2.1. Mathematical formulation

The present study employs the well-established dynamic model to simulate flow past the cylinders. The grid-filtered Navier–Stokes and continuity equations are the governing equations for large eddy evolution under the present assumption of the incompressible constant property flow:

where $u_{i}$ are the velocity components along the Cartesian coordinates of $x_{i}$ , $t$ is the time, $p$ is the pressure, and the overbar denotes the large (or resolved) scale obtained from grid filtering. The subgrid-scale stress tensor ( ${\it\tau}_{ij}$ ) is defined as ${\it\tau}_{ij}=\overline{u_{i}u_{j}}-\overline{u_{i}}\overline{u_{i}}$ , and accounts for the influence of subgrid scales on the resolved scales. In order to obtain a closure for (2.1), it must be modelled in terms of the resolved scales. Commonly, the subgrid-scale model is based on the eddy-viscosity model, where the anisotropic part of the subgrid-scale stress tensor is assumed to take the following form:

The most commonly used eddy-viscosity model proposed by Smagorinsky (Reference Smagorinsky1963) is

where ${\it\Delta}=({\it\Delta}_{1},{\it\Delta}_{2},{\it\Delta}_{3})^{1/3}$ is the filter width (Deardoff Reference Deardoff1970), where ${\it\Delta}_{1}$ , ${\it\Delta}_{2}$ and ${\it\Delta}_{3}$ are the dimensions of the computational cell, $C_{s}$ is the Smagorinsky constant and $|\overline{S}|=(2\overline{S_{ij}}\,\overline{S_{ij}})^{1/2}$ , with $\overline{S_{ij}}=((\partial \overline{u}_{i}/\partial x_{j})+(\partial \overline{u}_{j}/\partial x_{i}))/2$ .

The present study employs the dynamic eddy-viscosity approach (see Germano et al. Reference Germano, Piomelli, Moin and Cabot1991; Lilly Reference Lilly1992). Here, a second filter (a test filter is denoted by the tilde symbol) is defined and applied to the large eddy equation (2.1). The subgrid-scale stress and the resolved turbulent stress in the new test filter are given by

The stresses are characterized by the Germano identity,

where

Here, $\tilde{{\it\Delta}}$ is the characteristic filter width associated with the test filter. Eventually, a space- and time-dependent Smagorinsky coefficient can be obtained from the following equation:

Here, $\tilde{{\it\Delta}}$ is the characteristic filter width associated with the test filter. As is well known, in the dynamic eddy-viscosity model, the coefficient vanishes in laminar flow, where $L_{ij}$ is identically zero, which results in better prediction of the transition without ad hoc adjustments (Piomelli Reference Piomelli1999). However, such a procedure can be numerically ill conditioned as $M_{ij}$ approaches zero. Furthermore, the Smagorinsky coefficient can be locally negative, and this can give rise to serious numerical difficulties. To avoid this, an averaging of the model coefficient over statistically homogeneous directions or an ad hoc clipping procedure has been considered (Breuer Reference Breuer1998; Jordan & Ragab Reference Jordan and Ragab1998; Breuer Reference Breuer2000; Jordan Reference Jordan2002). However, these prescriptions would not work in flows that have no homogeneous directions. To overcome this problem, therefore, we simply average over the entire volume and consider the Smagorinsky coefficient to be a function of time, with no requirement for any ad hoc numerical stabilization or clipping operations. For this reason, this or a similar method has been widely adopted for solving various problems such as turbulent channel flow, flow around a cylinder and flow in an unbaffled stirred tank (Park et al. Reference Park, Lee, Lee and Choi2006; You & Moin Reference You and Moin2007, Reference You and Moin2009; Yoon, Balachandar & Ha Reference Yoon, Balachandar and Ha2009; Singh & You Reference Singh and You2013). Eventually, by contracting (2.9) and an averaging procedure over the entire computational volume (denoted by $\langle \ \rangle$ ), we obtain

This averaging procedure was successfully used for the complex problem of a mixing tank with blades (Yoon et al. Reference Yoon, Balachandar and Ha2009). This complex problem has strong vortex dynamics associated with the flow separation and the shear layers, even the interior flow. The method of LES used in this paper is basically the same as that used in Yoon et al. (Reference Yoon, Balachandar and Ha2009) which can be referred to for some more details on the method.

