Control of boundary layer flow and lock-on of wake behind a circular cylinder with a normal slit

European Journal of Mechanics B/Fluids 59 (2016) 99–114

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Control of boundary layer flow and lock-on of wake behind a circular cylinder with a normal slit Huai-Lung Ma ∗ , Cheng-Hsiung Kuo 1 Department of Mechanical Engineering, National Chung Hsing University, No. 250 Kuo-Kuang Road, Taichung 40227, Taiwan

highlights • This study discloses the upstream induction of the wake to the velocity fluctuation at the slit opening. • The modified phase-averaged method is employed to demonstrate the scenarios of periodic blowing/suction at the slit opening, the evolution of boundary-layer flow along the rear surface and the wake flows in the near-wake region.

• Frequency, amplitude responses and the slight phase shift of the wake flow structures confirms the one-sided lock-on characteristics of the wake. • This study provides some flow physics to widen the application range of vortex flowmeters.

article

info

Article history: Received 12 November 2015 Received in revised form 19 April 2016 Accepted 6 May 2016 Available online 26 May 2016 Keywords: Circular cylinder Normal slit Primary lock-on passive wake control Separation delay

abstract A two-dimensional flow simulation around a circular cylinder with a normal slit at a Reynolds number of 200 is conducted using Ansys/Fluent software. The slit ratio (S /D) ranges from 0.03 to 0.3. The scenarios of the characteristics of the slit flow, the evolution of the boundary-layer flow, and the wake flow are investigated and disclosed by modified phase-averaged method. Some critical outcomes are outlined. Induced by the shedding vortex street with shorter formation length, the pressure difference between the two slit openings drives the fluid at each slit opening, in form of periodic blowing/suction with zero net mass flux. Periodic blowing/suction at the slit opening serves as a perturbation that significantly delays the boundary-layer flow separation along the rear surface of the slit cylinder and reduces the vortex formation length. With proper slit ratio, this perturbation synchronizes with the evolution of the boundary-layer flow and the formation of vortex-street, leading to the primary lock-on of the wake. In this study, the optimal slit ratio is S /D = 0.18 at Re = 200. For all slit ratios studied, the frequency and amplitude responses, as well as the slight phase shift of the wake flow structures clearly demonstrate the primary lock-on of the wake. In this study, the one-sided lock-on frequency range lies only within fa /fso > 1 because the primary lock-on is caused by the synchronization between the periodic blowing/suction at the slit opening and the shedding vortex street with higher frequency (or reduced vortex formation length) behind the slit cylinder than that of a base-line (or non-slit) cylinder. © 2016 Elsevier Masson SAS. All rights reserved.

1. Introduction Flow control techniques are used to modify flows in such ways that are beneficial to engineering applications [1]. For instance, the vortex flowmeter, based on the principle of shedding vortices behind a bluff body, has been widely used in flow rate measurements [2]. As mentioned in the review of the vortex flowmeter

∗

Corresponding author. Tel.: +886 4 2284 0433x208; fax: +886 4 2287 7170. E-mail addresses: [email protected] (H.-L. Ma), [email protected] (C.-H. Kuo). 1 Tel.: +886 4 2284 0433x314; fax: +886 4 2287 7170. http://dx.doi.org/10.1016/j.euromechflu.2016.05.001 0997-7546/© 2016 Elsevier Masson SAS. All rights reserved.

[3], the advantage of a vortex flowmeter is the insensitivity to the fluid properties that makes its applications versatile. However, at a Reynolds number (Re) > 1000, the separated shear layers of a circular cylinder will be modulated due to the formation of streamwise vortices ; and the wake will become turbulent at a much higher Re. Thus, the Strouhal number of shedding frequency is a constant only within a limited range of Re, and this limits the application range of the flow meters. A cylinder with a normal slit could intensify vortex shedding [4], and the Strouhal number can be maintained nearly constant within a wider range of Re. The literature is full of discussions about unsteady actuation through the slit cylinder [5–8], and scholars have debated its nature for decades. In discussing the operating range for vortex flowmeters, the geometrical shapes of the vortex shedder are most emphasized. A

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Nomenclature CD

2 Drag coefficient, CD = FD / 12 ρ Uin D

2 D Lift coefficient, CL = FL / 12 ρ Uin Normalized CL , CL∗ ∈ [−1, 1] 2 Base-pressure coefficient, Cpb = (Pb − P∞ ) / 12 ρ Uin D Volumetric flux coefficient, CQ = Q s /Uin S Normalized CQ , CQ∗ ∈ [−1, 1] Diameter of the cylinder, m Oscillating frequency of slit flow, Hz (1/s) Vortex shedding frequency behind slit cylinder, Hz (1/s) fso Vortex shedding frequency behind non-slit cylinder, Hz (1/s) Lw Vortex formation length, m N Total number of cycles p Pressure, Pa pb Base pressure, Pa p∞ Freestream static pressure, Pa Qs (t ) flux through the slit/depth, Qs (t ) = Volumetric S /2 V ( X , t )dX ,m3 /s s −S /2 Re Reynolds number, Re = Uin D/ν REA Abbreviation of reattachment of boundary-layer flow St Strouhal number, St = fs D/Uin S Slit width, m SEP Abbreviation of separation of boundary-layer flow t Time, s t∗ Dimensionless time, t ∗ = t /Ts Ts Shedding period of slit cylinder, s U X-velocity component, m/s Uin Velocity of incident flow, m/s Uθ Tangential velocity at specific angular position θ , m/s Uθ∗ Dimensionless tangential velocity U ∗θ = Uθ /Uθ ,max , Uθ ,max Maximum Uθ of the boundary layer, m/s V Y-velocity component, m/s Vs (X , t ) Velocity at slit opening, m/s V¯s (X ) Time-averaged Vs (X , t ), m/s V˜s (X , t ) Coherent velocity fluctuation at slit opening, m/s ⟨Vs ⟩ Modified phase-averaged velocity component, m/s Vs,av e Average velocity across the slit, Vs,av e = Qs /S · 1, m/s Vs,rms (fa ) Root mean square value of V˜s (X , t ) at frequency fa , m/s y+ Dimensionless wall distance ZNMF Abbreviation of Zero Net Mass Flux ∆f Frequency resolution, Hz (1/s) ∆p Pressure difference between the top and bottom openings, Pa |CL (f )| Spectral density of the CL , 1/s CQ (f ) Spectral density of the CQ , 1/s

