3.1. The effect of the boundary layer on the end walls

The effect of the boundary layer on the end walls is investigated by carrying out computations in the high subcritical regime, with and without the end plates for $L_z=1D$. The comparison of the time-averaged coefficient of drag ($\overline {C}_D$), r.m.s. of coefficient of lift ($C_{Lrms}$), r.m.s. of coefficient of drag ($C_{Drms}$) and non-dimensional vortex shedding frequency obtained from the two sets of computations is presented in table 1. It is observed that the results from the two sets of boundary conditions are in good agreement and also close to the data reported by Szepessy & Bearman (Reference Szepessy and Bearman1992). Further comparison of the flows, for $Re=4.0\times 10^4$, obtained from computations with the two boundary conditions are presented in figure 4.

Figure 4. The $L_z=1D$, $Re=4.0\times 10^4$ flow past a cylinder with and without end plates at the lateral boundaries: surface distribution of time- and span-averaged (a) coefficient of pressure ($\overline {C}_P$) and (b) skin friction ($\overline {C}_f$) for slip, no-slip conditions on the end plates and no-slip conditions on the end plates but excluding the sidewall boundary layer (BL) while span averaging. Shown in panel (c) is the spanwise variation of $\overline {C}_P$ at the shoulder ($\theta =90^{\circ }$) via broken line and base of the cylinder ($\theta =180^{\circ }$) via solid line. Streamlines for the time- and span-averaged are shown for (d) slip wall, (e) no-slip wall and (f) no-slip wall excluding the sidewall boundary layer while span averaging.

Table 1. Flow past a circular cylinder of $L_z=1D$ with and without end plates at the lateral boundaries: time-averaged coefficient of drag ($\overline {C}_D$); r.m.s. of coefficient of lift ($C_{Lrms}$) and non-dimensional vortex shedding frequency ($St$) at $Re=2.0\times 10^4$ and $4.0\times 10^4$. The abbreviation ‘S & B’ stands for Szepessy & Bearman (Reference Szepessy and Bearman1992); BC denotes boundary condition.

Two sets of results for a cylinder with no-slip sidewalls are shown in the figure. In the first set of results, referred to as ‘no-slip wall’, the flow is averaged over the entire span. Results are also shown for span averaging that excludes the regions corresponding to boundary layers on the sidewalls. This is referred to as ‘no-slip wall excluding sidewall boundary layer’. The time- and span-averaged streamlines as well as surface distribution of the coefficients of pressure ($\overline {C}_P$) and skin friction ($\overline {C}_f$) for all the cases are in very good agreement. The time- and span-averaged streamlines as well as surface distribution of the coefficients of pressure ($\overline {C}_P$) and skin friction ($\overline {C}_f$) for all the cases are in very good agreement. The spanwise variation of time-averaged coefficient of pressure at shoulder ($\theta =90^{\circ }$) and base point ($\theta =180^{\circ }$), plotted in figure 4(c), confirms that the changes to the flow caused by the boundary layer on the end walls are very local and have little effect on the bulk of the flow. The displacement thickness ($\delta _1$) due to the boundary layer on the end wall at $(x/D, y/D)=(0, 3)$, estimated from the time-averaged velocity profile, is $0.0151 D$. This shows that the combined viscous region of the two end plates, for this $Re$, is restricted to a mere $3\,\%$ of the span which is expected to become even smaller for larger $Re$. For this reason, computations at larger $Re$ are carried out with slip boundary conditions on the lateral walls. The confinement effect of the lateral walls in restricting three-dimensionality of the flow will be investigated in a later section.

3.2. Time-averaged drag: variation with Re and classification of regimes

Figure 5 shows the variation of time-averaged coefficient of drag ($\overline {C}_D$) with $Re$ for cylinder with span lengths, $L_z=1D$ and $3D$. Also shown is the data from earlier studies for various $L_z$. To enable further discussion, we first identify the various flow regimes. Achenbach (Reference Achenbach1968) classified the flow regimes based on variation of time-averaged coefficient of drag ($\overline {C}_D$) with $Re$. Later, Schewe (Reference Schewe1983) showed that the classification is unambiguous if carried out on the basis of the variation of drag force ($F_x$) with $Re$. The maxima and minima of $F_x$, respectively, correspond to the onset and end of the critical regime. We utilize the method proposed by Schewe (Reference Schewe1983). We recall that the mean drag force is related to the time-averaged coefficient of drag as $\overline {F}_x=\frac {1}{2} \rho U^2_{\infty } L_z D \overline {C}_D$. This expression may be rewritten, in terms of $Re$, as $\overline {F}_x=Q\overline {C}_D Re^2$, where, $Q={\mu ^2 L_z}/{2 \rho D}$. Here, $Q$ is a constant for a given physical model and fluid. Therefore, the variation of $\overline {F}_x$ can be studied via the variation of $\overline {C}_D Re^2$ and used for classifying the flow regimes.

Figure 5. Flow past a circular cylinder: variation of time-averaged coefficient of drag ($\overline {C}_D$) with Reynolds number. The abbreviations are: EP, cylinder with side end plates; S & B, Szepessy & Bearman (Reference Szepessy and Bearman1992); S, Schewe (Reference Schewe1983); A, Achenbach (Reference Achenbach1968); L, Lehmkuhl et al. (Reference Lehmkuhl, Rodríguez, Borrell, Chiva and Oliva2014); C, Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017); B, Bearman (Reference Bearman1969) and D: Desai et al. (Reference Desai, Mittal and Mittal2020).

The onset of transition of boundary layer causes a reduction in drag with an increase in $Re$, while its increase with a further increase in $Re$ marks the end of the transition.