2.2. Numerical methodology

A second-order central difference scheme based on the finite volume method was used for spatial discretization of the governing equations. A two-step time-split scheme was adopted in order to advance the flow field in time, and continuity was satisfied by the solution of Poisson’s equation. Further details of this scheme are given in Kim & Moin (Reference Kim and Moin1985) and Zang, Street & Koseff (Reference Zang, Street and Koseff1994). The nonlinear terms are treated explicitly using the second-order Adams–Bashforth scheme, and the diffusion terms are treated implicitly using the Crank–Nicolson scheme. We derived the governing equations on a generalized curvilinear coordinate system. Near the boundary, there is no additional treatment for the LES model.

2.3. Definition of the body geometries

The three different cylindrical structures (smooth, wavy and twisted) are shown in figure 1. The geometry of the wavy cylinder is described by

where $D_{z}$ denotes the local diameter of the wavy cylinder and varies in the spanwise direction $z$ , and $a$ and ${\it\lambda}$ represent the amplitude and wavelength of the wavy surface, respectively. The mean diameter of the wavy cylinder $D_{m,W}$ is defined by

where $D_{max}$ and $D_{min}$ represent the maximum and minimum local diameters of the wavy cylinder. Moreover, $D_{max}$ , $D_{m,W}$ and $D_{min}$ are called the ‘node’, ‘middle’ and ‘saddle’, respectively, as shown in figure 2(a).

Figure 1.

The geometries of the cylinders: (a) smooth, (b) wavy and (c) twisted.

Figure 2.

Definition of the geometries for (a) the wavy and (b) the twisted cylinders.

We designed the twisted cylinder by rotating the elliptic cross-section along the spanwise direction so that the cylinder surface had a twisted spiral pattern, as shown in figure 1(c). The cross-section definition and its parameters are shown in figure 2(b), where $A$ and $B$ are the major and minor diameters of the twisted cylinder, respectively. The mean diameter of the twisted cylinder, $D_{m,T}$ , is defined as

It should be noted that the mean diameters of the wavy and the twisted cylinders are the same in the present study. Moreover, the major and minor diameters of the twisted cylinder are equal to the maximum and minimum local diameters of the wavy cylinder, respectively. The twisting phase angle ( ${\it\Phi}$ ) ranges from 0 to 360° and varies linearly along the spanwise coordinates of $z$ . The pitch of the twisted cylinder, $P_{t}$ , has a distance between turns ( ${\it\Phi}=0{-}360^{\circ }$ ) of an elliptic cross-section with twice the wavelength of the wavy cylinder ( ${\it\lambda}$ ). All types of cylinder have the same spanwise length, $L$ . The parameters used in this study were selected based on a typical experiment for flow over a wavy cylinder called ‘WY-A4’ (Lam et al. Reference Lam, Wang, Li and So2004a ,Reference Lam, Wang and So b ).

2.4. Computational domain and boundary conditions

The computational domain and boundary conditions are shown in figure 3. The grids are of the O-type with a far-field boundary at a distance of $20D$ from the cylinder in the $x$ – $y$ plane. The spanwise extent of the domain was chosen to be $4.55D$ , which corresponds to the size used by Lam & Lin (Reference Lam and Lin2008), as shown in figure 3. The inflow and outflow boundary conditions were applied to the outer surface. A uniform velocity profile was used at the inlet boundary, and a convective boundary condition, $(\partial u_{i}/\partial t)+(c\partial u_{i}/\partial x)=0$ , was imposed on the outflow region, where $c$ is the space-averaged streamwise outlet velocity. The displacement effect from the body is negligible when the inflow boundary is far enough from the body, and thus the flow is very close to potential flow. On the surface of the body, no-slip and no-penetration conditions were adopted. A Neumann-type boundary condition was used for the pressure. A periodic boundary condition was adopted in the spanwise direction by considering an infinitely long cylindrical structure.

Figure 3.

Schematic of the computational domain and boundary conditions.

2.5. Grid dependence test and validation

To validate the simulation, we first considered flow over a smooth cylinder and compared our numerical results and experimental data (Norberg Reference Norberg1987; Lam et al. Reference Lam, Wang and So2004b ; Lam & Lin Reference Lam and Lin2008) in terms of the mean streamwise velocity $U/U_{\infty }$ and the root mean square (r.m.s.) velocity $u^{\prime }/U_{\infty }$ . Moreover, three different grid systems (coarse, medium and fine grids), as listed in table 1, were used to test the grid dependence of the solutions for the smooth cylinder.

Table 1.

Grid dependence test for a smooth cylinder at $\mathit{Re}=3000$ compared with experimental and LES results: Number of grids $N$ , mean drag coefficient $\overline{C_{D}}$ , r.m.s. lift coefficient $C_{L,rms}$ , Strouhal number $\mathit{St}$ .

