Flow around a circular cylinder attached with a pair of fin-shaped strips

Ocean Engineering 190 (2019) 106484

Contents lists available at ScienceDirect

Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Flow around a circular cylinder attached with a pair of fin-shaped strips Hongjun Zhu a, b, *, Tongming Zhou b a b

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, 610500, China Department of Civil, Environmental and Mining Engineering, The University of Western Australia, Crawley, WA, 6009, Australia

A R T I C L E I N F O

A B S T R A C T

Keywords: Fin-shaped strips Placement angle Hydrodynamic coefficient Flow structure Vortex shedding

The effect of two symmetrically distributed fin-shaped strips on hydrodynamic forces and flow structures of a circular cylinder is numerically investigated at low Reynolds numbers of 60–180. The angular position (θ), defined as the angle between the starting point of the strips and the front stagnation point of the cylinder, is varied from 30� to 90� . It is observed that the boundary layer separation is tripped by the strips, altering the pressure distribution and forming a recirculation region behind the strips. As a result, the drag and lift co­ efficients are increased significantly. At θ ¼ 30� and 40� , the boundary layer reattaches on the cylinder surface after the first separation from the strip tip, followed by the second separation. However, when θ is larger than 50� , the separation point is fixed at the sharp corner of the strips. Compared to θ � 40� and θ � 70� , shorter wake formation length and larger wake width are observed at θ ¼ 50–60� , resulting in the larger drag force and faster vortex shedding. The strips placed at θ ¼ 40–70� have more contribution to the alteration of flow structure than those positioned at θ ¼ 30� or 80� –90� . At Re ¼ 180, placing the fin-shaped strips at 60� maximally increases the lift and drag coefficients by 40.03% and 16.98%, respectively, in comparison with the bare cylinder.

1. Introduction The unsteady viscous fluid flow past a circular cylinder has long been a topic of interest in the fields of fluid mechanics and fluid-structure interaction due to its fundamental and practical significance. Both experimental measurements and numerical simulations have confirmed that the instability of the flow wake behind a circular cylinder occurs when Reynolds number, Re (�UD/ν, where U is the free stream velocity, D is the diameter of the cylinder and ν is the fluid kinematic viscosity), is higher than 47 (Henderson, 1997). The vortex shedding identified by definite frequencies leads to time-dependent drag and lift forces acting on circular cylinders. Consequently, vibration, known as vortex-induced vibration (VIV), is induced over a range of Re, which is termed as the lock-on region or the synchronization region. As VIV influences the stability and shortens the structure life, it is usually treated as a destructive phenomenon in the past decades. A variety of methods have been proposed to suppress VIV actively and passively (Zdravkovich, 1981; Rashidi et al., 2016; Zhou et al., 2011; Zhu and Yao, 2015; Zhu et al., 2015, 2017). Instead of VIV suppression, the vibration energy of cylinders can be converted to electrical energy. The first vibration energy harvester based

on the concept of VIV was proposed by Bernitsas et al. (2008), which was termed as a VIVACE (vortex-induced vibration for aquatic clean energy) converter that can be used in flow speed as slow as 0.4 m/s (Lee and Bernitsas, 2011). After that, Bernitsas and his group (Ding et al., 2016; Kinaci et al., 2016; Park et al., 2013) improved the performance by symmetrically attaching a pair of rough strips on the windward surface of a circular cylinder as passive turbulence controls (PTC). It was observed that the vibration amplitude was significantly increased, which is beneficial for energy harvesting. Chang et al. (2011) investi­ gated the effects of the location, surface coverage and size of rectangular strips on the response amplitude at high Reynolds numbers. Both the suppression and amplification of vibration were observed via changing the location of strips. In our recent numerical study (Zhu et al., 2018a), it was found the response amplitude was sensitive to the geometry and the placement angle of strips. At Reynolds number of 30,480–304800, a significant vibration enhancement was achieved by placing rectangular strips at angular position of 20� –60� . The similar method of tripping boundary layer is achieved by attaching wires or rods on the cylinder surface as protrusions. Alam et al. (2003) experimentally investigated the effects of tripping rods on flow characteristics and fluid forces acting on a circular cylinder at a

* Corresponding author. State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, 610500, China. E-mail addresses: [email protected], [email protected] (H. Zhu). https://doi.org/10.1016/j.oceaneng.2019.106484 Received 17 March 2019; Received in revised form 1 September 2019; Accepted 20 September 2019 Available online 25 September 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 1. (a) Computational domain, (b) definitions of symbols, (c) zoom-in view of grids around the cylinder, and (d) grid structure (o-xy grid system).

