Numerical analysis of the flow topology around two rectangular cylinders in a staggered arrangement

Flow around bluff structures plays a very important role in real-life and engineering applications. Many applications of bluff body flows can be seen in our surroundings, like chimneys, cooling towers, high-rise buildings, bridges, micro-electromechanical devices, etc. The flows around bluff bodies have been the subject of several experimental and numerical studies. Most of the past studies were performed for exploration of a solitary body (cylinder) flow characteristics. Grove et al.¹ experimentally studied the flow around a circular cylinder at various Reynolds numbers (Re). They reported that, at the backside of the cylinder, the pressure coefficient remained constant for the range of Re from 25 to 177. Their findings also indicated that the Re has a direct impact on the wake length. Tritton² performed the experiments to observe the transitions in wake and variations in the drag coefficient around a circular cylinder for Re = 0.5–100. They reported that the transition mechanism in the vortex street behind the cylinder occurred at Re = 90. Rajani et al.³ delved into the 2D and 3D flow around a circular cylinder under various laminar flow conditions within the range of 0.1 ≤ Re ≤ 400. They utilized an implicit finite volume method (FVM) to analyze incompressible fluid flow, uncovering three distinct flow regimes: (a) creeping flow regime, (b) stable closed near-wake regime, and (c) laminar vortex shedding. Their observations indicated that the wake became unstable after Re = 49, leading to transverse wake oscillations near its end. The wake region for Re > 250 was found to be less regular with shorter wavelengths of streamwise vortices. Furthermore, it was noted that the highest values for lift coefficient and average drag coefficient (CDmean) in 2D simulations tend to be lower compared to those in 3D simulations. Erturk and Gokcol⁴ analyzed the incompressible, steady, viscous flow around a circular cylinder up to Re = 500. They found significant change in the flow behavior at Re = 100 and 300. Specifically, at the midpoint of the wake bubble, around Re = 300, there was a rapid linear increase in vorticity, accompanied by a swift bubble expansion. Zhang and Zhang⁵ addressed the incompressible viscous fluid dynamics at low Re for flow around a square cylinder within a channel using the numerical manifold method (NMM). Their results indicate that the pressure distributions were consistent for the case of steady flow while variations in pressure were observed beyond the cylinder due to the appearance of vortices. Numerical simulations were conducted by Erturk and Gokcol⁶ to model the flow of incompressible viscous fluid around a square cylinder in a channel using the finite difference method (FDM) at Re up to 410. They studied the influence of mesh, the blockage ratio (B), inflow, and the outflow boundaries on flow characteristics. Narayanan et al.⁷ numerically explored the impact of simultaneous transverse and rotational oscillations of a square cylinder on flow patterns and force coefficients by using FVM. Variations in non-dimensional parameters like frequency ratio (f = 0.5, 0.8), Re = 50–200, phase difference (φ), and rotational amplitude (θ₀) were adjusted to assess their influence on the flow characteristics and force coefficients. They found that the effect of Re on the drag coefficient (CD) disappears when the motion is completely anti-phase. Additionally, the CD exhibited low sensitivity to changes in other parameters at low Re values. Xu et al.⁸ studied the flow characteristics around a square cylinder under the influence of lateral boundaries. The impact of boundaries was analyzed for different flow states, including the laminar, transitional phase, vortex shedding phase, and turbulent phase. They found that the boundary constraints have a significant impact on the vortex generation mechanism in the wake of the cylinder. Norberg⁹ experimentally studied the flow and pressure forces around a fixed rectangular cylinder with side ratios (B/A) = 1, 1.62, 2.5, and 3 at different angles of occurrence (θ) and Re. They found significant impact of θ and B/A on reattachment and shear layer interface. Chiarini et al.¹⁰ studied the turbulent flow past a rectangular cylinder and found that the flow dynamics was influenced by the Kelvin–Helmholtz instability, and huge von Karman–like vortices were shed from the trailing edge. Lim¹¹ numerically investigated the surface pressure distribution around a rectangular cylinder in a turbulent boundary layer under the influence of wind direction by rotating the cylinder at different angles. He found that the change in surface pressure was highly dependent on the top and side faces of the cylinder as well as the wind direction. Thete et al.