2.1 Numerical method

DNS have been carried out in this study. The governing equations are the continuity and incompressible Navier–Stokes equations:

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{i}}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}t}+u_{j}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{i}}+v\frac{\unicode[STIX]{x2202}^{2}u_{i}}{\unicode[STIX]{x2202}x_{j}\unicode[STIX]{x2202}x_{j}}, & \displaystyle\end{eqnarray}$$

where $(x_{1},x_{2})=(x,y)$ are Cartesian coordinates, $u_{i}$ is the velocity component in the direction $x_{i}$ , $t$ is time, $\unicode[STIX]{x1D70C}$ is fluid density, $p$ is pressure and $\unicode[STIX]{x1D708}$ is kinematic viscosity. The DNS are carried out with the spectral/hp element method by using an open-source code $\text{Nektar}++$ (Cantwell Reference Cantwell2015). The embedded unsteady Navier–Stokes solver is selected, together with a second-order implicit–explicit (IMEX) time-integration scheme, a velocity correction scheme and a continuous Galerkin projection. Hereafter, the streamwise and transverse velocities are respectively denoted as $u_{x}$ and $u_{y}$ , which are normalised with the free-stream velocity $U$ already.

2.2 Computational domain and mesh

The numerical simulations adopt a rectangular computational domain with the centre of the cylinder located at $(x,y)=(0,0)$ . The computational domain size is $-60\leqslant x/D\leqslant 420$ in the streamwise direction and $-60\leqslant y/D\leqslant 60$ in the transverse direction. Figure 1 shows the macro-elements of the computational mesh near the cylinder. The macro-elements are further subdivided using fifth-order Lagrange polynomials ( $N_{p}=5$ ) on the Gauss–Lobatto–Legendre points for the quadrilateral expansions. In figure 1, the cylinder perimeter is discretised with 48 macro-elements, and the radial size of the first layer of macro-elements next to the cylinder is $0.0055D$ . To capture detailed near-wake and far-wake flow structures, a relatively high mesh resolution is used in the wake region by specifying the streamwise size of the macro-elements at the wake centreline ( $y=0$ ) varying linearly from $0.1875D$ at $x/D=2$ to $2.5\times 0.1875D$ at $x/D=400$ . The computational mesh consists of a total of 46 091 macro-elements, of which approximately 44 000 macro-elements are placed in the region of $x/D>2$ because the transition to the secondary vortex street (especially at low $Re$ values such as $Re=200$ ) is extremely disturbance-dependent (Thompson et al. Reference Thompson, Radi, Rao, Sheridan and Hourigan2014) and therefore mesh-dependent. A detailed mesh independence check is reported in Appendix A.

Figure 1.

Macro-elements of the computational mesh near the cylinder.

The boundary conditions are specified as follows. At the inlet and transverse sides, a uniform velocity ( $U$ , 0) is specified. At the outlet, the Neumann boundary condition (i.e. a zero normal gradient) is applied for the velocity. At the cylinder surface, the no-slip condition is imposed for the velocity. The pressure at the outlet is specified as a reference value of zero, while the pressure at all other boundaries uses a high-order Neumann boundary condition.

The internal flow follows an impulsive start at the beginning of the simulation. The time step size is chosen based on the criterion that the Courant–Friedrichs–Lewy (CFL) number is kept below 0.5.

With a focus on the near-wake primary vortex street, Henderson (Reference Henderson1997) found that the two-dimensional flow in the wake of a circular cylinder is perfectly time-periodic for $Re$ up to at least 1000. However, as will be shown in § 3.4, the flow starts to become slightly aperiodic when the primary vortex street is about to break down to form the two-layered vortices. In the present study, each case is first calculated for at least 1800 non-dimensional time units (defined as $t^{\ast }=tU/D$ ) to ensure that the entire wake flow has become fully developed. Since the intermediate- and far-wake flows are aperiodic, each case is run for at least another 900 non-dimensional time units (i.e. at least 200 primary vortex shedding cycles) to achieve meaningful results that are statistically stationary.

3.1 Time-averaged flow

Figure 2(b–e) shows some flow characteristics for $Re=600$ that are averaged over 950 non-dimensional time units. The time-averaged contours are generally symmetric about the wake centreline, which suggests that the statistical range is sufficient. Very slight asymmetries about the wake centreline are observed in figure 2(b,d,e) at $x/D\gtrsim 38$ , which is because the secondary vortex street in the far wake is highly spatially irregular (figure 2 a) and therefore a perfectly symmetric time-averaged flow would require a significant amount of statistical time period. Since the exact symmetry of the secondary vortex street is not the main focus of the study, a further increase in the statistical range is not attempted. Dynnikova et al. (Reference Dynnikova, Dynnikov and Guvernyuk2016) also presented similar patterns of the time-averaged velocity fields for $Re=600$ .

In the present study, the streamwise locations for the first and second transitions are determined by the time-averaged $u_{y}$ field shown in figure 2(c). Two pairs of local maxima are observed in figure 2(c) at $x/D=19.1$ and 29.1 (as marked by two vertical dashed lines). The positive and negative maxima in each pair are symmetric about the wake centreline. The first location $x/D=19.1$ corresponds to the first transition (see figure 2 a), where the flow is diverted away from the wake centreline. Hence a positive maximum in the $u_{y}$ field is observed above the wake centreline while a negative maximum is observed below the wake centreline (figure 2 c). The second location $x/D=29.1$ corresponds to the second transition (see figure 2 a), where the flow starts to reoccupy the wake centreline. Hence a negative maximum in the $u_{y}$ field is observed above the wake centreline while a positive maximum is observed below the wake centreline (figure 2 c). The local maxima in the $u_{y}$ field are the locations where the flow diversions are most pronounced, and therefore these locations are defined as the transition locations.

