# Passive Flow Control around Two Side-by-Side Square Cylinders Using a Control Plate

Authors: Waqas Sarwar Abbasi; Sumaira Nadeem; Hamid Rahman
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

This article is concerned with the numerical analysis of reduction of fluid forces (drag and lift) and suppression of vortex shedding around two side-by-side square cylinders using the lattice Boltzmann method. For this purpose, a passive flow control methodology is adopted in which a flat plate (termed as the control plate) is placed at the upstream position of cylinders. The gap ratio (g) between the cylinders and the plate is varied from 0 to 4 while the height of the control plate is varied from 0.2 to 0.8 at fixed Reynolds number of 150. Different flow patterns are found in this study depending on the gap ratios and different heights of the control plate. These flow patterns are single bluff body flow, flip-flopping flow, and antiphase vortex shedding flow patterns. The narrow gap ratio (g = 0.5) is found to have more impact in terms of force reduction and flow control as compared to larger gap ratios. This article also shows that the fluid flow parameters such as the mean drag coefficient, the Strouhal number, and rms values of drag and lift coefficients of both cylinders reduce considerably due to the control plate as compared to without the control plate. In this regard, it is found that the mean drag coefficient of upper cylinder reduces upto 45.94%, the Strouhal number of both cylinders reduces upto 18.66%, while a maximum reduction of 83.47% and 95.39% is observed in root-mean-square values of drag and lift coefficients of upper cylinder, respectively. Another important finding of this article is that the control plate does not suppress forces and control flow at all gap ratios and heights of control plate considered in this article. In some cases, the flow-induced forces were found to be increasing as compared to without the control plate, and this generally happened at higher gap ratios.


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## Body

## 1. Introduction

The analysis of flows around bluff bodies, having different shapes and sizes, by experimental and numerical techniques is very popular among researchers from many decades because flow past a bluff body is a direct application to many structures such as the offshore structures, components of heat exchanger, buildings, bridges, road vehicles, submarine structures, and aircrafts. [1–3]. The most valuable thing in such analysis is the flow-induced vibration which results as fluids interact with a solid structure. Such fluids can generate large unsteady forces which may cause to damage the bluff bodies. To handle such scenarios, in the past decades, a variety of vortex-induced vibration suppression methods, commonly termed as active and passive control methods, have been offered by the researchers. Usually, active control techniques require the external energy input, such as suction or blowing on the surface of cylinder and electromagnetic forcing. In contrast, passive control techniques include modification of the shape of structure or attaching the additive devices, which are simple, cheap, and easily applicable in practice.The fluid flows around cylindrical structures are influenced by many parameters such as shape, size, number of structures and their arrangement, angle of attack, and Reynolds number (Re). Sohankar et al. [4] carried out the analysis of the unsteady flow around a square cylinder at different angle of incidence = 0°–45° for Re = 45–200. They found a significant impact of Re on the drag coefficient (Cd) and the stagnation pressure coefficient. Saha [5] analysed the transitions in flow past a square cylinder under the influence of Re ranging from 100 to 325. They observed two-dimensional (2D) flow till Re = 160 while it became 3D at Re = 163.5. They found that the values of the Strouhal number (St) and Cd reduced by increasing Re. Chaitanya and Chatterjee [6] discussed the effects of the blockage ratio on fluid flow past a square cylinder for 0 ≤ Re ≤ 30 with 0.0 ≤ blockage ratio ≤ 0.9. They observed increment in St with an increase in blockage ratios. Sen et al. [7] discussed the flow characteristics around three different cylindrical shapes (square, circular, and elliptical) for Re ≤150. They observed that the square cylinder, flow separated at much low Re as compared to the circular cylinder. They got the highest Cd reduction in the case of square cylinder whereas the lift coefficient (Cl) was higher than Cd for the square cylinder as compared to circular and elliptical cylinders. Jiang and Chiang [8] observed the flow separation around a square cylinder for low Re in the range of 10–400. They found the flow separated near the leading edge at Re = 100.36, whereas the corresponding reattachment of flow was found at Re = 155. The saddle point appeared for Re ≥156 that was away from the surface of the cylinder.To moderate the unsteady wake region and to reduce the adverse effects of fluctuating forces acting on the bluff bodies, researchers proposed different controlling measures and analysed their efficacy through numerical or experimental simulation tests [9–27]. Dash et al. [9] analysed the Cd and Cl reduction and wake regime control around a square cylinder by attaching dual splitter plates at rear sides for a low Re = 100. They found 20.2% reduction in Cd and 98% reduction in Cl by this flow controlling technique. Gupta and Saha [10] controlled the vortex shedding by placing a small control cylinder of three different sizes (20%, 25%, and 30% of the main square cylinder) in the wake at Re = 75, 100, 125. They found that the mean drag coefficient (Cdmean) decreased about 10%–15% as compared to the isolated cylinder. Furthermore, they achieved complete suppression of flow at Re = 100 by controlling the cylinder of size 20% of the main cylinder. Bruneau and Mortazavi [11] investigated the flow around a square cylinder using porous media through the penalization method. They got 13% reduction in Cd and 35% in the root-mean-square value of Cl (Clrms). Sun et al. [12] analysed the transition process of flow-induced vibration of a circular cylinder by the use of a control plate at Re = 100. They noticed the behaviour of Clrms was the same at the plate size (L/d) = 0.75, 1, 1.5, whereas Cdmean first increased and then reduced and finally flattened at L/d = 0.75. Kwon and Choi [13] analysed the control of fluctuating forces behind a circular cylinder by attaching splitter plates having length L/d = 0–2 and Re = 80–160. They found St sharply decreased with the increase in the plate length until L/d = 1. Wu et al. [14] investigated the effects of a flexible filament whose leading edge was attached with the base of a circular cylinder while its trailing edge was free to flap to control flow at Re = 150. It was observed that relatively long filament, with low flexibility, efficiently suppressed fluctuations of Cl and vortex shedding. Gao et al. [15] carried out the investigation of the flow around a circular cylinder with two rigid control plates adjusted parallel to the incoming airflow for L/d = 1-2 at Re = 3.33×104. They found 51.5% reduction in Cd and 72.5% reduction in Cl. Mahato et al. [16] controlled the flow around a square cylinder by placing a cowl-shaped splitter plate at the rear corners with the radial position 0.8–1.5 at Re = 100. The location of the cowl plates at 1.4 was found to be most effective in reducing the sound pressure level as compared to the no-cowl case. Zhu and Wang [17] investigated vortex induced vibration suppression of a circular cylinder with a C-shaped plate placed in the wake at Re = 100. The best suppression was achieved at a distance of 1.5d between the cylinder and the plate. The Cl and cross-flow amplitude reduced 85.5% and 94.5%, respectively. Narendran and Jaiman [18] studied the effect of a near wake jet, positioned at the centre part of the base side of a freely vibrating mounted square cylinder, on the lock-in phenomenon at Re = 40, 60, 100, and 150. They found 90% reduction in the peak vortex-induced vibration amplitude of Clrms and 28% in Cdmean as compared to that of the isolated cylinder.In addition to solo bodies, there have been many studies concerning the flow physics around multiple cylinders. In the case of multiple cylinders, the flow structure mechanism tends to be more complex as compared to a single cylinder [19–23]. Han et al. [19] analysed the turbulent flow around two square cylinders placed in side-by-side arrangement at Re = 22,000 with the gap ratio (g = s/d) = 1.5–3.5. They observed that the cylinders’ spacing significantly affected the flow patterns. Agrawal et al. [20] investigated the effects of the gap ratio on two square cylinders in side-by-side arrangement. They observed different flow structures, naming, synchronized regime and flip-flop regime. They also found that the distribution of mean pressure was symmetric for synchronized and asymmetric for flip-flop regime. Kang [21] performed the simulation of flow around two side-by-side arranged square cylinders for 40 ≤ Re ≤ 160 and g < 5 and categorized the flow in six different wake patterns depending on Re and g. Liu et al. [22] investigated the fluid flow past a pair of side-by-side circular cylinders for 30 ≤ Re ≤ 100 and g < 3. Furthermore, they identified the steady wake behaviour at Re = 60 and g = 1.6. In the steady flow case, they found no vorticity and fluctuating forces that behaved as constant. Alam et al. [23] found four different wake patterns for the flow behind two side-by-side square cylinders.The suppression of flows around multiple cylinders is of significant interest as this not only enhances the structural performance but also saves energy. However, there have been very few studies addressing the suppression of vortex shedding and fluid forces control around multiple cylinders in general and for side-by-side square cylinders’ arrangement in particular. The studies [24–26] attempted to address these shortcomings, but these were mostly related to circular cylinders and only one of these addressed the square cross-sectional cylinders. Keeping these facts in view, the current study is conducted which mainly focuses on the analysis of flow modes around two side-by-side placed square cylinders and their suppression. Our main objective is to analyse the effects of an upstream placed control plate by focusing the gap ratio between cylinders and the plate, as well as between both cylinders with different plate heights. These effects will be discussed in terms of vorticity; fluid forces variations with time and statistical analysis. This article will provide information for the design of flow control devices regarding side-by-side arranged square-shaped bodies.The rest of the article is arranged as follows: In Sections2 and 3, the lattice Boltzmann method, geometry of the problem, and corresponding boundary conditions are presented. Section 4 describes the validation of the code and grid independence study. The computed results and a detail discussion are given in Section 5, and concluding remarks are drawn in Section 6.


