#### Abstract

In this study, the influence of the T-shaped control plate on the fluid flow characteristics around a square cylinder for a low Reynolds numbers flow is systematically presented. The introduction of upstream attached T-shaped control plate is novel of its kind as T-shaped control plate used for the first time rather than the other passive control methods available in the literature. The Reynolds numbers (Re) are chosen to be Re = 100, 150, 200, and 250, and the T-shaped control plate of the same width with varying length is considered. A numerical investigation is performed using the single-relaxation-time lattice Boltzmann method. The numerical results reveal that there exists an optimum length of T-shaped control plate for reducing fluid forces. This optimum length was found to be 0.5 for Re = 100, 150, and 200 and 2 for Re = 250. At this optimum length, the fluctuating drag forces acting on the cylinder are reduced by 134%, 1375, 133%, and 136% for Re = 100, 150, 200, and 250, respectively. Instantaneous and time-averaged flow fields were also presented for some selected cases in order to identify the three different flow regimes around T-shaped control plate and square cylinder system.

#### 1. Introduction

Controlling of flow and suppression of fluid forces around bluff bodies is an important research area for engineers and scientists because of its practical importance in mechanical engineering, structures and buildings, aeronautical engineering etc., at high Reynolds number (Re). Applications at very low Reynolds number can be found in microdevices, such as in micro-electro-mechanical systems (MEMS), computer equipment’s, and cooling of electronic devices. The flow past circular cylinders was mostly encountered in the earlier investigations. Among other bluff structures, square structure plays an important role in various engineering fields. The flow wake around the bluff structures can generate unsteady forces which have the potential to damage the structural integrity. Therefore, it is important to fully understand the flow characteristics and their resulting effects on the structure in order to control the structure integrity. Successful numerical simulation can show valuable flow characteristics and information which can be very complicated to attain experimentally.

Successful flow control remarkably reduces the magnitude and effects of the fluctuating forces directly acting on the surfaces of the bluff body. One can use either active or passive techniques to control the wake and reduce the fluid forces. The passive technique does not require any external energy like the active technique. The previous experimental measurements and numerical studies for flow control include splitter plates [1–10] and control cylinders/rods [11–18] for Re between 80 and 2000. The development of passive methods of reduction of fluid forces and wake control is an active area of research.

Abdi et al. [1] numerically studied the flow characteristics over a circular cylinder using the commercial software COMSOL Multiphysics. It was concluded that single splitter plate and two splitter plates reduce the drag by about 15% and 23%, respectively. Islam et al. [2] studied numerically the reduction of fluid forces for flow past a square cylinder in the presence of upstream, downstream, and dual splitter plates. The authors found that the drag force was reduced by up to 62.2%, 13.3%, and 70.2% for upstream, downstream, and dual splitter plates, respectively. Zhou et al. [3] experimentally observed that the upstream rigid splitter plate does not considerably reduce forces. Sharma and Eswaran [4] experimentally studied the flow past a square cylinder in presence of an attached flexible splitter plate at Re = 400. The authors observed that the drag coefficient varies nonmonotonically due to various flow regimes. The experimental investigation was conducted to study the effect of an attached splitter plate on the rear side of the square cylinder at Re = 485 by Chauhan et al. [5]. They found 23% drag reduction and also found that the vortex shedding starts to suppress for the long length of the splitter plate. Dash et al. [6] numerically examined the fluid forces for flow past a square cylinder in presence of dual attached splitter plates. It was found that the drag was reduced by 21% for Re = 100. Soumya and Prakash [7] numerically studied the effect of downstream attached splitter plate with elliptic cylinder for Re between 50 and 200 using Streamline Upwind/Petrov–Galerkin (SUPG) based finite element method (FEM). It was concluded that the drag forces were significantly reduced as the length of the splitter plate increased. Several two-dimensional numerical investigations have thus been adopted in studies for the reduction of fluid forces using rigid, flexible, and dual splitter plates (e.g., Barman and Bhattacharyya [8], Sharma and Dutta [9], and Sarioglu [10]).

A numerical study on a flow past a circular cylinder in the presence of a control cylinder was carried out by Kim et al. [11]. They discussed in detail the flow characteristics by changing the positions of the control cylinder.