We test the proposed scheme by applying it to the data from Schewe (Reference Schewe1983) for $\overline {F}_x$ and $\overline {C}_D$. Figure 6(a) shows the variation of $F_x$ and $\overline {C}_D Re^2$ with $Re$ for the data reported by Schewe (Reference Schewe1983). Both variables show similar variation with $Re$, including the location of local maxima and minima. This confirms that $\overline {C}_D Re^2$ can indeed be used as a proxy for $\overline {F}_x$ for classifying the flow regime. Figure 6(b) shows the variation of $\overline {C}_D$ and $\overline {C}_D Re^2$ with $Re$ for the data from the present computations for $L_z=1D$. The flow regimes, namely subcritical, critical and supercritical are marked in the figure. It can be observed that $\overline {C}_D Re^2$ achieves a local maxima at $Re=1 \times 10^5$. This marks the onset of the critical regime. In the critical regime both $\overline {C}_D$ and $\overline {C}_D Re^2$ decrease with an increase in $Re$. A local minima of $\overline {C}_D Re^2$ at $Re=1.5\times 10^5$ marks the end of critical regime. In the supercritical regime, $\overline {C}_D$ continues to decrease with $Re$; $\overline {C}_D Re^2$, however, increases with increase in $Re$.

Figure 6. Flow past a circular cylinder: (a) the variation of time-averaged drag force $\overline {F}_x$ and $\overline {C}_D Re^2$ with $Re$ for the data from Schewe (Reference Schewe1983) for cylinder with $L_z=10D$ and (b) the variation of time-averaged coefficient of drag ($\overline {C}_D$) and $\overline {C}_D Re^2$ with Reynolds number from present numerical simulations on cylinder with $L_z=1D$.

We note from figure 5 that the variation of $\overline {C}_D$ with $Re$, from the present study, for $L_z=1D$ is in good agreement with the experimental results of Szepessy & Bearman (Reference Szepessy and Bearman1992). It is also in good agreement with the computational results of Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017) in the subcritical regime. Their computations were carried out for $L_z=3D$ in the subcritical regime and $L_z=1D$ at higher $Re$. We note that the variation in $\overline {C}_D$ with $Re$, from the present study, is very similar for $L_z=1D$ and $3D$.

In addition, the boundary layer on the end plates does not have any significant effect on $\overline {C}_D$. In the supercritical regime, at $Re=4.0\times 10^5$, the $\overline {C}_D$ from present computation is very close to that from Schewe (Reference Schewe1983) and Bearman (Reference Bearman1969). Schewe (Reference Schewe1983) reported that the onset of drag crisis takes place at $Re_c\approx 2.8\times 10^5$ for a cylinder with $L_z=10D$. Desai et al. (Reference Desai, Mittal and Mittal2020) carried out experiments on cylinder of $L_z=22.5D$ and observed that $Re_c$ is $3.3\times 10^5$. Experiments by Achenbach (Reference Achenbach1968) were carried out on cylinders with $L_z=3.3D$ and $6.6D$. The drag crisis in their study was observed to be more gradual compared with other studies including the present one. The $Re_c$ was not reported in computational studies by Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017) and Lehmkuhl et al. (Reference Lehmkuhl, Rodríguez, Borrell, Chiva and Oliva2014). This figure clearly brings out the effect of span length of the cylinder on the $Re_c$. Also, $Re_c$ is relatively larger for cylinders of large span. It varies between $2.8\times 10^5\text {--}3.3\times 10^5$ for $10D \leq L_z \leq 22.5D$. On the other hand, it is relatively low for shorter span lengths. Figure 5 shows that $Re_c$ is approximately $1\times 10^5$ for $1D \leq L_z \leq 3D$.

3.3. Root mean square of force coefficients versus $Re$ and effect of span length ($L_z$)

Other quantities that are sensitive to the span length of the cylinder are the r.m.s. of the coefficient of lift ($C_{Lrms}$) and drag ($C_{Drms}$). The variation of $C_{Lrms}$ and $C_{Drms}$ with $Re$ for various $L_z$ are shown in figure 7. Their trend is similar to the variation of $\overline {C}_D$ with $Re$, shown in figure 5. In general, both $C_{Lrms}$ and $C_{Drms}$ increase with an increase in $Re$ in the subcritical regime and achieve a maxima prior to the flow entering the critical regime. The peak value of $C_{Lrms}$ and $C_{Drms}$ decreases with an increase in span length of the cylinder. The overall variation of $C_{Lrms}$ with $Re$, for the present study with $L_z=1D$, is in very good agreement with the data reported by Szepessy & Bearman (Reference Szepessy and Bearman1992).

Figure 7. Flow past a circular cylinder: variation of r.m.s. of coefficient of (a) lift ($C_{Lrms}$) and (b) drag ($C_{Drms}$) with Reynolds number. The abbreviations are: EP, cylinder with end plates; S & B, Szepessy & Bearman (Reference Szepessy and Bearman1992); K, Keefe (Reference Keefe1962); F, Fung (Reference Fung1960); S, Schewe (Reference Schewe1983); R, Rodríguez et al. (Reference Rodríguez, Lehmkuhl, Chiva, Borrell and Oliva2015); D, Desai et al. (Reference Desai, Mittal and Mittal2020).

The boundary layer on the end plate does not appear to have any significant effect on either $C_{Lrms}$ or $C_{Drms}$. The peak value, as well as the $Re_c$ where $C_{Lrms}$ experiences a steep decrease with increase in $Re$, show good match. Furthermore, $C_{Lrms}$ and $C_{Drms}$ decrease sharply with an increase in $Re$ in the critical regime. In the supercritical regime, they undergo a gradual decrease up to $\mbox {{Re}}=3.0\times 10^5$ followed by an increase with further increase in $\mbox {{Re}}$. A similar increase in the supercritical regime was also observed in experimental studies by Fung (Reference Fung1960), Schewe (Reference Schewe1983) and Desai et al. (Reference Desai, Mittal and Mittal2020). Keefe (Reference Keefe1962) reported data for the variation of $C_{Lrms}$ with $Re$ in the subcritical regime for $L_z=3D$. The results from the present study are in very good agreement with this data. The very significant effect of $L_z$ on variation of $C_{Lrms}$ with $Re$ is clearly seen from figure 7. For example, for $L_z=1D$, the flow becomes critical at $Re\approx 1 \times 10^5$ beyond which $C_{Lrms}$ decreases sharply (present results and Szepessy & Bearman Reference Szepessy and Bearman1992). It becomes critical at $Re\approx 1.4\times 10^5$ for $L_z\approx 6D$ (Fung Reference Fung1960; Szepessy & Bearman Reference Szepessy and Bearman1992), and at $Re \approx 2.8\times 10^5$ (Schewe Reference Schewe1983) for $L_z=10D$. Szepessy & Bearman (Reference Szepessy and Bearman1992) and Cadot et al. (Reference Cadot, Desai, Mittal, Saxena and Chandra2015) proposed that the drop in $C_{Lrms}$ in the critical regime is due to weakening of vortex shedding. This and the very interesting variation of $\overline {C}_D$ and $C_{Lrms}$ with $Re$ in the subcritical regime is explored, in a later section in this work.

