4.1. Classification of flow patterns

After careful examination of the fluid forces (including time-mean drag, fluctuating lift and surface pressure distributions) and the near-wake structures (including instantaneous vortices, vortex shedding frequency, recirculation bubble or wake closure length and three-dimensional topologies of time-mean streamlines), five distinct flow patterns (I–V) are identified when λ/D_m is increased from 0.43 to 8.59 (figure 4). The λ/D_m-ranges of the flow patterns (I–V) are marked in figure 4, with the red dots denoting representative wavelengths that are selected for further discussion in the following.

Figure 4. Dependence of flow patterns on wavelength λ/D_m. Red dots refer to the wavelength of the wavy cylinder selected for discussion.

Pattern I: at a relatively short wavelength λ/D_m (<2.58), the wavy elliptic cylinder generates a two-dimensional near wake, similar to that of a straight elliptic cylinder with an aspect ratio of the cross-section AR ≥ 0.37 and a zero angle of attack (Shi et al. Reference Shi, Alam and Bai2020, Re = 100). A representative flow pattern in this short wavelength range is given in figure 5(a1,a2), where instantaneous spanwise and streamwise vortical structures in the near wake for λ/D_m = 0.86 are shown. The spanwise vortical structures in figure 5(a1) are indicated by the iso-surfaces (λ ₂ = −0.1) detected via the λ ₂-criterion (Jeong & Hussain Reference Jeong and Hussain1995) and coloured by the magnitude of streamwise vorticity $(\omega _x^\ast )$, while the streamwise vortical structures are depicted by the iso-surfaces $(\omega _x^\ast = \pm 0.1)$ of ω_x. Clearly, the iso-surfaces of λ ₂ = −0.1 exhibit a street of von Kármán vortices downstream (figure 5a1), and no iso-surfaces of $\omega _x^\ast = \pm 0.1$ are detected around the cylinder and in the wake (figure 5a2). This observation indicates that the wake is two-dimensional. This flow behaves like the flow over a uniform cylinder, hence it is termed as 0λ flow. This is further corroborated by the distributions of the time-mean streamlines in the xz-plane (at y = 0) and xy-planes (at the saddle and node sections, denoted by ‘S’ and ‘N’, respectively), shown in figure 6(a) in a perspective view. It can be seen that a recirculation bubble is generated behind the cylinder, starting from the flow separation line (indicated by the blue dashed line) on the cylinder surface. The streamwise extent of the recirculation bubble at the node and saddle locations is the same, indicated by the bifurcation (saddle point) line of the streamlines (indicated by the red dashed line) in the xz-plane (at y = 0). That is, the recirculation bubble is spanwise invariant, suggesting a two-dimensional wake. As indicated by the blue dashed line over the cylinder surface, flow separation occurs slightly earlier at the saddle (148.3°) than at the node (154.3°), with the near-surface flow laterally moving from the saddle to the node.

Figure 5. Representative instantaneous structures for the five flow patterns. (a1–e1) Spanwise vortices, in terms of iso-surface of λ ₂ = −0.1 (coloured by streamwise vorticity $\omega _x^\ast $); (a2–e2) streamwise vortices, in terms of iso-surface of $\omega _x^\ast = 0.1$ (red) and −0.1 (blue). Images (a1,a2) show λ/D_m = 0.86 (flow pattern I); (b1,b2) 3.43 (II); (c1,c2) 4.58 (III); (d1,d2) 6.01 (IV); and (e1,e2) 8.59 (V). The cylinder in each is of two spanwise wavelengths.

Figure 6. Distributions of time-mean streamlines in the xy-plane (at saddle and node sections, denoted by ‘S’ and ‘N’, respectively) and in the xz-plane (at y = 0) for (a) λ/D_m = 0.86 (flow pattern I), (b) 3.43 (II), (c) 4.58 (III), (d) 6.01 (IV) and (e) 8.59 (V). The red dashed line denotes the streamwise extent of the recirculation bubble. The separation angles for node and saddle sections are provided.