The transition locations are marked in all the time-averaged flow fields shown in figure 2. Distinct features at the transition locations are observed in each of the panels. For the time-averaged Reynolds stress $(u_{x}^{\prime }u_{y}^{\prime })$ field shown in figure 2(e), local maxima also appear at the transition locations, where the flow shearing is the strongest. The $u_{x}^{\prime }$ and $u_{y}^{\prime }$ values are defined as

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle u_{x}^{\prime }=\sqrt{\frac{1}{N}\mathop{\sum }_{i=1}^{N}(u_{x,i}-\bar{u}_{x})^{2}}, & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle u_{y}^{\prime }=\sqrt{\frac{1}{N}\mathop{\sum }_{i=1}^{N}(u_{y,i}-\bar{u}_{y})^{2}}, & \displaystyle\end{eqnarray}$$

where $N$ is the number of values in the time history, and the overbar denotes the time-averaged value. For the time-averaged $u_{x}$ field shown in figure 2(b), large gradients in $u_{x}$ are observed at the wake centreline at approximately the two transition locations, which is due to the flow diversion in the transverse direction. Between the two transition locations (where the two-layered vortices take place), the streamwise velocity near the wake centreline decreases to extremely small values, as indicated by the blue area in figure 2(b). This area was called the ‘calm region’ in Durgin & Karlsson (Reference Durgin and Karlsson1971). The flow and vortices largely travel around this calm region rather than through it. Consistently, a rapid recovery (reduction) of the negative pressure at the location of the first transition is observed in figure 2(d).

3.2 Frequencies in the wake

Figure 3 shows the frequency spectra of the time histories of $u_{y}$ sampled at $y=0$ and various streamwise locations. The frequency spectra are calculated from the fast Fourier transform (FFT) of the time history of $u_{y}$ over a period of 215 primary vortex shedding periods. Two sharp peaks $f_{1}$ and $f_{2}$ are observed in the frequency spectra shown in figure 3. The first peak $f_{1}$ , which contains only a single frequency value, represents the primary vortex shedding frequency. The second peak $f_{2}$ , which consists of a sharp peak together with small-scale broad-band frequencies at the two sides of the peak, is associated with the development of the secondary vortex street.

Figure 3.

Frequency spectra of $u_{y}$ for $Re=600$ . The time histories of $u_{y}$ are sampled at $y=0$ and various streamwise locations. The frequency spectra are calculated from the FFT of the time history of $u_{y}$ over a period of 215 primary vortex shedding cycles.

In figure 3, the amplitude of $u_{y}$ is normalised by $1/(\sqrt{2}u_{y}^{\prime })$ . Figure 4(c) shows the normalised peak amplitude of $u_{y}$ at each of the two frequencies $f_{1}$ and $f_{2}$ as a function of the streamwise location. The reason for the normalisation is explained with the results before normalisation in figure 4(b). The amplitude of $u_{y}$ shown in figure 4(b) is controlled by two aspects. One is the fluctuation amplitude of the $u_{y}$ signal, and the other is the relative strengths of the two frequencies. The fluctuation amplitude of the $u_{y}$ signal can be quantified by the $u_{y}^{\prime }$ value shown in figure 4(a). The $u_{y}^{\prime }$ curve sampled at $y=0$ displays an obvious trough over the calm region, while the $u_{y}^{\prime }$ curve sampled at $y=1$ (out of the calm region) is much steadier. Different features in the $u_{y}^{\prime }$ versus $x/D$ relationship (figure 4 a) would influence the amplitude of $u_{y}$ versus $x/D$ relationship shown in figure 4(b) significantly. For example, the rapid drop in the amplitude of $u_{y}$ at $f_{1}$ over $x/D\sim 18$ –22 (along $y=0$ ) is completely due to a corresponding drop in the $u_{y}^{\prime }$ value over the same range of $x/D$ . To eliminate such an influence, the amplitude of $u_{y}$ is normalised by $1/(\sqrt{2}u_{y}^{\prime })$ in figure 4(c). After normalisation, the normalised amplitude becomes 1.0 if the time history of $u_{y}$ is perfectly sinusoidal (with an arbitrary amplitude and frequency).

Figure 4.

Variation of the amplitude of $u_{y}$ with the streamwise location for $Re=600$ . The amplitude of $u_{y}$ is examined with (a) root-mean-square of $u_{y}$ , (b) amplitude of $u_{y}$ at the two frequencies $f_{1}$ and $f_{2}$ , and (c) normalised amplitude of $u_{y}$ at the two frequencies $f_{1}$ and $f_{2}$ . The two transition locations are marked by the vertical dashed lines.

As shown in figure 4(c), the normalised amplitude at $f_{1}$ (sampled at $y=0$ ) is very close to 1.0 for the range of $x/D\lesssim 22$ . This is because for $x/D\lesssim 22$ the frequency spectra of $u_{y}$ are dominated by a single frequency value $f_{1}$ (figure 3). The small deviation from 1.0 is because the time history of $u_{y}$ is not perfectly sinusoidal. It is also seen in figure 4(c) that $f_{2}$ emerges at $x/D=21$ , which is slightly after the transition to the two-layered vortices. For $x/D$ from the first detection of $f_{2}$ to the onset of the secondary vortex street, there is a gradual increase in the normalised amplitude at $f_{2}$ . Correspondingly, there is a gradual decrease in the normalised amplitude at $f_{1}$ , since the fluctuation amplitude of the $u_{y}$ signal is increasingly contributed by $f_{2}$ rather than $f_{1}$ . After normalisation, similar variation trends are observed in figure 4(c) for the results sampled at $y=0$ and 1.

After the development of the secondary vortices over a short streamwise distance, the normalised amplitudes at $f_{1}$ and $f_{2}$ decay with distance downstream at $x/D\gtrsim 45$ . The two peaks at $f_{1}$ and $f_{2}$ would eventually disappear into the noise level. For example, figure 3 shows that $f_{1}$ is no longer observed at $x/D\sim 60$ .

