Computational fluid dynamics (CFD) is an interesting, complex, state-of-the-art field due to the rapid development of computer technology and newly developed numerical techniques and programming languages. One of the most significant aspects of CFD is the analysis of fluid–solid interactions and resulting outcomes. Much research has examined fluid flow around bluff bodies (commonly termed “cylinders”). The study of the wake structure mechanism behind cylindrical objects is important to many practical applications in various engineering disciplines. For example, such cylindrical objects appear in most civil designs, such as high-rise buildings, chimneys, suspension bridges, and many internal and external supporting components of such structures. Electronic chips are mostly rectangular/square in cross section, thus resembling rectangular cylinders. The flow mechanism and characteristics behind these objects depend on parameters such as the blockage ratio (B), Reynolds number (Re), inflow velocities, shape, size, arrangement, and quantity of structures in the flow field.

Many numerical and experimental studies relevant to circular and rectangular/square cylindrical structure flow characteristics are apparent from the literature. Mittal and Raghuvanshi (2001) conducted numerical analysis of vortex shedding behind circular cylinders for laminar flows by considering Re ranging from 60 to 100. They found that, as the Re increases the average drag coefficient (CD _mean ) decreases while the Strouhal number (St) increases. Belloli et al. (2014) experimentally analyzed flow around circular cylinders at high Re up to 6 × 10⁵ and found that drag was approximately 0.85 for the flat surface conditions at Re = 10⁵ and dropped to 0.3 at Re = 2.5 × 10⁵. Kuzmina and Marchevsky (2021) simulated flow interactions with circular cylinders by considering Re ranging from 20 to 200. The two flow regimes they observed were i) stable, where the flow was symmetric, and ii) vortex shedding, where they observed the von Karman vortex street. Furthermore, the separation point changed with a variation in Re. Comparing the fluid flow features, it is apparent that the circular cylinder displayed an unfixed boundary layer detachment, but the square/rectangular cylinder displayed a stable detachment point only from its edges. Due to this detachment phenomenon, the flow modes and wake structure mechanism differ in both geometries. Zhang and Zhang (2012) analyzed low-Re flow past a square cylinder that ranged from 25 to 150. They found two stable symmetric vortices behind the cylinder at Re = 25 and 50, indicating steady flow. They also found that pressure gradually rose while moving toward the cylinder from the inlet and dropped in the down-wake area. Perumal et al. (2012) studied the flow around a single square cylinder by considering Re ranging from 4 to 150. They found laminar, steady, and slightly separated flow from the cylinder at extremely low Re. At higher Re, the flow split into a pair of symmetrical vortices about the channel central line. They also observed that the flow appeared uniform at low B while, for moderated B = 10, an instability developed within the flow field for Re = 51. Ahmad et al. (2021) analyzed the flow characteristics around a rectangular cylinder at low aspect ratios (AR) and Re. They found that, for AR = 0.05, the vortex formed instantly behind the back surface of the cylinder. They also reported the sinusoidal nature of drag coefficient (CD) for all AR. Islam et al. (2012) conducted numerical simulations of rectangular cylinder flows with distinct aspect ratios by considering Re = 100, 150, 200, and 250 and AR varying from 0.15 to 4. They observed that the results for AR = 0.15 at Re = 100 indicated dual parallel rows of clockwise and anticlockwise vortices. At AR = 0.5, the vortices appeared at the top and bottom sides of the cylinder for a shorter duration. Octavianty et al. (2016) experimentally studied the radiation of sound and flow structure around a rectangular cylinder at various Re and AR with Mach numbers ((Ma) below 0.16. According to this research, the vortex formation region was extremely close to the back surface of the cylinder where the maximum spectrum line (SPL) occurred.