CL CL∗ Cpb CQ CQ∗ D fa fs

Greek Symbols

β θs θ¯s ν ρ φa φs φf ω

√

Dimensionless parameter, β = S /2 ω/2ν Separation angle of boundary-layer flow, rad Mean separation angle, rad Kinematic viscosity of fluid, m2 /s Fluid density, m3 /kg Phase difference between ∆p and CQ by analytic solution, rad Phase difference between ∆p and CQ by simulation, rad Phase difference between CQ∗ and CL∗ , rad Angular frequency, rad /s

cylinder with a slit was the first breakthrough design for the vortex flowmeter and was first conducted by Tsuchiya et al. [9] in 1970. Detailed empirical studies can be traced back to Igarashi [10–12]; the papers in Igarashi [10,11] provide extensive discussions on the flow characteristics around a circular cylinder with a slit. He found that, at small slit angles, the ejection of fluid into the near-wake region can significantly increase the base pressure coefficient and reduce the drag coefficient. When the slit angle is between 80° and 90°, Igarashi observed suction and blowing occurring alternatively at both ends of the slit with a periodic vortex shedding. Later, Igarashi [12] experimentally studied different bluff bodies (e.g., a trapezoidal cylinder, a cylinder with a slit, and a triangularsemicircular cylinder prism) in a wind tunnel with an Re ranging from 1.9 × 104 to 2.5 × 105 . He found that the circular cylinder with a normal slit greatly improved the two-dimensionality of vortex shedding. Also, the pressure loss of the circular cylinder with a normal slit was reduced by 50% compared to that of the trapezoidal cylinder for the same blockage ratio and the Re range. Later on, a few researchers further modified the cylinder with a slit by introducing a concave rear face and improved the performance of the vortex shedding (e.g., Turner et al. [13]). A flow visualization and a spectral analysis in the water channel showed that the slit cylinder with a concave rear surface generated the strongest and most regular vortex shedding among all the shapes examined. The benefits from this shape can be explained in terms of the flow structure in the near wake. With the same model, Popiel et al. [14] did extensive work in observing a vigorous swinging movement of the fluid along the rear surface of the cylinder. They found that the vortices shed from this specially shaped cylinder were stronger than that of other bluff cylinders, but were very vulnerable to endwall conditions. In the case of a cylinder with a normal slit, the shed vortices induce an alternating blowing/suction of fluids at the slit openings, and the slit acts as an information passage for vortex shedding. Pankanin et al. [15] used a two-color dyed flow visualization to determine the differences between a Karman vortex generated by a non-slit circular cylinder and that of a cylinder with a slit. The stagnation regions behind the slit cylinder were much shorter than that of the non-slit circular cylinder. The role of stagnation region seems to be crucial and is related to the oscillating fluid in the slit. Olsen and Rajagopalan [16] discussed the change of the ratio of St /Re and the drag coefficient CD ; and also observed that the ratio of St /Re for a cylinder with a normal slit was always higher than that of a non-slit circular cylinder over the range 60 < Re < 1000. The effect of the slit increased the strength of the vortices and the addition of a concave rear notch stabilized vortex shedding. More recently, Peng et al. [17] employed a cylinder with a normal slit to experimentally study the effect of the slit width on the vortex shedding signal for 2.4 × 103 ≤ Re ≤ 8.5 × 104 and S /D = 0–0.3 in the water channel and wind tunnel. For S /D = 0.1–0.15, the error of linearity in St and the three-dimensional appearance of vortex shedding structures be the least. The largest nonlinearity occurred in the case of S /D = 0.05. Based on the above review, the unsteady flows around the circular cylinder with a normal slit indeed exhibit several features significantly different from that behind a non-slit cylinder. However, the scenarios relating the alternate blowing/suction at each slit opening to the evolution of boundary-layer flow along the rear surface and the wake flows in the near-wake region have not yet been clearly revealed. In this study, we provide detailed evidences showing the scenarios or mechanisms that link these flows by modified phase-averaged method. Furthermore, the lockon of the wake flow in relation to the periodic perturbation at the slit opening is disclosed. Scientists and technicians would benefit from understanding the mechanism to acquire an optimal performance.

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Fig. 1. (a) Sketch of flow around a circular cylinder with a slit normal to the stream. (b) The cross-sectional view, coordinate system, and parameters of the slit circular cylinder studied. Table 1 Root-mean-squared amplitude of velocity fluctuation Vs,rms (fa ) /Uin , the frequencies in the slit (fa ) and of the wake (fs ) behind a slit cylinder as functions of the slit ratio (S /D). The vortex shedding frequency of the non-slit cylinder is fso = 0.099 Hz at Re = 200. D (m)

S /D

fa (Hz),

fa /fs

fa /fso

Vs,rms (fa ) /Uin

0.02

0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

0.100 0.101 0.103 0.105 0.108 0.110 0.112 0.114 0.116 0.117

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.010 1.020 1.040 1.061 1.091 1.111 1.131 1.152 1.171 1.182

0.007 0.018 0.035 0.056 0.076 0.094 0.106 0.111 0.113 0.114

2. Simulation methodology 2.1. Governing equations and parameters The flow simulation is conducted using Ansys/Fluent software [18]. Fig. 1 depicts the coordinate system and the flow parameters employed for the simulation, where D, S, and Uin denote the cylinder diameter, the slit width, and the free-stream velocity, respectively. The continuity and the momentum equations for unsteady, two-dimensional, viscous and incompressible flow are:

∂V ∂U + = 0, ∂X ∂Y  2  ∂ U ∂U ∂U ∂U 1 ∂p ∂ 2U +U +V =− +ν + 2 ∂t ∂X ∂Y ρ ∂X ∂ 2X ∂ Y  2  ∂V ∂V ∂V 1 ∂p ∂ V ∂ 2V +U +V =− +ν + ∂t ∂X ∂Y ρ ∂Y ∂ 2X ∂ 2Y

Fig. 2. Computational domain for the circular cylinder with a slit normal to the stream; a full domain of 90,882 grids for S /D = 0.15. (a) Full-flow domain; (b) near the slit region and a closeup view.

