Numerical investigation of flow features for two horizontal rectangular polygons

Over the past few decades, the study of flow over bluff bodies has witnessed significant advancements in research, primarily driven by the rapid and continuous evolution of science and technology. Bluff bodies, representing non-streamlined shapes with blunt front or rear faces, continuously interact with fluid in our environment, resulting in a complex and fascinating phenomenon. This interaction between fluid and solid materials can be observed everywhere in our daily life, showcasing the diverse applications and implications of such interactions as flows past an airplane, a submarine, an automobile, telecommunication, heat exchanger systems, electrical and mechanical chips, gas turbine blades, etc. Researchers have been intrigued by the unsteady incompressible flow phenomenon occurring around bluff bodies, particularly the wake formation behind the bodies. In addition, they have explored the impact of different parameters, such as Reynolds number (Re) and geometry, on wake formations. The subsequent change in flow behavior is observed through experimental and numerical methods.

Al-Mdallal¹ studied the two-dimensional (2D) flow of a viscous, incompressible fluid past a circular cylinder moving in a circular motion within a uniform stream. The simulations are carried out at a fixed Re = 200 with a varying aspect ratio (AR) = 0.05–0.25. They observed that in-line vortex vibrations occur when vortices are shed symmetrically on the surfaces of the bluff body. On the contrary, crossflow vortex vibrations arise when the shedding is asymmetric. Hybrid vortex vibrations combine both in-line and crossflow vibrations. Zhang et al.² analyzed the flow characteristics and investigated flow control mechanisms by examining the flow around shape-modified circular cylinders using the large eddy simulation method at Re = 5000. They found that the flow separation was delayed at the node section, resulting in a narrower and shorter wake. On the other hand, at the saddle section, the wake was broader and longer due to forward separation. Ma et al.³ carried out a two-dimensional numerical study to understand the effect of the upstream circular bar and downstream splitter plate with a suitable distance and splitter length. They found that correctly placing the circular bar can increase the shield of squares. In addition, the splitter plate downstream can suppress the vortex shedding completely, and the length of the splitter plate significantly affects the flow.

Krishan et al.⁴ investigated the flow-field characteristics of a synthetic jet (SJ) impinging on a circular cylinder at eight different radial pressure taps positioned at 45° intervals in the center. Additionally, hotwire anemometry was utilized to measure velocities in the slot lateral plane along the cylinder sides and the wake. Beyond pressure and velocity measurements, the study also involved a fog-based flow analysis. Zhang et al.⁵ conducted unsteady flow simulations to study flow past a circular and a square cylinder at high Re. The simulations utilized the unsteady Reynolds-Averaged Navier–Stokes (URANS) equations with different turbulence models. They noted that when the flow encounters bluff bodies, vortices are formed on the downstream side, causing periodic shedding, and this phenomenon results in an oscillatory behavior of the flow field. Kumar and Ray⁶ studied the effect of shear rate (K) on the vortex shedding phenomenon in terms of stream function and vorticity contours via a square cylinder using a higher order compact (HOC) scheme at Re = 100 and 200 and K = 0.0–0.1. They observed that as K increases, the alternately shed vortices differ in size and strength, making their effectiveness more pronounced. Mahdhaoui et al.⁷ discussed the numerical investigation of flow structure and heat transfers in a channel with a porous square cylinder using the Darcy–Brinkman–Forchheimer model. They found that the aspect ratio slightly influences heat transfers, while increased Reynolds numbers result in increased heat transfers.