To investigate the effect of span length, we consider the flow at $Re=0.5\times 10^5$ for $L_z=1D$ and $3D$. Figure 8(a,b) shows $\overline {ww}$ on $x$–$z$ plane at $y=0.05D$ for $L_z=1D$ using slip and no-slip boundary conditions on the plate. The same for $L_z=3D$, and with slip boundary conditions on the end walls, is shown in figure 8(c). The $\overline {ww}$ field is a measure of three-dimensionality in the flow. We utilize it to study the confinement effect of the lateral walls. It is observed that the three-dimensionality in the flow is significantly higher for $L_z=3D$, compared with $L_z=1D$. On the other hand, the boundary layer on the end plates do not appear to have a significant effect on the three dimensionality as indicated by the images for $L_z=1D$. In fact, as seen in § 3.1, they do not have any significant effect on the flow. This is further confirmed by the span-averaged, $\overline {u'u'}$ component of the Reynolds stress on the $x$–$y$ plane shown in figure 8(d,e). The same for $L_z=3D$, in figure 8(f) shows relatively lower stress in the near wake, indicating lower level of activity related to the instability of the shear layer, compared with $L_z=1D$. The confinement of the flow by closing in of lateral walls, therefore, leads to formation of LSB at lower $Re$ and early onset of transition. We also note that, as observed by Kravchenko & Moin (Reference Kravchenko and Moin2000), periodic boundary conditions on the lateral walls are not suitable for studying the effect of $AR$ of the cylinder on the flow. Perhaps this explains the prediction of delayed transition in the computational studies of Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017), Rodríguez et al. (Reference Rodríguez, Lehmkuhl, Chiva, Borrell and Oliva2015) and Lehmkuhl et al. (Reference Lehmkuhl, Rodríguez, Borrell, Chiva and Oliva2014) who employ a relatively short span but impose periodic conditions on the lateral boundaries.

Figure 8. Flow past a $Re=0.5\times 10^5$ circular cylinder: (a–c) $\overline {ww}$ in the $x$–$z$ plane at $y/D=0.05$ and (d–f) span-averaged $\overline {u'u'}$ in the $x$–$y$ plane where panels (a,d) are for a cylinder with $L_z=1D$ and slip condition on lateral boundaries, panels (b,e) are for $L_z=1D$ and no-slip condition on the end plates and panels (c,f) are for $L_z=3D$ and with slip condition on lateral boundaries.

3.4. SV and LSB

Figure 9 shows the time- and span-averaged streamlines for various $Re$. Following the separation of flow, a pair of counter-rotating standing vortices form the wake separation bubble, which is observed at all $Re$. In addition, a smaller region of recirculation appears downstream of the separation point for $Re>0.05\times 10^5$. It can be clearly seen in figure 9(b,c) for $Re=0.2\times 10^5$ and $Re=0.4\times 10^5$. In fact this bubble continues to exist in the critical and supercritical regimes but in a modified form. Son & Hanratty (Reference Son and Hanratty1969) also observed such a circulation region in the subcritical regime and referred to it as an SV. For consistency, we use the same nomenclature. Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017) also observed the secondary vortex but only in the subcritical regime. In contrast, Ono & Tamura (Reference Ono and Tamura2008) observed the SV even in the supercritical regime.

Figure 9. Flow past a circular cylinder with $L_z=1D$: time- and span-averaged streamlines for (a) $Re=0.02\times 10^5$, (b) $Re=0.2\times 10^5$, (c) $Re=0.4\times 10^5$, (d) $Re=1.4\times 10^5$, (e) $Re=3.0\times 10^5$, and (f) $Re=3.5\times 10^5$. The insets shows the close-up views of the flow to bring out the SV and LSB.

An LSB appears in the flow for $Re\geq 1.2\times 10^5$ as shown in figure 9(d,f) for certain $Re$. The presence of LSB and its role in the transition of the flow has been reported in several earlier studies (Achenbach Reference Achenbach1968; Singh & Mittal Reference Singh and Mittal2005; Lehmkuhl et al. Reference Lehmkuhl, Rodríguez, Borrell, Chiva and Oliva2014; Cheng et al. Reference Cheng, Pullin, Samtaney, Zhang and Gao2017; Chopra & Mittal Reference Chopra and Mittal2017; Pandi & Mittal Reference Pandi and Mittal2019). We identify the separation and attachment points associated with the SV and LSB via the variation of the time- and span-averaged surface skin friction distribution ($\overline {C}_f$) along the surface of the cylinder; $\overline {C}_f$ changes sign at these points. Based on the presence of SV and LSB, the flow can be broadly classified in three regimes. A representative flow in each of the three regimes is shown in figure 10 along with the surface distribution of $\overline {C}_P$ and $\overline {C}_f$.

Figure 10. Flow past a circular cylinder with $L_z=1D$: the left-hand column of the figure shows the close-up view of the time- and span-averaged streamlines near the upper shoulder of the cylinder and the right-hand column shows the time- and span-averaged coefficient of pressure and skin friction distribution of the upper surface of cylinder ($0\leq \theta \leq 180$) for $Re=$ (a) $0.02\times 10^5$, (b) $0.2\times 10^5$, and (c) $4.0\times 10^5$. The close-up view of the upper shoulder to enlarge the SV and LSB for $Re=4.0\times 10^5$ is shown in panel (d). The LS, secondary attachment (SA), secondary separation (SS), turbulent attachment (TA) and turbulent separation (TS) points are also shown in the figure.