3.3 Time evolution of the vortices

Figure 5 shows a typical sequence of the instantaneous vorticity fields for $Re=600$ (see also the supplementary movie ‘Movie1.mp4’ available online at https://doi.org/10.1017/jfm.2019.167). The vorticity fields are shown at $t^{\ast }=1860+nT_{1}$ , where $T_{1}~(=1/f_{1})$ is the primary vortex shedding period, and $n=1$ , 2, 3 and odd integers up to 15. To facilitate tracing of the time evolution of the vortices, the same vortices that appear in different snapshots are labelled with the same letters.

Figure 5.

A sequence of the instantaneous vorticity fields at $Re=600$ . The vorticity contours are shown from $-3$ to 3 with an interval of 0.3. The same vortices that appear in different snapshots are labelled with the same letters. The single vortices are marked by a rectangular frame around the vortex letter. The time evolution of the vorticity field can also be seen from the online supplementary movie ‘Movie1.mp4’ accompanying this article.

Based on flow visualisation, it is seen in figure 5 that the vorticity fields are highly repeatable for $x/D<24$ . In particular, the first transition at $x/D=19.1$ is a highly repeatable alternate rearrangement of the positive and negative vortices into two rows of vortices with the same sign. For $21<x/D<24$ , although $f_{2}$ is detected, its amplitude is still at least approximately one order of magnitude lower than that at $f_{1}$ (figure 4 b), such that its influence is expected to be rather small. The flow becomes visually irregular when the normalised amplitudes at $f_{1}$ and $f_{2}$ exhibit obvious decrease (from 1.0) and increase (from 0), respectively (figure 4 c).

The vortices start to become irregular when they start to rearrange themselves for the second transition. The second transition at $x/D=29.1$ is an irregular pairing and merging process. A typical pairing and merging cycle, as illustrated in figure 5(b,c), takes a period of approximately $2T_{1}$ . It starts with the pairing of a positive vortex B and a negative vortex C in figure 5(b). The two vortices become stretched after encountering each other. Meanwhile, two vortices D and E are located at the two sides of the B  $+$  C pair. The pairing of B  $+$  C blocks the path for the subsequent pairing of D and E. Hence instead of pairing, D and E further squeeze the B  $+$  C pair into a highly stretched pattern, as shown by the dashed blue line in figure 5(c). The B  $+$  C pair is then dismembered through the following processes:

1. (i) With the rotation of D and E, B is partly merged into D while C is partly merged into E. The merging starts from the lower left end of the B  $+$  C pair. The positive and negative vortices may not merge simultaneously. For example, the merging of the positive vortices happens earlier in the three pairing and merging cycles shown in figure 5(c–e).

2. (ii) During the above process, B and C may partly cancel each other.

3. (iii) The remaining part may form a shear layer (e.g. the shear layer between vortices D and E in figure 5 d).

Eventually, vortices B and C disappear while vortices D and E remain in the far wake.

The second and third pairing and merging cycles shown in figure 5(d,e) are similar to the first cycle shown in figure 5(b,c), while the fourth cycle in figure 5(f) is slightly different. Through examining a number of phases of the four cycles (see the online supplementary movie ‘Movie1.mp4’), it is found that with the evolution of the cycles there is a gradually enhanced spatial mismatch of the two vortices at pairing. The positive vortex becomes increasingly downstream of the negative one. For the fourth cycle, there is an obvious mismatch of the pairing of vortices N and O. Consequently, the positive vortex N is barely merged into vortex P, while the negative vortex O is largely merged into vortex Q. The majority of vortex N becomes an obvious shear layer (or even a vortex) in the far wake (figure 5 g).

The mismatch at the pairing becomes more significant in the fifth cycle shown in figure 5(g). Since vortex R is increasingly downstream of vortex S (figure 5 f), it becomes a single vortex (without being paired) at the formation of the secondary vortex street (figure 5 g). The single vortices shown in figure 5 are marked by a rectangular frame around the vortex letter. Vortex S is then paired with vortex T, and this vortex pair is subsequently merged into vortices U and V (figure 5 g).

The mismatch at the pairing that results in a single vortex would induce a subsequent mismatch that leads to another single vortex with the opposite sign. For the sixth cycle shown in figure 5(h), a negative single vortex W is observed prior to the pairing of X and Y. In addition, Y is slightly downstream of X, such that Y is barely merged into AA, while X is largely merged into Z. The majority of Y becomes an obvious vortex in the far wake (figure 5 i). It is worth noting that the formation mechanisms for vortices W and Y are different. Vortex W is a single vortex originated from the mismatch at the point of pairing and merging, while vortex Y is a remaining part after the pairing and merging of vortices X, Y, Z and AA.

For the seventh cycle shown in figure 5(i), there is no mismatch over the pairing of AB and AC. This cycle is similar to the first three cycles shown in figure 5(b–e).

The time evolution of the vortices is further visualised for a period of 501 non-dimensional time units. This time period is equal to both $114T_{1}$ and $61T_{2}$ (where $T_{2}=1/f_{2}$ ). The starting point of the time period is the pairing of B and C in the first cycle shown in figure 5(b). In other words, the time period covers $t^{\ast }=1860+2T_{1}$ to $1860+116T_{1}$ . Over this time period, a total of 53 pairing and merging cycles are observed, together with the emergence of an extra eight positive and eight negative single vortices due to the mismatch (e.g. vortices R and W in figure 5). In total, the two-layered vortices, which contain 114 positive and 114 negative vortices, are paired and merged into 61 ( $=53+8$ ) positive and 61 negative vortices, which is identical to $61T_{2}$ for this time period.

In addition to flow visualisation, the time evolution of the vortices can also be quantified by the time histories of $u_{y}$ sampled in the wake. Figure 6(a,b) shows the time histories of $u_{y}$ sampled along $y=0$ at $x/D=22.5$ and 27.5. Figure 6(c) shows the period $T$ of each cycle of the time histories shown in figure 6(a,b) over a range of $1860+2T_{1}$ to $1860+116T_{1}$ . At $x/D=22.5$ , the two-layered vortices are stably formed (figure 5). The flow is dominated by the primary vortex shedding frequency $f_{1}$ , while the influence of $f_{2}$ is negligibly small (see figure 3). As shown in figure 6(c), there are 114 cycles for the time history of $u_{y}$ sampled at $x/D=22.5$ , and their periods are all very close to $T_{1}$ , such that a single frequency peak $f_{1}$ is observed in figure 3.