In the case of multiple obstacles, gap spacing (G) is another important parameter which significantly affects flow characteristics. Several studies have examined the combined effects of Re and G on fluid flows around two inline circular or square or rectangular cylinders (Shiraishi et al., 1986; Su et al., 2002; Su et al., 2004; Kuo et al., 2008; Huang et al., 2012; Mithun and Tiwari, 2014; Gnatowska et al., 2020; Rajpoot et al., 2021; Shui et al., 2021; Wang et al., 2022). Important findings of these studies are highlighted in Table 1. From fluid–solid interaction analyses in the literature, it is well-known that bluff bodies of similar dimensions, if arranged differently, such as side-by-side, staggered, or tandem to the incoming flow, have significant differences in fluid force behavior as well as near wakes formed behind them. Chakraborty et al. (2022) analyzed the influence of gap spacing on flow past two circular cylinders placed side by side by considering Re = 5 × 10⁵ and G = 2 to 14. They observed that the pressure at the front of both cylinders was greater than the pressure on the back, resulting in a positive drag force for all G. Sarvghad et al. (2011) also conducted a numerical simulation of flow over similarly arranged cylinders by considering Re = 100 and 200 and 1.5 ≤ G ≤ 4. They categorized the flow in different patterns they termed biased flow, flip-flopping, synchronized anti-phase vortex shedding, and in-phase asymmetric. Adeeb et al. (2018) observed the flow around two side-by-side square cylinders by considering Re = 100, G = 1.5 to 5, and corner radii (R/D) = 0 to 0.5. They observed that, for 0 ≤ R/D ≤ 0.5, at G ≤ 1.5, the wake of both cylinders joined and functioned as an isolated wake structure for all R/D choices. Ma et al. (2017) analyzed the wake of two side-by-side square cylinders at Re in the range of 16–200 and G = 0 to 10. It was found that, when Re was low, the wake flow after both cylinders was steady state. When Re increased, the recirculation bubbles appeared within the downstream region attached with cylinders. These bubbles grew as unstable modes and dominated the steady mode. The flow became transient and vortex shedding was then observed. Burattini and Agrawal (2013) observed the interaction of wakes of two square cylinders placed side by side by considering Re = 73 and G = 0.5 to 6. It was reported that, with changing G, vortex shedding was much affected. Lee et al. (2019) reported an isolated bluff-body flow pattern for G ≤ 0.5, an asymmetric wake flow pattern for 0.75 ≤ G ≤ 1.25, a transitional flow pattern for 1.5 ≤ G ≤ 1.75, and a dual street flow pattern for G ≥ 2 for flow-past rectangular cylinders arranged side by side. Fluid flow-past staggered cylinders strongly depended on G and inclination angle ( α ) between cylinders. Ye et al. (2019) observed flow induced vibrations in the case of two circular cylinders in a staggered configuration at Re = 200, G = 5 to 7 with α = 0 ° to 90 ° . They reported that, at G = 5, the vortices’ generation and attachment with the second cylinder was much influenced by an inclined angle of 0 ° to 35 ° ; however, for an inclined angle beyond these values, the flow structure was not much varied. Aboueian and Sohankar (2017) numerically analyzed the shedding frequency in the case of flow past two staggered square cylinders by considering Re = 150, G = 0.1 to 6, and α = 45 ° . They reported that the variations in gap spacing are directly related to the shedding frequencies. According to Alam et al. (2016), the wake flow around two staggered square cylinders is very dependent on the angle of inclination between cylinders. Small angles greatly impact flow characteristics, while larger angles more weakly influence flow characteristics.

When multiple objects (more than two) appear in the fluid flow stream, the resulting forces and flow structure may significantly differ from those seen in case of two or a single body even at the same Reynolds number or gap spacings. Alam et al. (2017) observed the flow around three tandem circular cylinders by considering Re = 200, gap spacing between first and second cylinder (G ₁ ) = 3.5 to 5.25, and gap spacing between the second and third cylinder (G ₂ ) in the range 3.6–5.5. They categorized the flow structures as in-phase, antiphase, and intermediate, depending on G ₁ and G ₂. Eizadi et al. (2022) analyzed the transitions in the wake of six circular obstacles placed inline at Re = 40 to 180 and G = 0.5 to 18. They concluded that wake transitions of multiple cylinders depend not only on Re but also on G. Song et al. (2017) observed the flow patterns and force variations over four inline square cylinders by considering Re = 300 and G = 1.5 to 8. They reported that, when G < 3.5, CD _mean values for the downstream cylinders increased sharply with increasing G. For G > 3.5, the CD _mean values of the downstream cylinders decreased gradually with increasing G. Islam et al. (2018) investigated different aspect ratio effects on the flow around three inline cylinders at Re = 150, AR = 0.25 to 3, and G = 0.5 to 7. They observed that, at all G, the CL _rms of all cylinders decreased with increasing AR. Rahman et al. (2021) numerically computed the forces on three rectangular cylinders by considering 60 ≤ Re ≤ 180, AR = 0.25 to 4, and G = 1.5. They reported that the shedding frequencies of all cylinders increased while CD _mean decreased due to the increase in AR at all Re.

