Large eddy simulation of passive jet flow control on the wake of flow around a circular cylinder

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Large Eddy Simulation of Passive Jet Flow Control on the Wake of Flow around a Circular Cylinder Feng Xu , Wen-Li Chen , Zhong-Dong Duan , Jin-Ping Ou PII: DOI: Reference:

S0045-7930(19)30301-9 https://doi.org/10.1016/j.compfluid.2019.104342 CAF 104342

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Computers and Fluids

Received date: Revised date: Accepted date:

30 January 2019 1 October 2019 21 October 2019

Please cite this article as: Feng Xu , Wen-Li Chen , Zhong-Dong Duan , Jin-Ping Ou , Large Eddy Simulation of Passive Jet Flow Control on the Wake of Flow around a Circular Cylinder, Computers and Fluids (2019), doi: https://doi.org/10.1016/j.compfluid.2019.104342

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Highlights:    

LES was employed to study suppressing the cylinder wake using a passive jet flow control method. Influence parameters of the jet holes on aerodynamic forces and vortex shedding were studied. Airflow exhausted from backward holes can make the main vortex move to the farther downstream wake. Movement of the main vortex to downstream markedly reduced fluctuations of aerodynamic forces.

Large Eddy Simulation of Passive Jet Flow Control on the Wake of Flow around a Circular Cylinder Feng Xua,, Wen-Li Chenb, c #, Zhong-Dong Duana, Jin-Ping Oua, c a

School of Civil and Environmental Engineering, Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China

b

Key Lab of Structures Dynamic Behavior and Control (Harbin Institute of Technology), Ministry of Education, Harbin, 150090, China c

School of Civil Engineering, Harbin Institute of Technology, Harbin, 150090, China

Abstract: This study presents a passive jet flow control method to suppress the wake of a circular cylinder based on a computational fluid dynamics (CFD) numerical simulation at a high Reynolds number of 5.0×10 4. The investigation mainly focuses on the control effectiveness of the pattern of jet holes on the hollow pipe. First, the three-dimensional (3D) flow past a circular cylinder at a Reynolds number of 3900 is addressed based on the large-eddy simulation (LES) method to validate the feasibility of a numerical simulation. Then, the control effects of the passive jet flow on the aerodynamic forces and the wake flow of the circular cylinder are studied at different directions, angles, and heights of the jet holes. An optimal parameter scheme for suppressing the wake of a circular cylinder is then determined according to a comparison of the control effectiveness of the aerodynamic forces. The results indicate that the jet flow from the backward holes of the circular cylinder effectively separates the shear layers rolled up on both sides of the circular cylinder, which forces the vortex formation region downstream. This control measure can dramatically reduce the aerodynamic forces and suppress the wake of the circular cylinder. Keywords: Passive jet flow; wake; large-eddy simulation; circular cylinder; aerodynamic forces

 #

Corresponding author, Tel/fax.: +86-755-26033506. Email address: [email protected] (F. XU). Co-Corresponding author, Tel/fax.: +86-451-86282068. Email address: [email protected] (W. L. CHEN).

1. Introduction The circular cylinder is a common construction form in the engineering of structures such as bridge cables, chimneys, cooling towers, structures of offshore engineering, and marine pipelines. The flow around a circular cylinder involves flow separation, vortex generation, and shedding, which leads to vortex-induced vibration (VIV). The VIV not only causes long-term fatigue damage to the structure, but also results in a serious resonance effect that could instantly destroy the structure. Therefore, suppressing the wake of the circular cylinder and then eliminating the vortex-induced vibration is of great significance to improve the design and service life of a circular cylindrical structure. In recent decades, the flow around a stationary circular cylinder was investigated in a large number of experimental and numerical studies. Norberg (2003) summarized the previous experimental results of flow around a circular cylinder, and analyzed the lift coefficient of the circular cylinder in the range of Re  47 ~ 2 105 . Norberg (1987), Lourenco and Shih (1993), Ong and Wallace (1996), and Parnaudeau et al. (2008) studied the flow characteristics of a single circular cylinder through experimental tests at Re  3900 , and presented the time-averaged velocity at 0.5  x /D  10 in the near wake centerline and the mean pressure distribution on the surface of the circular cylinder, which provided a classical reference for later numerical simulation of the flow around the circular cylinder. Lehmkuhl et al. (2014) employed the large-eddy simulation (LES) with second-order spectra-consistent numerical schemes on an unstructured grid to study the flow past a circular cylinder at Reynolds numbers in the range 2.5×105~6.5×105. The research results give the phenomenon of “ drag crisis”, and analyze the formation and transformation mechanism of laminar separation bubble (LSB) on both sides of the cylinder. Jin et al. (2016) adopted the large-eddy simulation (LES) and detached-eddy simulation (DES) to analyze the flow characteristics of a circular cylinder at Re  3900 . It was found that the simulation results by LES were in good agreement with the experimental results. The suppression of vortex shedding in the near wake that leads to a reduction in the aerodynamic forces and the VIV of the cylinder has gradually become an extensively researched topic. Feng and Wang (2010, 2010, 2012), and Feng et al. (2011) conducted extensive wind tunnel experiments to study the role of the synthetic jet in delaying flow separation and reducing the drag force acting on a cylinder. Wang et al. (2013) employed two rotating control little cylinders symmetrically placed behind the cylinder to change the mode of vertex shedding. It was found that this method could suppress the vortex shedding and decrease the drag force, when the upper and lower cylinders were rotated clockwise and counterclockwise. Chen et al. (2013, 2014, 2015(a)) proposed an

active suction flow control method to control the VIV response of a circular cylinder by experiments and numerical simulations, and the results showed that steady suction flow could reduce the aerodynamic force and VIV of the cylinder. Xu et al. (2014, 2017) numerically investigated the control effect of the traveling wave wall (TWW) on the aerodynamic forces and the mode of vortex shedding at Re  200 . It was observed that the type of downstream propagating TWW can effectively eliminate the alternating wake. When the ratio of wave velocity to oncoming velocity was 2.0, the RMS value of the lift coefficients decreased to the lowest point. The mean drag coefficients decreased with an increase in wave velocity and decreased to the lowest point when the ratio of the wave velocity to the oncoming velocity was 5.0. These control methods require an external energy supplement, and they belong to the active flow control methods. Compared with the active flow control methods, the passive flow control method does not consume external energy and achieves the purpose of flow control, mainly by changing parameters such as flow conditions, boundary conditions, and pressure gradient. Yeo and Jones (2011) conducted an investigation on the flow around a yawed cylinder with various strake patterns by using a three-dimensional DES method at Re  1.4 105 . The results demonstrated that particular strake patterns could mitigate low-frequency and large-amplitude vibrations of stay cables activated by oblique wind. Lim and Lee (2002) studied the aerodynamic forces and wake flow around circular cylinders with different grooved surfaces (U and V shaped) by a series of experiments in which

