2.1. Strouhal number measurements

Hot-wire measurements were conducted in a closed-circuit wind tunnel with a square test section (0.6 m × 0.6 m) of 2.4 m length. The wind speed in the test section is from 1.5 to 50 m s⁻¹. The free-stream velocity was measured using a Pitot-static tube connected to a Furness micro-manometer (FCO 510, Furness Controls Ltd. Bexhill, UK). The experimental uncertainty in the measurement of U _∞ is estimated to be ±1.0 % and the longitudinal turbulent intensity T_u is 0.13 % in the absence of any cylinders. Two identical aluminium square cylinders of d = 19.1 mm, placed in tandem with the upstream cylinder mounted at 0.2 m downstream of the exit plane of the contraction, were installed perpendicularly in the midplane and spanned the full height of the test section, resulting in a maximum blockage of 3.18 % and an aspect ratio of 31.6. The blockage ratio was small enough and no correction was made for measurements (West & Apelt Reference West and Apelt1982), while the aspect ratio was considered to be adequately large to neglect the end effect of the cylinders.

Two Cartesian coordinate systems, (x-O-y) and (x′-O′-y′), are defined such that the origins O and O′ are at the centres of the upstream and downstream cylinders, respectively, with the x- or x′-axis and the y- or y′-axis along the streamwise and lateral directions, respectively. Two hot-wire probes (55P01, Dantec), HT1 and HT2, placed at (x*, y*) = (1.5, 1.5) and (L/d + 1.5, −1.5), respectively, were used to measure the streamwise fluctuating velocities (u ₁ and u ₂), which provided the information on the vortex shedding from the upstream and downstream cylinders, respectively (please refer to figure 2). The superscript asterisk in this paper denotes normalization by d and/or U _∞. The sensing element of the hot-wire was made of 0.5 μm tungsten wire, approximately 1.25 mm in length, and was connected to constant temperature anemometers (Dantec Streamline) operated at an overheat ratio of 1.8. The signals from the wires were low-pass filtered at a cutoff frequency of 1 kHz and then digitized at a sampling frequency f_sampling of 2 kHz on a personal computer using a 16-channel A/D board (NI PCI-6143). The sampling duration was 60 s for each combination of L/d and Re. The hot-wire was not calibrated since the measured u₁ and u₂ are used only to determine St. The St is determined from the power spectral density function E_u of the u signal, and E_u is calculated from a fast Fourier transform (FFT) algorithm. The FFT window size N_w is 4096. The frequency resolution Δƒ (= f_sampling/N_w) in the spectral analysis (Zhou et al. Reference Zhou, Du, Mi and Wang2012) is 0.49 Hz. The St behind an isolated cylinder was also measured to set a benchmark for comparison. Measurements were made for L/d = 1.0 ~ 5.0 and Re = 2.8 × 10³ ~ 2.8 × 10⁴.

Figure 2. Experimental arrangement and definitions of symbols. The parameter L is the spacing between the cylinder centres, and d is the side width of a square cylinder. The hot-wire probes are placed at x* (= x/d) = 1.5, L/d + 1.5 and y* (= y/d) = ±1.5.

2.2. Flow structure measurement

A Dantec high-frequency PIV system (Litron LDY 304-PIV, Nd:YLF) was used to measure the flow in the (x, y) plane. The flow was seeded with particles approximately 1 μm in diameter, generated from paraffin oil using a TSI 9307-6 particle generator. Flow illumination was provided by two pulsed laser sources of 527 nm wavelength, each with a maximum power of 30 mJ per pulse. The interval between two successive pulses was 75 μs. One charge-coupled device camera (Phantom V641, double frames, with a resolution of 2560 × 1600 pixels) was used to capture particle images. An interrogation window of 32 × 32 pixels was applied for image processing with a 50 % overlap in both the x and y directions. A total of 1000 images were captured for each flow configuration, i.e. one combination of L/d and Re, which is considered to be adequate for the root-mean-square values $u_{rms}^\ast $ of streamwise velocities to be converged. As demonstrated by Alam et al. (Reference Alam, Zhou and Wang2011), the maximum $u_{rms}^\ast $ in the wake of an isolated single cylinder converged to its asymptotic value, with a maximum departure of 2.8 %, once the number of images reached 500.

The same PIV system was also used for LIF flow visualization. The four surfaces of each cylinder were painted black to minimize reflection noise. Note that the laser source was placed above the cylinders, which were not transparent, thus partially blocking illumination below the cylinders. This should not adversely affect the capturing of the flow structure given symmetric or anti-symmetric flow about the x-axis (Zhou & Yiu Reference Zhou and Yiu2006). Seven pinholes in a row, each 1 mm in diameter, with a 1.5 mm centre-to-centre distance between two adjacent pinholes, were drilled on the midspan front surface of the upstream cylinder. The first and seventh holes were 1.5 mm away from the lower and upper edges of the cylinder, respectively. Since the cylinder wall was 1 mm in thickness, the distance between the hole centre and the cylinder edge should not be less than 1.5 mm. To measure pressure as close to the cylinder edge as possible, the holes were made 1.5 mm away from the lower or upper edge. Smoke was released from the pinholes to seed the flow. A total of 500 images was recorded for each combination of Re and L/d, although representative results only at Re = 1.3 × 10⁴ and L/d = 1.2, 1.4, 2.0, 2.6, 2.8 and 4.0 are presented in this paper.