2

Ocean Engineering 190 (2019) 106484

H. Zhu and T. Zhou

Reynolds number of 5.5 � 104. It was found that a stronger vortex was created behind the cylinder when the rods were placed at 45� –60� , while the fluid forces were reduced at the angular position of 30� . The sup­ pression of vortex shedding was further confirmed in their later exper­ iments with tripwires placed at the same angular position (Alam et al., 2016). Kim et al. (2018) experimentally studied the vibration of an elastically supported circular cylinder with a control rod installed in its wake. They reported that the vibration response was highly sensitive to the rod position, and galloping was generated by placing the rod at xc ¼ 0.4D and yc ¼ 0–0.4D or xc ¼ 0.4–0.8D and yc ¼ 0.4D (xc and yc are the streamwise and lateral distances measured from the rear stagnation point, respectively). Quadrante and Nishi (2014) also verified the importance of the control wires’ position. The hydrodynamic forces were diminished when tripping wires were placed at angular position ranging from 20� to 52.5� or greater than 97.5� , while the forces were increased with the angular position ranging from 52.5� to 97.5� . Nevertheless, the aforementioned numerical and experimental studies were carried out at higher Reynolds numbers of 103–106. Although PTC was proved effective in enhancing cylinder vibration, the intrinsic links between the flow characteristics and the associated changes in hydrodynamic forces need to be further revealed, especially at lower Reynolds numbers. This knowledge could be important in creating new controls for vibration amplification. Additionally, the laminar boundary layer is relatively hard to be altered or tripped due to its larger thickness as compared to a turbulent layer. Thus, to some extent, laminar flow can be treated as an extreme condition for test. If the boundary layer of a circular cylinder is obviously affected at a suf­ ficiently low Reynolds number, the alteration would be more vigorous for high Re numbers. In this work, fin-shaped strips are proposed with the intent of increasing the hydrodynamic forces and altering the vortex shedding. The primary aim is to examine the effects of fin-shaped strips on fluid forces, flow structure and vortex shedding at low Reynolds number range of 60–180 to get further insight into the interaction between fluid and structure under a controlled situation. It is expected that numerical simulations at low Reynolds numbers can provide detailed flow struc­ tures in the boundary layers of the cylinder surface, which is difficult to measure experimentally. This knowledge is invaluable in explaining the effect of control on the flow structure. To this end, the flow around a circular cylinder with a pair of fin-shaped strips is investigated using direct numerical simulation (DNS). The remainder of this paper is organized in the following manner. In Section 2, the governing equations, computational domain and model validation are introduced. The hydrodynamic coefficients and Strouhal number are discussed in Section 3, while the vortex formation and flow structures are analyzed in Section 4. Finally, major conclusions are drawn in Section 5.

2. Numerical model 2.1. Governing equations and numerical method As the wake transition of a circular cylinder occurs beyond Re ¼ 180 (Williamson, 1996a, 1996b; Jiang et al., 2016), the two-dimensional (2D) simulation was applied in capturing the 2D vortex shedding in this work. In 2D simulation, the cylinder is seemed as a rigid body so that the numerical results cannot be directly extrapolated to a flexible cyl­ inder, whose ambient flow is more complicated and is not considered in this work. However, the 2D numerical results could identify the char­ acteristics of flow over an infinite rigid cylinder in this low Reynolds number range of 60–180. Thus, the actual aspect ratio can be deter­ mined by the combined parameters such as the flow velocity, the required power density and the array configuration, as reported in previous literature (Zhu and Gao, 2017; Zhu et al., 2018a, 2018b). The DNS were carried out with the open source Computational Fluid Dynamics (CFD) solver OpenFOAM. The unsteady, incompressible Navier-Stokes (NS) equations, including the continuity and momentum equations expressed as follows, were directly solved.

Description

Number of cells

CD

CLrms

St

M1

the first layer height 0.009D cell expansion ratio of 1.05 the first layer height 0.003D cell expansion ratio of 1.05 the first layer height 0.003D cell expansion ratio of 1.02 the first layer height 0.001D cell expansion ratio of 1.02

87468

1.5721

0.6272

0.2362

128654

1.5683

0.6219

0.2315

152040

1.5657

0.6154

0.2289

186240

1.5650

0.6151

0.2285

M2 M3 M4

(1)

∂ui ∂ui 1 ∂p ∂2 ui ¼ þν þ uj ∂t ∂xj ρ ∂xi ∂xj ∂xj

(2)

where xi is the Cartesian coordinate in i direction, ui is the velocity component in the direction xi, t is the flow time, p is the pressure, ρ is the density of fluid and ν is the kinematic viscosity. Finite volume method (FVM) was employed to discretize Eqs. (1) and (2). The pressure-velocity coupling was solved with the PISO (pressure implicit with splitting of operators) algorithm (Jiang et al., 2016). The spatial discretization of the convective term and diffusion term were performed by the fourth-order cubic scheme and second-order linear scheme, respectively. The time derivative term was discretized using a blended scheme consisting of the second-order Crank–Nicolson scheme and a first-order Euler implicit scheme (Jiang et al., 2017). 2.2. Computational domain and boundary conditions A rectangle computational domain of 50D (in the streamwise direc­ tion) � 40D (in the transverse direction), as shown in Fig. 1, is adopted for 2D DNS. The domain size is determined based on the 2D domain size dependence study for a circular cylinder conducted by Jiang et al. (2016). The distance between the upstream boundary and the center of the circular cylinder (the original point) is 20D. The two lateral boundaries are also 20D away from the cylinder center so that the blockage ratio is around 2.5%. As depicted in Fig. 1, the circular cylinder is symmetrically attached by a pair of fin-shaped strips. Apart from the arc surface covering part of the circular cylinder, the fin-shaped strip is enclosed by a cambered surface and the subsequent plane perpendicular to the cylinder surface. Each strip, facing the incoming flow, covers 30� surface on each side of the cylinder, and the radius of the cambered surface is R/D ¼ 3, resulting in the maximum thickness of δ/D ¼ 0.062 for each strip. The location of strips is defined by the placement angle (θ) which is measured from the forward stagnation point to the upstream edge of the cambered surface. It is worth noting that the projected width (Dp) of the equipped cylinder perpendicular to the incoming flow varies with the location of strips. The maximum projected width (Dp ¼ 1.124D) occurs at θ ¼ 60� , while the projected width equals to the cylinder diameter at θ ¼ 30� and θ ¼ 90� . At the inlet boundary, a uniform flow with velocity of u∞ is specified in the x direction. At the outlet, a zero normal gradient condition and a reference value of zero are applied for the velocity and the pressure,