Islam et al. [12] numerically observed that the upstream control plate considerably reduced the drag forces and the shedding frequency for Re varying between 80 and 200 for flow past a square cylinder. Sharma and Dutta [13] experimentally examined the flow past a square cylinder in the presence of attached flexible foil, for different values of flexible foil length. Firat et al. [14] numerically examined the influence of a small control cylinder at the front of the square cylinder for various Reynolds numbers (50 ≤ Re ≤ 200), and it was found that the drag reduction occurs at gap spacing, = *s*/*D* = 2 and 3. Here, *s* is the spacing between the control cylinder and the main square cylinder, and *D* is the size of the square cylinder. Yen et al. [15] experimentally examined the flow regimes, drag coefficient, lift coefficient, turbulence intensity, and vortex shedding frequency behind a square cylinder in the presence of an upstream control bar by varying the values of Re, rotation angles, and values. They found that the upstream control bar reduces drag by about 57%. Zhu and Yao [16] numerically analyzed the effect of surrounding control cylinders for flow over a circular cylinder at intermediate Reynolds numbers. Different small control rods with the same diameter are placed around the main circular cylinder with uniform angle interval (*θ*) and gap spacing. For example, for four control rods, the angle interval is 9°. The authors found that the main circular cylinder attached to nine control rods with *θ* = 40° and = 0.6 can achieve a considerable vortex-induced vibration suppression effect for a wide range of Reynolds numbers. A numerical study on a flow past a square cylinder in the presence of control rods was carried out by Zhu et al. [17] and Chauhan et al. [18]. They discussed in detail the suppression of vortex shedding and reduction of fluid forces. When one bluff body is placed in the wake of another bluff body, the fluid flow characteristics considerably depend on the gap spacing between the bluff bodies. Some representative numerical studies are those of Abbasi et al. [19] and Ahmad and Islam [20]. Abbasi et al. [19] numerically examined the flow fields and fluid forces around three inline cylinders and found considerable drag reduction for the downstream cylinders. Ahmad and Islam [20] observed various flow regimes and sensitivity of fluid forces for four cylinders in diamond arrangements. For a review of the various passive and active methods of drag reduction, the reader is referred to Rashdi et al. [21].

The flow past a square cylinder in the presence of an upstream attached T-shaped control plate is not studied yet. It is important to know how to control the wake and reduce the fluid forces with the different length of the upstream attached T-shaped control plate. The main motivation for the current work is to examine in detail whether the T-shaped control plate considerably reduced the fluid forces and suppressed the vortex shedding? Also, we aim to characterize wake structure behavior, as a function of the length of T-shaped control plate and Reynolds numbers. The other important aim is to identify the suitable length of the T-shaped control plate that is associated with minimum drag and maximum suppression of vortex shedding. To get reasonably reliable knowledge of important design parameters such as drag and lift forces, vortex shedding frequency, and wake size is very important. We believe that this study will further enrich the drag reduction database using a new passive technique (T-shaped control plate).

The paper is organized as follows. In Section 2, lattice Boltzmann method, problem description, boundary conditions, grid independence study, domain independence study, and validation of the developed code are presented with brief discussion. Section 3 includes the numerical results in detail. Finally, in Section 4, conclusions are drawn based on the present numerical results.

#### 2. Lattice Boltzmann Method and Computational Details

Recently, the lattice Boltzmann method (LBM) has been applied successfully to a number of flow problems (see [19, 20] and some references cited in those works). The LBM is a compressible method, the Mach number, Ma = *U*_{∞}/ is set to less than 0.3 so that the compressible effect is negligible [22]. The reason is that we are interested in isothermal flow. In case of isothermal flow, RT = 1/3 is to be chosen in the units of *c* = *δx/δt* = 1. Here, *c* = *δ**x*/*δ**t*, *δ**x*, and *δ**t* are the lattice constant and the time step size, respectively. The governing differential equations for such fluid flow problems mainly consist of the equations of continuity and momentum, as follows: Continuity: Momentum:

In equations (1) to (3), *u*_{x} and *u*_{y} are the dimensionless velocity components along the *x*-direction and *y*-direction of a Cartesian coordinate system. Here, *p* is the pressure.

There are many different lattice Boltzmann methods. For a detailed study of the various LBM methods, the reader is referred to [22, 23]. This section briefly introduces the LBM. The lattice Bhatnagar–Gross–Krook (BGK) model [22] with single-relaxation-time (SRT) is given by [22]

Here, *f*_{i}(** x**,

*t*),

*f*

_{i}

^{(eq)}(

**,**

*x**t*),

*e*_{i}, and

*τ*are the particle distribution function at (

**,**

*x**t*), equilibrium distribution function (the Maxwell–Boltzmann distribution function) at (

**,**

*x**t*), the particle velocity along the

*i*th direction, and the single-relaxation-time parameter, respectively. Note the two-sides of equation (4) represent the solution. The left hand side applies as a streaming step, and the right hand side gives the collision between the particles. The derivation of Navier–Stokes (N-S) equations from LBM is well known nowadays and can be found in some recently published books (e.g., see [22, 23]). The solution of N–S equations can be calculated iteratively from equation (4). Collision step: Streaming step:

Here, represents the postcollision state. One advantage of this is that the streaming step is local in LBM and no need for any computation.