2.3. Surface pressure, time-averaged and fluctuating drag and lift measurements

Surface pressure distributions around the two square cylinders were measured at L/d = 1.1 ~ 5.0 with an increment of 0.1 at a representative Re = 1.3 × 10⁴. Each cylinder was furnished with one row of 24 pressure taps (6 taps on each surface) near the midspan. The centre-to-centre distance between two adjacent taps was 3.8 mm. Each tap was offset by 1.5 mm spanwise from its preceding one to minimize possible interference between them. The taps were connected to a pressure scanner (ESP-32HD, 32 ports, with a measurement range of ±1 KPa and a resolution of ±1.5 Pa) through the tubes of 0.8 mm inner diameter. The scanner was placed outside the test section of the wind tunnel, very close to the cylinder ends in order to improve the signal response. The pressure signals were sampled at a rate of 650 Hz to a data acquisition unit (DTC Initium) before being sent via an ethernet cable and stored in a computer. The data record duration was 60 s. The surface pressure around an isolated cylinder was also measured to set a benchmark and validate our results. Integrating the simultaneously captured pressures at 24 pressure taps around a cylinder leads to the time-averaged drag coefficient ${\bar{C}_D}$ and fluctuating drag and lift coefficients ${C^{\prime}_D}$ and ${C^{\prime}_L}$. In this paper, overbar and superscript prime denote time-averaged and root-mean-square values, respectively.

2.4. Surface-oil-flow visualization

A surface-oil-flow visualization experiment is employed to capture the mean shear stress pattern, including various singular points, e.g. stagnation, reattachment, separation, etc. (Younis, Alam & Zhou Reference Younis, Alam and Zhou2016; Alam et al. Reference Alam, Rastan, Wang and Zhou2022). The surfaces of the two cylinders were firstly sprayed with black paint as a background, and then a solution of silicone oil (100 cs), titanium dioxide, oleic acid and kerosene with a ratio of 4:1:2:2 in weight was painted onto the surfaces. The cylinders were thereupon installed in the test section of the wind tunnel, and the solution was distributed on the cylinder surface after at least 20 min of exposure to uniform incident flow. The photographs of the solution distributions on the cylinder surfaces were captured by a digital camera with a resolution of 3024 × 4032 pixels. Surface-oil-flow visualizations were performed at the same Reynolds number as the measurements of fluid forces, Strouhal numbers and pressures. The experimental uncertainty in the positions of flow separation or reattachment obtained from surface-oil-film visualization was estimated to be ±0.01d based on the camera resolution.

3.1. Extended-body regime

At a small gap, two square cylinders act like one single rectangular cylinder or extended body (Sakamoto et al. Reference Sakamoto, Hainu and Obata1987; Sakamoto & Haniu Reference Sakamoto and Haniu1988; Alam et al. Reference Alam, Rastan, Wang and Zhou2022), and an increase in L/d implies a longer after body, which acts to reduce the wake width w* and shear layer curvature, causing an increased St (Otsuki et al. Reference Otsuki, Washizu, Tomizawa and Ohya1974; Liu & Chen Reference Liu and Chen2002). From a different perspective, the increased L/d is equivalent to an increased ratio, from 2 to 3, of the longitudinal to lateral width of the ‘rectangular’ cylinder, again leading to a larger St (Okajima Reference Okajima1982). As such, St rises rather rapidly, e.g. from 0.086 at L/d = 1.0 to its maximum 0.142 at L/d = 1.5 for Re = 1.3 × 10⁴ (figure 3). The flow characteristics around two tandem cylinders indeed exhibit many similarities to those around a single square cylinder. Bai & Alam (Reference Bai and Alam2018) noted that, for a single square cylinder, the position of transition to turbulence significantly shifts toward the flow separation point when Re is increased from 10³ to 10⁴, but insignificantly for Re = 10⁴–10⁵. The shift led to a contracted vortex formation length. Presently, the extended-body regime shrinks from Re = 4500 to 13 000 (figure 3), which is linked to the gradual shift in the transition to turbulence toward the flow separation point. The value of E_u displays one prominent peak at St = 0.086 for L/d = 1.0 (figure 4) and St = 0.118 for L/d = 1.2 (figure 5a). Additional peaks are discernible at the second and third harmonics of St. Figure 5(b) illustrates the flow structure captured from flow visualization in this regime. The shear layers separating from the leading edge of the upstream cylinder roll up behind the downstream cylinder. There is no vortex shedding in the gap between the two cylinders.

Figure 4. Power spectral density function E_u of streamwise velocity u measured by the moveable hot-wires HT1 and HT2. Here, L/d = 1.0, Re = 1.3 × 10⁴.

Figure 5. (a) Power spectral density function E_u of streamwise velocity u measured by the moveable hot-wires HT1 and HT2; (b) image of typical flow structure from smoke visualization. Here, L/d = 1.2, Re = 1.3 × 10⁴.

3.2. Reattachment regime

In this regime, the shear layers separating from the upstream cylinder may reattach to the side surfaces of the downstream one and then separate again from the trailing edges of the cylinder to roll up periodically (Sakamoto & Haniu Reference Sakamoto and Haniu1988). The value of E_u displays one pronounced peak (figure 6a), indicating the occurrence of a single vortex-shedding frequency, and St drops initially rather quickly with increasing L/d (figure 3).

Figure 6. (a) Power spectral density function E_u of streamwise velocity u measured from the moveable hot-wires HT1 and HT2; (b) image of typical flow structure from smoke visualization. Here, L/d = 2.0, Re = 1.3 × 10⁴.

At L/d = 1.5, where the extended-body regime changes to the reattachment regime, the shear layer may reattach to the trailing edge of the downstream cylinder. A comparison in the flow visualization photographs between L/d = 2.0 and 2.6 reveals that the reattachment position shifts toward the leading edge with increasing L/d (figures 6b and 7b). This position influences the shear layer velocity, wake width w*, formation length $L_f^\ast $, etc. (Wang, Alam & Zhou Reference Wang, Alam and Zhou2018), thus producing a great impact upon St. For example, the shear layer will be subjected to much less obstruction if reattaching near the trailing edge (figure 6b) than near the leading edge (figure 7b). The reverse flow in the gap between the cylinders is thus stronger at L/d = 2.6 than at L/d = 2.0 (figures 6b and 7b). The shear layer instability frequency f_sl is inversely proportional to the shear layer thickness in the case of a single circular cylinder (Bauer Reference Bauer1961; Gerrard Reference Gerrard1966; Monkewitz & Nguyen Reference Monkewitz and Nguyen1987; Williamson & Brown Reference Williamson and Brown1998). Based on the data in the literature and their own measurements, Prasad & Williamson (Reference Prasad and Williamson1997) proposed an empirical correlation between Re and f_sl for a single circular cylinder wake, namely, f_sl/f_s = 0.0235 ⋅ Re ^0.67. Given a fixed Re, a thick shear layer corresponds to a lower f_sl and hence a decreasing f_s. An increase in L/d may provide more room for the reattached shear layer to develop before rolling up, implying a thicker layer when separating and hence a lower St. Liu & Chen (Reference Liu and Chen2002) also reported that an increase in L/d in the reattachment regime caused St to decrease, suggesting a wider wake width.