Re  8 103 to 1.4  105 . The results indicated that the drag reduction of the U-type grooved cylinder increased with increasing Re, and the U-type groove surface reduced the drag coefficient by 18.6% compared with the smooth cylinder at Re  1.4 105 , while the drag reduction of the V-type grooved cylinder was only 2.5%. Bao and Tao (2013) achieved wake control by dual plates symmetrically attached at the rear surface of a circular cylinder using the finite element method. They showed that the properly positioned dual plates device yields greater wake control than the traditional splitter plate, and the most effective range associated with the maximum drag reduction was between 40°–50°. Yu et al. (2015) used fairings fitted along the axis of long circular rises to reduce the drag force and suppress the VIV at Re  100, 500 and 1000 . Gao et al. (2017) experimentally researched the flow around a modified cylinder with a slit parallel to the incoming airflow at Re  2.67 104 . It was found that the RMS value of lift coefficients acting on the circular cylinder with a slit ratio of S D  0.05 was suppressed by up to 81.78% compared with that acting on the smooth cylinder, and the best drag reduction was as high as 14.64% at S D  0.125 . Chen et al. (2015(b), 2017) proposed a new passive control flow measure that destroyed the vortex shedding behind the cylinder. A hollow pipe with jet holes was firmly attached to the

circular cylinder, and two arrangement schemes for the holes were employed. The numerical simulation results demonstrated that this passive flow control method could effectively decrease the aerodynamic forces of the cylinder. Based on the passive jet flow control method of Chen et al. (2015(b), 2017), the present study concentrated on understanding the influence of the parameters of the jet holes on the wake control effect of flow around a circular cylinder under the fully open hole scheme by using the CFD numerical simulation at a high Reynolds number. The 3D flow past a circular cylinder at Re  3900 was first addressed with the LES method, and then the aerodynamic forces, the mean streamwise velocity of the wake centerline, and steady pressure coefficients on the middle surface of the circular cylinder were compared with the existing experimental and numerical results. The influence of passive jet flow control on the aerodynamic forces and the wake flow of the circular cylinder was studied by considering the opening direction, the opening angle, and the opening height of the jet holes. Thus, the optimal opening hole scheme of the passive jet flow control for suppressing the wake of the circular cylinder was found.

2. Numerical calculation model 2.1 Governing equations of the fluid flow For three-dimensional incompressible flow, the governing equations of fluid flow include the continuity equation and the Navier–Stokes equation. In the LES method, each instantaneous variable is divided into large-scale mean and small-scale components by using a filter function. The large-scale mean component is also called the filtered variable, which is directly calculated in the LES, and the small-scale component needs to be represented by a certain model. The filtered instantaneous continuity equation (1) and Navier–Stokes equations (2) can be obtained using the filter function to deal with the instantaneous governing equations. In a Cartesian coordinate system, the governing equations of the LES can be written as ui 0 xi

(1)

(ui u j  ui u j )   1 p  μ ui (ui )  (ui u j )    ( ) t x j ρ xi x j ρ x j x j

(2)

where ui is the velocity component in the i direction ( i  1, 2,3 ),  is the fluid density,  is the kinematic viscosity coefficient, and p is the pressure in the flow field. The variables with the overlines are the filtered variables, which are still the instantaneous values rather than the mean values. The subgrid scale stress



 ij   ui u j  ui u j



is an unknown quantity that reflects the effect of the motion of the small-scale vortex on the

governing equations. In solving the governing equation of the LES, the subgrid scale (SGS) model must be employed to construct the mathematical expression of subgrid scale stress with the relevant physical quantities. The dynamic Smagorinsky-Lilly model of was chosen for the calculation in this paper.

2.2 Numerical model and solution setting The computational domain and the lantern ring of the cylindrical surface are shown in Fig. 1 (a). The computational domain is a cuboid region of 28D 16D  0.5D , and the cylinder center is located at the coordinate origin. The distances from the upstream inlet, the downstream outlet, and the two sides to the cylinder center are 8D, 20D, and 8D, respectively. The diameter of the circular cylinder D is 0.1 m, and the spanwise length of the circular cylinder H p is 0.5D . A hollow pipe is placed around the circular cylinder; and the radial thickness of the hollow pipe D0 is 0.05D. The holes with a height of H h are equidistantly set on the pipe surface and are symmetrically arranged along the circumference direction of the cylinder, as shown in Fig. 1 (b). The central angle is set as α for each hole, which is half of the central angle between the two adjacent opening holes, as shown in Fig. 1 (c). The computational domain is discretized by structured hexahedral meshes, and the "O-type" block is used to generate a structured mesh near the cylinder surface in order to better capture the characteristics of flow around the circular cylinder. The grids near the cylinder surface are locally refined and the side length of the refined core region is 4D . The spanwise mesh of the cylinder is divided equally with a uniform mesh size 0.0625D . The smallest grid size near the surface of the circular cylinder and the lantern ring is 1.2  104 D in order to ensure

y   1 . The total grid number of the computational region is approximately 3  105 , which satisfies the requirement of LES for mesh accuracy. The mesh of the entire computational domain and the surrounding area of the circular cylinder are shown in Fig. 2. The flow direction is from the left to the right along the x direction, and the boundary conditions are set as follows: the front side consists of a velocity inlet with a uniform velocity U  ; the back side consists of an outflow boundary; the upper, lower, left, and right sides are set as symmetry boundaries; and the lantern ring and the cylinder surface are set as no slip wall boundaries, while the holes are set as interior boundaries so as to ensure that the airflow passes smoothly through the opening holes. The flow field is numerically calculated using the general commercial software FLUENT 17.0 based on the

finite volume method, and the turbulence model of LES with a SGS model is employed to simulate the turbulent flow around the cylinder and lantern ring. The SIMPLEC algorithm is used to calculate the coupling between the pressure and velocity fields. The format of the pressure interpolation is chosen as “Standard.” The bounded central differencing scheme is utilized for discretization of the convective term, and the bounded second order implicit scheme is adopted for the time-derivative term discretization. The time step used in the calculation is 0.001 s, which is approximately 1/100 of the period of vortex shedding.

