Numerous natural processes and engineering applications involve fluid flows. These applications include high-rise structures, bridges, submarines, and chimneys, among many others. The dynamics of such flows are so complex that understanding them requires the development and implementation of efficient flow models. Researchers have reported on several experimental and computational approaches to understanding flow action under various flow conditions. The study of flow across a bluff body is mainly concerned with the analysis of flow-induced vibrations and their effects on bluff structures. In addition, the dependency of wake patterns and force characteristics on various parameters such as Reynolds numbers ( R e ) and the size, shape, orientation, and number of bodies in the flow stream is also important. Most published research relating to bluff-body flow dynamics focuses on the flow structure mechanism, vorticity behavior, velocity and pressure fluctuations, variations in fluid forces, and so on. Several experimental and numerical studies regarding the fluid flow characteristics of a single bluff body are worth mentioning [1–7]. Gera et al. [1] conducted a numerical study for two-dimensional (2 d ) unsteady flow around a square cylinder considering R e in the range from 50 to 250 to keep flow laminar. They found that the flow was steady up to R e = 50, minor unsteadiness occurred between R e = 50–55, and after this, range flow became completely unsteady. Golani and Dhiman [2] numerically investigated 2 d flow and heat transfer around a circular cylinder at R e = 50–180. Their findings showed that drag coefficient ( C D ), lift coefficient ( C L ), Strouhal number ( S t ), and vortex shedding frequency f s depended strongly on the R e . Zdravkovich [3] experimentally measured forces, in a wind tunnel, on a circular cylinder near a plane wall at a range of R e values, 4.8 × 10⁴ ≤ R e ≤ 3 × 10⁵. He examined the C D and C L and found that the C L was affected by the gap-to-diameter ratio ( S / D ), while the C D was affected by the gap-to-boundary layer thickness ( δ / D ). Park et al. [4] numerically examined flow past a circular cylinder with R e up to 160. Flow quantities including S t , C D , and C L , and base pressure around the cylinder at low R e were recorded. They noted that, as R e increased, the total drag coefficient and pressure drag coefficient C D P also increased. Saha et al. [5] investigated the transitions and chaos in the wake of a square cylinder. They observed steady flow at R e = 40, and periodicity was seen in the wake at R e > 45, while chaos was expected to occur in the range of R e = 500–600. Sohankar et al. [6] investigated 2 d and 3 d unsteady flow across a square cylinder at R e = 150–500. They observed stable 2d laminar shedding flow at R e = 150, while at R e = 200, the effects of 3 d flow appeared. Perumal et al. [7] investigated the properties of 2d flow past an elliptical cylinder. They found that at R e = 78, periodicity appeared more quickly as the cylinder was positioned closer to the entrance of the flow domain. Additionally, it was observed that f s increased with increase in the blockage ratio.