Krishan et al.⁸ studied the flow-field measurements of a slotted synthetic jet with and without sidewalls. Velocity measurements for both configurations were conducted using a single-probe hot wire. They observed that when sidewalls are present, the flow is confined by attaching no-slip sidewalls to one side of the slot. In addition, the relative difference in the magnitude of distinct peaks in near-field spanwise velocity profiles indicated that the vortex doesn’t curl toward the centerline in the synthetic jet with sidewalls, attributed to the presence of the no-slip wall. Rahman et al.⁹ conducted numerical simulations using the lattice Boltzmann method to study the flow characteristics of a laminar, incompressible, and Newtonian fluid passing through three square cylinders arranged in a triangular configuration. The results indicate that as the gap spacing (G) increases, the mean drag coefficient of all three cylinders tends to converge to the mean drag coefficient of a single cylinder. Additionally, it is observed that increasing G diminishes the influence of a square cylinder, whether positioned upstream or downstream of two side-by-side square cylinders. Tang et al.¹⁰ studied the numerical simulation of the flow around twin circular cylinders arranged in tandem near a wall at Re = 200 with different combinations of G/D (a gap between the lowest surface of the twin cylinders and the plane wall) and L/D (center-to-center distance between the two cylinders). They noted that the upstream cylinder has a higher mean drag coefficient than the downstream cylinder for both combinations. However, in the two-wake mode, the downstream cylinder exhibits significantly larger root mean square values for both drag and lift coefficients compared to the upstream cylinder. Nepali et al.¹¹ investigated the vortex-induced vibration characteristics of two square cylinders arranged in tandem with two degrees of freedom at Re ≤ 120. They reported different vibration characteristics, such as the absence of galloping for specific mass ratios and the presence of different vortex-shedding modes based on vibration amplitudes. Additionally, the effect of gap spacing between cylinders has been extensively discussed, with critical gap spacing ratios identified for tandem cylinder pairs. Rajpoot et al.¹² examined the unsteady flow characteristics of tandem square cylinders near a moving wall at low Re. Their research provided detailed information on how the vortex shedding behind the cylinders was affected by different G/D and L/D values. They found that the drag coefficient values were highest for the single cylinder case, followed by the upstream cylinder in the tandem arrangement, and were lowest for the downstream cylinder. Shui et al.¹³ studied numerical simulations of flow around two square cylinders in tandem arrangement at Re = 100. They explored the flow structures in the far-wake regions behind the downstream square cylinders and identified their formation mechanisms. Their research indicates six distinct flow regimes, including two-layered vortex formation (TVF) and secondary vortex formation (SVF) modes. Hosseini et al.¹⁴ performed 2D simulations for flow past a row of cylinders in tandem arrangement with varying gap spacing from 1.1 to 10 and for Re ≤ 200. They identified three main regimes based on the flow behavior in the gap between the two most upstream cylinders. Their findings contribute to understanding flow characteristics in cylinder arrays and provide insights into the effects of pitch and Reynolds number on flow behavior. Pang et al.¹⁵ investigated the hydrodynamics of flow around two circular cylinders arranged in a side-by-side (SBS) configuration employing a two-dimensional pure-Lagrangian vortex method based on the instantaneous vorticity conserved boundary conditions (IVCBCs). The main objective of their study was to investigate the hydrodynamic coefficients and flow patterns for different 1.1 ≤ G/D ≤ 7 at Re = 6 × 10⁴. Through the investigation of biased flow, the researchers identified an intermediate frequency between narrow and wide wake frequencies that approaches the single cylinder. Singha et al.¹⁶ numerically investigated the flow characteristics across two SBS circular cylinders, describing the steady and unsteady flow regimes, wake patterns, and phase transition phenomena. The analysis is carried out for Re and G/D ranging from 20 to 160 and 0.2 to 4.0, respectively. The researchers observed two types of wakes in the steady flow regime. The first type exhibited attached vortices, like in the case of an isolated cylinder, while the second type showed detached standing vortices downstream of the cylinders.

Lee et al.¹⁷ utilized the immersed boundary approach to numerically examine the flow across parallel rectangular cylinders perpendicular to the flow direction at a fixed Re = 100. The range of gap spacing is set between 0.1 and 2. They observed four flow regimes based on the arrangement and gap spacing between the cylinders. They concluded that the flow characteristics are sensitive to the arrangement and distance between the cylinders. Rosa da Silva¹⁸ analyzed the flow dynamics across two SBS circular cylinders subjected to alternating movements at fixed Re and different gap spacings. They observed that the rotation of the cylinders influences the drag at small gap spacings. At the same time, its impact becomes negligible as the spacing between the cylinders increases.

Yang et al.¹⁹ numerically investigated the flow characteristics and vortex suppression regions for three staggered circular cylinders with two different arrangements. The simulations were performed using the immersed boundary method based on the lattice Boltzmann method at Re, which varied from 100 to 200 while maintaining fixed gap spacing between the cylinders. The findings showed that the arrangement of the cylinders had a significant impact on the wake structures and vortex shedding suppression.

Yan et al.²⁰ used the multiple-relaxation-time (MRT) based lattice Boltzmann method (LBM) to quantitatively examine the distinctive flow zones for three stationary circular cylinders arranged in a staggered configuration, with one cylinder placed in front of the others in a SBS arrangement. They conducted simulations at various gap spacing rangings from 1 to 10 and a constant Re = 200. The investigation focused on studying the interaction between the structure and the shedding of vortices behind the cylinders. They identified two distinct flow regions that they named steady and unsteady flows. Fezai et al.²¹ numerically studied the fluid flow over three staggered square cylinders at two symmetrical arrangements by varying 1 ≤ Re ≤ 180 and analyzed the effects of geometry on the flow behavior. They reported that the wake behind the cylinders exhibited unsteady periodic behavior characterized by the Strouhal number (St), which varies with the Re and the obstacle geometry. In addition, the spacing in the wake affects the pressure distribution and enhances the fluid flow velocity, with higher Re leading to stronger wake spacing. Fezai et al.²² numerically analyzed the flow characteristics around three staggered square cylinders arranged in two triangular configurations. The analysis is carried out for Re ranging from 1 to 110 to investigate three different flow regimes. The study determines the critical Reynolds number for both triangular arrangements and identifies the strong influence of the arrangement type on the bifurcation point. It is observed that both arrangements lead to a significant reduction in the value of the critical Reynolds number.

