2.1 Flow configuration and coordinate system

A sketch of the $D/d=2$ step cylinder geometry is shown in figure 2(a), where $L$ and $l$ represent the lengths of the large and small parts of the cylinder, respectively. In figure 2(b), the computational domain and coordinate system are shown, where $x$ -, $y$ - and $z$ -directions correspond to the streamwise, cross-flow and spanwise directions, respectively. The origin is located in the centre of the interface between the small and large cylinders. The inlet plane is $10D$ upstream from the centre of the step cylinder, while the outlet plane is $20D$ downstream. The spanwise height of the domain is $45D$ , of which the small and large cylinders occupy $15D$   $(l)$ and $30D$   $(L)$ , respectively. The width of the domain is $20D$ . This domain is larger than that used by Morton & Yarusevych (Reference Morton and Yarusevych2010b ) for the same $D/d$ and $Re_{D}$ . Boundary conditions applied in the present study are as follows.

Figure 2. (a) A sketch of the step cylinder geometry ( $D/d=2$ ). (b) Computational domain, origin and coordinate system illustrated from two different viewpoints. The diameter of the large cylinder, $D$ , is the length unit. The origin is located in the centre of the step, i.e. the interface between the small and large cylinders.

1. (i) The inlet boundary: uniform velocity profile, $u=U$ , $v=0$ , $w=0$ .

2. (ii) The outlet boundary: Neumann boundary condition for velocity components ( $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x=\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x=\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}x=0$ ) and constant zero pressure condition.

3. (iii) The other four sides of the computational domain: free-slip boundary conditions (for the two vertical sides, $v=0$ , $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y=\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}y=0$ ; for the two horizontal sides, $w=0$ , $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z=\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}z=0$ ).

4. (iv) The step cylinder surfaces: no slip and impermeable wall.

2.2 Computational method

For all cases in the present investigation, a thoroughly validated finite-volume-based numerical code MGLET (Manhart Reference Manhart2004) is used to directly solve the incompressible Navier–Stokes equations. The midpoint rule is used to approximate the surface integral of flow variables over the faces of the discrete volumes, leading to second-order accuracy in space. A third-order explicit low-storage Runge-Kutta scheme (Williamson Reference Williamson1980) is used for time integration with a constant time step $\unicode[STIX]{x0394}t$ that ensures a Courant–Friedrichs–Lewy (CFL) number smaller than 0.65. The pressure–velocity coupling is handled by solving a Poisson equation with Stone’s strongly implicit procedure (SIP) (Stone Reference Stone1968). The same code has recently also been used to investigate other complex flows, such as the spheroid wake (Jiang et al. Reference Jiang, Andersson, Gallardo and Okulov2016) and the curved cylinder wake (Jiang, Pettersen & Andersson Reference Jiang, Pettersen and Andersson2019).

All simulations are conducted on a staggered Cartesian mesh, while the solid surface of the step cylinder is handled by an immersed boundary method (IBM) (Peller et al. Reference Peller, Duc, Tremblay and Manhart2006). The computational domain is divided into cubic Cartesian grid boxes, named level-1 boxes. In each of them, $N\times N\times N$ cubic Cartesian grid cells are uniformly distributed. In order to refine the grid regions in which complex flow phenomena take place, such as the regions close to the step cylinder geometry, the region around the ‘step’, the regions where vortex dislocations happen, etc., all the grid boxes (the level-1 boxes) are equally split into eight smaller cubic boxes (the level-2 boxes). In each level-2 box, there are also $N\times N\times N$ cubic grid cells. Hence, the grid resolution on level 2 is two times finer than that on level 1. This splitting process goes on automatically until the finest grid level is reached. The overall properties of the grids for all simulations can be found in table 1. A schematic illustration of the mesh design is shown in figure 3.

Figure 3. An illustration of the multi-level grids: (a) a slice of the computational domain in the $x{-}z$ plane at $y/D=0$ , and (b) a slice of the computational domain in the $x{-}y$ plane at $z/D=0^{-}$ (at the large diameter $D$ region). Each square represents the slice of a corresponding cubic Cartesian grid box that contains $N\times N\times N$ grid cells. Here, there are five levels of grid boxes, where the first four levels are indicated by numbers. Owing to different minimum grid sizes, different cases have either five or six levels of grid boxes. (c) A zoom-in plot of the grid cells in the step region (red rectangle in panel (b)) for case 2.

Table 1. Detailed mesh information of all $D/d=2$ cases. The Reynolds number for all cases is $Re_{D}=UD/\unicode[STIX]{x1D708}=150$ .

2.3 Grid convergence study

Table 2 shows the Strouhal number ( $St$ ) of the three dominating vortex cells ( $St_{S}=f_{S}D/U$ , $St_{N}=f_{N}D/U$ and $St_{L}=f_{L}D/U$ ) behind the step cylinder calculated by a fast Fourier transform (FFT) of the time series of the streamwise velocity $u$ along a vertical sampling line positioned at $(x/D,y/D)=(0.6,0.2)$ . For these four cases, the differences between $St$ numbers of the same vortex cell are small. In figure 4(a), the distributions of mean streamwise velocity along the line AB (as indicated in inset figure 4( $a_{1}$ )) for all four cases are plotted to illustrate the flow variation on the ‘step’ just in front of the small cylinder. The curves in figure 4(a) and a zoom-in view in the inset figure 4( $a_{2}$ ) clearly show a convergent tendency from case 1 to case 4, and there are only minor differences between case 3 and case 4. Moreover, figure 4(b) shows time traces of the spanwise velocity ( $w$ ) in the N-cell formation region where the velocity varies dramatically with time. The fluctuations and the mean values of $w$ from case 3 and case 4 almost coincide. However, the computational cost of case 4 is significantly higher than that of case 3, due to the large number of grid cells and smaller time step. The long-period features of the flow that we will discuss in later sections require exceptionally long simulations (more than 3000 $D/U$ ). All discussions are therefore based on data from case 3. Case 4 was run only for a limited time for this convergence test.

