Effect of a Detached Bi-Partition on the Drag Reduction for Flow Past a Square Cylinder

. The objective of this research is to study the fluid flow control allowing the reduction of aerodynamic drag around a square cylinder using two parallel partitions placed downstream of the cylinder using the lattice Boltzmann method with multiple relaxation times (MRT-LBM). In contrast to several existing investigations in the literature that study either the effect of position or the effect of length of a single horizontal or vertical plate, this work presents a numerical study on the effect of Reynolds number (Re), horizontal position (g), vertical position (a), and length (Lp) of the two control partitions. Therefore, this work will be considered as an assembly of several results presented in a single work. Indeed, the Reynolds numbers are selec ted from 20 to 300, the gap spacing (0 ≤ g ≤ 13), the vertical positions (0 ≤ a ≤ 0.8d) , and the lengths of partitions (1d ≤ Lp ≤ 5d). To identify the different changes appearing in the flow and forces, we have conducted in this study a detailed analysis of velocity contours, lift and drag coefficients, and the root-mean-square value of the lift coefficient. The obtained results revealed three different flow regimes as the gap spacing was varied. Namely, the extended body regime for 0 ≤ g ≤ 3.9, the attachment flow regime for 4 ≤ g ≤ 5.5 , and the completely developed flow regime for 6 ≤ g ≤ 13. A maximal percentage reduction in drag coefficient equal to 12.5%, is given at the critical gap spacing (g cr = 3.9). Also, at the length of the critical partition (Lpcr = 3d), a Cd reduction percentage of 12.95% was found in comparison with the case without control. Moreover, the position of the optimal partition was found to be equal to 0.8d i.e. one is placed on the top edge of the square cylinder and the second one is placed on the bottom edge. The maximum value of the lift coefficient is reached for a plate length Lp = 2d when the plates are placed at a distance g = 4. On the other hand, this coefficient has almost the same mean value for all spacings between the two plates. Similarly, the root means the square value of the lift coefficient (Cl rms ) admits zero values for low Reynolds numbers and then increases slightly until it reaches its maximum for Re = 300.