¹² presented a numerical study for fluid flow over a rectangular prism under the effect of Re. They found the flow separation and vortex shedding phenomena were largely governed by the change in Re. They also demonstrated distinct vortex characteristics with increasing Re. Carmo et al.¹³ analyzed the wake characteristics for the flow around two staggered circular cylinders and identified three distinct flow modes named as mode A, mode B, and mode Gu and Sun et al.¹⁴ through wind tunnel investigation studied the interference effect on the wakes of two circular cylinders arranged in staggered configurations at high subcritical Re. They identified three flow patterns in terms of pressure distributions and flow visualizations. Furthermore, an in-depth analysis was conducted on the phenomenon of switching between distinct pressure patterns at critical angles, leading to a significant discontinuity in the lift force generated by the cylinders. Sumner et al.¹⁵ demonstrated wind tunnel experiments to measure the average aerodynamics forces and vortex shedding frequencies for two staggered circular cylinders of the same diameter in a cross-flow ranging from Re = 32 000 to 72 000, G = 1.125, 4.0, and θ = 0° to 90°. They examined the CDmean, CLmean, and St for both upstream and downstream cylinders. Based on the G, they categorized the staggered configuration as closely spaced, moderately spaced, and widely spaced. The closely spaced and moderately spaced setups exhibited intricate variations in the mean aerodynamic forces acting on the downstream cylinder, whereas the widely spaced configuration resulted in relatively consistent forces. The researcher discovered that the peak of Lift coefficient (CL) on the downstream cylinder was linked to the presence of shed Karman vortices from the upstream cylinder, either due to proximity or impingement. Fallah et al.¹⁶ conducted a numerical study of the flow field around two circular rotating cylinders in a staggered arrangement by using the lattice Boltzmann method (LBM). The simulations encompassed a range of absolute rotational speeds, G and θ, at Re = 100. Their findings indicated a significant impact of these parameters on the streamlines, vorticity, and pressure distribution. Furthermore, they also analyzed the influence of these parameters on CD and CL. Aboueian and Sohankar¹⁷ numerically investigated the flow features of two square cylinders arranged in a staggered fashion at Re = 150 with G = 0.1 to 6. They examined vorticity, pressure, and velocity magnitudes to reveal different flow patterns and their impact on the flow field. The variation in G led to five flow regimes: Single body flow (SBF), periodic gap flow, aperiodic flow, modulated periodic flow, and synchronized vortex shedding flow. They identified that the diverse shedding frequencies seen in the case of the downstream cylinder during the modulated periodic phase were more precisely linked to changes in the vortex-shedding frequencies of each cylinder. It was observed that the downstream cylinder encountered a higher drag force in comparison to the upstream cylinder. Zhao et al.¹⁸ employed direct numerical simulation to analyze the flow around two staggered square cylinders at Re = 500 and G = 1 to 8 with a height-to-width ratio (H) of 4 and a staggered distances-to-diameter ratio (S) of 0.5 and 1. They found that the shifting of the downstream cylinder sideways for S values of 0.5 and 1 significantly alters flow features and force coefficients. At G = 1 to 3, the shear layer from one side of the upstream cylinder trapped in the gap ultimately resulted in the shedding of vortices. Additionally, they also reported the appearance of a pronounced horseshoe vortex in front of the downstream cylinder for all G at S = 0.5 and 1. Dwivedi et al.¹⁹ deliberated the 3D viscous flow around a pair of square cylinders at Re = 100, with different configurations using different FDM schemes. In this study, they examined different flow features, including vorticity profiles and the variations of CD and CL. Nadeem et al.²⁰ examined how attached control plates enhance the reduction of hydrodynamic forces around two staggered square cylinders by changing the plate lengths from 0.1 to 10 times the size of the cylinder. They found that different flow patterns emerge based on plate length, linked to change in the Strouhal number (St = f × d/U_int), where f is vortex shedding frequency of both cylinders. Significant reductions in the fluid force parameters were observed with increasing plate length in this study. Furthermore, the plates were found to be more effective in controlling flow around the second cylinder. Fezai et al.²¹ presented a numerical study of the flow over three-square cylinders in two different triangular arrangements with Re = 1 to 110 by using FVM. They found three different flow patterns by systematically varying the Re. According to their findings, both triangular alignments significantly influenced the point of bifurcation, leading to a substantial reduction in the critical Re for the onset of instability.