Figure 4. (a) Distributions of mean streamwise velocity $\bar{u}/U$ along a sampling line AB in the $x{-}z$ plane at $y/D=0$ . Insets: ( $a_{1}$ ) a sketch of the sampling line AB of length $0.8D$ , positioned $0.15D$ in front of the small cylinder; and ( $a_{2}$ ) a zoom-in plot of the upper part of the curves in panel (a) (black rectangle in panel (a)). (b) Time traces of the spanwise velocity $w$ at point $(x/D,y/D,z/D)=(1,0,-2.5)$ in the N-cell region. The green line is obtained from Morton & Yarusevych (Reference Morton and Yarusevych2010b );  $T$ is the period of one N-cell cycle, which is the same time scale as Morton & Yarusevych (Reference Morton and Yarusevych2010b ) used.

Table 2. Strouhal numbers of the three dominating vortex cells (S-cell, $St_{S}=f_{S}D/U$ ; N-cell, $St_{N}=f_{N}D/U$ ; and L-cell, $St_{L}=f_{L}D/U$ ) for all cases studied. Results from Morton & Yarusevych (Reference Morton and Yarusevych2010b ) are from their numerical simulations for a step cylinder with $D/d=2$ at $Re_{D}=150$ . The result from Norberg (Reference Norberg1994) is calculated by (2.1), which was derived by Norberg based on laboratory experiments. (Note that, in our cases, $St_{S}$ is calculated based on the large-cylinder diameter, so a factor 2 should be used when using Norberg’s equation.)

2.4 Comparison with previous studies

An overview of the vortical structures in the wake of the step cylinder is illustrated in figure 1(a) by plotting the isosurface of $\unicode[STIX]{x1D706}_{2}=-0.05$ (Jeong & Hussain Reference Jeong and Hussain1995). By comparing figures 1(a), (b) and (c), one can see that the overall wake structures from the present study compare well with the previous numerical simulations by Morton & Yarusevych (Reference Morton and Yarusevych2010b ) and experiments by Dunn & Tavoularis (Reference Dunn and Tavoularis2006). Behind the step cylinder, as mentioned in § 1, the shedding of S-cell vortices is barely influenced, which makes it reasonable to introduce the correlation derived by Norberg (Reference Norberg1994),

(2.1) $$\begin{eqnarray}\displaystyle St=0.1835-3.458/Re+1.51\times 10^{-4}\,Re, & & \displaystyle\end{eqnarray}$$

to validate our $St_{S}$ . From the data in table 2, we see that $St_{S}$ from the present study is slightly lower than that from Morton & Yarusevych (Reference Morton and Yarusevych2010b ), but compares better with the experimental value reported in Norberg (Reference Norberg1994). In addition, we have obtained spanwise velocity data from Morton & Yarusevych (Reference Morton and Yarusevych2010b ) and displayed them in figure 4(b). The match between the present study and Morton & Yarusevych (Reference Morton and Yarusevych2010b ) is convincing. Based on all these careful comparisons, we believe that the grid resolution in case 3 is good enough to accurately simulate this flow.

3.1 Overview of the flow development

In figure 5(a), the streamwise velocity spectrum is obtained by means of a discrete Fourier transform (DFT) of continuous velocity data along a vertical sampling line parallel to the $Z$ -axis at position $(x/D,y/D)=(0.6,0.2)$ , over a long period of 2500 time units ( $D/U$ ). As in the previous studies (Dunn & Tavoularis Reference Dunn and Tavoularis2006; Morton & Yarusevych Reference Morton and Yarusevych2010b ; Tian et al. Reference Tian, Jiang, Pettersen and Andersson2017a ), the three dominating frequency components ( $St_{S}=f_{S}D/U$ , $St_{N}=f_{N}D/U$ and $St_{L}=f_{L}D/U$ ) and the beat frequency ( $St_{beat}=f_{beat}D/U$ ) are dominating.

Figure 5. (a) Streamwise velocity spectra along a spanwise line behind the step cylinder at $(x/D,y/D)=(0.6,0.2)$ . (b) Power spectrum plotted at position $(x/D,y/D,z/D)=(0.6,0.2,-4.4)$ , in which $St_{N}$ , $St_{L}$ , $St_{beat}$ and an exceptionally low-frequency $St_{1}$ are marked by small black circles. (Note that there is no S-cell vortex in the wake flow at spanwise position $z/D=-4.4$ , so $St_{S}$ does not show up in this figure.)

Figure 6. Isosurface of $\unicode[STIX]{x1D706}_{2}=-0.05$ showing developments of vortex structures on the $-Y$ side. The time $t$ is set to $t=t^{\ast }-2378.1D/U$ ( $t^{\ast }$ is the actual time). Solid and dashed curves in panels (g) and (h) indicate the loop structures on the $-Y$ and $+Y$ side, respectively. The red and green curves point to different NL-loop structures. In panels (a) and (b), the black solid line at $z/D=-2.9$ and the black dashed line at $z/D=-14$ indicate the positions of vorticity contours given in figure 9.

