instantaneous 3D vortex structure and vorticity contours at the midspan section in the TWW cylinder wake with different wave velocities. (a) $c / U = 0.5$ (b) $c / U = 1.0$ (c) $c / U = 1.5$ (d) $c / U = 2.0$ .

The time-averaged velocity contours and streamlines at the midspan section of the TWW cylinder with different velocity ratios are shown in Figure 20. When $c / U = 0.5$ , there are main vortices in the wake that are immediately adjacent to the rear edge of the TWW cylinder. As the velocity ratio increases to $c / U = 1.0$ , the time-averaged main vortex in the cylinder wake is elongated, and the center of the main vortex is farther away from the cylinder, so the aerodynamic force on the cylinder is weakened. It can be seen from Figure 20(c) and (d) that when $c / U \geq 1.5$ , the streamlines in the cylinder wake become straight, no longer curl, and are symmetrically distributed on both sides of the cylinder wake centerline. The main vortex in the cylinder wake region completely disappears, which is also the reason for the disappearance of the fluctuation degree of the lift coefficient. The low-velocity region and negative pressure region in the cylinder wake gradually decrease, so the drag in the downwind direction also decreases gradually.

Figure 20.

Time-averaged velocity contours and streamlines at the midspan section of the TWW cylinder with different wave velocities. (a) $c / U = 0.5$ (b) $c / U = 1.0$ (c) $c / U = 1.5$ (d) $c / U = 2.0$ .

Conclusions

In this study, LES is used to examine the TWW flow control method for 3D cylinder wake, and the influence laws of the main control parameters of the traveling wave on the aerodynamic force of the 3D cylinder are analyzed. The main conclusions are as follows.

1. For the calculation with high Reynolds number in this study, the optimal control parameter combination can be obtained through the control effect of the TWW on the cylinder aerodynamic force, that is, the ratio of traveling wave maximum amplitude to cylinder diameter is 0.02, the number of waves of the traveling wave is 4, and the ratio of wave velocity to incoming velocity is 1.5.

2. With the optimal control parameter combination, the backflow in the near-wake region of the TWW cylinder completely disappears, and the streamlines no longer curl and gradually become straight. The low-velocity region and negative pressure region in the cylinder wake gradually decrease, so the drag coefficient mean value in the downwind direction also decreases significantly. The drag coefficient fluctuation value increases significantly, but its high-frequency fluctuation originates from the disturbance frequency induced by the traveling wave motion, which reflects the active control characteristic of the TWW method.

3. With the optimal control parameter combination, the TWW control method completely eliminates the 3D flow characteristic in the spanwise direction of the cylinder, the complex 3D vortex structure in the cylinder wake region disappears, and the instantaneous vorticity at the cylinder midspan section is strap-shaped and distributed symmetrically and stably on both sides of the wake centerline. The lift coefficient fluctuation of the TWW cylinder disappears completely, approaching a straight line with a mean value of zero and achieving the purpose of eliminating the alternating vortex shedding in the 3D cylinder wake and suppressing the vortex-induced vibration.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the National Key Research and Development Program of China (2021YFC3100702), the National Natural Science Foundation of China (NSFC) (52078175, 51778199), the Natural Science Foundation of Guangdong Province (2019A1515012205), the Fundamental Research Funds of Shenzhen Science and Technology Plan (JCYJ20190806144009332), the Shenzhen Science and Technology Program (KQTD20210811090112003), and the Stability Support Program for Colleges and Universities in Shenzhen (GXWD20201230155427003-20200823134428001).

ORCID iDs

Feng Xu

Wen-Li Chen

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