Figure 4 shows distributions of $U/U_{\infty }$ and $u^{\prime }/U_{\infty }$ along the vertical line at different positions compared with the experimental results (Norberg Reference Norberg1987; Lam et al. Reference Lam, Wang and So2004b ) and the LES results of Lam & Lin (Reference Lam and Lin2008) using the fine-grid model. Consequently, the present results for $U/U_{\infty }$ and $u^{\prime }/U_{\infty }$ agree well with the experimental and numerical results, regardless of the different grid systems, as shown in figure 4.

Figure 4.

Mean streamwise velocity $U/U_{\infty }$ ( $a$ ) and r.m.s. streamwise velocity $u^{\prime }/U_{\infty }$ ( $b$ ) distributions of a smooth cylinder at different positions in the wake centreline compared with experimental and LES results at $\mathit{Re}=3000$ .

In order to quantitatively verify the present numerical method and grid systems, the present results for the time-averaged drag coefficient ( $\overline{C_{D}}$ ), the r.m.s. value of lift fluctuation ( $C_{L,rms}$ ) and the Strouhal number ( $\mathit{St}$ ) were compared with results from previous studies. In the present study, the drag coefficient ( $C_{D}$ ) and lift coefficient ( $C_{L}$ ) are defined by

where ${\it\rho}$ is the fluid density, $D_{m}$ is the mean diameter of each cylinder, $L$ is the spanwise length of the cylinder, and $F_{D}$ and $F_{L}$ are the total drag and lift forces, respectively. The Strouhal number is the non-dimensional vortex shedding frequency (  $f_{s}$ ) and is defined as

The frequency of vortex shedding, $f_{s}$ , is obtained by fast Fourier transform of the time history of the lift coefficient.

Table 1 compares the results with experimental data (Norberg Reference Norberg1987, Reference Norberg2003) and LES results (Lam & Lin Reference Lam and Lin2008). The results of our computations are found to be in fairly good agreement with the experimental data (Norberg Reference Norberg1987, Reference Norberg2003) and with the 3D simulation results obtained by Lam & Lin (Reference Lam and Lin2008).

As mentioned previously, the wavy cylinder is considered to be another comparison case. Thus, our results for the wavy cylinder for $U/U_{\infty }$ and $u^{\prime }/U_{\infty }$ are compared with the experimental results (Lam et al. Reference Lam, Wang and So2004b ) and LES results of Lam & Lin (Reference Lam and Lin2008), as shown in figure 5. There is excellent agreement between the present results and the experimental measurements. However, in the saddle planes at the position $x/D_{m}=3$ , a small discrepancy in the values of $U/U_{\infty }$ can be observed between our results and the experimental (Lam et al. Reference Lam, Wang and So2004b ) and LES results (Lam & Lin Reference Lam and Lin2008). Lam et al. (Reference Lam, Wang and So2004b ) reported that the end point of the reverse flow position is near the position $x/D_{m}=3$ . Therefore, $x/D_{m}=3$ is a highly sensitive region for the streamwise velocity distributions at a similar subcritical Reynolds number (Lam & Lin Reference Lam and Lin2008). This gave rise to a small difference in the velocity profile between our results and the experimental results. In general, the present results for $U/U_{\infty }$ and $u^{\prime }/U_{\infty }$ agree reasonably well with previous experimental and computational results, thus confirming the validity of our computation.

Figure 5.

Mean streamwise velocity $U/U_{\infty }$ (a: node) and r.m.s. streamwise velocity $u^{\prime }/U_{\infty }$ (b: saddle) distributions of a wavy cylinder at different positions in the wake centreline compared with experimental and LES results at $\mathit{Re}=3000$ .

For the twisted cylinder, a grid dependence test was performed, and the results are summarized in table 2. Even though experimental data on the flow over the twisted cylinder are not yet available, assessments of the effect of grid resolution on the calculated solutions were conducted. The numbers of coarse, medium and fine grids were 2.5, 3.5, and 4.5 million, respectively. A number of uniform layers from 80 to 100 was considered with increasing total grid numbers. Grid stretching was employed to increase the grid resolutions near the surface and in the wake region, ensuring that there were at least 40 nodes in the vorticity thickness over the surface in the attached boundary-layer region. Curve-fitting coordinates were used. The different grid systems passed the grid dependence tests for small variations in the values of $\overline{C_{D}}$ , $C_{L,rms}$ and $\mathit{St}$ . As a result, the grid dependence test for the twisted cylinder was satisfied, and the medium-grid system is employed to discuss the results.

Table 2.