Table 1 Mesh dependence test for a cylinder with 60� fin-shaped strips at Re ¼ 180. Mesh

∂ui ¼0 ∂xi

3

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

1.05. A grid independence test was undertaken first. Four different grids for the cylinder with 60� fin-shaped strips at Re ¼ 180 were checked. Table 1 summarized the test results, illustrating the dependence of the time-averaged drag coefficient (CD ), the root-mean-square lift coeffi­ cient (CLrms) and the Strouhal number (St) on the grid size. The drag and lift coefficients are computed from the surface distribution of pressure and shear stress. Obviously, the hydrodynamic coefficients and the Strouhal number St converge at M3. The differences between M3 and M4 is within 0.2%, demonstrating that further increase of mesh reso­ lution has negligible effect on the results. Therefore, the grid size of M3 was chosen in simulation. The non-dimensional time-step (u∞Δt/D) is kept below 0.0005. As a result, the Courant number at such mesh res­ olution and time step is less than 0.2 in the whole domain. 2.4. Model validation The numerical model used in this study was validated with the simulation results of flow past a bare circular cylinder over a Re range of 60–180. The predicted time-mean drag coefficient, base pressure coef­ ficient and Strouhal number of lift coefficient are shown in Fig. 2, together with the experimental and numerical results reported previ­ ously. The base pressure coefficient is defined as: Cpb ¼

pb p∞ � ρu2∞ 2

(3)

where pb is the time-averaged pressure at the rear stagnation point of the cylinder and p∞ is the reference pressure at the inlet. It is seen from Fig. 2 that the drag coefficient CD decreases sharply as Re increases from 60 to 120, then remains a relatively stable value as Re further increases to 180. It agrees well with the numerical results re­ ported by Henderson (1995). The difference between the present results and that of Rajani et al. (2009) is within 1.8%. As the Reynolds number is between the onset of vortex shedding (Re ¼ 47) and the onset of three dimensionality (Re ¼ 188.5 � 1), the absolute value of the base pressure coefficient increases gradually, which is consistent with the reported numerical results (Henderson, 1995; Barkley and Henderson, 1996; Rajani et al., 2009) and the experimental data (Williamson and Roshko, 1990). The predicted St-Re relationship, showing an increase with growth rate gradually reduced in the considered Re range, is also in good agreement with the experimental and numerical results in previous literature (Williamson and Roshko, 1990; Henderson, 1995; Barkley and Henderson, 1996; Rajani et al., 2009). 3. Hydrodynamic coefficients and strouhal number 3.1. Drag and lift coefficients Fig. 3 shows the temporal evolution of the hydrodynamic coefficients of the equipped cylinder at Re ¼ 160 compared to the bare cylinder. In all the considered cases, the drag and lift coefficients attain a stable fluctuation at t* ¼ 100 (t* is the dimensionless time defined as t* ¼ tu∞/ D), indicating the flow reaches a periodic but statistically stable state. It is clearly seen that both the drag and lift forces increase when a pair of fin-shaped strips are placed in the considered placement angles. At angular position of θ ¼ 40� –80� , the larger hydrodynamic coefficients may be attributed to the increased projected width (Dp) facing the incoming flow. The larger the projected width, the more obvious in­ crease in the drag and lift coefficients. Nevertheless, there is a slight increase of hydrodynamic coefficients at θ ¼ 30� and 90� whose pro­ jected width is not changed. This result indicates that the flow around the cylinder is influenced by the strips, which will be discussed in Sec­ tion 4. The dependence of hydrodynamic coefficients on Re at different placement angles is shown in Fig. 4. The time-averaged drag coefficient

Fig. 2. Comparison of the present results with those reported previously: (a) time-mean drag coefficient, (b) time-mean base pressure coefficient, and (c) Strouhal number.

respectively. The slip-wall boundary condition (∂u =∂y ¼ 0 and v ¼ 0) is applied at the two lateral boundaries. A non-slip boundary condition is applied on the surface of the cylinder and the strips. 2.3. Computational mesh As shown in Fig. 1, an enlarged concentric circle around the cylinder is employed to create an O-xy grid system near the cylinder, while the rest regions are tessellated with rectangular grids. The perimeter of the cylinder is equally discretized with 218 nodes. The radial size of the first layer of mesh next to the cylinder is 0.002D, meeting the requirement of yþ<0.25. The cell expansion ratio in the whole domain is kept below 4

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 3. Time histories of (a) the drag coefficient and (b) the lift coefficient for Re ¼ 160 at different placement angles.

the projected width, which was also observed by Quadrante and Nishi (2014) in their experimental study on the vibration amplification of a circular cylinder by attaching tripping wires. Moreover, compared to θ ¼ 90� where the strips cover the cylinder surface from 90� to 120� , placing strips at θ ¼ 30� with the cylinder surface 30� –60� covered has more distinct influence on the drag force. When the strips are located between 40� and 80� , beyond the turning point on the CD -Re curve, CD