Researchers have primarily focused on circular and square cylinders by examining their works. The current study aims to contribute to understanding the flow past rectangular polygons in a staggered arrangement, thus opening new avenues for drag reduction and flow control in practical engineering applications.

Fluid motion occurring in nature can be described by the NSEs, which can cover almost 12 orders of magnitude in length in meters, from bacteria swimming and microfluidic devices for droplet generation (of order 10⁻⁶), to wind turbines, oceans, and the atmosphere (of order 10⁶). NSEs are a set of partial differential equations that essentially depict the momentum conservation of a continuum for fluids where you have a viscous term. To identify the appearance of the flow field, the velocity term appears quadratically, and complex boundary conditions are present. The nonlinear nature and presence of complex boundaries are significant limitations to solving NSEs analytically. Therefore, we have two choices, i.e., experimental methods and computational modeling. If you are interested in describing a fluid on a small scale, you would typically resort to molecular dynamics (MD) domestic simulations governed by Newton’s equations of motion. If you are interested in larger scales, let us say you want to know how fluid moves in a bucket of water where the scale may be 20–30 cm; then you would use continuum simulations, i.e., computational fluid dynamics (CFD). Between these two scales, something interesting is happening. You want to understand the statistical behavior, which can be accomplished by using the lattice Boltzmann method to deal with the mesoscopic scale.

The SRT-LBM is a particle-based method²⁷ whose algorithm is divided into two main steps: collision and streaming of particles with discrete set velocities ξ_i. These steps make up most of the algorithm. By applying the multi-scale Chapman–Enskog expansion to the standard lattice Boltzmann equation,²⁷ one can retrieve the classical (NSEs) for nearly compressible fluids in the range of low Mach numbers.^25,28

In Subsections IV A and IV B, we will analyze the problem with its grid distribution and check the validity of the numerical code to conduct the following study.

This numerical study is designed to explore the change in flow features past two rectangular polygons, R1 and R2, when different physical parameters of practical importance are varied. The geometry is sketched so that two rectangular polygons, R1 and R2, are aligned horizontally along the center line of a rectangular flow stream (see Fig. 1). Here, horizontal alignment clearly defines the difference in length and width of the polygons. The long side, which is parallel to the X-axis, is the length “L1” of the polygon, while the short side, which is parallel to the Y-axis, is the width “L2” of the polygon. When L1 > L2, the polygons are positioned in horizontal alignment because the separation region of the flow is much smaller than the re-attachment region of the flow. L1 is 40 and L2 is 20, with an aspect ratio (AR) of 2, where AR is the ratio of length to width. D indicates the size of a rectangular polygon. The “L” is the total length of the channel, and the “W” is the total width of the rectangular flow stream. The span from the inlet to the R2 polygon is called the upstream distance “L^U,” where the flow is laminar and steady. The span from the R1 polygon to the outlet is called the downstream distance “L^D,” where the flow is rotational and turbulent. The gap between the polygons is represented by “G” in general. For this study, both a gap in the X-direction “G^X” and a gap in the Y-direction “G^y” have the same value. G^XY = G/D gives the non-dimensional gap between two polygons. Now, the total length of the computing rectangular channel is L = L^U + L1 + G^X + L1 + L^D. The mesh distribution for the proposed study is presented in Table I.

The flow enters the system with a uniform inflow velocity (u = U_o, v = 0), which after the fluid–solid interaction phenomenon with two rectangular polygons, experienced no-slip boundary conditions (u = v = 0) on the walls of the polygons. Due to the viscous nature of the fluid, it gets separated. It rotates beyond the polygons and moves to the exit of the channel, where the fluid comes across the convective boundary conditions (⁠$∂u∂t+Uo∂u∂x=0$⁠), which are used to restrict the converse flow in the channel. The width of the rectangular flow stream is chosen so that the lower and upper walls have a negligent effect on the flow before and after the fluid-structure interaction, so periodic boundary conditions are implemented.

The numerical code of the lattice Boltzmann Method is already employed to study flow features along different configurations of bluff bodies, ranging from single to multiple aligned in tandem, SBS, and staggered configurations. To check the validity of the present code, simulations are conducted for an isolated square polygon and an isolated rectangular polygon at a Reynolds number equal to 150. Two parameters of practical importance, i.e., an average drag coefficient (Cdmean) and Strouhal number (St), which describe the oscillatory nature of the fluid, are calculated. The outcomes of simulations are compared with the results present in the open literature^29–33 (see Table II). The results are in good agreement with the existing experimental and numerical results. Minor discrepancies are observed, which is evident because different scientists used different experimental and numerical setups that can alter the outputs of problems like boundary conditions, channel length, width, mesh resolution, and fluid chosen. Such parameters can impact the Cdmean and St of the proposed problem. To further verify the code, simulations are performed for two square polygons arranged in SBS and a staggered arrangement at Re = 150 and G = 4. Outcomes are compared with the findings of Agarwal³⁴ (see Table II), which demonstrate that the different configurations of bluff bodies in a crossflow significantly impact the fluid’s physical characteristics. This discussion made it conclusive that the 2D lattice Boltzmann code is an accurate method to investigate the flow past two rectangular polygons arranged in a staggered arrangement.