The above-mentioned literature review indicates that the previous experimental as well as numerical studies were mostly concentrated on the flow behavior around individual bluff bodies, particularly circular and square cylinders. Compared to single bluff body flow dynamics, studies regarding multibody flow dynamics are limited in the open literature. The past studies also indicate that the important parameters concerning the bluff bodies’ flow dynamics include the vorticity patterns, pressure variations, streamline behaviors, wake dynamics, the variations in fluid-induced drag and lift forces under the impact of Re, object shape, spacing, alignment, etc. Furthermore, the numerical methods for such types of problems have gained increasing attention from researchers due to the simplicity of computational procedures and increased simulation accuracy, which makes it easy to visualize the results by investing less effort as compared to the experimental measurements. Although extensive literature is available for square or circular body flow dynamics, rectangular bluff bodies have been less explored in this regard. Specifically, the flow dynamics analyses surrounding vertically aligned rectangular cylinders in a staggered configuration are rarely seen in the available literature. Such bodies have a direct relevance to practical scenarios like tall buildings, towers, etc, which are mostly seen in vertical alignments. Therefore, keeping these facts in view, the objective of the current study is to enhance our comprehension of the flow over rectangular cylinders in a staggered alignment by using LBM. It will be focused in this work that how the flow around staggered rectangular cylinders is influenced by the variations in G between the cylinders. How do the wake transitions occur by growing G gradually from lower to higher values? To what extent does the jet flow influence the near-wake flow dynamics? How do the fluid forces and the wake of the upstream rectangular body influence the downstream one when the bodies are placed in a staggered fashion? In addition, the flow structures will also be compared to those of circular or square cylinders in order to discuss the major differences.

This study utilizes the LBM, a widely used discrete numerical approach for modeling fluid flow problems. Originally stemming from lattice gas cellular automata (LGCA), the LBM was pioneered by McNamara and Zanetti²² to overcome the statistical noise challenges associated with LGCA. The choice of the LBM for this study is supported by several advantages that make it particularly suitable for simulating fluid flow around complex geometries, such as bluff bodies in various arrangements. LBM is known for its relatively simple implementation compared to other computational fluid dynamics methods, making it a good tool for computations. LBM is well-suited for parallel computations, allowing for efficient simulations that can handle large-scale problems and complex flow scenarios. Originating from lattice gas cellular automata, LBM effectively mitigates statistical noise issues, enhancing the reliability of the results. The lattice Boltzmann equation (LBE) is known for its explicit and quasi-nonlinear nature, making it efficient for both steady and unsteady flow simulations. These advantages collectively justify the selection of LBM for the research, enabling a comprehensive analysis of the fluid dynamics surrounding the bluff bodies. A comparative investigation was carried out by Guo et al.²³ to assess the performance of both the gas-kinetic scheme (GKS) and LBM models in simulating the flow around a solitary isolated square cylinder within a channel. They noted that the lattice Boltzmann equation (LBE) demonstrates ∼10 times faster convergence than GKS for steady flows and is around three times faster for unsteady flows.

The schematic of the proposed scenario is depicted in Fig. 2, illustrating two rectangular cylinders (C1 and C2) with a common aspect ratio (⁠$AR=hd$⁠, where h represents the cylinder’s height and d is their width) of 3:2 arranged in a staggered configuration within a rectangular channel having the length X and height Y. Both the cylinders are at a gap distance G apart in the horizontal as well as vertical directions. The total length of the rectangular computing channel is X = X_u + d + G + d + X_d, with the upstream boundary at a distance X_u = 10d and the downstream boundary at a distance X_d = 20d. The side boundaries are located at Y_u = 8d upside and Y_d = 8d downside from the upper and lower cylinders, respectively. The total height of the computational domain is Y = Y_u + h + G + h + Y_d. This size of the domain has been validated to offer a balance among solution correctness and computational efficiency, unaffected by domain-related influences, when simulating uniform flow around an obstacle. Furthermore, the domain dimensions are either closer to or exactly matching with the values taken in studies.^17,27–29

In this study a special emphasis is paid while dealing with the domain boundaries. It is focused on expressing the flow entrance, exit, and side wall boundaries with appropriate mathematical forms in order to ensure the accuracy of results at domain boundaries. Note that in LBM computations, all the boundary conditions are to be expressed in terms of the distribution functions. Details of boundary conditions selected in the current study are given below.

