Secondary vortex street in the intermediate wake of a circular cylinder

Abstract

Particle image velocimetry measurements are carried out in the wake of a circular cylinder at two values of the Reynolds number (3460 and 5800), based on the free stream velocity and the cylinder diameter, to investigate the spatial organization of vortical motions in the intermediate wake. The proper orthogonal decomposition method (POD) is used to extract information from the vorticity data. While the coherent motion associated with the von Kármán vortex street is well reflected in the first two POD modes which account for about 8% of the total enstrophy, the motion associated with the secondary vortex street is captured in the third and fourth POD modes which account for only about 2.4% of the total enstrophy. The measurements show that the secondary vortex street only alternates in space. This is significantly different from the far wake where there is continuous switching between symmetric and antisymmetric arrangements about the centreline (e.g. Bisset et al., J Fluid Mech 218:439–461, 1990), the alternative regime being nearly twice as frequent as the opposite regime.

Graphical abstract

1 Introduction

The quasi-organized large structures (or secondary vortex street) in the far-wake, which are much weaker than the von Kármán vortex street in the near-wake, have been well-established by various measurement techniques, e.g. flow visualization (Keffer 1965; Papailiou and Lykoudis 1974; Cimbala 1984), hot-wire (Keffer 1965; Townsend 1979; Antonia et al. 1987; Browne et al. 1989; Zhou and Antonia 1995) or particle image velocimetry (PIV) measurements (Tang et al. 2015b). However, although at least three possible mechanisms for the formation of the far-wake coherent structures have been proposed, there is a lack of agreement between different proposals. First, based on results at a Reynolds number $R_d={U_\infty }d/\nu = 1700$, where $\nu$ is the kinematic viscosity of the fluid, $U_\infty$ is the free stream velocity and d is the cylinder diameter, Keffer (1965) argued that the horseshoe or hairpin-shaped vortex structure generated at the cylinder may hold the key to the genesis of the organized large structures in the far-wake. Second, at $R_d=160$, Matsui and Okude (1981, 1983) proposed that the organized large structures in the far-wake are caused by the amalgamation or pairing of the von Kármán vortices in the near-wake. Third, Taneda (1983) suggested that the secondary vortex street in the far-wake is caused by a hydrodynamic instability in the wake profile; this proposal received some support from the experiments of Cimbala (2006) for $R_d=90$–2200. Using a linear stability analysis method, Kumar and Mittal (2012) found a region around $x/d\approx 46.5$, where the mean velocity U on the flow centreline has a minimum, is most significant for the appearance of the secondary vortex street. Dynnikova et al. (2016) further highlighted this region using the time-averaged integral of $y{\overline{\omega }}_z$, $M(x)=\int _{ - \infty }^ \infty {y{\overline{\omega }}_z } \mathrm{d}y$, which represents the density of the dipole moment, where $\overline{\omega }_z$ is the mean spanwise vorticity (the overbar denotes time averaging). These criteria used to identify the origin of the secondary vortex street have been well confirmed at relatively low $R_d$ ($\le 600$) (Kumar and Mittal 2012; Mizushima et al. 2013; Dynnikova et al. 2016). However, whether or not they are applicable to high $R_d$ (say, $>1000$) is yet to be ascertained; this will be examined in the present paper. It should also be mentioned that, the results of Browne et al. (1989), Kumar and Mittal (2012), Dynnikova et al. (2016) over the range of $R_d=150$–5580 showed that the intermediate region ($20< x/d < 50$ or so) of the wake appears to be important in the development of the secondary vortex street. The present study draws on this idea and focuses mainly on this region of the wake, which is dominated by relatively strong coherent structures, whose interaction is believed to influence the dynamics of the flow. The main objective of the study is to extract these coherent structures or at least their signature, and analyse them with a view to gaining insight into the dynamics of the intermediate region of the wake and hence their likely influence on the development of the far-wake.