## 2. Lattice Boltzmann Method (LBM)

The lattice Boltzmann method (LBM) is a well-known numerical scheme which is used for simulation of complex problems related to heat, mass, and momentum transfers (Kruger et al. [27]). The main idea of LBM is to bridge the gap between micro- and macroscales of simulation by not considering each particle behaviour alone but behaviour of a collection of particles as a unit. The LBM has a mesoscopic scale-based approach which is the intermediate of the macroscopic and microscopic scales [28, 29].The governing differential equations for laminar, incompressible, and unsteady fluid flow, considered in current work, are as follows:(1)∇.u=0,(2)∂u∂t+u.∇u=−1ρ∇P+ν∇2u,where ρ is the density, u is the velocity, ν is the kinematic viscosity, and P is pressure. Equation (1) expresses the law of conservation of mass, and equation (2) expresses the conservation of momentum. These equations are commonly termed as Navier–Stokes Equations and can be derived from the lattice Boltzmann equation (Bhatnagar et al. [30]).The Boltzmann equation, used in statistical mechanics, is(3)∂f∂t+c.f=Ω,where f is the distribution function which acts as a representative for a collection of particles, c is the velocity direction, t is time, and Ω is a collision term which describes the rate of change of fi resulting from the collision. The collision operator introduced by Bhatnagar et al. [30] has the following form:(4)Ω=1τfeq−f,where τ is a relaxation time factor which is associated with the kinematic viscosity as ν=cs2τ−1/2 and cs is the speed of sound propagation with a constant value cs=1/3 and τ>0.5 for the D2Q9 model (Ma et al. [31]). Three important features are introduced by the kinetic nature of the LBM that differentiate and make it prominent from other numerical approaches. First, the convective operator of the LBM in phase space is linear; second, Navier–Stokes equations can be obtained from the lattice Boltzmann equation and pressure can be found by the equation of state P=ρcs2. Third, the LBM operates a minimal set of velocities such as only one or two speeds and a few moving directions in the phase space (Chen and Doolen [32]). Although the LBM is rapidly gaining popularity among researchers for the simulations of complex fluid flow systems, this approach has some limitations. As for example, the computations at high-Mach number flows in the aerodynamics are quite difficult through the LBM due to compressibility constraints; the lack of thermo-hydrodynamic scheme is another drawback of the LBM, and the LBM demands much memory for the highly refined grids; in some cases, the boundary conditions also become difficult to handle because in LBM boundary conditions are written in terms of the distribution function and not in the macroscopic quantities (Boubaker [33]).In equation (4), feq is the local equilibrium distribution function which is also known as the Maxwell–Boltzmann distribution function [29]. By using the approximation of Bhatnagar, Gross, and Krook approximation, equation (3) becomes(5)∂f∂t+c.∇f=1τfeq−f.The discrete form of equation (5), with a specified velocity direction ci, is(6)∂fi∂t+ci.∇fi=1τfieq−fi.Discretizing the above equation by finite difference method,(7)fix+∆x,t+∆t=fix,t−∆tτfix,t−fieqx,t.The equilibrium distribution function for the D2Q9 model is(8)fieq=ρwi1+1cs2ci.u+12cs4ci.u2−12cs2u2,where wi are weighting coefficients and u is the velocity vector.The macroscopic quantities such as the fluid densityρ and the fluid velocity vector u are computed using the following equations:(9)ρ=∫fdc=∑i=08fi,ρu=∫cfdc=∑i=08cifi.The LBM different models, depending on nature of problems to be solved, are expressed generally asDnQm, where n indicates dimensions of the problem and m is the number of the particles or directions. In the current work, we have used the D2Q9 lattice model, that is, the two-dimensional and nine directions model [29]. In this model, each lattice node is connected by eight directions and each particle moves one lattice unit only along x-axis, y-axis, and diagonal directions.The velocity directionsci for the D2Q9 model are(10)ci=c0,0,i=0,ccosπ2j−1,sinπ2j−1,i=1,2,3,4,2ccosπ42j−2,sinπ42j−1,i=5,6,7,8.Weighting coefficients are different according to the models. For the D2Q9 model, the weighting coefficients are given in Table1, where w0 stands for weighting coefficients of rest particles, wa stands for weighting coefficients of short or perpendicular direction moving particles, and wd stands for weighting coefficients of particles moving diagonally.Table 1
Weighting coefficients for the D2Q9 model.
ConstantsValuesw04/9wa1/9wd1/36


## 3. Geometry of the Problem and Boundary Conditions

The schematic configuration of the problem under consideration is given in Figure1. Two side-by-side square cylinders (c1 and c2) of the same size “d” with a control plate of length “d” and height “w” placed upstream are exposed to a uniform incoming flow (U∞). The size of the cylinder and the control plate length is taken as 20, i.e., d = 20. The domain area (W × H, where W and H are the width (length) and height of the computational domain, respectively) for different gap ratios is given in Table 2. The distance from the inlet to the front surface of the plate is UL = 7.5d, while the distance from the rear surfaces of cylinders to the outlet of the computational domain is DL = 30d. The surface-to-surface distance between c1 and c2 as well as between the plate and cylinders is denoted by s. The dimensionless gap ratio (g = s/d) and the height of the control plate (h = w/d) are systematically varied between 0.5 and 4 and 0.2 and 0.8, respectively.Figure 1
Schematic configuration of the proposed problem.Table 2
Size of computational domain for different cases.
CasesW × HCasesW × Hg = 0791 × 261g = 2831 × 301g = 0.5801 × 271g = 2.5841 × 311g = 1811 × 281g = 3851 × 321g = 1.5821 × 291g = 4871 × 341The boundary conditions used in this article are uniform inflow velocity at inlet, convective boundary condition at outlet, periodic boundary condition at lateral boundaries, and no-slip boundary condition at the surfaces of obstacles (cylinders and control plate).


## 4. Code Validation Study and the Grid Independence Study

### 4.1. Code Validation Study

Before performing the numerical simulations for flow behind two side-by-side square cylinders with an upstream control plate, the code is validated for flow around a single square cylinder and the results are presented in Table3 along with the absolute errors (enclosed in parentheses) for each case. Table 3 shows the comparison of the mean drag coefficient and the Strouhal number computed at Re = 150, with the numerical data of Gera et al. [34], Sohankar et al. [35], Rubichaux et al. [36], and Guo et al. [37] and experimental measurements of Okajima [38] and Sohankar [35]. It can be seen from Table 3 that our Cdmean value is in a good agreement with the experimental data of Okajima [38] and the numerical data of Gera et al. [34] and Sohankar et al. [35]. Also, St agrees well with Sohankar et al. [35], Rubichaux et al. [36], and Norberg [39] data (see Table 3). From this comparison, it can be deduced that our code works well for capturing the important flow features around square cylinders.Table 3
Code validation study at Re = 150.
CdmeanStPresent1.43270.1579Gera et al. [34]1.4110 (0.01514)0.1410 (0.1070)Sohankar et al. [35]1.4080 (0.0172)0.1610 (0.0196)Rubichaux et al. [36]1.5600 (0.0888)0.1640 (0.0385)Guo et al. [37]1.3740 (0.0409)0.1443 (0.0861)Okajima [38]1.4920 (0.0413)0.1420 (0.1006)Norberg [39]—0.1500 (0.0500)


### 4.2. Grid Independence Study

This section is devoted to the grid refinement study to ensure that our results do not depend on a particular grid size. For this purpose, we have analyzed the effects of 5, 10, 20, and 40 grid points around the surface of the cylinder and on the physical parameter’s values. The computed values of Cdmean, St, and Clrms are given in Table4 corresponding to each grid size. It can be seen from the table that, by increasing the number of grid points on the surface of the cylinder from 5 to 40, the Cdmean and St decrease, while Clrms increases upto d = 10 and then shows decreasing behavior. Furthermore, at smaller grid points, the physical parameters’ values are higher (d = 5 and 10) and at d = 40, a larger computational domain is needed, and the simulations are cost-effective, and that is why we have selected 20 grid points for the current study. Another reason for choosing this grid size is that the values given in Table 3, which agree well with the literature results, are also computed at d = 20. The domain independence study can be found in our previous work [40].Table 4
Physical parameters data for different sizes’ square cylinders.
Size of cylinder indCdmeanStClrms51.53590.54460.3011101.49120.30100.3286201.43270.15790.2790401.37360.08000.2190


## 4.1. Code Validation Study

Before performing the numerical simulations for flow behind two side-by-side square cylinders with an upstream control plate, the code is validated for flow around a single square cylinder and the results are presented in Table3 along with the absolute errors (enclosed in parentheses) for each case. Table 3 shows the comparison of the mean drag coefficient and the Strouhal number computed at Re = 150, with the numerical data of Gera et al. [34], Sohankar et al. [35], Rubichaux et al. [36], and Guo et al. [37] and experimental measurements of Okajima [38] and Sohankar [35]. It can be seen from Table 3 that our Cdmean value is in a good agreement with the experimental data of Okajima [38] and the numerical data of Gera et al. [34] and Sohankar et al. [35]. Also, St agrees well with Sohankar et al. [35], Rubichaux et al. [36], and Norberg [39] data (see Table 3). From this comparison, it can be deduced that our code works well for capturing the important flow features around square cylinders.Table 3
Code validation study at Re = 150.
CdmeanStPresent1.43270.1579Gera et al. [34]1.4110 (0.01514)0.1410 (0.1070)Sohankar et al. [35]1.4080 (0.0172)0.1610 (0.0196)Rubichaux et al. [36]1.5600 (0.0888)0.1640 (0.0385)Guo et al. [37]1.3740 (0.0409)0.1443 (0.0861)Okajima [38]1.4920 (0.0413)0.1420 (0.1006)Norberg [39]—0.1500 (0.0500)


## 4.2. Grid Independence Study

This section is devoted to the grid refinement study to ensure that our results do not depend on a particular grid size. For this purpose, we have analyzed the effects of 5, 10, 20, and 40 grid points around the surface of the cylinder and on the physical parameter’s values. The computed values of Cdmean, St, and Clrms are given in Table4 corresponding to each grid size. It can be seen from the table that, by increasing the number of grid points on the surface of the cylinder from 5 to 40, the Cdmean and St decrease, while Clrms increases upto d = 10 and then shows decreasing behavior. Furthermore, at smaller grid points, the physical parameters’ values are higher (d = 5 and 10) and at d = 40, a larger computational domain is needed, and the simulations are cost-effective, and that is why we have selected 20 grid points for the current study. Another reason for choosing this grid size is that the values given in Table 3, which agree well with the literature results, are also computed at d = 20. The domain independence study can be found in our previous work [40].Table 4
Physical parameters data for different sizes’ square cylinders.
Size of cylinder indCdmeanStClrms51.53590.54460.3011101.49120.30100.3286201.43270.15790.2790401.37360.08000.2190


## 5. Results and Discussion

Numerical simulations are performed for the flow around two side-by-side square cylinders in the presence of an upstream control plate. The results are presented and discussed, in this section, in terms of vorticity contour visualization, time-signal variations of Cd and Cl, power spectrum of Cl, fluid forces’ variation, and their reduction. The heights of the control plate and gap ratios are varied between 0.2 to 0.8 and 0 to 4, respectively.