Pattern II: when the wavelength is increased to λ/D_m = 3.43 (pattern II, 2.58 < λ/D_m < 4.44), its effects on the near-wake structures emerge, the wake being three-dimensional and wavy along the spanwise direction (figure 5b1). Alternate streamwise vortices are now generated (figure 5b1,b2), albeit spanwise vortices are dominant, shedding alternately from the two sides of the cylinder and appearing staggered in the wake. The spanwise vortex tubes undulate in the spanwise direction, following the cylinder wavelength. The wavy formation of the spanwise vortical structures can be ascribed to the presence of the counter-rotating streamwise vortices additionally generated downstream of the nodes (figure 5b2), which was also observed by Ahmed & Bays-Muchmore (Reference Ahmed and Bays-Muchmore1992), Zhang et al. (Reference Zhang, Daichin and Lee2005) and Lam & Lin (Reference Lam and Lin2008) for wavy circular cylinders. The spanwise vortex tubes, albeit undulating, are continuous along the spanwise direction and alternate in the wake, suggesting that the spanwise vortex shedding from one side synchronously occurs along the whole cylinder span while alternating with that from the other side. The vortex shedding happens synchronously from the whole cylinder span, or from the entire wavelength, thus the flow here is called 1.0λ flow. The formation of the three-dimensional undulation in the near wake is further manifested by the distributions of time-mean streamlines in the two orthogonal planes (figure 6b). Different from the observation made in pattern I (figure 6a), the streamwise size of the recirculation bubble differs between the node and saddle sections, longer at the node than at the saddle, which is evidenced by the sine-like variation of the bifurcation line in the xz-plane at y = 0. The near-surface flow heads from the node to the saddle, resulting in an earlier flow separation at the node (146.6°) than at the saddle (157.0°), which is again opposite to that in pattern I (figure 6a). It can thus be said that delayed or advanced flow separation is dictated by the near-surface flow heading to the saddle or the node. If the near-surface flow is directed toward the node, the flow separation is delayed at the node and vice versa.

Pattern III: with a further increase in the wavelength to λ/D_m = 4.44–5.01, the alternate spanwise and streamwise vortices are suppressed, the wake is stabilized and the flow becomes steady (figure 5c1,c2). It is referred to as steady flow. Unlike patterns I and II, pattern III does not exhibit large-scale Kármán vortices; instead, the iso-surfaces of spanwise and streamwise vortices appear in the vicinity of the cylinder. These observations indicate that shear layers separating from the cylinder are stable, without rolling up into large-scale spanwise or streamwise vortical structures. The recirculation bubble size changes dramatically along the cylinder span, being smallest at the saddle section (figure 6c). Again the flow separation is more delayed at the saddle (157.0°) than that at the node (148.1°), with the near-surface flow heading from the node to the saddle.

Pattern IV: in the range of 5.01 < λ/D_m < 6.40, the wake of the wavy elliptic cylinder recovers to be three-dimensional, unsteady and characterized by strong, dominant streamwise vortices and disconnected spanwise vortices (figure 5d1,d2). As seen in figure 5(d2), the iso-surfaces of $\omega _x^\ast = \pm 0.1$ display large-scale $\varLambda$-like structures inclined towards the downstream direction, each spanning approximately one spanwise wavelength. These streamwise $\varLambda$-like structures of oppositely signed vorticities spread out along both streamwise and spanwise directions. The presence of these dominant streamwise vortical structures is a reason for disconnected spanwise vortical structures along the spanwise direction, as indicated by the broken iso-surfaces of λ ₂ = −0.1 (figure 5d1). Another distinct and interesting feature of this flow pattern is that spanwise vortex shedding from a half-wavelength (e.g. from the top node to the adjacent saddle) is out of phase (phase lag 180°) with that from the next half-wavelength, see figure 5(d1,d2); more information of the phase lag will be provided later. That is, vortex shedding synchronously occurs only for a half-cylinder wavelength. This is the main reason why spanwise vortex tubes are disconnected along the span for pattern IV, different from those (i.e. an entire wavelength synchronization of vortex shedding) for patterns I and II. This flow can be termed as a half-wavelength flow or 0.5λ flow. Such a half-wavelength vortex shedding (out-of-phase vortex shedding from the two successive half-wavelengths) provides room for a streamwise vortex to grow up in the spanwise direction. The vortex shedding of 0.5λ flow is identified as the first of its kind, which may yield zero instantaneous lift and zero fluctuating drag on the cylinder of one wavelength, although alternating vortex shedding from the cylinder is still manifest. It may thus have a great implication for near-wake flow control and FIV suppression. The corresponding forces generated will be presented later. In the time-mean sense, the flow separation is delayed at the saddle that is accompanied by the smallest recirculation bubble size (figure 6d).