Grid dependence test for a twisted cylinder at $\mathit{Re}=3000$ : mean drag coefficient $\overline{C_{D}}$ , r.m.s. lift coefficient $C_{L,rms}$ , Strouhal number $\mathit{St}$ .

3.1. Drag, lift coefficients and the Strouhal number

Figure 6 shows drag and lift coefficients ( $C_{D}$ and $C_{L}$ ) as a function of time for three different cylinders. The values of $C_{D}$ and $C_{L}$ for the wavy cylinder are much lower than those of the smooth cylinder over the whole computational time, which can be understood by observing the time histories of $C_{D}$ and $C_{L}$ as shown in figures 6(a) and (b), respectively. These results are consistent with the findings of Lam et al. (Reference Lam, Wang, Li and So2004a ,Reference Lam, Wang and So b ), Lam & Lin (Reference Lam and Lin2007, Reference Lam and Lin2008), Lee & Nguyen (Reference Lee and Nguyen2007) and Xu et al. (Reference Xu, Chen and Lu2010), who confirmed that the wavy cylinder achieves a significant reduction of drag and lift in comparison with a smooth cylinder.

Figure 6.

Time histories of (a) the drag coefficients and (b) the fluctuating lift coefficients; (c) phase-space plot of the force coefficients; (d) power spectral density (PSD) of the time-dependent lift coefficient for different cylinders at $\mathit{Re}=3000$ .

Furthermore, the twisted cylinder gives a greater reduction in $C_{D}$ and $C_{L}$ than the reduction due to the wavy cylinder, as compared with the smooth cylinder. In particular, the oscillating amplitude of the lift coefficient for the twisted cylinder is greatly decreased compared with the smooth and wavy cylinders, as shown in figure 6(b).

We considered the phase diagram plot of the time history of lift versus drag to clearly compare the time-dependent forces of the different types of cylinders, as shown in figure 6(c). Among the three types of cylinders, the twisted cylinder has the narrowest region wherein the time variation of the force coefficients is limited. In addition, this region has the smallest values, indicating force reduction and a considerable suppression of force fluctuation.

The power spectral densities of the time-dependent lift coefficients of different cylinders are plotted in figure 6(d) in order to identify the frequency of vortex shedding. The Strouhal number ( $\mathit{St}$ ) of the wavy cylinder is approximately 0.21, similar to that of the smooth cylinder at the same Reynolds number. However, the twisted cylinder has a smaller value of $\mathit{St}$ than the smooth and wavy cylinders ( $\mathit{St}=0.190$ ). This lower value of $\mathit{St}$ is associated with a much longer vortex formation length and the nearly suppressed oblique vortex shedding in the near wake behind the twisted cylinder. These wake structures will be discussed in §§ 3.2 and 3.3 in terms of the instantaneous vortical structures and vortex formation length. In addition, the spectral peaks for the twisted cylinder are relatively weak, indicating an almost complete suppression of flow unsteadiness.

Table 3 summarizes the time-averaged drag coefficient ( $\overline{C_{D}}$ ), the r.m.s. value of the lift coefficient ( $C_{L,rms}$ ) and $\mathit{St}$ for the three types of cylinders. The $\overline{C_{D}}$ and $C_{L,rms}$ values of the wavy cylinder are less than those of the smooth cylinder, with reduction rates of approximately 8 and 78 %, respectively. For the conditions of waviness used in this study, the present reduction rate is quite consistent with that of Lam & Lin (Reference Lam and Lin2008). The twisted cylinder achieves approximately 13 and 5 % reductions in $\overline{C_{D}}$ compared with the smooth and wavy cylinders, respectively. Furthermore, the value of $C_{L,rms}$ for the twisted cylinder is much less than that for the smooth cylinder, with a reduction rate of up to 96 %. Compared with the values of $C_{L,rms}$ , we note that the fluctuating force on the twisted cylinder is significantly suppressed.

Figure 7.

Perspective view (left column) and side view (right column) of the isosurface of swirl strength of (a) the smooth, (b) the wavy and (c) the twisted cylinder at $\mathit{Re}=3000$ .

Table 3.

Comparison of the force coefficients and Strouhal number for the three different cylindrical structures (smooth, wavy and twisted). These mean values are obtained from the medium-grid system at $\mathit{Re}=3000$ : mean drag coefficient $\overline{C_{D}}$ , r.m.s. lift coefficient $C_{L,rms}$ , Strouhal number $\mathit{St}$ .