### 5.1. Flow Structures

The flow around the cylinders exhibits different patterns under the influence of plate’s height and gap ratios. The results show three main flow patterns which are categorized as the single bluff body flow, flip-flopping flow, and antiphase vortex shedding flow patterns. Each of these flow patterns have certain characteristics which are discussed below.


#### 5.1.1. Single Bluff Body Flow

When the gap ratio between the cylinders and control plate is small, the triad of objects behaves like a single body and the associated flow behavior is termed as single bluff body flow. In the present case, the single bluff body flow is found at (g, h) = (0, 0.2–0.8) and (0.5, 0.2–0.8). To discuss the characteristics of the single bluff body flow, (g, h) = (0, 0.8) and (0.5, 0.6) are considered as representative cases and vorticity contour visualization, and temporal variations of Cd and Cl and power spectrum for these cases are shown in Figure 2. The shear layers detaching from the plate immediately interact with the cylinders and roll up to form vortices at the down wake region of the cylinders (Figure 2(a)). No gap flow is seen due to closely packed bodies. With a slight increment in the gap ratio, it can be seen that the shear layers detached from the plate move inside the gaps and create a jet flow, but the flow remains similar to that of the single bluff body in which vortices are formed only at the down wake region (see Figure 2(b)). This phenomenon was also reported by Bao et al. [41]. It is clear from Figures 2(c) and 2(d) that, due to resemblance with the single bluff body flow, both drag and lift forces on the cylinders illustrate periodic behavior. Similar variation of force coefficients was observed at (g, h) = (0.5, 0.6) and other cases of the single bluff body flow (not shown here). Also, the drag coefficients of both cylinders are antiphase to each other while the lift coefficients of both cylinders are in phase to each other with the same amplitude of consecutive cycles. This can be attributed to the generation mechanism of vortices from the cylinders.Figure 2
Flow structure mechanism for the single bluff body flow pattern.
(a)(b)(c)(d)(e)(f)From the power spectrum graphs for the single bluff body flow, minor secondary peaks along with primary peak can be seen (Figures2(d) and 2(e)). Furthermore, the power spectrum energy of both cylinders is almost the same which indicates the smooth passage of the flow in the down wake region. Bao et al. [42] reported the single bluff body flow at g = 1.5 for the flow around three circular cylinders in the equilateral triangular arrangement. Rahman et al. [43] also reported the single bluff body flow pattern for the flow behind two side-by-side square cylinders in the presence of a downstream control plate.


#### 5.1.2. Flip-Flopping Flow

Another flow pattern spotted in this article is the flip-flopping flow pattern which spans over (g, h) = (1, 0.2–0.8) and (1.5, 0.2–0.8). The representative case of this flow structure is depicted in Figure 3 at (g, h) = (1, 0.6). The vorticity contour shows that the fluid flows inside the gaps after detaching from the plate due to increase in the gap ratio. The vortices generated from the plate move in the gaps as well as on the outer sides of the cylinders which results in irregular vortex shedding and complex flow structure in the downstream wake region (Figure 3(a)). This complexity in flow structure is due to strong interaction of jet between the cylinders. In this flow pattern, there is no definite shape or size of the vortices contrary to those observed in case of the single bluff body flow.Figure 3
(a) Vorticity contour visualization, (b, c) signals of CD and CL, and (d, e) FFT spectrum of Cl for the flip-flopping flow pattern.
(a)(b)(c)(d)(e)The shed vortices are not apparent at near downstream; while at far downstream of the computational domain, the vortices shattered, and a dispersed flow structure can be witnessed from Figure3(a). Due to modulation in flow structure, the resulting Cd and Cl exhibit irregularity (Figures 3(b) and 3(c)) because, owing to insufficient gap space, the jet flow strongly affects the flow structure at the immediate down wake region and this effect of the jet flow transports in the whole down wake region. Yan et al. [44] also reported the irregularity of vortices at downstream due to the passage of vortices of the upstream cylinder through gaps. Furthermore, due to the irregularity of vortices, the variations in drag and lift were also reported in their work. Multiple diffusive peaks in power spectrum graphs also appear due to irregularity in flow structure, and that is why it is difficult to compute St from these diffused peaks (see Figures 3(d) and 3(e)).


#### 5.1.3. Antiphase Vortex Shedding Flow

The antiphase vortex shedding flow structure is seen at larger gap ratiosg = 2–4 and plate heights h = 0.2–0.4. For discussion, (g, h) = (2.5, 0.4) is considered as a representative case, and its corresponding vorticity contour and force coefficients graphs are shown in Figure 4. It is clear from vorticity contour that, as the gap between cylinders is large enough, the vortices shed from the control plate and interact with the front sides of the cylinders, as well as move inside the gap between the cylinders. In the wake of the cylinders, the vortices are irregular with different shapes and sizes (Figure 4(a)). Yan et al. [44] also reported such type of flow structure. According to Moussaoui et al. [45], when three square cylinders are arranged in V-shape and the gap between the cylinders is large (g = 2), then both the objects form their own vortex street of vortices like a single object. The same behavior can also be seen in Figure 4(a). Furthermore, the vortices generated from the cylinders are antiphase to each other, and this behavior can also be verified from the lift signals given in Figure 4(c). From time-signal graphs of Cd and Cl, it is apparent that both cylinders experience the same drag force (Figures 4(b) and 4(c)). The amplitude of oscillating lift force on both cylinders is identical, but it acts in opposite direction (antiphase) on both cylinders. Due to the periodic variation of Cl, a single dominant peak can be observed from power spectrum with identical spectrum energy (Figures 4(d) and 4(e)).Figure 4
(a) Snapshot of vorticity, (b, c) temporal variations of drag and lift coefficients, and (d, e) power spectrum of the lift coefficient at (g, h) = (2.5, 0.4).
(a)(b)(c)(d)(e)


### 5.2. Force Statistics

The effect of the control plate with varying height (h = 0.2–0.8) on fluid flow parameters such as the mean drag coefficient, the Strouhal number, and Cdrms and Clrms of both cylinders is discussed in this section at small (g = 0.5), intermediate (g = 1.5), and large (g = 4) gap ratios. The data of two square cylinders without control plates are also provided for comparison purpose.


#### 5.2.1. At Small Gap Ratio (g = 0.5)

At smallg, the Cdmean of both cylinders decreases with an increment in h (see Figure 5(a)). The local maxima for the mean drag coefficient of c1 and c2 are 1.7772 and 1.7882, respectively, and can be seen at h = 0.2. Whereas the local minima for the mean drag coefficient of c1 and c2 is identical and its value is 1.0953, which is observed at h = 0.8. The values of Cdmean1 and Cdmean2 of the cylinders without the control plate are 1.9801 and 2.0260, respectively. Without the control plate, Cdmean of c1 is smaller than c2. The values of Cdmean1 and Cdmean2 of both cylinders with the control plate are smaller than Cdmean1 and Cdmean2 of both cylinders without the control plate (Figure 5(a)). Also, by increasing the height of the control plate, the mean drag coefficient values decrease which shows that Cdmean reduced by placing a control plate at the upstream position, and for larger height of the control plate, this reduction is more significant (see Figure 5(a)).Figure 5
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at a small gap ratio.
(a)(b)(c)(d)The St values of both cylinders with the control plate increase in the range fromh = 0.2 to 0.6, but for h = 0.8, St of both cylinders exhibit slight decreasing behaviour (Figure 5(b)). The St values of both cylinders with the control plate are identical to the St values of the cylinders without the control plate at h = 0.6 and smaller for other heights of the control plate (Figure 5(b)). From this discussion, it can be deduced that, for control plate heights h = 0.2–0.6, the effect of the control plate is almost negligible in terms of the Strouhal number. The maximum St1 and St2 values for the cylinders with control plates are 0.1264 at h = 0.6, while the minimum St1 and St2 values are the same, which is 0.1033 and can be seen at h = 0.8. Cylinders without a control plate have identical St1 and St2 value of 0.1270 (Figure 5(b)).The root-mean-square value of the drag coefficient for both cylinders with the control plate shows decreasing behaviour with increment in the plate height (Figure5(c)). For cylinders with a control plate, the local maxima for Cdrms of c1 and c2 are 0.0984 and 0.0929 observed at h = 0.2 and the local minima for Cdrms of c1 and c2 are 0.0219 and 0.0275 observed at h = 0.6 and 0.8, respectively. Without the control plate, Cdrms of c1 and c2 are 0.1639 and 0.1664, respectively. For cylinders without the control plate, the root-mean-square value of c1 is higher than c2. The maximum and minimum values of root-mean-square values of c1 and c2 Cdrms values of the cylinders with the control plate are smaller than the Cdrms values of the cylinders without the control plate, which shows the Cdrms values reduced by placing the control plate at the upstream position, and this reduction increases as the control plate height increases. With increment in h, Clrms shows decreasing trend (see Figure 5(d)). The maximum Clrms1 and Clrms2 values for the cylinders with control plates are 0.1469 and 0.1335 at h = 0.2, respectively, and the minimum Clrms1 and Clrms2 values are 0.0118 and 0.0130 at h = 0.6 and h = 0.8, respectively. The Clrms1 and Clrms2 values of the cylinders without the control plate are 0.2735 and 0.2820, respectively. For without the control plate case, the Clrms value of c1 is slightly smaller than c2. From abovementioned discussion, it can be concluded that the maximum and minimum Clrms1 and Clrms2 values of the cylinders with the control plate are lower than the Clrms values of the cylinders without the control plate. This demonstrates that reduction occurs as the control plate is positioned upstream of the cylinders, and this reduction increases as the control plate height increases.