2.3 Validation of the numerical simulation Before investigating the passive jet flow control method, the flow characteristics around a standard circular cylinder are studied by adopting the LES method, and the mean streamwise velocity along the wake centerline and the steady pressure coefficients on the middle surface of the circular cylinder are compared with the previous experimental and numerical results. The diameter of the circular cylinder D equals 0.1 m, and the spanwise length of the circular cylinder is  D , which is consistent with the length employed in the numerical simulations of Kravchenko and Moin (2000) and Jin (2016). The fluid medium is assumed to be water, and its density  and dynamic viscosity coefficient  are 1000 kg/m3 and 0.025641 N  s/m2 , respectively. The smallest grid size near the cylinder surface is 2.0  103 D , and the total grid number is approximately 1.01 106 . The boundary conditions and the solution format of the numerical calculation model are the same as those described in section 2.2. The distributions of the y  values for all the positions and the three representative cross sections on the cylinder surface at a time are shown in Fig. 3 (a) and Fig. 3 (b). As shown in Fig. 3 (a), the y  values at most positions on the cylinder surface are less than 1.0 and the mean value of y  is approximately 0.43, which satisfies the requirements of the LES method for the accuracy of the computational mesh. The y  value of the cylinder-pipe model in the next study are shown in Fig. 3 (c) and Fig. 3 (d), from which it can be seen that the y  values of the various surfaces of the model are below 0.6. Fig. 4 shows the results of the time-histories of the lift and drag coefficients and the spectral analysis of the lift coefficients obtained by the LES method. As the effect of the fluctuation of the small-scale vortices on the large-scale vortices can be considered in the LES, the amplitude of the lift coefficients and the mean value of the drag coefficients appear to fluctuate significantly, which is evidently different from the results of the RANS methods. In the present study, the simulation results of the mean drag coefficient value Cd , the root-mean-square

(RMS) values of lift and drag coefficients C'l and Cd' , and the Strouhal number St are 1.015, 0.097, 0.039 and 0.214, respectively. A comparison with the experimental and numerical results is shown in Table 1. The mean streamwise velocity profiles of the wake centerline on the middle position of the circular cylinder are chosen for comparison with the experimental results (Lourenco and Shih (1993); Ong and Wallace (1996); Parnaudeau et al. (2008)) and the numerical results of Jin et al. (2016), as shown in Fig. 5. It can be seen that the trend of the present result is consistent with the existing research results, especially the PIV result ( x D  2 ) and the constrained large-eddy simulation (CLES) result ( x D  2 ). With an increase in the distance x from the point on the wake centerline to the center of the circular cylinder, the mean streamwise velocity u decreases first and then increases, but changes minimally with increasing x for x /D  8 . A recirculation zone exists in the near weak and the recirculation length Lr corresponds to the distance between the leeward endpoint of the circular cylinder and the point at which a sign change of the mean streamwise velocity u occurs. From Fig. 5 (a), it can be obtained that the Lr is approximately 1.539D . Fig. 5 (b) shows the comparison between the steady pressure coefficients on the middle section of the circular cylinder and the corresponding experimental or numerical results (Norberg (1987); Lourenco and Shih (1993); Jin et al. (2016)). The present numerical result is in good agreement with the references. The mean pressure coefficient CP is 1.017 at the front stagnation point ( θ = 0 ), which is greater than 1.0. It is noted that the maximum values of the pressure coefficients calculated by Jin et al. (2016) and Wornom et al. (2011) was slightly larger than 1.0. The value of CP decreases with an increase in θ , and the value of CP decreases to the lowest point of –1.166 at θ = 69.4 . As θ continues to increase, CP gradually increases and basically remains constant when θ  100 . The time-averaged streamlines in the near wake on the middle section of the circular cylinder are shown in Fig. 6 (a). From this figure, it is seen that two large-scale vortices are formed in the near wake behind the circular cylinder. The distances from the center of the upper and lower vortex to the center of the circular cylinder are 1.371D and 1.352D , respectively, while the vertical distance between the upper and lower vortex centers is 0.636D . Fig. 6 (b) shows the instantaneous vorticity contour on the middle surface of the circular cylinder at

t  31s . It can be seen that the cylinder wake contains vortices with different scales and complex vortex shedding modes, and the length of the free shear layer separated from the lower surface of the circular cylinder is approximately 1.448D .

A comparison of typical flow parameters between the present numerical simulation and the existing research results is presented in Table 1. The typical flow parameters include the recirculation length Lr , the maximum backflow velocity U m , the mean pressure coefficients C pb at the leeward of the cylinder, the Strouhal number St , the mean drag coefficient Cd , and the RMS value of lift and drag coefficient C'l . According to Table 1, it

can be concluded that the present numerical results lie between the experimental results and the results obtained by CLES. The C'l is close to the experimental and DES simulation results, and is about 0.56 times of LES results of Zhang et al. (2015). In particular, the Strouhal number St and the mean drag coefficient Cd are in good agreement with the previous results, which indicates that the accuracy of the present mesh and the numerical solution settings can accurately capture the characteristics of flow around the 3D circular cylinder.

3. Results and discussion The numerical calculations are carried out at Re  5.0 104 , and the fluid medium used in this paper is air. By comparing the aerodynamic forces of the uncontrolled and controlled cylinders, the control effectiveness of the passive jet flow in suppressing the wake of the circular cylinder is studied by changing the opening direction, angle α and height H h of the jet holes. The two dimensionless parameters ECl  (Cl' _ origin  Cl' _ fullhole ) / Cl' _ origin and ECd  (Cd _ origin  Cd _ fullhole ) / Cd _ origin are defined, which represent the relative variation in the RMS value of the lift coefficient and the mean value of the drag coefficient, respectively. These two parameters are used to evaluate the control effectiveness of the passive jet flow on the wake of flow around the circular cylinder and determine the optimal parameters of the jet holes. When ECl and ECd are close to 1.0, the suppression effect on the wake is optimal. In contrast, when ECl and ECd are close to 0, no control exists.

3.1 The direction of the jet holes When the angle of the jet hole α is fixed at 7.5° and the ratio of the height of the jet hole to the height of the hollow pipe H h H p is set as 0.5, the influence of the opening or not opening the hollow pipe at the front stagnation point of the circular cylinder on the control effect is analyzed. Fig. 7 shows the schematic diagrams of the models under the various cases described in this section. In the A0 case, the surface of the circular cylinder is smooth and without a hollow pipe, and the cases F_ON and F_OFF represent the cylinder with a hollow pipe that