(2.1a) (2.1b)

(2.1c)

where U is the X -velocity component, V is the Y -velocity component, p is the pressure, ν is the kinematic viscosity of the fluid, and ρ is the fluid density. The parameters were ρ = 998.2 kg/m3 and ν = 1.005 × 10−6 m2 /s throughout the study. From Table 1, the values of fa /fso and Vs,rms /Uin (fa ) at S /D = 0.3 are 18.2% and approximately 16 times higher than those at S /D = 0.03. Moreover, the values of fa /fs are persistently 1.0. These results indicate that the wider the slit is, the larger fa /fso and Vs,rms /Uin (fa ) are. The values of fa are exactly equal to those of fs regardless of the variations in S /D. Briefly speaking, the frequencies of the slit flow and the shedding vortex street are synchronized. 2.2. Flow domain and boundary conditions A C-type mesh is generated in the flow domain of a bullet shape. The full domain and the close-up view near the slit are

Fig. 3. Time history of the drag, lift, and base pressure coefficients at Re = 200. The results from other simulations are CD = 1.34 ± 0.045 (Palma) [21], CL = 0 ± 0.69 (Linnick) [22], and Cpb = −0.956 (Liu) [23]. The results from other experiments are CD = 1.3 (Wille) [24] and Cpb = −0.96 (Williamson) [25].

shown in Fig. 2(a) and (b), respectively. The origin is defined at the center of the cylinder, the inlet boundary is located at 25D upstream, and the outlet boundary is 75D downstream. The upper and lower boundaries of the flow domain are approximately 25D from the wake center line (Y /D = 0). No slip and no penetration conditions are imposed on the cylinder surface and the slit walls. The boundary conditions of U = Uin and V = 0 are specified

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a

b

c

Fig. 5. (a) Phase shift between the pressure difference ∆p and the volume flow rate Qs for the slit circular cylinder for Re = 200 and S /D = 0.15. (b) Comparison of the phase delay between the numerical results and the analytic solution for S /D ranging from 0.03 to 0.3 and Re = 200.

Fig. 4. (a) Pressure difference (∆p) between the mean static pressure at the top opening [X /S] ∈ [−0.5, 0.] , [Y /D] = [0.48] and at the bottom opening [X /S] ∈ [−0.5, 0.5] , [Y /D] = [−0.48] for Re = 200 and S /D = 0.15. (b) The streamline pattern at instant t1∗ when ∆p attained its maximum value. (c) The streamline pattern at instant t2∗ when ∆p attained its minimum value.

at the inlet boundary. The boundary conditions of ∂ U /∂ Y = 0 and V = 0 are imposed on the upper and lower boundaries. The downstream boundary condition was defined as p∞ = 0 (gauge) at the outlet boundary. Initially, a uniform velocity field (U = Uin , V = 0) is imposed everywhere in the flow domain. After calculations, the pressure and velocity signals at each grid point are extracted from the saved sequential frames. The conservation equations using the two-dimensional laminar viscous model are discretized by employing a finite volume method [19]. For time discretization, an implicit second-order scheme is employed, and a second-order least squares cell based scheme is applied for temporal discretization. In addition, pressure–velocity coupling employs the semi-implicit method for pressure linked equationsconsistent (SIMPLEC) algorithm [20]. In all flow domains, the grid cells are quadrilateral. Within the slit region, the grid cells are uniform in size. The height of the first mesh cell near the cylinder wall achieves a desired y+ = 0.12 using flat-plate boundary layer theory, and there are about ten grid cells in the boundary layer. Such grid sizes in the boundary-layer flow around the cylinder

are considered small enough to ensure accurate simulation of flow dynamics. Further, the −5/3 slope in the velocity spectra of Fig. 6(b,d) can also verify that the grid size in this study can resolve the fine-scale vortices at higher frequency range. In this study, the frequency of interest equals approximately 0.1 Hz. A sampling frequency of 200 Hz and a sampling period of 500 s are chosen to have a frequency resolution of ∆f = 0.002 Hz. In this study, Table 2 lists the grid convergence study for the flow past a single circular cylinder at Re = 200. 2.3. Code validation The benchmark case for code validation is the flow past a single circular cylinder at Re = 200. Fig. 3 shows the time histories of the drag coefficient (CD ), the lift coefficient (CL ) and the base-pressure coefficient (Cpb ). As mentioned by Liu et al. [23], the frequency of the base-pressure coefficient was actually the frequency of vortex shedding. Among the abundant examples found in the literature, Table 2 Grid convergence study for Re = 200. y+

Mesh

CD

CL

Cpb

St

0.24 0.12 0.08

25,800 89,620 138,900

1.36 ± 0.048 1.35 ± 0.045 1.35 ± 0.045

±0.70 ±0.69 ±0.69

−0.976 −0.971 −0.970

0.194 0.197 0.197

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a

b

c

d

103

Fig. 6. (a,c) Spatial distributions of time-mean slit velocity across the top and bottom openings of the slit with a block at the center and the slit, for Re = 200 and S /D = 0.15. (b,d) Corresponding spectra for the dimensionless slit velocity at [X /D, Y /D] = [0, 0.49]. Table 3 Comparison of results for Re = 200. Reference

CD

CL

Cpb

St

Present study Liu et al. [23] Belov et al. [26] Rogers & Kwak [27] Palma et al. [21] Linnick & Fasel [22] Chen et al. [28] Farrant et al. [29] Miyake et al., reported in [26] Rosenfield et al., reported in [27] Henderson, reported in [26] Kovaznay (Exp.), reported in [26] Roshko (Exp.) [30] Williamson (Exp.) [25] Wille (Exp.) [24]

1.35 ± 0.045 1.31 ± 0.049 1.19 ± 0.042 1.23 ± 0.05 1.34 ± 0.045 1.34 ± 0.044 1.33 ± 0.04 1.36 (mean) 1.34 ± 0.043 1.46 ± 0.05

±0.69 ±0.69 ±0.64 ±0.65 ±0.68 ±0.69 ±0.72 ±0.76 ±0.67 ±0.69

−0.971 −0.956 −0.936

0.197 0.192 0.193 0.185 0.190 0.197 0.197 0.196 0.196 0.211 0.197 0.19 0.19 0.197