This numerical study is proposed to investigate the flow features past two rectangular polygons placed in horizontal alignment in the 2D flow field. Two rectangular polygons are placed in a staggered arrangement. The Reynolds number is fixed at 150, while the gap between the rectangular polygons is changing (increasing) from 0 to 6. Results are presented as vortex shedding mechanisms (VSMs), interpretation of time trace history of drag and lift coefficients, and interpretation of power spectrum energy of lift coefficient. The vortex-shedding mechanisms are calculated by using the 2D vorticity equation. Mathematically, the vorticity at a point in a flow field is defined as the curl of the linear velocity vector: ω = ∇ ×v. Since the fluid is viscous, two important fluid forces, drag and lift, appear. As the interaction between the fluid and the solid takes place, the drag force {F^D} and the lift force {F^L} along their respective coefficients (Cd = 2F^D/ρU_o²D2 and Cl = 2F^L/ρU_o²D2, where Cd is a drag coefficient and Cl is a lift coefficient) are calculated by using the momentum exchange method.³⁵ Finally, the power spectrum is evaluated by using the Fast Fourier transform.

Before exploring the flow features around two staggered rectangular polygons, we investigated the flow interacting with the square and rectangular polygons at Re = 150. Results in the form of vortex snapshots, time histories of Cd and Cl plots, and power spectrum plots are presented in Figs. 2(a)–2(h).

The shedding behavior of vortices is clearly seen in Figs. 2(a) and 2(b), where both square and rectangular polygons are shedding vortices in an alternate fashion. The left-side top corner of polygons is shedding a negative vortex (represented by solid lines), while the left-side bottom corner is shedding a positive vortex (represented by dashed lines). These vortices move vertically and in an alternate fashion in the far domain creating a regular pattern called von Kármán vortex street. This flow regime is specific to a particular set of velocities used because that restricts the range of the Reynolds number used.

The time-history plots of Cd and Cl [see Figs. 2(c)–2(f)] show the periodic behavior confirming the alternate shedding of vortices. It is observed that the time of Cl is always twice that of Cd. The spectrum plots clearly show the presence of a single peak for both square and rectangular polygons. This highest peak is called single cylinder interaction frequency (SCIF), which ensures the oscillatory nature of the flow after FSI [see Figs. 2(g) and 2(h)]. The laminar flow exposed to square and rectangular polygons has St values of 0.1608 and 0.1630, respectively.

In Subsections V A–V G, we are going to investigate distinct flow features that appear for flow past two rectangular polygons placed horizontally in a staggered configuration. The main aim of this study is to discuss the effect of flow separation and re-attachment behavior after interaction.

Solo bluff body flow features appear at Re = 150 and at G = 0, 0.1, and 0.2, respectively. Only selected cases, i.e., (Re, G) = (150, 0) and (150, 0.2), are discussed in detail [see Figs. 3(a)–3(j)]. Due to the staggered alignment of the rectangular polygons, there is always a time lapse between the FSI phenomena. Fluid, which is moving in a uniform and laminar fashion, first interacts with R2 and gets separated in the form of upper and lower shear layers. However, due to the viscous nature of the fluid, it sticks to the lateral walls of the first polygons. These separated shear layers move to the second polygon, where we see no flow separation from the upper corner of R1, but the lower corner shows separation because the edge of the rectangular can separate the flow [see Figs. 3(a) and 3(b)]. No jet flow is observed since there is no gap between R1 and R2. In the far domain of the channel, we observed spike-shaped positive vortices and oval-shaped negative vortices moving in an alternate trend, generating a regular von Kármán vortex street. This flow is exactly a copy of single bluff body flow. Here, two polygons are acting as a single body with an irregular shape, even at a small gap of G = 0.2. The irregularity in the shape has affected the shape and movement of vortices, as the vortex street is very much deflected to the bottom boundary. However, when the gap is increased from 0 to 0.2, the spiky nature, size of the vortex, and deflection of the vortex street increase due to the presence of jet flow between R1 and R2 [see Fig. 3(b)]. Plots of the time-history of drag show a sinusoidal nature for R2, while a modulated wave is observed for R1. This happens because the flow is already disturbed when it interacts with the second polygon [see Figs. 3(c)–3(f)]. The time-history plots of lift are periodic for R1 and R2, respectively. The oscillation frequency for both drag coefficients and lift coefficients is the same. This discovery is consistent with the data suggesting that the presence of the highest peak in lift is linked to the maximum amount of drag. The phenomena can be further discussed through the power spectrum analysis of Cl1 and Cl2, as depicted in Figs. 3(g)–3(j). When the values of (Re, G) are set to (150, 0.0), the interaction frequency between the single cylinder and both polygons is monitored. The peak has the highest values for R1 and small values for R2 due to the time-lapse of flow interaction. The flow when exposed to two horizontally placed staggered rectangular polygons has St values of (Re, G, St1, St2) = (150, 0.0, 0.1009, 0.1006) and (Re, G, St1, St2) = (1, 0.0.098, 0.0.0977), respectively. Similar flow was also observed by Sumner et al.³⁶ for a pair of circular polygons placed in a staggered configuration at Re = 900, angle of attack (α) = 10°, and gap spacing of 1.0, respectively.