At $x/D=27.5$ , the two-layered vortices are about to pair and merge and no longer in a parallel pattern (figure 5). The time history of $u_{y}$ shown in figure 6(b) is modulated by $f_{2}$ significantly, and the corresponding frequency spectrum shown in figure 3 is dominated by $f_{2}$ rather than $f_{1}$ . As shown in figure 6(c), there are 61 cycles for the time history of $u_{y}$ sampled at $x/D=27.5$ , which is consistent with the visualisation of 61 positive and 61 negative vortices at the pairing and merging process. The variation of $T$ for the time history of $u_{y}$ sampled at $x/D=27.5$ is larger than that sampled at $x/D=22.5$ . Hence the frequency range of $f_{2}$ is wider than that of $f_{1}$ (figure 3). The peak frequency $f_{2}$ corresponds to the period $T_{2}$ shown in figure 6(c). It is also noted that $T_{2}~(=8.21)$ is very close to the averaged period of 8.19. For an ideal condition where the pairing and merging process does not generate mismatched single vortices, one would expect $T_{2}=2T_{1}$ (i.e. $2f_{2}=f_{1}$ ). As shown in figure 6(c), the periods for the time history of $u_{y}$ sampled at $x/D=27.5$ are capped at approximately $2T_{1}$ . Owing to the frequent generation of the single vortices, the periods are reduced from $2T_{1}$ . In particular, eight obvious troughs are observed in the time history of $T$ , which is consistent with the visualisation of eight positive and eight negative single vortices. It is also noticed from the time history of $T$ that the single vortices are generated every few pairing and merging cycles, but this process is not exactly regular.

Figure 6.

Time histories of $u_{y}$ sampled at $y=0$ and $x/D=22.5$  (a) and 27.5 (b). In addition, panel (c) shows the period $T$ of each cycle of the time histories over a range of $1860+2T_{1}$ to $1860+116T_{1}$ .

3.4 Irregularity of the flow

As mentioned in § 3.3, the instantaneous vorticity fields shown in figure 5 become visually irregular at $x/D>24$ . The irregularity of the flow is further shown in figure 7 by the spatial distribution of the vortex centres. The vortex centres are extracted from 22 snapshots of the vorticity field at different phases (in contrast to figure 5 for the same phase). For $x/D<24$ , the centres of positive and negative vortices follow two clear trajectories on the two sides of the wake centreline. The spatial locations of the vortices start to become visually irregular at $x/D>24$ when they start to rearrange themselves for the irregular pairing and merging process. After the second transition at $x/D=29.1$ , the positive vortex centres may enter the upper half-domain, while the negative vortex centres may enter the lower half-domain. As the vortices propagate along the streamwise direction, the spatial regions of positive and negative vortices (as shaded in figure 7) both expand along the two sides of the transverse direction. The spatial expansion ratio is approximately $y:x=0.112$ . Within the two shaded regions for the positive and negative vortices, the spatial distribution of the vortices is irregular.

Figure 7.

Spatial distribution of the vortex centres for $Re=600$ . The vertical dashed lines mark the two transition locations, while the horizontal dashed line marks the wake centreline.

For the vortex centres identified in figure 7, the absolute value of the vorticity at the vortex centre is plotted in figure 8 as a function of the streamwise location. Within the first few cylinder diameters of the near wake, the primary vortices undergo intensive rotation after being shed from the cylinder surface, and a nonlinear drop of the vorticity with $x/D$ is observed in figure 8. After that, a linear decay of the vorticity is observed as the vortex propagates in the range of $7\leqslant x/D\leqslant 23$ . This range includes the first transition at $x/D=19.1$ , which is because at this transition the vortices are not subjected to splitting or merging.

Figure 8.

Absolute value of the vorticity at the vortex centre as a function of the streamwise location for $Re=600$ . The two transition locations are marked by the vertical dashed lines.

However, after the second transition at $x/D=29.1$ , some vorticity values appear to be lower than the main curve, as circled in the three groups in figure 8. Group 1 at around $x/D=29.1$ is induced by the stretched vortices at pairing (e.g. vortices B and C in figure 5). The vorticity is decreased because the vortex pair may partly cancel each other over the vortex pairing process, and partly merged into subsequent vortices over the vortex merging process. Group 2 consists of the single vortices (e.g. vortex A in figure 5). The vorticity is slightly smaller because the single vortices are generated without vortex merging. Group 3 consists of the remaining parts of the stretched vortices after the pairing and merging process (e.g. vortex N in figure 5).

The irregularity of the wake flow is further quantified by the irregularity of the time histories of $u_{y}$ sampled at various streamwise locations. The irregularity of a time history can be quantified by the coefficient of variation ( $c_{v}$ ) of the positive and negative peak values extracted from each period of the time history. The positive and negative peak values form two series of data $a_{i}$ and $b_{i}$ . The coefficient of variation for the positive and negative peaks are calculated as

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle c_{v,positive}=\frac{\sqrt{\displaystyle \frac{1}{N}\mathop{\sum }_{i=1}^{N}(a_{i}-\bar{a})^{2}}}{|\bar{a}-\bar{u}_{y}|}, & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle c_{v,negative}=\frac{\sqrt{\displaystyle \frac{1}{N}\mathop{\sum }_{i=1}^{N}(b_{i}-\bar{b})^{2}}}{|\bar{b}-\bar{u}_{y}|}, & \displaystyle\end{eqnarray}$$

where $N$ is the number of values in $a_{i}$ and $b_{i}$ , and the overbar denotes the averaged value. The coefficient of variation for the time history is determined as

(3.5) $$\begin{eqnarray}c_{v}=\frac{c_{v,positive}+c_{v,negative}}{2}.\end{eqnarray}$$

The variation of $c_{v}$ of the time history of $u_{y}$ (sampled at $y=0$ ) with the streamwise location is shown in figure 9. The results are sampled at $y=0$ because the wake centreline $y=0$ goes through the calm region directly and would be most sensitive to the flow transition.