is opened and not opened at the front stagnation point of the circular cylinder, respectively. Table 2 shows the aerodynamic statistical parameters (the RMS value of the lift coefficient Cl' and the mean value for the drag coefficient Cd ), the frequency characteristics (the vortex shedding frequency f s and the Strouhal number St ) and the control effectiveness of the passive jet flow control pipe ( ECl and ECd ) under various cases. The time histories of the lift and drag coefficients of the circular cylinder and hollow pipe under different cases are given in Fig. 8. It can be clearly seen that the fluctuation of the lift and drag coefficients is significantly decreased in both the F_ON case and F_OFF case. In comparison with the results of the uncontrolled smooth cylinder, the dominant vortex shedding frequency f s was increased by 5.89% and 6.87% for the F_ON case and F_OFF case, respectively. At the same time, the mean drag coefficient Cd was decreased by 17.03% and 15.0% for the F_ON case and F_OFF case, respectively, and the RMS value of the lift coefficients Cl' was obviously decreased by 91.99% and 89.93% for the F_ON case and F_OFF case, respectively. It is shown that for the Reynolds number calculated in the present study, the passive jet flow control pipe with full holes significantly reduces the fluctuation of the lift coefficient, but the control effectiveness of the mean drag coefficient is not obvious. The fluctuation of the lift coefficient is caused by the alternating vortex shedding in the near wake. Therefore, a decrease in the RMS value of the lift coefficient can indicate that the passive jet flow control pipe has a better control effect on suppressing the wake of flow around the circular cylinder. Compared with the F_OFF case, the ECl and ECd increased by 2.16% and 2.03% for the F_ON case, respectively, which shows that the control effectiveness values in the F_ON and F_OFF cases are very close and insensitive to wind direction. Even in the case of the passive jet flow control pipe with full holes, whether the pipe is opened or not on the front stagnation point of the circular cylinder will also affect the control effectiveness, which demonstrates that the quantity of the effective holes in the flow direction can ensure adequate front intake and rear exhaust, which is the key to controlling the wake of the cylinder. In addition, the flow direction is changeable and unpredictable in many practical engineering applications; thus, the passive jet flow control pipe with full holes equidistantly distributed along the entire pipe could be a better way to deal with this obstacle. Consequently, the influence of the number of holes on the hollow pipe is not considered, and the passive jet flow control pipe with full holes is selected in the present study. This study focuses on the control effectiveness of different angles and heights of the jet holes on the wake of the circular cylinder.

Fig. 9 shows the time-averaged streamlines and vorticity contours on the middle section of the model for each case. For the A0 case, a pair of counter-rotating main vortices, which have a greater strength and are close to the rear surface of the cylinder, are formed on both sides of the centerline on the middle section of the cylinder wake. When the hollow pipe is tightly fixed on the cylinder surface, the pipe holes have sufficient blowing capacity, and the jet airflow can roll up a series of small-scale vortices in the near wake of the cylinder in either the F_ON case or F_OFF case. Besides, a pair of large-scale and low-intensity main vortices exist at the downstream behind the small-scale vortices. As shown in the time-averaged streamlines for the A0 case, the distance from the center of the main vortices to the center of the cylinder is approximately 0.85D . Because of the role of the blowing airflow in the F_ON case and F_OFF case, the forming region of the main vortices is moved downstream for a distance, and the distances from the center of the main vortices to the center of the cylinder are 1.75D and 1.65D , respectively. For the F_ON case, because the pipe is opened at the front of the F point, as

shown in Fig. 7, the air inflow from the front holes and the exhaust outflow from the rear holes of the pipe are little larger than those in the F_OFF case, which results in a slightly larger distance from the center of the main vortices to the center of the cylinder. Obviously, as a result of the blowing action of the hollow pipe, the shedding position of the main vortices in the wake of the cylinder is moved farther downstream, which can greatly decrease the fluctuation of the aerodynamic force and suppress the VIV of the circular cylinder. Fig. 10 shows the instantaneous vorticity contours and streamlines on the middle section of the model for each case, where the red and blue regions represent positive vorticity (counterclockwise vortex) and negative vorticity (clockwise vortex). In the uncontrolled A0 case, the shear boundary layers are separated from the upper and lower sides of the cylinder for the flow around the circular cylinder and curled in the near wake of the cylinder to form vortices. Furthermore, the vortices are alternately shed from the upper and lower surfaces of the cylinder and moved downstream in one period of vortex shedding. As the vortex moves downstream, the vortex scale increases and its intensity decreases, eventually dissipating in the far field of the wake. Compared with the streamline results of the A0 case, it can be found that the streamline pattern has similar characteristics at different times for the F_ON case and F_OFF case: there are multiple small-scale vortices in the near wake of the cylinder, and large-scale vortices in the downstream wake are formed owing to the downstream movement of the upstream outer vortices, as shown in Fig. 10 (b) and Fig. 10 (c). Take the F_ON case as an example; at the T 4 moment, when the lift coefficient reaches its maximum value, the main vortices are formed in the wake that is further downstream of the cylinder, and there are multiple

small-scale vortices between the cylinder and the main vortices. The inner small vortices are formed by the air blowing from the holes, while the outer small vortices are formed by separation of the shear boundary layers on both the upper and lower sides of the cylinder. At the T 2 moment of the lift coefficient of 0, the small vortices in the near wake behind the cylinder merge into a pair of vortices. Because of the limited blowing airflow from the holes and insufficient energy, the inner pair of vortices is not alternately shed from the rear surface of the cylinder. However, the lower outermost vortex develops continuously into the main vortex, which is capable of obtaining sufficient energy from the boundary layer, and eventually, alternately shedding occurs in the downstream wake of the inner vortices. At the 3T / 4 moment when the lift coefficient reaches the maximum negative value, a pair of reverse vortices develop between the blowing airflow from the jet holes and the upper shear boundary layer of the cylinder, where the outer vortex is gradually moved downstream and develops into the main vortex at the T moment. On the whole, the results at the T / 4 and 3T / 4 moments are not completely symmetrical, and the results of the T / 2 and T moments are not identical. This is because the LES used in the present study can take into account the influence of small-scale vortices on the flow field, and can better simulate the intermittent and burst of turbulence. The aerodynamic force is not a simple harmonic vibration curve, as shown in Fig. 8. It can be seen that the alternately shedding vortices are formed in the wake farther away from the cylinder owing to the influence of the blowing airflow from the jet holes. Furthermore, the vortex shedding region is moved a certain distance in the downstream direction, which will dramatically reduce the fluctuating pressure on the cylinder surface, unlike in the uncontrolled A0 case. As a result, the fluctuation of the aerodynamic force of the cylinder is greatly decreased. In the A0 case, there is an obvious low pressure region in the vortex shedding area of the wake behind the cylinder. When the hollow pipe is firmly fixed on the cylinder surface, the energy is injected into the wake by the blowing airflow from the jet holes, which can reduce the pressure difference between the front and rear of the cylinder. Therefore, the purpose of reducing the streamwise drag force of the cylinder is achieved.