Figure 7. (a) Power spectral density function E_u of streamwise velocity u measured from the moveable hot-wires HT1 and HT2; (b) image of typical flow structure from smoke visualization. Here, L/d = 2.6, Re = 1.3 × 10⁴.

Note a broadband bump of relatively high frequencies (f* > 0.4) in E_u measured from hot-wire 1 (figures 6a and 7a), which is probably due to the instability or Kelvin–Helmholtz vortices of the shear layers separating from the upstream cylinder (figures 6b and 7b). This bump is appreciably stronger for L/d = 2.6 than for L/d = 2.0, suggesting that the reattachment incurs stronger shear layer instability near the leading edge (figure 7b) than near the trailing edge (figure 6b).

3.3. Transition regime

The transition involves a change in the flow structure from reattachment to co-shedding or vice versa, and naturally two distinct St values. The higher St corresponds to the reattachment regime, and the lower to the co-shedding (figure 8b). The L/d range where the transition takes place depends on Re, 3.0 ~ 3.2 for Re = 4.5 × 10³, 2.8 ~ 2.9 for Re = 1.3 × 10⁴ and 2.7 ~ 2.9 for Re = 2.8 × 10⁴. In this regime (e.g. L/d = 2.8, Re = 1.3 × 10⁴), the intermittent occurrence of both flow states is captured in the time histories of u₁ and u₂ from HT1 and HT2, respectively (figure 8a). Both signals are characterized by small- and large-amplitude fluctuations, corresponding to the reattachment and co-shedding flow structures, respectively (figure 8a). The value of E_u, calculated from u₁ or u₂, shows two prominent peaks at St = 0.100 and 0.131 (figure 8b). The peak is less pronounced at St = 0.131 than at St = 0.100. The higher St results from vortex shedding when reattachment occurs and the lower from vortex shedding when co-shedding takes place. It seems plausible that the vortices associated with the co-shedding state are characterized by a larger strength than those formed in the reattachment state. Figure 8(c,d) shows the corresponding flow structures in the two states. In the reattachment case, the gap shear layer reattaches on the leading edge of the downstream cylinder and splits into two, partly moving into the gap between the cylinders and partly flowing over the side surface, and then rolls up to form vortices behind the downstream cylinder (figure 8c). In the co-shedding case, the shear layer appears to roll up entirely in the gap between the cylinders, forming the gap vortices (figure 8d). As seen in figure 8(c,d), the curvature of the gap shear layer appears larger in the co-shedding case than in the reattachment flow, resulting in a smaller St in the latter case (Nakaguchi, Hasimot & Muto Reference Nakaguchi, Hasimot and Muto1968; Alam et al. Reference Alam, Zhou and Wang2011). Furthermore, being obstructed by the downstream cylinder, the gap vortices are convected at a small velocity, also contributing to a smaller St (figure 8b). Sakamoto & Haniu (Reference Sakamoto and Haniu1988) observed at Re = 3.32 × 10⁴ the bistable flow involving reattachment and co-shedding at L/d = 4.0 for their smallest T_u (= 0.19 %) examined. Their transition took place at a considerably larger L/d than the present L/d. This difference is not surprising since their blockage ratio, inter alia, reached 9.8 %, considerably larger than the present 3.18 %.

Figure 8. (a) Time histories of streamwise velocity u from hot-wires HT1 and HT2; (b) power spectral density function E_u of streamwise velocity u and measured from the moveable hot-wires HT1 and HT2; (c,d) images of typical flow structures from smoke visualization. Here, L/d = 2.8, Re = 1.3 × 10⁴.

3.4. Co-shedding regime

In this regime, the quasi-periodical vortices are generated both between and behind the cylinders at the same frequency (figure 9). The value of St rises slowly with increasing L/d, which is linked to the gradually diminishing effect of the downstream cylinder on the upstream cylinder, and eventually approaches that in an isolated cylinder wake (St_o = 0.132). One may expect a different vortex-shedding frequency in the gap between the cylinders from that behind because of the very different initial conditions of incident flow. However, the former may trigger the latter, producing a ‘lock-in’ phenomenon (Alam & Sakamoto Reference Alam and Sakamoto2005).

Figure 9. (a) Power spectral density function E_u of streamwise velocity u measured from the moveable hot-wires HT1 and HT2; (b) image of typical flow structure from smoke visualization. Here, L/d = 4.0, Re = 1.3 × 10⁴.

4.1. Near-wake characteristic parameters

The near wake of a cylinder is characterized by the wake width w* and vortex formation length $L_f^\ast $, which can be determined from the distribution of $u_{rms}^\ast $. Figure 10 presents the iso-contours of PIV-measured $u_{rms}^\ast $ (Re = 1.3 × 10⁴) in the wake for the extended-body (L/d = 1.2, 1.4) and reattachment (L/d = 1.8, 2.2, 2.4 and 2.7) regimes. Two maxima occur rather symmetrically about y* = 0, which are frequently considered to be the mean position where the shear layer rollup takes place (Bloor Reference Bloor1964; Gerrard Reference Gerrard1966). The maximum $u_{rms}^\ast $ declines from 0.24 to 0.22 when L/d increases from 1.2 to 1.4 in the extended-body regime but remains almost constant, at approximately 0.2, in the reattachment regime from L/d = 1.8–2.4 before dropping further to 0.18 from L/d = 2.4 to 2.7. The observation suggests that the vortices formed become weaker in strength from the extended-body regime to the reattachment regime. Compared with the extended-body regime at L/d = 1.4, the location of the maximum $u_{rms}^\ast $ changes a little for L/d = 1.8–2.4 in the reattachment regime but shifts appreciably downstream at L/d = 2.7. This location is correlated with the base pressure of the downstream cylinder, as will be discussed later. As L/d is increased to 3.2 (co-shedding flow), the maximum $u_{rms}^\ast $ rises sharply to 0.3, and its position shifts toward the cylinder base, indicating stronger vortex shedding when the flow changes from the reattachment to co-shedding regime. An increase in L/d in the co-shedding regime, e.g. from 3.2 to 4.2, leads to a decrease in the maximum $u_{rms}^\ast $ to 0.24.