3.2. Instantaneous vortical structures

Figure 7 shows the instantaneous vortical structures at a Reynolds number of 3000 for the twisted, smooth and wavy cylinders. In order to define these 3D vortical structures, we adopted the method given by Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999). They defined a vortical region as a region with negative ${\it\lambda}_{2}$ , the second largest eigenvalue of $\mathsf{S}_{ij}^{2}+{\it\Omega}_{ij}^{2}$ , where $\mathsf{S}_{ij}^{2}$ and ${\it\Omega}_{ij}^{2}$ are the strain rate tensor and rotation rate tensor, respectively.

Our smooth cylinder results clearly reveal that large-scale Kármán vortices are formed in the near-wake region, as shown in figure 7(a). However, the vortex roll-up of the separated shear layer from the wavy cylinder is delayed to a downstream position. Therefore, the vortex shedding nearly disappears in the near wake of the wavy cylinder and reappears weakly in the far wake, compared with the smooth cylinder, as shown in figure 7(b). This modification of the vortical structures due to waviness is consistent with the findings from other computational (Lam & Lin Reference Lam and Lin2008; Xu et al. Reference Xu, Chen and Lu2010) and experimental studies (Lam et al. Reference Lam, Wang and So2004b ; Zhang et al. Reference Zhang, Dai and Lee2005).

For the twisted cylinder, the shear layer is more elongated downstream, and the vortex is shed farther from the body than that of the wavy cylinder, as shown in figure 7(c). The region of non-existence of small-scale eddies in the near wake is much wider than that of the wavy cylinder in figure 7(b). Consequently, the initial formation of the vortical structure is delayed in the downstream direction, and the Kármán vortex rollers are significantly weakened farther downstream, as shown in figure 7(c). This elongation of the shear layer from the body, and the near disappearance of vortex shedding in the near wake with weak vortical strength, contribute directly to a reduction in drag and lift oscillation. Moreover, 3D forcing as the active control, as considered by Kim & Choi (Reference Kim and Choi2005), caused about the same modification of vortical structures in the wake of the smooth cylinder with shape modification as the wavy and twisted cylinders, thus leading to reductions in drag and lift oscillation.

Elongation of the shear layer from the body and the near disappearance of vortex shedding in the near wake are also clearly identified by the vorticity components. Thus, representatively, we note that the transverse vorticity ${\it\omega}_{y}$ confirms the elongation of the shear layer from the body. It also confirms the near disappearance of vortex shedding in the near wake observed from the isosurface of swirling strength, as shown in figure 7.

Figure 8 shows the isosurfaces of the instantaneous transverse vorticities ( ${\it\omega}_{y}=\pm 1.5$ ) for the smooth, wavy and twisted cylinders. The isosurfaces of the instantaneous transverse vorticities immediately behind the wavy cylinder reveal regimes of zero vorticity centred adjacent to the saddle planes shown in figure 8(b), which can be clarified by comparison with the dense distribution of ${\it\omega}_{y}$ behind the smooth cylinder, as shown in figure 8(a). Regimes of zero vorticity (the recovery region) behind the wavy cylinder were also reported by Lam & Lin (Reference Lam and Lin2008). For the twisted cylinder, the patterns of near-wake transverse vorticity appear to be well organized and coherent with significantly elongated isosurfaces of ${\it\omega}_{y}$ , as distinctly seen in the top view of the isosurfaces of ${\it\omega}_{y}$ shown in figure 8(c). In particular, the twisted cylinder forms much wider regimes of zero vorticity than the wavy cylinder.

Figure 8.

Top (left column) and side (right column) views of the isosurface of transverse vorticity for (a) the smooth, (b) the wavy and (c) the twisted cylinder at $\mathit{Re}=3000$ .

Figure 9 shows the instantaneous velocity vectors and transverse vorticity ( ${\it\omega}_{y}$ ) contours in the horizontal ( $x$ – $z$ ) plane near the cylinder’s surface $(\,y/D_{m}=-0.6)$ for the three different cylinders. Behind the smooth cylinder, the formation locations of the vorticities shown in figure 9(a) correspond to those of the isosurface of the swirl strength as shown in figure 7(a). Although positive and negative vorticities appear in the instantaneous velocity field of the smooth cylinder, they are distributed in a seemingly random manner, especially near the rear stagnation point (Lin, Towfighi & Rockwell Reference Lin, Towfighi and Rockwell1995; Zhang et al. Reference Zhang, Dai and Lee2005) (not shown here for the sake of brevity). However, the positions of the vorticities are locked spatially in the streamwise direction as shown in figure 9(a). These results are in good agreement with those of Wu, Sheridan & Welsh (Reference Wu, Sheridan and Welsh1996) and Zhang et al. (Reference Zhang, Dai and Lee2005).

Figure 9.