Pattern V: at λ/D_m > 6.40, the wake is still highly three-dimensional although interactions of the additionally generated streamwise vortices become weakened because of the large distance between the nodes (figure 5e1,e2). The spanwise vortex tube in the vicinity of the cylinder spans the entire cylinder span, albeit in a wavy fashion. It indicates that the vortex shedding from one side of the cylinder occurs for the whole cylinder span, not like that observed for pattern IV. Different from the $\varLambda$-like structures spanning one wavelength (figure 5d2), the streamwise vortices grow and mainly congregate downstream of the node sections (figure 5e2). As a result, the spanwise vortex tubes break downstream as occurs for a uniform cylinder involving streamwise vortices (Bai & Alam Reference Bai and Alam2018). This flow can be called a very large wavelength flow or ∞λ flow. This is also reflected by the distributions of time-mean streamlines (figure 6e) where the recirculation bubble has a longer streamwise extent at the node section than at the saddle section. The difference in the streamwise size of the recirculation bubble between the node and saddle sections becomes smaller than those for patterns III and IV.

4.2. Recirculation bubble length

Figure 7(a) shows variations with λ/D_m of the wake closure or recirculation bubble length (L_c) at the saddle and node sections, as well as their ratios ($L_c^S/L_c^N$, superscripts ‘S’ and ‘N’ denoting saddle and node sections, respectively). The L_c is defined as the streamwise extent of the recirculation bubble from the cylinder centre. It can be seen in figure 7(a) that the saddle and node sections have different trends of $L_c^\ast $ vs λ/D_m. In the range of λ/D_m < 2.58, where flow pattern I occurs, $L_c^\ast $ for the saddle section is close to that for the node section, yielding a ratio $L_c^S/L_c^N$ nearly equal to one (figure 7a). This is linked to the two-dimensional flow (figure 5a1). The value of $L_c^\ast $ for the node section escalates with increasing λ/D_m whilst that for the saddle section declines gradually in 2.58 < λ/D_m < 4.44. The value of $L_c^S/L_c^N$ thus declines rapidly. With a further increase in λ/D_m from 4.44 to 5.01, $L_c^\ast $ at the node section keeps increasing while that at the saddle section reaches its minimum. The value of $L_c^S/L_c^N$ reaches a low value of about 0.5 in this range. When flow pattern III is modified into flow pattern IV in the range of 5.01 < λ/D_m < 6.40, $L_c^\ast $ values for both sections fluctuate, maintaining $L_c^S/L_c^N$ nearly constant. The large deviations of $L_c^\ast $ for the saddle and node sections, associated with the low $L_c^S/L_c^N$, manifest the three-dimensional characteristics of the wake observed for flow pattern IV (figure 5d). The value of $L_c^\ast $ for the node section jumps at the border between patterns IV and V. At λ/D_m > 6.40 (pattern V), with increasing λ/D_m, $L_c^\ast $ drops for the node section and increases for the saddle section to reach that for the straight elliptic cylinder. Overall, $L_c^\ast $ is more dependent on λ/D_m at the node plane than at the saddle plane.

Figure 7. Variations with wavelength λ/D_m of the global parameters: (a) recirculation bubble length $L_c^\ast $ at the saddle and node sections and their ratio $L_c^S/L_c^N$ ; and (b) $2(L_c^N - L_c^S)/\lambda $.

To further investigate the effects of λ/D_m on the wake structures, we calculated the difference in $L_c^\ast $ between the node to saddle sections, i.e. $2(L_c^N - L_c^S)/\lambda $ and present its variation with λ/D_m in figure 7(b). The value of $2(L_c^N - L_c^S)$ is presumably connected with the capability of λ/D_m to modify the time-mean flow structures. The published data for the wavy circular cylinder (Lam & Lin Reference Lam and Lin2008, Reference Lam and Lin2009; Lin et al. Reference Lin, Bai, Alam, Zhang and Lam2016) are included for comparison. Clearly, $2(L_c^N - L_c^S)/\lambda $ is large in patterns III and IV characterized by the most stabilized near wake and the largest force reduction, respectively. This observation suggests that the wavy elliptic cylinder with λ/D_m = 4.44–6.40 can alter the near-wake structures in a way that the coherent vortical structures are completely suppressed (4.44 < λ/D_m < 5.01, see figure 5c1,c2) or the additionally generated $\varLambda$-like structures by the node make the near wake stable (with 5.01 < λ/D_m < 6.40). Interestingly, the published data of $2(L_c^N - L_c^S)/\lambda $ for the wavy circular cylinder display the maximum in the range of 6 < λ/D_m < 7.5, where the most stabilized wake is also observed. Therefore, $2(L_c^N - L_c^S)/\lambda $ may play a dominant role in modifying the wake of the wavy elliptic or circular cylinder.