The two-dimensional nine-velocity lattice model (*d*2*q*9, where *d* is the dimensions and *q* is the numbers of particles) [22] is used in this study. In the *d*2*q*9 (see Figure 1) model, *e*_{i} denotes the nine discrete velocity set, as follows:

The equilibrium distribution function (*f*_{i}^{(eq)} (** x**,

*t*)) can be solved from

Here, *ω*_{i} is the weighting coefficient. The weighting coefficient, is given by

The density and momentum fluxes in the discretized velocity space can be obtained asand

The pressure can be calculated through the equation of state and is

The speed of sound for the *d*2*q*9 model is [22]. The corresponding kinematic viscosity in the N-S equations (2) and (3) derived from equation (4) is [22].

It is also known that the single relaxation time LBM is simple and good for parallel systems. Its difficulty lies in the necessity of taking the value of the relaxation time parameter. The stability of the single relaxation time LBM mostly appears at high Reynolds numbers. In such a situation, we need to refine the grids with various relaxation times and to check the results (note that viscosity in lattice units is correlated to the relaxation time). This single relaxation time LBM is conditionally stable and is valid for > 0.5. Here, in this problem, the values of for Re = 100, 150, 200, and 250 are 0.5263, 0.5175, 0.5132, and 0.5105, respectively. One feature of LBM is that the pressure can be calculated through the equation of state instead of Poisson equation [22]. In LBM, we can find the pressure from the equation of state directly once knowing the density instead of using Poisson equation. For the LBM, the CFL (Courant–Friedrichs–Lewy) number is CFL = *e*_{i}*δ**t*/*δ**x* = 1, which is fixed at unity.

Figure 2 presents the computational domain of flow past a square cylinder in the presence of an upstream attached T-shaped control plate. A square cylinder of size *D* is placed within the computational domain. A T-shaped control plate of length *L* is attached to the front surface of the square cylinder. = 0.1*D* is the width of the head of the T-shaped control plate. The distance between the left boundary of the domain and the front surface of the T-shaped control plate is *L*_{u} = 10*D*, whereas the downstream distance of the domain from the rear surface of the square cylinder to the right boundary of the domain is *L*_{d} = 39*D*. The distance between the upper and lower walls is kept as *L*_{y} = 13*D*, resulting in a blockage ratio (*β* = *L*_{y}/*D*) = 13. *L*_{x} = 50*D* is the length of the computational domain. A Cartesian grid was employed, and the origin (0, 0) is set at the center of the main square cylinder. The details of the simulation parameters are summarized in Table 1. *C*_{D} and *C*_{L} are the drag forces and lift forces in the streamwise and transverse directions, respectively.

The following boundary conditions are incorporated in the present study.(i)At inlet, uniform inflow velocity is applied, *u* = 0.04385, = 0(ii)At outlet, convective boundary condition is applied [24](iii)At the upper and lower walls of the domain, no-slip (*u* = = 0) is applied(iv)At the square cylinder and T-shaped control plate surfaces, no-slip boundary condition is imposed

The forces acting on the surfaces of the square cylinder can be calculated from the momentum-exchange method [25].

The implementation of the LBM is simple and straightforward. The following steps are presented for calculating forces and fluid properties:(i)Specify the streaming time step *δ**t.* Calculate the single relaxation time parameter *τ*(ii)The local distribution function must be updated through collision step (equation (5a))(iii)The fluid particles are streamed to neighboring streaming lattice nodes through the streaming step (equation (5b))(iv)Implement suitable initial and boundary conditions for the distribution function(v)Calculate macroscopic variables (equations (9) and (10))(vi)Repeat steps (ii) to (iv) until the convergence criteria or the assigned maximum iteration numbers are reached

It is noticed that the lattice Boltzmann equation only requires the streaming step and collision step to evolve the fluid filled with complex nonlinearities. No special treatment is required for nonlinear terms in Navier–Stokes equations. The LBM explicitly calculates the pressure from the density. For computations, the simulation parameters are shown in Table 1.

The vorticity (dimensional) is defined as

Here, *u* and are the velocity components in the streamwise direction and transverse direction, respectively. It is to be noted that the vorticity is calculated by using a second-order central difference scheme and is then nondimensionalized by the uniform inflow velocity (*U*_{∞}) and the side length of the square cylinder (*D*):

For analysis, we have defined the following nondimensional parameters as given in equations (15)–(18)

Here, Re, *C*_{D}, *C*_{L}, and *St* are the Reynolds number, drag coefficient, lift coefficient, and Strouhal number, respectively. It is to be noted here, *F*_{x} and *F*_{y} are the drag and lift forces experienced by the square cylinder along with the streamwise and transverse directions, respectively. These forces are calculated using the momentum-exchange method [25]. *f*_{s} is the vortex shedding frequency calculated using the fast Fourier transformation (FFT) of the time series of the lift coefficient, *C*_{L}.