(CD ) at θ ¼ 30� and 90� reveal the same trend as the bare cylinder, i.e. a rapid reduction occurring over 60 � Re < 120 followed by a relatively stable value over 120 � Re � 180. The CD -Re curves of these three cases (bare cylinder, θ ¼ 30� and θ ¼ 90� ) are almost parallel, with the largest magnitude at θ ¼ 30� followed by that at θ ¼ 90� . As the projected width at θ ¼ 30� and 90� is the same as the bare cylinder, it further demon­ strates that the drag coefficient could be increased without increasing 5

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 4. Variation of (a) the time-mean drag coefficient and (b) the root-meansquare lift coefficient with Reynolds number at different placement angles. Fig. 5. The Lissajous figure of the hydrodynamic coefficients for (a) cylinders with fin-shaped strips at Re ¼ 160 and (b) the cylinder with 60� strips at different Reynolds numbers.

increases gradually instead of keeping a stable value. It indicates that with increasing Reynolds number the growth of pressure drag is larger than the drop in shear stress in this Re range (Henderson, 1997). The maximum growth rate appears at θ ¼ 60� , followed θ ¼ 50� , but the drag coefficients of the two cases are very close when Re � 140. At Re ¼ 180, CD of the cylinder with strips attached at θ ¼ 50� and 60� reaches the maximum value of 1.566, which is 16.98% larger than that of bare cylinder, followed by θ ¼ 70� and θ ¼ 40� and then θ ¼ 80� . The cylinder with 60� strips has the maximum projected width Dp ¼ 1.124D, resulting in the maximum pressure difference and hence the maximum drag. The cylinder with 50� strips has the same projected width as θ ¼ 70� (Dp (50� ) ¼ Dp(70� )), and similarly Dp(40� ) ¼ Dp(80� ). However, the drag coef­ ficient at θ ¼ 50� is larger than that at θ ¼ 70� , and the drag coefficient at θ ¼ 40� is larger than that at θ ¼ 80� . As the coverage angle of each strip is 30� , the sharp corner appears at 90� for the 60� strips (60� þ30� ¼ 90� ). Thus the sharp corners of 40� strips and 50� strips locate at the forward surface of the circular cylinder, while the corners of 70� strips and 80� strips appear at the rear surface. It illustrates that the location of sharp corner has a significant impact on the drag force, which coincides well with the observation of rectangular strips in Zhu et al. (2018a). As shown in Fig. 4, the root-mean-square (RMS) lift coefficient in­ creases gradually with Re. Compared to the bare cylinder, the growth rate becomes larger after the cylinder is attached with fin-shaped strips. Similar to the drag coefficient, the maximum growth rate occurs at θ ¼ 50� and 60� , followed by θ ¼ 70� , 40� , 80� , 30� and 90� in sequence.

For 60� strips at Re ¼ 180, the lift coefficient is increased by 40.03% as compared to the bare cylinder. Even for 90� strips, there is an increase of 4.1% in the lift force. Therefore, at Re ¼ 180, placing the fin-shaped strips at 60� maxi­ mally increases the lift and drag coefficients by 40.03% and 16.98%, respectively. In contrast, Quadrante and Nishi (2014) reported that positioning tripping wires at 75� achieved the maximum increases of lift and drag coefficients by 63% and 44%, respectively. It is noted that the diameter of the control wires is 0.12D, about twice the maximum thickness of the employed strips in this work. Additionally, their ex­ periments were carried out at higher Reynolds number of 3.45 � 103–2.04 � 104. Thus, although the increases of hydrodynamic forces are relatively low, the performance could be improved by thick­ ening the strips and it has a great potential at higher Re as the hydro­ dynamic coefficients keep growing with increasing of Re. Fig. 5 depicts the Lissajous figure of the lift coefficient with the drag coefficient, which was usually used to infer the motion trajectory of the structure (Sahin and Owens, 2004; Wu and Shu, 2011; Wu et al., 2014). Figure eight trajectories are clearly observed, but they are not symmetric with respect to the y-axis, indicating an existence of phase angle be­ tween the hydrodynamic coefficients along the two directions. As the frequency ratio of drag coefficient to lift coefficient is 2:1, the phase 6

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 6. Comparison of time-averaged pressure coefficient at different Reynolds numbers.

7

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 7. Time-averaged base pressure coefficient at different Reynolds numbers.

angle is in range of π/8–π/4. The minimum phase angle occurs at 60� strips, which is favorable to the vibration response (Vandiver et al., 2009). As Re increases, the growth of CD is faster than CL so that the figure eight becomes more asymmetric (Fig. 5b). The phase angle is nearly increased to π/4 at Re ¼ 180. Overall, both the drag and lift forces attain the greatest promotion by attaching a pair of fin-shaped strips to the cylinder surface at placement angle of 60� , followed closely by the 50� strips. These results provide a reference to enhance the flow-induced vibration and thereby the improvement of energy harvesting. 3.2. Pressure coefficient Fig. 6 shows the variation of the time-mean pressure coefficient. The position angle along the cylinder surface is measured from the front stagnation point, i.e. β ¼ 180� represents the rear stagnation point. It is clearly seen that the pressure distribution around the cylinder surface is altered by the introduction of fin-shaped strips. It is embodied in three aspects: the occurrence of the minimum pressure, the pressure recovery behind the strips and the turning point of Cp -β curve in the downstream of sharp corners. For the bare cylinder, the minimum pressure appears around β ¼ 86� at Re ¼ 60, corresponding to the maximum velocity close to the cylinder surface, and it moves forward slightly to β ¼ 82� as Re increases to 180. It is related to the shift of boundary layer separation point, which moves upstream with increasing Reynolds number (Braza et al., 1986; Rajani et al., 2009). By placing the strips, the occurrence of the pressure trough is affected. When θ is smaller than 70� , the minimum pressure occurs at the sharp corners of strips, indicating that the boundary layer develops along the arc surface of strips and reaches the maximum velocity at the corners. Nevertheless, when the placement angle is increased to 80� , the occurrence of the minimum pressure shifts back to 83� , and there is another turning point presenting at the sharp corners with a relatively large pressure value. At θ ¼ 90� , the pressure trough occurs at the same location as the bare cylinder, and the transition of pressure at the sharp corners becomes smooth. Compared to the bare cylinder, the pressure recovery rate becomes faster after the pressure valley, especially for 30� strips. As the place­ ment angle increases from 30� to 80� , this recovery rate is gradually reduced to the same level as the bare cylinder. The main reason is the pressure drop generated by the strips is weakened as strips move back. It indicates that the strips trip the boundary layer and hence alter the flow structure at the backward-facing step, while the change is decreased with the increase of the angular position. It is noted from Fig. 6 that the Cp -β curve has an obvious turning point after the rapid recovery of pressure for the cylinder with 30� strips.