The vortex structures in the near wake are illustrated by consecutive snapshots of the isosurface of $\unicode[STIX]{x1D706}_{2}$ in figure 6. The time $t$ is set to $t=t^{\ast }-2378.1D/U$ , where $t^{\ast }$ is the actual time in the simulation. This applies all through §§ 3–5. All N- and L-cell vortices are labelled by a combination of capital letters and numbers: ‘N’ and ‘L’ represent N- and L-cell vortices, respectively, while the number indicates the shedding order. To differentiate vortices shed from different sides of the step cylinder, we use capital letters with primes (N $^{\prime }$ and L $^{\prime }$ ) to represent vortices shed from the $+Y$ side; and only capital letters (N and L) to represent vortices shed from the $-Y$ side. In figure 6(a–f), every N-cell vortex has one corresponding L-cell vortex with the same direction of rotation (e.g. N0 and L0; N $^{\prime }$ 1 and L $^{\prime }$ 1, etc.). Owing to different shedding frequencies of N- and L-cell vortices, loop structures appear when corresponding N- and L-cell vortices are out of phase. From figure 6(g,h), loop structures (N8–L $^{\prime }$ 9) and (N $^{\prime }$ 9–L10) form, and are indicated by green and red curves, respectively. Details of the formation processes of those loop structures were described in Tian et al. (Reference Tian, Jiang, Pettersen and Andersson2017a ). Based on the order of their appearances, the green and red curves are named the NL-loop 1 and NL-loop 2, respectively.

Figure 7. Schematic topology sketches illustrating the long-time history of the vortex connection topology and the vortex shedding. The thick black and grey straight lines represent vortices on the $-Y$ and $+Y$ sides, respectively. Only the N- and L-cell vortices are shown by short and long straight lines. The connections between them are depicted by thin solid curves. Black and grey solid curves indicate the connections between an N-cell vortex and its counterpart L-cell vortex. The red and green curves reveal different NL-loop structures (same colour code as used in figure 6). The dashed curves, on the other hand, indicate broken connections that are not able to persist due to dislocations. We define a new term ‘long N-cell cycle’ containing several conventional N-cell cycles, while the conventional N-cell cycle was firstly defined by Morton & Yarusevych (Reference Morton and Yarusevych2010b ) and adopted also in the present study.

Based on long-time observations ( $2500D/U$ ), a schematic topology sketch is shown in figure 7. This will be used to introduce some important concepts. In figure 7, the short and long straight lines represent the N- and L-cell vortices, respectively. Between them, the curved solid lines connect the N-cell vortex and its counterpart L-cell vortex. The dashed curves indicate broken connections that are not able to persist due to dislocations. Detailed visualizations of vortex connections and dislocations in the first N-cell cycle are shown in figure 6. To ease the observation, we only show the connections between the N- and L-cell vortices. The L–L and N–N loops (Tian et al. Reference Tian, Jiang, Pettersen and Andersson2017a ) are not shown in this figure.

We define the side of the N-cell vortex in an NL-loop structure as the side of the loop itself. For example, the NL-loop N8–L $^{\prime }$ 9 (shown by green curves) in figure 6(g) is identified to form at the $-Y$ side. As shown in figure 7, from the first to the seventh N-cell cycle, the NL-loop 1 (the green curves) appears alternately at the $+Y$ and $-Y$ side between subsequent N-cell cycles. This is what we called the antisymmetric vortex interactions in Tian et al. (Reference Tian, Jiang, Pettersen and Andersson2017a ). However, an unexpected interruption of this antisymmetry is observed between the seventh and eighth N-cell cycles. Figure 7 shows, in both the seventh and eighth N-cell cycles, that the NL-loop 1 appears at the $-Y$ side (green curves connect to black short lines which represent the N-cell vortex on the $-Y$ side). We introduce the term ‘long N-cell cycle’ to identify the uninterrupted series of antisymmetric N-cell cycles. Within one long N-cell cycle, antisymmetric vortex interactions appear between subsequent N-cell cycles. However, at the boundary between two long N-cell cycles, this antisymmetry is interrupted. Our long-time observation covering eight long N-cell cycles shows that there are either seven or eight N-cell cycles in one long N-cell cycle. In fact, an exceptionally low frequency ( $St_{1}$ ) is captured in figure 5(b) where a power spectrum at position $(x/D,y/D,z/D)=(0.6,0.2,-4.4)$ is shown. The value of $St_{1}$ is 0.0032, and is around $St_{beat}/7.5$ . This coincides well with our observation that one long N-cell cycle contains either seven or eight N-cell cycles. We believe that this low-frequency component is related to the long N-cell cycles. More detailed information on this long-period phenomenon, and the unexpected interruption, will be discussed in § 5. The other visible frequency components in figure 5(b) are combinations of the basic frequency components, i.e.  $St_{S}$ , $St_{N}$ , $St_{L}$ and $St_{1}$ (Gerich & Eckelmann Reference Gerich and Eckelmann1982).