Figure 10. Iso-contours of the root mean square streamwise velocity $u_{rms}^\ast = {u_{rms}}/{U_\infty }$ for (a) L/d = 1.2, (b) L/d = 1.4, (c) L/d = 1.8, (d) L/d = 2.2, (e) L/d = 2.4, (f) L/d = 2.7, (g) L/d = 3.2 and (h) L/d = 4.2. Re = 1.3 × 10⁴. The contour increment is 0.02.

The wake width w* can be defined as the transverse distance between the two maximum $u_{rms}^\ast $ values where the two shear layers roll up to form vortices (Griffin & Ramberg Reference Griffin and Ramberg1974; Roshko Reference Roshko1993). By the same token, following Bloor (Reference Bloor1964) and Gerrard (Reference Gerrard1966), we may define the vortex formation length L_f by the streamwise distance from the downstream cylinder base to the $u_{rms}^\ast $ maximum. The variations in w* and $L_f^\ast $, along with St, with L/d are presented in figure 11. With increasing L/d, the rise in St in the extended regime is accompanied by a decrease in w*, while the decrease in St from L/d = 1.5 to 2.5 in the reattachment regime is associated with an increase in w*. A further increase in L/d from 2.5 to 2.7 in the reattachment regime results in a decline in w* but a rise in St. With the flow changing from reattachment (L/d = 2.7) to co-shedding (L/d = 3.2), the sharp drop in St is accompanied by a rise in w* from 1.13 to 1.36. The w* gradually decreases in the co-shedding regime, reaching 1.2 at L/d = 4.2. Overall, St is inversely linked with w*. It has been well established in a single cylinder wake that, for a fixed Re, the smaller the w*, the higher the St (Bearman & Trueman Reference Bearman and Trueman1972; Griffin & Ramberg Reference Griffin and Ramberg1974). It seems plausible that this conclusion is also valid for a two-cylinder system. The variation in $L_f^\ast $ with L/d is, however, different from that in w*. The $L_f^\ast $ contracts from 1.65 at L/d = 1.2 to 1.08 at L/d = 1.4 in the extended-body regime and remains almost unchanged in the reattachment regime, albeit displaying a rise from L/d = 2.4 to 2.7. The transition regime is characterized by a drastic drop in $L_f^\ast $ from 1.04 at L/d = 2.7 to 0.38 at L/d = 3.2. The value of $L_f^\ast $ remains more or less constant in the co-shedding regime. The observation suggests that St is more closely correlated with or influenced by w* than by $L_f^\ast $.

Figure 11. Dependence on cylinder spacing ratio L/d of (a) wake width w* and Strouhal number St and (b) vortex formation length $L_f^\ast $ from the centre of upstream cylinder at Re = 1.3 × 10⁴.

4.2. Velocity field in the wake

It is important to understand the shear layer development, vortex shedding and flow structure (both instantaneous and time averaged) behind the cylinders in the different flow regimes. As such, we present the time-averaged sectional streamlines (referred to as streamlines hereinafter for simplicity), superimposed with the iso-contours of time-averaged streamwise velocity ${\bar{U}^\ast }$ or spanwise vorticity $\bar{\omega }_z^\ast $ in figures 12(a), 13(a) and 14(a), along with the phase-averaged streamlines, superimposed with phase-averaged vorticity $\langle \omega _z^\ast \rangle$ in figures 12(b), 13(b) and 14(b). The phase-averaged streamlines are viewed on a reference frame moving at the vortex convection velocity $\langle U_c^\ast \rangle$, estimated as the streamwise displacement of the vortex between two successive PIV snapshots divided by the corresponding time interval. One thousand time-resolved PIV snapshots were captured over a duration of 1 s, out of which the time histories of u and v may be extracted. As in Chen et al. (Reference Chen, Zhou, Zhou and Antonia2016), a fourth-order Butterworth filter, centred at the vortex-shedding frequency, was used to filter the signal of thus obtained v at (x*, y*) = (L/d + 1.5, −1.5), producing a sinusoidal signal. This signal was used as the reference signal to determine the phase. In the phase-averaging process, approximately 20 data points are available for each phase.

Figure 12. (a) Time-averaged streamlines superimposed with the normalized iso-contours: (ai) time-mean streamwise velocity ${\bar{U}^\ast }$, (aii) time-mean spanwise vorticity $\bar{\omega }_z^\ast $, (aiii) flow structure sketch; (b) phase-averaged streamlines after subtracting convection velocity, superimposed with phase-averaged vorticity $\langle \omega _z^\ast \rangle$ contours; (c) flow structure sketch based on the phase-averaged data (L/d = 1.2, Re = 1.3 × 10⁴) obtained in the extended-body flow regime. $\bullet$, maximum magnitude of $\langle \omega _z^\ast \rangle$.

Figure 13. (a) Time-averaged flow streamlines superimposed with the normalized iso-contours: (ai) time-mean streamwise velocity ${\bar{U}^\ast }$, (aii) time-mean spanwise vorticity $\bar{\omega }_z^\ast $, (aiii, aiv) flow structure sketches; (b) phase-averaged streamlines after subtracting convection velocity, superimposed with phase-averaged vorticity $\langle \omega _z^\ast \rangle$ contours (L/d = 2.2); (c,d) flow structure sketches based on phase-averaged data (L/d = 1.6~2.4 and 2.5~2.7, Re = 1.3 × 10⁴) obtained in the reattachment flow regime. $\bullet$, maximum magnitude of $\langle \omega _z^\ast \rangle$.