In addition to the analysis of fluid flow physics around cylinders, many researchers have used various control mechanisms to prevent vortex shedding and reduce fluid forces. It is a well-known fact that vortex shedding and flow-induced forces may result in detrimental effects on structures placed in flow streams. Therefore, through use of various strategies to control these effects, the drastic effects of fluids on solid bodies can be prevented. Flow-control methods are generally classified as either active or passive methods of control. In active control methods, flows around bluff bodies are controlled in various ways, for example, through rotation of the bluff bodies, oscillations, jet-blowing, or suction. The main objective of active control techniques is fluid flow control through the provision of external energy. Active control techniques require relatively complex procedures that supply external power to the flow. In contrast, passive flow control methods involve shape modification or the introduction of additional devices, such as thin plates or small control cylinders of different dimensions and sizes. These techniques are, therefore, energy-free and often easier to implement. Saha and Shrivastava [8] observed the suppression of vortex shedding in a case of flow past a square cylinder at R e = 100 using blowing. At various jet velocities, they found that vortex shedding vanished along with major reductions in C D . Abograis and Alshayji [9] studied the reduction of the fluid forces acting on a square cylinder in a laminar flow using a passive control method at R e = 160. They found that placing control plates at both upstream and downstream positions resulted in more reduction in the C L and C D on the cylinder as compared to placement of plates at one side only. Chen et al. [10] tested different shapes of 2 d riser fairing for the suppression of vortex-induced vibration (VIV) faced by marine risers. These included water-drop-shaped fairings and caudal fin-like fairings. They concluded that all optimal shapes played important roles in VIV suppression at some range of fluid velocity. Dey and Das [11] analyzed the reduction of forces in the case of flow around a square cylinder by attaching a triangular-shaped thorn. They observed that C D and C L were reduced by 16 % and 46%, respectively, for R e = 100, while for R e = 180, the reduction was 22 % and 60 % , respectively. Furquan and Mittal [12] performed numerical simulations for flow around two square cylinders, placed side by side, with attached flexible splitter plates at R e = 100. The effect of separation between the cylinders was also examined. It was found that greater separation between the two cylinders reduced the oscillation amplitude of the plates and increased the oscillation frequency. Ghadimi et al. [13] numerically investigated vortex shedding around a square cylinder with a detached splitter plate at R e = 100–200. They concluded that C D and f s were significantly reduced by the splitter plate and that vortex shedding was suppressed in the wake. Chauhan et al. [14] experimentally investigated flow over a square cylinder with an attached splitter plate at R e = 485. They observed a reduction in C D and S t with increasing plate length. Gallegos and Sharma [15] studied the effects of channel blockage on the dynamic behavior of a flexible plate attached to a circular cylinder. It was observed that the frequency and oscillation amplitude of the flexible plate were greatly affected by the size of the cylinder and the effects of blockages. Ali et al. [16] numerically examined the effect of the control plate on flow past a square cylinder with fixed R e = 150. They identified three flow regimes depending on the length of the control plate: the first flow regime was observed for L ≤ 1.0 D , where S t decreased with an increase in plate length; the second flow regime was observed for 1.25 D ≤ L ≤ 4.75 D , where S t first increased and then started decreasing with an increase in plate length; and the third flow regime was observed for L ≥ 5 D , where S t remained unchanged when the plate length increased. Mansy et al. [17] studied the effect of a splitter plate, placed upstream, on flow around a square cylinder. They considered different velocity ratios at R e = 56 and 200 and found that the splitter plate stabilized the incoming flow to a greater extent at low velocity ratios than at high velocity ratios. Barman and Bhattacharyya [18] studied the effect of dual splitter plates on flow around a square cylinder at R e = 150. They found that splitter plates controlled both drag force and f s . In the case of the plate placed at the upstream location, drag decreased with increased plate length; in the case of the downstream location, vortex shedding suppression was observed. Bao and Tao [19] attached dual splitter plates at the rear surface of a circular cylinder to investigate wake flow behavior. Their findings showed that the use of dual splitter plates caused higher drag reduction and powerful wake suppression at comparatively shorter plate lengths as compared to the use of a single plate. Kumar et al. [20] also investigated the control of drag force on a square cylinder using two detached splitter plates at Re = 150–200. In addition to these studies, other studies have also used new active methods such as a modulated pulse jet (MPJ) [21–23] or plasma operators [24–28] to control fluid flows around solid structures. Several further studies indicating the use of surface acoustic waves and other flow control mechanisms include [29–34].

From the above mentioned discussion and to the best of our knowledge, it can be concluded that fluid flow and wake behavior of a square cylinder with an attached splitter at the rear side have been less investigated and analyzed in the available literature. In particular, there is no corresponding work employing the lattice Boltzmann method (LBM) in the case of such geometry. Therefore, the current investigation was carried out from the perspective of these applications and limitations in the literature. More precisely, the current study emphasized five main points: 1) the development of a flat plate-based control mechanism that is more effective and useful for bluff bodies; 2) understanding of the extent to which such a plate suppresses flow-induced forces and controls the vortex shedding process; 3) comparison of the flow structure around a square cylinder with an attached plate to that around a single square cylinder without a plate in order to thoroughly observe the changes in the flow structure mechanism due to the plate; 4) analysis of the optimum conditions for maximum reduction in fluid forces; and 5) applicability of the lattice Boltzmann method for simulating such flow control problems. It is important to mention here that flow control in the laminar regime is of particular relevance in the case of low Re flows, such as those occurring in electronic devices [35].

The governing continuity and momentum equations for the laminar, incompressible, and unsteady fluid flow models considered in the current study are as follows: ∇ . u = 0 , (1) ∂ u ∂ t + u . ∇ u = − 1 ρ ∇ p + ν ∇ 2 u , (2) where u is the velocity vector, ρ is the fluid density, p is pressure, ν is kinematic viscosity, and t is time.

In this study, the well-known mesoscopic-level-based technique, the lattice Boltzmann method, is used to capture the important fluid flow characteristics. The LBM is a non-traditional method, as it replaces the Navier–Stokes equations with the discrete Boltzmann equation in fluid flow simulations. Furthermore, the salient features of the LBM include second-order accuracy, ease of implementation for complex boundaries, parallel computations, quasi-non-linearity, and a relatively easy process for computation of the pressure term; these make it more prominent than other CFD approaches [36]. It is important to mention here that the compressibility and discretization errors play an important role in the overall accuracy of the method. The Mach number (Ma) controls the compressibility error in the LBM, which is of the order O (Ma ² ). The discretization error in the LBM is of the order O (Ma ⁻¹ h ^δ), where δ represents the order of the underlying finite difference scheme. For stable and accurate computation, we have chosen e = 1, leading to Ma = 0.1. It should be noted that all the results in subsequent sections were obtained using code developed in-house in the FORTRAN language.