−1.0

−0.96 1.3

we collect some typical results, summarized by Belov et al. [26] and Rogers et al. [27], in Table 3. Some other simulation results using different methods [21,22] are also compared. In Table 3, CD , CL , Cpb , and the Strouhal number St of the shedding vortex are also listed. In Table 3, the present results are in excellent agreements with those in the literature, verifying that the numerical scheme and results in the present simulation are adequate and reliable. 3. Results and discussion 3.1. Characteristics of slit flow 3.1.1. Driving source of perturbation Since the pressure distributions across the slit opening are almost uniform (not shown here), the averaged pressure across

each slit opening is employed to calculate the pressure difference ∆p between two openings of the slit. Positive ∆p indicates that the pressure at the bottom opening is greater than that at the top opening, and vice versa. Fig. 4(a) depicts the variation of the pressure difference ∆p within an oscillating cycle for S /D = 0.15 and Re = 200. The corresponding streamline patterns are illustrated in Fig. 4(b) and (c), respectively, while ∆p attains the maximum and minimum values. In Fig. 4(b), when the pressure difference reaches the maximum positive value, all the streamlines in the slit are directed from the bottom to the top openings, except for a small vortex near the leading edge of the bottom opening. A mirror-image streamline pattern is observed when the pressure difference reaches its minimum value. This demonstrates that the positive value of ∆p expels the fluid out of the top opening of the slit (i.e., the blowing phase), and the negative value of ∆p sucks fluid into the slit opening (i.e., the suction phase). In other words, the pressure difference induced by the shedding vortex street will be the driving source to move the fluid at the slit opening in form of alternate blowing/suction. 3.1.2. Phase shift between the pressure difference and volumetric flow rate This section describes the relationship between the pressure difference ∆p and the dimensionless flow rate CQ for S /D = 0.15 and Re = 200. It is evident in Fig. 5(a) that the cyclic variations of both ∆p and CQ are nearly sinusoidal functions; however, the curve of ∆p leads that of CQ by an amount of phase shift (φs ). This phase shift demonstrates that the fluid inside the slit will be driven a certain time after the pressure difference (i.e., driving source) is applied.

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a

b

c

d

Fig. 7. (a,d) Coherent velocity component across the slit obtained by the modified phase-averaged method for Re = 200 and S /D = 0.15. (b,c) Instant X -velocity profiles in the boundary layer within a completed cycle for Re = 200, S /D = 0.15 and X /S = 0.5.

Furthermore, from Fig. 5(b), the amount of phase lag (φs ) increased as the slit ratio (S /D) increased. The analytical solution of the phase delay φa is derived in the following and compared in Fig. 5(b). Based on the results shown in Fig. 4(b) and (c), the flow inside the slit can be assumed to be incompressible and fully developed. The gravity effect is excluded. Thus, the Navier–Stokes equations reduce to 0=−

1 ∂p

(3.1a)

ρ ∂X

∂V s 1 ∂p =− +ν ∂t ρ ∂Y



∂ Vs ∂Y 2 2



.

(3.1b)

From Fig. 4(a), the periodic unsteady pressure difference (or gradient) and the unsteady flow in the slit can be expressed as

−

1 ∂p

ρ ∂Y

iω t

= Ge

(3.2) iω t

Vs (X , t ) = f (X )e

(3.3)

where G is a constant, f (X ) is a function of X , and ω denotes the oscillating (or angular) frequency of the slit flow. Substitution of Eqs. (3.2)–(3.3) into eq. (3.1b) yields a second order, linear, ordinary differential equation (3.4). f ′′ −

ω 2ν

G

(1 + i)2 f = − . ν

(3.4)

Applying no-slip boundary conditions at the slit walls, Vs = 0 at X = ±S /2, the general solution of Eq. (3.4) is Vs ( X , t ) =

G iω

 1−

cosh [α (1 + i) X ] cosh [α (1 + i) (S /2)]



eiωt .

(3.5)

Integrating Eq. (3.5) with respect to X , we have the volume flow rate/depth in the Z direction (Qs ) across the slit opening in the following form Qs (t ) = Heiωt .

(3.6)

The real and the imaginary parts of H are arranged in the following forms     2β e − e−2β − 2 sin β SG     Re (H ) = − (3.7a) ω β 2 e2β + e−2β + 4 cos 2β Im (H )

=

SG

ω



(2β − 1) e2β + (2β + 1) e−2β + 4β cos 2β − 2 sin 2β     β 2 e2β + e−2β + 4 cos 2β

 . (3.7b)

In Eqs. (3.7a) and (3.7b), the dimensionless parameter (β) is defined as

 β = S /2 ω/2ν.

(3.8)

The analytical phase lag (φa ) between the pressure difference and the volume flow rate can be expressed as   (2β − 1) e2β + (2β + 1) e−2β + 4β cos 2β − 2 sin 2β −1   φa = tan . (3.9) e2β − e−2β − 2 sin 2β In Fig. 5(b), the simulation values of the phase delay agree very well with those of the analytical results at small values of β . At high β (or large S /D), there exists the maximum deviation which may be caused by violating the assumption of fully-developed flow in the slit at high values of β . The parameter β depends upon the fluid density, the slit width and the oscillating frequency. In this study, the oscillating frequency differs slightly for various slit widths; thus, as the fluid density is the same, the phase delay is mainly caused by the variations of the slit width. In Fig. 5(b), the amount of phase delay increases as the slit width increases.

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Fig. 8. Contours of pressure coefficients and streamlines around the circular cylinder with a normal slit within the blowing half cycle for Re = 200 and S /D = 0.15.

3.1.3. Time-periodic blowing and suction at slit opening In this section, the velocity distributions across the slit opening are characterized. According to Hussain and Reynolds [31], a triple decomposition of the velocity component Vs (X , t ) at the slit

opening can be written as: Vs (X , t ) = V¯s (X ) + V˜s (X , t ) + Vs′ (X , t )

(3.10)

where V¯s (X ) is the time-mean velocity defined in Eq. (3.11), V˜s (X , t ) is the coherent velocity fluctuation, and Vs′ (X , t ) is the

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Fig. 8. (continued)

random turbulent fluctuation. V¯s (X ) =

1  NTs

Vs (X , t )

(3.11)

where N is the number of the oscillating cycle employed, Ts is the period of the shedding vortices behind the slit cylinder, and NTs is the total length of the recorded time series. ⟨Vs (X , t )⟩ denotes the velocity component obtained by the modified phase-averaged technique via Eq. (3.12).