3.2 The height of the jet holes When the angle of the jet hole α is fixed to 7.5°, and the hollow pipe is opened in the front of the stagnation point (F point) of the circular cylinder, the influence of different heights of the jet holes on the control effect is then analyzed. The height of the pipe ( H h ) and the height of the holes ( H p ) are shown in Fig. 1 (b). According to the research results of Chen et al. (2015(b), 2017), the ratios of the heights of the jet holes to the heights of the pipe H h H p are 0.25, 0.5, 0.75, and 1.0, and the H25, H50, H75, and H100 are used to represent the corresponding cases, respectively. Fig. 11 shows the schematic diagrams of the models with different height of

the jet holes under the four cases. Table 3 shows the aerodynamic statistical parameters and the control effect under various cases; the meaning of each parameter is the same as that given in Table 2. From the table, we can see that the main frequency of the vortex shedding of the H25 case is similar to the result of the uncontrolled A0 case, and the main frequency of the vortex shedding of the H25 case is 0.98% higher than that of the A0 case. With an increase in the height of the holes, the control effect of the RMS value of the lift coefficient ECl increases first and then decreases, and reaches a maximum value of 91.99% in the H50 case, which indicates that the best effect of the passive jet flow control method on suppressing the wake is achieved when the height of the holes is half of the height of the pipe. Furthermore, with an increase in the height of the holes, the control effect of the mean value of the drag coefficient ECd increases gradually from 8.68% to 24.90%, which demonstrates that the greater the height of the holes, the smaller the effective obstruct area of the pipe to the flow, which results in a decrease in the drag force. In the H50 case, in which the best control effect to the lift coefficient is observed, the control effect of the drag coefficient is 17.03%. The amplitude and the frequency characteristics of the aerodynamic force directly determine the VIV response of the cylinder. Fig. 12 shows the time histories of the lift and drag coefficients compared with the results of the A0 case under different cases with various heights of the holes. The lift and drag coefficients of the H25–H100 cases are the sum of the lift and drag coefficients of the three walls, which include the inner and outer surfaces of the hollow pipe, and the surface of the inner circular cylinder. In the H25 case, the fluctuations of the lift and the drag coefficients of the model are reduced, and the reduction in the aerodynamic force fluctuation is more obvious in the H50 and H75 cases. As the opening height of the holes is increased, reaching a maximum in the H100 case, the fluctuation degree of the aerodynamic force shows an increasing trend. Fig. 13 shows the results of the spectral analysis obtained by the FFT transformation of the dimensionless lift coefficients. The corresponding abscissa of the curve peak is the Strouhal number St . The St of the flow for the uncontrolled cylinder in the A0 case is 0.210, and the St of the model in the H25 case is close to that of the A0 case, but the peak value of the amplitude spectrum has been significantly reduced, which indicates that the passive jet flow control method has begun to play a role in reducing the fluctuation of the lift coefficient. In the H50 and H75 cases, it can be clearly seen that the results of the spectral analysis for the aerodynamic force have no obvious peak, and the amplitude is remarkably reduced. Owing to the increase in the height of the holes, the

intake airflow at the front side and the exhaust airflow at the rear side of the hollow pipe are increased, and the alternating vortex shedding of flow around the uncontrolled cylinder is destroyed by the jet flow control method. As a consequence, the aerodynamic force in the cross wind direction is completely eliminated, which leads to a significant decrease in the amplitude of the aerodynamic spectral analysis and no obviously main frequency. For the H100 case, the peak value of the aerodynamic amplitude spectrum for the model is increased, but it is still much smaller than that of the A0 case, and the St of the model is increased to 0.239, which shows that the frequency of vortex shedding in the wake of the model is slightly increased. Fig. 14 and Fig. 15 show the time-averaged streamlines and vorticity contours on the middle section of the model for each case. For the H25 case, the approximate symmetrical main vortices in the wake region of the cylinder are moved in the downstream direction for a distance by the influence of the blowing airflow from the jet holes. For the H50 case, there is single pair of reverse vortices in the wake of the cylinder, and the presence of near reverse flows causes the main vortex to move to a further downstream position. At the same time, the corresponding distance of the H50 case from the center of the main vortices to the center of the cylinder is about 1.75D . As can be seen from the time averaged streamlines of the H75 case, the symmetric single pair of vortices

appears in the cylinder wake region, and the distance from the center of the main vortices to the center of the cylinder is reduced to about 1.70D . That is, the vortex formation region is close to the rear surface of the cylinder. For the H100 case, only one pair of symmetrical vortices occurs in the cylinder wake, and the blowing airflow from the rear jet holes combines with the boundary layers that separated from the surface of the model. Unlike in the uncontrolled A0 case, the position of the main vortex formation region is still moved a smaller distance downstream, resulting in a longer length of the main vortex formation region, but less than the result of the H25 case. Fig. 16 shows the instantaneous vorticity contours and streamlines on the middle section of the model for the H100 case, while the results for the H50 case are shown in Fig. 10 (b). For the H50 case, stable inner vortices appear in the near wake of the cylinder owing to the blowing action, and the outer vortices exhibit alternate shedding in the farther downstream region, resulting in a significant reduction in the fluctuation of the lift coefficient. For the H100 case, the airflow from the front holes to the inside of the hollow pipe is lost during the flow process owing to the larger height of the holes, which results in a reduction in the blowing airflow from the rear holes of the cylinder; as a result, the blowing airflow cannot roll up to form the inner vortices. However, the blowing airflow still has a control effect: the vortices formed by separating shear layers from the surface of the

model occur in the slightly further downstream wake. The control effect ECl of the RMS value for the lift coefficient in the H100 case is 19.11% lower than that of the H50 case.

3.3 The angle of the jet holes When the ratio of the height of the jet holes to the height of the hollow pipe H h H p is fixed to 0.5, and the hollow pipe is opened in the front of the stagnation point (F point) of the circular cylinder, the influence of the different angles of the jet holes on the control effect can be finally analyzed. As shown in Fig. 1 (c), the holes are uniformly arranged on the surface of the hollow pipe. In this section, the angles of the jet holes are 3.75°, 7.5°, 11.25°, 15°, and 22.5°, and the corresponding numbers of holes are 48, 24, 16, 12, and 8. The corresponding cases are represented by A3.75, A7.5, A11.25, A15, and A22.5, respectively. Fig. 17 shows the schematic diagrams of the models with different angles of the jet holes for the five cases. The time-histories, statistical parameters, frequency characteristics of the aerodynamic forces and vortex shedding pattern in the wake for each case are obtained by numerical simulation, and then the optimal angle of the jet holes is determined. Table 4 shows the aerodynamic statistical parameters and the control effect for various cases. As shown in the table, the main frequencies of the vortex shedding of the controlled cases are 2.94%–5.89% higher than the results of the uncontrolled A0 case. For one thing, with an increase in the angle of the holes, the control effect of the RMS value of the lift coefficients increases first and then decreases, and reaches a maximum value of 91.99% and a minimum value of 75.81% in the A7.5 case and A22.5 case, respectively. This indicates that optimal suppression of the wake when the passive jet flow control method is applied is achieved when the angle of the holes is 7.5° (24 holes on the pipe). Obviously, when the hollow pipe is opened in the front of the stagnation point (F point) of the circular cylinder and the height of the holes is fixed, and even the surface of the hollow pipe has a minimum number of holes (A22.5 case), the RMS value of the lift coefficient will be 75.81% lower than that in the A0 case, and the control effect will also be remarkable. Meanwhile, the control effectiveness of the mean value of the drag coefficient increases first and then decreases, and reaches a maximum value of 17.03% and a minimum value of 15.90% in the A7.5 case and A22.5 case, respectively. In general, a hollow pipe with different opening angles can effectively suppress the wake of the cylinder and significantly reduce the fluctuation in the lift coefficient, at the same time, the control effect on the mean value of the drag force is not obvious. Fig. 18 shows the time histories of the aerodynamic coefficients of the A0 case and five control cases with various angles of the jet holes. In contrast to the results of the A0 case, it can be clearly seen from each figure that the fluctuation of the lift and drag coefficients are significantly decreased. Only the aerodynamic fluctuation of