The stopping criteria once we get the steady state is

It is noticed that *U*_{∞} = *δ**x*/*δ**tU*_{lb} represents the velocity in a system of lattice units. Here, *U*_{lb} represents the lattice Boltzmann velocity. In LBM, *U*_{lb} is proportional to the Mach number of the fluid. In Tables 2–5, the percentage deviation is also given in brackets. For *D* = 10, 20, 30, and 40, the mean drag coefficient (*C*_{Dmean}), Strouhal number (*St*), root-mean-square value of drag coefficient (*C*_{Drms}), and lift coefficient (*C*_{Lrms}) values are as shown in Table 2. Here, *D* (= *δ**x*) represents the lattice units on each side of the square cylinder and T-shaped control plate. The numerical results for *C*_{Dmean}, *St*, *C*_{Drms}, and *C*_{Lrms} in Tables 3–5 for different values of blockage ratio (*β* = *L*_{y}/*D*), *L*_{u}, and *L*_{d}, respectively. The results presented in Tables 2–5 have been performed at Re = 250 and *L*/*D* = 5. Inspection of Table 2 shows, however, that the *D* effect dies away as *D* ≥ 20. Guo et al. [24] proposed and used that *D* = 20 is reasonably good to achieve good numerical results for the square cylinder. The blockage ratio *β* is 7.7%. It is observed that the *β*-value beyond 13 does not have any considerable influence on the integral parameters presented in Table 3. It was previously expressed that a good two-dimensional result can be achieved for flow of a square cylinder with a blockage ratio greater than about 5% [39]. One can also see the effect of *L*_{u} and *L*_{d} in Tables 4 and 5, respectively. In the present numerical study, the value of the blockage ratio is already more than 5%. *D* = 20, *β* = 13, *L*_{u} = 10*D*, and *L*_{d} = 39*D* is, therefore, used for all computations in the present numerical study.

In order to ensure the validity of the code, we calculate the integral parameter values for Re = 100, 150, 200, and 250 for comparison with available data for flow past a square cylinder without T-shaped control plate. A comparison of the present results with the available data is given in Table 6. Our numerical results are generally in good agreement with available published data. The present results almost fell within the range of available numerical values, even some experimental values (Okajima [27], Norberg [28]). There are some slight differences between the present and experimental results. Their differences arise from experimental uncertainties, using various boundary conditions, the effect of blockage, grid structures, etc.

#### 3. Results and Discussion

It is clear from previous investigations that the reduction of fluid forces and wake control depend on active and passive techniques. Keeping in view these importance, the present study was conducted to systematically analyze the importance of upstream attached T-shaped control plate length, ranging from *L/D* = 0.5 to 8 at Reynolds numbers, Re = 100, 150, 200, and 250. The present numerical results are calculated after the dynamic steady-state condition is reached (see equation (19)). The most important findings on the basis of our numerical investigation will be presented in this section in detail.

Figures 3(a)–3(e) presents the vorticity contours visualization around the square cylinder with and without a T-shaped control plate. Five different cases, one is without T-shaped control plate (*L*/*D* = 0) and another four cases with different T-shaped control plate lengths (*L*/*D* = 2, 4, 6, 8) are considered for the various flow pattern investigation. The solid and dotted lines represent the positive and negative vortices, respectively. It can be seen from the figure that there is no vortex shedding from the T-shaped control plate at any length due to its attachment with the main square cylinder. Strouhal number (*St*) of the isolated cylinder and the square cylinder with upstream attached T-shaped control plate at lengths of 2, 4, 6, and 8 are 0.1510, 0.1618, 0.1316, 0.0896, and 0.1316, respectively. It is observed from the *St* values that the length of the T-shaped control plate caused a decrease and increase in the vortex shedding frequency of the main square cylinder. This decrease and increase can be further confirmed from the vorticity graphs. Qualitatively, the number of vortices that appeared behind the main square cylinder decreased from 13 to 10 for *L*/*D* = 2 and 4, to 6 for *L*/*D* = 6 and then increased from 6 to 11 for *L*/*D* = 6 and 8. It is observed that the formation of vortices is quite sensitive to the length of the upstream attached T-shaped control plate. From these figures, we can draw some important conclusions about flow characteristics. For all *L*/*D* values in Figures 3(a)–3(e), the alternate generation of shed vortices can be clearly seen behind the main square cylinder. This flow is called the single bluff-body flow regime. Shed vortices travel in alternate fashion throughout the flow field. We further divided the single bluff-body flow regime into two different flow regimes. One is called the single bluff body with primary vortex shedding frequency (regime-I), and the second one is called the single bluff body with secondary frequencies together with primary vortex shedding frequency (regime-II). In the case of regime-I, one can see only one dominant shedding frequency peak in the power spectra (see Figures 4(a) and 4(b)). While in the case of regime-II, one can see some minor extra peaks called secondary frequencies together with the dominant vortex shedding frequency (Figures 4(c) and 4(d)). It can also be seen from Figures 3(b)–3(e), the generated shear layers from the bottom and top side of the head of the T-shaped control plate attach to the front and bottom and top sides of the main square cylinder, and then, they combined with the generated shed vortices from the main square cylinder.