Figure 9.

Variation of $c_{v}$ of the time history of $u_{y}$ (sampled at $y=0$ ) with the streamwise location for $Re=600$ . The two transition locations are marked by the vertical dashed lines. The insets show examples of the time histories sampled at various streamwise locations.

According to (3.3)–(3.5), $c_{v}=0$ indicates that the time history is perfectly time-periodic. Figure 9 shows that the time histories of $u_{y}$ are perfectly time-periodic ( $c_{v}<0.001$ ) for the near-wake region of $x/D\lesssim 15$ , which supports Henderson’s (Reference Henderson1997) argument that the near-wake flow is perfectly time-periodic.

However, the flow field becomes aperiodic ( $c_{v}$ increases gradually) at the first transition. The bottom inset in figure 9 shows the time history of $u_{y}$ sampled at $(x/D,y/D)=(19.5,0)$ . The aperiodicity of the flow is solely due to a gradual variation in the velocity amplitude, while the frequency is unchanged. The unchanged frequency is evidenced by the frequency spectra of $u_{y}$ shown in figure 3, where a single frequency value, namely the primary vortex shedding frequency $f_{1}$ , is observed for $x/D\leqslant 20.5$ . The aperiodicity of the flow (i.e. the variation in the velocity amplitude) recovers slightly at $x/D\sim 22$ when the two-layered vortices are stably formed. It is worth noting that such a subtle aperiodicity of the flow at the first transition is difficult to identify through pure flow visualisation (e.g. figure 5).

With the emergence and increase in the amplitude of $u_{y}$ at $f_{2}$ after $x/D=21$ , the time history of $u_{y}$ is modulated. For example, the middle inset in figure 9 shows the time history of $u_{y}$ sampled at $(x/D,y/D)=(25,0)$ , where two frequencies $f_{1}$ and $f_{2}$ are observed in the corresponding frequency spectrum shown in figure 3. At $(x/D,y/D)=(25,0)$ , the aperiodicity of the flow is attributed to both the variation in the velocity amplitude (which is similar to the bottom inset in figure 9) and the modulation of an additional frequency $f_{2}$ . The existence of two frequencies leads to a pattern of alternate high and low peaks in the time history of $u_{y}$ . Such a pattern becomes increasingly pronounced as the amplitude of $u_{y}$ at $f_{2}$ increases. The growing amplitude at $f_{2}$ with increase in the streamwise location from $x/D=21$ (figure 4 b) leads to an increasing modulation of the time history of $u_{y}$ , and results in a rapid increase in $c_{v}$ (figure 9). After $f_{2}$ becomes dominant (at $x/D>26.5$ ), the low peaks become so small that they gradually disappear, resulting in a new pattern of the time history as shown in the upper left inset of figure 9. As the small low peaks are no longer considered in the determination of $c_{v}$ , a moderate reduction in $c_{v}$ is observed.

After the generation of the secondary vortices, the $c_{v}$ value increases with $x/D$ for $x/D>34$ (figure 9). An example of the time history of $u_{y}$ sampled at $(x/D,y/D)=(50,0)$ is shown in the upper right inset in figure 9. The increasing irregularity of the time history of $u_{y}$ with distance downstream suggests that the secondary vortex street becomes increasingly irregular with distance downstream.

4.1 Time-averaged flow

Figure 10(a,b) shows the instantaneous vorticity field for $Re=300$ at $t^{\ast }=3150$ and 3250, while figure 10(c,d) shows the $u_{y}$ and $u_{x}^{\prime }u_{y}^{\prime }$ fields averaged over 1400 non-dimensional time units. Based on the time-averaged $u_{y}$ field shown in figure 10(c), the two transition locations are determined at $x/D=24.9$ and 60.4.

Figure 10.

Flow characteristics for $Re=300$ : (a) instantaneous vorticity field at $t^{\ast }=3150$ , (b) instantaneous vorticity field at $t^{\ast }=3250$ , (c) time-averaged transverse velocity field, and (d) time-averaged Reynolds stress field. The two transition locations are marked by the vertical dashed lines.

Figure 11.

Frequency spectra of $u_{y}$ for $Re=300$ . The time histories of $u_{y}$ are sampled at $y=0$ and various streamwise locations. The frequency spectra are calculated from the FFT of the time history of $u_{y}$ over a period of 292 primary vortex shedding cycles.

Figures 2(e) and 10(d) show that the local maxima of the $u_{x}^{\prime }u_{y}^{\prime }$ field match well the first transition location for both $Re=300$ and 600, but may not always match the second transition location. For $Re=600$ –1000, two local maxima of the $u_{x}^{\prime }u_{y}^{\prime }$ field are observed within $0.5D$ downstream of the second transition location (e.g. figure 2 e). However, with decreasing $Re$ these two local maxima become increasingly downstream of the second transition location (e.g. figure 10 d). In other words, the most intensive velocity fluctuation happens after the secondary vortex street develops for a certain distance. Therefore, the transition locations are determined based on the transverse velocity field rather than the Reynolds stress field.

It is noted that the transition locations of $x/D=24.9$ and 60.4 represent time-averaged results. Based on visualisation of the time evolution of the flow, it is found that the first transition location is stationary while the second varies with time (see e.g. figure 10 a,b). Kumar & Mittal (Reference Kumar and Mittal2012) also found that the second transition location for $Re=200$ moves back and forth in time. The moving location for the second transition for $Re=200$ –300 appears to be a major difference with respect to those of $Re=400$ –1000 where both transition locations are stationary. This point will be further investigated in § 4.3.