4.3. Wake evolution

We further investigate the wake behaviour of the wavy elliptic cylinder, based on time evolutions of the lateral velocity fluctuation (v*) and streamwise vorticity fluctuation $(\omega _x^\ast )$ recorded at (x*, y*) = (5, 0) along the entire span, as well as on instantaneous streamlines in the near wake. Typical wavelengths of λ/D_m = 0.86 (pattern I), 3.43 (II), 5.44 and 6.01 (IV) and 8.59 (V) are chosen. Figures 8 and 9 show respectively the time evolutions of v* and $\omega _x^\ast $ for the span of two wavelengths, with the nodes located at z/λ = 0, 1.0 and 2.0 and the saddles at z/λ = 0.5 and 1.5. The contours are scaled identically in each figure to facilitate comparison, except in figure 8(c2), where a small scale is used to explore the evolution explicitly. As the flow is two-dimensional and steady for patterns I and III, respectively, the v* contours for pattern III and $\omega _x^\ast $ contours for patterns I and III are not shown. For λ/D_m = 0.86 (pattern I), the v* signal (figure 8a) is uniformly distributed along the span while exhibiting a periodic switch between positive and negative with time. In pattern II (figure 8b), the distributions of v* appear wavy along the span, and the temporal distributions remain periodic with time. The waviness comes up with streamwise vortices (figure 9a) in a periodic fashion in space and time domains. The v* evolution in pattern IV is different from the others, positive or negative v* occupies one cylinder wavelength from the saddle to saddle, appearing alternately in both time and space domains (figure 8c2). Similarly, the $\omega _x^\ast $ topology shows patterns of $\varLambda$-like vortices spanning one cylinder wavelength (figure 9b). In pattern V, the value of v* vigorously varies along the time axis around the node plane, and $\omega _x^\ast $ is dramatically strong, echoing the highly three-dimensional flow. Let us pay more attention to pattern IV where v* contours reflect that, in one cylinder wavelength (e.g. z/λ = 0–1.0), v* is negative and positive (the first column) for z/λ = 0–0.5 and z/λ = 0.5–1.0, respectively, which indicates that the vortex shedding is from the +y (>0) side of the cylinder for z/λ = 0–0.5 but from the −y (<0) side for z/λ = 0.5–1.0 (figure 8c2). That is, vortex shedding from one half of the wavelength is out of phase with the other half for pattern IV. This is, however, not the case for patterns I, II and V where vortex shedding occurs synchronously from the whole cylinder span.

Figure 8. Time evolutions of lateral velocity v* recorded at (x*, y*) = (5, 0); (a) λ/D_m = 0.86 (flow pattern I), (b) 3.43 (II), (c1,c2) 5.44 (IV) and (d) 8.59 (V).

Figure 9. Time evolutions of streamwise vorticity $\omega _x^\ast $ recorded at (x*, y*) = (5, 0); (a) λ/D_m = 3.43 (flow pattern II), (b) 5.44 (IV) and (c) 8.59 (V).