Fig. 8. Power spectral density functions of CL at Re ¼ 160 for different place­ ment angles.

This turning point is also observed from the curves of 40� strips except Re ¼ 180. It is mainly attributed to the reattachment of boundary layer, which will be discussed later. Fig. 7 shows the dependence of the base pressure on Reynolds number. These results reveal the same trend as those of lift coefficient (Fig. 4b). The minimal base pressure occurs at θ ¼ 60� (Cp ¼-1.2318 at Re ¼ 180), followed by θ ¼ 50� (Cp ¼-1.2305 at Re ¼ 180). The low base pressure explains the large time-mean drag, as shown in Fig. 4. Addi­ tionally, as compared to the bare cylinder, the reduction of base pressure is accelerated after a critical Re (80 for 50� and 60� strips), illustrated by a steeper slope of the curves. Henderson (1995, 1997) pointed out that the decrease in shear stress of the bare cylinder is almost equal to the increase in pressure drag so that the CD -Re curve is relatively flat in 120 � Re � 180 (Fig. 2a). As the reduction of base pressure becomes larger, the pressure drag acting on the cylinder with 60� and 50� strips is increased more than the decrease of shear stress, resulting in the in­ crease in the mean drag coefficient. Similarly, the trend of Cp explains the variation of CD in other cases.

3.3. Strouhal number The time history of lift coefficients is analyzed by FFT (Fast Fourier 8

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 9. Dependence of Strouhal number on Reynolds number for different placement angles.

Transform) algorithm and the peak frequency of the spectrum is defined as the Strouhal number (St ¼ fsDp/u∞, where fs is the peak frequency of the spectrum and Dp is the projected width). Fig. 8 plots the spectra of the lift coefficient for the cylinder with strips located at different angles at Re ¼ 160. For the bare cylinder, the Strouhal frequency is 0.186, consisting well with the value in previous literature. When the cylinder is attached with fin-shaped strips, the St is increased but the growth varies with the placement angle. The largest St occurs at θ ¼ 60� , indi­ cating the fastest vortex shedding, followed by the cylinder with 50� strips with f* ¼ 0.218. It illustrates that the strips placed at θ ¼ 60� and 50� not only achieve the largest fluid forces but also gain the highest shedding frequency, which are favorable to energy harvesting. A small peak at 3f* represents the third harmonic of St, which becomes more obvious as the strips move from 30� to 60� and reaches the largest one at θ ¼ 60� . The emergence of the third harmonic is associated with the recirculation region formed behind the strips and the alternating shearlayer reattachment (Younis et al., 2016; Yao and Jaiman, 2017). Fig. 9 compares the St-Re relationship for the cylinder with strips at different placement angles. For each case, the shedding frequency in­ creases smoothly with Reynolds number, indicating the installation of fin-shaped strips does not change the growth trend of St for the twodimensional time-periodic flow in this Re range. The magnitude of St at θ ¼ 30� is relatively small, as the shear layers are tripped by the strips and retarded due to the reattachment. The small St presenting at θ ¼ 90� is because the boundary layer separation point moves backward to the sharp corner of strips (120� ). Conversely, the forward transfer of boundary layer separation point leads to the large St at θ ¼ 50� and θ ¼ 60� . 4. Vortex formation and flow structures 4.1. Vortex formation

Fig. 10. Dependence of (a) vortex formation length, (b) boundary-layer sepa­ ration point and (c) wake width on Reynolds numbers for the bare cylinder.

The time-averaged flow field is analyzed to quantify the vortex for­ mation. The formation length of vortex Lf and the separation point of boundary layer are obtained from the contour line of u ¼ 0, as shown in Fig. 10 and Fig. 11. The wake width D0 is defined by the largest vertical distance between the positive and negative vorticity peaks (Jiang and Cheng, 2017) because the vorticity field directly reveals the vertical extend for the vortices to be rolled up to trigger vortex shedding (Fig. 10), where the normalized vorticity ω is defined as: � � ∂v ∂u Dp ω¼ (4) ∂x ∂y u∞

bare cylinder are consistent with the simulation results in Jiang and Cheng (2017), and the boundary-layer separation point agrees with the numerical results in Braza et al. (1986) and Rajani et al. (2009). Fig. 11 depicts the contour line of u ¼ 0 for the time-averaged flow around the cylinder with fin-shaped strips. It is worth noting that the contour line behind the 30� strips is separated by the cylinder surface, indicating the reattachment of boundary layer. The same phenomenon is observed for the cylinder with 40� strips and the cylinder with 50� strips at small Reynolds numbers. Due to the reattachment, the boundary layer separation occurs twice: the first one at the sharp corner of strips and the