The code validation and grid independence can be found in our recently published article, Tahir et al.²⁹ 

In the current work, we scrutinized the impact of G on the flow dynamics surrounding two rectangular cylinders in staggered arrangement. Various G values ranging from 0 to 10 were considered while maintaining a constant Re = 150. Here we characterized the flow patterns mainly based on the vortex structures and the Strouhal numbers. Furthermore, the variations in pressure distributions and streamline structures, time-dependent variations of lift and drag coefficients, and the phase portraits are also described for different flow patterns that appeared in this study. The detailed analysis of each flow pattern is presented in Secs. IV A–IV D.

In the current work, the isolated bluff structure (IBS) flow pattern, consisting of a single street of vortices, appeared at Re = 150 and G = 0 [see Fig. 3(a)]. G = 0 among the cylinders indicates that both cylinders act as an extended IBS with adjacent corners. The initial elongated vortices developed in IBS flow adopt a circular shape while traveling to the downwake region as shown in Fig. 3(a). In addition, the dimensions of vortices differ from those of a single cylinder case due to the vertically aligned rectangular cylinder structure. Aboueian and Sohankar¹⁷ also found the single bluff body flow pattern for two staggered square cylinders but with different characteristics. The initial separated flow seems to deflect upward, which ultimately transforms to negative and positive vortices after complete detachment from cylinders. Maximum flow pressure seems to be in the region covering the front middle side of C1 to the near upper corner location of C2, while minimum pressure is at the bottom front corner of C1, as well as it alternately varies in the wake region of both cylinders [Fig. 3(b)]. The rotating streamlines exhibit an upward stretched elliptical structure attached with both cylinders because of the upright arrangements of cylinders. In the region away from cylinders, the wake is concentrated and broadened. The IBS pattern mentioned earlier can be further verified by observing the time series data of CL and CD for both cylinders [Figs. 3(c) and 3(d)]. It is obvious from the temporal variations that both the CD and CL of cylinders have significant modulations in the amplitudes of curves due to the varying size of vortices appearing in the wake. Figures 3(e) and 3(f) show the power spectrum graphs indicating the same St values (0.0825, 0.0825) due to similarity in the vortex shedding frequencies. Figures 3(g) and 3(h) depict the phase portrait, i.e., the graphical representation of CL vs CD. The phase portraits clearly depict the modulations in amplitudes of CD and CL. In addition, the phase portraits indicate that both force coefficients repeat their values after some time. This repetition often manifests as closed curves or loops in the phase portrait but in different directions. It appears that the magnitude of the forces influenced by different factors or the parameters governing the system do not appear to be identical in both cylinder cases.

