3.1. The F3SNL mode

In the present section, the formation of the F3SNL mode will be discussed based on the $l/D=10$ case. The mode F3SNL also occurs for $l/D=5$ and 3; the corresponding wakes are shown in Appendix B. When the F3SNL mode occurs, the wake behind an I-shaped step cylinder can be treated as a combination of two single-step cylinder wakes. Figure 4(a) shows that the three main spanwise vortices (S-, N- and L-cell vortices) which were originally identified by Dunn & Tavoularis (Reference Dunn and Tavoularis2006) behind the single-step cylinder (figure 1c) form in the wakes behind both the top part ($z>0$) and the bottom part ($z<0$) of the I-shaped step cylinder. Figure 5(a) shows that the S-cell vortex sheds behind the small cylinder with the largest shedding frequency $St_S$; the L-cell vortex sheds behind the large cylinder with the shedding frequency $St_L$; and the N-cell vortex sheds between the S- and L-cell vortices with the smallest shedding frequency $St_N$.

The formation of F3SNL is due to the length of the small cylinder ($l$) being so large that the interaction between the two steps can be neglected. The three-dimensional flow structures induced by each step are similar to those induced by the step of a single-step cylinder. Figure 7(a–c) shows the spanwise distribution of the time-averaged recirculation length $L_r$, which is obtained from the position where the time-averaged streamwise velocity crosses zero from negative to positive. Figure 7(d–f) shows the time-averaged base pressure coefficient $\overline {C_{pb}}$ along a spanwise sampling line located 0.02$D$ downstream of the I-shaped step cylinder wall at $y=0$. Here, $\overline {C_{pb}}=2(\overline {p_b}-p_0)$, where $\overline {p_b}$ is the time-averaged base pressure and $p_0$ is the pressure at the outlet boundary. The distance $h=0.02D$ is selected because it is slightly larger than the smallest cell's diagonal ($\sqrt {2}\varDelta < h=0.02D<1.5\sqrt {2}\varDelta$, where $\varDelta =0.01D$) such that we safely avoid wiggles possibly caused by cells directly cut by the cylinder surface and still stay as close as possible to the surface. It should be noted that the sampling line is located at different streamwise positions behind the small and large cylinders due to the different diameters of the small and large cylinders. As a reference, the distribution of $\overline {C_{pb}}$ and $L_r$ along a single-step cylinder with $D/d=2.0$ at $Re_D=150$ is obtained in the region $-15< z<5$ from Tian et al. (Reference Tian, Zhu, Holmedal, Andersson, Jiang and Pettersen2023) and plotted as the black dash line in figure 7(a–f). The comparison of $\overline {C_{pb}}$ and $L_r$ between the $l/D=10$ case (the red curve in figure 7) and the single-step cylinder (the red dashed curve in figure 7) shows tiny differences, with a maximum deviation of 7 % appearing at $z_d=1$ in figure 7(d). As a result, the formation of the S-, N- and L-cell vortices remains almost the same when the configuration changes from the single-step cylinder to the I-shaped step cylinder with $l/D=10$.

Figure 7. Distribution of time-averaged base pressure coefficient $\overline {C_{pb}}$ (a,c) on the large cylinder part and (b) on the small cylinder part. Distribution of recirculation length $\overline {L_r}$ (d, f) on the large cylinder part and (e) on the small cylinder part. The black, red, green and purple curves represent the corresponding distributions for $l/D=10$, 2, 1.5 and 0.2. The red dashed curves represent the distributions along a single-step cylinder with $D/d=2$ at $Re_D=150$. The corresponding data are obtained from Tian et al. (Reference Tian, Zhu, Holmedal, Andersson, Jiang and Pettersen2023). The horizontal axis for (a,c,d, f) is set as $z_D=z-z_{step}$ (where $z_{step}$ represents the coordinate of the step on the corresponding part of the cylinder; subscripts $\it {b}$ and $\it {t}$ indicate the bottom and top parts of the I-shaped step cylinder), indicating the distance between the sampling position and the step. The horizontal axis for (b,e) is set as $z_d = zD/l$, indicating the relative sampling position on the small cylinder.

Contrary to the single-step cylinder wake, where the S-N vortex dislocation occurs at the bottom of the small cylinder (figure 1c), the S-N vortex dislocation occurs at both the top and bottom of the small cylinder for $l/D=10$ (as shown in figure 4a). This causes the topology of the S-S vortex loop to change from the half-loop for the single-step cylinder (reported by Dunn & Tavoularis (Reference Dunn and Tavoularis2006), Tian et al. (Reference Tian, Jiang, Pettersen and Andersson2017a) and visualized by the red dash-dot-dot curve in figure 1c) to the full loop for the I-shaped step cylinder (visualized by the red dash-dot-dot curve in figure 4a). The vortex interaction between the S- and N-cell vortices is further illustrated by consecutive snapshots of the isosurface of $\lambda _2=-0.01$ in figure 8. All vortices are labelled by a combination of capital letters and numbers: ‘N’ and ‘S’ represent N- and S-cell vortices, respectively, while the number indicates the vortex shedding order. To differentiate between vortices shed from different sides of the cylinder, the capital letters with primes represent vortices shed from the $+Y$ side and pure capital letters represent vortices shed from $-Y$. When the S- and N-cell vortices just shed from the same cylinder side, they connect directly (e.g. $S_2$ connects directly to ${N}_{t2}$ and ${N}_{t2}$ in figure 8a). As they convect downstream, since the N-cell vortex contains a larger vortex strength than the S-cell vortex, all N-cell vortices split into at least two filaments (Morton & Yarusevych Reference Morton and Yarusevych2010). One connects to the subsequent N-cell vortex rotating in the opposite direction (e.g. a filament of ${N'_{b1}}$ connects to ${N_{b2}}$ from figure 8a to 8b); the other connects to the S-cell vortex rotating in the opposite direction (e.g. a filament of ${N'_{b1}}$ connects to ${S_2}$ in figure 8b,c) or the same direction (e.g. a filament of ${N_{b2}}$ connects to ${S_4}$ in figure 8b,c), depending on the phase alignment between the S- and N-cell vortices. The S-cell vortex that does not connect to any N-cell vortex connects to other S-cell vortices rotating in the opposite direction, forming the new SS-loop structure (e.g. ${S'_3}$ connects to $S_4$ in figure 8b,c).