Figure 14. (a) Time-averaged flow streamlines superimposed with the normalized iso-contours: (ai) time-mean streamwise velocity ${\bar{U}^\ast }$, (aii) time-mean spanwise vorticity $\bar{\omega }_z^\ast $, (aiii) flow structure sketch; (b) phase-averaged streamlines after subtracting the convection velocity, superimposed with phase-averaged vorticity $\langle \omega _z^\ast \rangle$ contours; (c) flow structure sketches based on phase-averaged data (L/d = 4.2, Re = 1.3 × 10⁴) obtained in the co-shedding flow regime. $\bullet$, maximum magnitude of $\langle \omega _z^\ast \rangle$.

Extended-body regime: the time-averaged streamlines and ${\bar{U}^\ast }$-contours (L/d = 1.2) show two recirculation bubbles behind the cylinders and on the side surfaces of the downstream cylinder, in addition to two shear layers (figure 12a). The wake behind the downstream cylinder resembles that of a single square cylinder. The negative values of ${\bar{U}^\ast }$ indicate the presence of flow recirculation, and the magnitude of the minimum value within the recirculation region serves as a measure of its strength. The wake for L/d = 1.2 exhibits more pronounced and stronger flow recirculation than that for the reattachment (L/d = 2.2, figure 13ai) and co-shedding regimes (L/d = 4.2, figure 14ai). Due to the relatively small gap width for L/d = 1.2, the flow recirculation in the gap is notably weaker compared with that for L/d = 2.2, where a relatively strong recirculation occurs. Figure 12(aii) reveals that the shear layers are characterized by high vorticities. The shear layers exhibit larger vorticity magnitudes for L/d = 1.2 than for L/d = 2.2 (figure 13aii) while L/d = 4.2 exhibits the smallest vorticity magnitudes (figure 14aii). Figure 12(b) exhibits the $\langle \omega _z^\ast \rangle$-contours superimposed with phase-averaged streamlines at phases φ = 0, 0.5π, π and 1.5π (Chen et al. Reference Chen, Zhou, Zhou and Antonia2016). The streamlines are obtained after subtracting the maximum $\langle U_c^\ast \rangle = 0.61$ at φ = 0 identified at the location of the maximum $\langle \omega _z^\ast \rangle$ (Zhou, Zhang & Yiu Reference Zhou, Zhang and Yiu2002), where the vortex centre is marked by a black circle. The flow separates from the leading edges of the upstream cylinder and forms shear layers over the lateral surfaces of both cylinders. At phase φ = 0, the lower shear layer rolls up into the wake without reattachment and then grows by entraining the surrounding fluid with a pronounced vortex centre at x* = 3.5 at φ = 0.5π (figure 12bii), while the upper shear layer grows and penetrates the wake from φ = 0.5π to 1.5π (figure 12bii–biv). The vortex from the lower shear layer pitches off during φ = π − 1.5π. Figure 12(c) displays corresponding simplified flow sketches.

Reattachment regime: the data (L/d = 2.2) indicate that the shear layer from the upstream cylinder reattaches onto the downstream cylinder, producing two strong recirculation regions over the leading corners of the downstream cylinder (figure 13ai). The reattached shear layer splits into two, one reversed upstream and the other flowing downstream and separating, forming vortices. The wake recirculation region contracts in size and appears weaker in strength (figure 13ai), as indicated by the maximum magnitude of the negative ${\bar{U}^\ast }$, than the extended-body regime (figure 12ai). The observed drop in St from L/d = 1.6 to 2.4 in this regime (figure 3) is due to a shift of the reattachment position toward the leading corners (figure 13aiii,aiv). As the shear layer hits the cylinder corner (figure 13div), its low-velocity or inner slice goes to the inter-cylinder gap and the high-velocity or outer slice flows over the downstream cylinder, separating and forming vortices (Alam, Sakamoto & Zhou Reference Alam, Sakamoto and Zhou2005). The associated high-velocity shear layer accounts for the small rise in St from L/d = 2.5 to 2.7 (figure 3). In addition, the shear layer hitting the corner leads to a decrease in w* from 1.17 at L/d = 2.5 to 1.13 at L/d = 2.7 (figure 11a), which conforms to the corresponding small rise in St. The flow structure (figure 13b) is similar to that in the extended-body regime (figure 12b). However, due to a larger cylinder gap and the interaction between the shear layer and the cylinder corner, more shear layer impinges directly upon the rear surface of the upstream cylinder. For φ = 0–1.5π, the lower gap recirculation bubble is enhanced substantially in strength (figure 13b). The value of $\langle U_c^\ast \rangle$ for the lower wake vortex increases from 0.21 to 0.49 in φ = 0–0.5π, then to 0.72 at φ = π and finally to 0.76 at φ = 1.5π. At φ = 0, the lower shear layer rolls up and reattaches to the lower side surface of the downstream cylinder, producing noticeable recirculation on the lower side of the gap. From φ = 0 to 0.5π (figure 13bi–bii), the vortex from the upper side pinches off, and $\langle U_c^\ast \rangle$ increases from 0.49 to 0.64 while the lower shear layer stretches and forms a vortex from φ = 0.5π to 1.5π (figure 13bii–biv), in a similar fashion to that shown in figure 12(bii–biv). The value of $\langle U_c^\ast \rangle$ of the lower vortex increases from 0.21 to 0.76 as φ increases from 0 to 1.5π. The sketches in figure 13(c,d) illustrate the flow structures in this regime for L/d = 1.6–2.4 and 2.5–2.7, where the shear layers reattach on the side surfaces and the leading corners for L/d = 2.5–2.7, respectively.