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Figures 5(a)–5(e) shows the instantaneous streamlines at different T-shaped control plate lengths at Re = 200. It is observed that the T-shaped control plate length will affect flow features, especially in the wake region.

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At *L/D* = 0 (Figure 5(a)) where it is the single bluff body flow pattern, one bigger eddy is located behind the rear surface of the square cylinder without T-shaped control plate. In Figure 5(b) as *L/D* = 2, there is a small eddy at the bottom of the main square cylinder. The streamline graph further confirms that the shear layers separated from the top and bottom surfaces of the head of the control plate directly reattach to the main square cylinder. Some waviness was also observed in the wake region for all cases presented in Figures 5(a)–5(e). It is clear from Figure 5(c) that there is no close recirculation zone behind the main square cylinder. The streamline graphs further confirm that the separation point changes due to T-shaped attached control plate as compared to the square cylinder (without T-shaped control plate).

Figures 6(a)–6(e) shows the instantaneous pressure contours at different T-shaped control plate length at Re = 200. The use of an upstream attached T-shaped control plate, irrespective of what the flow regime is, reduces the pressure difference between the front surface of the T-shaped control plate and the rear surface of the main square cylinder. As a result, a reduction of the mean drag coefficient in comparison with an isolated occurs. Furthermore, as the *L*/*D* value increases, the pressure distribution above and below the length of the T-shaped control plate also changes.

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As we have seen from the vorticity contours and streamlines that flow changes its characteristics from single bluff body to steady flow and then from steady flow to unsteady flow by changing the values of *L*/*D*. This can also be analyzed from Figures 7(a) and 7(b) which presents the time histories of *C*_{D} and *C*_{L} at various values of *L*/*D* and shows that the amplitude of forces changed by varying *L*/*D* values. It is seen from Figure 6(a) that the *C*_{D} variation is very sensitive to the T-shaped control plate length. The variation of *C*_{D} for *L*/*D* = 4, 6, and 8 are more sensitive than that for *L*/*D* = 2. The oscillations amplitude for the *C*_{L} in comparison with an isolated cylinder (without T-shaped control plate) decreases.

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The power spectrum of lift coefficients at *L/D* = 2, 4, 6, and 8 are shown in Figures 4(a)–4(d). The highest peak in the graphs refers to the primary vortex shedding frequency (*Stp*), and the other small peaks represent the secondary frequencies (*Sts*). In the power spectra graph, ‘*E*’ stands for energy. The single dominant peak confirms the periodic nature of lift coefficients at *L*/*D* = 2 and 4 (Figure 6(b)). Some small modulation exists for *L*/*D* = 6 and 8 in lift coefficients, and as a result, one can see one or two extra small peaks together with the primary vortex shedding frequency.

The instantaneous vorticity contours visualization shown in Figures 8(a)–8(d) clearly illustrates the influence of Re on the flow characteristics. It should be noted that, in the case *L*/*D* = 5 at Re = 100, we observed complete suppression of vortex shedding. This flow regime is called the steady flow regime (regime-III). In steady flow regime only, the streamline can be seen behind the cylinder without any recirculation or shed vortices. These figures clearly illustrate the basic difference between steady flow and single bluff-body flow. Flow behind the main square cylinder, identified by Karman vortex street at Re = 150, has almost similar vortices in alternating fashion from upper and lower surface of the main square cylinder (Figure 8(b)). This flow is called regime-I. In addition, as Reynolds number increased, we observed regime-II. Figures 8(c) and 8(d) show that periodic vortex shedding is further maintained when we increased the Reynolds number Re to 250. As Re increases above 100, the number of shed vortices increases, and consequently, the drag force decreases; see Figure 9(a). It is also noted that the vortices are shed alternatively, and the sizes of the shed vortices from the top and bottom surfaces of the main square cylinder are different. The latter tends to elongate in the near wake, and the former is almost round. This is because of the upstream attached T-shaped control plate length and increased Reynolds numbers. Furthermore, its strength increases as the value of Re increases. As a result, the Strouhal numbers of the square cylinder with the upstream attached T-shaped control plate at Reynolds numbers of 150, 200, and 250 are 0.1024, 0.1056, and 0.1104, respectively.

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Figures 10(a)–10(d) show the pressure distributions along the front and back sides of the main square cylinder with different Reynolds numbers at *L*/*D* = 5. As shown, the pressure distribution on the back side for all Reynolds numbers is considerably changed; thus, the remarkable differences in drag forces are mainly due to the differences in the pressure distribution along the surfaces of the length of the T-shaped control plate. It is observed that the maximum value of pressure exists at the center point of the front surface. However, as the value of Re changes, the pressure distribution on the surfaces of the length of the control plate changes considerably.