The trajectories of instantaneous streamlines are shown in figure 10 for λ/D_m = 3.43 (pattern II), 4.58 (III) and 6.01 (IV). Here, the results are presented for one cylinder wavelength only. To facilitate interpretation, the streamlines are introduced from the symmetric xz plane (y = 0) upstream of the cylinder at the node and saddle, coloured in red and blue, respectively, while the streamlines in the middle planes between the node and saddle are coloured in green. Consider the case of λ/D_m = 3.43 (pattern II, figure 10a) first, the streamlines issuing at different sections all separate from the same side (y > 0) of the cylinder at this instant. The streamlines from the node plane spiral towards the saddle, entraining those from the middle planes. The streamlines then extend downstream of the saddle. The spatial characteristics of the streamlines are linked to the wavy wake of alternating spanwise vortices (figure 5b). The streamline at the saddle does not spiral but displays a signature of vortex shedding. For λ/D_m = 4.58 (pattern III, figure 10b), the streamlines also separate from the same side (y > 0) of the cylinder at this instant. However, different from those in figure 10(a), the streamlines mainly roll up into the recirculation bubble with a long streamwise extent at the node (figure 10b1). The streamline near the middle plane (green line) forms a steady recirculation bubble, spiralling from the middle plane to the node plane (figure 10b2,b4). Meanwhile, the recirculation bubble between the middle and saddle planes spirals toward the centre, small in size and strong in circulation, which can be observed clearly from the zoomed-in view in figure 10(b3). That is, the spiral flow, having a bifurcation at the middle plane, heads toward the node and saddle planes. According to the green streamlines in figure 10(b2), the large-scale spiral flow can be detected only at two cylinder ends (node sections). The spiral flow toward the nodes is due to the large-scale recirculation bubble at the nodes while that toward the saddle, albeit weak, is the same as that for pattern II. A sketch showing the bifurcation of the spiral flow is provided in figure 10(b5) with the yz- plane view. In fact, the spiral flow has similar features in xz- and yz-planes (figure 10b3,b4); therefore, only the yz- plane is provided. Recall that $L_c^S/L_c^N \approx 0.5$ with λ/D_m = 4.58 (figure 7a), which leads to a 45° inclination angle of the recirculation bubble core between the node and saddle planes (figure 10b5). The high inclination may be the reason for the bifurcation. When the streamlines exit the bubble at node and saddle planes, they become parallel with the free-stream flow (figure 10b2).

Figure 10. Instantaneous and time-mean streamlines; (a1–a3) λ/D_m = 3.43 (flow pattern II), (b1–b5) 4.58 (III) and (c1–c3) 6.01 (IV), where (a1–c1) provide the top views (xy- plane), (a2–c2) provide the global view, (a3) is the side view of (a2), (b3,b4) display the zoom-in view of the specific streamline, (b5) is a sketch showing the spiral flow of the streamline marked with dashed square in (b2) with yz-plane and (c3) is the time-mean streamline.

For the cylinder with λ/D_m = 6.01 (pattern IV, figure 10c2), the instantaneous streamlines are more complicated than those with shorter wavelengths. It is noted that the streamlines at the two nodes (black) or middle sections (green) issue from the opposite sides of the upper and lower half of the cylinder while those (blue) in the saddle plane separate from both sides of the cylinder. This further confirms that the vortex sheddings from the upper and lower halves are antiphased, see also the top views (figure 10c1) of the streamline patterns in figure 10(c2). Such characteristics are internally linked to the formation of $\varLambda$-like streamwise vorticity structures.

Due to the significant difference in the wake sizes between node and saddle planes (figure 6d), a spiral flow behind the cylinder along the cylinder span is generated, which could be more clearly understood from the time-mean streamlines shown in figure 10(c3). This time-mean spiral flow is similar to that in the steady flow pattern (λ/D_m = 4.58, figure 10b2). With the spiral flow being stronger near the node plane, the bifurcation of the spiral flow now occurs further close to the node plane (figure 10c3).

Furthermore, the instantaneous vortex shedding for five distinct flow patterns is sketched in figure 11 to have a clear understanding of the three-dimensional flow topologies. Meanwhile, the time-mean spiral streamline marked by the red arrows provides a picture of the flow three-dimensionality. Flow pattern I features two-dimensional flow (figure 11a), where vortex shedding and streamlines are both two-dimensional, regardless of the wavelength. In other words, the flow is 0λ flow. The wavelength effect appears for flow pattern II, generating the spiral flow from the node plane to the saddle plane (figures 10a2 and 11b), leading to the waviness of the vortex structure (figure 11b). Vortex shedding alternately occurs from the two sides for the entire wavelength. For steady flow (pattern III), the stable recirculation bubble is generated in the wake, having bifurcated spiral flows heading toward the saddle and node planes (figure 11c). Flow pattern IV is characterized by the antiphase vortex shedding from the two halves of the cylinder (figure 11d). The resulting instantaneous lift force is thus zero over one complete wavelength. Again, flow associated with vortex shedding is bifurcated, yielding spiral flow toward the node and saddle planes. The size of the time-mean recirculation bubble is much smaller in the saddle plane than in the node plane for patterns III and IV (figures 6c,d and 10b2,c3), leading to a shorter formation length (figure 7a). Lin et al. (Reference Lin, Bai, Alam, Zhang and Lam2016) supposed that stable shear layers render a longer recirculation bubble, which is not the case here. For this flow pattern, the recirculation bubble is approximately steady at the saddle plane as the saddle plane is the common plane for the vortex shedding from the upper and lower halves of the cylinders. Flow pattern V still undergoes bifurcation but now the vortex shedding synchronously occurs from the whole span (figure 11e). It is understood that bifurcation occurs when the difference in recirculation bubble sizes between node and saddle planes is significant $(L_c^N/L_c^S \le 0.63)$, for λ/D_m ≥ 4.30, patterns III–V (figure 7a).