Studies describing the fluid flow around other geometries like airfoils have also been conducted (Bajalan et al., 2011; Mirzaei et al., 2012; Rangan and Santanu Ghosh, 2022; Abdolahipour, 2023). Another aspect of fluid–solid interactions is flow control around bluff bodies. Utilizing flow control strategies, efficient devices have been designed to save energy by minimizing flow-induced forces and controlling wake flow structures. Among various available flow controlling strategies, flow control through modulated pulse jet (Abdolahipour et al., 2021; Abdolahipour et al., 2022) and plasma actuator (Salmasi et al., 2013; Mohammadi and Taleghani, 2014; Taleghani et al., 2018) are frequently studied by researchers.

It can be concluded from this literature that the flow characteristics of multiple bodies are influenced by many parameters, including Re, G, B, AR, and α . Among these, fluid flows around rectangular bodies have been less investigated, especially vertically mounted bodies. Information about cases when both cylinders have different AR is very rare. From an application point of view, bluff bodies with rectangular cross sections are particularly important because, for example, most civil structures have rectangular cross sections and are vertically assembled. Electronic devices mostly have rectangular cross-sectional internal components. Flow around such bodies differs from flow around circular/square cross-sectional bodies. Hence, the current study will explore the fluid flow characteristics around two tandem rectangular bodies with different aspect ratios. The main focus will be on the effects of aspect ratios and varying gap spacing between cylinders on vorticity patterns, pressure variations, streamline behavior, and variations in fluid forces behavior. This study will also enhance our understanding of the dependence of the flow characteristics of both bodies on each other. Furthermore, it will help in designing flow control strategies for similar bodies placed in cross flows.

The current analysis utilizes the well-known numerical methodology, the lattice Boltzmann method (LBM). The simplified nature, easy implementation, and accuracy features of LBM make it more suitable than conventional methods for simulating fluid dynamics problems (Mohammad, 2011). This method involving two main steps, streaming and collision, has several advantages over the Navier–Stokes (NS) solvers. It has an explicit nature with conditional stability conditions (Chen and Doolen, 1998). The nonlinearity appearance in the case of NS equations does not appear in this method because the Boltzmann equation (BE) is quasi-linear. Pressure can be obtained through a simple procedure from the equation of state instead of dealing with the Laplace equation in each time step. LBM contains a variety of discrete models for simulating fluid flows. The current study is based upon the well-known two-dimensional nine-velocity directions (D2Q9, D indicates dimensions and Q the number of velocity directions) model (Figure 1) (Sukop and Throne, 2006).

The discrete BE along a specified direction is ∂ f ∂ t + e i ∇ f i = 1 τ f i e q − f i , (1) where t = time, f i = distribution function, τ = parameter for relaxation time, and f i e q = the equilibrium distribution function.

After discretization through finite differencing, the lattice Boltzmann equation (LBE) takes the following form: f i x + ∆ x , t + ∆ t = − ω f i x , t − f i e q x , t , (2) where ω = ∆ t τ , τ is the relaxation time, and ∆ t = ∆ x = 1 (Wolf-Gladrow, 2000).

For the current study, the following form of f i e q is considered: f i e q = ρ w i 1 + 3 e i . u − 3 2 u 2 + 9 4 e i . u 2 , (3) where ρ is density, u is fluid velocity, and w i are weighting coefficients (see Table 2).

Here, w ₀ are weight-associated with rest particle, w _a are weights for particles moving along the axis, and w _d are weights for particles moving diagonally.

Density and velocity are expressed in terms of f i as ρ = ∑ i = 0 i = n f i , (4) ρ u = ∑ i = 0 i = n e i f i . (5)

The fluid kinematic viscosity is expressed as ν = 2 τ − 1 6 . (6)

The values of e i are given as e i = 0 , 0 , i = 0 , c − ⁡ cos i π 2 , − ⁡ sin i π 2 , i = 1   t o   4 , 2 c − ⁡ cos 2 i + 1 π 2 , − ⁡ sin 2 i + 1 π   2 ,   i = 5   t o   8 .   (7)

For some recent developments of LBM schemes regarding applications in different fields, readers are referred to Noori et al. (2019) and Noori et al. (2020).

The problem’s schematic diagram considered in this study is shown in Figure 2. This figure depicts a channel containing the two tandem rectangular cylinders placed in a fluid stream to be analyzed here. The height of the first cylinder (C ₁ ) is h ₁ and of the second cylinder (C ₂ ) is h ₂ , and the width of both cylinders is denoted by D. In this study, h ₁ and h ₂ are discretized so as to have 40 and 30 lattices, respectively, while the width D has 20 lattices. Each cylinder is of a different aspect ratio (AR = height of cylinder/width of cylinder): the first cylinder has AR = 2:1 while the second has AR = 3:2. X _u = 10D is the distance from the channel entrance position to C ₁ , and X _d = 20D is the distance from C ₂ to the domain outlet. The distance from the upper surface of C ₁ to the upper boundary is Y _u = 8D, while Y _b = 8D is the distance from the lower surface of C1 to the lower boundary of the domain. These lengths are selected based on the recommendations of previous research in order to have a minimal effect of domain size on results (Perumal et al., 2012). Channel length L varies as the gap between cylinders (G = s/D) varies, while the height H is fixed. All lengths in this study are non-dimensionalized using the characteristic length (D = 20).