4.2 Frequencies in the wake

Figure 11 shows the frequency spectra of the time histories of $u_{y}$ sampled at $y=0$ and various streamwise locations. Besides the single frequency $f_{1}$ for the primary vortex shedding, the second frequency region $f_{2}$ is broad-band, in the absence of a distinct peak as observed in figure 3 for $Re=600$ . The physical mechanism for a broad-band range of $f_{2}$ is discussed in § 5.2.

In figure 11, the amplitude of $u_{y}$ is normalised by $1/(\sqrt{2}u_{y}^{\prime })$ . Figure 12(c) shows the normalised peak amplitude of $u_{y}$ at each of the two frequencies $f_{1}$ and $f_{2}$ as a function of the streamwise location, while figure 12(b) shows the results before normalisation. As shown in figure 12(c), the largest normalised amplitude at $f_{2}$ is 0.27, which is much smaller than that of 0.68 for $Re=600$ (figure 4 c). This is because at $Re=300$ the frequency spectra at $f_{2}$ are distributed over a broad-band range.

Figure 12.

Variation of the amplitude of $u_{y}$ (sampled at $y=0$ ) with the streamwise location for $Re=300$ . The amplitude of $u_{y}$ is examined with (a) root-mean-square of $u_{y}$ , (b) amplitude of $u_{y}$ at the two frequencies $f_{1}$ and $f_{2}$ , and (c) normalised amplitude of $u_{y}$ at the two frequencies $f_{1}$ and $f_{2}$ . The two transition locations are marked by the vertical dashed lines.

4.3 Time evolution of the vortices

The time evolution of the vortices at $Re=300$ is illustrated by a typical sequence of the instantaneous vorticity fields shown in figure 13 (see also the online supplementary movie ‘Movie2.mp4’). The vorticity fields are shown at $t^{\ast }=3200+nT_{1}$ , where $n=2$ , 3, 4 and even integers up to 14. Based on flow visualisation, it is seen that the vorticity fields are highly repeatable for $x/D<45$ , which includes the first transition. Similar to the results of $Re=600$ shown in § 3, the range of $x/D$ for the highly repeatable flow is that in which the amplitude at $f_{2}$ is at least approximately one order of magnitude lower than that at $f_{1}$ (figure 12 b), such that the influence of $f_{2}$ is too weak to be visually detected.

Figure 13.

A sequence of the instantaneous vorticity fields at $Re=300$ . The vorticity contours are shown from $-1$ to $+1$ with an interval of 0.1, plus $-0.64$ to $-0.56$ as well as 0.56–0.64 with an interval of 0.02 (to highlight the two-layered vortices near merging). The same vortices that appear in different snapshots are labelled with the same letters. The single vortices are marked by a rectangular frame around the vortex letter. The time evolution of the vorticity field can also be seen from the online supplementary movie ‘Movie2.mp4’ accompanying this article.

The vortices start to become irregular when they begin to rearrange themselves for the second transition. The second transition begins with the merging of two same-sign vortices (figure 13), rather than the pairing of two opposite-sign vortices as observed for $Re=600$ (figure 5). At $Re=300$ , when the two opposite-sign vortices (e.g. vortices B and C in figure 13 b) are about to pair, they are however not strong enough to break the restriction of the shear layer in between. Hence instead of pairing, they are each merged with the subsequent same-sign vortex directly (figure 13 c). Similar to the pairing and merging cycle for $Re=600$ , the merging of the same-sign vortices for $Re=300$ would also reduce the number of vortices by half. Therefore, a typical merging cycle for $Re=300$ also takes a period of approximately $2T_{1}$ .

The second merging cycle shown in figure 13(d) is similar to the first cycle shown in figure 13(b,c), while the third cycle shown in figure 13(e,f) is different. As vortices L and N merge together, vortex J is left out to be a single vortex. This phenomenon also happens for the subsequent two merging cycles, where single vortices O and T are observed in figure 13(g,h).

The moving location for the second transition for $Re=300$ originates from the moving location for the merging of the same-sign vortices shown in figure 13. In addition, the streamwise length occupied by the vortex merging process is not fixed. Compared with $Re=600$ , the flow at $Re=300$ is subjected to lower disturbance levels, such that vortex merging may not be triggered at the same streamwise location.

In addition to the flow visualisation in figure 13, the moving location for the vortex merging is also evidenced in figure 14 by the time histories of $u_{y}$ sampled in the wake. Figure 14(a–c) shows the time histories of $u_{y}$ sampled at $y=0$ and three streamwise locations, while figure 14(d) shows the period $T$ of each cycle of the time histories shown in figure 14(a–c). At $x/D=40$ , the two-layered vortices are highly repeatable with time since the amplitude of $u_{y}$ at $f_{2}$ is only 3.3 % of that at $f_{1}$ (figure 12 b). Therefore, the time history of $u_{y}$ shown in figure 14(a) is largely regular with time, and the periods are all very close to $T_{1}$ (figure 14 d). At $x/D=45$ , the amplitude at $f_{2}$ increases to 18.6 % of that at $f_{1}$ (figure 12 b). The amplitude at $f_{2}$ for the time history shown in figure 14(b) is intensively contributed by some short ranges of time, for example $t^{\ast }\sim 2700$ and 3130, rather than generally evenly contributed by the entire range of the time history (as is the case for $Re=600$ ). For example, the FFT of the time history of $u_{y}$ over the short range of $t^{\ast }=2650$ –2730 gives the amplitude at $f_{1}$ identical to that shown in figure 12(b), but the amplitude at $f_{2}$ increased by approximately five times, reaching 93.8 % of that at $f_{1}$ . These short ranges of time are clearly identified in figure 14(d), where a drastic increase in the period from approximately $T_{1}$ to $2T_{1}$ is observed. These are the ranges of time when vortex merging happens closest to the cylinder. At $x/D=50$ , the amplitude at $f_{2}$ increases to 90.8 % of that at $f_{1}$ (figure 12 b), because more periods are increased from approximately $T_{1}$ to $2T_{1}$ (figure 14 d). Owing to the frequent generation of the single vortices, the periods after vortex merging are generally below $2T_{1}$ , which is similar to the results of $Re=600$ shown in figure 6(c). The remaining ranges of time, for example $t^{\ast }\sim 2770$ and 2960, are the ranges of time when vortex merging happens furthest away from the cylinder.