In the present paper, the main focus is on the intermediate wake, which is analysed using the proper orthogonal decomposition method (POD; Lumley 1967). Relatively recently, Tang et al. (2015b) have carried out a systematic comparison between the velocity-based POD and the vorticity-based POD in the three characteristic regions of the flow downstream of a two-dimensional circular cylinder, namely the near, intermediate and far wakes. They showed that the vorticity-based POD is quite efficient at segregating the von Kármán vortex street from the secondary vortex street in the region $x/d=36$–47, where x and d are the downstream distance from the cylinder centre and the cylinder diameter, respectively. Since this study aims to elucidate the development and the spatial arrangement of the secondary vortex street in the intermediate wake, Tang et al.’s (2015b) vorticity-based POD approach is used.

2 Experimental details

Experiments were carried out in a closed circuit wind tunnel with a 5.6-m-long working section ($0.8\,\hbox {m}\times 1.0\,\hbox {m}$). The flow non-uniformity was within $\pm \,0.3$ % (r.m.s.) within the central cross-sectional area of about $0.74\,\hbox {m} \times \,0.95\,\hbox {m}$, and the longitudinal turbulence intensity was about 0.5%. A circular cylinder of $\textit{d}=10\,\hbox {mm}$ in diameter was mounted horizontally across the working section, resulting in an aspect ratio of 80. The origin of the coordinate system is defined at the centre of cylinder, with x and y along the streamwise and transverse directions, respectively (Fig. 1). Two sets of measurements have been conducted. The first was carried out at a free stream velocity ${U_\infty }=8.7\,\hbox {m/s}$, corresponding to a Reynolds number $R_d={U_\infty }d/\nu = 5800$. The measured region is $x/d=4$–170 (each view field covered a distance ranging from 15 to 23d in the x direction) of the wake. The second was carried out at a free stream velocity ${U_\infty }=5.2\hbox {m/s}$, corresponding to a Reynolds number $R_d=3460$ and the measurements were carried out in the region $x/d=0$–47, with a view field of about 23d in the x direction.

A two-component time-resolved Dantec PIV system was used to measure the velocity field in the (x, y) plane of the wake. The flow was seeded with smoke generated from olive oil by a TSI 9307-6 particles generator. The seeding particle average diameter is approximately 1 $\upmu$m. Flow illumination was provided by a dual-beam laser source of 30 mJ/pulse (Litron LDY304-PIV, Nd:YLF) with spherical and cylindrical lenses to form a thin light sheet of approximately 1 mm thickness. The wavelength of the laser source is 532 nm. The time separations between the laser pulse were 100 and 60 $\upmu$s for $R_d = 3460$ and 5800, respectively. The trigger rates for the double frame mode were fixed at 700 Hz for $R_d= 3460$ and 727 Hz for $R_d=5800$, respectively. The PIV images were captured using a high-speed camera (Phantom V641, 4-megapixel sensors and 2560 pixels $\times$ 1600 pixels resolution). A total of 2000 PIV image pairs were captured for each measurement. Convergence tests were conducted for the mean spanwise vorticity $\overline{\omega _z}$ and the variance (or mean spanwise enstrophy) $\overline{\omega _z^2}$ at $x/d = 25$ and $y/d = 0$. The two quantities were found to converge to within an uncertainty of 2% when the number of sampling images reached 500. Accordingly, a total of 2000 image pairs was deemed to be adequate for the convergence of the statistics.

Fig. 1

Schematic arrangement and coordinate axis

3 POD method

The POD method has become a well established tool for extracting information on the large scale structures since it was first introduced by Lumley (1967). Note that, while the POD modes are not the actual coherent structures, they are often identified as (or associated with) coherent structures for convenience; we will adopt this common practice here. The basic idea behind the POD method when applied to a fluctuation field, here the vorticity field $\omega ({x,y,t})$, is to project $\omega ({x,y,t})$ onto an orthogonal coordinate system $\varPhi (x{,y})$ which maximizes the following expression

$$\begin{aligned} \frac{{ \left\langle {{[\varPhi (x{,y}),\omega (x{,y},t)]}^2} \right\rangle }}{{[\varPhi (x{,y}),\varPhi (x{,y})]}}. \end{aligned}$$

(1)

