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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.

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 [

Abdi et al. [

A numerical study on a flow past a circular cylinder in the presence of a control cylinder was carried out by Kim et al. [

Islam et al. [

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

Recently, the lattice Boltzmann method (LBM) has been applied successfully to a number of flow problems (see [_{∞}/

Continuity:

Momentum:

In equations (_{x} and _{y} are the dimensionless velocity components along the

There are many different lattice Boltzmann methods. For a detailed study of the various LBM methods, the reader is referred to [

Here, _{i}(_{i}^{(eq)} (_{i}, and

Collision step:

Streaming step:

Here,

The two-dimensional nine-velocity lattice model (_{i} denotes the nine discrete velocity set, as follows:

A two-dimensional nine-velocity lattice model ((d2q9).

The equilibrium distribution function (_{i}^{(eq)} (

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

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

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

The speed of sound for the

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 _{i}

Figure _{u} = 10_{d} = 39_{y} = 13_{y}/_{x} = 50_{D} and _{L} are the drag forces and lift forces in the streamwise and transverse directions, respectively.

Schematic diagram of the proposed problem.

Simulation parameters.

_{∞} | ||
---|---|---|

100 | 0.04386 | 0.5263 |

150 | 0.04386 | 0.5175 |

200 | 0.04386 | 0.5132 |

250 | 0.04386 | 0.5105 |

The following boundary conditions are incorporated in the present study.

At inlet, uniform inflow velocity is applied,

At outlet, convective boundary condition is applied [

At the upper and lower walls of the domain, no-slip (

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 [

The implementation of the LBM is simple and straightforward. The following steps are presented for calculating forces and fluid properties:

Specify the streaming time step

The local distribution function must be updated through collision step (equation (

The fluid particles are streamed to neighboring streaming lattice nodes through the streaming step (equation (

Implement suitable initial and boundary conditions for the distribution function

Calculate macroscopic variables (equations (

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

The vorticity (dimensional) is defined as

Here, _{∞}) and the side length of the square cylinder (

For analysis, we have defined the following nondimensional parameters as given in equations (

Here, Re, _{D}, _{L}, and _{x} and _{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 [_{s} is the vortex shedding frequency calculated using the fast Fourier transformation (FFT) of the time series of the lift coefficient, _{L}.

The stopping criteria once we get the steady state is

It is noticed that _{∞} = _{lb} represents the velocity in a system of lattice units. Here, _{lb} represents the lattice Boltzmann velocity. In LBM, _{lb} is proportional to the Mach number of the fluid. In Tables _{Dmean}), Strouhal number (_{Drms}), and lift coefficient (_{Lrms}) values are as shown in Table _{Dmean}, _{Drms}, and _{Lrms} in Tables _{y}/_{u}, and _{d}, respectively. The results presented in Tables _{u} and _{d} in Tables _{u} = 10_{d} = 39