As the G between cylinders is gradually enlarged, the flow pattern transits from IBS to chaotic flow (CF). This flow pattern is found at spacing values ranging from G = 0.5 to 2.5, while its representative cases, G = 0.5 and 2.5, are presented only in Fig. 4. Figure 4(a) indicates that the wake structure experiences notable disruption due to the pronounced stimulus of the jet flow emanating from the spaces among the cylinders. In the proximity of the cylinders, the interaction of the lower shear layer from C1 with the upper shear layer of C2 through intermixing outcomes in the generation of a CF. Figure 4(b) depicts that the jet flow impact minimizes, and initially separate vortices are generated but with time intermingle with each other, resulting in chaos and amalgamations. In the downwake regions, the vortices trailing both cylinders lack regularity; instead, merged structures are observed, slanted toward the upper region of the domain. The vorticity contours clearly indicate multifaceted separation, reattachment, and shedding of positive and negative vortices from the cylinders. Unlike the IBS flow, the regions of maximum pressure distributions appear separately at the front sides of both cylinders [Figs. 4(c) and 4(d)]. Moreover, for this case, the low-pressure zones gradually shift toward the lower parts of the domain in the far wake regions. The streamlines randomly switch in the lower and upper regions of the wake due to chaos in the wake structures. At G = 0, the structures of the swirling eddies adjacent to cylinders and in the wake appear randomly, having diverse and random patterns, influenced by the jet flow. Due to chaotic flow, the CD and CL of both the cylinders randomly fluctuate between low and high amplitude curves [Figs. 4(e)–4(h)]. Figures 4(e) and (g) depict anti-phase variation in CD for both cylinders, while Figs. 4(f) and 4(h) indicate in-phase variations in CL of both cylinders influenced by the development of initial vortices. The magnitude of CD signals of the upper cylinder is high for both cases as compared to the lower cylinder. Similar is the case for lift coefficient signals as well. This can be attributed to the upward deflection of vortex structures appearing in the wake. Figures 4(i) and 4(j) indicate the appearance of several peaks in the power spectrum due to random fluctuations in the CL curves at G = 0. However, as G increases and lift signals stabilize, this multiple peaks phenomenon gradually disappears [Figs. 4(k) and 4(l)]. Note that for the CF pattern case, the St of both the cylinders differs from each other, indicating the appearance of multiple frequencies beyond G = 0.5. Figures 4(m)–4(p) show the phase portrait resembling a tangled thread and threaded tube, indicating the random fluctuations of force coefficients due to chaotic wake flow. The phase portraits for both the cylinders significantly differ, possibly indicating different underlying dynamics or parameter values for fluid forces. These phase portraits also indicate that the randomness in force coefficient values gradually minimizes as the spacing value increases from G = 0.5 to 2.5.

The modulated synchronized (MS) flow is observed in this study at G = 3–8 (see Fig. 5). For discussion purposes, we considered G = 3 and 8 as representative cases. The resulting vorticity contour, pressure streamline variations, time variations of CD and CL, the CL spectrum, and the phase portrait are shown in Fig. 5. Figures 5(a) and 5(b) show that each cylinder generates its separate vortex street in which vortices travel alternately like the von Kármán vortex street in a synchronized pattern without crossing each other. The vorticity snapshots indicate the detachment of the flow from the cylinders, followed by some variations in size and shape of vortices, which indicate the flow modulation. The pressure streamlines contours shown in Figs. 5(c) and 5(d) indicate a small tiny recirculating eddy produced at the rear side of both cylinders, which gets smaller as G increases. The pressure is higher at the front surface of the lower cylinder as compared to the upper one at G = 3, while it appears to be similar for both cylinders at G = 8. Along with varied pressure zones inside the gap region, the figure also demonstrates the appearance of discrete low-pressure patches following each cylinder. The variation over time indicates modulations in CD signals for both cylinders because of the variation in shape and size of the vortices, while the CL is exhibiting a fully periodic pattern [Figs. 5(e)–5(h)]. The periodic lift is due to the distinct vorticity street of each cylinder as represented in Figs. 5(a) and 5(b). The amplitude variation in drag of both cylinders is more prominent at G = 2.5, which gradually stabilizes as the spacing reaches G = 8. Furthermore, the C2 experiences higher drag compared to C1 at G = 3, while this difference minimizes at G = 8 [Fig. 5(e)]. CL of both cylinders are in-phase with each other at G = 3 and anti-phase at G = 8 [Fig. 5(h)]. The power spectrum graphs indicate that both the cylinders have identical spectra with the same value of St [Figs. 5(i)–5(l)]. At G = 3, a minor secondary peak can be seen in the power spectrum of CL because of the variation in shape and size of the vortices, while for increasing G, the secondary peak disappears and the identical St value of both cylinders decreases [Figs. 5(k) and 5(l)]. Aboueian and Sohankar¹⁷ categorized the flow pattern with similar characteristics as the modulated periodic flow pattern in the case of two staggered square cylinders. Figures 5(m)–5(p) depict the phase portrait, which follows the smooth closed path. This pattern of phase portraits indicates that the force coefficients (CD and CL) repeat their values after a certain period. This is influenced by the periodicity of forces being exerted on cylinders.