Distorted solo bluff body flow appears at Re = 150 and G = 0.5, 0.75, and 1, respectively. Only selected cases, i.e., (Re, G) = (150, 0.5) and (150, 1.0), are discussed in detail [see Figs. 4(a)–4(j)]. It is observed that there is enough gap between R1 and R2 so that flow can pass between the two polygons. Flow separation is independently occurring in both polygons. However, due to small gaps, shear layers separating the lower corner of R2 and the upper corner of R1 are mixed. At G = 0.5, a negative vortex sheds at 300D, while at G = 1, the shear layer grows, and shedding is observed at 400D in the far domain. We observe two positive vortices damaging the alternate fashion of shedding. It is also observed that vortices are distorting at the end of the channel, so this flow is called distorted solo bluff body flow. Initially, the flow in the near domain shows SBBF, but in the far domain, it shows disturbed behavior. In addition, the size, shape, and movement of vortices are disturbed. Vortices are oval-shaped, circular-shaped spikes shaped positioned both vertically and laterally, making the flow distorted. Time-history plots of drag show a sinusoidal nature for G = 1, while a modulated wave is observed for R1 at G = 0.5. Cd1 is modulated, and Cd2 is sinusoidal at G = 0.5 but has the same range of drag coefficient. Modulated waves appear due to the small gap and presence of jet flow [see Figs. 4(c)–4(f)]. The time-history plots of lift are periodic for both polygons at G = 0.5 and 1, respectively. The drag and lift coefficients both show anti-phase behavior. It is observed that the single cylinder interaction frequency occurs for both polygons when the values of (Re, G) are set to (150, 0.1) [see Figs. 4(g)–4(j)]. The presence of these peaks provides evidence that a relatively tiny gap has a considerable impact on flow because of jet flow, which influences flow and redirects the wake toward the lower boundary. The flow when exposed to two horizontally placed staggered rectangular polygons has St values of (Re, G, St1, St2) = (150, 0.5, 00.0892, 0.0889) and (Re, G, St1, St2) = (1, 0.1009, 0.1006), respectively. Similar flow was also observed by Sumner et al.³⁶ for a pair of circular polygons placed in a staggered configuration at Re = 1270, angle of attack (α) = 50°, and gap spacing of 1.0, respectively.

Jumbled flow appears at Re = 150 and G = 1.5, respectively [see Figs. 5(a)–5(e)]. This flow is characterized by a complete loss of flow structure [see Fig. 5(a)]. Flow separation takes place between R1 and R2 in the form of upper and lower shear layers that are reattached to the horizontal surfaces of both polygons. These shear layers are growing horizontally. The lower shear layer of R2 is intermixing with the upper shear layer of R1, creating a chaotic flow in the far domain of the channel. Independent negative and positive vortices are shedding near 400D, after which we observe the mixture of these vortices. The shear layer enlargement and amalgamation effect the proper shedding of vortices, which leads to chaos. Therefore, due to this feature, this flow is called jumbled flow. The validity of the prior observations regarding the shedding pattern can be confirmed by examining the drag and lift coefficients of the R1 and R2 time trace series, respectively. Figures 5(b) and 5(c) demonstrate that both Cd1 and Cd2 exhibit modulated signals, but CL1 and CL2 display sinusoidal behaviors that fluctuate continuously with changing amplitude. The highest observed amplitude is found in Cd1, with a value of 0.745, when the values of (Re, G) are equal to (150, 1.5). Figure 5(b) illustrates a time series of the lift and drag coefficients for two cylinders that are positioned next to each other with a gap spacing of 1.5. Additionally, the lift coefficient of the second polygon is greater than that of the first polygon. The oscillation frequencies of the drag and lift coefficients are not identical during motion. This finding aligns with the observed data, indicating that the presence of a peak in lift is also associated with the highest level of drag. The power spectrum analysis of Cl1 and Cl2 provides additional insight into the occurrence [see Figs. 5(d) and 5(e)]. The rate of interactions between individual cylinders is observed for both polygons with the conditions (Re, G) = (150, 1.5). The coalescence of shear layers can be identified by the presence of two minor peaks on the right side of the single-cylinder interaction frequency for Cl1. There are baby peaks seen for Cl2, which confirms that it exhibits a single polygon behavior and supports the sinusoidal pattern of the lift coefficient shown in Figs. 5(d) and 5(e). The flow when exposed to two horizontally placed staggered rectangular polygons has St values of (Re, G, St1, St2) = (150, 1.5, 0.1243, 0.1937), respectively. Jumbled flow also appeared in the study of Burattini and Agrawal³⁷ for flow past two side-by-side square polygons at Re = 73 and G = 1.5, respectively.