The boundary conditions in this study are applied in terms of distribution functions. At the inlet position, uniform incoming flow ( u = U i n , v = 0 ) is assumed (Islam et al., 2012). The convective boundary condition in terms of distribution functions given as ∂ f i ∂ t + U i n ∂ f i ∂ x = 0 is used at the outlet boundary (Ahmad et al., 2021). The surface of each cylinder as well as the upper and lower walls of the domain is treated through the no-slip boundary condition, mathematically expressed as ( u = v = 0 ). In LBM simulations, the no-slip condition is applied in terms of the bounce back rule in which the particles which stream into the wall are bounced back in the opposite direction inside the fluid stream. For example, the particles with distribution functions f ₇ , f ₄ , and f ₈ streaming toward the lower wall bounce back as f ₇ = f ₅ , f ₄ = f ₂ , and f ₈ = f ₆ . A similar procedure was also adopted for other directions for fluid particles hitting the solid walls (Rahman et al., 2021). Although the geometry considered in the current study has smooth boundaries, LBM is also a suitable choice for curved or moving boundaries (Tao et al., 2018; Marson et al., 2021).

The flow around two tandem rectangular cylinders with distinct aspect ratios was simulated at Re = 100 by considering different values of G progressively varying from 0.25 to 20. From previous studies, we can conclude that the flow structure mechanism around multiple rectangular cylinders appears to be a complicated phenomenon that depends on several parameters, including Re, G, and AR (Islam et al., 2018; Rahman et al., 2021). In the current study, the resulting flow regimes are specified into different patterns in terms of the creation of different shape vortices, wake structure mechanism, and the behavior of shear layers detaching from cylinder corners and interacting with each other. These different flow patterns depend on increasing values of G in this study. The prominent flow patterns observed in this study are the single slender body (SSB) flow found in the range G = 0.25–0.75, the shear layer reattachment (SLR) flow found in the range G = 1–4, the intermittent shedding (IS) flow mode found in the range G = 4.25–5.25, and the binary vortex street (BVS) flow found for G = 5.5–20, except at G = 8 where the single-row vortex street (SRVS) flow was observed. Similar flow patterns have been reported by Zdravkovich (1987) for flow around tandem bodies with different characteristics. In the following subsections, a comprehensive illustration for each flow pattern is presented and discussed.

Variations of various fluid force parameters acting on both cylinders with varying gap spacing is presented in this section in order to analyze the influence of G on the forces. The parameters considered for this purpose are CD _mean , St, CD _rms , CL _rms , amplitude of the drag coefficient (CD _amp ), and amplitude of the lift coefficient (CL _amp ). Figure 8 presents the effect of G on variations of these parameters at Re = 100.