The operator $\langle \cdots \rangle$ denotes a time average. The projections are optimal in the sense that the first few projections capture most of the energy (here the enstrophy). In the present case, this is achieved by solving the following integral equation

$$\begin{aligned} \int {{R_{ij}}} (x,y,{x'},{y'}){\varPhi _j}({x'},{y'})\mathrm{d}{x'} \mathrm{d}{y'}= \lambda {\varPhi _i}(x,y), \end{aligned}$$

(2)

where $R_{ij}$ is the two-point vorticity correlation tensor

$$\begin{aligned} {R_{ij}}(x,y,{x'},{y'}) = \left\langle {\omega _i}(x,y,t)\omega _j^*({x'},{y'},t) \right\rangle . \end{aligned}$$

(3)

The solution to Eq. (2) can be found by the Hilbert–Schmidt theory since the kernel of Eq. (2) is Hermitian symmetric and the flow field has finite enstrophy [or mathematically speaking, square integrable (Berkooz et al. 1993)].

For a snapshot method of POD (Sirovich 1987) the integral equation becomes

$$\begin{aligned} {Q^\mathrm{T}}Qv = \lambda v, \end{aligned}$$

(4)

where ${Q^\mathrm{T}}Qv$ is the two-point space-correlation tensor in matrix form given by the product of the fluctuation part of the vorticity at different spatial locations, and $\lambda$ and $\nu$ are the eigenvalues and the eigenvectors of ${Q^\mathrm{T}}Qv$, respectively. The variable Q is generally arranged as follows

$$\begin{aligned} Q = \left[ \begin{array}{llll} \omega _{i1}^1&{}\quad \omega _{i1}^2&{}\quad \cdots &{}\quad \omega _{i1}^N\\ \omega _{i2}^1&{}\quad u_{i2}^2&{}\quad \cdots &{}\quad \omega _{i2}^N\\ \vdots &{}\quad \vdots &{}\quad \cdots &{}\quad \vdots \\ \omega _{iM}^1&{}\quad \omega _{iM}^2 &{}\quad \cdots &{}\quad \omega _{iM}^N, \end{array} \right] \end{aligned}$$

(5)

where $i\,(={x,y,z})$ denotes the fluctuating part of each of the three vorticity components. Note that only $\omega _z$ has been used in the present paper. The indexes N and M are the number of snapshots (from $1, 2, \ldots , N$) and the positions of velocity vectors in a given snapshot (from $1, 2,\ldots , M$), respectively. Ordering the solutions of Eq. (5) according to the size of the eigenvalues ${\lambda ^k}$, the POD modes can be calculated as follows

$$\begin{aligned} {\varPhi _k}(x{,y}) = \frac{1}{{\sqrt{{\lambda _k}} }}A{v_k}, \end{aligned}$$

(6)

where the first few modes are normally assumed to be the most energetic modes. Projecting the fluctuating field $\omega (x,y)$ onto the POD modes, we can obtain the POD coefficients that express the relative importance of the different modes.

$$\begin{aligned} {a_k}(t) = \varPhi _k^T(x{,y})\omega (x,y,t). \end{aligned}$$

(7)

Then, the expansion of the fluctuating part of a snapshot k is

$$\begin{aligned} \omega ^k(x{,y},t)= \varPhi _k{a_k}(t). \end{aligned}$$

(8)

4 Results and discussion

We first carry out measurements to characterise the shear layer. To this end, the growth rate of the shear layer may be assessed by determining the vorticity thickness ${\delta _w}$ defined as

$$\begin{aligned} {\delta _w}({x/d}) = \frac{{{U_\infty }}}{{\partial { U}/\partial y{|_{\max }}_y}}, \end{aligned}$$