Instantaneous velocity vectors (left column) and transverse vorticity contours (right column) in the $x$ – $z$ plane at $y/D_{m}=-0.6$ for (a) the smooth, (b) the wavy and (c) the twisted cylinder at $\mathit{Re}=3000$ . Here, the same levels are used for the transverse vorticity contours in each figure.

For the wavy cylinder, the magnitude of the velocity vectors’ near wake is smaller than that of the smooth cylinder. Successively, the instantaneous transverse vorticity behind the wavy cylinder is distributed farther downstream at the approximate location of $x/D_{m}=2$ , as shown in figure 9(b). Thus, the existence of zero-vorticity regimes, as shown in figure 8(b) for the isosurfaces of ${\it\omega}_{y}$ , is confirmed by the velocity vectors and ${\it\omega}_{y}$ contours in this plane. In contrast to the smooth cylinder, due to the presence of waviness, the flow above the node has a low speed because the node of maximum diameter plays a blocking role with respect to the approaching flow. Therefore, the relatively low-speed flow above the node entrains into the high-speed flow above the saddle with the minimum diameter. Consequently, above the wavy cylinder, the transverse vorticity reveals a well-organized pattern that is not observed for the case of the smooth cylinder. This vorticity above the surface has large values at the locations above each nodal point of the wavy cylinder, and the incline toward the saddle. These results are consistent with the observations reported in Zhang et al. (Reference Zhang, Dai and Lee2005).

The twisted cylinder produces periodic distributions of instantaneous velocity vectors and transverse vorticity along the spanwise direction up to a location farther downstream, as shown in figure 9(c), compared with the wavy cylinder. The dependence of the flow field’s near wake on the twisted shape is similar to that of the wavy cylinder. Namely, the high-speed flow above the trough with the minor axis rushes downstream and the low-speed flow appears above the crest corresponding to the major axis. We note that a cylinder with a helical surface perturbation reveals a spatially periodic pattern of wake-like flows along the span of the cylinder (Chyu & Rockwell Reference Chyu and Rockwell2002). The present twisted cylinder also produces a spatially periodic sequence of wake-like patterns that have characteristics common to the very near wake of a 2D bluff body. The elongated vorticity layers bound a large defect in velocity and they fluctuate from their tails with small-scale vorticity concentrations.

Figure 10.

Contours of the time-averaged spanwise vorticity in the $x$ – $y$ plane for three cylinders at $\mathit{Re}=3000$ . (a) Smooth cylinder and wavy cylinder: (b) node, (c) middle and (d) saddle. Twisted cylinder: (e) ${\it\Phi}=0^{\circ }$ , ( f) ${\it\Phi}=45^{\circ }$ , (g) ${\it\Phi}=90^{\circ }$ and (h) ${\it\Phi}=135^{\circ }$ .

These wake-like flows could also be interpreted as jet-like flows. It is known from hydrodynamic stability theory that a simple planar jet has two modes of instability: sinuous and varicose (Drazin & Reid Reference Drazin and Reid1981). As shown by the schematic in figure 9(c), the velocity vectors exhibit a series of vortex pairs along the oscillating jet centreline (high-speed flow) that are connected to the periodic (plus–minus) oscillation of the transverse velocity and the fluctuation in the longitudinal velocity. This results in jet-like flows in the near wake of the twisted cylinder. From the patterns behind the twisted cylinder, the separated free-shear layers are relatively more stable, and then roll up into vortices farther downstream, leading to a reduction of the suction near the base region of the twisted cylinder.

3.3. Vortex formation length, pressure coefficient and 3D separation

3.4. Turbulent kinetic energy

Turbulent fluctuations around a body are associated with the fluctuating forces exerted on it (Wu, Lu & Zhuang Reference Wu, Lu and Zhuang2007). The contours of the turbulent kinetic energy (TKE) in the $x$ – $z$ plane for different cylinders are plotted in figure 15. The wavy cylinder exhibits negligibly small TKE in the near wake (from cylinder base to $x/D_{m}\approx 1.4$ ) in comparison with the smooth cylinder, which can be clarified by comparing the smooth and wavy cylinders shown in figures 15(a) and (b), respectively. The TKE distribution behind the wavy cylinder shows periodic variations along the spanwise direction and resembles the geometric shape of the cylinder, in contrast to the uniform distribution of the smooth cylinder. The maximum TKE in the nodal plane is higher than the TKE in the saddle and middle planes. Moreover, the positions of maximum TKE are farther from the back of the wavy cylinder than from the back of the smooth cylinder. These characteristics of the TKE distribution of the wavy cylinder are consistent with the findings of Zhang et al. (Reference Zhang, Dai and Lee2005) and Lam & Lin (Reference Lam and Lin2008).