#### 5.2.2. At the Intermediate Gap Ratio (g = 1.5)

Cdmean of both cylinders with the control plate at the intermediate gap ratio decreases with increasing height of the control plate (Figure6(a)). The local maxima for mean drag coefficients of c1 and c2 with the control plate are 1.8172 and 1.8289 occur at h = 0.2, while the local minima for mean drag coefficients of c1 and c2 with the control plate are 1.7284 and 1.7286 which occur at h = 0.8. In the case “cylinders without plate,” the mean drag coefficients of c1 and c2 have values of 1.8764 and 1.864, respectively. The mean drag coefficient of c1 is smaller than c2 in the case without the control plate. By comparing the values of mean drag coefficients of both cylinders with and without the control plate, it can be observed that mean drag coefficients of both cylinders with the control plate are smaller than that of without the control plate case. This observation indicates that Cdmean of both cylinders reduced significantly by placing the control plate at the upstream position, and this reduction increases for increasing control plate height which confirms the effect of the control plate at the intermediate gap ratio. The St values of both cylinders with the control plate show decreasing behaviour for increasing h (Figure 6(b)). The Strouhal value of c1 and c2 and St2 with the control plate have maximum values of 0.1701 and 0.1695 and minimum values of 0.1529 and 0.1524 at h = 0.2 and 0.8, respectively. St1 and St2 of cylinders without the control plate have the same value and are equal to 0.1671. Furthermore, it can be observed that the St values of both cylinders, with the control plate case, are larger than the St values of arrangement without the control plate till h < 0.4, and for other heights, it is smaller (see Figure 6(b)). This shows that, upto h < 0.4, the placement of the control plate at the upstream position is not useful because, instead of reduction, it results in increment of the St values; for the remaining control plate heights, it minimizes the St values (Figure 6(b)).Figure 6
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at the intermediate gap ratio.
(a)(b)(c)(d)Cdrms1 and Cdrms2 of the cylinders, with the control plate case, show decreasing behaviour with increasing height of the control plate except that Cdrms1 slightly increases fromh = 0.4 to 0.6 (Figure 6(c)). The local maxima for the root-mean-square value of the drag coefficient of c1 and c2 with the control plate are 0.1337 and 0.1350, respectively, examined at h = 0.2, and the local minima for root-mean-square value of drag coefficient of c1 and c2 with control plate are 0.1141 and 0.1169, respectively, investigated at h = 0.8. In case of without the control plate, the root-mean-square values of drag coefficient of c1 and c2 are 0.1648 and 0.1646, respectively. The Cdrms value of c1 is smaller than the Cdrms value of c2 in without the control plate case. By comparing the Cdrms of both cylinders with and without the control plate, it is observed that Cdrms values of both cylinders with the control plate are smaller than the Cdrms values of cylinders without the control plate (Figure 6(c)), which again confirm that the control plate is effective for the reduction of Cdrms values. The Clrms values variation for both cylinders are shown in (Figure 6(d)). Clrms1 shows decreasing behaviour for increasing height of the control plate, while Clrms2 shows decreasing behaviour for h = 0.2 to 0.6, while from h = 0.6 to 0.8, it increases. The local maxima for root-mean-square values of lift coefficient of c1 and c2 are 0.6010 and 0.6095, respectively, which occurred at h = 0.2, while the local minima for root-mean-square values of lift coefficient of c1 and c2 are 0.5527 and 0.5440, which are observed at h = 0.8 and 0.6, respectively. The root-mean-square values of lift coefficient of c1 is smaller than c2, in without control plate case. It can be seen from Figure 6(d) that the control plate is not effective at g = 1.5 for all chosen heights in terms of Clrms values because the Clrms values of both cylinders with control plate are higher than the Clrms values without control plate case.


#### 5.2.3. At the Large Gap Ratio (g = 4)

At largeg, the mean drag coefficient of both c1 and c2 with the control plate shows increasing behaviour with an increment in h (Figure 7(a)). The local maxima for mean drag coefficients of c1 and c2 with the control plate are 1.6882 and 1.6881, respectively, observed at h = 0.8 and the local minimum value is 1.5873 at h = 0.2, while Cdmean1 and Cdmean2 of the cylinders without the control plate have the same value of 1.5794 (Figure 7(a)). It is obvious that, for g = 4, the control plate is not effective in terms of Cdmean reduction as these values of Cdmean become higher than those of without the control plate case after placing the plate. This indicates that the control plate is not beneficial in terms of Cdmean for g = 4. The St values for the large gap ratio of cylinders with the control plate illustrate a declining behaviour with an increment in h (Figure 7(b)). The local maxima for the Strouhal number of c1 and c2 with the control plate are 0.1667 and 0.1695 investigated at h = 0.2, and the local minima for the Strouhal number of c1 and c2 with the control plate are 0.1598 and 0.1592 investigated at h = 0.8. In without the control plate case, the Strouhal number of c1 and c2 are 0.1667 and 0.1661, respectively. The Strouhal number of c1 is higher than the Strouhal number of c2. It is obvious from this graph that, at h = 0.2, St2 with the control plate is higher than St2 without the control plate while St1 of both the arrangements is identical, and for remaining chosen heights, the St values, for with the control plate case, are smaller than the St values of without the control plate (see Figure 7(b)). This shows that the control plate is useful for reduction of St for h = 0.4–0.8 (Figure 7(b)).Figure 7
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at the large gap ratio.
(a)(b)(c)(d)The Cdrms values of both cylinders with the control plate show increasing behaviour for increasingh (Figure 7(c)). The local maxima for the root-mean-square value of drag coefficient of c1 and c2 are 0.0549 and 0.0552, respectively, observed at h = 0.8, and the local minima for the root-mean-square value of drag coefficient of c1 and c2 have identical value of 0.0251 observed at h = 0.4. By comparing the Cdrms values of both cylinders with and without the control plate, it is observed that the values with the control plate are smaller than without the control plate case for h < 0.6, while this trend reverses for h > 0.6 (Figure 7(c)). Like Cdrms values, Clrms1 and Clrms2 also show increasing behaviour for increasing height of the control plate (Figure 7(d)). The local maxima for the root-mean-square value of lift coefficient of c1 and c2 are 0.5105 and 0.5109 for cylinders with the control plate case at h = 0.8. The local minima for the root-mean-square value of lift coefficient of c1 and c2 with the control plate have an identical value of 0.3568 and can be seen at h = 0.2. Furthermore, from figure, it is apparent that Clrms1 and Clrms2 of the cylinders with the control plate are like Clrms1 and Clrms2 of the cylinders without the control plate at h = 0.2, while Clrms1 and Clrms2 of the cylinders with the control plate are greater than Clrms1 and Clrms2 of the cylinders without the control plate at other values of h (Figure 7(d)). This indicate that the control plate is not useful in terms of Clrms reduction at g = 4 in the present case.


### 5.3. Percentage Differences in Fluid Forces

The effect of the control plate, with varying height, in terms of percentage differences of fluid forces without and with the control plate is presented and discussed in this section. The percentage differences of Cdmean, St, Cdrms, and Clrms for both cylinders are computed at small, intermediate, and large gap ratios. The computed values are given in Table5. In Table 5, the positive sign of percentage differences indicates the reduction, while the negative sign in percentage differences indicates increment in the resulting fluid forces parameters. It can be seen from Table 5 that, for g = 0.5 and 1.5, the mean drag coefficients of both cylinders reduced considerably as compared to that without the control plate case, and a maximum reduction of 45.94% can be seen in Cdmean2 at (g, h) = (0.5, 0.8). While surprisingly, for g = 4, the mean drag coefficients of both cylinders increased, and a maximum percentage increase of 6.89% occurred for both the cylinders at (g, h) = (4, 0.8). For the Strouhal number values, the reduction can be seen for all gap ratios except (g, h) = (1.5, 0.2) where both St1 and St2 increased 1.80% and 1.44%, respectively, and at (g, h) = (4, 0.2), there is 2.05% increase in St2 which is the maximum one for all the chosen cases. On the contrary, the maximum reduction in St1 and St2 is seen at (g, h) = (0.5, 0.8) which is 18.66%. Furthermore, Table 5 also shows that the reduction in St values is less than the Cdmean values, which indicates that the control plate has more influence on Cdmean as compared to St.Table 5
Percentage differences of Cdmean, St, Cdrms, and Clrms.
GhAge Cdmean1 (%)Age Cdmean2 (%)Age St1 (%)Age St2 (%)Age Cdrms1 (%)Age Cdrms2 (%)Age Clrms1 (%)Age Clrms2 (%)0.50.27.1311.747.487.4839.9644.1746.2952.660.414.0919.785.914.0962.2963.8864.3166.740.628.8632.820.470.4786.6482.8195.6992.550.842.5945.9418.6618.6683.2283.4795.2195.391.50.23.161.88−1.80−1.4418.8717.98−14.41−19.790.45.203.140.240.602524.24−8.05−15.270.66.296.004.372.6321.7223.94−6.15−6.920.86.297.268.498.8030.7628.98−5.22−7.9840.2−0.5−0.50−2.0511.6811.68−0.73−0.730.4−1.48−1.482.102.0513.7413.74−5.33−5.330.6−3.32−3.322.102.05−8.93−8.93−17.24−17.240.8−6.89−6.894.144.15−88.66−89.69−43.08−43.19In terms of percentage differences, the root-mean-square values of drag and lift coefficients are very sensitive. Cdrms of both cylinders reduced substantially atg = 0.5, and a maximum reduction of 83.47% can be seen for Cdrms2 at (g, h) = (0.5, 0.8). For g = 1.5, Cdrms of both cylinders reduced, but compared to g = 0.5, these reductions are less (see Table 5). While, in the case of g = 4, at small plate heights (0.2 and 0.4), reduction in Cdrms occurs, while for h = 0.6 and 0.8, Cdrms increased and a maximum increment of 89.69% can be seen at (g, h) = (4, 0.8). The Clrms of both the cylinders reduces at the small gap ratio g = 0.5 and opposite trend can be seen for g = 1.5 and 4 for all chosen heights of the control plate. The maximum reduction and increment in Clrms2 is 95.39% and 43.19% which occurs at (g, h) = (0.5, 0.8) and (4, 0.8), respectively.From the abovementioned discussion, it can be deduced that the small gap ratio (g = 0.5) is favorable in terms of reduction in flow-induced forces and flow control for two side-by-side cylinders for all chosen heights of the control plate. And as the gap ratio increased, the effect of the control plate minimizes gradually. At sufficiently higher gap ratios, the opposite trend occurs due to the control plate; that is, instead of control, it upsurges the fluid forces parameters. Furthermore, the plate height h = 0.8 is also critical in terms of reduction and increase in fluid forces parameters of two side-by-side cylinders.