(9)

where U is the local mean velocity. Figure 2 shows the variation of the vorticity thickness along the downstream distance x / d for $R_d=3460$ and 5800, respectively. Also shown on the figure are the best fits for the data over the ranges $x/d=3$–20 (${\delta _w}({x/d})\sim -\,x^{0.3}$) and 50–120 (${\delta _w}({x/d})\sim -\,x^{0.91}$). It can be observed from this figure that ${\delta _w}({x/d})$ exhibits a power-law behaviour over the range $x/d=50$–120. Interestingly, The Kármán-vortex-street-dominated flow region ends at $x/d \simeq 46$ and 48 for $R_d= 5800$ and 3460, respectively (Mi and Antonia 1999), a region just before the power-law behaviour occurs. Beyond this region, the flow becomes approximately self-preserving (Thiesset et al. 2013; Tang et al. 2015a). Further, one may identify an approximate scaling range over the range $x/d=3$–20 at $R_d= 5800$ and 3460 respectively, where the contributions of the coherent structures (Kármán vortices) are significant. Indeed, Zhou et al. (2002) showed that the averaged contributions from the coherent motion to $\overline{v^2}$ on the flow centreline are 64.9, 33.6 and 5.9% at $x/d=10$, 20 and 40, respectively ($R_d= 5800$). It is also observed that $x/d\approx 20$–50 turns out to be the range where the Kármán vortex street and secondary vortex street co-exist (Tang et al. 2015b; Browne et al. 1989); this region is the focus of this study.

Fig. 2

${\delta _w}/d$ as a function of x / d. Black and blue symbols correspond to $R_d=3460$ and 5800, respectively; pink line, $\sim x^{0.91}$; red lines, $\sim x^{0.3}$

The view window of the camera covers the area delimited by $x/d = 22$–47 and $y/d= -\,7.6$ to 7.6 at $R_d= 3460$. The POD analysis is performed only for the vorticity data over the area delimited by $x/d = 22$–47 and $y/d= -\,3$ to 3. Note that all the following results are from POD analysis on the data in this region. Figure 3 shows the distribution of the POD mode contributions for the enstrophy and its cumulative summation. Although slow, the convergence of the cumulative contribution of POD modes reaches 100% as expected. The first two modes are the most important, each contributing about 4% (to the total enstrophy). This is consistent with results found in the literature for the near-wake. For example, Feng et al. (2011) carried out PIV measurements in the region $x/d \le 3$ and showed that the first two modes dominate the POD mode distribution for the enstrophy. However, their contribution to the enstrophy is larger than the present one, about 60% for $R_d =950$, which is expected because the vortical structures are more coherent and more energetic in the region $x/d \le 3$ than in the region $x/d = 22$ to 47; note that Tang et al. (2015b) have provided a systematic comparison between velocity- and vorticity-based POD methods in a turbulent wake and found that the first two modes of both methods account for about 65% of the total energy/enstrophy in the near-wake while in the intermediate wake, the contributions from the first two modes decreases significantly (less than 20% for both methods). It is largely due to the decay of the Kármán vortex street since the coherent contribution from the Karman vortices to the turbulent kinetic energy decreases significantly when x / d increases from 10 to 40 (see for example figure 10 of Chen et al. 2016). On the other hand, Chen et al. (2016) have estimated the turbulent energy production at $x/d=10,\,20$, and 40, respectively, and found that the maximum concentrations of the three-dimensional turbulent energy production at $x/d= 10$ and 40 are 16 and 22% larger than the corresponding contributions of two-dimensional cut. This implies that the three-dimensionality of the flow is enhanced as x / d increases (see also Williamson 1996). This enhancement should also partially explain why, for the present POD analysis, the first two POD modes contribute only 8% to the total enstrophy. Further, Feng et al. (2011) showed that these two modes are associated with the Kármán vortex street. We, too, will show later that, for the present vorticity-based POD results, the first two modes are associated with the Kármán vortex street. We will also show that the third and fourth modes, each accounting for about 1.2% of the total enstrophy, represent the secondary vortex street.