⟨Vs (X , t )⟩ =

N 1 

N n=1

Vs (X , t + nTs ).

(3.12)

In Eq. (3.12), the phase-average quantities are computed from N = 50 cycles with increments of 0.0005Ts (i.e., 0.5 ms), and the time-average quantities are calculated from 100,000 instants. The coherent component V˜s (X , t ) is then given in Eq. (3.13). V˜s (X , t ) = ⟨Vs (X , t )⟩ − V¯s (X ).

(3.13)

Fig. 6(a) illustrates the distributions of V¯s (X )/Uin across the top and bottom openings for S /D = 0.15 but the slit is blocked at the center. Evidently, the velocity distributions across the slit openings are not uniform. At the top opening, the velocity directs upwards within the leading portion and downwards in the trailing portion of the slit opening. At the bottom opening, the velocity distribution is the exact mirror image of that at the top opening. At the walls of the slit (X /S = ±0.5), the velocity satisfies the no-slip condition. In addition, the average of V¯s (X )/Uin across the opening gives zero values, indicating that the net mass flux across each slit

opening is zero. In Fig. 6(b), the corresponding velocity spectrum is acquired at [X /D, Y /D] = [0, 0.49] for the same condition as that in Fig. 6(a). The velocity spectra show a predominate peak centered at the fundamental frequency (fa /fs = 1 ); and the magnitudes of other higher harmonics show a monotonous decreasing trend. Fig. 6(c) shows the distributions of V¯s (X )/Uin across the top and bottom openings of the slit for S /D = 0.15 and Re = 200. The distributions greatly resemble that in Fig. 6(a) except the magnitude is 1.3 times larger than that in Fig. 6(a). The average of V¯s (X )/Uin across each slit opening also gives zero values. Fig. 6(d) shows the corresponding velocity spectra acquired at [X /D, Y /D] = [0, 0.49] for S /D = 0.15. In Fig. 6(d), the velocity spectrum shows a predominate peak at the fundamental frequency (fa /fs = 1 ) with an enhanced magnitude of the third harmonic (fa /fs = 3 ). Note that the spectral amplitudes of the fundamental and the third harmonics are at least two orders of magnitude larger than those in Fig. 6(b). Since the wake has the self-excited nature and can be modeled as a Van-der Pol oscillator with cubic nonlinearity [32,33], the remarkably enhanced spectral amplitudes of the fundamental and the third harmonics in Fig. 6(d) are evidently induced by the shedding vortex street. Namely, alternate vortex shedding induces a periodic pressure difference between two openings of the slit which drives the fluid inside the slit to move back and forth at the same frequency. The upstream induction of the shedding vortex street is in synchronization with the perturbation at each slit opening. This synchronization leads to much enlarged amplitude of the fundamental and the third harmonics in Fig. 6(d). Once the slit is blocked at the center, the amplitudes of the fundamental and the third harmonics in Fig. 6(b)

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a

b

c

Fig. 9. (a) Time-mean streamline pattern in the upper half-flow domain. (b–c) Velocity profiles of the boundary layer to show the flow separation for Re = 200 and S /D = 0. R is the radial distance measured from the origin.

are negligibly small. This indicates that the fluid movement inside the slit opening almost stops and the communication between two openings is terminated. Thus, synchronization between the slit flow and the shedding vortex is terminated. Fig. 7(a) and (d) depict the spatial distributions of V˜s (X , t ) and U˜s (X , t ) across the top opening of the slit at eight instants within one oscillating cycle, respectively. In Fig. 7(a), all the coherent velocities V˜s (X , t ) across the top opening of the slit are positive, indicating that the fluid at the slit opening is blown out in the first half cycle. The magnitude of V˜s (X , t ) increases appreciably within t ∗ = [0, 1/8], reaches the maximum out-blowing velocity at t ∗ = 2/8; but increases mildly in t ∗ = [1/8, 3/8] and reduces appreciably within t ∗ = [3/8, 1/2]. During the second half cycle, all the velocities are negative, implying that the fluid is sucked into the slit opening. Similar variations are also observed in the second half cycle with the maximum suction velocity occurred at t ∗ = 6/8. Besides, the amplitudes of V˜s (X , t ) are small near the leading edge and increase appreciably toward the trailing edge of the slit opening. Fig. 7(b) and (c) provide the velocity profiles in the boundary layer at X /S = 0.5 and S /D = 0.15. In the outer region of the boundary-layer of Fig. 7(b), the magnitude increases progressively almost in phase with the blowing phase and the maximum velocity overshoot occurred at t ∗ = 2/8. However, the changes of magnitude in the inner region of the boundary layer may experience certain phase lag. On the contrary, the variations of velocity in the outer region of boundary layer are in phase with and those in the inner region of the boundary layer take certain phase lead during the suction phase. Fig. 7(d) and (a) share

certain development trend in common within one oscillating cycle. However, the maximum magnitude of V˜s (X , t ) is triple times that of U˜s (X , t ). This indicates that V˜s (X , t ) is the primary flow at each slit opening. In Figs. 7(a) and 6(a), the cyclic average of V˜s (X , t ) and spatial average of V¯s (X ) are all zero across the slit opening. This clearly demonstrates that the nature of the periodic blowing/suction at the slit opening is analogous to that of a zero-net-mass-flux (ZNMF ) jet [34,35]. Though the periodic blowing/suction at the slit opening provides zero net mass, it introduces a net advancing momentum into the boundary-layer flow during the blowing and suction phases [5]. 3.2. Boundary-layer flow characteristics To phase-lock the boundary layer and wake flows with the periodic blowing/suction at the slit opening, it is necessary to calculate the cyclic variation of the dimensionless volumetric flow rate (Eq. (3.14)) as a Ref. [36] CQ =

Qs Uin S

.