## 5.1. Flow Structures

The flow around the cylinders exhibits different patterns under the influence of plate’s height and gap ratios. The results show three main flow patterns which are categorized as the single bluff body flow, flip-flopping flow, and antiphase vortex shedding flow patterns. Each of these flow patterns have certain characteristics which are discussed below.


### 5.1.1. Single Bluff Body Flow

When the gap ratio between the cylinders and control plate is small, the triad of objects behaves like a single body and the associated flow behavior is termed as single bluff body flow. In the present case, the single bluff body flow is found at (g, h) = (0, 0.2–0.8) and (0.5, 0.2–0.8). To discuss the characteristics of the single bluff body flow, (g, h) = (0, 0.8) and (0.5, 0.6) are considered as representative cases and vorticity contour visualization, and temporal variations of Cd and Cl and power spectrum for these cases are shown in Figure 2. The shear layers detaching from the plate immediately interact with the cylinders and roll up to form vortices at the down wake region of the cylinders (Figure 2(a)). No gap flow is seen due to closely packed bodies. With a slight increment in the gap ratio, it can be seen that the shear layers detached from the plate move inside the gaps and create a jet flow, but the flow remains similar to that of the single bluff body in which vortices are formed only at the down wake region (see Figure 2(b)). This phenomenon was also reported by Bao et al. [41]. It is clear from Figures 2(c) and 2(d) that, due to resemblance with the single bluff body flow, both drag and lift forces on the cylinders illustrate periodic behavior. Similar variation of force coefficients was observed at (g, h) = (0.5, 0.6) and other cases of the single bluff body flow (not shown here). Also, the drag coefficients of both cylinders are antiphase to each other while the lift coefficients of both cylinders are in phase to each other with the same amplitude of consecutive cycles. This can be attributed to the generation mechanism of vortices from the cylinders.Figure 2
Flow structure mechanism for the single bluff body flow pattern.
(a)(b)(c)(d)(e)(f)From the power spectrum graphs for the single bluff body flow, minor secondary peaks along with primary peak can be seen (Figures2(d) and 2(e)). Furthermore, the power spectrum energy of both cylinders is almost the same which indicates the smooth passage of the flow in the down wake region. Bao et al. [42] reported the single bluff body flow at g = 1.5 for the flow around three circular cylinders in the equilateral triangular arrangement. Rahman et al. [43] also reported the single bluff body flow pattern for the flow behind two side-by-side square cylinders in the presence of a downstream control plate.


### 5.1.2. Flip-Flopping Flow

Another flow pattern spotted in this article is the flip-flopping flow pattern which spans over (g, h) = (1, 0.2–0.8) and (1.5, 0.2–0.8). The representative case of this flow structure is depicted in Figure 3 at (g, h) = (1, 0.6). The vorticity contour shows that the fluid flows inside the gaps after detaching from the plate due to increase in the gap ratio. The vortices generated from the plate move in the gaps as well as on the outer sides of the cylinders which results in irregular vortex shedding and complex flow structure in the downstream wake region (Figure 3(a)). This complexity in flow structure is due to strong interaction of jet between the cylinders. In this flow pattern, there is no definite shape or size of the vortices contrary to those observed in case of the single bluff body flow.Figure 3
(a) Vorticity contour visualization, (b, c) signals of CD and CL, and (d, e) FFT spectrum of Cl for the flip-flopping flow pattern.
(a)(b)(c)(d)(e)The shed vortices are not apparent at near downstream; while at far downstream of the computational domain, the vortices shattered, and a dispersed flow structure can be witnessed from Figure3(a). Due to modulation in flow structure, the resulting Cd and Cl exhibit irregularity (Figures 3(b) and 3(c)) because, owing to insufficient gap space, the jet flow strongly affects the flow structure at the immediate down wake region and this effect of the jet flow transports in the whole down wake region. Yan et al. [44] also reported the irregularity of vortices at downstream due to the passage of vortices of the upstream cylinder through gaps. Furthermore, due to the irregularity of vortices, the variations in drag and lift were also reported in their work. Multiple diffusive peaks in power spectrum graphs also appear due to irregularity in flow structure, and that is why it is difficult to compute St from these diffused peaks (see Figures 3(d) and 3(e)).


### 5.1.3. Antiphase Vortex Shedding Flow

The antiphase vortex shedding flow structure is seen at larger gap ratiosg = 2–4 and plate heights h = 0.2–0.4. For discussion, (g, h) = (2.5, 0.4) is considered as a representative case, and its corresponding vorticity contour and force coefficients graphs are shown in Figure 4. It is clear from vorticity contour that, as the gap between cylinders is large enough, the vortices shed from the control plate and interact with the front sides of the cylinders, as well as move inside the gap between the cylinders. In the wake of the cylinders, the vortices are irregular with different shapes and sizes (Figure 4(a)). Yan et al. [44] also reported such type of flow structure. According to Moussaoui et al. [45], when three square cylinders are arranged in V-shape and the gap between the cylinders is large (g = 2), then both the objects form their own vortex street of vortices like a single object. The same behavior can also be seen in Figure 4(a). Furthermore, the vortices generated from the cylinders are antiphase to each other, and this behavior can also be verified from the lift signals given in Figure 4(c). From time-signal graphs of Cd and Cl, it is apparent that both cylinders experience the same drag force (Figures 4(b) and 4(c)). The amplitude of oscillating lift force on both cylinders is identical, but it acts in opposite direction (antiphase) on both cylinders. Due to the periodic variation of Cl, a single dominant peak can be observed from power spectrum with identical spectrum energy (Figures 4(d) and 4(e)).Figure 4
(a) Snapshot of vorticity, (b, c) temporal variations of drag and lift coefficients, and (d, e) power spectrum of the lift coefficient at (g, h) = (2.5, 0.4).
(a)(b)(c)(d)(e)


## 5.1.1. Single Bluff Body Flow

When the gap ratio between the cylinders and control plate is small, the triad of objects behaves like a single body and the associated flow behavior is termed as single bluff body flow. In the present case, the single bluff body flow is found at (g, h) = (0, 0.2–0.8) and (0.5, 0.2–0.8). To discuss the characteristics of the single bluff body flow, (g, h) = (0, 0.8) and (0.5, 0.6) are considered as representative cases and vorticity contour visualization, and temporal variations of Cd and Cl and power spectrum for these cases are shown in Figure 2. The shear layers detaching from the plate immediately interact with the cylinders and roll up to form vortices at the down wake region of the cylinders (Figure 2(a)). No gap flow is seen due to closely packed bodies. With a slight increment in the gap ratio, it can be seen that the shear layers detached from the plate move inside the gaps and create a jet flow, but the flow remains similar to that of the single bluff body in which vortices are formed only at the down wake region (see Figure 2(b)). This phenomenon was also reported by Bao et al. [41]. It is clear from Figures 2(c) and 2(d) that, due to resemblance with the single bluff body flow, both drag and lift forces on the cylinders illustrate periodic behavior. Similar variation of force coefficients was observed at (g, h) = (0.5, 0.6) and other cases of the single bluff body flow (not shown here). Also, the drag coefficients of both cylinders are antiphase to each other while the lift coefficients of both cylinders are in phase to each other with the same amplitude of consecutive cycles. This can be attributed to the generation mechanism of vortices from the cylinders.Figure 2
Flow structure mechanism for the single bluff body flow pattern.
(a)(b)(c)(d)(e)(f)From the power spectrum graphs for the single bluff body flow, minor secondary peaks along with primary peak can be seen (Figures2(d) and 2(e)). Furthermore, the power spectrum energy of both cylinders is almost the same which indicates the smooth passage of the flow in the down wake region. Bao et al. [42] reported the single bluff body flow at g = 1.5 for the flow around three circular cylinders in the equilateral triangular arrangement. Rahman et al. [43] also reported the single bluff body flow pattern for the flow behind two side-by-side square cylinders in the presence of a downstream control plate.