Fig. 3

Percentage contribution of the single POD modes (circles) and its cumulative summation (squares) in term of the POD modes

Aside from the percentage contribution to the total enstrophy in Fig. 3, the importance of the different modes is also reflected by the POD coefficients that are determined by projecting the vorticity fields onto the POD modes as illustrated by Eq. (7) for a given vorticity field. The relation between two modes can be shown in a scatter plot of the two coefficients. Figure 4 shows such a scatter plot for the coefficients of the first four modes: $a_2^+$ as a function of $a_1^+$ and $a_4^+$ vs. $a_3^+$. The superscript ”+” denotes normalization by $\sum _{k = 1}^{1000} { \langle a_k^2(t) \rangle }$, which represents the total energy. The scatter plots for the distributions of $a_4^+$ vs. $a_3^+$ and $a_2^+$ vs. $a_1^+$ show a circular pattern but with different radii $r = \sqrt{a_1^{ + 2} + a_2^{ + 2}}$. For the distribution of $a_2^+$ vs. $a_1^+$, most points are located near a circle with the radius $r= 0.28$ and the centre at $(a_1^+, a_2^+) = (0,\,0)$. The distribution of $(a_3^+, a_4^+)$ shows a similar circular pattern but with a smaller radius $r=0.15$. Note that $r^2$, which is proportional to the enstrophy contribution, is about 3.3 times larger for $a_2^+$ vs. $a_1^+$ ($r^2=0.077$) than for $a_4^+$ vs. $a_3^+$ ($r^2=0.023$). This is because the contribution to the total enstrophy from the first two modes is significantly higher than that from the third and fourth modes (Fig. 3). The circular patterns indicate a strong connection between modes 1 and 2 (also modes 3 and 4), which will be discussed in some detail in the following paragraphs.

Fig. 4

POD coefficients of the first four modes. The black symbols correspond to $a_2^+$ vs. $a_1^+$; the red symbols correspond to $a_4^+$ vs. $a_3^+$. The superscript ”+” denotes normalization by $\sum _{k = 1}^{1000} { \langle a_k^2(t) \langle }$, which is equivalent to the total energies. The black and red curves correspond to the circles with radiuses $r= 0.28$ and 0.15, respectively

Fig. 5

First five POD modes. a Mode 1, b mode 2, c mode 3, d mode 3, e mode 5

Fig. 6

Time history of the first five POD coefficients (left) and the corresponding power spectra (right). The vertical dashed line(s) in a and b corresponds to $St=0.210$, in c and d correspond to $fd/U_\infty =0.145$ and 0.185, and in e corresponds to $fd/U_\infty =0.255$

The first five modes are shown in Fig. 5; the time history of the first five POD coefficients and the corresponding power spectra are shown in Fig. 6. It can be seen from Fig. 5 that the first two modes exhibit a pattern of alternating regions of positive and negative vorticities along $y/d=0$ and reflecting the organized nature of the von Kármán vortex street in this region (Tang et al. 2015b). The averaged vortex wavelength ${\lambda _c} = {U_c}/{f_v}$ increases from 4.2d to 4.4d when x / d increases from 20 to 40 ($U_c$ is the convection velocity of the vortices, which varies from 0.87${U_\infty }$ to 0.92${U_\infty }$ when x / d increases from 20 to 40 (Zhou and Antonia 1992; Zhou et al. 2002) and $f_v$ is the vortex shedding frequency). These values of ${\lambda _c}$ are identical to those estimated from the first two modes (Fig. 5). Further, the time history of the POD coefficients for the first two modes varies periodically, and the power spectra show the dominant frequency (Fig. 6), corresponding to the Strouhal number $(St=f_vd/U_\infty =0.21$), which is exactly the shedding frequency of the Kármán vortices (e.g. Browne et al. 1989).