3.2 Necessity of monitoring the phase information of each N- and L-cell vortex

All the interesting physical phenomena, i.e. the formation of NL-loops, the unexpected interruption of the antisymmetry, etc., are directly related to the vortex dislocations in the wake behind the step cylinder. A consensus from the literature (Williamson Reference Williamson1989; Lewis & Gharib Reference Lewis and Gharib1992; Morton et al. Reference Morton, Yarusevych and Carvajal-Mariscal2009) is that vortex dislocations are attributed to different shedding frequencies. In the present configuration, if both N- and L-cell vortices shed regularly, it is natural to directly use $f_{L}$ and $f_{N}$ to measure the phase difference ( $\unicode[STIX]{x1D6F7}$ ) between N- and L-cell vortices. However, the actual wake flow is more complicated. In figure 8(a,b), we plot time traces of the instantaneous cross-flow velocity $v$ at two locations in the L- and N-cell regions in the symmetry plane. Two dashed sinusoidal curves with constant frequencies $f_{L}$ and $f_{N}$ are also plotted in figure 8. By comparing figures 8(a) and (b), it is clear that, unlike the regularly shed L-cell vortices, the shedding frequency of the N-cell vortex slightly fluctuates during every N-cell cycle. This was also briefly mentioned by Morton & Yarusevych (Reference Morton and Yarusevych2010b ), but not investigated further. The irregularity of the N-cell shedding makes it challenging but necessary to monitor the phase information of every N-cell vortex. Therefore, we developed a method to obtain the phase information ( $\unicode[STIX]{x1D711}$ ) and the phase difference ( $\unicode[STIX]{x1D6F7}$ ) of vortices. Details can be found in appendix A.

Figure 8. Time trace of the oscillating cross-flow velocity $v$ is plotted as the solid line: (a) at the sampling point $(x/D,y/D,z/D)=(1.4,0,-14.8)$ in the L-cell region, and (b) at the sampling point $(x/D,y/D,z/D)=(1.4,0,-2.8)$ in the N-cell region. For comparison, pure sinusoidal curves are plotted as dashed lines with frequency $f_{L}$ in panel (a) and frequency $f_{N}$ in panel (b); $f_{L}$ and $f_{N}$ are calculated by FFT obtained from figure 5.

4.1 Two different phase difference accumulation mechanisms

From figure 6, one can see that both the N- and L-cell vortices are spanwise vortices. This means that the variation of the streamwise distance between corresponding N- and L-cell vortices can reflect the changes in their phase difference ( $\unicode[STIX]{x1D6F7}$ ). In the present study, we use the location of the most concentrated spanwise vorticity ( $\unicode[STIX]{x1D714}_{z}$ ) to indicate the position of the corresponding vortex. In figure 9, we plot instantaneous spanwise vorticity $\unicode[STIX]{x1D714}_{z}$ contours at an $(x,y)$ plane in the N-cell region $z/D=-2.9$ and L-cell region $z/D=-14$ . Four black lines indicate the positions of vortices N0 and L0. One can see that from $tU/D=4.5$ to 9, the streamwise distance between N0 and L0 increases from $1.8D$ ( $3.7D-1.9D$ ) to $2.3D$ ( $7.7D-5.4D$ ) as they convect downstream. This means that, even after both N0 and L0 disconnect from the shear layer, as shown in figure 6(a), $\unicode[STIX]{x1D6F7}$ between them continues to accumulate. By marking the moment when the N-cell vortex just forms as an individual wake-type vortex, we divide the process of $\unicode[STIX]{x1D6F7}$ accumulation into two parts. Before this moment, $\unicode[STIX]{x1D6F7}$ between the N- and L-cell vortex is dominated by their different shedding frequencies, called $\unicode[STIX]{x1D6F7}_{f}$ . After this moment, $\unicode[STIX]{x1D6F7}$ is caused by different convective velocities in the N- and L-cell regions, and called $\unicode[STIX]{x1D6F7}_{c}$ . Detailed descriptions of monitoring $\unicode[STIX]{x1D6F7}_{f}$ and $\unicode[STIX]{x1D711}$ can be found in appendix A. Owing to the spatial inhomogeneity of the convective velocity, it is difficult to accurately assess its effect on $\unicode[STIX]{x1D6F7}$ . Yet, the distributions of mean streamwise velocity ( $\bar{u}$ ) in different vortex cell regions can roughly indicate the influence.

Figure 9. Instantaneous spanwise vorticity $\unicode[STIX]{x1D714}_{z}$ ( $\unicode[STIX]{x1D714}_{z}=\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x-\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y$ ) contour plots in an ( $x,y$ ) plane in (a,c) the N-cell region $z/D=-2.9$ (black solid line in figure 6(a,b)) and (b,d) in the L-cell region $z/D=-14$ (black dotted line in figure 6(a,b)). By detecting the location of concentrated vorticity, the positions of vortices N $^{\prime }$ 9 and L $^{\prime }$ 9 are marked by black lines. (Note that we have compared the position of the centre of the concentrated vorticity and the centre in the region defined by $\unicode[STIX]{x1D706}_{2}$ isolines, and confirmed only tiny differences.)

Figure 10. (a) Distributions of the mean streamwise velocity ( $\overline{u}/U$ ) along spanwise lines with the same $x$ -coordinate ( $x/D=1.6$ ), but for different $y$ -coordinates. (b) Distributions of mean streamwise velocity $\bar{u}/U$ along spanwise lines with different $x$ -coordinates in the symmetry plane ( $y/D=0$ ).

In figure 10, the spanwise distributions of $\bar{u}$ are plotted at several downstream positions. As shown in figure 10(a), the mean streamwise velocity in the N-cell region in the symmetry plane $(y/D=0)$ is nearly 0.2 $U$ less than that in the L-cell region. At the side plane $y/D=0.8$ , this difference still reaches 0.1 $U$ . From figure 10(b), we see that the difference in mean streamwise velocity is clear until a downstream position $x/D=4$ . In other words, at least until $x/D=4$ in the wake, the convective velocity distribution is distinctly non-uniform in the spanwise direction. This non-uniformity induces an additional $\unicode[STIX]{x1D6F7}$ when the vortices convect downstream. We note that this role of the non-uniform convection velocity and its effects have never been addressed before.