## 5.1.2. Flip-Flopping Flow

Another flow pattern spotted in this article is the flip-flopping flow pattern which spans over (g, h) = (1, 0.2–0.8) and (1.5, 0.2–0.8). The representative case of this flow structure is depicted in Figure 3 at (g, h) = (1, 0.6). The vorticity contour shows that the fluid flows inside the gaps after detaching from the plate due to increase in the gap ratio. The vortices generated from the plate move in the gaps as well as on the outer sides of the cylinders which results in irregular vortex shedding and complex flow structure in the downstream wake region (Figure 3(a)). This complexity in flow structure is due to strong interaction of jet between the cylinders. In this flow pattern, there is no definite shape or size of the vortices contrary to those observed in case of the single bluff body flow.Figure 3
(a) Vorticity contour visualization, (b, c) signals of CD and CL, and (d, e) FFT spectrum of Cl for the flip-flopping flow pattern.
(a)(b)(c)(d)(e)The shed vortices are not apparent at near downstream; while at far downstream of the computational domain, the vortices shattered, and a dispersed flow structure can be witnessed from Figure3(a). Due to modulation in flow structure, the resulting Cd and Cl exhibit irregularity (Figures 3(b) and 3(c)) because, owing to insufficient gap space, the jet flow strongly affects the flow structure at the immediate down wake region and this effect of the jet flow transports in the whole down wake region. Yan et al. [44] also reported the irregularity of vortices at downstream due to the passage of vortices of the upstream cylinder through gaps. Furthermore, due to the irregularity of vortices, the variations in drag and lift were also reported in their work. Multiple diffusive peaks in power spectrum graphs also appear due to irregularity in flow structure, and that is why it is difficult to compute St from these diffused peaks (see Figures 3(d) and 3(e)).


## 5.1.3. Antiphase Vortex Shedding Flow

The antiphase vortex shedding flow structure is seen at larger gap ratiosg = 2–4 and plate heights h = 0.2–0.4. For discussion, (g, h) = (2.5, 0.4) is considered as a representative case, and its corresponding vorticity contour and force coefficients graphs are shown in Figure 4. It is clear from vorticity contour that, as the gap between cylinders is large enough, the vortices shed from the control plate and interact with the front sides of the cylinders, as well as move inside the gap between the cylinders. In the wake of the cylinders, the vortices are irregular with different shapes and sizes (Figure 4(a)). Yan et al. [44] also reported such type of flow structure. According to Moussaoui et al. [45], when three square cylinders are arranged in V-shape and the gap between the cylinders is large (g = 2), then both the objects form their own vortex street of vortices like a single object. The same behavior can also be seen in Figure 4(a). Furthermore, the vortices generated from the cylinders are antiphase to each other, and this behavior can also be verified from the lift signals given in Figure 4(c). From time-signal graphs of Cd and Cl, it is apparent that both cylinders experience the same drag force (Figures 4(b) and 4(c)). The amplitude of oscillating lift force on both cylinders is identical, but it acts in opposite direction (antiphase) on both cylinders. Due to the periodic variation of Cl, a single dominant peak can be observed from power spectrum with identical spectrum energy (Figures 4(d) and 4(e)).Figure 4
(a) Snapshot of vorticity, (b, c) temporal variations of drag and lift coefficients, and (d, e) power spectrum of the lift coefficient at (g, h) = (2.5, 0.4).
(a)(b)(c)(d)(e)


## 5.2. Force Statistics

The effect of the control plate with varying height (h = 0.2–0.8) on fluid flow parameters such as the mean drag coefficient, the Strouhal number, and Cdrms and Clrms of both cylinders is discussed in this section at small (g = 0.5), intermediate (g = 1.5), and large (g = 4) gap ratios. The data of two square cylinders without control plates are also provided for comparison purpose.


### 5.2.1. At Small Gap Ratio (g = 0.5)

At smallg, the Cdmean of both cylinders decreases with an increment in h (see Figure 5(a)). The local maxima for the mean drag coefficient of c1 and c2 are 1.7772 and 1.7882, respectively, and can be seen at h = 0.2. Whereas the local minima for the mean drag coefficient of c1 and c2 is identical and its value is 1.0953, which is observed at h = 0.8. The values of Cdmean1 and Cdmean2 of the cylinders without the control plate are 1.9801 and 2.0260, respectively. Without the control plate, Cdmean of c1 is smaller than c2. The values of Cdmean1 and Cdmean2 of both cylinders with the control plate are smaller than Cdmean1 and Cdmean2 of both cylinders without the control plate (Figure 5(a)). Also, by increasing the height of the control plate, the mean drag coefficient values decrease which shows that Cdmean reduced by placing a control plate at the upstream position, and for larger height of the control plate, this reduction is more significant (see Figure 5(a)).Figure 5
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at a small gap ratio.
(a)(b)(c)(d)The St values of both cylinders with the control plate increase in the range fromh = 0.2 to 0.6, but for h = 0.8, St of both cylinders exhibit slight decreasing behaviour (Figure 5(b)). The St values of both cylinders with the control plate are identical to the St values of the cylinders without the control plate at h = 0.6 and smaller for other heights of the control plate (Figure 5(b)). From this discussion, it can be deduced that, for control plate heights h = 0.2–0.6, the effect of the control plate is almost negligible in terms of the Strouhal number. The maximum St1 and St2 values for the cylinders with control plates are 0.1264 at h = 0.6, while the minimum St1 and St2 values are the same, which is 0.1033 and can be seen at h = 0.8. Cylinders without a control plate have identical St1 and St2 value of 0.1270 (Figure 5(b)).The root-mean-square value of the drag coefficient for both cylinders with the control plate shows decreasing behaviour with increment in the plate height (Figure5(c)). For cylinders with a control plate, the local maxima for Cdrms of c1 and c2 are 0.0984 and 0.0929 observed at h = 0.2 and the local minima for Cdrms of c1 and c2 are 0.0219 and 0.0275 observed at h = 0.6 and 0.8, respectively. Without the control plate, Cdrms of c1 and c2 are 0.1639 and 0.1664, respectively. For cylinders without the control plate, the root-mean-square value of c1 is higher than c2. The maximum and minimum values of root-mean-square values of c1 and c2 Cdrms values of the cylinders with the control plate are smaller than the Cdrms values of the cylinders without the control plate, which shows the Cdrms values reduced by placing the control plate at the upstream position, and this reduction increases as the control plate height increases. With increment in h, Clrms shows decreasing trend (see Figure 5(d)). The maximum Clrms1 and Clrms2 values for the cylinders with control plates are 0.1469 and 0.1335 at h = 0.2, respectively, and the minimum Clrms1 and Clrms2 values are 0.0118 and 0.0130 at h = 0.6 and h = 0.8, respectively. The Clrms1 and Clrms2 values of the cylinders without the control plate are 0.2735 and 0.2820, respectively. For without the control plate case, the Clrms value of c1 is slightly smaller than c2. From abovementioned discussion, it can be concluded that the maximum and minimum Clrms1 and Clrms2 values of the cylinders with the control plate are lower than the Clrms values of the cylinders without the control plate. This demonstrates that reduction occurs as the control plate is positioned upstream of the cylinders, and this reduction increases as the control plate height increases.


### 5.2.2. At the Intermediate Gap Ratio (g = 1.5)

Cdmean of both cylinders with the control plate at the intermediate gap ratio decreases with increasing height of the control plate (Figure6(a)). The local maxima for mean drag coefficients of c1 and c2 with the control plate are 1.8172 and 1.8289 occur at h = 0.2, while the local minima for mean drag coefficients of c1 and c2 with the control plate are 1.7284 and 1.7286 which occur at h = 0.8. In the case “cylinders without plate,” the mean drag coefficients of c1 and c2 have values of 1.8764 and 1.864, respectively. The mean drag coefficient of c1 is smaller than c2 in the case without the control plate. By comparing the values of mean drag coefficients of both cylinders with and without the control plate, it can be observed that mean drag coefficients of both cylinders with the control plate are smaller than that of without the control plate case. This observation indicates that Cdmean of both cylinders reduced significantly by placing the control plate at the upstream position, and this reduction increases for increasing control plate height which confirms the effect of the control plate at the intermediate gap ratio. The St values of both cylinders with the control plate show decreasing behaviour for increasing h (Figure 6(b)). The Strouhal value of c1 and c2 and St2 with the control plate have maximum values of 0.1701 and 0.1695 and minimum values of 0.1529 and 0.1524 at h = 0.2 and 0.8, respectively. St1 and St2 of cylinders without the control plate have the same value and are equal to 0.1671. Furthermore, it can be observed that the St values of both cylinders, with the control plate case, are larger than the St values of arrangement without the control plate till h < 0.4, and for other heights, it is smaller (see Figure 6(b)). This shows that, upto h < 0.4, the placement of the control plate at the upstream position is not useful because, instead of reduction, it results in increment of the St values; for the remaining control plate heights, it minimizes the St values (Figure 6(b)).Figure 6
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at the intermediate gap ratio.
(a)(b)(c)(d)Cdrms1 and Cdrms2 of the cylinders, with the control plate case, show decreasing behaviour with increasing height of the control plate except that Cdrms1 slightly increases fromh = 0.4 to 0.6 (Figure 6(c)). The local maxima for the root-mean-square value of the drag coefficient of c1 and c2 with the control plate are 0.1337 and 0.1350, respectively, examined at h = 0.2, and the local minima for root-mean-square value of drag coefficient of c1 and c2 with control plate are 0.1141 and 0.1169, respectively, investigated at h = 0.8. In case of without the control plate, the root-mean-square values of drag coefficient of c1 and c2 are 0.1648 and 0.1646, respectively. The Cdrms value of c1 is smaller than the Cdrms value of c2 in without the control plate case. By comparing the Cdrms of both cylinders with and without the control plate, it is observed that Cdrms values of both cylinders with the control plate are smaller than the Cdrms values of cylinders without the control plate (Figure 6(c)), which again confirm that the control plate is effective for the reduction of Cdrms values. The Clrms values variation for both cylinders are shown in (Figure 6(d)). Clrms1 shows decreasing behaviour for increasing height of the control plate, while Clrms2 shows decreasing behaviour for h = 0.2 to 0.6, while from h = 0.6 to 0.8, it increases. The local maxima for root-mean-square values of lift coefficient of c1 and c2 are 0.6010 and 0.6095, respectively, which occurred at h = 0.2, while the local minima for root-mean-square values of lift coefficient of c1 and c2 are 0.5527 and 0.5440, which are observed at h = 0.8 and 0.6, respectively. The root-mean-square values of lift coefficient of c1 is smaller than c2, in without control plate case. It can be seen from Figure 6(d) that the control plate is not effective at g = 1.5 for all chosen heights in terms of Clrms values because the Clrms values of both cylinders with control plate are higher than the Clrms values without control plate case.