In contrast, for $Re=600$ the vortex pairing and merging process happens at a fixed streamwise location. Hence the time histories of $u_{y}$ shown in figure 6(a,b) do not contain sudden changes of the pattern with time. The amplitude of $u_{y}$ at $f_{2}$ for the time history shown in figure 6(b) is generally evenly contributed by the entire range of the time history rather than some short bursts.

Figure 14.

Time histories of $u_{y}$ sampled at $y=0$ and $x/D=40$  (a), 45 (b) and 50 (c). In addition, panel (d) shows the period $T$ of each cycle of the time histories.

Figure 15.

Spatial distribution of the vortex centres for $Re=300$ . The vertical dashed lines mark the two transition locations, while the horizontal dashed line marks the wake centreline. The inclined dashed lines mark the upper and lower envelopes for the majority of the vortex centres.

4.4 Irregularity of the flow

Figure 15 shows the spatial distribution of the vortex centres for $Re=300$ . The vortex centres are extracted from 19 snapshots of the vorticity field. Two clear trajectories for the centres of positive and negative vortices are observed for $x/D<45$ , while irregular spatial distributions of the vortices are observed for $x/D>45$ when they start to rearrange themselves for the irregular vortex merging process. The inclined dashed lines in figure 15 mark the upper and lower envelopes for the majority of the vortex centres (while extreme conditions may occur because at $Re=300$ the vortex merging process is highly irregular with time). The spatial expansion ratio is approximately $y:x=0.0182$ , which is approximately one-sixth of that at $Re=600$ (figure 7), since the vortices and vortex interactions are weaker at lower $Re$ values. Because of this slow transverse expansion, the positive and negative vortices generally stay in their respective half-domain up to $x/D\sim 120$ , which is much further downstream than the second transition location of $x/D=60.4$ . Figure 16 shows the absolute value of the vorticity for the vortex centres identified in figure 15. As the vortices propagate downstream, the vorticity decays gradually. However, the decay rate is not linear. After the vortex centres display irregularity at $x/D>45$ (figure 15), slight scatters in the vorticity are observed in figure 16.

Figure 16.

Absolute value of the vorticity at the vortex centre as a function of the streamwise location for $Re=300$ . The two transition locations are marked by the vertical dashed lines.

Figure 17 shows the variation of $c_{v}$ of the time history of $u_{y}$ (sampled at $y=0$ ) with the streamwise location for $Re=300$ . For $x/D\leqslant 21$ , the $c_{v}$ values are smaller than 0.001, which suggests that the near-wake flow is perfectly time-periodic. A very slight increase in the $c_{v}$ value is observed at the first transition location, which suggests that the first transition is slightly aperiodic. The aperiodicity comes from a slight variation in the velocity amplitude, while the frequency is a single value. A major increase in the $c_{v}$ value is observed after $x/D=30$ , where $f_{2}$ emerges and increases in the amplitude with increase in $x/D$ . Different from the results at $Re=600$ , this major increase in $c_{v}$ for $Re=300$ is partly due to the moving location for the vortex merging process. For example, the large $c_{v}$ values observed at $x/D=45$ and 50 are due to a mixing of the pre- and post-merging $u_{y}$ signals (figure 14 b,c). With a further increase in the proportion of the post-merging $u_{y}$ signals, the $c_{v}$ value reduces gradually after $x/D=50$ . Eventually, as the secondary vortex street becomes increasingly irregular with distance downstream, an increase in the $c_{v}$ value is observed for $x/D>85$ .

Compared with flow visualisation, the irregularity of the flow is better detected by the $c_{v}$ value. For example, while the flow visualisations in figures 13 and 15 suggest that the flow is highly repeatable up to approximately $x/D=45$ , a large $c_{v}$ value is observed at $x/D=45$ (figure 17). This is partly because the flow irregularity is detected more sensitively by the time histories of the velocity components in the wake (e.g. figure 14) than flow visualisation, and partly because figures 13 and 15 may not always include the short ranges of time when vortex merging happens closest to the cylinder.

Figure 17.

Variation of $c_{v}$ of the time history of $u_{y}$ (sampled at $y=0$ ) with the streamwise location for $Re=300$ . The two transition locations are marked by the vertical dashed lines.

4.5 Results with a truncated wake

It has been indicated in §§ 4.3 and 4.4 that the amplitude of $u_{y}$ at $f_{2}$ and the flow irregularity in the two-layered vortex street are induced by the vortex merging process. These findings are further supported by an additional case with the computational domain truncated at $x/D=50$ to exclude vortex merging. An instantaneous vorticity field of the fully developed flow is shown in figure 18. For this case, the time histories of $u_{y}$ sampled at various streamwise locations along both $y=0$ and $y=1$ are perfectly time-periodic, with the $c_{v}$ values smaller than $10^{-4}$ , which indicates that the flow is perfectly time-periodic. In particular, the $c_{v}$ value at the location of the first transition remains below $10^{-4}$ , which suggests that the slightly aperiodic first transition indicated in figure 17 is also induced by vortex merging.

Figure 18.

Instantaneous vorticity field of the fully developed flow for $Re=300$ at $t^{\ast }=3900$ with the computational domain truncated at $x/D=50$ .

Figure 19 shows the amplitude of $u_{y}$ in the wake (sampled at $y=0$ ) for the case with the truncated domain, together with the results with a sufficiently long domain up to $x/D=420$ (with the results adapted from figure 12 a,b). Figure 19(a) shows that without the influence of the second transition an exponential decay of the $u_{y}^{\prime }$ value is observed for $x/D>34$ for the truncated case. The slightly smaller $u_{y}^{\prime }$ values for the truncated case result in slightly smaller amplitudes at $f_{1}$ (figure 19 b). For the truncated case, only a single frequency value $f_{1}$ is observed in the frequency spectra of $u_{y}$ . The absence of $f_{2}$ for the truncated case proves that the $f_{2}$ signal detected in the two-layered vortex street indeed originates from vortex merging.