4.2 Effects of two phase difference accumulation mechanisms

4.2.1 Differences in formation positions of the NL-loop 1 and NL-loop 2

The formation process of the NL-loop 1 in each N-cell cycle is repetitive. An example of this process is presented in figure 11, where the vortex structures are shown from both $+Y$ and $-Y$ sides of the step cylinder. In figure 11(a,b), and the corresponding zoom-in plots (figure 11 f,g), the foot of vortex N8 completely disconnects from the shear layer at $x/D=2.8$  (marked by a black line in figure 11(b,g)). At this moment ( $tU/D=33.3$ ) the NL-loop 1 structure has not yet formed, because there is still no direct connection between N8 and L $^{\prime }$ 9. It takes some more time for N8 to convect downstream and eventually develop into the NL-loop 1 with L $^{\prime }$ 9 at $x/D=3.3$ and $tU/D=34.5$ . This process is indicated in figure 11(b–e) and the corresponding zoom-in plots (figure 11 g–j). By following the same process as described in § 4.1, we found that from $tU/D=33.3$ to 34.5, the streamwise distance between vortex N $^{\prime }$ 9 and L $^{\prime }$ 9 increases from $5.3D$ to $6.1D$ as they move downstream. When $\unicode[STIX]{x1D6F7}$ between vortex N $^{\prime }$ 9 and L $^{\prime }$ 9 increases, L $^{\prime }$ 9 gradually disconnects from its counterpart N $^{\prime }$ 9 and forms the NL-loop 1 with N8 (see figure 11 k–o).

Figure 11. The formation process of the NL-loop 1 structure in the first dislocation process (defined in figure 7) is shown from the $-Y$ and $+Y$ sides in (a–e) and (k–o) rows, respectively. In (f–j), zoom-in plots of vortex structures at N-cell region (black rectangle in panel a) are shown. The black circles highlight the position where the NL-loop 1 forms.

Figure 12. The formation process of the NL-loop 2 structure in the first dislocation process is shown from both $+Y$ and $-Y$ sides. The black circles highlight the position where the NL-loop 2 structure is formed.

Unlike the NL-loop 1 structure, which has a distinct formation position, it is difficult to pinpoint where the NL-loop 2 forms. As shown in figure 12(a–e), in the black circle area, it is not clear how the foot of vortex N $^{\prime }$ 9 completely separates from the shear layer and subsequently connects to L10 as they move downstream. The connection between N $^{\prime }$ 9 and L10 forms in the very near wake before N $^{\prime }$ 9 completely disconnects from the shear layer. In order to compare with the formation process of the NL-loop 1 structure shown in figure 11, we use the same method to monitor the variation of the streamwise distance between vortices N10 and L10. At $tU/D=36.9$ (the corresponding instantaneous isosurface of the vortex structure is shown in figure 12(g)), the distance between vortex N10 and L10 reaches $5.9D$ . This is very close to the distance between N $^{\prime }$ 9 and L $^{\prime }$ 9 at $tU/D=34.5$ (figure 11 n), when L $^{\prime }$ 9 successfully induces N8 to connect to itself and together form the NL-loop 1 structure. At $tU/D=36.9$ , as shown in figure 12(b), the leg of vortex N $^{\prime }$ 9 is at position $x/D=2.8$ . At the same downstream position, the foot of vortex N8 already disconnects from the shear layer, as shown in figure 11(b). It is reasonable to speculate that $\unicode[STIX]{x1D6F7}$ between N10 and L10 becomes sufficiently large to attract N $^{\prime }$ 9 to connect to L10 before it disconnects from the shear layer. As a consequence, the formation position of NL-loop 2 is not so clear.

Figure 13. (a) Time trace of $\unicode[STIX]{x1D6F7}_{f}$ between corresponding N-cell and L-cell vortices in the first (LNC1) and second (LNC2) long N-cell cycles, i.e. from the first to the 15th N-cell cycle. The circles represent $\unicode[STIX]{x1D6F7}_{f}$ between an N-cell vortex and its counterpart L-cell vortex. The green and red circles indicate $\unicode[STIX]{x1D6F7}_{f}$ , which eventually causes formation of the NL-loop 1 and NL-loop 2 structures, respectively. From the first to the eighth N-cell cycle, the green and red circles are numbered. (b) A zoom-in plot of the time trace of $\unicode[STIX]{x1D6F7}_{f}$ in the first N-cell cycle (the black dashed rectangle in panel a). From the left to the right, circles represent $\unicode[STIX]{x1D6F7}_{f}$ between the vortex pair N $^{\prime }$ 1–L $^{\prime }$ 1 to the vortex pair N10–L10, respectively. (Note that the detailed calculation processes can be found in appendix A. All detailed data about $\unicode[STIX]{x1D6F7}_{f}$ and the longer-time trace of $\unicode[STIX]{x1D6F7}_{f}$ are included in the supplementary material, file 1, available at https://doi.org/10.1017/jfm.2020.110. The trigger value and the threshold value are estimated based on 55 N-cell cycles, as shown in supplementary file 3.)

Figure 14. The just-formed NL-loop 1 structures in the first to the eighth N-cell cycle are plotted from both the $-Y$ and $+Y$ sides (the first long N-cell cycle consists of the first to the seventh N-cell cycles). The black line marks the formation position of NL-loop 1. The red circle in panel (f) highlights an irregular absence of the NL-loop 1 structure, which will be discussed later.