Co-shedding regime: both streamlines and ${\bar{U}^\ast }$-contours (figure 14ai) display two recirculation bubbles between and behind the cylinders. The bubbles appear shorter in length and weaker in recirculation than in the other two regimes. A shorter recirculation bubble enhances the drag, as will be shown later. The shear layer around the upstream cylinder is associated with a higher concentration of vorticity than that around the downstream cylinder; but the recirculation bubbles exhibit a smaller size and strength between the cylinders than behind (figure 14aii). This is reasonable because of a very limited space for the vortices to develop between the cylinders. The vortices alternately separated from the upstream cylinder impinge on the frontal surface of the downstream cylinder, producing turbulent buffeting in the developing shear layers (figure 14b). The value of $\langle U_c^\ast \rangle$ for the two upper vortices decreases from φ = 0 to π and increases for φ = π − 1.5π, suggesting that vortex shedding from the downstream cylinder is approximately in phase with that from the upstream. The gap vortex from the lower side grows from φ = 0 to 0.5π and approaches from φ = 0.5π to π and then impinges upon the downstream cylinder from φ = π to 1.5π, which triggers the vortex shedding from the lower side of the downstream cylinder. The impinging vortex is essentially part of the large vortex (figure 14biv), thus referred to as a binary vortex (Sobczyk et al. Reference Sobczyk, Wodziak, Gnatowska, Stempka and Niegodajew2018). The same evolution takes place with the gap vortices separated from the upper side of the upstream cylinder, which is schematically shown in figure 14(c).

4.3. Surface flow field

To gain an in-depth physical understanding of flow separation and reattachment, we examine the surface-oil-flow patterns in figure 15 for the two cylinders at L/d = 1.2, 2.2, 2.7, 2.8 and 4.0 (Re = 1.3 × 10⁴).

Figure 15. Surface-oil-flow patterns for cylinder spacing ratio L/d = 1.2 (extended-body regime), 2.2 (reattachment regime), 2.7 (reattachment regime), 2.8 (transition regime) and 4.0 (co-shedding regime). Re = 1.3 × 10⁴. ▴, reattachment lines. $\blacklozenge$, separation lines.

Extended-body regime: there are obviously narrow oil strips taking place near the trailing and leading corners of both cylinders at L/d = 1.2, apparently resulting from the crawling of the reverse flow under the shear layer (figure 12b). The scratch of the oil on the upper surface of the downstream cylinder originates from the flow reversal of the wake recirculation (figure 12aiii).

Reattachment regime: the reverse flow impinges on the trailing corner of the upstream cylinder at L/d = 2.2, producing a stripe near the trailing corner on the upper surface of the upstream cylinder (figure 13b). The upper surface of the downstream cylinder displays one reattachment line, as indicated by ▾, and two separation lines, marked by $\blacklozenge$. Flow separation near the leading corner is due to the reverse flow originating from the shear layer reattachment (figure 13aiii). The separation line near the leading corner is often referred to as secondary flow separation, reported in previous studies on bluff bodies with sharp leading edges (Shang et al. Reference Shang, Zhou, Alam, Liao and Cao2019; Abdelhamid, Alam & Islam Reference Abdelhamid, Alam and Islam2021). On the other hand, flow separation near the trailing corner is linked to the reattached flow that separates and rolls up before forming vortices behind the cylinders (figure 13aiii,aiv). The reattachment line coincides well with that from the PIV data in figure 13(ai). When L/d is increased to 2.7, the shadows around the leading corner (y* = 0.5, x′* = −0.5) on the upper and front surfaces of the downstream cylinder result from the impingement of the shear layer from the upstream cylinder. The impinging flow may be characterized by two stable states, that is, the outer slice of higher velocity reattaches upon the upper surface of the downstream cylinder or the inner slice of lower velocity rolls up, forming vortices, in the gap between cylinders. The former leads to a rise in St because of the higher flow velocity over the downstream cylinder surface, whilst the latter yields a considerably smaller St, and one state may overwhelm the other at one instant but may be overwhelmed at another instant (figure 8a). This is fully consistent with the previous discussion based on the PIV data (figure 13) and the occurrence of the bi-stable state, as highlighted by the red symbols in figure 3. It also accounts for the increase in St from L/d = 2.5 to 2.7 (figure 3).

Co-shedding regime: the oil film appears washed out from the trailing corner toward the middle of the upper surface of the upstream cylinder at L/d = 4.0, resulting from the separated reverse flow associated with vortex shedding from this cylinder (figure 14aiii). The frontal and upper surfaces of the downstream cylinder also appear distinct from the other cases because of the gap vortex impingement and shear layer development over this cylinder.

Transition regime: the dyed-oil film on the cylinder surfaces shows a mix of reattachment and co-shedding flows, with the reattachment line near the leading corner, as at L/d = 2.7, and a corrugated oil accumulation on the frontal surface similar to L/d = 4.0. Additionally, the upper surface mostly bears resemblance to that for L/d = 4.0.

5.1. Force measurement validation

A comparison is made between the present measurements in an isolated square cylinder wake and those in the literature to ensure the measurement technique deployed is reliable. Figure 16(a) compares the ${\bar{C}_{Po}}$ distribution around the isolated cylinder with those previously reported. The present ${\bar{C}_{Po}}$ agrees well with Hasan's (Reference Hasan1989) and Lee's (Reference Lee1975) measurements but exhibits an appreciable departure from Sakamoto et al.'s data, which is not unexpected, as will be seen from following discussion on ${\bar{C}_D}$. The present ${C^{\prime}_{Po}}$ (figure 16b) is slightly larger and smaller on the side and rear surfaces than its counterparts by Sakamoto et al. (Hasan and Lee did not measure ${C^{\prime}_{Po}}$), which is ascribed to a difference in Re between the two measurements. Table 2 summarizes ${\bar{C}_D}$, ${C^{\prime}_D}$ and ${C^{\prime}_L}$ and those obtained by others. The present ${\bar{C}_D}$ is in good agreement with Lee's (Reference Lee1975) and Reinhold et al.'s (Reference Reinhold, Tieleman and Maher1977) measurements, with a maximum departure of 8.4 %. Sakamoto et al.'s estimate is on the large side, as reflected in their ${\bar{C}_{Po}}$ distribution (figure 16a). The larger ${\bar{C}_D}$ in Sakamoto et al. (Reference Sakamoto, Hainu and Obata1987). is attributed to their larger T_u (= 0.19 %) and blockage ratio (= 9.8 %) than the present data (T_u = 0.13 %, the blockage ratio = 3.18 %). It has been well established that aerodynamic forces acting on the cylinder are highly dependent on experimental conditions (e.g. Surry Reference Surry1972; Laneville, Gartshore & Parkinson Reference Laneville, Gartshore and Parkinson1975). A large blockage ratio may lead to an overestimate of ${\bar{C}_D}$ (West & Apelt Reference West and Apelt1982). On the other hand, a large T_u causes the shear layers to reattach intermittently to the side surface of the cylinder, again resulting in an increase in ${\bar{C}_D}$ (Lee Reference Lee1975). The present ${C^{\prime}_D}$ and ${C^{\prime}_L}$ agree well with the others, except ${C^{\prime}_L}$, which exceeds Reinhold et al.'s estimate by 10.8 %. Reinhold et al.'s T_u is as high as 12 %, which may account for their underestimated ${C^{\prime}_L}$. The above comparison provides a validation for the present force measurement.