### 5.2.3. At the Large Gap Ratio (g = 4)

At largeg, the mean drag coefficient of both c1 and c2 with the control plate shows increasing behaviour with an increment in h (Figure 7(a)). The local maxima for mean drag coefficients of c1 and c2 with the control plate are 1.6882 and 1.6881, respectively, observed at h = 0.8 and the local minimum value is 1.5873 at h = 0.2, while Cdmean1 and Cdmean2 of the cylinders without the control plate have the same value of 1.5794 (Figure 7(a)). It is obvious that, for g = 4, the control plate is not effective in terms of Cdmean reduction as these values of Cdmean become higher than those of without the control plate case after placing the plate. This indicates that the control plate is not beneficial in terms of Cdmean for g = 4. The St values for the large gap ratio of cylinders with the control plate illustrate a declining behaviour with an increment in h (Figure 7(b)). The local maxima for the Strouhal number of c1 and c2 with the control plate are 0.1667 and 0.1695 investigated at h = 0.2, and the local minima for the Strouhal number of c1 and c2 with the control plate are 0.1598 and 0.1592 investigated at h = 0.8. In without the control plate case, the Strouhal number of c1 and c2 are 0.1667 and 0.1661, respectively. The Strouhal number of c1 is higher than the Strouhal number of c2. It is obvious from this graph that, at h = 0.2, St2 with the control plate is higher than St2 without the control plate while St1 of both the arrangements is identical, and for remaining chosen heights, the St values, for with the control plate case, are smaller than the St values of without the control plate (see Figure 7(b)). This shows that the control plate is useful for reduction of St for h = 0.4–0.8 (Figure 7(b)).Figure 7
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at the large gap ratio.
(a)(b)(c)(d)The Cdrms values of both cylinders with the control plate show increasing behaviour for increasingh (Figure 7(c)). The local maxima for the root-mean-square value of drag coefficient of c1 and c2 are 0.0549 and 0.0552, respectively, observed at h = 0.8, and the local minima for the root-mean-square value of drag coefficient of c1 and c2 have identical value of 0.0251 observed at h = 0.4. By comparing the Cdrms values of both cylinders with and without the control plate, it is observed that the values with the control plate are smaller than without the control plate case for h < 0.6, while this trend reverses for h > 0.6 (Figure 7(c)). Like Cdrms values, Clrms1 and Clrms2 also show increasing behaviour for increasing height of the control plate (Figure 7(d)). The local maxima for the root-mean-square value of lift coefficient of c1 and c2 are 0.5105 and 0.5109 for cylinders with the control plate case at h = 0.8. The local minima for the root-mean-square value of lift coefficient of c1 and c2 with the control plate have an identical value of 0.3568 and can be seen at h = 0.2. Furthermore, from figure, it is apparent that Clrms1 and Clrms2 of the cylinders with the control plate are like Clrms1 and Clrms2 of the cylinders without the control plate at h = 0.2, while Clrms1 and Clrms2 of the cylinders with the control plate are greater than Clrms1 and Clrms2 of the cylinders without the control plate at other values of h (Figure 7(d)). This indicate that the control plate is not useful in terms of Clrms reduction at g = 4 in the present case.


## 5.2.1. At Small Gap Ratio (g = 0.5)

At smallg, the Cdmean of both cylinders decreases with an increment in h (see Figure 5(a)). The local maxima for the mean drag coefficient of c1 and c2 are 1.7772 and 1.7882, respectively, and can be seen at h = 0.2. Whereas the local minima for the mean drag coefficient of c1 and c2 is identical and its value is 1.0953, which is observed at h = 0.8. The values of Cdmean1 and Cdmean2 of the cylinders without the control plate are 1.9801 and 2.0260, respectively. Without the control plate, Cdmean of c1 is smaller than c2. The values of Cdmean1 and Cdmean2 of both cylinders with the control plate are smaller than Cdmean1 and Cdmean2 of both cylinders without the control plate (Figure 5(a)). Also, by increasing the height of the control plate, the mean drag coefficient values decrease which shows that Cdmean reduced by placing a control plate at the upstream position, and for larger height of the control plate, this reduction is more significant (see Figure 5(a)).Figure 5
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at a small gap ratio.
(a)(b)(c)(d)The St values of both cylinders with the control plate increase in the range fromh = 0.2 to 0.6, but for h = 0.8, St of both cylinders exhibit slight decreasing behaviour (Figure 5(b)). The St values of both cylinders with the control plate are identical to the St values of the cylinders without the control plate at h = 0.6 and smaller for other heights of the control plate (Figure 5(b)). From this discussion, it can be deduced that, for control plate heights h = 0.2–0.6, the effect of the control plate is almost negligible in terms of the Strouhal number. The maximum St1 and St2 values for the cylinders with control plates are 0.1264 at h = 0.6, while the minimum St1 and St2 values are the same, which is 0.1033 and can be seen at h = 0.8. Cylinders without a control plate have identical St1 and St2 value of 0.1270 (Figure 5(b)).The root-mean-square value of the drag coefficient for both cylinders with the control plate shows decreasing behaviour with increment in the plate height (Figure5(c)). For cylinders with a control plate, the local maxima for Cdrms of c1 and c2 are 0.0984 and 0.0929 observed at h = 0.2 and the local minima for Cdrms of c1 and c2 are 0.0219 and 0.0275 observed at h = 0.6 and 0.8, respectively. Without the control plate, Cdrms of c1 and c2 are 0.1639 and 0.1664, respectively. For cylinders without the control plate, the root-mean-square value of c1 is higher than c2. The maximum and minimum values of root-mean-square values of c1 and c2 Cdrms values of the cylinders with the control plate are smaller than the Cdrms values of the cylinders without the control plate, which shows the Cdrms values reduced by placing the control plate at the upstream position, and this reduction increases as the control plate height increases. With increment in h, Clrms shows decreasing trend (see Figure 5(d)). The maximum Clrms1 and Clrms2 values for the cylinders with control plates are 0.1469 and 0.1335 at h = 0.2, respectively, and the minimum Clrms1 and Clrms2 values are 0.0118 and 0.0130 at h = 0.6 and h = 0.8, respectively. The Clrms1 and Clrms2 values of the cylinders without the control plate are 0.2735 and 0.2820, respectively. For without the control plate case, the Clrms value of c1 is slightly smaller than c2. From abovementioned discussion, it can be concluded that the maximum and minimum Clrms1 and Clrms2 values of the cylinders with the control plate are lower than the Clrms values of the cylinders without the control plate. This demonstrates that reduction occurs as the control plate is positioned upstream of the cylinders, and this reduction increases as the control plate height increases.


## 5.2.2. At the Intermediate Gap Ratio (g = 1.5)

Cdmean of both cylinders with the control plate at the intermediate gap ratio decreases with increasing height of the control plate (Figure6(a)). The local maxima for mean drag coefficients of c1 and c2 with the control plate are 1.8172 and 1.8289 occur at h = 0.2, while the local minima for mean drag coefficients of c1 and c2 with the control plate are 1.7284 and 1.7286 which occur at h = 0.8. In the case “cylinders without plate,” the mean drag coefficients of c1 and c2 have values of 1.8764 and 1.864, respectively. The mean drag coefficient of c1 is smaller than c2 in the case without the control plate. By comparing the values of mean drag coefficients of both cylinders with and without the control plate, it can be observed that mean drag coefficients of both cylinders with the control plate are smaller than that of without the control plate case. This observation indicates that Cdmean of both cylinders reduced significantly by placing the control plate at the upstream position, and this reduction increases for increasing control plate height which confirms the effect of the control plate at the intermediate gap ratio. The St values of both cylinders with the control plate show decreasing behaviour for increasing h (Figure 6(b)). The Strouhal value of c1 and c2 and St2 with the control plate have maximum values of 0.1701 and 0.1695 and minimum values of 0.1529 and 0.1524 at h = 0.2 and 0.8, respectively. St1 and St2 of cylinders without the control plate have the same value and are equal to 0.1671. Furthermore, it can be observed that the St values of both cylinders, with the control plate case, are larger than the St values of arrangement without the control plate till h < 0.4, and for other heights, it is smaller (see Figure 6(b)). This shows that, upto h < 0.4, the placement of the control plate at the upstream position is not useful because, instead of reduction, it results in increment of the St values; for the remaining control plate heights, it minimizes the St values (Figure 6(b)).Figure 6
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at the intermediate gap ratio.
(a)(b)(c)(d)Cdrms1 and Cdrms2 of the cylinders, with the control plate case, show decreasing behaviour with increasing height of the control plate except that Cdrms1 slightly increases fromh = 0.4 to 0.6 (Figure 6(c)). The local maxima for the root-mean-square value of the drag coefficient of c1 and c2 with the control plate are 0.1337 and 0.1350, respectively, examined at h = 0.2, and the local minima for root-mean-square value of drag coefficient of c1 and c2 with control plate are 0.1141 and 0.1169, respectively, investigated at h = 0.8. In case of without the control plate, the root-mean-square values of drag coefficient of c1 and c2 are 0.1648 and 0.1646, respectively. The Cdrms value of c1 is smaller than the Cdrms value of c2 in without the control plate case. By comparing the Cdrms of both cylinders with and without the control plate, it is observed that Cdrms values of both cylinders with the control plate are smaller than the Cdrms values of cylinders without the control plate (Figure 6(c)), which again confirm that the control plate is effective for the reduction of Cdrms values. The Clrms values variation for both cylinders are shown in (Figure 6(d)). Clrms1 shows decreasing behaviour for increasing height of the control plate, while Clrms2 shows decreasing behaviour for h = 0.2 to 0.6, while from h = 0.6 to 0.8, it increases. The local maxima for root-mean-square values of lift coefficient of c1 and c2 are 0.6010 and 0.6095, respectively, which occurred at h = 0.2, while the local minima for root-mean-square values of lift coefficient of c1 and c2 are 0.5527 and 0.5440, which are observed at h = 0.8 and 0.6, respectively. The root-mean-square values of lift coefficient of c1 is smaller than c2, in without control plate case. It can be seen from Figure 6(d) that the control plate is not effective at g = 1.5 for all chosen heights in terms of Clrms values because the Clrms values of both cylinders with control plate are higher than the Clrms values without control plate case.