Figure 8. Isosurface of $\lambda _2=-0.01$ in the region around the small cylinder for $l/D=10$ showing developments of vortex structures on the $-Y$ side (a) at $t=331.7$, (b) at $t=335.4$, (c) at $t=337.7$. The SS-loop is marked by the red dash-dot-dot line.

3.2. The F2MNL mode

In the present subsection, the formation of the F2MNL mode will be discussed based on the $l/D=2$ case. This mode also occurs for $l/D=2.5$, as shown in table 1; the wake for $l/D=2.5$ is shown in Appendix B. As the wake mode changes from F3SNL to F2MNL, a comparison between figures 4(a) and 4(b) shows that the formation of the N- and L-cell vortices and the vortex dislocations between them are similar for $l/D=10$ and $l/D=2$, except that the S-cell vortex is absent for $l/D=2$. Figure 5(a,b) further shows that, for $l/D=10$ and $l/D=2$, the shedding frequency of the N-cell vortex ($St_N$) and the L-cell vortex ($St_L$) does not change significantly (deviation <2 %), the frequency component $St_S$ disappears and the frequency component $St_N$ occurs both behind the small cylinder and behind the large cylinder close to the step ($-5< z<5$).

The formation of F2MNL is due to the fact that the decreasing length of the small cylinder ($l$) barely affects the flow characteristics behind the large cylinder, but significantly affects those behind the small cylinder due to the enhanced viscous suppression of the velocity fluctuations in the small cylinder near wake as the gap between the two steps decreases. Further details are discussed below. The comparison between the black and red solid curves in figure 7(a,d) shows that, as $l/D$ decreases from 10 to 2, $L_r$ and $C_{pb}$ only slightly decrease along the large cylinder with no significant change in their distribution, and thus the formation of the N- and L-cell vortices are rarely affected. Along the small cylinder, the strong wake flow interactions between the two steps for $l/D=2$ lead to a completely different distribution of $L_r$ and $C_{pb}$ compared with the $l/D=10$ case. Even for the spanwise position $z_d=0$ (i.e. $z=0$) located in the middle of the small cylinder, which is least affected by the steps, $L_r$ increases by around 95 % from 1.1$D$ to 2.1$D$ and $-C_{pb}$ increases by around 10 % from 0.7 to 0.78 when $l/D$ decreases from 10 to 2. This is because the length of the small cylinder becomes too short to let the step effect decay along the small cylinder when $l/D$ decreases from 10 to 2. For $l/D=10$, the black solid curve in figure 7(a–f) shows that, due to the different diameters of the small and large cylinders, the base pressure ($-\overline {C_{pb}}$) is weaker and the recirculation length ($\overline {L_r}$) is smaller behind the small cylinder than behind the large cylinder away from the step (Rajani, Kandasamy & Majumdar Reference Rajani, Kandasamy and Majumdar2009). However, the wakes behind the small and large cylinders mix, leading to a strengthened base pressure and increased recirculation length behind the small cylinder as the step is approached (referred to as the wake mixing effect). Unlike the $l/D=10$ case, the short small cylinder length in the $l/D=2$ case is not long enough for the wake mixing effect to vanish along the small cylinder. As a result, the red and black solid curves in figure 7(b,e) show that, in the vicinity of the step, $-\overline {C_{pb}}$ and $\overline {L_r}$ are similar for $l/D=10$ and 2; as away from the step, $-\overline {C_{pb}}$ and $\overline {L_r}$ decreases for $l/D=10$ but only exhibits a slight variation for $l/D=2$. For $l/D=2$, the recirculation length $L_r$ remains larger than for $l/D=10$ over the entire small cylinder, suppressing the instability in the near wake behind the small cylinder due to the increased viscous effects, leading to the disappearance of the S-cell vortex.

Figure 9(a–f) shows consecutive spanwise vorticity ($\omega _z =\partial v / \partial x - \partial u / \partial y$) contours in the $xy$-plane in the middle of the small cylinder ($z=0$) for $l/D=10$ (figure 9a–c) and for $l/D=2$ (figure 9d–f). The time history of the lift coefficient ($C_L$) is shown in figure 9(g,h), where the time instants for figure 9(a–f) are marked. Here, $C_L = 2 F_L(t)/\rho U^2 D_p L_c$, where $F_L$ is the sampled lift force; $D_P=D$ for the large cylinder and $D_P=d$ for the small cylinder; $L_c$ is the spanwise length of the part of the step cylinder where forces are sampled. Different from the staggered distribution of the vorticity $\omega _z$ behind the small cylinder for $l/D=10$ shown in figure 9(a–c), figure 9(d–f) shows that $\omega _z$ is almost distributed symmetrically on the $+Y$ and $-Y$ sides of the small cylinder in the near-wake region ($0.5< z<2$) for $l/D=2$. This implies that no typical staggered Kármán vortices are shed behind the small cylinder for $l/D=2$ (i.e. the vortex structures visualized by the isosurface of $\lambda _2$ behind the small cylinder in figure 4(b) are not the typical Kármán vortex). The underlying mechanism is that the shorter distance between the two steps for $l/D=2$, compared with the $l/D=10$ case, suppresses the velocity fluctuations in the near wake, leading to the absence of typical staggered Kármán vortex shedding behind the small cylinder.