A22.5 case is slightly increased, but it is still far less than that of the uncontrolled A0 case. Fig. 19 shows the results of the spectral analysis obtained by the FFT transformation of the dimensionless lift coefficient. As shown in Fig. 19 (a)–(d), the spectral analyses of the aerodynamic forces have no obvious peaks, and the amplitude is remarkably decreased and approximated to 0, which demonstrates that the lift force in the cross wind direction is basically eliminated by the passive jet flow control method. For the A22.5 case, an obvious peak appears in the amplitude spectrum of the aerodynamic force, and the St increases to 0.216, which indicates that the fluctuation in the aerodynamic forces is enhanced, and the frequency of the vortex shedding in the wake of the cylinder is increased. Fig. 20 and Fig. 21 show the time-averaged streamlines and vorticity contours on the middle surface of the model for each case. As shown in Fig. 20, there is one pair of vortices in the wake of the cylinder in the all cases. The formation region of the main vortex of the A7.5 case is longer than that of the A3.75 case. Besides, the distance from the center of the main vortex to the center of the cylinder in the A7.5 case is slightly larger than that of the A3.75 case. For the A11.25 case and A15 case, the opening angle of the pipe gradually increased, and the number of holes on the surface of the pipe gradually reduced. There is no fully developed vortex in the near wake of the cylinder, and the main vortex is formed in the farther downstream wake; moreover, the length of the main vortex formation region of the two cases is very close. For the A22.5 case, the blowing airflow from the uniform large holes is merged with the separated upper and lower boundary layers, and there is only one pair of main vortices in the wake of the cylinder. Compared with the other four cases, the main vortex center is closer to the center of the cylinder, but the distance is still significantly greater than that of the uncontrolled A0 case. In contrast, the control effectiveness ECl of the A7.5 case on the wake of the cylinder is the best, and the control effectiveness ECl of the A22.5 case is relatively poor. The instantaneous vorticity contours and streamlines on the middle section of the model under A7.5 case are given in Fig. 10 (b), and the corresponding results of the A22.5 case are shown in Fig. 22. In the A22.5 case, owing to the large angle of the holes, most of the airflow is exhausted from the holes on the upper and lower sides of the model rear part, which results in a reduction in the effective blowing airflow from the rear holes of the hollow pipe behind the cylinder. Therefore, the small-scale vortices generated at the holes on the upper and lower surfaces of the rear part of the pipe are merged with the vortices generated by the separated shear boundary layer to form a large-scale and high-energy main vortex that sheds into the farther downstream wake, which is different from the inside shedding vortex that does not alternate in the near wake of the cylinder in the A7.5 case. The distance between the alternately shedding

main vortex and the cylinder is reduced, which leads to an increase in the fluctuation of the aerodynamic forces. It can be seen that the vortex shedding pattern in the wake of the model changes owing to a change in the opening angle of the hollow pipe, which causes a change in the aerodynamic force of the model surface. Based on the analysis results of the above all various calculation cases, the optimal parameter scheme of the hollow pipe for suppressing the wake of a circular cylinder is as follow: the pipe at the front stagnation of the circular cylinder is open, the hole height is 50% of the pipe height and the angle of each jet hole is 7.5°.

4. Discussion The jet flow from the hollow pipe make the near wake develop into two inversed rotating vortices which are attached on the rear surface of the cylinder all the time and takes up most of the space. Consequently, the original vortices rolled up by the shear layer on two sides of the cylinder shed alternatingly and are pushed away from the cylinder to the downstream. Therefore, the influence of the alternating vortex shedding on aerodynamic forces is significantly reduced. In addition, the attached vortices on the cylinder surface plays a role of buffer layer, which segregate the unsteady wake and the cylinder successfully. As a result, the fluctuating components of aerodynamic coefficients can be suppressed effectively. Namely, the cylinder is protected by the second flows from the hollow pipe. The effect of the unsteady wakes on the cylinder is alleviated noticeably. The anti-symmetric wake pattern is transformed into a symmetric mode. The dissipation of energy in the wake is reduced. Hence, the negative pressure in the wake is mitigated so that the pressure difference between the windward side and leeward side of the cylinder decreases.

5. Conclusion In the present study, the control effectiveness of different parameters of the passive jet flow control method in suppressing the wake of a circular cylinder were investigated. The influences of the direction, height and angle of the jet holes on the aerodynamic forces, and the vortex shedding pattern in the wake were analyzed in detail. The conclusions obtained from this study are as follows: The velocity field in the wake and the distribution of the mean pressure for a 3D circular cylinder at

Re  3900 obtained by the LES method have been consistent with the existing experimental and numerical simulation results. It can be concluded that the present grid accuracy and numerical settings can accurately capture the characteristics of the flow field around a circular cylinder, which is the basis of further studies on suppressing the wake by using the passive jet flow control method. For the Re calculated in the present study, the optimal combination of the control parameters of the uniform

opening hollow pipe is as follows: the pipe is opened in front of the stagnation point of the circular cylinder, the ratio of the height of the jet holes to the height of the hollow pipe H h H p is set as 0.5 and the angle of the jet holes α is set as 7.5°. Under the optimal parameter combination, the fluctuations of the aerodynamic forces on the surface of the cylinder and hollow pipe model are reduced to a minimum. Moreover, the RMS value of the lift coefficient is dramatically reduced by 91.99% and the mean value of the drag coefficients is decreased by approximately 17.03%. The airflow entering from the front holes is exhausted from the rear holes of the hollow pipe, and the blowing airflow can effectively separate the shear boundary layer from the surface of the circular cylinder. As a result, the main vortex is moved to the farther downstream wake, and alternating vortex shedding occurs. The length of the main vortex formation region is greatly extended, and the influence of the alternating shedding main vortex on the cylinder surface is significantly reduced. Consequently, the wake of the circular cylinder can be effectively suppressed by the passive jet flow control method, which leads to significant reduction in the fluctuation of the aerodynamic force on the cylinder surface, and suppression of the vortex-induced vibration of the elastically mounted circular cylinder can be achieved.