In the near wake, the twisted cylinder forms a considerably wider wake region where negligible TKE is present, as shown in figure 15(c), compared with the smooth and wavy cylinders. The TKE distribution behind the twisted cylinder shows periodic variations along the spanwise direction; this is similar to the wavy cylinder. However, the position of the maximum TKE value is farther from the back of the twisted cylinder than from the backs of the smooth and wavy cylinders. The maximum TKE of the twisted cylinder occurs at approximately $x/D_{m}=4$ , and the maximum values of the TKE for the smooth and wavy cylinders appear at downstream locations of approximately $x/D_{m}=2$ and $x/D_{m}=2.7$ , respectively. The overall values of the TKE behind the twisted cylinder are smaller than those of the smooth and wavy cylinders at $\mathit{Re}=3000$ . These features are consistent with the elongation of the vortex formation length and the lower turbulence intensity shown in figures 11 and 12. Consequently, the significant reduction in TKE in the near wake of the twisted cylinder contributes to a reduction in the drag and fluctuating lift.

3.5. Effect of the cross-sectional aspect ratio on flow over the twisted cylinder

In this section, we simply discuss the effect and the optimal range of the aspect ratio for the range of $1.25\leqslant A/B\leqslant 2.25$ from the point of view of aerodynamic performance. Figure 16 shows the mean drag and fluctuating lift coefficients according to $A/B$ , which is defined as the ratio of cross-section height to width. Regardless of $A/B$ , all the values of the mean drag and r.m.s. lift coefficients are smaller than those of the smooth cylinder. The mean drag coefficient of the twisted cylinders drops to a minimum at a certain aspect ratio and increases again as $A/B$ increases, as shown in figure 16(a). A maximum drag reduction of up to 13 % is achieved for a value of $A/B=1.44$ , compared with the smooth cylinder. We conjecture that the optimal value of the aspect ratio will be around $A/B=1.44$ , indicating the minimum mean drag coefficient. In the case of the fluctuating lift coefficients, the values are almost constant, varying with $A/B$ except for $A/B=1.25$ , as shown in figure 16(b). A maximum reduction of the fluctuating lift coefficient of up to 96 % is obtained at a value of $A/B=1.44$ in comparison with the smooth cylinder.

Figure 16.

(a) Mean drag and (b) r.m.s. lift coefficients, according to the aspect ratio of the twisted cylinder at $\mathit{Re}=3000$ .

Figure 17 shows the instantaneous vortical structures according to $A/B$ . In the range of $1.25\leqslant A/B\leqslant 1.44$ , the positions of the small-scale eddies move rapidly farther away from the back of the twisted cylinder with increasing $A/B$ , as shown in figures 17(a,b). However, for the case in which $A/B$ is greater than 1.75, the positions of the small-scale eddies gradually approach the base of the cylinder with increasing Re, implying an augmentation of suction near the base region of the twisted cylinder, as shown in figures 17(c–e). Namely, these features are very closely associated with increasing the drag, which is confirmed by the results of the variation of the mean drag and r.m.s. lift coefficients with $A/B$ , as shown in figure 16. As a result, the aspect ratio of the twisted cylinder plays an important role in the control of the flow structures, affecting the mean drag and fluctuating lift forces.

Figure 17.

Isosurface of swirl strength according to $A/B$ value for (a) 1.25, (b) 1.44, (c) 1.75, (d) 2.0 and (e) 2.25 at $\mathit{Re}=3000$ .

3.6. Effect of Reynolds number on flow over the twisted cylinder

In this section, we discuss the effect of the Reynolds number on flow around the twisted cylinder. For $\mathit{Re}=3000$ , our evaluation of aerodynamic characteristics for the twisted cylinder in the range $1.25\leqslant A/B\leqslant 2.25$ indicates that the best aerodynamic performance is obtained at $A/B=1.44$ . Thus, for $A/B=1.44$ , the effect of the Reynolds number on the aerodynamic characteristics was investigated for the range $3\times 10^{3}\leqslant \mathit{Re}\leqslant 1\times 10^{4}$ .