Figure 9. Consecutive instantaneous vorticity $\omega _z$ contours in the $xy$-plane at $z=0$ showing the flow developments around the middle section of the cylinder: (a–c) for $l/D=10$, (d–f) for $l/D=2$. The corresponding time history of the lift coefficient for the centre part of the small cylinder at $z=0$: (g) for $l/D=10$ and (h) for $l/D=2$.

Figure 10(a,b) shows the distribution of the velocity fluctuations along a streamwise sampling line at $(y, z)=(0, 0)$ for $l/D=2$ (in red) and $l/D=10$ (in black), indicating that the velocity fluctuations are highly suppressed for $l/D=2$ compared with $l/D=10$. In addition to stabilizing the near-wake fluctuations, the reduced distance between the two steps may also lead to a decrease in the upstream velocity of the small cylinder due to an enhanced frictional effect, causing the vorticity generation around the small cylinder to become insufficient for the S-cell formation. Whether the disappearance of the S-cell vortex is caused by the suppression of the vorticity generation around the small cylinder will be discussed below. The time-averaged circulation ($\varGamma = \int \int _{A}^{B}u|\omega | \,{\rm d} y\,{\rm d} t/T$, where $T=100$ represents the integration time; the integration path (AB) is shown by the red solid line on the vertical axis in figure 9a,d) generated and convected into the wake from the small cylinder is estimated. The result shows that, from $l/D=10$ to $l/D=2$, $\varGamma$ only decreases slightly (3 % from 1.25 to 1.22). Now, it is safe to conclude that, as the recirculation length sharply increases, the wake instability is highly suppressed for $l/D=2$ compared with that for $l/D=10$, leading to the disappearance of the S-cell vortex.

Figure 10. Distribution of velocity fluctuations along a horizontal sampling line at $(y, z)=(0, 0)$ (a) $u'u'$, (b) $v'v'$.

A further detailed investigation shows that, although only one dominant shedding frequency $f_N$ is detected behind the small cylinder (figure 5b) for $l/D=2$, the corresponding vortex structure is not the N-cell vortex. Figure 11(a–e) shows consecutive snapshots of the isosurface of $\lambda _2=-0.01$ in a close-up region around the small cylinder for $l/D=2.0$ (as marked by the red rectangle in figure 4b). It appears that a vortex-loop structure forms in the near wake behind the small cylinder (i.e. in the middle part of the I-shaped step cylinder). We identify this structure as the M-cell vortex loop due to the following two reasons: (a) although this vortex cell has the same shedding frequency as the N-cell vortex, it has a different topology. (b) The shape of the M-cell vortex for F2MNL is similar to the M-cell vortex for F3MNL; whereas the M-cell vortex in the F3MNL mode has a different shedding frequency and shape compared with the N-cell vortex, which will be discussed in detail in § 3.3. Figure 11 shows that all M-cell vortex loops in the F2MNL mode consist of at least a top part M-cell vortex ($M_t$) and a bottom part M-cell vortex ($M_b$), located at $z>0$ and $z<0$, respectively. Based on a long-time observation (approximately 300 shedding periods), three different types of M-cell vortex loops are observed: (type 1) the top and bottom parts of the M-cell vortex loop are from the same cross-flow side (either $+Y$ or $-Y$ side) of the cylinder (as shown in figure 11a–c); (type 2) the top and bottom parts of the M-cell vortex loop are from the opposite cross-flow side of the cylinder (as shown in figure 11d–f); (type 3) besides a pair of $M_t$ and $M_b$ from the same side of the cylinder, another $M$ cell vortex from the opposite side of the cylinder also merges into the M-cell vortex loop (as shown in figure 11g–i).

Figure 11. Isosurface of $\lambda _2=-0.01$ in a close-up region around the small cylinder for $l/D=2$ (indicated by the red rectangle in figure 4b) shows three types of M-cell vortex loops: (a–c) type 1 M-cell vortex loop, (d–f) type 2 M-cell vortex loop, (g–i) type 3 M-cell vortex loop. ( j–l) Time history of the cross-flow velocity at $(x, y, z)=(1, 0, {\pm }2)$ for type 1–3 M-cell vortex loops. The top ($N_t$) and bottom ($N_b$) N-cell vortices and the top ($M_t$) and bottom ($M_b$) parts of the M-cell vortices are indicated in (a). The blue and red colours indicate positive and negative spanwise vorticity, respectively.