The time trace of $\unicode[STIX]{x1D6F7}_{f}$ accumulation proves our speculation. By using the method described in appendix A, the time trace of $\unicode[STIX]{x1D6F7}_{f}$ accumulation in the first N-cell cycle is shown in figure 13(b). Circles in this figure represent $\unicode[STIX]{x1D6F7}_{f}$ of corresponding N- and L-cell vortices, in which the green circle represents $\unicode[STIX]{x1D6F7}_{f}$ between N $^{\prime }$ 9 and L $^{\prime }$ 9, and the red circle represents $\unicode[STIX]{x1D6F7}_{f}$ between N10 and L10. Eventually, the dislocations of the vortex pairs corresponding to the green and red circles cause formation of the NL-loop 1 structure (N8–L $^{\prime }$ 9) and the NL-loop 2 structure (N $^{\prime }$ 9–L10), respectively.

One can see that the red circle represents a larger $\unicode[STIX]{x1D6F7}_{f}$ value than the green one, which means that $\unicode[STIX]{x1D6F7}_{f}$ between N10 and L10 is larger than that between N $^{\prime }$ 9 and L $^{\prime }$ 9. Therefore, compared to the vortex pair N10–L10, the vortex pair N $^{\prime }$ 9–L $^{\prime }$ 9 needs a larger contribution of $\unicode[STIX]{x1D6F7}_{c}$ to achieve a sufficiently large $\unicode[STIX]{x1D6F7}$ to trigger the vortex dislocation between them, and the subsequent formation of the NL-loop 1 (N8–L $^{\prime }$ 9). In other words, due to the reduced need of a contribution from $\unicode[STIX]{x1D6F7}_{c}$ , the vortex dislocation between N10 and L10 and the subsequent NL-loop 2 (N $^{\prime }$ 9–L10) is formed closer to the cylinder, which causes the unclear formation position.

4.2.2 Variation of formation positions of the NL-loop 1 structures

In figure 14, the NL-loop 1 structures in the first eight N-cell cycles are plotted. The black vertical lines show the positions where the NL-loop 1 structures just form. Except for the sixth N-cell cycle (figure 14 f), the NL-loop 1 structures alternately appear at the $+Y$ and $-Y$ side of the step cylinder from the first to the seventh N-cell cycles (the first long N-cell cycle). The shape and the formation position of the NL-loop 1 structure also vary, as depicted in figure 14(a–h).

Figure 15. Relation between $\unicode[STIX]{x1D6F7}_{f}$ and the formation position $(x/D)$ of corresponding NL-loop 1 structures in the first to the seventh long N-cell cycles (52 N-cell cycles are included). Based on the trends of the curves, one stable range and two unstable ranges are identified. A straight dashed line outlines the trend of the stable range. The occasionally absent NL-loop 1 structures in these seven long N-cell cycles are marked by red colour at the expected formation positions, which can be obtained by linear interpolation.

Table 3. Formation positions and $\unicode[STIX]{x1D6F7}_{f}$ values that cause the NL-loop 1 structures in the first to seventh N-cell cycle (the first long N-cell cycle). The corresponding vortex structures are shown in figure 14(a–g). In the sixth N-cell cycle, there is a loop formation failure.

In table 3, the relation between the formation positions of the NL-loop 1 structures in the first long N-cell cycle and their $\unicode[STIX]{x1D6F7}_{f}$ are presented. One can see that, except for the sixth N-cell cycle, the formation positions move downstream as the corresponding $\unicode[STIX]{x1D6F7}_{f}$ decreases. This can easily be ascribed to the already discussed two different phase difference accumulation mechanisms. Formation of the NL-loop 1 structures requires sufficiently large $\unicode[STIX]{x1D6F7}$ . Since $\unicode[STIX]{x1D6F7}_{f}$ decreases, $\unicode[STIX]{x1D6F7}_{c}$ must contribute more, which consequently leads to a longer formation time and further downstream formation position. To be more clear, the relations between $\unicode[STIX]{x1D6F7}_{f}$ and formation positions of the corresponding NL-loop 1 structures during the first–seventh long N-cell cycles are plotted in figure 15. Generally, except for several irregular points, the formation position shifts downstream as $\unicode[STIX]{x1D6F7}_{f}$ decreases. (Note that the appearances of the irregular points will be discussed in § 5.3.)

Based on the discussions above, we conclude that, for the NL-loop structures, $\unicode[STIX]{x1D6F7}$ is accumulated by the joint influence of different shedding frequencies and different convective velocities

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6F7}_{f}+\unicode[STIX]{x1D6F7}_{c}.\end{eqnarray}$$

As the relative magnitude of $\unicode[STIX]{x1D6F7}_{f}$ and $\unicode[STIX]{x1D6F7}_{c}$ varies, the formation processes of the NL-loop structures change.

5.1 Trend of $\unicode[STIX]{x1D6F7}_{f}$ variation

The time trace of $\unicode[STIX]{x1D6F7}_{f}$ accumulation between corresponding N- and L-cell vortices is plotted in figure 13(a). We use green and red circles to indicate $\unicode[STIX]{x1D6F7}_{f}$ of the pair of N- and L-cell vortices whose dislocation eventually causes the NL-loop 1 and NL-loop 2, respectively. Two solid lines with the same corresponding colours describe the decreasing tendency in $\unicode[STIX]{x1D6F7}_{f}$ of both NL-loop 1 and NL-loop 2 over time.