Figure 16. Distributions of (a) time-averaged pressure coefficient ${\bar{C}_{Po}}$ and (b) fluctuating pressure coefficient ${C^{\prime}_{Po}}$ on the surfaces of an isolated single cylinder.

Table 2. Comparison between present and previous measurements in the wake of an isolated cylinder.

5.2. Time-averaged forces

As shown in figure 17 (Re = 1.3 × 10⁴), ${\bar{C}_D}$ on the upstream cylinder is greatly larger than that on the downstream cylinder, irrespective of flow regime. In fact, the latter is negative in the extended and reattachment regimes. That is, the two cylinders attract each other. This is because the recirculation in the gap between the cylinders causes suction, which leads to a negative drag on the downstream cylinder. The value of ${\bar{C}_D}$ on the downstream cylinder changes sign after transition, due to the change of the downstream cylinder from being enclosed or partially enclosed within the separation bubble of the upstream cylinder to being impinged by the upstream-cylinder-generated vortices.

Figure 17. Variations with cylinder spacing ratio L/d in time-averaged drag coefficient ${\bar{C}_D}$ at Re = 1.3 × 10⁴: ○, upstream cylinder; , downstream cylinder; ---- isolated single cylinder (L/d = ∞).

In the extended-body regime (L/d ≤ 1.5), ${\bar{C}_D}$ grows in magnitude on each cylinder with increasing L/d (figure 17). This is because the lateral separation between the two free shear layers is smaller, yielding a more pronounced negative pressure in the gap between the cylinders. As shown in figure 18(a), ${\bar{C}_P}$ at L/d = 1.1 decreases from 1.0 at the front stagnation point (point 1) to 0.37 near the leading edge (point 3) of the upstream cylinder but remains almost unchanged on the rear surface. As a matter of fact, ${\bar{C}_P}$ on this surface is almost identical to that on the frontal surface of the downstream cylinder, suggesting stagnant fluid within the gap and hence constant pressure. With increasing L/d from 1.1 to 1.5, the base pressure of the upstream cylinder drops from −0.89 to −1.06, resulting in an increase in ${\bar{C}_D}$ on the upstream cylinder (figure 17). Meanwhile, ${\bar{C}_P}$ on the downstream cylinder drops on the frontal surface but rises on the rear surface, causing an increase in the magnitude of ${\bar{C}_D}$.

Figure 18. Distributions of time-averaged pressure coefficient (${\bar{C}_P}$) on the two cylinders at Re = 1.3 × 10⁴: (a) extended-body regime; (b) reattachment regime; (c) co-shedding regime. Open symbols: upstream cylinder; half-closed symbols: downstream cylinder.

In the reattachment regime, vortices formed behind the downstream cylinder are relatively weak (figures 12–14), and the magnitude of ${\bar{C}_D}$ on both cylinders declines (figure 17). The typical distributions of ${\bar{C}_P}$ on the upstream cylinder (L/d = 1.7 and 2.7, figure 18b) are quite similar to those in the extended-body regime (figure 18a). However, those on the downstream cylinder differ markedly between the two regimes. For example, compared with that in the extended-body regime (L/d = 1.1), ${\bar{C}_P}$ rises on the side surface of the downstream cylinder, especially near the trailing and leading edges at L/d = 1.7 and 2.7, respectively, due to the shear layer reattachment at these locations. The reattachment incurs a reverse flow in the gap, giving rise to an increase in ${\bar{C}_P}$ along the frontal surface of the downstream cylinder. This effect becomes more pronounced for a larger L/d, resulting in a rise in ${\bar{C}_P}$ on this cylinder. The decrease in ${\bar{C}_D}$ on the upstream cylinder with increasing L/d is due to the increase in pressure on the rear surface. Contrary to the observation in the extended-body regime (figure 18a), ${\bar{C}_P}$ on the rear surface of the upstream cylinder goes up with an increasing L/d (figure 18b), accounting for the drop in ${\bar{C}_D}$ on the upstream cylinder.

From the reattachment to co-shedding regime (L/d = 5.0), as shown in figure 18(b,c), ${\bar{C}_P}$ drops sharply on the rear surface of the upstream cylinder but rises rapidly on the frontal surface of the downstream cylinder. Accordingly, ${\bar{C}_D}$ experiences a rapid climb (figure 17). With a further increase in L/d, ${\bar{C}_P}$ changes little on the front and rear surfaces of the upstream cylinder. So does ${\bar{C}_D}$. On the other hand, ${\bar{C}_P}$ increases on the frontal surface of the downstream cylinder as much as on the rear surface. The combined effect makes ${\bar{C}_D}$ decline, although very slowly from L/d = 3.0 to 5.0.