For all three types of the M-cell vortex loops, figure 11 shows that complex vortex interactions occur as these M-cell loops form and convect downstream; however, no M-cell vortex is shed from (i.e. dislocates with) the corresponding N-cell vortex. In other words, all the M-cell vortices shed in phase with the corresponding N-cell vortices, thereby explaining why only one dominant frequency exists in the M- and N-cell regions ($-5< z<5$ in figure 5b). The different types of M-cell vortex loops result from variations in the phase relationship between $N_t$ and $N_b$ cells, which changes between in phase or out of phase. This phase difference can be measured by monitoring the time history of the cross-flow velocity ($v$) at $z_1=1$ in the $N_t$ region and $z_2=-1$ in the $N_b$ region. When $N_t$ and $N_b$ shed in phase, e.g. from $t=1292.4$ to $t=1296.2$ (figure 11a–c), the time history of $v$ at $z_1$ and $z_2$ peaks (with the same sign) almost simultaneously (figure 11j). Here, $M_t$ and $M_b$ form in phase at the same cross-flow side of the cylinder, inducing the type 1 M-loop. When $N_t$ and $N_b$ shed anti-phase, e.g. from $t= 901.4$ to $t=904.0$ (figure 11d–f), the time history of $v$ peaks with the opposite sign almost simultaneously (figure 11k). Here, $M_t$ and $M_b$ form in phase at the opposite cross-flow sides ($-Y$ and $+Y$ sides) of the cylinder, inducing the type 2 M-loop. When $N_t$ and $N_b$ shed partly out of phase (i.e. between in phase and anti-phase), the vortex strength between the corresponding $N_t$ and $N_b$ cells becomes unbalanced. To compensate for this, one more M-cell vortex on the weaker M-cell side joins the M-cell vortex loop (the type 3 M-loop). For example, $M_b$ forms earlier than $M_t$ in figure 11(g). This is also reflected by the velocity time history shown in figure 11(l): the black solid line (corresponding to $N_t$) shows an earlier peak than the black dashed line (corresponding to $N_b$). Therefore, during the formation of the corresponding M-loop, the strength of $M_b$ is larger than $M_t$, causing the centre of the M-loop to move slightly in the $+z$ direction and merge with $M_t$.

3.3. The F3MNL mode

In the present subsection, the formation of the F3MNL mode will be discussed based on the $l/D=1.5$ case. As shown in table 1, this mode also occurs for $l/D=1$; the corresponding wake is shown in Appendix B. A comparison between figures 4(b) and 4(c) shows that both the F2MNL and F3MNL modes contain the M-, N- and L-cell vortices. Figure 5(b,c) further show two changes as $l/D$ decreases from 2 to 1.5: (i) a new shedding frequency component $St_M$ occurs in the small cylinder wake. This frequency is lower than the shedding frequency of the S-cell vortex for $l/D \geq 3$ and higher than the shedding frequency of the N- and L-cell vortices for all cases; (ii) the dominant shedding frequency behind the large cylinder becomes blurred. The underlying mechanisms will now be explained.

Although the M-cell vortex exists for both F2MNL and F3MNL, the occurrence of a new frequency component $St_M$ in F3MNL is due to the fact that the M-cell vortex can dislocate from the corresponding N-cell vortex in F3MNL, making the M-cell vortex exhibit a different shedding frequency than the N-cell vortex. This is in contrast to the F2MNL mode, where the M-cell vortex always keep the one-to-one relationship with the N-cell vortex (as described in § 3.2). When dislocations occur between the M- and N-cell vortices, it appears that the M-cell vortex can either simultaneously dislocate from the top and bottom N-cell vortices or dislocate from only one of them, depending on the phase difference between the corresponding top and bottom N-cell vortices. Figure 12(a–e) shows consecutive snapshots of isosurfaces of $\lambda _2=-0.01$ for $l/D=1.5$ in the region around the small cylinder marked by the red rectangle in figure 4(c). At $t= 1200.2$, $M_{1}$ forms in figure 12(a). As $M_{1}$ convects downstream (figure 12a–c), it dislocates from the corresponding N-cell vortices $N_{t1}$ and $N_{b1}$. Another M-cell vortex $M_{2}$ forms between $N_{t1}$ and $N_{b1}$ in figure 12(d,e). Here, the M-cell vortex $M_1$ simultaneously dislocates from the corresponding top ($N_{t1}$) and bottom ($N_{b1}$) N-cell vortices, because $N_{t1}$ and $N_{b1}$ almost shed in phase. When the top and bottom N-cell vortices are out of phase, the M-cell vortex can only dislocate from one of them. An example is shown in figure 13, where the M-cell vortex $M_3$ only dislocates from the bottom N-cell vortex $N_{b3}$ but keeps connected with the top N-cell vortex $N_{t3}$. The dislocations between the M- and N-cell vortices persist in time. The contour of the cross-flow velocity $v$ is plotted along a spanwise sampling line at $(x,y)=(0.6, 0)$ in figure 14. The positive and negative values of $v$, shown in the contours, are induced by the vortices shed from the $+Y$ and $-Y$ sides of the I-shaped cylinder, respectively. The number of M-cell vortices is larger than the number of N-cell vortices shed from $t=1200$ to 1300, indicating a series of dislocations at the M-N boundary.

Figure 12. Isosurface of $\lambda _2=-0.01$ in a close-up region around the small cylinder for $l/D=1.5$ (indicated by the red rectangle in figure 4c) shows a vortex dislocation process when the M-cell vortex simultaneously dislocates from the corresponding top and bottom N-cell vortices.

Figure 13. Isosurface of $\lambda _2=-0.01$ in a close-up region around the small cylinder for $l/D=1.5$ (indicated by the red rectangle in figure 4c) shows a vortex dislocation process when the M-cell vortex dislocates from the bottom N-cell vortex while it keeps the connection with the top N-cell vortex.