Figure 11. Sketches of the flow structures for distinct flow patterns. (a–e) Flow patterns I–V. The red lines indicate bifurcation in the vortex shedding and recirculation.

4.4. Fluid forces

Fluid forces acting on the wavy elliptic cylinder (WEC) are examined in figure 12 showing the dependence of ${\bar{C}_D}$ and ${C^{\prime}_L}$ on λ/D_m. The values of ${\bar{C}_{D0}}$ and ${C^{\prime}_{L0}}$ of the straight elliptic cylinder (EC) is included in the figure for comparison. The flow patterns I, II, III, IV and V are marked in the figure to facilitate interpretations of the force behaviour. It can be seen that ${\bar{C}_D}$ and ${C^{\prime}_L}$ dip rapidly with increasing λ/D_m from 0 to 4.44 (figure 12a,b), being mostly larger in pattern I and smaller in pattern II than those of the straight cylinder. The force coefficients reach their minima at 4.44 < λ/D_m < 5.01 (pattern III). Compared with the straight cylinder counterpart, the reduction in ${\bar{C}_D}$ is up to 5 % for λ/D_m = 4.44–5.01 (figure 12a); on the other hand, ${C^{\prime}_L}$ is zero (figure 12b). The reduced ${\bar{C}_D}$ and zero ${C^{\prime}_L}$ result from the stabilized wake (steady flow). These observations on the fluid forces are consistent with those made on the near-wake structures (figure 5c1,c2). In the range of 5.01 < λ/D_m < 6.40 (pattern IV), ${\bar{C}_D}$ recovers slightly while ${C^{\prime}_L}$ remains zero (figure 12a,b). The zero ${C^{\prime}_L}$ results from 0.5λ flow as the lift generated in a half-wavelength of the cylinder is cancelled by the opposite lift generated in the other half-wavelength of the cylinder, which will be further clarified later. At λ/D_m > 6.40 (pattern V), both ${\bar{C}_D}$ and ${C^{\prime}_L}$ increase gradually with increasing λ/D_m and are expected to approach asymptotically that of the straight cylinder (with an infinite wavelength).

Figure 12. Dependence on wavelength λ/D_m of (a) mean drag coefficient ${\bar{C}_D}$ and (b) fluctuating lift coefficient ${C^{\prime}_L}$. The red dashed line indicates ${\bar{C}_{D0}}$ and ${C^{\prime}_{L0}}$ for the straight EC and ‘I’ to ‘V’ on the top of the plots denote identified flow patterns.

The optimal wavelength of the WEC, in terms of fluid force reduction, can be identified in figure 12. The optimal wavelength covers a range of λ/D_m = 4.44–6.40, corresponding to the minimum ${\bar{C}_D}$ and zero ${C^{\prime}_L}$. Note that, while ${\bar{C}_D}$ remains its minimum in pattern III and recovers slightly in pattern IV, ${C^{\prime}_L}$ maintains zero in the entire range of the optimal wavelength. This is essentially linked to the distinct features in patterns III and IV (figure 5c1,c2,d1,d2). For the wavy circular cylinder, two different optimal wavelengths (i.e. λ/D_m ≈ 2 and 6) were detected by Lam & Lin (Reference Lam and Lin2009) (Re = 100) as well as by Lam & Lin (Reference Lam and Lin2008) and Lin et al. (Reference Lin, Bai, Alam, Zhang and Lam2016) (Re = 3 × 10³). The reductions in ${\bar{C}_D}$ and ${C^{\prime}_L}$ are up to 16 % and 93 %, respectively. Distinct mechanisms of force reduction are associated with the two optimal wavelengths for the wavy circular cylinder (Lin et al. Reference Lin, Bai, Alam, Zhang and Lam2016). A detailed comparison between the wavy elliptic and circular cylinders will be given in § 5.