In-phase jumbled flow appears at Re = 150 and G = 2, and 2.5, respectively. Only one selected case, i.e., (Re, G) = (150, 2.5), is discussed in detail [see Figs. 6(a)–6(e)]. The gap between the polygons is quite sufficient that independent shedding of vortices is observed from both R1 and R2 [see Fig. 6(a)]. Due to the rectangular nature of polygons, shear layers are reattached to the horizontal surface, which delays the shedding mechanism and makes shear layers grow horizontally. In addition, both polygons are placed in staggered alignment, and the separation phenomenon faces the time lapse. Due to these two reasons, we observed that the initial flow is separating in the near domain behind the polygons, while in the far domain, we observe intermixing and distortion of shed vortices. It is clear from the vorticity snapshot that R1 is detaching the negative vortex at 400D, while R2 is detaching the negative vortex at 300D. Now the lapse in time for shedding is causing the amalgamation of vortices, leading to jumbled flow in the channel. Therefore, this flow is named an in-phase jumbled flow because, irrespective of its staggered nature, shedding is in-phase when observed simultaneously. These results are clearly verified by the magnitude of Cd1, which is larger than that of Cd2. Based on the provided information, the amplitude of the signal varies for Cl1 for the upstream rectangular polygon, whereas for the downstream polygon, Cl2 has a sinusoidal behavior. Furthermore, the lift coefficient of the first polygon intersects that of the second polygon. The oscillation frequencies of the drag and lift coefficients align while they are in motion. This study is consistent with the observed data, suggesting a relationship between the occurrence of a maximum lift and the highest degree of drag. An analysis of the power spectrum of Cl1 and Cl2 provides further clarification regarding the reported results [see Figs. 6(d) and 6(e)]. The frequency of interactions between individual cylinders is seen for both polygons under the conditions (Re, G) = (150, 2.5). The presence of separate shedding is indicated by the tallest peaks, while merging is seen by the presence of smaller peaks on both sides of the interaction frequency for Cl1, as shown in Figs. 6(d) and 6(e). The flow when exposed to two horizontally placed staggered rectangular polygons has St values of (Re, G, St1, St2) = (150, 2.5, 0.1418, 0.1735), respectively. Aboueian and Sohankar³⁸ investigated numerically the similar flow for two square polygons at Re = 150 and at G = 1 at α = 45°, respectively. They observed that a period causes the shedding pattern to change between in-phase and anti-phase flow patterns, which results in amplitude modulation, particularly in the lift coefficient of the downstream cylinder. This occurs due to the presence of gap spacing between the cylinders.

Anti-phase jumbled flow appears at Re = 150 and G = 3, respectively [see Figs. 7(a)–7(e)]. Again, the gap between the polygons is enough for independent shedding of vortices from both [see Fig. 7(a)]. Due to the rectangular nature of polygons, shear layers are reattached to the horizontal surface, which delays the shedding mechanism and makes shear layers grow horizontally. In addition, both polygons are placed in staggered alignment, which leads to a time lapse in the separation process. It is observed that initial flow is separating in the near domain behind the polygons, while in the far domain we observe amalgamation and distortion of shed vortices. It is clear from the vorticity snapshot that R1 is shedding a positive vortex near 400D, while R2 is shedding the negative vortex at 300D. Now, the lapse in time of shedding is causing amalgamation of vortices, leading to anti-phase jumbled flow in the flow domain. Therefore, this flow is named anti-phase jumbled flow because, irrespective of its staggered nature, shedding is anti-phase when observed simultaneously. Time-history plots of both drag and lift coefficients show modulated behavior for both R1 and R2 [see Figs. 7(b) and 7(c)]. Power spectrum energy plots show multiple peaks for R1 and a single peak for R2 [see Figs. 7(d) and 7(e)], which occur due to mixing of vortices and horizontal growth of shear layers. The flow when exposed to two horizontally placed staggered rectangular polygons has St values of (Re, G, St1, St2) = (150, 3.0, 0.1447, 0.1467), respectively. Similar results were reported by Aboueian and Sohankar,³⁸ who conducted a numerical investigation of the flow between two square polygons. The study was conducted at a Reynolds number (Re) of 150 and a G value of 1, with an angle (α) of 45°. It was noted that the presence of a period leads to a shift in the shedding pattern from in-phase to anti-phase flow patterns. This shift generates an amplitude modulation, specifically in the lift coefficient of the cylinder located downstream. This phenomenon arises because of the existence of gaps between the cylinders.