Figure 5 shows that modes 3 and 4 also display a similar pattern to that of modes 1 and 2 but with a larger averaged vortex wavelength ${\lambda _c}\approx 5.1d$–6.4d, which is in good agreement with the estimate using ${\lambda _c} = {U_c}/{f_v}=4.9d$–6.2d ($U_c$ is taken to be 0.90${U_\infty }$ and normalized ${f_v}$ (or St) varies from $St=0.145$ to 0.185, see Fig. 6). Browne et al. (1989) showed that (1) there is a broad peak in the spectra of the lateral velocity fluctuations corresponding to the secondary vortex street from about $x/d=14.2$ to 400, and (2) the location of the secondary peak shifts towards smaller frequencies as x / d increases. Figure 4 of Browne et al. (1989) also shows that, over the range $x/d = 22$–47, the peak frequency decreases from about 0.173 to 0.140 when x / d increases from about 20 to 47, which is also in good agreement with the present peak-frequency range $St=0.145$–0.185. Although not shown here, the POD method carried out over the area of $x/d = 22$– 34.5 and $x/d = 34.5$–47 ($y/d= -\,3$ to 3 for both cases) showed that the secondary peak-frequency decreases as x / d increases. Interestingly, we can see a clear transition in the isocontours of modes 3 and 4. Notice how the modes quickly split in two when $x/d \ge 35$; at $x/d\approx 40$, they are divided into two symmetrical parts. The range of x / d ($\approx 30$–40) over which this mode splitting occurs corresponds to the maximum rate of change of ${\delta _w}$ in the x direction (Fig. 2), possibly suggesting the effect of the shear layer on the evolution of the secondary vortex street. Figure 5e shows the contour of the fifth POD mode. It is of significantly smaller averaged size than the first four modes, reflecting less energetic coherent structures. The drop in the energy content as the order of modes increases is illustrated in Fig. 6 which shows the power spectra for the first five POD modes. Notice the decrease in magnitude of the high-frequency peak (Fig. 6) accompanied by a spreading of energy around the peak.

Fig. 7

a Arbitrary instantaneous vorticity field; b reconstructed vorticity field with the first two modes; c reconstructed vorticity field with the third and fourth modes. Note that for b and c, mean values of the vorticity are added

Fig. 8

Dependence of ${U^+=U/U_\infty }$ on the flow centreline at low $R_d$ (filled squares, Cimbala 2006, $R_d=500$; filled circles, Williamson and Prasad 2006, $R_d=150$; black dashed curve, Kumar and Mittal (2012), $R_d=150$; black solid curve, Dynnikova et al. 2016, $R_d=150$) and high $R_d$ (open circles, Antonia and Mi (1998), $R_d=3000$; open squares, Zhou et al. 2002, $R_d=5800$; cross symbols, Chen et al. (2016), $R_d=2500$; red dashed curve, present PIV data, $R_d=3460$; red solid curve, DNS at $R_d=2000$, see text)

Fig. 9

Dependence of $M(x)=\int _{ - \infty }^ \infty {y\overline{\omega }_z } dy$ on the flow centreline at low $R_d$ (blue dashed, blue solid, and blue dot-dashed curves correspond to $R_d=140$, 300, and 600, respectively; Dynnikova et al. 2016) and high $R_d$ (red solid curve, present PIV data, $R_d=3460$; red dashed curve, DNS at $R_d=2000$, see text)

The above analysis shows that the vorticity-based POD method has successfully separated the large scale energy containing motion associated with the von Kármán vortex street from the smaller structures associated with the secondary vortex street. The former is captured by the first and second modes, the latter by the third and the fourth modes. This is well illustrated in Fig. 7 which shows that the von Kármán vortex street and the secondary vortex street can be reconstructed separately using the first and second POD modes and third and fourth POD modes, respectively. As expected and verified in Fig. 7, alternating von Kármán vortices appear, which is consistent with the classical vortex shedding mechanism (e.g. Zhou et al. 2002). Furthermore, the secondary vortex street also shows a similar pattern to the alternating one, in contrast with that observed in the far-wake (e.g. Bisset et al. 1990) where the spatial arrangement of the secondary vortex street switches between opposing and alternating modes (the latter being nearly twice as frequent as the former). We recall that although the instantaneous secondary vortices in the far-wake are strongly three-dimensional, there is a reasonable similarity between the two-dimensional vector plots and flow visualisation which show the three-dimensional nature of the flow in this region (Antonia et al. 1987). The present paper has focused on the intermediate wake, where the three-dimensionality of the secondary vortices should be less pronounced than that in the far-wake and thus the present two-dimensional results should provide further insight into the nature of the three-dimensional structure of the secondary vortices.