4.5. Intrinsic mechanism and consequence of 0.5λ flow generation

It is worth investigating why ${C^{\prime}_L} = 0$ in pattern IV although there is alternate vortex shedding from the cylinder or why this pattern is named as 0.5λ flow. We choose two cross-sections A and B in one spanwise wavelength, with the same distance l away from the saddle plane (figure 13a). Each cross-section is probed with four surface locations Ai or Bi (i = 1–4), corresponding to a similar azimuthal angle $|{\theta}|$. The points are symmetric about the xz-plane at y = 0 and yz-plane at x = 0. Note that l = 0.4λ and θ = 30° are randomly selected in figure 13(a), without the loss of generality. Time histories of the lateral component ${C_{py}} = {C_p}\sin \theta $ (where ${C_p} = (P - {P_\infty })/0.5\rho U_\infty ^2$) of the surface pressure coefficients for Ai and Bi are presented for λ/D_m = 5.44 (pattern IV, figure 13b,c), 4.58 (pattern III, figure 13e, f) and 1.72 (pattern I, figure 13h,i). The value of ${C_{py}}$ is essentially connected to the lift force generated by the cylinder. Here, we discuss the case of pattern IV first (figure 13b,c). The ${C_{py}}$ signals at the two points on the same side (y > 0 or y < 0) of section A are almost in phase (e.g. A1 and A2 are indicated by the black curves, or A3 and A4 by red dashed curves) (figure 13b,c). The same happens in section B. Interestingly, on the same side (e.g. y < 0) of the cylinder, the ${C_{py}}$ signal at A1 (or A2) is out of phase with that at B1 (or B2), albeit both A1 and B1 (or A2 and B2) points have the same θ value but are located on different cross-sections. This suggests that vortex shedding from the same side (y > 0 or y < 0) of the two different cross-sections is out of phase, which is supported by the instantaneous flow structures, in terms of spanwise vorticity $(\omega _z^\ast )$, in sections A and B in figure 13(d). It can be seen that shear layers separating from the upper (or lower) side of section A (i.e. S-A in figure 13d) are oscillating in an out-of-phase fashion with that from the upper (or lower) side of section B (i.e. S-B), which is highlighted by the dashed ellipses. Meanwhile, shear layers separating from the saddle plane (i.e. S in figure 13d) are symmetric. Further note that, in figure 13(b), due to the out-of-phase and similar-magnitude fluctuations of ${C_{py}}$ in A1 and B1 (or A3 and B3), the summation of ${C_{py}}$ at A1 and B1, i.e. ${C_{py}}(\textrm{A}1 + \textrm{B}1)$ (which half is indicated by the red line) on the lower side is approximately equal to, in magnitude, that of ${C_{py}}$ at A3 and B3, i.e. ${C_{py}}(\textrm{A}3 + \textrm{B}3)$ (which half is indicated by the blue line) on the upper side. A similar observation can be made in figure 13(c) on ${C_{py}}$ at A2 and B2 (or A4 and B4). That is, ${C_{py}}(\textrm{A}1 + \textrm{B}1) ={-} {C_{py}}(\textrm{A}3 + \textrm{B}3)$ and ${C_{py}}(\textrm{A}2 + \textrm{B}2) ={-} {C_{py}}(\textrm{A}4 + \textrm{B}4)$. Therefore, the total lift forces on the two cross-sections (with similar distance l away from the saddle plane) must cancel out from each other, although the lift fluctuation (integrated by the surface pressure fluctuation) in each cross-section may not be necessarily zero. In other words, a zero total lift fluctuation on the entire wavelength of the cylinder is not unexpected, given that l is randomly selected.

Figure 13. Time histories of surface pressure coefficient C_py (= C_psinθ, θ is the azimuth angle) for locations Ai and Bi around the two cross-sections A (S-A) and B (S-B) as indicated in (a) and the instantaneous vorticity structure at S-A, S-B and saddle section. Panels (b–d) show λ/D_m = 5.44 (flow pattern IV), (e–g) 4.58 (III) and (h–j) 1.72 (I).