The gradual decrease of $\unicode[STIX]{x1D6F7}_{f}$ in each N-cell cycle can be expressed as

(5.1) $$\begin{eqnarray}S=\unicode[STIX]{x1D6FC}\frac{1}{2\,f_{L}}-\unicode[STIX]{x1D6FD}\frac{1}{2\,f_{N}},\end{eqnarray}$$

where $S$ (with dimension $D/U$ ) is a measure of the phase shift of every vortex pair in one N-cell cycle, as compared to the N-cell cycle before it. In this expression, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ represent the number of L- and N-cell vortices in one N-cell cycle; and $f_{L}$ and $f_{N}$ are the shedding frequencies of L- and N-cell vortices. In the present case, $\unicode[STIX]{x1D6FC}=15$ , $\unicode[STIX]{x1D6FD}=13$ , $f_{L}D/U=0.1780$ and $f_{N}D/U=0.1545$ (from figure 5), from which we obtain $S=0.064D/U$ . This means that, in one N-cell cycle, the duration of 13 N-cell and 15 L-cell vortices is not exactly the same. Although $S$ has a very small value, after a certain number of N-cell cycles, the accumulated difference becomes large enough to influence the dislocation process. Moreover, by checking figure 13(a) and the supplementary file 1, one can find that from R1 to R7 or from G1 to G7, $\unicode[STIX]{x1D6F7}_{f}$ decreases by approximately 0.4 $D/U$ , i.e.  $\unicode[STIX]{x1D6F7}_{f}$ decreases by around 0.067 $D/U$ after every N-cell cycle, which is close to the $S$ value from expression (5.1).

It is worth mentioning that the numbers $\unicode[STIX]{x1D6FC}=15$ , $\unicode[STIX]{x1D6FD}=13$ and $S=0.064D/U$ are related to the particular configuration studied. For different configurations, i.e. different $D/d$ and $Re_{D}$ , these numbers in expression (5.1) may vary. But what we observe in figure 13(a) will be a common feature, because it is extremely unlikely to attain an $S$ value exactly equal to zero.

5.2 Interruption of the antisymmetric phenomenon

From figure 14(a–h), one can clearly see that the NL-loop 1 structure alternately appears at the $-Y$ and $+Y$ side of the step cylinder in subsequent N-cell cycles. This is the antisymmetric phenomenon reported in Tian et al. (Reference Tian, Jiang, Pettersen and Andersson2017a ). In the present case, the long-time observation reveals that this antisymmetric phenomenon will be interrupted once in a while. As shown in figure 14(g,h), instead of being antisymmetric, the loop structures in the seventh and eighth N-cell cycles are symmetric. In these two N-cell cycles, both NL-loop 1 structures, i.e. N86–L $^{\prime }$ 99 and N100–L $^{\prime }$ 115, are formed at the $-Y$ side.

This interruption is caused by the decreasing tendency in $\unicode[STIX]{x1D6F7}_{f}$ , as we discussed in § 5.1. Normally, there are 13 N-cell and 15 L-cell vortices in one N-cell cycle. The odd number of N-cell vortices causes the antisymmetric phenomenon. However, as shown in figure 13(a), when $\unicode[STIX]{x1D6F7}_{f}$ continues to decrease along the green line from G1 to C8, it eventually becomes insufficient in point C8. Even by including the contribution of $\unicode[STIX]{x1D6F7}_{c}$ ,  $\unicode[STIX]{x1D6F7}$ is still not large enough to induce the formation of the expected NL-loop 1 (N $^{\prime }$ 99–L114). Therefore, in this N-cell cycle, one additional vortex pair shedding is needed to make $\unicode[STIX]{x1D6F7}$ sufficiently large to induce formation of the NL-loop 1. The additional one pair of N- and L-cell vortices makes the number of N-cell vortices in the eighth N-cell cycle become even, and thereby interrupts the antisymmetric phenomenon. In the supplementary file 3, the time trace of $\unicode[STIX]{x1D6F7}_{f}$ between N- and L-cell vortices in the first to the eighth long N-cell cycles (the first to the 55th N-cell cycles) is illustrated.

5.3 Trigger value and threshold value of vortex dislocations

Based on the results from earlier papers (Williamson Reference Williamson1989; Morton & Yarusevych Reference Morton and Yarusevych2010b ) and our discussions in §§ 5.2 and 4, it is clear that only when $\unicode[STIX]{x1D6F7}$ becomes sufficiently large can the vortex dislocation process be triggered. We call this value the ‘trigger value’. Based on (4.1), $\unicode[STIX]{x1D6F7}$ consists of two parts, $\unicode[STIX]{x1D6F7}_{f}$ and $\unicode[STIX]{x1D6F7}_{c}$ . Owing to the complexity of $\unicode[STIX]{x1D6F7}_{c}$ , an accurate trigger value is hard to obtain, but an approximate value is possible to estimate. As we discussed in § 4, unlike the NL-loop 1 structure, which has a clear formation position, the NL-loop 2 structure forms in the near wake. This makes it hard to define the exact formation position of the NL-loop 2. It means that the $\unicode[STIX]{x1D6F7}_{f}$ that induces this NL-loop 2 is very close to the trigger value, and only a modest contribution from $\unicode[STIX]{x1D6F7}_{c}$ is needed. Therefore, by considering all the largest $\unicode[STIX]{x1D6F7}_{f}$ corresponding to the NL-loop 2 structures, we can draw the blue line in figure 13(a) to approximate the trigger value (around 5.60 $D/U$ ).