As compared in Fig. 10, the formation length and wake width of a 9

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 11. The iso-contour of u ¼ 0 at different Reynolds numbers and placement angles.

second one at the cylinder surface behind the strips. In contrast, when the placement angle is larger than 60� , the separation occurs at the tips of strips although the Reynolds number is as small as 60. Since the size of the recirculation bubble behind the cylinder is determined by the sec­ ondary separation, the separation point after reattachment is used to determine the separation position α in Fig. 12 (a). It is clearly seen that the separation point shifts forward (i.e. reducing α) when the cylinder is attached with 30� strips though the boundary layer reattaches the cyl­ inder surface after the tripping of strips. Consequently, the formation length is reduced and the wake width is increased, as illustrated in Fig.12 (b) and Fig.12 (c), and hence the reduction of base pressure and the increase of drag force (Fig. 4). Similarly, the separation point of the cylinder with 40� strips moves forward except at Re ¼ 180 where the

secondary separation disappears. For the cylinder with 50� strips, the secondary separation occurs at Re ¼ 60 and Re ¼ 80, and disappears at higher Reynolds numbers. For other cases, the separation point is locked on the sharp corners of strips. Compared to θ � 70� , the separation points occurring at the tips of 50� and 60� strips locate forwardly, resulting in the shortest formation length and the widest wake width and thereby the largest drag force. For the cylinders with 70� strips and 80� strips, the boundary layer separation occurs earlier than that of the bare cylinder (Fig. 11a) so that the wake width is wider. Noted that the separation point of the cylinder with 90� strips appears later than the bare cylinder. Nevertheless, due to the outward push of shear layer by the strips, the wake width is still larger than the bare cylinder. Addi­ tionally, as shown in Fig. 12, an increase in Reynolds number leads to 10

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

velocity urms* (¼urms/u∞) for different placement angles at Re ¼ 160. There are two peaks of urms* symmetrically distributed about Y/D ¼ 0, indicating the roll-up positions of the shear layers on both sides. The lateral separation between the two peaks becomes wider after the cyl­ inder is attached with fin-shaped strips. The widest lateral separation occurs at 60� and 50� strips, followed by 70� and 40� strips, then 80� strips, and then 30� and 90� strips, consisting well with the trend of wake width. Thus this lateral separation is sometimes defined as the wake width for analysis in previous literature (Alam et al., 2018; Younis et al., 2016). The streamwise distance from the cylinder center to the peaks follows the trend of formation length, i.e. the shortest distance presents at 60� strips. While a large wake width corresponds to a large CD and CLrms, the formation length is inversely linked to these quantities (Alam et al., 2011; Wang et al., 2018). The maximum urms* increases with the increase of placement angle from 30� to 50� , but declines with further increasing from 60� to 90� . As a result, the rolling of shear layers grows strongly with the introduction of 50� or 60� strips, indicating more disturbance energy. Unlike urms*, the contours of the lateral velocity vrms* (¼vrms/u∞) only exhibit one peak on the wake centerline, as shown in Fig. 14. Nevertheless, the streamwise position of the peak vrms* moves forward and the magnitude of the peak vrms* grows correspondingly as θ in­ creases from 30� to 60� , the same trend as the peak of urms*. As θ further increases from 60� to 90� , the position of the peak of vrms* retreats along with the reduction of magnitude. This trend explains the variation of lift force (Younis et al., 2016). Fig. 15 shows the time-averaged Reynolds shear stresses u’ v’ =u2∞ for the cylinder with fin-shaped strips attached at different positions. The two large peaks, symmetrically located on each side of Y/D ¼ 0, arise from the roll-up of shear layers, while the two small peaks correspond to the reattachment of shear layers on the cylinder leeward surface. The magnitude of peak Reynolds shear stress increases, albeit slightly, as the strips move from 30� to 60� . It indicates that the momentum transport by the fluctuating velocity component is enhanced. Additionally, the position of the peaks moves closer to the cylinder base, illustrating the reduction of formation length. When the placement angle is beyond 60� , both the magnitude and position of the shear stress vary along the opposite direction. It is noted that another two tiny peaks occur near the cylinder base at θ ¼ 40� –80� , and the peaks are most apparent at θ ¼ 60� . These peaks are caused by the recirculation region formed behind the strips, which adheres to the cylinder surface. In order to further examine the separation of boundary layer, Fig. 16 shows the enlargement of the streamlines around the cylinder. It can be seen that at θ ¼ 30� and 40� , the flow separates from the corner of strips, forming a recirculation region behind the strips. The separated flow reattaches on the cylinder surface and then separates again from the cylinder surface. This phenomenon was also observed by Alam et al. (2003) in the experiments of flow past a pair of tripping rods positioned symmetrically in the windward surface of a circular cylinder. Compared to the 30� strips, the recirculation region at θ ¼ 40� elongates and the reattachment length dwindles, resulting in a relatively early separation, and thereby a wider wake width and a larger CD. At θ � 50� , the recir­ culation region integrates into the wake flow with the absence of reat­ tachment, leading to fixed separation point at the corner of strips. The instantaneous vorticity contours for different placement angles at Re ¼ 160 are compared in Fig. 17. All the considered cases display the typical 2 S (two vortices are shed per cycle, the upper vortex clockwise and the lower one counterclockwise) shedding mode, indicating that the disturbance of the strips on vortex shedding is limited in low Reynolds number. Nevertheless, the streamwise and lateral distances between two adjacent vortices vary with the placement angle of strips. As θ increases from 30� to 60� , the streamwise distance reduces whereas the lateral gap grows, in consistent with the increase of shedding frequency. The opposite trend occurring in 60� �θ � 90� also illustrates the reduction in St.