Acknowledgments This research was funded by the National Natural Sciences Foundation of China (NSFC) (51778199, 51578188 and U1709207), the National Key Research and Development Program of China (2018YFC0809403, 2016YFC0701107), the Natural Science Foundation of Guangdong Province (2017A030313324), and the fundamental research funds of Shenzhen science and technology plan (JCYJ20180306172123896).

Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Captions of figures list Fig. 1 Computational domain and the model of hollow pipe model on surface of the cylinder: (a) Computational domain, (b) Model of the hollow pipe on surface of the cylinder, (c) Diagram of the plane parameters of the hollow pipe model Fig. 2 The grid division of the computational domain and the local grid around the model: (a) The entire grid of the computational domain and the local refined grid, (b) The grids on surfaces of the cylinder and pipe Fig. 3 The y  value distributions on the surfaces for the single circular cylinder at Re  3900 and for the cylinder-pipe model at Re  5.0 104 : (a) The entire y  values of all grid points on the surface of the bare circular cylinder with length H   D at Re  3900 , (b) The y  values on three representative sections, z  H 3 , z  H 2 and z  2H 3 , on the bare circular cylinder with length H   D at Re  3900 , (c) The

y  values of all grid points on the surface of the circular cylinder with length H p  0.5D at Re  5.0 104 for the cylinder-pipe model, (d) The y  values of all grid points on the outer (blue triangles) and inner (pink dots) surfaces of the hollow pipe with length H p  0.5D at Re  5.0 104 for the cylinder-pipe model Fig. 4 Time-histories and spectrum analysis of aerodynamic forces of a circular cylinder at Re  3900 : (a) Time-histories of the lift and drag coefficients, (b) Spectrum analysis of Cl Fig. 5 Mean streamwise velocity profiles on the wake centerline and mean pressure coefficients on the middle surface of the cylinder: (a) Mean streamwise velocity profiles, (b) Mean pressure coefficients Fig. 6 Time-averaged streamlines in the near wake and instantaneous contour of Z-swirling on the middle surface of the cylinder: (a) Time-averaged streamlines, (b) Instantaneous contour of Z-swirling ( t =31s ) Fig. 7 Schematic diagrams of the cylinder with a hollow pipe under various directions of the jet holes: (a) A0, (b) F_ON, (c) F_OFF

Fig. 8 Time-histories of the aerodynamic coefficients of the model for each case: (a) F_ON, (b) F_OFF Fig. 9 Time-averaged streamlines and Z-swirling on the middle surface of the model for each case: (a) Time-averaged streamlines, (b) Time-averaged Z-swirling Fig. 10 Instantaneous Z-swirling and streamlines on the middle surface of the model for each case: (a) A0, (b) F_ON, (c) F_OFF Fig. 11 Schematic diagrams of the cylinder with the hollow pipe under various heights of the jet holes: (a) H25, (b) H50, (c) H75, (d) H100 Fig. 12 Time-histories of the aerodynamic coefficients of the model for each case: (a) H25, (b) H50, (c) H75, (d) H100 Fig. 13 Spectrum analysis of the lift coefficient Cl of the model for each case: (a) H25, (b) H50, (c) H75, (d) H100 Fig. 14 Time-averaged streamlines on the middle surface of the model for each case: (a) H25, (b) H50, (c) H75, (d) H100 Fig. 15 Time-averaged Z-swirling on the middle surface of the model for each case: (a) H25, (b) H50, (c) H75, (d) H100 Fig. 16 Instantaneous Z-swirling and streamlines on the middle surface of the model for the H100 case: (a) t  T /4 , (b) t  T /2 , (c) t  3T /4 , (d) t  T

Fig. 17 Schematic diagrams of the cylinder with the hollow pipe under various angles of the jet holes: (a) A3.75, (b) A7.5, (c) A11.25, (d) A15, (e) A22.5 Fig. 18 Time-histories of the aerodynamic coefficients of the model for each case: (a) A3.75, (b) A7.5, (c) A11.25, (d) A15, (e) A22.5 Fig. 19 Spectrum analysis of the lift coefficient Cl of the model for each case: (a) A3.75, (b) A7.5, (c) A11.25, (d) A15, (e) A22.5 Fig. 20 Time-averaged streamlines on the middle surface of the model for each case: (a) A3.75, (b) A7.5, (c) A11.25, (d) A15, (e) A22.5 Fig. 21 Time-averaged Z-swirling on the middle surface of the model for each case: (a) A3.75, (b) A7.5, (c) A11.25, (d) A15, (e) A22.5 Fig. 22 Instantaneous Z-swirling and streamlines on the middle surface of the model for the A22.5 case: (a) t  T /4 , (b) t  T /2 , (c) t  3T /4 , (d) t  T

(a) Computational domain

(b) Model of the hollow pipe on surface of the cylinder

(c) Diagram of the plane parameters of the hollow pipe model Fig. 1 Computational domain and the model of hollow pipe model on surface of the cylinder

(a) The entire grid of the computational domain and the local refined grid

(b) The grids on surfaces of the cylinder and pipe Fig. 2 The grid division of the computational domain and the local grid around the model

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(blue triangles) and inner (pink dots) surfaces of the

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hollow pipe with length H p  0.5D at

Re  5.0 104 for the cylinder-pipe model Fig. 3 The y  value distributions on the surfaces for the single circular cylinder at Re  3900 and for the cylinder-pipe model at Re  5.0 104

(a) Time-histories of the lift and drag coefficients

(b) Spectrum analysis of Cl Fig. 4 Time-histories and spectrum analysis of aerodynamic forces of a circular cylinder at Re  3900

1.2 1.0 0.8

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-2 0

100

200 300 tU∞/D (c) H75

H75

400

0 Cl A0

-2 500

0

100

200 300 tU∞/D (d) H100

Fig. 12 Time-histories of the aerodynamic coefficients of the model for each case

H100

400

500

0.6 A0 H25

0.5 0.4 0.3

0.212

0.2 0.1 0.0

0.210 A0 H50

0.5 Amplitude

Amplitude

0.6

0.210

0.4 0.3 0.2

0.222

0.1 0.05

0.10

0.15 0.20 St

0.25

0.0

0.30

0.05

0.10

(a) H25

0.6

0.4 0.3 0.220

0.1 0.0

0.30

0.210 A0 H100

0.5 Amplitude

Amplitude

0.6

A0 H75

0.2

0.25

(b) H50

0.210

0.5

0.15 0.20 St

0.4 0.3 0.239

0.2 0.1

0.05

0.10

0.15 0.20 St (c) H75

0.25

0.30

0.0

0.05

0.10

0.15 0.20 St (d) H100

Fig. 13 Spectrum analysis of the lift coefficient Cl of the model for each case

0.25

0.30

(a) H25

(b) H50

(c) H75

(d) H100

Fig. 14 Time-averaged streamlines on the middle surface of the model for each case