(3.14)

3.2.1. Evolution of boundary-layer flow Fig. 8 provides a sequence of color-coded contours of the pressure coefficient overlaid with the streamlines within the blowing half cycle for S /D = 0.15 and Re = 200. In Fig. 8(a), the variation of CQ across the top opening of the slit within one oscillating cycle is employed as a reference, and t ∗ = 0 is defined

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Fig. 10. (a) Time-mean streamline pattern in the upper half-flow domain. (b–c) Velocity profiles in the boundary layer to show the flow separation and reattachment for Re = 200 and S /D = 0.15. (d) Comparison between time-mean tangent velocity of slit cylinder with a block at the center and those of slit cylinder with no block to show the steady streaming effect at θ = 70°. R is the radial distance measured from the origin.

as the instant when the value of CQ is zero and about to blow out of the slit opening. The sequential flow structures in Fig. 8(b)–(j) are phase-locked with the variation of CQ (Fig. 8(a)) and demonstrate the evolution of the boundary-layer flow along the rear surface of the cylinder and vortex formation. Within the blowing half cycle (t ∗ = [0, 1/2]), the values of CQ are positive; and negative values of CQ correspond to the suction half cycle (t ∗ = [1/2, 1]). In Fig. 8(b)–(j), the angular positions of the boundary layer separation and reattachment (θs ) are measured in counterclockwise direction from the rear stagnation point of the cylinder. At t ∗ = 0 two standing recirculation regions are clearly observed at the top and the bottom openings of the slit. Meanwhile, the pressure at the top opening is lower than that at the bottom opening. This pressure

difference is the primary driving force pushing the fluid out of the top opening. Furthermore, a separation bubble A1 has been already formed at the upper rear surface (θs = 15°–54°) of the cylinder. During the blowing half cycle, the velocity magnitudes increase in the outer region of the boundary layer and decrease in the inner region of the boundary layer (Fig. 7(b)). In Fig. 8(c), the separation bubble A1 advances and extends further downstream (θs = 6°–56°) along the upper-rear surface. At the same moment, the fluid at the bottom slit opening undergoes a suction phase and a small-scale vortex B1 (θs = −25° to − 30°), of the same sign as vortex A1, has been formed near the bottom-rear surface. In Fig. 8(d), the vortex A1 continues to grow in size and strength; it also moves further downstream to interact with the vortex B1 near

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109

Fig. 11. Time-mean streamline patterns around cylinders for Re = 200 and S /D = 0–0.3. The vortex formation length (Lw ) and mean angle of the boundary-layer separation

(θ¯s ) are also depicted.

the rear stagnation point. In Fig. 8(e)–(f), CQ from the top opening reaches the maximum and the vortex A1 eventually merges with the vortex B1 into a large-scale CW rotating vortex H. Within t ∗ = [4/16, 8/16] (Fig. 8(f)–(j)), the amount of CQ at the top opening starts to decrease to zero; meanwhile the CW rotating vortex H starts to shed downstream. Along the lower-rear surface, a vortex of CW rotation is formed near θs = −45° to − 61°. The size of vortex Sb first reduces and then increases. This separation bubble Sb remains nearly stagnant but grows in size, extending in the range

of θs = −15° to − 53° along the rear lower surface within the rest of the first half cycle (t ∗ = [4/16, 8/16]). In the near-wake region, the low-pressure regime always accompanies the clockwise vortex H. In the second half cycle, the development and evolution of boundary-layer flow along the rear lower surface of the cylinder repeats exactly the same scenarios as that of the rear upper surface in the first half cycle. The average location of boundary-layer separation occurs at θ¯s = 67° for Re = 200 and S /D = 0.15, and

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that for S /D = 0 occurs at θ¯s = 90°–95°. It is clear from Fig. 8 that the blowing of the slit flow adds to the streamwise momentum and significantly delays the boundary-layer flow separation, and that the vortex formation length is largely reduced. 3.2.2. Incipience of separation and reattachment of boundary-layer flow Fig. 9 illustrates the time-mean streamlines and velocity profiles of the boundary-layer flow along the rear upper surface of the non-slit cylinder. The streamline pattern in Fig. 9(a) shows only one primary clockwise recirculation zone with a separation point (S3) and a three-way saddle point (S1). Fig. 9(b) and (c) depict the non-dimensional tangential velocity profiles Uθ∗ in the boundary layer along the rear upper surface. The tangential velocity was transformed from the streamwise and transverse velocity components at each grid point in the flow domain. Notably, Uθ∗ was normalized by the local maximum tangential velocity at each specific angular position, such that the dimensionless quantity of Uθ∗ approached unity at a distance far away from the cylinder surface. Fig. 9(b) and (c) present the velocity profiles during the processes of separation (SEP) and reattachment (REA) of the boundary layer, respectively. For instance, the average location of the incipient separation occurs at θ = 67°. In Fig. 9(c), all the velocity profiles share the common features of reversed flow near the cylinder surface. In Fig. 10(a), the corresponding time-mean streamline pattern shows one CW primary recirculation region accompanied by another small CCW recirculation region near the rear stagnation region. Three critical points are found in Fig. 10(a), namely, S3 denotes the separation point and two three-way saddle points S2 and S1 represent the time-mean reattachment point and the rear stagnation point, respectively. In Fig. 10(b) and (c), the average incipient location of the boundary-layer flow separation occurs at θ = 62°, and that of the reattachment occurs at θ = 34° for S /D = 0.15. Figs. 9 and 10 clearly indicate that the boundarylayer flow separation is delayed significantly by the presence of the normal slit. Further, the time-mean boundary-layer velocity profiles at θ = 70° are compared in Fig. 10(d) for the slit cylinders (S /D = 0.15) with and without a block at the center. Evidently, the velocity magnitude in the boundary layer is always large for the slit cylinder without a block. As shown in Fig. 6(b,d), the spectral amplitude of the fundamental harmonic is least two orders of magnitude larger for the slit cylinder without a block than that with a block. For the slit cylinder without a block, the remarkably large perturbation amplitude at the slit opening creates an oscillating boundary-layer velocity profiles in Fig. 7(b,c) which is analogue to that of the Stokes’ flow with oscillating flow over a stationary flat plate. As reviewed by Riley [37], for a viscous fluid, a steady streaming velocity component will be developed within the oscillatory boundary layers. In Fig. 10(d), the large magnitude of the velocity for the slit cylinder without a block is primarily caused by the steady streaming effect in the oscillatory boundary layer. 3.2.3. Mean location of the boundary-layer separation and vortex formation length In Fig. 11(a), the vortex formation length (Lw ) is defined as the streamwise distance from the cylinder center to the location where the four-way saddle point S4 is formed in the time-mean streamline pattern. The time-mean angular position of the boundarylayer flow separation is denoted as θ¯s . To minimize the errors of Lw and θ¯s , the time-mean streamline patterns are better to select complete cycles while the total time elapse is long enough during averaging. In Fig. 11, all the time-mean streamline patterns are symmetrically distributed about the wake center line (Y /D = 0). Though the streamline patterns along the rear surface of the