Figure 19.

Variation of the amplitude of $u_{y}$ (sampled at $y=0$ ) with the streamwise location for $Re=300$ . The amplitude of $u_{y}$ is examined with (a) root-mean-square of $u_{y}$ , and (b) amplitude of $u_{y}$ at the two frequencies $f_{1}$ and $f_{2}$ . The two transition locations are marked by the vertical dashed lines.

5.1 The transition locations

The flow characteristics for $Re=150$ –1000 are summarised in this section. The transition locations for $Re=150$ –1000, which are determined based on the time-averaged transverse velocity field, are shown in figure 20. The first transition occurs within a small range of approximately $x/D=20$ –30. The second transition moves upstream significantly with increasing $Re$ , as is also observed in Inoue & Yamazaki (Reference Inoue and Yamazaki1999) and Vorobieff et al. (Reference Vorobieff, Georgiev and Ingber2002).

Figure 20.

Transition locations determined based on the time-averaged transverse velocity field. The red vertical bars indicate approximate ranges for the movement of the second transition location over time.

It is noted that the transition locations shown in figure 20 represent time-averaged results. In addition, the time evolution of the flow is checked through flow visualisation. For $Re\geqslant 400$ , both transition locations do not vary with time. For $Re\leqslant 300$ , the first transition location is stationary while the second moves back and forth in time. The moving location for the second transition originates from the moving location for the merging of the same-sign vortices and a varying streamwise length occupied by the vortex merging process. The red vertical bars shown in figure 20 indicate approximate ranges for the movement of the second transition location over time. The range of movement increases with decreasing $Re$ .

5.2 Frequencies in the wake

As illustrated in figure 4(b,c) for $Re=600$ and figure 12(b,c) for $Re=300$ , the $f_{2}$ signal is first detected in a short streamwise distance after the first transition. Figure 20 summarises the streamwise locations for the first detection of $f_{2}$ for $Re=200$ –1000. For $x/D$ from the first detection of $f_{2}$ to the second transition (the shaded range in figure 20), there is a gradual changeover in the normalised amplitude of $u_{y}$ from $f_{1}$ to $f_{2}$ (e.g. figures 4 c and 12 c). The streamwise location for the same amplitude at $f_{1}$ and $f_{2}$ is also shown in figure 20. The $f_{1}$ -dominated range is larger than the $f_{2}$ -dominated range. This is because, when $f_{2}$ is first detected, its amplitude is only approximately $1\times 10^{-4}$ – $1\times 10^{-3}$ , and it takes a certain streamwise distance for the amplitude to grow over a few orders of magnitude.

The frequency spectra of $f_{2}$ for $Re=200$ –1000 are governed by the pairing and merging of the two-layered vortices. With increasing $Re$ , the pairing and merging of vortices become increasingly regular with time. At relatively low $Re$ of 200–300, the irregular movement of the streamwise location for the vortex merging and the irregular generation of the single vortices lead to large variations in the periods of the time histories of $u_{y}$ (e.g. figure 14 d), which result in broad-band frequency spectra of $f_{2}$ (e.g. figure 11). In contrast, for $Re=400$ –1000 the vortex pairing and merging process happens at a fixed streamwise location. Hence the periods of the time histories of $u_{y}$ (e.g. figure 6 c) are mainly concentrated in a relatively small range, and a sharp $f_{2}$ peak is observed in the frequency spectra (e.g. figure 3). For $Re=400$ –600, the sharp $f_{2}$ peak is accompanied by small-scale broad-band frequencies at the two sides of the peak (e.g. figure 3), because the irregular generation of the single vortices would result in irregular local reductions in the periods of the time histories of $u_{y}$ (e.g. figure 6 c). For $Re=800$ –1000, the generation of the single vortices becomes more regular with time. For example, an examination of the time evolution of the flow at $Re=1000$ over 94 primary vortex shedding cycles shows that there are 34 pairing and merging cycles, and each cycle is accompanied by one or two single vortices in the pattern of (i) a positive single vortex, (ii) a negative single vortex, or (iii) positive and negative single vortices. The patterns for each of the 34 pairing and merging cycles are summarised in figure 21. The shaded and clear grids indicate the existence and absence of a single vortex, respectively. The absence of a positive/negative single vortex generally happens every four to five cycles. In addition, figure 21 contains three identical ranges that repeat every nine cycles. The largely regular generation of the single vortices results in a sharp $f_{2}$ peak in the frequency spectrum with hardly any broad-band frequencies at the two sides (figure 22). Without the influence of the broad-band frequencies, some harmonics of $f_{1}$ and $f_{2}$ are also identified.

Figure 21.

Single vortex pattern for each of the 34 pairing and merging cycles observed at $Re=1000$ .

Figure 22.

Frequency spectrum of $u_{y}$ sampled at $(x/D,y/D)=(26,0)$ for $Re=1000$ . The frequency spectrum is calculated from the FFT of the time history of $u_{y}$ over a period of 199 primary vortex shedding periods.

Because the merging of the two-layered vortices is accompanied by frequent generation of the single vortices, the peak frequency $f_{2}$ for $Re=400$ –1000 and the majority of the broad-band range of $f_{2}$ for $Re=200$ –300 are larger than $0.5f_{1}$ . For $Re=400$ –1000, $f_{2}$ in the frequency spectra is determined by the number of vortices observed in the flow through the following expression:

(5.1) $$\begin{eqnarray}\frac{f_{2}}{\,f_{1}\,}=\frac{\text{number of vortices after the merging}}{\text{ number of vortices before the merging }}.\end{eqnarray}$$

For example, table 1 lists the numbers of vortices observed before and after the merging for certain non-dimensional time units for $Re=400$ , 600 and 1000. The results shown in table 1 satisfy equation (5.1).

Table 1.

Numbers of vortices observed before and after the merging for certain non-dimensional time units.