## 5.2.3. At the Large Gap Ratio (g = 4)

At largeg, the mean drag coefficient of both c1 and c2 with the control plate shows increasing behaviour with an increment in h (Figure 7(a)). The local maxima for mean drag coefficients of c1 and c2 with the control plate are 1.6882 and 1.6881, respectively, observed at h = 0.8 and the local minimum value is 1.5873 at h = 0.2, while Cdmean1 and Cdmean2 of the cylinders without the control plate have the same value of 1.5794 (Figure 7(a)). It is obvious that, for g = 4, the control plate is not effective in terms of Cdmean reduction as these values of Cdmean become higher than those of without the control plate case after placing the plate. This indicates that the control plate is not beneficial in terms of Cdmean for g = 4. The St values for the large gap ratio of cylinders with the control plate illustrate a declining behaviour with an increment in h (Figure 7(b)). The local maxima for the Strouhal number of c1 and c2 with the control plate are 0.1667 and 0.1695 investigated at h = 0.2, and the local minima for the Strouhal number of c1 and c2 with the control plate are 0.1598 and 0.1592 investigated at h = 0.8. In without the control plate case, the Strouhal number of c1 and c2 are 0.1667 and 0.1661, respectively. The Strouhal number of c1 is higher than the Strouhal number of c2. It is obvious from this graph that, at h = 0.2, St2 with the control plate is higher than St2 without the control plate while St1 of both the arrangements is identical, and for remaining chosen heights, the St values, for with the control plate case, are smaller than the St values of without the control plate (see Figure 7(b)). This shows that the control plate is useful for reduction of St for h = 0.4–0.8 (Figure 7(b)).Figure 7
Influence of the control plate on (a) Cdmean, (b) St, (c) Cdrms, and (d) Clrms at the large gap ratio.
(a)(b)(c)(d)The Cdrms values of both cylinders with the control plate show increasing behaviour for increasingh (Figure 7(c)). The local maxima for the root-mean-square value of drag coefficient of c1 and c2 are 0.0549 and 0.0552, respectively, observed at h = 0.8, and the local minima for the root-mean-square value of drag coefficient of c1 and c2 have identical value of 0.0251 observed at h = 0.4. By comparing the Cdrms values of both cylinders with and without the control plate, it is observed that the values with the control plate are smaller than without the control plate case for h < 0.6, while this trend reverses for h > 0.6 (Figure 7(c)). Like Cdrms values, Clrms1 and Clrms2 also show increasing behaviour for increasing height of the control plate (Figure 7(d)). The local maxima for the root-mean-square value of lift coefficient of c1 and c2 are 0.5105 and 0.5109 for cylinders with the control plate case at h = 0.8. The local minima for the root-mean-square value of lift coefficient of c1 and c2 with the control plate have an identical value of 0.3568 and can be seen at h = 0.2. Furthermore, from figure, it is apparent that Clrms1 and Clrms2 of the cylinders with the control plate are like Clrms1 and Clrms2 of the cylinders without the control plate at h = 0.2, while Clrms1 and Clrms2 of the cylinders with the control plate are greater than Clrms1 and Clrms2 of the cylinders without the control plate at other values of h (Figure 7(d)). This indicate that the control plate is not useful in terms of Clrms reduction at g = 4 in the present case.


## 5.3. Percentage Differences in Fluid Forces

The effect of the control plate, with varying height, in terms of percentage differences of fluid forces without and with the control plate is presented and discussed in this section. The percentage differences of Cdmean, St, Cdrms, and Clrms for both cylinders are computed at small, intermediate, and large gap ratios. The computed values are given in Table5. In Table 5, the positive sign of percentage differences indicates the reduction, while the negative sign in percentage differences indicates increment in the resulting fluid forces parameters. It can be seen from Table 5 that, for g = 0.5 and 1.5, the mean drag coefficients of both cylinders reduced considerably as compared to that without the control plate case, and a maximum reduction of 45.94% can be seen in Cdmean2 at (g, h) = (0.5, 0.8). While surprisingly, for g = 4, the mean drag coefficients of both cylinders increased, and a maximum percentage increase of 6.89% occurred for both the cylinders at (g, h) = (4, 0.8). For the Strouhal number values, the reduction can be seen for all gap ratios except (g, h) = (1.5, 0.2) where both St1 and St2 increased 1.80% and 1.44%, respectively, and at (g, h) = (4, 0.2), there is 2.05% increase in St2 which is the maximum one for all the chosen cases. On the contrary, the maximum reduction in St1 and St2 is seen at (g, h) = (0.5, 0.8) which is 18.66%. Furthermore, Table 5 also shows that the reduction in St values is less than the Cdmean values, which indicates that the control plate has more influence on Cdmean as compared to St.Table 5
Percentage differences of Cdmean, St, Cdrms, and Clrms.
GhAge Cdmean1 (%)Age Cdmean2 (%)Age St1 (%)Age St2 (%)Age Cdrms1 (%)Age Cdrms2 (%)Age Clrms1 (%)Age Clrms2 (%)0.50.27.1311.747.487.4839.9644.1746.2952.660.414.0919.785.914.0962.2963.8864.3166.740.628.8632.820.470.4786.6482.8195.6992.550.842.5945.9418.6618.6683.2283.4795.2195.391.50.23.161.88−1.80−1.4418.8717.98−14.41−19.790.45.203.140.240.602524.24−8.05−15.270.66.296.004.372.6321.7223.94−6.15−6.920.86.297.268.498.8030.7628.98−5.22−7.9840.2−0.5−0.50−2.0511.6811.68−0.73−0.730.4−1.48−1.482.102.0513.7413.74−5.33−5.330.6−3.32−3.322.102.05−8.93−8.93−17.24−17.240.8−6.89−6.894.144.15−88.66−89.69−43.08−43.19In terms of percentage differences, the root-mean-square values of drag and lift coefficients are very sensitive. Cdrms of both cylinders reduced substantially atg = 0.5, and a maximum reduction of 83.47% can be seen for Cdrms2 at (g, h) = (0.5, 0.8). For g = 1.5, Cdrms of both cylinders reduced, but compared to g = 0.5, these reductions are less (see Table 5). While, in the case of g = 4, at small plate heights (0.2 and 0.4), reduction in Cdrms occurs, while for h = 0.6 and 0.8, Cdrms increased and a maximum increment of 89.69% can be seen at (g, h) = (4, 0.8). The Clrms of both the cylinders reduces at the small gap ratio g = 0.5 and opposite trend can be seen for g = 1.5 and 4 for all chosen heights of the control plate. The maximum reduction and increment in Clrms2 is 95.39% and 43.19% which occurs at (g, h) = (0.5, 0.8) and (4, 0.8), respectively.From the abovementioned discussion, it can be deduced that the small gap ratio (g = 0.5) is favorable in terms of reduction in flow-induced forces and flow control for two side-by-side cylinders for all chosen heights of the control plate. And as the gap ratio increased, the effect of the control plate minimizes gradually. At sufficiently higher gap ratios, the opposite trend occurs due to the control plate; that is, instead of control, it upsurges the fluid forces parameters. Furthermore, the plate height h = 0.8 is also critical in terms of reduction and increase in fluid forces parameters of two side-by-side cylinders.


## 6. Concluding remarks

In this article, the numerical simulations were carried out to study the effect of a control plate, placed upstream, on fluid flow around two side-by-side square cylinders in terms of regulating the vortex shedding mechanism and reduction of fluid forces. The height of the control plate and the gap ratio were varied between 0.2 to 0.8 and 0 to 4, respectively, while the Reynolds number was fixed at 150. The main findings of this article are summarized below:(i)
The flow structure mechanism, behind the cylinders, was categorized as the single bluff body flow observed at (g, h) = (0–0.5, 0.2–0.8), flip-flopping flow observed at (g, h) = (1–1.5, 0.2–0.8), and antiphase vortex shedding flow observed at (g, h) = (2–4, 0.2–0.8).(ii)
The single bluff body flow was characterized by the fact that all three bodies in triad behaved like a single body and the drag and lift coefficients of the cylinders showed periodic behavior.(iii)
In flip-flopping flow due to strong interaction of jet and gap flows, an irregular and complex flow structure was observed and the flow was dominated by secondary cylinder interaction frequencies.(iv)
In the case of antiphase vortex shedding flow, the detachment of vortices from both cylinders was found to be in opposite direction which resulted in antiphase variation of lift coefficients. Furthermore, the flow in this case was found to be fully developed and dominated by primary vortex shedding frequency.(v)
It was observed that, atg = 0.5 (small gap ratio) for all chosen heights, the control plate reduced the fluid forces and this reduction increased as the height of control plate increased.(vi)
It was also observed that, along with reduction, accession in fluid forces was also investigated and this accession was found for largerg.(vii)
The maximum reduction 45.94% was observed in Cdmean2, 18.66% in St, 83.47% in Cdrms2, and 95.39% in Clrms2.(viii)
It was also found that the control plate, in some cases, fails to control the flow and to reduce the flow induced forces; instead, it results an increase in fluid forces. The maximum increase in such cases was found to be 6.89% in Cdmean1 and Cdmean2, 2.89% in St2, 89.69% in Cdrms2, and 43.19% in Clrms2.



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*Source: 1652994-2023-02-18.xml*