The synchronized flow (SF) is identified in the present investigation for the spacing range G = 8.5–10. For discussion, G = 8.5 is taken as a representative case. The vorticity contour, pressure and streamlines contour, temporal histories of CD and CL, and power spectrum of CL for this representative case are illustrated in Fig. 6. The vorticity contour in Fig. 6(a) reveals that each cylinder generates its distinct vortex street, where vortices move in a synchronized manner from the very beginning without intersecting each other. The influence of the jet flow diminishes due to the minimal proximity of cylinders, and no variation in shape and size can be seen contrary to that in the case of the MS flow pattern [Fig. 6(a)]. In the spacing range corresponding to SF, the shedding pattern transitions between in-phase and anti-phase due to initial vortices created by both cylinders. The alterations in streamlines also delineate the distinct vortex street behind each cylinder [Fig. 6(b)]. The streamlines contour shows that small eddies of different sizes appear behind each cylinder, indicating the convergence region of the detached shear layers. Furthermore, this flow demonstrates the emergence of distinct low-pressure regions in the wake of each cylinder with modulated pressure zones in the gap region. As the variations in shape and size of vortices are no more prevalent in the wake, and each cylinder owing their own vortex street, therefore both drag and lift are completely periodic [Figs. 6(c) and 6(d)]. These figures also indicate that the drag and lift force magnitudes on both cylinders are almost similar but comparatively lower than those in the case of other flow patterns. This suggests that the proximity effect of each cylinder on the wake dynamics of the other becomes minimal; that is why the wake of each cylinder behaves distinctly. The power spectrum plot displays a single dominant peak, with slightly different St values for both cylinders [Figs. 6(e) and 6(f)]. As the flow is fully synchronized with periodic drag and lift signals, therefore, the flow is dominated by primary vortex shedding frequency, and no secondary peak appears in power spectrum graphs. The phase portrait for the SF pattern given in Figs. 6(g) and 6(h) resembles that for the MS pattern shown in Figs. 5(o) and 5(p).

The complete spacing range of each flow pattern discussed earlier is described in Table I.

This section explores the variations in fluid flow parameters, including CDmean, root mean square value of drag coefficient (⁠$CDrms=∑i=1nCDi−CDmean2/n$⁠, where CD(i) represents the drag coefficient at the ith time step, CDmean is the mean drag coefficient, and n indicates the total number of iterations), root mean square value of lift coefficient (⁠$CLrms=∑i=1nCLi−CLmean2/n$⁠, where CL(i) is the lift coefficient at the ith time step, CLmean is the mean lift coefficient), St, amplitude of drag coefficient (⁠$CDamp=CDmax−CDmean$⁠, where CDmax is the largest value of CD), and amplitude of lift coefficient (⁠$CLamp=CLmax−CLmean$⁠) for both cylinders. The analysis focuses on the impact of various G values from 0 to 10 and fixed Re = 150 on these fluid flow parameters. Figure 7(a) illustrates a predominantly decreasing trend in CDmean for both cylinders as G increases from 0 to 1.5. The CDmean of C1 exhibits irregularities within the range of 0.5 to 2.5, followed by a subsequent decrease in response to an increase in G. Note that at value G = 0.5, the flow pattern transitions from IBS to CF. Beyond G = 8, the CDmean of C1 and C2 attain similar values due to similarity in wake structures. Owing to this range of G from 8.5 to 10, a synchronized flow pattern was witnessed as described in Sec. IV D. Notably, the CDmean for C1 and C2 has local maxima at G = 0, corresponding to the IBS pattern, while the local minimum value for C1 appears at G = 1.5 in the CF pattern and that for C2 appears in the case of the SF flow pattern. Figure 7(b) indicates that the CDrms of C2 is mostly higher than that of C1 due to the impact of the wake of C1, which reattaches to C2 after detaching from C1, thus modulating its wake at small values of G. In the CF flow pattern case, a rapid decrease in CDrms of both cylinders is obvious from Fig. 7(b). After that, it shows a linear trend for both cylinder cases with an overlap until G = 10. The CLrms values of both cylinders display almost similar trends of dips and peaks to those seen in the case of CDmean values [Fig. 7(c)]. Initially, CLrms of C1 is lower than C2, but for large G, both converge to nearly equal values. Figure 7(d) illustrates the fluctuation of St for both cylinders, determined through the application of FFT on the CL. Initially, both the cylinders have the same trend of reduction, with C1 attaining its minimum value at G = 1 and C2 at G = 0.5. After that, St of C1 suddenly jumps to a local maxima at G = 1.5, with a gradual decrease afterward. At G = 3, the St for both C1 and C2 converge to similar values, indicating the weaker impact of jet flow disturbing the wakes of both cylinders. Note that beyond G = 3, both cylinders exhibit synchronization with independent vortex streets. CDamp of both cylinders seems to have random low and high values in the IBF and CF flow patterns case (G = 0 to 3). Afterward, the CDamp of C1 has almost linear behavior, while that of C2 slightly increases and then settles to a constant value. Both CDamp values overlap in the range of G from 6.5 to 10, indicating negligible disturbances in the amplitudes. Figure 7(f) indicates sudden decreasing and increasing trends of CLamp of C1 resembling a W shape initially in the range G = 0 to 2. Then it exhibits gradually increasing behavior, while that for C2 has an opposite behavior to C1. Initially it has a smooth behavior, then it attains a V shape with a gradual decrease afterward. Until G = 8.5, both C1 and C2 overlap and show linearity in the SF pattern.