Fig. 12. (a) Vortex formation length and (b) mean angle of the boundary-layer separation as a function of the slit ratio (S /D) for the slit and non-slit circular cylinder at Re = 200.

cylinder are different at various S /D, the distributions of the primary recirculation region are basically the same except the locations of the saddle point S4. Some distinctions in comparison with the non-slit cylinder are outlined in Fig. 11. As the S /D becomes large, the size of the primary recirculation region decreases and the vortex formation length Lw becomes shorter. In addition, the time-mean angular position of the boundary-layer separation θ¯s is delayed farther downstream along the rear surface of the slit cylinder. In Fig. 12(a) and (b), the variations in the vortex formation length and the time-mean separation locations are illustrated as functions of S /D. Evidently, both the vortex formation length and the time-mean locations of boundary-layer separation decrease in the same trend as S /D increased. By definition in this study, a small angular position implies that the boundary-layer flow separation is delayed farther downstream along the rear surface of the slit cylinder. The same decreasing trend of Lw and θ¯s in Fig. 12 clearly demonstrates that the shorter vortex formation length is mainly caused by the remarkable delay of the boundary-layer separation along the upper-rear surface of the slit cylinder. 3.3. Primary lock-on of the wake at various slit widths Based on the discussions in Sections 3.1 and 3.2, the perturbation of periodic blowing/suction at the slit opening is analogous to ZNMF perturbation to the boundary layer to enhance the airfoil lift. The frequency and amplitude responses of the slit flow and the wake flow that links with the primary lock-on character of the wake flow behind the slit cylinder will be discussed in the following. 3.3.1. Frequency response within fa /fso > 1 Fig. 13 shows the typical frequency spectra of the CQ (left column) and the CL (right column) for S /D = 0.09, 0.15, and 0.3, and Re = 200. For the slit width studied herein, all the spectra show a predominant peak at the same fundamental frequency fs which are slightly higher than that behind the non-slit cylinder (Fig. 14(a)). Since the lift force is induced by the shedding vortex street, the CL represents the integrated performance of the wake on cylinder. Also, the CQ denotes the overall perturbation at the slit opening. The frequency (fa ) of periodic blowing/suction at the slit opening is considered as the exciting frequency and the shedding frequency (fs ) behind the slit cylinder is the responding frequency. Coincidence of the fundamental frequency indicates that the wake

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a

b

c

d

e

f

111

Fig. 13. (a,c,e) Spectra of the volumetric flux coefficient through the slit for S /D = 0.09, 0.15, and 0.3. (b,d,f) Corresponding spectra for the lift coefficient for S /D = 0.09, 0.15, and 0.3. The primary large-scale vortex shedding frequency is fs .

and the periodic perturbation at the slit opening are locked to each other [38,39]. Though the fundamental frequencies in the left column are exactly the same as those in the right column for all the slit ratios studied, the contents of the corresponding spectra are quite different and worthy discussing. Since the wake is considered as a Van-der Pol oscillator with cubic nonlinearity, all the spectra of CL (right column) exhibit discrete peak especially at the fundamental and the third harmonics. The magnitudes of other higher harmonics are negligibly small. On the left column, the spectra of CQ share common features as the velocity spectrum in Fig. 6(d). The remarkably enlarged spectral amplitude of CQ at the fundamental and the third harmonics are strong evidences showing the upstream induction coming from the shedding vortex street. The spectra in Fig. 13 further confirm two important issues. First, the perturbation frequency at the slit opening is locked to that of the shedding vortex street. Second, the spectra of CQ clearly reveal that the upstream induction is mainly caused by the shedding vortex street. The second point bridges the feed-

back path and completes the closed-loop synchronization for the primary lock-on of the wake depicted in Fig. 1(a). Fig. 14(b) illustrates the relation between fa /fs and fa /fso , where fso denotes the shedding frequency of a non-slit cylinder at the same Reynolds number. Plotting the frequency ratios fa /fs versus fa /fso listed in Table 1 will yield Fig. 14(b) in which all the values of fa /fs equal 1.0 while the values of fa /fso range from 1.01 to 1.182. This clearly indicates that the primary lock-on of the wake behind the slit cylinder occurs only within the range of fa /fso > 1.0 for all slit width studied herein. 3.3.2. Volumetric flow rate through slit Fig. 15(a)–(c) show the time histories of the dimensionless volume flow rate across the slit opening at S /D = 0.03, 0.15 and 0.3 for Re = 200. The cyclic variations of CQ are all periodic in nature, and the oscillating amplitude increases with increasing slit ratio (S /D). The oscillating amplitudes at S /D = 0.15 and 0.3 are approximately twenty times larger than that at S /D =

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a

b

Fig. 14. (a) Spectra of CL for the non-slit cylinder. (b) Spectra of fs of the wake under the periodic perturbation at frequency fa for S /D ranging from 0.03 to 0.3. The natural vortex shedding frequency of the non-slit cylinder is fso .

a

b

c

d

Fig. 15. Cyclic variations in the volumetric flux coefficient for Re = 200 and S /D of (a) 0.03, (b) 0.15, and (c) 0.3. (d) Phase diagram of CL and CD .

0.03. The remarkable differences in the oscillating amplitude result in different effects of primary lock-on; for instances, the amount of reduction in the vortex formation length and the delay of boundary-layer separation are insignificant at S /D = 0.03 in Fig. 12. 3.3.3. Amplitude responses In this section, the cyclic variations of CL and CD are employed to evaluate the amplitude responses subject to periodic blow-

ing/suction at the slit opening. In Fig. 15(d), the phase diagrams with CL and CD are presented for all the slit ratios studied. All the trajectories form closed curves in the form of a figure ‘‘8’’ oriented horizontally. Moreover, the directions of all the closed curves are the same. This implies that the wake flow structures will not show a π phase jump within this perturbation frequency range. For all the closed curves, CL oscillates around the zero value, implying that the cyclic average value of CL is zero. The oscillating amplitudes of CL increase from 0.71 to 0.81, for S /D = 0.12–0.18, then reduce down to 0.74 at S /D = 0.3. The maximum oscillating amplitude of CL occurs at S /D = 0.18. In this study, the optimal slit ratio is S /D = 0.18, which is the same as that reported in Peng et al. [17]. In addition, the cyclic average value of CD increased as S /D increased. The peak-to-peak amplitude of CD remained nearly constant at approximately 0.1.