Figure 14. Cross-flow velocity component $v$ as a function of the non-dimensional time, along the spanwise sampling line at $(x, y)=(0.6, 0)$ for $l/D=1.5$.

The two main reasons for the occurrence of the vortex dislocation between the M- and N-cell vortices when $l/D$ decreases from 2 to 1.5 are (i) the larger vortex strength of the M-cell for $l/D=1.5$ than for $l/D=2$ and (ii) the higher convective velocity in the M-cell region than in the N-cell region. Firstly, as $l/D$ decreases from 2 to 1.5, the decreased distance between the top and bottom N-cell vortices leads to more circulation energy being transferred from the N-cell region into the M-cell region, causing an increase in the circulation strength of the M-cell vortex. The total circulation strength ($\varGamma$) within a flow region in the present work is calculated based on $\varGamma _i = {\sum _{j=1}^{N} |\omega _{i}| A_j}$, where $\omega$ represents the vorticity, $i=x, y$ or $z$ is the direction of $\omega$, $A$ represents the corresponding area perpendicular to the direction of $\omega$ and $N$ is the total number of grid cells within the selected flow region. Then, the averaged circulation strength ($\bar {\varGamma }$) can be obtained by $\bar {\varGamma }=\sqrt {\varGamma _x^2+\varGamma _y^2+\varGamma _z^2}/VO$, where $VO$ represents the volume of the selected flow region. Figure 15(a) shows the averaged circulation strength in the M-cell region (marked by the black box in figure 15b) and in the shear layer region of the small cylinder (marked by the red box in figure 15b) for $l/D=2.0$ and 1.5. The black curves exhibit an oscillatory behaviour. Every time an M-cell vortex passes through the monitoring flow region, the black curves reach a peak. The red curve (shear region) remains almost unchanged since the corresponding monitoring flow region is located upstream of the position the M-cell vortex sheds. It appears that the circulation strengths within the shear layer around the small cylinder are comparable for $l/D=2$ and 1.5. However, the circulation strength in the M-cell region increases substantially (i.e. the oscillation amplitude along the black solid curve for $l/D=1.5$ is clearly higher than for the black dashed curve for $l/D=2$). These tendencies are due to more circulation energy being transferred from the N-cell region to the M-cell region as $l/D$ decreases from 2 to 1.5. This stronger circulation strength makes the top and bottom parts of the M-cell vortex able to sustain themselves while disconnecting from the corresponding N-cell vortices as the M-cell vortex convects downstream for $l/D=1.5$, instead of disconnecting from each other as for $l/D=2$. Secondly, the smaller blockage effect of the small cylinder (compared with the large cylinder) causes the averaged convective velocity to become larger behind the small cylinder (the M-cell region) than behind the large cylinder (the N-cell region). Due to the spatial inhomogeneity of the convective velocity in the present case, it is difficult to evaluate an accurate convective velocity in the N- and M-cell regions. Yet, the mean streamwise velocity in the different vortex cell regions can be used to roughly compare the convective velocities in these regions. Figure 16 shows the averaged streamwise velocity in the M-cell region and the N-cell region, based on $\bar {u} = {\sum _{j=1}^{N} u_j A_j} / \sum _{j=1}^{N} A_j$, where $N$ represents all grid cells within the sampling flow region, and $A$ is the streamwise projection area of the corresponding grid cell. The flow region selected to calculate the averaged velocity is $-0.4< y<0.4$ (covering the concentrated swirling region in the wake behind the small cylinder with $D/d=2$), $-l/2D< z< l/2D$ for the M-cell region and $-5< z<0$ for the N-cell region. Figure 16(a) indicates that the vortex in the M-cell region convects faster than that in the N-cell region. The higher convective velocity and stronger circulation strength cause the new shedding frequency of the M-cell vortex, $St_M$, to occur for $l/D=1.5$. Our explanation can be further confirmed by checking the variation of the shedding frequency of the M-cell vortex when $l/D$ decreases from 1.5 to 1.0. The corresponding mean streamwise velocity distribution and shedding frequencies are shown in figures 16(b) and 31(b) (Appendix B), indicating that, as the convective velocity behind the small cylinder decreases, the shedding frequency of the M-cell vortex decreases from 0.23 for $l/D=1.5$ to 0.21 for $l/D=1$.

Figure 15. (a) Time history of the circulation strength in the M-cell region (solid black line for $l/D=1.5$ and dashed black line for $l/D=2$) and the shear layer around the small cylinder (solid red line for $l/D=1.5$ and dashed red line for $l/D=2$). (b) The red and black boxes show the flow volumes used to integrate the circulation strength in the shear layer around the small cylinder and the M-cell region, respectively.

Figure 16. (a) Distributions of the mean streamwise velocity in the M-cell and N-cell regions for $l/D=1.5$. (b) Distributions of the mean streamwise velocity in the M-cell region for $l/D=1.5$ and 1.