Effect of different grids points for Re = 250, _{u} = 10_{d} = 39

_{Dmean} | _{Drms} | _{Lrms} | ||
---|---|---|---|---|

10 | −0.2985 (5.0%) | 0.1135 (4.6%) | 0.0359 (4.7%) | 0.3156 (4.1%) |

20 | −0.2846 | 0.1083 | 0.0342 | 0.3028 |

30 | −0.2804 (1.5%) | 0.1072 (1.0%) | 0.0336 (1.8%) | 0.2987 (1.4%) |

40 | −0.2812 (1.2%) | 0.1072 (1.0%) | 0.0337 (1.5%) | 0.2995 (1.1%) |

Influence of _{u} = 10_{d} = 39

_{Dmean} | _{Drms} | _{Lrms} | ||
---|---|---|---|---|

6 | −0.2972 (4.4%) | 0.1132 (4.3%) | 0.0356 (4.0%) | 0.3152 (4.0%) |

13 | −0.2846 | 0.1083 | 0.0342 | 0.3028 |

18 | −0.2832 (0.5%) | 0.1075 (0.7%) | 0.0338 (1.2%) | 0.3004 (0.8%) |

Influence of _{u} for Re = 250, _{d} = 39

_{u} | _{Dmean} | _{Drms} | _{Lrms} | |
---|---|---|---|---|

5 | −0.2698 (5.5%) | 0.1148 (5.7%) | 0.0359 (4.7%) | 0.3192 (5.1%) |

10 | −0.2846 | 0.1083 | 0.0342 | 0.3028 |

15 | −0.2798 (1.7%) | 0.1066 (1.6%) | 0.0337 (1.5%) | 0.2987 (1.4%) |

20 | −0.2803 (1.5%) | 0.1066 (1.6%) | 0.0338 (1.2%) | 0.2998 (1.0%) |

Influence of _{d} for Re = 250, _{u} = 10

_{d} | _{Dmean} | _{Drms} | _{Lrms} | |
---|---|---|---|---|

15 | −0.2739 (4.0%) | 0.1132 (4.3%) | 0.0356 (4.0%) | 0.3165 (4.3%) |

25 | −0.2846 | 0.1083 | 0.0342 | 0.3028 |

35 | −0.2802 (1.4%) | 0.1069 (1.3%) | 0.0337 (1.5%) | 0.2987 (1.4%) |

45 | −0.2814 (1.1%) | 0.1069 (1.3%) | 0.0339 (0.9%) | 0.3001 (0.9%) |

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

Comparison of present results of flow past a square cylinder with available results.

Re = 100 | _{Dmean} | _{Drms} | _{Lrms} | |
---|---|---|---|---|

Present | 1.4125 | 0.1450 | 0.0035 | 0.1780 |

Dash et al. [ | 1.460 | 0.1440 | … | 0.1840 |

Luo et al. [ | … | 0.142–0.145 | … | … |

Okajima [ | … | 0.140 | … | … |

Norberg [ | … | 0.140 | … | … |

Sohankar et al. [ | 1.444 | 0.1450 | 0.0019 | 0.130 |

Saha et al. [ | … | … | 0.0030 | 0.122 |

Re = 150 | _{Dmean} | _{Drms} | _{Lrms} | |

Present | 1.4012 | 0.1520 | 0.0172 | 0.2732 |

Okajima [ | … | 0.1420 | … | … |

Norberg [ | … | 0.1550 | … | … |

Sohankar et al. [ | 1.4080 | 0.1610 | 0.0061 | 0.1770 |

Saha et al. [ | … | … | 0.0170 | 0.2740 |

Re = 200 | _{Dmean} | _{Drms} | _{Lrms} | |

Present | 1.4268 | 0.1510 | 0.0294 | 0.3250 |

Okajima [ | … | 0.1440 | … | … |

Norberg [ | … | 0.1520 | … | … |

Sohankar et al. [ | 1.424 | 0.165 | 0.0121 | 0.240 |

Saha et al. [ | … | … | 0.0260 | 0.305 |

Dutta et al. [ | 1.410 | 0.154 | … | … |

Re = 250 | _{Dmean} | _{Drms} | _{Lrms} | |

Present | 1.4432 | 0.1482 | 0.0348 | 0.4050 |

Sohankar et al. [ | 1.4490 | 0.1510 | 0.0162 | 0.375 |

Saha et al. [ | … | … | 0.0320 | 0.150 |

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

Figures

Instantaneous vorticity contours visualization of square cylinder in presence of attached T-shaped control plate at Re = 200. (a) Square cylinder without T-shaped control plate. (b)

Power spectra analysis of fluctuating lift coefficient at Re = 200 for different T-shaped control plate length: (a)

Figures

Comparison of the streamlines in the near wake of the main square cylinder with and without an upstream T-shaped control plate at at Re = 200: (a) square cylinder without T-shaped control plate, (b)

At

Figures

Pressure contours visualization of square cylinder in presence of attached T-shaped control plate at Re = 200: (a) square cylinder without T-shaped control plate, (b)

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 _{D} and _{L} at various values of _{D} variation is very sensitive to the T-shaped control plate length. The variation of _{D} for _{L} in comparison with an isolated cylinder (without T-shaped control plate) decreases.

Time histories of (a) drag coefficients and (b) lift coefficients of square cylinder at Re = 200 for different T-shaped control plate length.

The power spectrum of lift coefficients at

The instantaneous vorticity contours visualization shown in Figures

Instantaneous vorticity contours visualization at

Time histories of (a) _{D} and (b) _{L} of square cylinder at

Figures

Pressure contours visualization at

The graphical representation of _{D} and _{L} shown in Figures _{D} and _{L} in Figures _{L} confirm the periodic nature of the flow except for the _{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 _{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 _{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 _{L}.

Spectra analysis of _{L} using the FFT at

The instantaneous vorticity contour visualization shown in Figures

Instantaneous vorticity contours visualization at Re = 100: (a)

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 _{D} and _{L} at various values of _{L} confirms the alternate shedding behaviour from cylinder at _{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 _{L} amplitude is somewhat increased in the case of

Time histories of (a) drag coefficients and (b) lift coefficients of square cylinder at Re = 100 for different T-shaped control plate length.