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The graphical representation of *C*_{D} and *C*_{L} shown in Figures 9(a) and 9(b) further indicate the flow transitions. The lift coefficient represents the transverse force component which represents the magnitude of the vortex shedding, since no vortices shed from the main square cylinder; that is why, the lift force is steady for Re = 100 at *L*/*D* = 5. It is noticed that due to the steady flow nature the straight constant line can be clearly seen for *C*_{D} and *C*_{L} in Figures 9(a) and 9(b). The profiles of *C*_{L} confirm the periodic nature of the flow except for the *C*_{D} at Re = 150, 200, and 250. The periodic nature confirms that the vortices shed from the upper and lower sides of the cylinder with the same frequency. It is also observed that the *C*_{L} amplitude is somewhat increased as the value of Re increased. The negative drag value is observed for Re = 150, 200, and 250. Due to the presence of T-shaped control plate no periodic nature is observed for Re = 150, 200, and 250.

Figures 11(a)–11(c) shows the spectra analysis of the fluctuating lift coefficient at *L*/*D* = 5 for different Reynolds numbers. It is to be noted that due to the constant nature of *C*_{L} for Re = 100, no vortex shedding is observed behind the cylinder. The power spectra show two minor peaks together with the dominant vortex shedding frequency in the case of regime-II. But still, the primary vortex shedding frequency is the dominant frequency. The dominant primary vortex shedding frequency peak is seen in all the chosen cases presented in Figures 11(a)–11(c), which confirms the single frequency observed in the time trace analysis of lift coefficient *C*_{L}.

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The instantaneous vorticity contour visualization shown in Figures 12(a)–12(e) clearly illustrates the influence of *L/D* on the flow characteristics. It is clear from Figures 12(a)–12(d) that the vortices are shed alternatively from the upper and lower surfaces of the main square cylinder. The spacing between the two shed vortices is almost constant. However, the size of the vortex shed from the upper side of the square cylinder is comparatively larger than the size of the vortex shed from the lower side of the square cylinder. This may be due to the length of the T-shaped control plate, but still alternating vortex shedding can be clearly seen behind the square cylinder. The T-shaped control plate length effect can be seen more clearly from streamlines in Figure 12(e) for *L*/*D* = 6. In Figure 12(e), the flow is completely suppressed, and no alternate vortex shedding can be seen behind the square cylinder. This is called the steady flow regime (regime-III). It is also seen that the vortex formation length becomes longer, and the transverse spacing between the shed vortices becomes smaller as the value of *L/D* increased from 0 to 4. Such observations were found by Zhou et al. [32] for flow past a circular cylinder using tripping rods at angles (*θ*) = 40° and Re = 200. This confirms that the T-shaped control plate can also be used to understand the flow characteristics behind the bluff body.

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As we have seen from the vorticity contours and streamlines that flow changes its characteristics from single bluff body to steady flow regime by changing the values of *L/D*. This can also be analyzed from Figures 13(a) and 13(b) which present the time histories of *C*_{D} and *C*_{L} at various values of *L/D* and show that the amplitude of forces changed by varying *L/D* values. The lift coefficient of *L*/*D* = 1, 2, and 4 has periodic nature with smaller amplitude cycle compared to those observed for flow past an isolated cylinder (*L*/*D* = 0) in Figure 13(b). At *L/D* = 6, the constant behavior can be observed for the square cylinder. The periodic nature of *C*_{L} confirms the alternate shedding behaviour from cylinder at *L/D* = 0, 1, 2, and 4. The periodic nature of *C*_{L} shows the periodic nature of the flow with almost constant amplitude. This periodic nature confirms that the vortices shed from the upper and lower surface of the main square cylinder with same frequency. One can also observe that the *C*_{L} amplitude is somewhat increased in the case of *L*/*D* = 2 than *L*/*D* = 4.

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The forces acting on the square cylinder in the streamwise and transverse directions by the fluid are the important criteria to analyze the flow characteristics quantitatively, and thus, the *C*_{Dmean}, *St*, *C*_{Drms}, and *C*_{Lrms} are depicted in Figures 14(a)–14(h), respectively, where the solid and dotted straight lines refer to the case of a square cylinder without the T-shaped control plate. It is seen from Figures 14(a) and 14(b) that *C*_{Dmean} of the square cylinder with T-shaped control plate is lower than that of an isolated cylinder without the T-shaped control plate (*L*/*D* = 0) for all considered Reynolds numbers. It is seen that the *C*_{Dmean} slightly increases with increasing Re. A maximum value of 0.1538 of *C*_{Dmean} is observed for (*L*/*D*, Re) = (8, 200) and a minimum value of −0.520 of *C*_{Dmean} is noticed for (*L*/*D*, Re) = (0.5, 150). As the vortex length increases with narrow wake, the pressure on the rear surface of the square cylinder increases, and as a result, the drag reduces.