Another interesting phenomenon for the F3MNL mode is that the frequency contours become blurred behind the large cylinder in the N- and L-cell vortex regions, as shown in figure 5(c). This is caused by the interactions between the M- and N-cell vortices described above. Figure 17(a,b) compares the time history of the downwash flow (spanwise flow) and the base pressure in the vicinity of the step behind the large cylinder for $l/D=2$, $l/D=1.5$ and $l/D=0.2$. As pointed out by Tian et al. (Reference Tian, Jiang, Pettersen and Andersson2020a), the formation mechanism of the N-cell vortex is closely related to the step-induced variations of the spanwise velocity ($w$) and the base pressure. As $l/D$ decreases from 2.0 to 1.5, the occurrence of the vortex dislocation between the M- and N-cell vortices disturbs the wake flow and induces a more irregular and sharp variation of the spanwise velocity and the base pressure for $l/D=1.5$ (the black solid line in figure 17a,b) than for $l/D=2.0$ (the black dashed line in figure 17a,b). This causes the shedding frequencies of the N- and L-cell vortices to be unstable for $l/D=1.5$, leading to a blurred frequency contour behind the large cylinder for $l/D=1.5$, as shown in figure 5(c). Another consequence of the unstable shedding frequencies ($St_N$ and $St_L$) is that the regularity of the vortex dislocations between the N- and L-cell vortices becomes weaker for $l/D=1.5$ than for $l/D=10$ and 2. As shown in figure 6(a–c), the formation position and the period of the NL-dislocation vary only for $l/D=1.5$.

Figure 17. (a) Time history of the spanwise velocity $w$ at $(x, y, z_D)=(0.6, 0, -0.1)$, where $z_D$ is defined in figure 7 for $l/D=2.0$, 1.5 and 0.2. (b) Time history of the base pressure at $(x, y, z_D)=(0.51, 0, -0.1)$. The period of the vortex dislocation between the N- and L-cell vortices for $l/D=2.0$ is marked.

3.4. The F1L mode

In the present section, the formation of the F1L mode will be discussed based on the $l/D=0.2$ case. A comparison between figures 4(c) and 4(d) shows two differences as the flow changes from the F3MNL to the F1L mode: (i) the N- and M-cell vortices disappear, while the L-cell vortex becomes the only dominant vortex; (ii) the streamwise vortices form regularly between the L-cell vortices shed from the different sides of the I-shaped step cylinder.

To be certain that the vortex in the small cylinder wake belongs to the L-cell vortex instead of the M-cell vortex, figure 18(a–c) shows consecutive spanwise vorticity $\omega _z$ contours in the $xy$-plane in the middle of the small cylinder ($z=0$) for $l/D=0.2$. The corresponding time history of $C_L$ is shown in figure 18(d). It appears that the vortices behind the small cylinder are generated from the shear layer located at $y=\pm 0.5$ (figure 18a–c). Since $y=\pm 0.5$ is located at the surface of the large cylinder, the vortices behind the small cylinder are mainly from the shear layer of the large cylinder, belonging to the L-cell vortex. The disappearance of the M-cell vortex as $l/D$ decreases from 2 to 0.2 is caused by the fact that the shorter distance between the two steps highly suppresses the surrounding velocity around the small cylinder, thus suppressing the generation of circulation. Figure 19 shows the contours of the time-averaged velocity magnitude ($U_{mag}=\sqrt {u^2+v^2+w^2}$) on the $xy$-plane in the middle of the small cylinder for $l/D=2$ and 0.2. It shows that the surrounding flow around the small cylinder (e.g. the region encircled by the white rectangle in figure 19) in the $l/D=0.2$ case is clearly weaker than that in the $l/D=2$ case. As a result, the circulation generated from the $+Y$ and $-Y$ sides of the small cylinder is clearly smaller for $l/D=0.2$ (as shown in figure 18a–c) than for $l/D=2$ (as shown in figure 9e–g), leading to the disappearance of the M-cell vortex.

Figure 18. (a–c) Consecutive instantaneous vorticity $\omega _z$ contours in the $xy$-plane at $z=0$ showing the flow developments around the middle section of the cylinder for $l/D=0.2$. (d) The corresponding time history of the lift coefficient for the centre part of the small cylinder at $z=0$.

Figure 19. Time-averaged magnitude of the velocity ($U_{mag}=\sqrt {u^2+v^2+w^2}$) on the $xy$-plane in the middle of the small cylinder ($z=0$): (a) for $l/D=2$, (b) for $l/D=0.2$.

The vanishing of the N-cell vortex is due to the fact that the three-dimensional effect of the step on the large cylinder wake becomes weak as the length of the small cylinder decreases to 0.2$D$. Figure 17(a,b) shows that the fluctuation of the downwash flow and the base pressure become much more regular and weaker for $l/D=0.2$ than for $l/D=1.5$ and 2.0; and these two flow quantities dominate the formation of the N-cell vortex (Tian et al. Reference Tian, Jiang, Pettersen and Andersson2020a). The formation of the regular streamwise vortices between the neighbouring L-cell vortices is due to the step disturbing the wake. Firstly, the spanwise velocity ($w$ shown in figure 17a) triggered by the step contributes to the formation of the streamwise vorticity ($\omega _x = \partial w/\partial y - \partial v/\partial z$). Secondly, the above-mentioned velocity suppression effect of the two steps, and the small diameter of the small cylinder, cause the circulation strength of the L-cell vortex to be smaller behind the small cylinder than behind the large cylinder away from the step. This is illustrated by the time history of the circulation strength ($\varGamma$) within the flow volume just behind the small cylinder (marked by the red box in figure 20b) and behind the large cylinder away from the step (marked by the black box in figure 20b). Only the L-cell vortices shed from the $-Y$ side of the cylinder are monitored. This difference in vortex strength induces part of the L-cell vortex to split and connect to the other L-cell vortices, which shed from the opposite side of the cylinder, as shown in figure 4(d).

Figure 20. (a) Time history of the circulation strength in the wake behind the small and large cylinders for $l/D=0.2$. (b) The red box ($3< x<4$, $-1< y<0$ and $-0.1< z<0.1$) and black box ($3< x<4$, $-1< y<0$ and $-3< z<2$) show the flow volumes used to calculate the circulation strength behind the small and large cylinders, respectively.