It is well-proven in the literature that Eqs 1, 2 can be derived from the discrete Boltzmann equation by using the Chapman–Enskog expansion [37], which is an asymptotic expansion relating statistical mechanics with the theory of continuum fluid dynamics. Due to this fact, in the lattice Boltzmann method, the discrete Boltzmann equation is solved, instead of directly dealing with Eqs. 1, 2, along different lattice links to analyze the fluid flow dynamics. The discrete form of the Boltzmann equation is f j x + v j Δ t , t + Δ t = f j x , t + 1 τ f j e q x , t − f j x , t , (3) where j ranging from 0 to n indicates the lattice links, the discrete velocity directions are v j = e Δ x Δ t , and the parameter τ > 0.5 is the dimensionless relaxation time, which is also termed the stability control parameter. In Eq. 3, f j is the particle distribution function, whereas f j e q is the equilibrium distribution function, defined for two-dimensional problems as f j e q = ρ w j 1 + 3 v . u − 3 2 u . u + 9 2 v . u 2 , (4) where w j indicate the weights whose values are associated with the nature of lattice models [38].

The macroscopic quantities are linked to f j by the following constraints: ρ x , t = ∑ j = 0 n f j = ∑ j = 0 n f j e q , (5) ρ u x , t = ∑ j = 0 n v j f j = ∑ j = 0 n = v j f j e q . (6)

Based on the nature and dimensions of the problem, different models are used in the LBM [38], which, in general form, are specified as DmQn, where m indicates the dimension and n indicates the number of lattice links. In the current study, the D2Q9 model is used, which is a two-dimensional model with a square lattice and nine velocity directions (Figure 1). As mentioned earlier, the weights are specific to the model and are derived by evaluating the moments of distribution functions satisfying certain isotropy conditions [38, 39]. For the D2Q9 model, the weights and velocity directions are w j = 4 / 9 , j = 0 1 / 9 , j = 1,2,3,4 1 / 36 , j = 5,6,7,8 . (7) and v j = v 0,0 , j = 0 v cos π 2 j − 1 , sin π 2 j − 1 , j = 1,2,3,4 2 v cos π 2 j − 9 2 , sin π 2 j − 9 2 j = 5,6,7,8 . (8)

The geometry of the problem under consideration is presented in Figure 2. A fixed square cylinder with an attached flat plate is placed inside a channel having length l and height H. D is the size of the cylinder, and L is the non-dimensional length of the plate. The length of the plate is systematically varied in the range 0.1–8.5. Specifically, plate length is varied with step size 0.1 in the range 0.1–2, with step size 0.2 in the range 2–6, and with step size 0.25 in the range 6–8.5. It is of note that the height of the plate is fixed at 0.2 throughout this study. The height H of the channel is fixed at 21, while the length l of the channel varies depending on the length of the plate. Furthermore, we selected an upstream distance, from the inlet to the cylinder, of L _U = 10 times the size of the cylinder and a downstream distance, from the plate rear end to the outlet, of L _D = 20 times the size of the cylinder. These domain locations are fixed for all chosen lengths of the attached plate. All the domain locations and plate lengths are non-dimensionalized using the size D of the cylinder.

The flow at the inlet is prescribed by the parabolic velocity ( u = 4U _in (y/H)(1 – y/H); v = 0). One reason for choosing this velocity profile is that parabolic incoming flows are of practical relevance in most fluid flow problems. A second is that, in most previously published work, a uniform inflow velocity was chosen for flow control-related problems. We attempted to address the flow control mechanism around a square cylinder when the inflow velocity is parabolic. At the upper and lower channel walls and the cylinder surface, a no-slip boundary condition ( u = v = 0 ) is applied using the bounce-back rule, in which the fluid particles hitting the solid walls bounce back in the opposite direction [38]. A convective boundary condition [40] is used at the outlet position of the channel to ensure that the fluid particles leave the channel without causing significant changes in the flow domain.