3.3.4. Phase shift between flow structures Fig. 16 depicts the pressure contours overlaid with streamline patterns at the same instant, for instance t ∗ = 4/16 for S /D ranging from 0.12 to 0.3 and Re = 200. In Fig. 16, the location of the minimum negative pressure (L) moves slightly downstream of the top opening of the slit as the slit ratio (S /D) increases. In addition, the centers of the vortex H in the near-wake region move closer to the cylinder with increasing S /D. Since all the flow structures are selected at t ∗ = 4/16, the insignificant differences in the locations L and the vortex H clearly indicate that a small phase difference exists between the flow structures with different S /D. Since the above flow structures are phase-locked with the cyclic variation of CQ∗ , the quantitative phase difference between flow structures at different S /D can also be related in terms of the cyclic variation of the lift coefficients CL∗ . The cyclic variations of CL∗ and CQ∗ are compared in Fig. 17(a). The variations of CL∗ and CQ∗ are all periodic in nature but the curve of CQ∗ lags that of CL∗ by certain amount of phase (φf ). Since the curve CQ∗ is the reference, the difference of φf between different S /D will represent the amount of phase shift between the flow structures of different S /D. Fig. 17(b) shows that the phase lag (φf ) increases dramatically as the dimensionless parameter (β) (or slit ratio, S /D) increases within β = 0.171.06. Correspondingly, within this range of β , the phase shift between flow structures in Fig. 16(a,b) is appreciable. Whereas the phase lag remains nearly constant for β = 1.061.81. In other words, the flow structures in Fig. 16(c,d) are nearly in phase. The maximum phase shift was close to that of the primary lock-on within the frequency range of fa /fso > 1.0.

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Fig. 16. Flow structures at t ∗ = 4/16 for S /D of (a) 0.12, (b) 0.18, (c) 0.24, and (d) 0.30.

Fig. 17. (a) Normalized volumetric flux coefficient CQ∗ and normalized lift coefficient CL∗ for the slit cylinders with various S /D at Re = 200. (b) Phase difference φf between CQ∗ and CL∗ for various S /D and β at Re = 200.

3.4. Scenarios leading to primary lock-on As illustrated in Fig. 1(a), the shedding vortex street provides a periodic upstream induction to the boundary-layer flow at each slit opening because the wake flow is self-excited in nature. The periodic pressure difference between the top and the bottom slit openings drives the fluid in the slit to move back and forth with zero mass flux at each slit opening. At each slit opening, the

periodic blowing/suction serves as a perturbation source applied at the location of the boundary-layer separation. This perturbation adds extra advancing momentum in the boundary layer flow and significantly delays the boundary-layer flow separation along the rear surface of the slit cylinder. While the boundary-layer flow separation is delayed, the vortex formation length reduces, and the strength of upstream induction will be enhanced. The

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above scenarios form a closed-loop synchronization leading to the primary lock-on of the wake behind the slit cylinder. Based on the results of the frequency, amplitude responses and the phase shift between the flow structures of different S /D (Figs. 13–17), the scenarios of closed-loop synchronization lead to the primary lock-on of the wake behind the slit cylinder. One exception is that the lock-on frequency range only lies within fa /fso > 1. Compared with the frequency range of the primary lock-on in active flow control, the other part of the frequency band of lock-on is missing because the primary lock-on of the slit cylinder is upstream-induced by the wake with higher shedding frequency (or shorter vortex formation length). Since the lock-on phenomena lie only within fa /fso > 1, the flow structures with π out of phase [40,41] and a sudden jump of CL and CD [42,43] across fa /fso = 1 are not found in this study. Based on the results in Figs. 15(d), 16 and 17(b), only small phase shift between the flow structures of different S /D (or different β ) indeed occurs within fa /fso > 1. 4. Conclusions In this study, a two-dimensional flow simulation is carried out on a circular cylinder with a normal slit at Re = 200 using Ansys/Fluent software. The slit ratio (S /D) ranges from 0.03 to 0.3. The characteristics of the slit flow, the evolution of the boundarylayer flow, and the wake flows are studied by modified phaseaveraged method. It is found that the shedding vortices in the wake periodically provide an upstream induction and create a periodic pressure difference between the top and the bottom slit openings. This pressure difference drives the fluid at the slit opening to move back and forth with zero mass flux. The periodic blowing/suction at each slit opening serves as a perturbation to the boundary layer to significantly delay the boundary-layer flow separation along the rear surface of the slit cylinder and reduces the vortex formation length. The pressure difference between the two slit openings is enlarged because of the reduced vortex formation length. The upstream induction by the shedding vortex street bridges the feedback path and completes the closed-loop synchronization. The above scenarios of a closed-loop synchronization lead to the primary lock-on of the wake behind the slit cylinder. The frequency range of primary lock-on lies only within fa /fso > 1 because this lock-on is upstream induced by the wake with higher shedding frequency (or shorter vortex formation length). The optimal slit ratio is S /D = 0.18 at Re = 200. Acknowledgments A significant proportion of the simulations is performed at the CFD Laboratory of Department of Mechanical Engineering in the National Chung Hsing University. We cordially thank our colleagues and classmates for their support and assistance. We also appreciate the financial support from the Ministry of Science and Technology (MOST) under Grant Number NSC-102-2221-E-005032-MY3. References [1] M. Gad-el-Hak, Flow Control: Passive, Active, and Reactive Flow Management, Cambridge University Press, 2000. [2] H. Zhang, Y. Huang, Z. Sun, A study of mass flow rate measurement based on the vortex shedding principle, Flow Meas. Instrum. 17 (2006) 29–38. [3] A. Venugopal, A. Agrawal, S. Prabhu, Review on vortex flowmeter—Designer perspective, Sens. Actuar. A Phys. 170 (2011) 8–23. [4] G. Pankanin, What is the role of the stagnation region in Karman vortex shedding? Metrol. Meas. Syst. 18 (2011) 361–370.

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