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**(h)**

It is noticed that the *St* number decreases with the increase in *L*/*D* and experiences sudden jump at *L*/*D* = 7 at Re = 100, 150, and 200, which confirms the changes in flow characteristics. The value of *St* for L/*D* ≥ 2.5 exceeds the value of the single cylinder for Re = 150, 200, and 250 (Figures 14(c) and 14(d)). It is to be noted that the variation trend of the *St* number with *L*/*D* is opposite to those of *C*_{Dmean}. There is a quick jump in *St* number for Re = 150 at *L*/*D* = 7. It was observed that two flow regimes (regime-I to regime-II) appear intermittently. The results further show that the Strouhal values are considerably lower than the isolated cylinder as the value of *L*/*D* increased. Generally, the *St* number of the square cylinder became lower than the isolated cylinder as the value of *L*/*D* increased. The slight increase in *C*_{Dmean} and quick decrease in Strouhal number is closely related to the wake structure changes behind the square cylinder.

It is seen from Figure 14(f) that, as the value of *L*/*D* increased, considerable changes in *C*_{Drms} are produced at Re = 250. It is noted that the effect of *L*/*D* on the *C*_{Drms} for Re = 200 and 250 (Figure 14(f)) as compared to Re = 100 and 150 (Figure 14(e)). The *C*_{Drms} for *L*/*D* = 6 to 8 at Re = 250 is much higher than that for a single square cylinder without T-shaped control plate (Figure 14(f)). In all other considered cases, the lift is smaller. As already discussed, flow features qualitatively above, when the *L*/*D* value increases, the *C*_{Lrms} increase. When the L/*D* value reaches a particular value, for example, 3.5 at Re = 150 (Figure 14(g)), there occurs a maximum value. As L/*D* increases, *C*_{Lrms} increases first and reaches their maximum values for all Re values and then decreases (Figures 14(g) and 14(h)). It is found that, for Re = 150, 200, and 250 the maximum value occurs at *L*/*D* = 3.5, 3, and 3.5, respectively. The *C*_{Lrms} associated with the square cylinder with the upstream attached T-shaped control plate for Re = 100 is lower than that of the isolated cylinder.

The *C*_{Lrms} values at Re = 250 (Figure 14(f)) are more scattered and either increased or decreased while changing the value of *L*/*D*. It can be seen from Figure 14(e) that the cylinder with the upstream attached T-shaped control plate had lower values of C_{Lrms} than the isolated cylinder at Re = 100 as the value of *L*/*D* increased. This can be attributed to the main reason that the shear layers downstream of the square cylinder are delayed by the use of the upstream attached T-shaped control plate. It can also be clearly seen that, for Re = 150 by increasing *L*/*D* up to 2.5, the *C*_{Lrms} had a fixed value and considerably lower than the isolated cylinder value. However, by further increasing the length (3 ≤ *L*/*D* ≤ 5.5), *C*_{Lrms} increases and attains its maximum value and then starts to decrease slowly. In some specific cases, the value is more than the isolated cylinder. In general, increasing the length of T-shaped control plate from *L*/*D* = 4 onwards did not have a considerable effect on integral parameters.

Figures 15(a) and 15(b) depicts the percentage variation of *C*_{Dmean} and *C*_{Lrms} for all the considered cases of T-shaped control plate length. It is noticed that, with increasing *L*/*D*, the percentage reduction in terms of *C*_{Lrms} increases. The previous studies suggest that the usually very small length of the splitter is good enough for maximum drag reduction and fluid forces suppression. Here we also observed that a small length (*L*/*D* ≥ 1.5) of T-shaped control plate considerably reduced the drag forces. For example, at Re = 250, the reduction in *C*_{Dmean} is 126%, 131%, 134%, 136%, 129%, 126%, 120%, 120%, 120%, 120%, 116%, 115%, and 113% for the T-shaped plate with *L*/*D* = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 7, and 8, respectively.

**(a)**

**(b)**

According to the values of *L*/*D* and Re and based on the investigation of the time-averaged and instantaneous vorticity contours visualization on the T-shaped control plate and square cylinder, three major flow regimes are found in this numerical study and shown in Figure 16. One flow regime, *L*/*D* = 4.5 to 6 at Re = 100, is a steady flow regime in which no vortex shedding is observed behind the square cylinder. Another flow regime is the single bluff body flow regime. In the case of a single bluff body flow regime, a regular von Karman vortex street is observed behind the main square cylinder. In the case of a single bluff body flow regime, we further observed two different flow regimes. Single bluff body with *Stp* and *Sts*. *Stp* and *Sts* stand for dominant primary vortex shedding frequency and secondary frequencies together with primary vortex shedding frequency, respectively. Single bluff bodies with *Stp* are found at (*L*/*D*, Re) = (0.5–4, 100), (0.5–5, 150), (0.5–4.5, 200), and (0.5–4, 250). Single bluff bodies with *Sts* are found at (*L*/*D*, Re) = (7-8, 100), (5.5–8, 150), (5–8, 200), and (4.5–8, 250).