In the first of the three regimes, the boundary layer is in a laminar state when it separates and does not reattach. The SV is also not observed in this case. This state is observed for $Re\leq 2\times 10^3$. A schematic of the flow in this regime is shown in figure 11(a). The acronym LS in the figure refers to the separation of the laminar boundary layer. An SV appears in the second regime. The flow, however, is devoid of the LSB. The flow exhibits this state for $5\times 10^3 \leq Re \leq 1\times 10^5$. The end of this regime marks the end of the subcritical regime. Figure 10(b) shows an example of the flow in this regime for $Re=2\times 10^4$. The SV is located downstream of the separation point of the laminar boundary layer. It is attached to the surface of the cylinder and embedded inside the wake separation bubble. The extent of SV can be identified by the region of positive $\overline {C}_f$ downstream of LS point. A schematic of the flow for this state is shown in figure 11(b). We refer to the separation and attachment points of the SV as SS and SA points, respectively. The direction of streamlines in the SV is opposite to those in the wake separation bubble. As a result, as shown in figure 11(b), SS is to the right of SA.

Figure 11. Flow past a circular cylinder with $L_z=1D$: schematic of time- and span-averaged streamlines to show various flows observed in present study. (a) Laminar separation of boundary layer without TA; neither SV nor LSB is observed. Panel (b) shows LS without TA; SV is observed downstream of LS. Panel (c) shows LS with TA; both SV and LSB are observed. Panel (d) shows the variation of time- and span-averaged LS, SA, SS, TA and TS points with $Re$. The regions of SV and LSB are shaded in purple and sky-black colours, respectively. The abbreviations are: C, Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017); A, Achenbach (Reference Achenbach1968); S & H, Son & Hanratty (Reference Son and Hanratty1969).

The formation of LSB marks the onset of third regime ($Re\geq 1.2\times 10^5$). Figure 10(c,d) shows the flow and $\overline {C}_f$ for a representative flow at $Re=4\times 10^5$. A schematic of the same is shown in figure 11(c). Both, SV and LSB are observed in the flow. The laminar boundary layer separates downstream of the shoulder. The separated shear layer undergoes transition to turbulent state and reattaches to the surface (Singh & Mittal Reference Singh and Mittal2005) at the TA point (see figure 11c). An LSB forms between LS and TA. Compared with regime 2, which is devoid of LSB, the peak negative value in the $\overline {C}_f$ is relatively large in regime 3. This variation of $\overline {C}_f$ can be used to check the existence of LSB in the flow. The turbulent boundary layer separates farther downstream at the TS point. As seen from figures 9(c,d) and 10(c), LSB is much larger than SV. In fact, SV is embedded inside the LSB. The size of SV and LSB have been exaggerated in the schematic, shown in figure 11(c), to bring out this feature.

Figure 11 shows the variation of LS, SS, SA, TA and TS points with $Re$. These points are identified from the variation of $\overline {C}_f$ on the surface the cylinder. We note from the figure that TA and TS are very close to each other for $1.2\times 10^5\leq Re\leq 2.5\times 10^5$. Therefore, time- and span-averaged streamlines, and not the surface distribution of $\overline {C}_f$, are utilized to identify TA and TS for this range of $Re$. Also shown in the figure are the results from earlier studies. The region on the surface of the cylinder occupied by SV and LSB are marked in purple and black colour, respectively. In the subcritical regime, the circumferential extent of the SV first increases with increase in $Re$ for $5\times 10^3\leq Re \leq 4\times 10^4$ and then decreases. The SV is much smaller in the critical and supercritical regimes, compared with that in subcritical regime. Its circumferential extent decreases slightly with increase in $Re$ in the critical and supercritical regimes. The circumferential extent of the LSB, decreases with increase in $Re$. This is consistent with observations made by Roshko (Reference Roshko1961) and Lehmkuhl et al. (Reference Lehmkuhl, Rodríguez, Borrell, Chiva and Oliva2014). The point of TS moves downstream with increase in $Re$. The location of TS from the present computations is in good agreement with those from Achenbach (Reference Achenbach1968).

Figure 12 shows the time- and span-averaged distribution of $\overline {C}_P$ for various $Re$ on the surface of cylinder with $L_z=1D$. The signature of LSB and SV in the $\overline {C}_P$ distribution is very interesting. Past studies have shown that LSB leads to a plateau in the $\overline {C}_P$ distribution (Achenbach Reference Achenbach1968; Lehmkuhl et al. Reference Lehmkuhl, Rodríguez, Borrell, Chiva and Oliva2014; Cheng et al. Reference Cheng, Pullin, Samtaney, Zhang and Gao2017; Chopra & Mittal Reference Chopra and Mittal2017). Figure 12 shows that the plateau is followed by a sharp dip in $\overline {C}_P$ and recovery, resembling a ‘kink’. Figure 10(c) shows a close-up of the same along with the structure of the flow. The ‘kink’ in the $\overline {C}_P$ distribution is between the points of SS and SA, and can therefore be attributed to the SV. Ono & Tamura (Reference Ono and Tamura2008) reported that the SV and LSB coexist in the supercritical regime. Although not pointed out by them, a ‘kink’ can be observed in their plot as well for the surface $\overline {C}_P$ distribution. Recently, Eljack et al. (Reference Eljack, Soria, Elawad and Ohtake2021) observed SV and LSB together in flow past a NACA 0012 airfoil at $Re=5\times 10^4$ and $9\times 10^4$. Although not pointed out by them, a kink can be seen in their plots as well for the surface $\overline {C}_P$ distribution. On the other hand, Pandi & Mittal (Reference Pandi and Mittal2019) reported an LSB, without the SV, on an Eppler 61 airfoil for $Re$ beyond $Re=2\times 10^4$. Consistent with our findings, the plateau in $\overline {C}_P$ distribution due to the LSB, is devoid of the kink. For $Re=1.2\times 10^5$, as shown in figure 12, the LSB is observed only on top half of the cylinder. The one sided transition has also been observed in earlier studies by Schewe (Reference Schewe1983), Bearman (Reference Bearman1969), Lehmkuhl et al. (Reference Lehmkuhl, Rodríguez, Borrell, Chiva and Oliva2014), Cadot et al. (Reference Cadot, Desai, Mittal, Saxena and Chandra2015) and Chopra & Mittal (Reference Chopra and Mittal2017). We further observe that the local variations of $\overline {C}_P$, within the LSB are stronger in the supercritical regime compared with the critical regime. Chopra & Mittal (Reference Chopra and Mittal2017) explained this via the intermittent nature of LSB during transition. A similar behaviour was reported by Deshpande et al. (Reference Deshpande, Kanti, Desai and Mittal2017) for the LSB on a sphere. In the subcritical regime, the SV is weak and as a result it does not cause significant variation in the surface $\overline {C}_P$ distribution. In the critical and supercritical regimes, the kink associated with the SV in the surface $\overline {C}_P$ distribution becomes sharper with an increase in $\mbox {{Re}}$ pointing to an increase in the strength of SV despite a decrease in its circumferential extent. Although not reported here, the extent and location of SV and LSB have been compared for $L_z=1D$ and $3D$ for the flow at $Re=3.0\times 10^5$. They are found to be in good agreement.