Fig. 12. Variation with Reynolds number of the separation point (a), vortex formation length (b) and wake width (c).

the reduction in the formation length and wake width, explaining the reduction of base pressure and the increase of vortex shedding frequency. 4.2. Flow structure Fig. 13 displays the contours of the RMS values of the streamwise 11

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 13. Iso-contours of root-mean-square streamwise velocities urms* at different placement angles at Re ¼ 160.

12

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 14. Iso-contours of root-mean-square transverse velocities vrms* at different placement angles at Re ¼ 160.

13

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 15. Iso-contours of time-averaged Reynolds stress u’ v’ =u2∞ at different placement angles at Re ¼ 160.

14

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 16. Streamlines and boundary-layer separation points at different placement angles at Re ¼ 160.

5. Conclusions

region is formed behind the strips. Moreover, this region at θ ¼ 30� and 40� is separated by the reattached boundary layer flow from the flow wake, leading to an obvious turning point after the pressure recovery. When the strips are placed at 40� –80� , the reduction of base pressure becomes larger as Re increases, and hence the uplift in drag force. The minimal base pressure occurs at θ ¼ 60� , resulting in the maximal drag. (3) Flow separation is locked on the sharp corner of strips for θ � 50� . The separation points at the corners of 50� and 60� strips locate forwardly, resulting in the shorter formation length and the wider wake width, and thereby the larger drag and the faster vortex shedding. Additionally, a third harmonic appears and reaches the peak at θ ¼ 60� , associated with the recirculation region and alternating shear-layer reattachment. (4) As the placement angle increases from 30� to 60� , the magnitudes of the peaks of streamwise velocity, lateral velocity and Reynolds shear stress increase and their locations move forward. A

A two-dimensional direct numerical simulation is conducted on flow around a circular cylinder with a pair of fin-shaped strips symmetrically located on its surface at Re of 60–180. The major findings are as follows: (1) Both the drag and lift coefficients are increased after the instal­ lation of strips. The larger the projected width, the higher the growth rate in fluid forces. Nevertheless, due to the tripping strips, the hydrodynamic forces rise slightly at θ ¼ 30� and θ ¼ 90� though their projected widths are the same as the bare cylinder. The maximum increases in the drag and lift, achieved by the 60� strips, are about 16.98% and 40.03% larger than that of a bare cylinder, respectively, and the phase angle between the two hydrodynamic coefficients attains the minimal value. (2) The fin-shaped strips trip the boundary layer and alter the pres­ sure distribution around the cylinder surface. A recirculation 15

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Fig. 17. Instantaneous spanwise vorticity contours at different placement angles at Re ¼ 160.

conversely trend is found as θ further increases from 60� to 90� . Therefore, the magnitude and position of the peak fluctuating velocities explain the variations in formation length and wake width, and hence the change in hydrodynamic forces.

Alam, M.M., Elhimer, M., Wang, L.J., Jacono, D.L., Wong, C.W., 2018. Vortex shedding from tandem cylinders. Exp. Fluid 59, 60. Braza, M., Chassaing, P., Minh, H.H., 1986. Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech. 165, 79–130. Barkley, D., Henderson, R.D., 1996. Three-dimensional floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241. Bernitsas, M.M., Raghavan, K., Ben-Simon, Y., Garcia, E.M.H., 2008. VIVACE (vortex induced vibration aquatic clean energy): a new concept in generation of clean and renewable energy from fluid flow. J. Offshore Mech. Arct. Eng.-Trans. ASME 130, 041101. Chang, C.C., Kumar, R.A., Bernitsas, M.M., 2011. VIV and galloping of single circular cylinder with surface roughness at 3.0�104�Re�1.2�105. Ocean Eng. 38, 1713–1732. Ding, L., Zhang, L., Bernitsas, M.M., Chang, C.C., 2016. Numerical simulation and experimental validation for energy harvesting of single-cylinder VIVACE converter with passive turbulence control. Renew. Energy 85, 1246–1259. Henderson, R.D., 1995. Details of drag curve near the onset of vortex shedding. Phys. Fluids 7 (9), 2012–2014. Henderson, R.D., 1997. Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech. 352, 65–112. Jiang, H.Y., Cheng, L., Draper, S., An, H.W., Tong, F.F., 2016. Three-dimensional direct numerical simulation of wake transitions of a circular cylinder. J. Fluid Mech. 801, 353–391. Jiang, H.Y., Cheng, L., Draper, S., An, H.W., 2017. Two- and three-dimensional instabilities in the wake of a circular cylinder near a moving wall. J. Fluid Mech. 812, 435–462. Jiang, H.Y., Cheng, L., 2017. Strouhal-Reynolds number relationship for flow past a circular cylinder. J. Fluid Mech. 832, 170–188.