(a) H25

(b) H50

(c) H75

(d) H100

Fig. 15 Time-averaged Z-swirling on the middle surface of the model for each case

(a) t  T /4

(b) t  T /2

(c) t  3T /4

(d) t  T

Fig. 16 Instantaneous Z-swirling and streamlines on the middle surface of the model for the H100 case

(a) A3.75

(b) A7.5

(d) A15

(c) A11.25

(e) A22.5

Fig. 17 Schematic diagrams of the cylinder with the hollow pipe under various angles of the jet holes

2

2

Cd

1 Cl , Cd

Cl , Cd

1

Cd

0 -1

-1

Cl A0

-2 0

100

200 300 tU∞/D

A3.75

400

0 Cl A0

-2

500

0

100

200 300 tU∞/D

(a) A3.75

400

500

(b) A7.5

2

2

Cd

Cd

1 Cl , Cd

1 0 -1 A0

-2 0

100

200 300 tU∞/D

A11.25

400

0 -1

Cl

Cl A0

-2

500

0

100

200 300 tU∞/D

(c) A11.25

(d) A15

2

Cd

1 Cl , Cd

Cl , Cd

A7.5

0 -1

Cl A0

-2 0

100

200 300 tU∞/D

A22.5

400

500

(e) A22.5 Fig. 18 Time-histories of the aerodynamic coefficients of the model for each case

A15

400

500

0.6 A0 A3.75

0.5 0.4 0.3 0.2

0.210

0.1 0.0

0.210 A0 A7.5

0.5 Amplitude

Amplitude

0.6

0.210

0.4 0.3 0.2

0.222

0.1 0.05

0.10

0.15 0.20 St

0.25

0.0

0.30

0.05

0.10

(a) A3.75

0.6 A0 A11.25

0.30

0.210 A0 A15

0.5 Amplitude

0.3 0.2

0.220

0.1

0.4 0.3 0.2

0.220

0.1 0.05

0.10

0.15 0.20 St

0.25

0.0

0.30

0.05

0.10

(c) A11.25

0.15 0.20 St (d) A15

0.6

0.210 A0 A22.5

0.5 Amplitude

Amplitude

0.6

0.4

0.0

0.25

(b) A7.5

0.210

0.5

0.15 0.20 St

0.4 0.3

0.216

0.2 0.1 0.0

0.05

0.10

0.15 0.20 St

0.25

0.30

(e) A22.5 Fig. 19 Spectrum analysis of the lift coefficient Cl of the model for each case

0.25

0.30

(a) A3.75

(b) A7.5

(c) A11.25

(d) A15

(e) A22.5 Fig. 20 Time-averaged streamlines on the middle surface of the model for each case

(a) A3.75

(b) A7.5

(c) A11.25

(d) A15

(e) A22.5 Fig. 21 Time-averaged Z-swirling on the middle surface of the model for each case

(a) t  T /4

(b) t  T /2

(c) t  3T /4

(d) t  T

Fig. 22 Instantaneous Z-swirling and streamlines on the middle surface of the model for the A22.5 case

Captions of tables list Table 1 Comparison of typical flow parameters between the present numerical simulation and the previous research results Table 2 Aerodynamic statistical parameters and control effect under various cases with different directions of the jet holes Table 3 Aerodynamic statistical parameters and control effect under various cases with different heights of the jet holes Table 4 Aerodynamic statistical parameters and control effect under various cases with different angles of the jet holes

Table 1 Comparison of typical flow parameters between the present numerical simulation and the previous research results Case Lourenco and Shih (1993), Experimental Parnaudeau et al. (2008), PIV HWA (Hot wire anemometry) LES Cardell (1993), Experimental Jin et al. (2016), Numerical (CLES) Kravchenko and Moin (2000), Numerical (LES) Zhang et al. (2015), Numerical (LES) D’Alessandro et al. (2016), Numerical (DES) Dong et al. (2006), PIV Numerical (DNS) Present, Numerical (LES)

Lr / D

Um / U

CPb

Cd

Cl '

St

1.3±0.1

-0.25±0.1

0.88±0.05

0.99±0.05

-

0.215±0.005

1.51 1.56

-0.340 -0.260

-

-

-

0.208±0.002 0.208±0.001

1.4±0.1

-

-

0.98±0.05

0.03~0.08

0.215±0.005

1.69

-0.249

-

-

-

0.206±0.001

1.35

-0.37

0.94

1.04

-

0.210

-

-

-

1.018

0.174

0.22

1.678

-

0.829

0.9857

0.1088

0.214

1.47 1.36

-0.252 -0.291

0.93

-

-

0.208

1.539

-0.320

0.869

1.015

0.097

0.214

Table 2 Aerodynamic statistical parameters and control effect under various cases with different directions of the jet holes Case

A0

F_ON

F_OFF

RMS value of lift coefficients, Cl'

0.649

0.052

0.066

Mean value of the drag coefficients, Cd

1.233

1.023

1.048

Frequency of the shedding vortex, f s (Hz)

15.338

16.241

16.391

Strouhal number, St

0.210

0.222

0.224

Control effect of the lift coefficient, ECl (%)

0

91.99

89.83

Control effect of the drag coefficient, ECd (%)

0

17.03

15.00

Table 3 Aerodynamic statistical parameters and control effect under various cases with different heights of the jet holes Case

A0

H25

H50

H75

H100

RMS value of lift coefficients, Cl'

0.649

0.181

0.052

0.073

0.176

Mean value of the drag coefficients, Cd

1.233

1.126

1.023

0.983

0.926

Frequency of the shedding vortex, f s (Hz)

15.338

15.489

16.241

16.090

17.444

Strouhal number, St

0.210

0.212

0.222

0.220

0.239

Control effect of the lift coefficient, ECl (%)

0

72.11

91.99

88.75

72.88

Control effect of the drag coefficient, ECd (%)

0

8.68

17.03

20.28

24.90

Table 4 Aerodynamic statistical parameters and control effect under various cases with different angles of the jet holes Case

A0

A3.75

A7.5

A11.25

A15

A22.5

RMS value of lift coefficients, Cl'

0.649

0.053

0.052

0.064

0.072

0.157

Mean value of the drag coefficients, Cd

1.233

1.024

1.023

1.030

1.027

1.037

Frequency of the shedding vortex, f s (Hz)

15.338

15.940

16.241

16.090

16.090

15.789

Strouhal number, St

0.210

0.218

0.222

0.220

0.220

0.216

Control effect of the lift coefficient, ECl (%)

0

91.83

91.99

90.14

88.91

75.81

Control effect of the drag coefficient, ECd (%)

0

16.95

17.03

16.46

16.71

15.90

48