The variation of CD _mean for flow around two tandem rectangular cylinders with increasing G is presented in Figure 8A. It can be observed that CD _mean of C ₁ is greater than CD _mean of C ₂ for all chosen values of G because the incoming flow initially interacts with C ₁ and thus exerts maximum drag force on C ₁ . Another possible reason that C ₁ experiences significantly higher drag force than C ₂ is that it has higher AR. Relating the variations of drag force coefficient, it is apparent that the change in flow patterns significantly affects the drag force. In the range G = 0.25 to 1, the average drag on C ₁ slightly increases and then shows decreasing behavior until G = 4. It then increases with increasing G and approaches its maximum value at G = 20. Note that, at G = 1, the flow pattern changes from SSB to SLR, while after G = 4, the flow pattern changes from SLR to IS. The CD _mean of C ₂ is negative and initially decreases until G = 4. Huang et al. (2012) also observed the negative value of CD _mean for C ₂ for flowing past 2:1 rectangular cylinders in tandem at Re = 200. After G = 4, it jumps from negative to positive values due to a change in flow pattern from SLR to IS. The negative values of CD _mean indicate that the drag force acts as a thrust force, thus generating a backflow due to narrow gaps between cylinders. The minimum value of CD _mean for both cylinders can be seen at G = 4, while the maximum for C ₂ occurs at G = 8, and CD _mean of C ₁ is maximum at G = 20. Figure 8A also shows that, after G = 10, the influence of gap spacing on CD _mean of both cylinders decreases. Figure 8B presents the variation of St of both cylinders with G. Both cylinders have same St for all G, indicating that the vortices shed from both cylinders with same frequency notwithstanding whether both cylinders have different ARs. Rahman et al. (2021) also found similar behavior in St for flow past three rectangular cylinders. Initially for G = 0.25 to 0.75, St shows increasing behavior. SSB flow was observed in this range of gap spacing. In the range G = 1.75 to 4, it shows decreasing behavior, which indicates that the shedding frequency decreases due to the push of shear layers inside the gaps between cylinders. After that it increases with increasing G and approaches the local maximum value for both cylinders at G = 14. The minimum value of St can be observed at G = 4 for both cylinders where the SLR flow pattern was seen. The variation of CD _rms of both cylinders at different values of G is presented in Figure 8C. Initially, when G = 0.25 to 1, the CD _rms of C ₁ slightly increased and then showed decreasing behavior in the range G = 1.25 to 1.75 where the SLR flow is observed. The CD _rms of C ₂ show increasing behavior in the range G = 0.25 to 1.75. The CD _rms of both cylinders jump to higher values at G = 2 and then again show decreasing behavior until G = 4. After G = 4, it jumps to higher values due to a change in flow pattern from SLR to IS. The CD _rms of both cylinders show a mix of increasing and decreasing behaviors as G increases further. The minimum value of CD _rms for both cylinders can be noticed at G = 4, while the maximum value of CD _rms of C ₁ appears at G = 18, and CD _rms of C ₂ is maximum at G = 8 where the SRVS flow pattern was reported. The effect of varying G on CL _rms of both cylinders is shown in Figure 8D. This graph depicts that, in the range G = 0.25 to 4, the CL _rms of C ₁ shows decreasing behavior. After that, it increases with increasing G and thus approaches its local maximum value at G = 6. The CL _rms of C ₂ shows decreasing behavior initially in the SSB flow pattern regime and, after that, slightly increases and then shows decreasing behavior until G = 4. The CL _rms curves then show an increasing trend to higher values for both cylinders. Note that the IS flow pattern is observed in this range. At G = 4.5 to 7.75, the value of CL _rms of C ₂ increases and then suddenly decreases, which can be observed at G = 8 because of flow pattern change from BVS to SRVS flow. The variation of CD _amp for both cylinders with varying G is shown in Figure 8E. It can be observed that, until G = 4, CD _amp for both cylinders is almost constant, indicating little change in amplitude of drag force in this spacing range. After G = 4, CD _amp jumps to the highest values for both cylinders due to the appearance of the IS flow pattern. Note that, in this flow pattern, the vortices appear to be in an irregular pattern in the wake of both cylinders. This graph also shows that the higher spacing values result in higher amplitude drag force than the smaller spacing values. The variation of CL _amp of both cylinders shown in Figure 8F indicates that the C ₂ bears higher amplitude than C ₁ at almost all spacing values. This is due to the wake interference effects of C ₁ on C ₂ . Shui et al. (2021) also reported similar results in CL _amp for flow around two tandem square cylinders. Initially, CL _amp corresponding to C ₁ decreased until G = 4 but, after G = 4, it suddenly increased to a higher value and became almost independent of G, showing negligible modifications. In the range G = 0.25 to 0.75, the CL _amp of C ₂ shows decreasing behavior. After G = 4, the CL _amp of both cylinders has sudden jumps (Figure 8F). The minimum value of CL _amp of both cylinders can be observed at G = 4 where the SLR flow pattern is observed. Its maximum value for C ₁ appears at G = 4.25 and the maximum value for C ₂ at G = 9.

Numerical calculations were performed to analyze the fluid flow around two vertically positioned rectangular cylinders in tandem arrangement using the lattice Boltzmann method. The cylinders considered in this study were of different aspect ratios. The main goal of this study was to determine the wake structures under the effect of gap spacing in the range G = 0.25 to 20 at Re = 100. The results were presented and discussed in the form of vorticity contour visualizations, pressure streamline contours, variation of drag, and lift coefficient against time. Fluid force parameters of average drag coefficient, Strouhal number, rms values of drag, lift coefficients, and the amplitudes of these force coefficients were also analyzed under the impact of changing gaps between cylinders. The important findings of this study are:

(1) Bearing various characteristics, five different wake flow patterns were observed in this study depending on various ranges of gap spacings: i) single slender body, ii) shear layer reattachment, iii) intermittent shedding, iv) binary vortex street, and v) single-row vortex street.