Acknowledgements The research work was supported by National Natural Science Foundation of China (No. 51979238), Youth science & technology foundation of Sichuan Province (No. 2017JQ0055) and Youth scientific and technological innovation team foundation of Southwest Petroleum University (group name: the safety of deep-water pipe strings, No. 2017CXTD06). The authors appreciate the support from the Pawsey Supercomputing Center. References Alam, M.M., Sakamoto, H., Moriya, M., 2003. Reduction of fluid forces acting on a single circular cylinder and two circular cylinders by using tripping rods. J. Fluids Struct. 18, 347–366. Alam, M.M., Zhou, Y., Wang, X.W., 2011. The wake of two side-by-side square cylinders. J. Fluid Mech. 669, 432–471. Alam, M.M., Kim, S., Maiti, D.K., 2016. Flow interference between two tripped cylinders. Wind Struct. 23, 109–125.

16

H. Zhu and T. Zhou

Ocean Engineering 190 (2019) 106484

Kim, S., Alam, M.M., Maiti, D.K., 2018. Wake and suppression of flow-induced vibration of a circular cylinder. Ocean Eng. 151, 298–307. Kinaci, O.K., Lakka, S., Sun, H., Fassezke, E., Bernitsas, M.M., 2016. Computational and experimental assessment of turbulence stimulation on flow induced motion of a circular cylinder. J. Offshore Mech. Arct. Eng.-Trans. ASME 138, 041802. Lee, J.H., Bernitsas, M.M., 2011. High-damping, high-Reynolds VIV tests for energy harnessing using the VIVACE converter. Ocean Eng. 38, 1697–1712. Park, H., Kumar, R.A., Bernitsas, M.M., 2013. Enhancement of flow-induced motion of rigid circular cylinder on springs by localized surface roughness at 3�104�Re�1.2�105. Ocean Eng. 72, 403–415. Quadrante, L.A.R., Nishi, Y., 2014. Amplification/suppression of flow-induced motions of an elastically mounted circular cylinder by attaching tripping wires. J. Fluids Struct. 48, 93–102. Rajani, B.N., Kandasamy, A., Majumdar, S., 2009. Numerical simulation of laminar flow past a circular cylinder. Appl. Math. Model. 33, 1228–1247. Rashidi, R., Hayatdavoodi, M., Esfahani, J.A., 2016. Vortex shedding suppression and wake control: a review. Ocean Eng. 126, 57–80. Sahin, M., Owens, R.G., 2004. A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder. Phys. Fluids 16, 1305. Vandiver, J.K., Jaiswal, V., Jhingran, V., 2009. Insights on vortex-induced, traveling waves on long risers. J. Fluids Struct. 25, 641–653. Wang, X.W., Alam, M.M., Zhou, Y., 2018. Two tandem cylinders of different diameters in cross-flow: effect of an upstream cylinder on wake dynamics. J. Fluid Mech. 836, 5–42. Williamson, C.H.K., Roshko, A., 1990. Measurement of base pressure in the wake of a cylinder at low Reynolds numbers. Zeits. Flugwiss Weltraum. 14, 38–46. Williamson, C.H.K., 1996. Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.

Williamson, C.H.K., 1996. Three-dimensional wake transition. J. Fluid Mech. 328, 345–407. Wu, J., Shu, C., 2011. Numerical study of flow characteristics behind a stationary circular cylinder with a flapping plate. Phys. Fluids 23, 073601. Wu, J., Qiu, Y.L., Shu, C., Zhao, N., 2014. Flow control of a circular cylinder by using an attached flexible filament. Phys. Fluids 26, 103601. Yao, W., Jaiman, R.K., 2017. Model reduction and mechanism for the vortex-induced vibrations of bluff bodies. J. Fluid Mech. 827, 357–393. Younis, M.Y., Alam, M.M., Zhou, Y., 2016. Flow around two non-parallel tandem cylinders. Phys. Fluids 28, 125106. Zdravkovich, M.M., 1981. Review and classification of various aerodynamic and hydrodynamic means for suppressing vortex shedding. J. Wind Eng. Ind. Aerodyn. 7, 145–189. Zhou, T., Razali, S.F.M., Hao, Z., Cheng, C., 2011. On the study of vortex-induced vibration of a cylinder with helical strakes. J. Fluids Struct. 27, 903–917. Zhu, H.J., Yao, J., 2015. Numerical evaluation of passive control of VIV by small control rods. Appl. Ocean Res. 51, 93–116. Zhu, H.J., Yao, J., Ma, Y., Tang, Y.B., 2015. Simultaneous CFD evaluation of VIV suppression using smaller control cylinders. J. Fluids Struct. 57, 66–80. Zhu, H.J., Gao, Y., 2017. Vortex induced vibration response and energy harvesting of a marine riser attached by a free-to-rotate impeller. Energy 134, 532–544. Zhu, H.J., Liao, Z.H., Gao, Y., Zhao, Y., 2017. Numerical evaluation of the suppression effect of a free-to-rotate triangular fairing on the vortex-induced vibration of a circular cylinder. Appl. Math. Model. 52, 709–730. Zhu, H.J., Gao, Y., Zhou, T.M., 2018. Flow-induced vibration of a locally rough cylinder with two symmetrical strips attached on its surface: effect of the location and shape of strips. Appl. Ocean Res. 72, 122–140. Zhu, H.J., Zhao, Y., Zhou, T.M., 2018. CFD analysis of energy harvesting from flow induced vibration of a circular cylinder with an attached free-to-rotate pentagram impeller. Appl. Energy 212, 304–321.

17