Introduction
Fluid-structure interaction is a very frequent phenomenon in our daily life. Therefore, the study of fluid flow in the presence of rough bodies represents an interesting topic of different types of investigation in various engineering applications such as flows in aircraft, submarines, automobiles, cooling of electronic components, buildings, etc. In these areas, the structures become bluff bodies characterized by loads (aerodynamic forces) caused by the strong wind-structure interaction. This can lead to malfunctioning and complex flow problems and cause structural damage to some of the systems used in these domains, which motivates the development of flow control instruments to suppress or eliminate the vortex shedding (Mooneghi et al., 2016;Fatahian et al., 2019;Gilliéron, 2002;Li et al., 2016;Loh et al., 2013).
Active control (using external energy) and passive control (without external energy) represent the two existing types of flow control devices. The passive control requires only simple instruments (flat plates, splitter plates, square rods, circular rods, etc.) to control the fluid flow and is, therefore, more economical. The partition or splitter plate is a fine rigid plate arranged parallelly to the flow. Numerous experimental and numerical studies can be found where the detached flat plate is applied as a control instrument by varying its position or length (Aabid et al., 2019;Ali et al., 2012;Anderson et al., 1997;Apelt et al., 1973Apelt et al., , 1975Bruneau et al., 2014;Dehkordi et al., 2011;Ding et al., 2021;Doolan, 2009;Hassanzadeh Saraei et al., 2021;S. Ul Islam et al., 2015;Shams Ul Islam et al., 2014;Liu et al., 2016;Maruai et al., 2018;Mat Ali et al., 2011;Nidhul et al., 2015;Ozono, 1999;Park et al., 2013;Rashidi et al., 2016;Roshko, 1954;Sakamoto et al., 1997;Turki, 2008;Zhou et al., 2005). These studies included a large gap of Reynolds numbers, positions, and lengths of the detached flat plate showing that the vortex shedding mechanism and the physical parameters are significantly impacted. Roshko (Roshko, 1954) carried out an experimental study where a detached flat plate is inserted to eliminate the periodic vortex formation behind a circular cylinder. The author found a critical position of a detached flat plate is 2.7d.
The impact of a separated flat plate on the fluid flow past around circular cylinders is studied experimentally by Apelt et al. (Apelt et al., 1973) for a range of Reynolds number 10 4 < Re < 5 × 10 4 . In the case where the plate length d1 is the same as the length of cylinder d, the authors observed a noticeable reduction in drag coefficient where a minimum value has been reached. In another experimental investigation realized by Apelt and West (Apelt et al., 1973), the vortex shedding was suppressed and the drag coefficient takes a constant value for d1 > 5d. A numerical study that used a flat plate to control the vortex shedding behind a square cylinder in a channel was realized by Zhou et al. (Zhou et al., 2005) who they studied the impact of the location and the height of the plate on the fluid-structure. Turki (Turki, 2008) used the Finite Volume Method to examine the impact of a separated flat plate on the vortex shedding control behind a square cylinder for a Reynolds number ranging from 110 to 200. A slight increase in the value of the Strouhal number "St" is obtained as the gap spacing "g" increases. Likewise, they found that the St decreases with increasing g until it reaches a local minimum at about g = 2.82d for Re = 200, then, it is an increase. An investigation was carried out by , which used bipartition to control the flow past a square cylinder placed in a channel using the LBM method at Re = 150. The results obtained by the authors show that the implementation of the bi-partition was beneficial in all the cases studied. This was felt in the reduction of the drag value and fluctuating lateral forces due to vortex shedding behind the square block. Islam et al. (S. Ul Islam et al., 2015) performed a numerical study on the impact of a downstream control plate on different flow regimes around a square rod for Re = 150 and different spacings (0 ≤ g ≤11).
In this study, the authors classified the flow into three different regimes: the extended body flow (0 ≤ g ≤ 1.53), the reattachment flow (1.9 < g < 4) and the completely developed flow (4.8 ≤ g ≤ 11) flow regimes. They also indicated that the optimal gap between the square rod and the control plate is g = 0, where the drag force coefficient admits maximum reduction. Kwon and Choi (You et al., 1998) performed a numerical study on the length of the detached control plate effect on the vortex shedding around circular cylinders at lower Reynolds numbers (80 ≤ Re ≤ 160). They found that the elimination of vortex shedding after a circular cylinder depended strongly on the detached flat plate and the Reynolds number and observed that the vortex shedding disappears totally when the length of the control plate exceeds the critical length. Also, experimental investigations in two-dimensional and three-dimensional flows were carried out by Anderson and Szewczyk (Anderson et al., 1997) to examine the impact of a splitter plate on the nearest wake of a circular cylinder and they deduced the superposition principle. They found that the combination of some 3-D geometries and flow configuration produces a nominal two-dimensional wake. Rashidi et al. (Rashidi et al., 2016) performed a paper review on the existing numerical and experimental studies concerning the suppression of vortex shedding and wake control methods. They classified these methods into two groups and presented the benefits, limitations, power efficiency, and specific applications of the two methods. Ali et al. (Ali et al., 2012) varied the length of the splitter plate from 0.5 to 6 in their numerical investigation at a low Reynolds number.
Another numerical investigation was realized by Doolan (Doolan, 2009) owning on the effect of a flat plate positioned downstream of a square cylinder. The authors noted that the shear layers delivered by the edges of the square cylinder interact strongly with the flat plate. Doolan also found a significant reduction of the root-meansquare value of the lift coefficient (Clrms) compared with the case of a single square cylinder (i.e without a flat plate). Although this investigation does not include the study of the impact of the spacing between the block and the plate and the length of the flat plate. Away from the square and circular cylinders, Ozono (Ozono, 1999) carried out a numerical investigation to control the vortex shedding around the rectangular cylinder utilizing a detached flat plate. A remarkable variation of the Strouhal number was observed when varying the position of the plate.
In recent years, many researchers have used one or more rods (square or circular) to control the flow past cylinders (Alonzo-Garcia et al., 2021;Bao et al., 2013;Chauhan et al., 2019;Chiarini et al., 2021;Gupta et al., 2019;Shams Ul Islam et al., 2017;Vamsee et al., 2014;Yu et al., 2020;Zhong et al., 2020;Zhu et al., 2020). A reduction in the range of 10% to 15% in drag coefficient has been reported by Gupta (Gupta et al., 2019) by using a small control rod for Re = 100. Bao and Tao (Bao et al., 2013) used two control plates disposed parallel to decrease the fluid force on a circular rod. The authors noted that the retarding of vortex shedding on the principal rod is more important compared to the case of a single control plate. They concluded that the position and length of the plate have a significant role in the flow control behind the principal rod. These findings are confirmed by the numerical simulations performed by Vamsee et al. (Vamsee et al., 2014) related to the influence of one or two control plates situated upstream and/or downstream of a single square rod. Islam et al. (Shams Ul Islam et al., 2017) used a 2D lattice Boltzmann method to simulate flow past on the principal cylinder with separate control positioned at various positions for Re equal 160, whereas the spacing ratio between these cylinders was taken in the range from 0.5 to 8. A reduction of 8.3% was obtained by the authors for the mean value of the drag coefficient (Cdmean). This result was obtained when both control rods were located at an upstream position. Whereas, when these rods were situated at a downstream location, the reduction became 51%. For four control rods situated upstream and downstream of a square rod, a reduction of 50.8% was obtained.
In the literature, many investigations have been performed on the flow through a square cylinder with one passive control device at various gap spacing and different lengths of control devices. Thus, the main objective of this investigation is to present a detailed study on the reduction of vortex shedding and the reduction of the fluctuation amplitude of the drag and lift of a square cylinder by adding two passive control devices.