5.1. Structural load along the large cylinder

Figure 24(a,c) shows the distributions of $\overline {C_D}$ and $\overline {C^{\prime }_L}$ along the large cylinder, while figure 24(b,d) shows the distributions of $\overline {C_D}$ and $\overline {C^{\prime }_L}$ along the small cylinder; the structural load distribution along a single-step cylinder with $D/d=2$ for $Re_D=150$ (adopted from Tian et al. Reference Tian, Zhu, Holmedal, Andersson, Jiang and Pettersen2023) is given for comparison. As $l/D$ decreases from 10 to 1.5, figure 24(a,c) shows that the distributions of $\overline {C_D}$ and $\overline {C^{\prime }_L}$ along the large cylinder are qualitatively similar to those for the single-step cylinder. This is because the dominant vortex cells (the N- and L-cell vortices) are similar for the F3SNL, F2MNL and F3MNL modes, as well as for the single-step cylinder (see figures 1c and 4a–c). In figure 24(a,c), the local extremes of drag coefficient $EX_{DL}$ and lift coefficient $EX_{LL}$ are marked by triangles and circles, respectively, for all cases.

As $l/D$ decreases to 0.2, an apparent change of $\overline {C^{\prime }_L}$ is the disappearance of the local minimum $EX_{LL}$ (see figure 24a). The occurrence of this minimum for $l/D=10$, 2 and 1.5 is caused by the vortex dislocations between the N- and L-cell vortices: Tian et al. (Reference Tian, Zhu, Holmedal, Andersson, Jiang and Pettersen2023) reported that the formation of such vortex dislocations can cause a 90 % reduction in the local lift amplitude. The series of vortex dislocations at the N-L cell boundary for $l/D=10$, 2 and 1.5 shown in figures 4(a–c) and 6(a–c), are underpinning the local minima $EX_{LL}$. As the vortex dislocations between the N- and L-cell vortices disappear for $l/D=0.2$ due to the absence of the N-cell vortex (see figure 6d), the local minimum $EX_{LL}$ also disappears. As a result, $\overline {C^\prime _L}$ becomes larger along the small cylinder for $l/D=0.2$ than for $1.5< l/D<10$.

For all the cases, $\overline {C_D}$ decreases towards the step along the large cylinder until reaching the local minimum $EX_{DL}$, and then increases towards to the step. This is similar to the behaviour observed for the single-step cylinder (Tian et al. Reference Tian, Zhu, Holmedal, Andersson, Jiang and Pettersen2023). However, a detailed investigation reveals that the mechanism underpinning the increase of $\overline {C_D}$ from $EX_{DL}$ towards the step is different for the $l/D=0.2$ case than for the other cases.

This is further detailed in table 2, showing the total drag coefficient $\overline {C_D}$, and the pressure-induced drag coefficient over the fore part and aft part of the cylinder, as well as over the entire cylinder at $EX_{DL}$ and at $z_D/D=-0.1$ (i.e. 0.1$D$ away from the step). The ratio of the change in $\overline {C_D}$ due to the pressure is also given as $R = \Delta \overline {C_{Dp}}/ \Delta \overline {C_D}$. Firstly, $R$ is larger than 78 % for all cases, indicating that the pressure-induced drag coefficient $\overline {C_{Dp}}$ dominates the variation of the total drag coefficient. Secondly, the increase of the pressure-induced drag coefficient $\overline {C_{Dp}}$ is mainly due to the increased $\overline {C_{Dp}}$ on the aft part of the cylinder for the F3SNL, F2MNL and F3MNL modes. This is similar to the mechanism reported for the single-step cylinder (Tian et al. Reference Tian, Zhu, Holmedal, Andersson, Jiang and Pettersen2023). However, for the F1L mode, the increase of pressure-induced drag coefficient $\overline {C_{Dp}}$ is caused by the increased $\overline {C_{Dp}}$ on the fore part of the cylinder; the pressure-induced drag coefficient on the aft part of the cylinder decreases slightly from $EX_{DL}$ to the step for the F1L mode, as opposed to the other modes. These different behaviours between the F3SNL, F2MNL and F3MNL modes on one side, and the F1L mode on the other side, are due to the different wake modes around the I-shaped step cylinders. For the F3SNL, F2MNL and F3MNL modes, there are three vortex cells behind the I-shaped step cylinder. Figure 7(d) shows that the recirculation length in the N-cell vortex region is larger than in the neighbouring L-cell and S-cell (or M-cell) vortex regions, leading to a non-monotonic variation of the recirculation length along the large cylinder. The time-averaged streamlines in the $xy$-planes reveal that the circulation centre moves upstream as the step is approached, thus strengthening the pressure behind the large cylinder, leading to an enhanced pressure force on the aft part of the cylinder for the F3SNL, F2MNL and F3MNL modes (see table 2). For the F1L mode, however, the L-cell vortex is the only dominant vortex cell behind the step cylinder, leading to a monotonic variation of the recirculation length along the large cylinder, as shown in figure 7(d). Figure 25(d,g) shows that the recirculation centre moves slightly downstream as the step is approached, thereby weakening the pressure behind the large cylinder, thus leading to the slightly decreased pressure force on the aft part of the cylinder (see table 2).