5.3. Fluctuating forces

The dependence of ${C^{\prime}_D}$ and ${C^{\prime}_L}$ on L/d is distinct from one regime to another (Figure 19, Re = 1.3 × 10⁴) and is closely correlated with that of ${C^{\prime}_P}$ (figure 20). With increasing L/d, ${C^{\prime}_D}$ and ${C^{\prime}_L}$ on the upstream cylinder diminish in the extended-body and reattachment regimes and jump in the transition regime before declining again in the co-shedding regime. The values of ${C^{\prime}_D}$ and ${C^{\prime}_L}$ on the downstream cylinder both increase for a larger L/d in the extended-body regime because of a marked drop in $L_f^\ast $ (figure 11b). The value of ${C^{\prime}_D}$ keeps rising up to L/d = 2.5 in the reattachment regime, which is again accompanied by a further decline in $L_f^\ast $. Interestingly, ${C^{\prime}_D}$ and ${C^{\prime}_L}$ on the downstream cylinder show a small drop and rise, respectively, from L/d = 2.5 to 2.7, which is internally consistent with a small rise in St (figure 3). The high-velocity slice flowing on the side surfaces of the downstream cylinder gives rise to the increasing ${C^{\prime}_L}$, while a growth in $L_f^\ast $ (figure 11b) accounts for the drop in ${C^{\prime}_D}$. In the co-shedding regime, the downstream cylinder undergoes a declining ${C^{\prime}_D}$ with increasing L/d but more or less constant ${C^{\prime}_L}$.

Figure 19. Variations with cylinder spacing ratio L/d in (a) fluctuating drag coefficient ${C^{\prime}_D}$ and (b) lift coefficient ${C^{\prime}_L}$ at Re = 1.3 × 10⁴: ○, upstream cylinder; , downstream cylinder; and ----, isolated single cylinder (L/d = ∞).

Figure 20. Distributions of fluctuating pressure coefficient (${C^{\prime}_P}$) on the two cylinders at Re = 1.3 × 10⁴: (a) extended-body regime; (b) reattachment; (c) co-shedding. Open symbols: upstream cylinder; half-closed symbols: downstream cylinder.

In the extended-body regime, ${C^{\prime}_P}$ changes little with increasing L/d on the front and rear surfaces of the upstream cylinder, although exhibiting a discernible rise on the side surface (figure 20a). The rise is connected to an increase in the effective after-body length, which acts to reduce the separation between the separated free shear layer and the side surface. As such, ${C^{\prime}_D}$ on the upstream cylinder changes little but ${C^{\prime}_L}$ drops appreciably from L/d = 1.1 to 1.5. The value of ${C^{\prime}_P}$ on the downstream cylinder increases on the rear surface but drops on the front surface from L/d = 1.1 to 1.5, the former accounting largely for the increased ${C^{\prime}_D}$. In the reattachment regime, ${C^{\prime}_P}$ on the downstream cylinder behaves differently, displaying a drop on the side surface but an increase on the front surface with increasing L/d. A shift in the reattachment point towards the leading edge acts to strengthen the shear layer instability. Accordingly, ${C^{\prime}_P}$ on the side surface contracts and, as a result, ${C^{\prime}_L}$ declines. For the upstream cylinder, as L/d increases, ${C^{\prime}_P}$ on all surfaces shrinks in both the extended and reattachment regimes (figure 20a,b). So do ${C^{\prime}_D}$ and ${C^{\prime}_L}$. In the extended-body and reattachment regimes, ${C^{\prime}_P}$ and also ${C^{\prime}_L}$ on the downstream cylinder are higher than their counterparts upstream, due to the fact that vortex shedding occurs only behind the downstream cylinder.

In the co-shedding regime, ${C^{\prime}_P}$ on both cylinders becomes very large compared with that in the reattachment or extended-body regime (figure 20c), because vortices are shed from the upstream cylinder as well as from the downstream. Under the impingement of the upstream-cylinder-generated vortices, ${C^{\prime}_P}$ values on the front surface of the downstream cylinder and on the side surface and near the leading edge are significantly larger than their counterparts on the upstream cylinder. The value of ${C^{\prime}_P}$ on the rear and side surfaces of the upstream cylinder does not change appreciably with L/d, while ${C^{\prime}_P}$ on the front surface of the downstream cylinder declines slightly with L/d because of the weakened gap vortices impinging on the downstream cylinder.

Three significant findings can be made from the data of ${C^{\prime}_D}$, ${C^{\prime}_L}$ and ${C^{\prime}_P}$. Firstly, ${C^{\prime}_D}$ on the downstream cylinder is larger than that on the upstream, regardless of the flow regime. The physics behind this, however, differs between the co-shedding regime and the extended or reattachment regime. So does that between the reattachment and co-shedding flows in the transition regime. Note that ${C^{\prime}_D}$ is the consequence of the pressure fluctuations on the front and rear surfaces of the cylinder. In the extended-body and reattachment regimes, the flow between the cylinders is largely stagnant, and alternate vortex shedding occurs behind the downstream cylinder. The large ${C^{\prime}_D}$ on the downstream cylinder is mainly contributed by the large fluctuating pressure on its rear surface. In the co-shedding regime, the upstream-cylinder-generated vortices impinge on the front surface of the downstream cylinder and trigger vortex shedding from the downstream cylinder. As such, the large ${C^{\prime}_D}$ on the downstream cylinder results from large ${C^{\prime}_P}$ on its front and rear surfaces. Secondly, ${C^{\prime}_L}$ is greater on the downstream cylinder than on the upstream in the extended-body and reattachment regimes. However, this can be sometimes reversed in the co-shedding regime. The unsteady shear layers from the upstream cylinder overshoot or reattach on the side surfaces of the downstream cylinder in the extended-body regime and cause a large fluctuating pressure on its side surfaces, thus resulting in large ${C^{\prime}_L}$. In the co-shedding regime, the upstream-cylinder-generated vortices impinge upon the downstream cylinder, acting to impair the strength of the ensuing vortex shedding. As a result, ${C^{\prime}_L}$ may drop. Finally, the maximum ${C^{\prime}_D}$ on each cylinder occurs at L/d = 3.0. This may result from the synchronization of vortex shedding from the downstream cylinder with that from the upstream cylinder, albeit with a phase lag of 2π (Sakamoto et al. Reference Sakamoto, Hainu and Obata1987; Alam & Zhou Reference Alam and Zhou2007).