To ensure that the results of the current study were not influenced by the number of grid points on the surface of the square cylinder, a grid independence analysis was performed for flow past a single square cylinder at Re = 150. For this analysis, the mean drag coefficient, Strouhal number, and root-mean-square values of the drag coefficient were computed by dividing the cylinder’s surface into 10, 20, and 40 grid points (Table 1). In comparison to the other schemes, the results for the 20-point grid were more appropriate in terms of computation time and accuracy. Although a 40-point grid may produce more accurate results, it takes a great deal of computation time to reach convergence, and there is not much difference in results between the 20- and 40-point grids. On the other hand, a 10-point grid suffers from a lack of accuracy. Thus, all computations in this study were performed on a 20-point grid, D = 20. Islam et al. [40] and Guo et al. [41] also proposed a 20-point grid for convergence.

To confirm the validity of the code, the computations were performed for flow around a single square cylinder without a plate at Re = 150 to test whether our code is capable of capturing the important physics of flow around bluff bodies. The results are illustrated in terms of instantaneous contours of vorticity, streamlines, drag and lift coefficients, and power spectrum in Figure 3. In the vorticity graph (Figure 3A), the dashed lines indicate the clockwise vortices and the solid lines show the anticlockwise vortices. It can be noted that the incoming flow detaches from the front edges of the cylinder and rolls up to form vortices in the wake of a cylinder. These vortices move alternately in the domain, exhibiting the well-known Kármán vortex street behavior. In the streamlines graph (Figure 3B), a recirculating eddy can be observed adjacent to the rear lower corner of the cylinder, indicating vortex formation due to the merging of shear layers detaching from the upper and lower sides of the cylinder. The alternate movement of vortices is indicated by the waviness of streamlines in the wake region of the cylinder. Due to the unsteadiness in flow, C D and C L exhibit periodic behavior, as shown in Figures 3C, D. The C D value initially oscillates randomly, jumps to a higher value, and then stabilizes after reaching a steady oscillating state over time. Figure 3D shows that C L at Re = 150 starts with minor oscillations, which can be attributed to parabolic inflow velocity, and as time passes, CL signals oscillate in the range [−0.4, 0.4] with a mean value of zero, which confirms the complete unsteadiness in the flow produced. To determine the vortex shedding frequency, the power spectrum analysis of C L is carried out by applying a fast Fourier transformation technique. The resulting S t is characterized by a single dominant peak indicating uniform unsteadiness in flow without any chaos (Figure 3E). This flow behavior is commonly observed in the case of bluff bodies [2, 8, 11, 13]. Further information on qualitative and quantitative validation of the code in terms of statistical parameters such as average drag coefficient, Strouhal number, and RMS values of drag and lift coefficients can be found in most of our previously published papers [42–47]. These observations indicate the capacity of our code to capture the important characteristics of fluid flows around bluff bodies.

Flow around a square cylinder exhibits different modes with fluid force modifications in a case in which a controlling device is introduced in the wake [16, 48]. In the present case, when the length of the plate attached to the cylinder is systematically varied from 0.1 to 8.5, three major flow modes are observed. In accordance with certain characteristics of these flow modes, they are named the unsteady, transitional, and steady flow modes. These classifications depend on the flow structure observed in vorticity contours, behavior of streamlines behavior, temporal variations in the drag and lift coefficients, power spectrum analysis of the lift coefficient, Strouhal number, and RMS values of the drag and lift coefficients. This section presents only representative cases for each flow mode, keeping in mind the fact that all simulation results cannot be included in the manuscript for purposes of brevity and to avoid repetition. The effect of plate length on important flow parameters such as C D m , S t , C D r m s , and C L r m s is also analyzed and compared to the values of these parameters in the case of a single cylinder to examine the trend in fluid force suppression.

The current study investigated the capacity of a flat plate attached to the rear side of a square cylinder to control the fluctuating forces and regulate vortex shedding using the lattice Boltzmann method. The plate length was varied in the range 0.1 ≤ L ≤ 8.5, and the Reynolds number was fixed at 150. The effects of the plate on flow characteristics were observed in terms of vorticity contours, streamline visualization, variation of time-dependent force coefficients, energy spectrum, and velocity variations. In addition, the influence of plate length on major flow parameters such as C D m , C D r m s , C L r m s , and S t was also investigated. Furthermore, the percentage difference in these values was calculated by comparing the values of these parameters to those in the case of a single square cylinder without the plate. Some of the key findings of this study are as follows:

1  Depending on the plate length, three different flow modes were observed: unsteady flow ( L = 0.1–6.5), transitional flow ( L = 6.75–7.5), and steady flow ( L = 7.75–8.5). In the unsteady flow mode, vortices exhibited von Kármán street behavior in the wake of the cylinder starting from the trailing edge of the plate. For the transitional flow mode, minor transverse oscillations in the wake were observed near the exit of the computational domain. However, vortex shedding in the wake region was observed to be completely suppressed for the steady flow mode.