One further observation associated with the difference between the von Kármán vortex street and the secondary vortex street needs to be made. It is well-known that the strength of the von Kármán vortex decreases as x / d increases in the intermediate wake of a circular cylinder (as well illustrated in Fig. 5a,b or Fig. 7b). However, the secondary vortex street does not appear to follow the same trend as the von Kármán vortex street. It seems to maintain its relatively strong coherence and strength over the range $x/d\approx 22$–35; it weakens beyond this range. We cannot yet explain this behaviour.

Kumar and Mittal (2012) found that the intermediate wake, in which the mean velocity U on the flow centreline has a minimum, is most significant for the appearance of the secondary vortex street at $R_d=150$. The appearance of the secondary vortex street is most likely due to the instability of the time-averaged flow of the resultant simple shear layer after the primary vortex street is annihilated (Mizushima et al. 2013). Dynnikova et al. (2016) further highlighted that, for $R_d =140$–600, both U on the flow centreline and the quantity $M(x)=\int _{ -\, \infty }^ \infty {y\overline{\omega }_z } \mathrm{d}y$, which represents the density of the dipole moment, exhibit local minima near the location where the Kármán street breaks down which is believed to mark the formation of the secondary vortex street. It is thus interesting to examine the dependence of $U^+=U/U_\infty$ and M(x) on the flow centreline for the present PIV data. They are shown in Figs. 8 and 9, respectively. Also reported on the figures are the available data in the literature and unpublished DNS data for a circular cylindrical wake at $R_d=2000$ (details of the simulation are similar to those found in Lefeuvre et al. (2014)). We can observe that $U^+$ and M exhibit completely different behaviours at low $R_d\,(=140$–600) than at high $R_d\, (=2000$–5800). On one hand, for $x/d>10$, $U^+$ at low $R_d$ decreases as x / d increases before reaching a local minimum at $x/d\approx 46$, whereas for $R_d \ge 2000$, it increases monotonically as x / d increases. On the other hand, for the same x / d range, M at low $R_d$ also has a local minimum which becomes more obvious as $R_d$ increases. However, at high $R_d$ it does not exhibit such a behaviour. The difference between $U^+$ and M at low and high $R_d$ suggests that, at high $R_d$, it is not applicable to use the local minima of $U^+$ and M to determine the origin of the secondary vortex street. This is not surprising since, at low $R_d\, (=140$–600), the origin of the secondary vortex street moves upstream as $R_d$ increases while for high $R_d$ ($>\,1000$) figure 4 of Browne et al. (1989) shows that the secondary vortex street can be traced back to the near wake at $x/d=14$, where the contribution of the von Kármán vortex street to the total turbulent kinetic energy is significantly larger than that of the secondary vortex street (see figure 2 of Browne et al. 1989, which shows that the peak is considerably sharper than that of the secondary vortex street at $x/d=14$).

5 Conclusions

Particle image velocimetry measurements are carried out in the wake of a circular cylinder, with the aim to investigate the development and spatial arrangement of the secondary vortex street in the intermediate wake. It is found that the vorticity thickness approximately follows a power-law behaviour in the regions of $5 \le x/d \le 20$ and $50 \le x/d \le 120$.

A vorticity-based POD analysis shows that the von Kármán vortex street and the secondary vortex street are well captured and segregated from each other in the intermediate wake. The von Kármán vortex street is captured by the first and second modes, which accounts for about 8% of the total enstrophy while the secondary vortex street is captured by the third and fourth modes which account for only about 2.4% of the total enstrophy. Furthermore, the POD analysis shows that a transition occurs in the secondary vortex street in the region $30 \le x/d \le 40$, where the third and fourth POD modes split horizontally into two symmetrical parts. Interestingly, this region corresponds to the region where the vorticity thickness starts to transition from one power-law to another. It was also observed that the spatial arrangement of the secondary vortex street exhibits only one type of mode, the alternating mode, similar to that of the von Kármán vortex street. This is different from that observed in the far-wake (Bisset et al. 1990) where the secondary vortex street switches between opposing and alternating modes. Furthermore, we found that the behaviours of $U^+$ on the flow centreline and that of the density of the dipole moment M are significantly different between low and high $R_d$. This implies that it is not appropriate to use $U^+$ and M to determine the origin of the secondary vortex street at high $R_d$.