Besides the trigger value, there is another interesting value of $\unicode[STIX]{x1D6F7}_{f}$ that should be noted. As we discussed in § 5.2, when $\unicode[STIX]{x1D6F7}_{f}$ continues to decrease from G1 to C8 (figure 13 a), the expected formation of the NL-loop 1 (N $^{\prime }$ 99–L118) fails. We can speculate that there is a threshold value for $\unicode[STIX]{x1D6F7}_{f}$ , such that when $\unicode[STIX]{x1D6F7}_{f}$ becomes less than this value, a vortex dislocation will not occur, even when taking the contribution from $\unicode[STIX]{x1D6F7}_{c}$ into account. By connecting all of the smallest values of $\unicode[STIX]{x1D6F7}_{f}$ in green ( $\unicode[STIX]{x1D6F7}_{f}$ inducing the NL-loop 1), the threshold value can be estimated by the yellow line in figure 13(a), with a value around $\unicode[STIX]{x1D6F7}_{f}U/D=4.30$ .

Owing to the complexity of $\unicode[STIX]{x1D6F7}_{c}$ , the formation of NL-loop structures becomes quite unstable when $\unicode[STIX]{x1D6F7}_{f}$ is close to the trigger value or to the threshold value. As shown in figure 15, one stable $\unicode[STIX]{x1D6F7}_{f}$ range and two unstable $\unicode[STIX]{x1D6F7}_{f}$ ranges can be identified. A dashed straight line outlines the trend in the stable range ( $4.45\lesssim \unicode[STIX]{x1D6F7}U/D\lesssim 4.65$ ), in which the distribution of markers is concentrated, and the formation position decreases almost linearly as $\unicode[STIX]{x1D6F7}_{f}$ increases. Outside this stable range, the tendency becomes unclear. When $\unicode[STIX]{x1D6F7}_{f}$ is smaller than 4.45 $D/U$ , i.e. much smaller than the trigger value 5.60 $D/U$ and close to the threshold value 4.30 $D/U$ , the contribution of $\unicode[STIX]{x1D6F7}_{c}$ becomes significant and determines whether the NL-loop 1 is able to be formed or not. Figure 15 shows that, in the unstable range ( $4.30\lesssim \unicode[STIX]{x1D6F7}_{f}U/D\lesssim 4.45$ ), all NL-loop 1 structures are generated beyond a downstream position $x/D=4$ , i.e.  $\unicode[STIX]{x1D6F7}$ of these NL-loop 1 structures cannot exceed the trigger value upstream of $x/D=4$ . In this situation, how much $\unicode[STIX]{x1D6F7}_{c}$ can accumulate downstream of $x/D=4$ determines whether these NL-loop 1 structures will appear or not. In figure 10(b), we see apparent differences between the convective velocity in the N- and L-cell regions, while these differences diminish as we move downstream. Downstream of $x/D=4$ , the differences become very small. In other words, the accumulation of $\unicode[STIX]{x1D6F7}_{c}$ is modest downstream of $x/D=4$ . Based on the above observation, we can conclude that, for the NL-loop 1 structures with $\unicode[STIX]{x1D6F7}_{f}$ in the range ( $4.30D/U$ – $4.45D/U$ ), their phase differences are around a critical value, and a small variation in $\unicode[STIX]{x1D6F7}_{c}$ can influence their formation positions, and even lead to unsuccessful formations. The fact that all the red markers in figure 15, which represent the occasionally absent NL-loop 1 structures, are located in this range also supports this conclusion.

Another unstable range appears when $\unicode[STIX]{x1D6F7}_{f}$ corresponding to NL-loop 1 becomes larger than 4.65 $D/U$ , which is close to the approximate level of the trigger value. In this situation, only a small contribution from the convective velocity is needed for NL-loop 1 to form. Meanwhile, as shown in figure 10(b), when $x/D$ is smaller than 3 ( $x/D=1.4$ , 1.6, 1.8, 2, 3) the differences between the convective velocities in the N- and L-cell regions are obvious. These differences are able to increase $\unicode[STIX]{x1D6F7}_{c}$ rapidly. Variations in the convective velocity could also affect the formation position, and lead to a less clear tendency.

Detailed investigations show that many interesting and important features of vortex dislocation in a step cylinder wake may easily be overlooked if the observation time is too short. In the present case, during one long N-cell cycle, most N-cell cycles need an odd number of N- and L-cell vortices to accumulate sufficient $\unicode[STIX]{x1D6F7}$ , and to trigger the vortex dislocation, which leads to antisymmetry between subsequent N-cell cycles. However, between two long N-cell cycles, the antisymmetry is interrupted. In our earlier paper (Tian et al. Reference Tian, Jiang, Pettersen and Andersson2017a ), we did not foresee this interruption, because we tried to conclude that the antisymmetric dislocation process is a result of $f_{L}/f_{beat}\approx 7.5$ , such that two dislocations are needed to compensate for the frequency differences between the N- and L-cells, and that the wake could return to normal one-to-one shedding. However, more detailed observations based on substantially longer simulations show that the small differences (i.e.  $S$ in expression (5.1)), which were ignored when we obtain the $f_{L}/f_{beat}\approx 7.5$ relationship, are continuously accumulating and cause a decreasing tendency in the time trace of $\unicode[STIX]{x1D6F7}_{f}$ . This is exactly what causes the interruption of the antisymmetry discussed in this section. We note again that these results require exceptionally long simulations. The value of $S$ is so small that it may easily be ignored in short-term observations. But it turns out to lead to very interesting vortex dislocation phenomena. We mentioned that McClure et al. (Reference McClure, Morton and Yarusevych2015) also reported similar small differences in the vortex dislocations behind a dual step cylinder. However, instead of investigating how different vortex dislocations vary, they focused on when two identical vortex dislocations appear.