The wake length (*L*_{r}/*D*) is quantitatively calculated and is given in Table 7. The data confirms that wake lengths changed as the value of *L*/*D* increased from 0.5 to 8. As shown, the *L*_{r}*/D* for all the considered *L/D* values is changed. For cases with Re = 100 and 4.5 ≤ *L/D* ≤ 7, the more *L*_{r}*/D* values are observed due to steady flow behavior behind the square cylinder.

#### 4. Conclusions

A two-dimensional unsteady fluid flow around a square cylinder with an upstream attached T-shaped control plate was investigated numerically, and the important findings were reported in the present study. The flow characteristics and fluid forces were examined at Re = 100, 150, 200, and 250. The length of the T-shaped control plate is varied. A considerable effect of T-shaped control plate on the fluid forces around the square cylinder is observed, and accordingly, the following important conclusions can be drawn from the present computation:(i)It was found from the present computations that the vortex street behind the square cylinder was still maintained, but there was a significant decrease in drag. Decreasing the length of the upstream attached T-shaped control plate had a considerable effect on reduction of drag coefficient and rms value of lift coefficient. However, for *L*/*D* ≥ 5, it did not have a considerable effect on the fluid forces. The minimum *C*_{Dmean} was observed at *L*/*D* = 0.5 for Re = 100, 150, and 200 and at *L/D* = 2 for Re = 250. At L/*D* = 0.5, 134%, 137%, 133%, and 136% reduction of *C*_{Dmean} was achieved for Re = 100, 150, 200, and 250, respectively. It was also found that when the length of T-shaped control plate is short, for instance, *L*/*D* ≤ 3, the effect of the length of the T-shaped control plate on controlling the flow past a square cylinder becomes noticeable.(i)A complete vortex shedding suppression for T-shaped control plate was achieved, at a T-shaped control plate length of *L*/*D* = 4.5 to *L*/*D* = 6 at Reynolds number 100. A maximum *C*_{Lrms} reduction of about 96% compared to the isolated cylinder was achieved at Re = 100 and *L*/*D* = 7 using T-shaped control plate. It was found that, as *L*/*D* increased, rms values of both drag coefficient and lift coefficient of square cylinder without T-shaped control plate considerably reduced compared with the isolated cylinder.(ii)Three different flow regimes were found in this study. The first one is the single bluff body flow regime with primary vortex shedding frequency (regime I). The second one is the single bluff body flow regime with secondary vortex shedding frequencies together with the primary vortex shedding frequency (regime II). The third one is the steady flow regime (regime III). In the case of regime-I, we found the dominance of primary vortex shedding frequency. On the other hand, in the case of regime-II, some extra minor peaks also exist in the power spectra.

#### Abbreviations

C_{D}: | Drag coefficient |

C_{L}: | Lift coefficient |

C_{Dmean}: | Mean drag coefficient |

C_{Drms}: | Root-mean-square of drag coefficient |

C_{Lrms}: | Root-mean-square of lift coefficient |

c_{s}: | Speed of sound |

Re: | Reynolds number (Re = U_{∞}D/υ) |

St: | Strouhal number (St = f_{s}D/U_{∞}) |

Stp: | Strouhal number based on primary vortex shedding |

Sts: | Strouhal number based on secondary vortex shedding frequencies |

D: | Size of the main square cylinder |

L: | Length of the T-shaped control plate |

: | Width of the T-shaped control plate head |

f_{s}: | Vortex shedding frequency |

U_{∞}: | Uniform inflow velocity |

L_{u}: | Upstream distance from the inlet position |

L_{d}: | Downstream distance from the rear surface of the square cylinder |

L_{x}: | Length of the computational domain |

L_{y}: | Height of the computational domain |

Ma: | Mach number |

f_{i}(, xt): | Particle distribution function |

f_{i}^{(eq)} (, xt): | Equilibrium distribution function |

p: | Pressure |

ρ: | Density of the fluid |

τ: | Single-relaxation-time parameter |

δt: | Computational time-step |

δx: | Lattice spacing |

υ: | Kinematic viscosity of the fluid |

ω_{i}: | Weighting coefficients |

e_{i}: | Discrete particle velocity |

d: | Dimensions |

q: | The number of particles |

ω_{z}: | Vorticity |

BGK: | Bhatnagar–Gross–Krook |

LBM: | Lattice Boltzmann method |

LGA: | Lattice gas automata |

N-S: | Navier–Stokes |

rms: | Root mean square |

SRT: | Single-relaxation-time. |

#### Data Availability

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

#### Conflicts of Interest

The authors declare that they have no known conflicts of interest that could have appeared to influence the numerical work reported in this study.

#### Acknowledgments

The second author Dr. Shams-ul-Islam is specially grateful to Higher Education Commission (HEC) Pakistan for providing funds under project no. 9083/Federal/NRPU/R&D/HEC/2017.