Figure 12. Flow past a circular cylinder with $L_z=1D$: (a) surface distribution of time- and span-averaged coefficient of pressure ($\overline {C}_P$) for various $Re$ and (b) variation of time- and span-averaged coefficient of peak suction pressure ($-\overline {C}_{Ppeak}$) and its location ($\theta _{C_{Ppeak}}$) with $Re$. The abbreviations are: P, present; T, Tani (Reference Tani1964); C, Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017); S & B, Szepessy & Bearman (Reference Szepessy and Bearman1992).

The variation of peak suction coefficient ($-\overline {C}_{Ppeak}$) and its location on the surface of the cylinder ($\theta _{C_{Ppeak}}$) with $Re$ are shown in figure 12(b). The average of the values from the lower and upper surface of the cylinder is plotted. The peak surface suction increases with $\mbox {{Re}}$ up to $\mbox {{Re}} \sim 4 \times 10^4$ and then decreases with further increase in $\mbox {{Re}}$ in the subcritical regime. Its location is close to $70^{\circ }$ in this regime which is well upstream of the point of separation (see figure 11). The variation of $-\overline {C}_{Ppeak}$ with $\mbox {{Re}}$ correlates well with the variation of points of LS and SA as shown in figure 11. Furthermore, $-\overline {C}_{Ppeak}$ increases and $\theta _{C_{Ppeak}}$ moves downstream with an increase in $Re$ in the critical and supercritical regimes. The sharp increase in peak suction in the critical regime correlates well with the formation of LSB and the associated delay in the flow separation (see figure 11). Also shown in figure 12(b) is the data from earlier studies (Tani Reference Tani1964; Szepessy & Bearman Reference Szepessy and Bearman1992; Cheng et al. Reference Cheng, Pullin, Samtaney, Zhang and Gao2017). The data from the various studies are in very good agreement in the subcritical regime. We note that the study by Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017) predicts a much larger peak suction, and its location is closer to the shoulder of the cylinder, in the supercritical regime compared with the other studies. The results from the present study are in good agreement with those from Tani (Reference Tani1964). The diagnostics on the velocity field that demonstrate the transition of the flow are presented in Appendix D.

3.5. The symmetric and antisymmetric modes

Figure 13 shows the space–time plot of $C_P(\theta, t)$ at midspan for three $Re$. Vortex shedding appears as the periodic activity with alternating values of low and high pressure. Amongst the three $Re$, for which the pictures are shown, it is most prominent at $Re=6 \times 10^4$. We note from figure 7 that $C_{Lrms}$ is maximum at this $\mbox {{Re}}$, indicating the strongest vortex shedding in the entire range of $\mbox {{Re}}$ investigated in this study. Desai et al. (Reference Desai, Mittal and Mittal2020), in their laboratory experiments, observed that the flow in the high subcritical regime is associated with ‘irregular shedding’, i.e. there are random occurrences of weakened vortex shedding. The same was reported earlier by Perrin et al. (Reference Perrin, Braza, Cid, Cazin, Chassaing, Mockett, Reimann and Thiele2008) and Cao & Tamura (Reference Cao and Tamura2015). Desai et al. (Reference Desai, Mittal and Mittal2020) observed that during such instances the lift coefficient varies with smaller amplitude, and base pressure is relatively higher. It was shown that the low frequency modulation in the time histories of $C_L$ is related to the expansion–contraction of the vortex formation region. They identified the various modes in the flow via POD of the surface pressure data. However, their analysis is restricted to the high subcritical and critical regimes and the experiments were conducted for a cylinder of relatively large span ($L_z/D\sim 22$). We perform a similar analysis for a cylinder with low $AR$ ($L_z/D=1$) and extend it flow in the supercritical regime.

Figure 13. Flow past a circular cylinder: space–time variation of the coefficient of pressure ($C_P(\theta,t)$) on the surface of the cylinder (left) and time variation of $C_P(\theta =90^{\circ },t)$ (right) at midspan for $Re=$ (a) $0.6\times 10^5$, (b) $1.5\times 10^5$ and (c) $3.5\times 10^5$. Additionally, $\tilde {C}_P(\theta =90^{\circ },t)$ is shown in the line plots on the right.