In the case of pattern III, there is no fluctuation in ${C_{py}}$ at any of the points (figure 13e, f), confirming the occurrence of the steady flow that is shown by the spanwise vorticity $(\omega _z^\ast )$ in sections A, B and the saddle planes (figure 13g). The scenario in pattern I (figure 13h,i) is opposite to that in pattern IV. The ${C_{py}}$ signals at two similar points (e.g. A1 and B1, A2 and B2) are perfectly in phase (figure 13h,i), suggesting the synchronous occurrence of vortex shedding from the whole cylinder span (figure 13j).

The C_p fluctuation is linked to the flow separation and hence the flow separation occurring from the whole span or a half-span can be understood from the cross-correlations of C_p fluctuation along the span. The cross-correlation analysis is performed for λ/D_m = 6.01 (flow pattern IV) and λ/D_m = 3.43 (flow pattern II), as shown in figure 14. The value of C_p at z/λ = 1.0 is chosen as the reference signal to carry out the cross-correlation for the whole span. For λ/D_m = 6.01 (figure 14a), at phase ϕ = 0, the cross-correlation coefficient (Cro) is 1.0 for the upper half-span (z/λ > 0.5) and −1.0 for the lower half (z/λ < 0.5) while the peaks and valleys are changing alternately and oppositely between the two halves. It suggests that vortex sheddings from the two halves are out of phase, consistent with the flow structure (figure 10c2). The cross-correlation coefficient approaches 0 at z/λ = 0.4–0.6, suggesting that the flow is more or less steady near the saddle plane. When λ/D_m = 3.43, the value Cro = 1.0 appears for the whole span (figure 14b), with peaks and valleys materializing alternately. In other words, alternate vortex shedding occurs from the whole span, similarly to pattern II (figure 10a2).

Figure 14. Cross-correlation coefficient of sectional pressure coefficient (C_P) with that at Z/λ = 1.0 (node); (a) λ/D_m = 6.01 (flow pattern IV), (b) λ/D_m = 3.43 (flow pattern II).

4.6. Strouhal number

Vortex shedding behaviour in the wake is investigated, based on fluctuating lateral velocities (v), together with their power spectral density (PSD) functions, sampled at different spanwise locations in the wake. Representative λ/D_m = 3.43, 6.01 and 8.59 in patterns II, IV and V, respectively, are considered, as they are associated with highly three-dimensional characteristics, particularly along the spanwise direction. Figure 15 presents the time histories of v signals, along with their corresponding PSDs. The v signals were sampled at (x*, y*) = (3, 1) in the node, saddle and middle planes. In the case of pattern II, the v signals at the three planes fluctuate periodically with time, having negligible phase shifts (figure 15a). The amplitude of oscillation is, however, relatively small at the saddle plane and large at the node plane. The corresponding PSDs show pronounced peaks at the normalized frequency of f* = 0.144 (figure 15b). The observation indicates an identical frequency of vortex shedding at different spanwise locations.

Figure 15. Time histories (a,c,e) and PSD functions (b,d, f) of the lateral velocity v* sampled at (x*, y*) = (3, 1). (a,b) λ/D_m = 3.43 (flow pattern II); (c,d) 6.01 (IV); and (e, f) 8.59 (V).

For patterns IV and V (figure 15c–f), the v signals at the different planes behave distinctively. First, the amplitude of the v signal in the saddle plane is substantially smaller than that in the node and middle planes, essentially zero for pattern IV (figure 15c). Second, the v signal in the saddle plane is not exactly out of phase with that in the node for pattern IV as we expect. This is because the formation length of vortex shedding differs between the saddle and node planes. Although the amplitude of fluctuations differs between planes, the PSDs of the signals at different spanwise locations exhibit pronounced peaks at f* = 0.117 and 0.122 for patterns IV and V, respectively (figure 15d, f).

Figure 16 shows the variation in St with λ/D_m. The red horizontal line represents St ₀, Strouhal number of the straight EC. The value of St ≈ St ₀ for pattern I, indicating straight-cylinder-like flow, i.e. 0λ flow. When pattern I is modified to pattern II, St is considerably reduced. This is due to the presence of streamwise vortices and their role in undulating the primary spanwise vortices (figure 5b1,b2). As such, the vortex shedding is retarded to some extent. The flow is steady in pattern III, hence $St = 0$. The value of St is more or less constant in pattern IV and grows with λ/D_m in pattern IV.

Figure 16. Variations of Strouhal number (St) with λ/D_m; St ₀ represents the Strouhal number of the straight EC.