In-phase two rows vortex street appears at Re = 150 and G = 5, and 6, respectively. Only one selected case is presented for discussion, i.e., (Re, G) = (150, 6) [see Figs. 8(a)–8(e)]. At G = 6, two polygons are far enough apart that one has no impact on the other. As fluid-polygon interaction takes place, independent shedding of positive and negative vortices is observed. Two von Kármán vortex streets are observed in the far domain of the channel [see Fig. 8(a)]. Both polygons shed negative vortices simultaneously, leading to in-phase behavior. Due to these flow features, this type of flow is called in-phase two rows vortex street flow. Initially, both negative and positive vortices are spike-shaped, but as flow moves in the channel, they become oval and round.

Examining the time series of drag and lift coefficients for R1 and R2 can provide more evidence to support the conclusions about the shedding pattern. Figures 8(b) and 8(c) illustrate that Cd1 and Cd2 exhibit a periodic pattern. Both Cl1 and Cl2 exhibit sinusoidal patterns characterized by a consistent amplitude. The results clearly indicated that the magnitude of Cd1 is greater than that of Cd2. Furthermore, the lift coefficient of the upstream polygon exceeds that of the downstream polygon. The oscillation frequencies of the drag and lift coefficients align while they are in motion. This discovery is consistent with the observed data, suggesting a relationship between the maximum lift and the highest level of drag force.

Power spectrum energy plots show multiple peaks for R1 and a single peak for R2 [see Figs. 8(d) and 8(e)], which occurs due to the mixing of vortices and horizontal growth of shear layers. Both staggered polygons experience fluctuating forces because of the shedding mechanism, and energy lost from the flow field is converted into a regular vortex motion. A sinusoidal behavior can be seen in the time-trace analysis of lift and drag in the figure. Analysis of the power spectrum energy reveals the presence of a dominant peak, which can be attributed to periodicity in the lift signal. The power spectrum has a similar profile resembling a single polygon, providing a strong indication of the influence of von Kármán Street on the surrounding area. The separation is sufficiently large to ensure that the wake of one polygon does not impact the wake of another. The flow when exposed to two horizontally placed staggered rectangular polygons has St values of (Re, G, St1, St2) = (150, 6.0, 0.1506, 0.156), respectively. Aboueian and Sohankar³⁸ numerically investigated the similar flow for two square polygons at Re = 150 and at G = 4, 5, and 6 at α = 45°, respectively.

Anti-phase two rows vortex street appears at Re = 150 and G = 3.5, 4, 4.5, and 5.5, respectively. Only one selected case, i.e., (Re, G) = (150, 5.5), is discussed in detail [see Figs. 9(a)–9(e)]. At G = 5.5, one polygon has no impact on the other, due to which independent shedding of vortices occurs [see Fig. 9(a)]. Since the flow is named anti-phase, it is observed that R1 is shedding a negative vortex while R2 is shedding a positive vortex simultaneously. Two independent von Kármán vortex streets are observed in the far domain. Initially, the shed vortices are of a pointed nature, but as the flow develops at the far end of the computational domain, it becomes oval and rounded in shape. Due to these flow features, this flow is named anti-phase two rows vortex street flow. The validity of the prior observations regarding the shedding pattern can be confirmed by examining the drag and lift coefficients of the R1 and R2 time series plots, respectively. Figures 9(b) and 9(c) demonstrate that both Cd1 and Cd2 exhibit modulated signals, whereas Cl1 and Cl2 display sinusoidal behaviors that continually vary in amplitude. The highest observed amplitude is found in Cd1. Figures 9(b) and 9(c) illustrate a time series of the lift and drag coefficients for two polygons that are positioned next to each other with a gap spacing of less than 5.5. Cd1 has been demonstrated to possess a greater magnitude than Cd2. Additionally, the lift coefficient of the first polygon is greater than that of the second polygon. The oscillation frequencies of the drag and lift coefficients are not identical during motion. This finding aligns with the observed data, indicating that the presence of a peak in lift is also connected to the highest level of drag. The power spectrum analysis of Cl1 and Cl2 provides additional insight. The occurrence rate of interactions between single cylinders is observed for both polygons with the conditions (Re, G) = (150, 5.5). The convergence of shear layers can be identified by the presence of two smaller peaks on the right side of the single-cylinder interaction frequency for Cl1. There are no baby peaks seen for CL2, which confirms that it exhibits a single polygon behavior. This further supports the sinusoidal behavior of the lift coefficient, as shown in Figs. 9(d) and 9(e). The flow when exposed to two horizontally placed staggered rectangular polygons has St values os (Re, G, St1, St2) = (150, 5.5, 0.1506, 0.1589), respectively. Similar flow also appeared in the study of Burattini and Agrawal³⁷ for flow past two side-by-side square polygons at Re = 73 and G = 6, respectively.