Figure 18 shows the variation of the mean drag and r.m.s. fluctuating lift coefficients as a function of Re for the smooth and twisted cylinders, compared with previous experimental and simulation results by various researchers (Keefe Reference Keefe1962; Bishop & Hassan Reference Bishop and Hassan1964; Tanida, Okajima & Watanabe Reference Tanida, Okajima and Watanabe1973; White Reference White1974; Moeller Reference Moeller1982; Moeller & Leehey Reference Moeller, Leehey, Paidoussis, Griffin and Sevik1984; Norberg Reference Norberg1987; Tadrist et al. Reference Tadrist, Martin, Tadrist and Seguin1990; Gopalkrishnan Reference Gopalkrishnan1993; West & Apelt Reference West and Apelt1993; Lu, Dalton & Zhang Reference Lu, Dalton and Zhang1997; Jordan Reference Jordan2002; Norberg Reference Norberg2003; Dong et al. Reference Dong, Karniadakis, Ekmekci and Rockwell2006; Lam & Lin Reference Lam and Lin2008). In the case of the smooth cylinder, the mean drag and r.m.s. lift coefficients rapidly increase with increasing Re. These features are consistent with experimental and simulation results, as previously mentioned, which indicates that the shear layer instability from the separation point of the body develops with increasing Re. Similarly to the characteristics of the statistical quantities for the smooth cylinder, the mean drag and r.m.s. fluctuation lift coefficients of the twisted cylinder also increase as Re increases. However, the maximum augmentation of the mean drag coefficient of the twisted cylinder is within 1 %, whereas a maximum augmentation of the mean drag coefficient of the smooth cylinder of up to 21 % is obtained with increasing Re, as shown in figure 18(a). Moreover, the increase in the r.m.s. lift coefficient of the twisted cylinder seems to be negligible in comparison with that of the smooth cylinder, as shown in figure 18(b). These trends suggest that regardless of increasing Re, the geometry of the twisted cylinder effectively controls the flow structures, which contributes directly to the reduction of the drag and the lift oscillation compared with the smooth cylinder. The increments of the mean drag and r.m.s. lift coefficients are governed by the linear function $f\approx a\cdot \mathit{Re}$ , where $f$ and $a$ are the variables of the force coefficient and slope, respectively. For the mean drag and r.m.s. lift coefficients, the slopes of the smooth cylinder are about one order of magnitude higher than those of the twisted cylinder.

Figure 18.

(a) Mean drag and (b) r.m.s. lift coefficient as a function of Re for the smooth and twisted cylinders compared with experimental and simulation results.

Figure 19 shows the normalized mean spanwise vorticity of the smooth cylinder at $\mathit{Re}=3000$ and $\mathit{Re}=10\,000$ . As Re increases, the spanwise vorticity length and width are smaller and thicker, respectively. These features are characterized by the location of the onset of transition, which moves upstream towards the separation point. Moreover, the large-scale Kármán vortices are formed closer to the base of the cylinder, which is consistent with the experimental and direct numerical simulation results of Dong et al. (Reference Dong, Karniadakis, Ekmekci and Rockwell2006) and other studies (Bloor Reference Bloor1964; Norberg Reference Norberg1987; Unal & Rockwell Reference Unal and Rockwell1988; Norberg Reference Norberg, Eckelmann, Graham, Huerre and Monkewitz1993, Reference Norberg1994; Lin et al. Reference Lin, Towfighi and Rockwell1995; Norberg Reference Norberg1998).

Figure 19.

Contours of mean spanwise vorticity in the $x$ – $y$ plane for smooth cylinders: (a) $\mathit{Re}=3000$ and (b) $\mathit{Re}=10\,000$ . Data for $\mathit{Re}=5000$ and 8000 have been intentionally omitted for clarity.

For the twisted cylinder, regardless of the pitch angle, shrinkages of the shear layer are observed, and the distributions near the tip of the vorticity layer tend to broaden with increasing Re, as shown in figure 20. Namely, in the subcritical regime, the overall patterns of the vorticity distributions of the twisted cylinder are similar to those of the smooth cylinder with increasing Re. However, the spanwise vorticity length of the twisted cylinder is much longer than that of the corresponding smooth cylinder, irrespective of any increase in Re.

Figure 20.

Contours of mean spanwise vorticity in the $x$ – $y$ plane for twisted cylinders. Here, $\mathit{Re}=3000$ (a–d): (a) ${\it\Phi}=0^{\circ }$ , (b) ${\it\Phi}=45^{\circ }$ , (c) ${\it\Phi}=90^{\circ }$ and (d) ${\it\Phi}=135^{\circ }$ ; $\mathit{Re}=10\,000$ (e–h): (e) ${\it\Phi}=0^{\circ }$ , ( f) ${\it\Phi}=45^{\circ }$ , (g) ${\it\Phi}=90^{\circ }$ and (h) ${\it\Phi}=135^{\circ }$ . Data for $\mathit{Re}=5000$ and 8000 have been intentionally omitted for clarity.