The current work examined the fluid flow characteristics surrounding two staggered arranged rectangular cylinders through numerical computations. The main goal of this study was to examine the influence of G between cylinders on both the flow structures and the forces induced by the fluid. Various G were considered within the range of G = 0–10 while maintaining a fixed Re = 150. The study investigates the behavior of viscous fluid when interacting with two vertically positioned rectangular cylinders in staggered arrangements. Main findings of the current study are summarized below:

1. For different ranges of G, four flow patterns were identified depending on the wake structures, St variations, and variations in CD and CL: isolated bluff structure (IBS), chaotic flow (CF), modulated synchronized (SF) flow, and synchronized flow (SF).

2. The IBS flow was observed solely at G = 0, where both cylinders act as one unit. The combination of positive and negative vortices creates a unified wake with round shaped vortices in the wake. For this flow pattern, both cylinders were found to have similar St, indicating the unique frequencies.

3. CF was seen at 0.5 $≤$ G $≤$ 2.5, displaying intricate wake structures lacking defined vortex shapes and sizes. These complex structures significantly influenced flow-induced forces with notable fluctuations. In this case, both cylinders have different values of St, indicating multiple vortex shedding frequencies.

4. In the spacing range 3 $≤$ G $≤$ 8, the MS pattern was observed. In this flow pattern, each cylinder produced its distinct vortex street, with vortices moving independently without interacting with other cylinders’ wakes. Over time this flow caused amplitude modulation in CD for both cylinders, while CL remained consistently periodic. In this flow pattern, similar St values were observed for both cylinders.

5. Between 8.5 $≤$ G $≤$ 10, the flow was identified as fully synchronized with no modulation in drag, where each cylinder had its independent wake flow features. The flow properties around each cylinder closely resembled those of a single square cylinder as well as the MS pattern. In this flow pattern, St of both cylinders slightly differs from each with periodic drag and lift.

6. Vortex structures were found to be disturbed at low G but settled to regular behavior at higher G values. Different force parameters, such as CDmean, CDrms, CLrms, St, CDamp, and CLamp, were found to be greatly influenced by wake structures, specifically by the CF pattern.

7. The results of this research enhance the comprehension of the flow around staggered arrangements of rectangular cylinders, potentially paving the way for advancements in controlling flows around civil structures.

The authors have no conflicts to disclose.

Neelam Tahir: Investigation (equal); Writing – original draft (equal). Waqas Sarwar Abbasi: Supervision (equal). Hamid Rahman: Formal analysis (equal). Arshad Riaz: Formal analysis (equal). Ghaliah Alhamzi: Resources (equal); Software (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.