The right-hand panels of figure 13 show the time histories of the $C_P$ at midspan and $\theta =90^{\circ }$ for three $Re$. They are associated with three distinct time scales. The high frequency, and somewhat random, variation ($C_P'$) is primarily due to the shear layer vortices while the relatively time periodic variation ($C_{Pv}$) is caused by the von Kármán vortex shedding. The low frequency variations ($\hat {C}_P$) that result in modulations in the amplitude of variations due to vortex shedding have been attributed to expansion–contraction of the vortex formation region (Desai et al. Reference Desai, Mittal and Mittal2020). It is possible to estimate these components related to disparate time scales by using appropriate filtering. Chopra & Mittal (Reference Chopra and Mittal2017) proposed a double decomposition, $C_P(\theta,t)=\tilde {C}_P(\theta,t)+C_P'(\theta,t)$. They estimated $\tilde {C}_P(\theta,t)$ via a simple procedure akin to low pass filtering. The procedure involves a moving average of $C_P(\theta,t))$ over a few shear layer cycles, $\tilde {C}_P(\theta,t)=({1}/{T})\int _{t-T/2}^{t+T/2} C_P(\theta,t)\,\mathrm {d}t$. They utilized a window size of $T=T_k/10$ for the averaging, where $T_k$ is the time period of the von Kármán vortex shedding. We utilize the same procedure in the present work. In addition to $C_P(\theta =90^{\circ },t)$ at midspan, figure 13 also shows $\tilde {C}_P(\theta =90^{\circ },t)$. The difference of the two may be utilized to estimate $C_P'(\theta =90^{\circ },t)$.

Proper orthogonal decomposition of the coefficient of pressure at the surface of the cylinder is carried out at various $\mbox {{Re}}$, to identify the coherent flow structures. We are primarily interested in flow features related to vortex shedding and expansion–contraction of the recirculation bubble (Desai et al. Reference Desai, Mittal and Mittal2020). Therefore, we conduct POD on the span-averaged $\tilde {C}_P(\theta,t)$. Now 15 000–25 000 snapshots during approximately $60$ non-dimensional time units, for the fully developed unsteady flow, are used to carry out the analysis at each $\mbox {{Re}}$. To reduce the computational effort, not all grid points on the surface of the cylinder are used; rather every fourth point is chosen. Therefore, the spanwise-averaged data corresponding to only $200$, of the $800$, points of $\tilde {C}_P(\theta,t)$ on the surface of the cylinder is used for the POD analysis.

Figure 14 shows the results from the POD analysis of the surface $\tilde {C}_P(\theta,t)$ for $\mbox {{Re}}=0.6\times 10^5$. The percentage energy content of the first 10 modes and the eigenmodes corresponding to the first four modes are shown in the figure. We note that the first two modes account for nearly $94\,\%$ of the energy. The most dominant modes is antisymmetric with respect to $\theta =0^{\circ }$ and contains $87\,\%$ of the energy; we refer to this mode as $AS_1$, where $AS$ points to its antisymmetric nature. It corresponds to the von Kármán vortex shedding. The second mode is symmetric about the wake axis and contains approximately $7\,\%$ of the energy. We refer to it as mode-$S_1$. Desai et al. (Reference Desai, Mittal and Mittal2020) showed that while the non-dimensional time frequency associated with mode-$AS_1$ is $0.2$ approximately and corresponds to the Strouhal number of von Kármán vortex shedding, mode-$S_1$ has a much lower frequency. They further showed that mode-$S_1$ is associated with expansion–contraction of vortex formation region. With its low frequency variation, it is responsible for amplitude modulations in time history of pressure, and force coefficients. Figure 14 shows that modes 3 and 4 have relatively low energy. Owing to their symmetry property about the wake centreline, we refer to them as $S_2$ and $AS_2$, respectively.

Figure 14. The POD of moving and span-averaged surface pressure ($\tilde {C}_P(\theta,t)$) for $\mbox {{Re}}=0.6\times 10^5$: (a) the energy content of the first 10 modes; (b,c) the top two antisymmetric and symmetric modes, respectively.

The POD analysis was carried out at other $\mbox {{Re}}$. Mode-$AS_1$ is found to be the most dominant mode at all $\mbox {{Re}}$ in the range studied. Figure 15(a) shows the percentage energy carried by the leading four modes at various $\mbox {{Re}}$. As also found by Desai et al. (Reference Desai, Mittal and Mittal2020), energy of mode-$AS_1$ decreases, while that of mode-$S_1$ increases with an increase in $\mbox {{Re}}$ in the high subcritical and critical regimes. The decrease in energy of mode-$AS_1$ points to weakening of vortex shedding in these regimes, with increase in $\mbox {{Re}}$. On the other hand, it again picks up in the supercritical regime, as shown by the increase in percentage energy of the mode-$AS_1$.

Figure 15. POD of low-pass filtered span-averaged surface pressure ($\tilde {C}_P(\theta,t)$): (a) variation of percentage energy content of $AS_1$, $S_1$, $S_2$ and $AS_2$ modes with $\mbox {{Re}}$. Surface pressure distribution of eigenmodes corresponding to mode $AS_1$ (b) and $S_1$ (d) for various $\mbox {{Re}}$. Panels (c,e) show the close-up views of panels (b,d), respectively, to bring out the variations in $AS_1$ and $S_1$ modes due to SV and LSB at $Re=1.5\times 10^5$ and $3.5\times 10^5$. Here, LS, TA and TS correspond to the points of laminar separation, TA and turbulent separation, respectively. The variation within the SV is shown with a thicker line in panels (c,e).

Figures 15(b) to 15(e) show the evolution of the POD modes $AS_1$ and $S_1$ with an increase in $Re$. The points of LS, TA and TS are marked in figure 15(c,e) for $Re=1.5\times 10^5$ and $3.5\times 10^5$. We recall that the LSB forms between LS and TA. Also marked in the figures is the region of SV via thick solid lines. We note that both $AS_1$ and $S_1$ have imprint of the LSB and SV. The effect is very similar as observed in the time- and span-averaged $C_P$ distribution (see figures 10c and 12). The local peaks in the modes increase with an increase in $Re$. Except for $Re=1.2 \times 10^5$, the LSB forms symmetrically downstream of the two shoulders. It is quite interesting that its imprint is seen not just in the mode $S_1$, but also mode $AS_1$ that is responsible for the vortex shedding.