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 _{Dmean}, _{Drms}, and _{Lrms} are depicted in Figures _{Dmean} of the square cylinder with T-shaped control plate is lower than that of an isolated cylinder without the T-shaped control plate (_{Dmean} slightly increases with increasing Re. A maximum value of 0.1538 of _{Dmean} is observed for (_{Dmean} is noticed for (

Variation of integral parameters of square cylinder with and without attached T-shaped control plate as a function of _{Dmean}, (c, d) _{Drms}, and (g, h) _{Lrms}.

It is noticed that the _{Dmean}. There is a quick jump in _{Dmean} and quick decrease in Strouhal number is closely related to the wake structure changes behind the square cylinder.

It is seen from Figure _{Drms} are produced at Re = 250. It is noted that the effect of _{Drms} for Re = 200 and 250 (Figure _{Drms} for _{Lrms} increase. When the L/_{Lrms} increases first and reaches their maximum values for all Re values and then decreases (Figures _{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 _{Lrms} values at Re = 250 (Figure _{Lrms} than the isolated cylinder at Re = 100 as the value of _{Lrms} had a fixed value and considerably lower than the isolated cylinder value. However, by further increasing the length (3 ≤ _{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

Figures _{Dmean} and _{Lrms} for all the considered cases of T-shaped control plate length. It is noticed that, with increasing _{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 (_{Dmean} is 126%, 131%, 134%, 136%, 129%, 126%, 120%, 120%, 120%, 120%, 116%, 115%, and 113% for the T-shaped plate with

Percentage variation of (a) _{Dmean} and (b) _{Lrms} as a function of

According to the values of

Flow regimes map as a function of

The wake length (_{r}/_{r}_{r}

Wake length (_{r}/

Re = 100 | Re = 150 | Re = 200 | Re = 250 | |
---|---|---|---|---|

0.0 | 1.0 | 1.0 | 0.5 | 2.0 |

0.5 | 2.0 | 1.0 | 1.0 | 1.5 |

1.0 | 2.5 | 1.5 | 1.0 | 0.5 |

1.5 | 0.75 | 1.25 | 0.5 | 0.5 |

2.0 | 1.25 | 1.5 | 1.0 | 1.0 |

2.5 | 1.0 | 0.5 | 0.75 | 1.0 |

3.0 | 0.5 | 1.0 | 1.25 | 1.0 |

3.5 | 0.5 | 1.25 | 1.25 | 0.5 |

4.0 | 1.0 | 1.0 | 2.5 | 0.75 |

4.5 | 2.75 | 0.5 | 1.5 | 0.75 |

5.0 | 2.5 | 0.5 | 1.75 | 2.0 |

5.5 | 2.0 | 1.0 | 1.0 | 1.5 |

6.0 | 2.5 | 1.5 | 0.75 | 1.0 |

7.0 | 2.0 | 0.75 | 1.0 | 1.0 |

8.0 | 1.5 | 0.5 | 1.0 | 1.5 |

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:

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 _{Dmean} was observed at _{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,

A complete vortex shedding suppression for T-shaped control plate was achieved, at a T-shaped control plate length of _{Lrms} reduction of about 96% compared to the isolated cylinder was achieved at Re = 100 and

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.

_{D}:

Drag coefficient

_{L}:

Lift coefficient

_{Dmean}:

Mean drag coefficient

_{Drms}:

Root-mean-square of drag coefficient

_{Lrms}:

Root-mean-square of lift coefficient

_{s}:

Speed of sound

Reynolds number (Re = _{∞}

Strouhal number (_{s}_{∞})

Strouhal number based on primary vortex shedding

Strouhal number based on secondary vortex shedding frequencies

Size of the main square cylinder

Length of the T-shaped control plate

Width of the T-shaped control plate head

_{s}:

Vortex shedding frequency

_{∞}:

Uniform inflow velocity

_{u}:

Upstream distance from the inlet position

_{d}:

Downstream distance from the rear surface of the square cylinder

_{x}:

Length of the computational domain

_{y}:

Height of the computational domain

Mach number

_{i}(

Particle distribution function

_{i}

^{(eq)}(

Equilibrium distribution function

Pressure

Density of the fluid

Single-relaxation-time parameter

Computational time-step

Lattice spacing

Kinematic viscosity of the fluid

_{i}:

Weighting coefficients

_{i}:

Discrete particle velocity

Dimensions

The number of particles

_{z}:

Vorticity

Bhatnagar–Gross–Krook

Lattice Boltzmann method

Lattice gas automata

Navier–Stokes

Root mean square

Single-relaxation-time.

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

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

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.