This section discusses the variation of different physical entities, like Cdmean, Cdrms, Clrms, and St, with gap spacing G at a fixed value of Re = 150. The results of this discussion are presented in Figs. 10(a)–10(d). Here, Cdmean1 and Cdmean2 are the mean drag coefficients, Cdrms1 and Cdrms2 are the root mean square values of drag coefficients, Clrms1 and Clrms2 are the root mean square values of lift coefficients, and St1 and St2 are the Strouhal values of two horizontally placed rectangular polygons.

Figure 10(a) shows that Cdmean1 is increasing until G = 0.75. After that, it shows a decreasing behavior until G = 6. On the other hand, Cdmean2 has its maximum value at G = 0.1, after which we observe a decreasing trend until G = 6. The possible variation in Cdmean1 and Cdmean2 is observed at small gap spacings between G = 0–2. After that, the results are like isolated rectangular polygons for both Cdmean1 and Cdmean2, respectively. This outcome shows that Cdmean is approaching an isolated rectangular polygon with moderate and extensive gaps. When a comparison is made with an isolated square polygon, the Cdmean of a pair of two staggered polygons and the Cdmean of the isolated rectangular polygon are far less than that of the square polygon at any gap spacing. The minimum value of the mean drag coefficient is observed for Cdmean1 at G = 0.

Cdrms and Clrms of both upstream and downstream polygons show similar behavior for all gap spacings [see Figs. 10(b) and 10(c)]. The maximum value of Cdrms1 and Cdrms2 is observed at G = 0, and that of Clrms1 and Clrms2 at G = 0. The minimum value of Cdrms for both polygons occurs at G = 6. The overall trend of Cdrms and Clrms increases and decreases with increasing gap spacing between the polygons. The values of Cdrm1, Cdrms2, Clrms1, and Clrms2 are sandwiched between the isolated square and rectangular polygons with the highest and lowest polygons. The values of Clrms of both polygons are approaching the values of a single rectangular polygon at large gap spacings [see Fig. 10(c)].

Based on the lift coefficient, the Fast Fourier transform can be used to determine the Strouhal number, which is a measure that describes the behavior of the oscillatory flow. A power spectrum is a signal that describes the energy distribution in terms of frequency units. It is a statistical average of the signal’s frequency used to calculate the power spectrum’s energy. The highest peak, the single cylinder interaction frequency, is displayed against all gap spacings in Fig. 10(d). The most prominent peak was observed at G = 1.5 for the upstream cylinder R2, while at G = 0.5, the lowest value for St2 was observed. After G = 1.5, St1 shows an increasing behavior, while St2 shows a decreasing behavior until G = 6. It is also observed that Strouhal’s number of square, rectangular polygons is approached by two rectangular polygons placed horizontally in a staggered arrangement at large gap spacings. When there are considerable gaps, the Strouhal number does not change. At large gap spacings, all the physical entities, such as the Cdmean, Cdrms, Clrms, and St of both staggered cylinders, are getting very close to the values of the single cylinder Cdmean, Cdrms, Clrms, and St.

A two-dimensional numerical study is conducted for flow past two rectangular polygons placed in horizontal and staggered alignment. The Reynolds number is fixed at 150, while the gap between the polygons varies from 0 to 6. Simulations are carried out using SRT-LBM.

• Seven flow regimes are identified and named solo bluff body flow, distorted solo bluff body flow, jumbled flow, in-phase jumbled flow, anti-phase jumbled flow, in-phase two rows vortex street flow, and anti-phase two rows vortex street flow.

• It is observed that the shape of the bluff body has the ability to alter the course of the flow field.

• In addition, when two rectangular polygons are placed in very close proximity, the gap between the polygons has a significant impact on the flow features, which leads to the development of seven distinct flow regimes with their own unique features.

• The horizontal alignment of polygons restricts the immediate shedding rather than favoring the re-attachment and growth of shear layers.

• Time histories of drag and lift coefficients move between modulated and sinusoidal behavior for different gaps and among the same gaps for different polygons.

• Power spectrum energy shows single or multiple peaks, ensuring flow oscillations.

• Cdmean, Clrms, and St of two horizontal polygons approach the single rectangular polygon data at large gap spacings as compared to a single square polygon.

The authors have no conflicts to disclose.

Farheen Gul: Data curation (equal); Formal analysis (equal). Ghazala Nazeer: Writing – original draft (equal); Writing – review & editing (equal). Madiha Sana: Data curation (equal). Sehrish Hassan Shigri: Investigation (equal). Shams UlIslam: Conceptualization (equal).

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