3.6. Variation of vortex shedding with $Re$

We further investigate the evolution of vortex shedding with increase in $Re$. Alternate shedding of vortices from the cylinder causes it to experience time varying lift force. The r.m.s. of $C_L$ for various $Re$ is shown in figure 7. A very significant decrease in $C_{Lrms}$, with increase in $Re$, occurs in the high subcritical and critical flow regime. The variation of $C_{Lrms}$ in the supercritical regime is very interesting. It decreases very gradually with an increase in $\mbox {{Re}}$ up to $\mbox {{Re}}=3.0\times 10^5$ followed by a gradual increase with a further increase in $\mbox {{Re}}$. The unsteadiness in the pressure near the two shoulders of the cylinder is the main contributor to $C_{Lrms}$. Therefore, we also study the variation of r.m.s. of $C_P$ at the shoulder of the cylinder for various $Re$. Here, $C_{Prms}$ are estimated at $\theta =\pm 90^{\circ }$ and their average is plotted in figure 16(a). Here, values for $C_{Lrms}$, shown in figure 7, are replotted in the figure for comparing the variations with $\mbox {{Re}}$. As expected, the variation of $C_{Lrms}$ and $C_{Prms}$ with $Re$ are very well correlated. This information can be useful in experimental investigations wherein $C_{Prms}$ at the shoulder can be utilized to study vortex shedding.

Figure 16. Flow past a circular cylinder with $L_z=1D$: (a) variation, with $Re$, of r.m.s. of $C_P$ ($=C_{Prms}$) averaged on the two shoulders ($\theta =\pm 90^{\circ }$) of the cylinder and r.m.s. of $C_L$ ($C_{Lrms}$). Shown in panel (b) is the variation of the vortex formation length ($L_f$) with $Re$. The abbreviations are: P, present; S & B, Szepessy & Bearman (Reference Szepessy and Bearman1992); B, Bloor (Reference Bloor1964); D, Desai et al. (Reference Desai, Mittal and Mittal2020).

The vortex formation length ($L_f$) from the present computations, for various $Re$, is shown in figure 16(b) along with that from Szepessy & Bearman (Reference Szepessy and Bearman1992), Bloor (Reference Bloor1964) and Desai et al. (Reference Desai, Mittal and Mittal2020). It is the streamwise location where $\overline {u'u'}$ is maximum along the wake centreline. The $L_f$ from present computations is in good agreement with that reported by Szepessy & Bearman (Reference Szepessy and Bearman1992). Figure 16 shows that the variation of $C_{Lrms}$ with $Re$, in the subcritical and critical regimes, is inversely proportional to $L_f$. In the subcritical regime, $L_f$ decreases with an increase in $Re$ up to $Re=0.4\times 10^5$. The trend of $L_f$ from the present computations is in good agreement with that from Bloor (Reference Bloor1964). The experiments of Desai et al. (Reference Desai, Mittal and Mittal2020) were conducted with a model of large $AR$ ($=22$). Therefore, the critical $Re$ for the onset of drag crisis is much larger in their experiments (${\approx }3.9\times 10^5$). All the data points shown in figure 16(b) from their experiments, are for the high subcritical regime. We observe that the variation of $L_f$ from their experiments follows the same trend as that from the present computations in the critical and high subcritical regimes. These results show that the reduction in $C_{Lrms}$ in the high subcritical and critical regimes is primarily due to two phenomena: (i) decrease in the strength of vortices and (ii) increase in vortex formation length. Figure 16(b) further shows that $L_f$ decreases with increase in $\mbox {{Re}}$ in the supercritical regime. The variation of Strouhal number ($St$) with $\mbox {{Re}}$, along with results from earlier studies, is shown in figure 17. It is estimated from the dominant frequency in the time variation of the lift coefficient. The $St$ estimated from the time histories of the cross-flow component of velocity in the near wake leads to very similar values; $St$ is close to $0.2$ in the subcritical regime. It increases sharply with an increase in $Re$ in the critical and early supercritical regime. For the present computations, it is $0.43$ at $Re=4\times 10^5$, which is almost twice the value that is observed in subcritical regime. As seen from figure 17, it can be as large as $0.46-0.48$ for higher $\mbox {{Re}}$ in supercritical regime (Bearman Reference Bearman1969; Schewe Reference Schewe1983).

Figure 17. Flow past a circular cylinder: variation of non-dimensional vortex shedding frequency, Strouhal number ($St$), with Reynolds number. The abbreviations are: S & B, Szepessy & Bearman (Reference Szepessy and Bearman1992); R, Rodríguez et al. (Reference Rodríguez, Lehmkuhl, Chiva, Borrell and Oliva2015); S, Schewe (Reference Schewe1983); B, Bearman (Reference Bearman1969).

3.6.1. Spanwise correlation of vortex shedding

Figure 15(a) shows that the vortex shedding persists through the entire $\mbox {{Re}}$ range, including the critical regime. How, then, does its coherence across the span vary with $\mbox {{Re}}$? To investigate the same we utilize the Pearson's correlation coefficient that is a measure of the linear correlation between two variables $\phi (t)$ and $\psi (t)$. It is defined as $R_{\phi \psi }={\overline {(\phi -\bar {\phi })(\psi -\bar {\psi })}}/{\sqrt {\overline {(\phi -\bar {\phi })^2}}\sqrt {\overline {(\psi -\bar {\psi })^2}}}$, where an overbar represents the time-average of the corresponding quantity. Following the work of Szepessy (Reference Szepessy1994) we utilize $C_P(\theta =90^{\circ }, t)$ on the surface of the cylinder to estimate $R_{pp}$ between a point fixed at midspan and another at a different span location. Further, we attempt to filter out the variation in $C_P$ due to vortex shedding and use it to estimate $R_{pp}$.