Table 2. Time-averaged pressure-induced drag force coefficient ($\overline {C_{Dp}}$) and total drag force coefficient ($\overline {C_D}$) on the fore part ($x<0$) and the aft part ($x>0$) of the cylinder for different wake modes at $EX_{DL}$ and the position 0.1$D$ away from the step surface. The shaded columns represent the position 0.1$D$ away from the step. From $EX_{DL}$ to the position 0.1$D$ below the step, the ratio of the change in total $\overline {C_D}$ due to pressure is shown in the last line, $R = \Delta \overline {C_{Dp}}/ \Delta \overline {C_D}$.

Figure 25. Time-averaged streamlines on $xy$-planes behind the large cylinder: (a–d) at $z_D=-0.1$ for $l/D=10$, 2, 1.5 and 0.2; (e–h) at the local minimum point $EX_{DL}$ shown in figure 24(a) for $l/D=10$, 2, 1.5 and 0.2. The red reference line indicates the location of the circulation centre on the $xy$-plane.

5.2. Structural load along the small cylinder

As the configuration changes from the single-step cylinder to the I-shaped dual-step cylinder with different gaps between the two steps, the structural load is more affected along the small cylinder than along the large cylinder. First, the structural load (i.e. $\overline {C_D}$ and $\overline {C^{\prime }_L}$) on the I-shaped step cylinder with $l/D=10$, 2 and 1.5 will be discussed. Then, the structural loads will be discussed for $l/D=0.2$.

Figure 24(b,d) shows that the lift coefficient and the drag coefficient in the middle of the small cylinder for $l/D=10$ are higher than for the single-step cylinder. This is because the symmetric shape of the I-step cylinder leads the S-cell vortex to shed in parallel behind the middle of the small cylinder for $l/D=10$ (see figure 4a), while an oblique shedding angle of around 13 degrees was found for the single-step cylinder (Tian et al. Reference Tian, Zhu, Holmedal, Andersson, Jiang and Pettersen2023) at the same distance away from the step. This oblique vortex shedding will re-direct part of the spanwise vortex circulation ($\omega _z$) into circulation in the other two directions. This will in turn decrease the base pressure strength, causing smaller structural load coefficients in the single-step cylinder case.

When $l/D$ decreases from 10 to 2 (and 1.5), figure 24(b,d) shows that the distribution of $\overline {C_D}$ does not change significantly, while $\overline {C^{\prime }_L}$ changes substantially along the small cylinder. The smaller $\overline {C^{\prime }_L}$ as $l/D$ decreases is due to the fact that the vorticity distribution in the near-wake region behind the small cylinder changes from asymmetric to almost symmetric (around the streamwise centre line) as $l/D$ decreases from 10 to 2 (see figure 9a–f). During this process, no clear decrease occurs in the magnitude of the spanwise vorticity $|\omega _z|$ (the dominant vorticity component in that region), leading to the almost unchanged $\overline {C_D}$.

As $l/D$ further decreases from 2 to 0.2, $\overline {C^{\prime }_L}$ increases and $\overline {C_D}$ decreases. Firstly, $\overline {C_D}$ is discussed in the light of the pressure on the $+X$ and $-X$ parts of the cylinder; figure 26(a) shows the circumferential distribution of the time-averaged pressure ($\overline {C_{p}}=2(\bar {p}-p_0)$) along the small cylinder at $z=0$ for $l/D=2$ (red curve) and 0.2 (black curve). It appears that the time-averaged impinging pressure on the $-X$ part of the small cylinder ($\theta <45$) and pressure on the $+X$ part of the small cylinder ($\theta >90$) are weaker for $l/D=0.2$ than for $l/D=2$, leading to a smaller pressure difference between the $-X$ and the $+X$ parts of the cylinder, i.e. between fore and aft, and thus an overall smaller drag distribution along the small cylinder for $l/D=0.2$ than for $l/D=2$ (as shown in figure 24b). This is because the velocity surrounding the small cylinder is suppressed as the distance between the step surfaces decreases from $l/D=2$ to $l/D=0.2$, due to friction.

Figure 26. The circumferential distribution of pressure along a slice in the middle of the small cylinder at $z=0$: (a) the time-averaged pressure and (b) the instantaneous pressure when $C_L$ peaks at t1 (in figure 18d) for $l/D=0.2$ and at t4 (in figure 9h) for $l/D=2$. Notice that the continuous and broken lines coincide in (a).

Secondly, $\overline {C^\prime _L}$ is discussed in the light of the pressure on the $+Y$ and $-Y$ sides of the cylinder. Figure 26(b) shows the circumferential distribution of the instantaneous pressure along the small cylinder at $z=0$ for $l/D=2$ (red curve) and 0.2 (black curve) when $C_L$ peaks for $l/D=0.2$ (i.e. at t1 in figure 18d) and when $C_L$ peaks for $l/D=2$ (i.e. at t4 in figure 9h). It appears that the pressure difference between the $+Y$ and $-Y$ sides of the cylinder is visible for $l/D=0.2$, but almost negligible for $l/D=2$ when $C_L$ peaks. Thus, the fluctuation amplitude of $C_L$ for the $l/D=0.2$ case is much larger than for $l/D=2$, as shown in figures 9(h) and 18(d). This is due to the disappearance of the M-cell vortex and the appearance of the L-cell vortex behind the small cylinder, as $l/D$ decreases from 2 to 0.2, which in turn makes the near wake change from symmetric to asymmetric, as shown by the vorticity contours in figures 9(e–g) and 18(a–c). This explains why $\overline {C^\prime _L}$ is larger for $l/D=0.2$ than for $l/D=2$.