Time histories of $C_P$ at midspan and $\theta =90^{\circ }$ for three $Re$ are shown in the right-hand panels of figure 13. We propose a triple decomposition of the time series to enable us to estimate the time variations due to vortex shedding, $C_P(\theta,t)=\hat {C}_P(\theta,t)+C_{Pv}(\theta,t)+C'_P(\theta,t)$. Here, $C_P'$ is the contribution due to activity of vortices generated via instability of the shear layer; $C_{Pv}$ is the contribution from vortex shedding/mode-$AS_1$ and $\widehat {C_P}$ is the low frequency modulation arising from the expansion–contraction of the vortex formation region (Desai et al. Reference Desai, Mittal and Mittal2020) due to mode-$S_1$. Let $\tilde {C}_P(\theta,t)=\hat {C}_P(\theta,t)+C_{Pv}(\theta,t)$ be the moving average over a few shear layer cycles as described in the previous subsection. Therefore, $C_P(\theta,t)$ may also be expressed as $C_P(\theta,t)=\tilde {C}_P(\theta,t)+C'_P(\theta,t)$. In a similar manner, $\hat {C}_P(\theta,t)$ can also be estimated via a moving average of $C_P(\theta,t)$, but over a window of larger time period that spans a few vortex shedding cycles, $\hat {C}_P(\theta,t)=({1}/{T}) \int _{t-T/2}^{t+T/2} C_P(\theta,t)\,\mathrm {d}t$. We choose $T=2T_k$, where $T_k$ is the time period of vortex shedding. The contribution to ${C}_P(\theta,t)$ from the vortex shedding is then estimated via $C_{Pv}(\theta,t)=\tilde {C}_P(\theta,t)-\hat {C}_P(\theta,t)$.

Figure 18 shows the span-time variation of $C_P(t,z)$, $\tilde {C}_P(t,z)$ and $C_{Pv}(t,z)$ at $\theta =90^{\circ }$ for three $\mbox {{Re}}$ in the subcritical ($=0.6\times 10^5$), critical ($=1.5\times 10^5$) and supercritical (=$3.5\times 10^5$) regimes. Vortex shedding can be identified in all the panels by time variations that alternate between low and high values. Now $\tilde {C}_P$ filters out the shear layer activity from $C_P$. As expected, the difference between the two fields is significant in the critical and supercritical regimes. Additionally, $C_{Pv}$ shows the estimate of the time variations in $C_P$ due to vortex shedding. The range of the colourmap has been suitably adjusted for each $\mbox {{Re}}$. Amongst the three $\mbox {{Re}}$ for which data is presented in the figure, the shedding is strongest at $\mbox {{Re}}=0.6\times 10^5$ and weakest at $3.5\times 10^5$. Figure 19(a) shows the spanwise variation of the two-point correlation coefficient $R_{pp}(C_P)$ between the $C_P$ at midspan and another spanwise location at various $\mbox {{Re}}$. The variations corresponding to $\mbox {{Re}}$ in the subcritical and critical regimes are shown in solid lines, while those in the supercritical regime are shown in broken lines. Consistent with the symmetry of the problem set-up about midspan, $R_{pp}$ exhibits symmetry with respect to $z=0$. The correlation decreases as one moves away from midspan. The decrease is quite rapid in the critical and supercritical regime. It becomes as low as $0.24$ in the supercritical regime at the far end of the cylinder. Furthermore, $R_{pp}(C_P)$ decreases with an increase in $\mbox {{Re}}$, at any span location, as is also evident from the panels in the first row of span-time variation of $C_P(t,z)$.

Figure 18. Flow past a circular cylinder: space–time plot of coefficient of pressure ($C_{P}(\theta =90^{\circ },t,z)$) (a,d,g), moving averaged coefficient of pressure ($\tilde {C}_{P}(\theta =90^{\circ },t,z)$) (b,e,h) and fluctuations associated with vortex shedding ($C_{Pv}(\theta =90^{\circ },t,z)$) (c,f,i) for (a–c) $Re=0.6\times 10^5$, (d–f) $Re=1.5\times 10^5$ and (g–i) $Re=3.5\times 10^5$.

Figure 19. Flow past a circular cylinder: variation of spanwise correlation coefficient $R_{pp}$ with $z/D$ for different $\mbox {{Re}}$ at $\theta =90^{\circ }$ based on (a) $C_{P}$, (b) $\tilde {C}_{P}$ and (c) ${C}_{Pv}$.

Next, the correlation coefficient is computed by utilizing the low-pass filtered time series, $\tilde {C}_P$. Figure 19(b) shows $R_{pp}(\tilde {C}_P)$ along the span, for various $\mbox {{Re}}$. The vortices that form due to the rolling up of the separated shear layer are relatively farther in the wake in the subcritical regime. Therefore, they do not have a significant effect on the surface pressure at the shoulder of the cylinder. Consequently, $R_{pp}(\tilde {C}_P)$ is very similar to $R_{pp}(C_P)$ in this regime. However, the differences are significant in the critical regime and beyond. In general, for a given $\mbox {{Re}}$ and at any span location, $R_{pp}(\tilde {C}_P)$ is higher than $R_{pp}(C_P)$. Unlike $R_{pp}(C_P)$, which decreases with an increase in $\mbox {{Re}}$, $R_{pp}(\tilde {C}_P)$ shows little variation with $\mbox {{Re}}$ beyond $\mbox {{Re}}=1.5\times 10^5$. We note that the results presented by Szepessy (Reference Szepessy1994) are based on similarly filtered time series of surface static pressure.

The correlation coefficient computed using $C_{Pv}(t,z)$ is referred to as $R_{pp}(C_{Pv})$. Its spanwise variation for various $\mbox {{Re}}$ is shown in figure 19(c). The correlation coefficient for the subcritical $\mbox {{Re}}$ ($=0.6\times 10^5$) is virtually identical for the three cases that utilize either $C_P$ or $\tilde {C}_P$ or $C_{Pv}$; the vortex shedding across the span is highly correlated. Now $R_{pp}(C_{Pv})$ decreases with an increase in $\mbox {{Re}}$ in the critical regime. It increases with a further increase in $\mbox {{Re}}$ beyond $3.0\times 10^5$. This indicates that the spanwise coherence of vortex shedding decreases with an increase in $\mbox {{Re}}$ in the critical regime and then increases in the supercritical regime. This is consistent with the observation from the panels showing the span-time variation of $C_{Pv}$ at the shoulder of the cylinder in figure 18. The increase in spanwise coherence in the supercritical regime is not captured by the correlation coefficient based on either $C_P$ or $\tilde {C}_P$.