Presentation of the physical problem and boundary conditions
The physical problem under consideration is drawn in Figure1. It consists of a 2D channel of length-height dimension "L = 37d"-"H = 11d", including a square cylinder followed by two control partitions. The square cylinder of size "d" is situated at Lu = 6.0d from the entry and Ld = 30.0 d from the outlet. The two controlling partitions of length "Lp" and thickness "h=0.02d" are placed horizontally behind the cylinder with gap space noted "g". The vertical distance between the two partitions is "a".
Consider that the fluid enters with a parabolic velocity in the horizontal direction while the vertical component is assumed to be zero (v= 0). The implementation of boundary conditions is very essential for the stability and precision of the LBM numerical approach. Indeed, the Bounce-Back boundary conditions (Bouzidi et al., 2001; are applied to define the inconnu distribution functions at solid boundaries, from the known functions, by the following relation: Where ( ⃗ , ) is the inconnu distribution function at the wall node ( ⃗ ) and ̅ ( ⃗ , ) is the knowing function in the contrary direction to ( ⃗ , ).
At the inlet of the channel, the boundary conditions of Zou and He are applied (Zou et al., 1997) since the flow is completely developed with a parabolic velocity profile.

Description of the numerical method
The numerical approach employed for the simulation of the wake structures and aerodynamic forces exerted on the square block in the presence of the dual detached partition is the lattice Boltzmann method with multi-relaxation time (MRT-LBM) with the uniform grids (740×220). The choice of this method is based on its flexibility, ease of implementation, parallel computational advantages, and its diverse applications. Indeed, many researchers Admi et al., 2020Admi et al., , 2022cAdmi et al., , 2022aAdmi et al., , 2022bBenhamou et al., 2020;Bhatnagar et al., 1954;D'Humières et al., 2002;Lahmer, Benhamou, et al., 2022;Lallemand et al., 2000;Mezrhab et al., 2010;Mohamad, 2011;Moussaoui et al., , 2011Moussaoui et al., , 2019Moussaoui et al., , 2021Qian et al., 1992;Admi et al, 2022) performed to simulate diverse physical phenomena such as fluid flow, wave propagation, heat exchange, etc by using the LBM method. For precision and convergence reasons, the D2Q9 model ( Figure 2) is utilized in this paper (Mohamad, 2011). The spatial-temporal evolution of the LBE is defined by the development of the function fi representing the density distribution of particles: Where Ω represents the operator of the collision developed by D'Humières based on the LBM-BGK model. The development proposed by D'Humières (D' Humières et al., 2002) permits to write the previous equation as follows: Where S, and represente respectively the relaxation matrix, the moment and the equilibrium moment vectors. The −1 denotes the inverse matrix of the transformation matrix . −1 and are (9 * 9) matrices. They permit to link the vector = ( 0 , 1 , 2 , … , ) to the vector = ( 0 , 1 , 2 , … , ) via the linear transformations: The matrix M of order 9 is explicitly given by (Mohamad, 2011). After the collision, two moments are locally conserved: the density (m0) and the quantity of movement (m3, m5). The other moments (m1, m2, m4, m6, m7, m8), named respectively, the kinetic energy, the energy square, the components of the energy flux, and the components of the viscous stress tensor, relax linearly towards their equilibrium values. These non-conserved moments are defined by: Where m c is the moments after the collision and m eq is the equilibrium moments such as: The equilibrium moments m eq are defined by (Mohamad, 2011). In compact notation, the S matrix can be expressed as: = diag( 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ) The relaxation times used are those mentioned in the reference (Frisch et al., 1986).We have chosen 0 = 3 = 5 = 1, 1 = 2 = 1.4, 4 = 6 = 1.2 and 7 = 8 = 1/(3 + 0.5), where ν is the kinematic viscosity of the simulated fluid. The density ρ and the momentum ρu are calculated by:

Validation
To assure the reliability and the exactitude of our LBM-MRT code, several validation results with existing numerical and experimental results in the literature are presented in this paper. Firstly, a numerical simulation of the fluid flow around a square cylinder without control and with a blocking ratio H/d = 8 has been performed. The results obtained in this validation work are compared with those found by Breuer et al. (Breuer et al., 2000), for which they applied two different methods, namely the finite volume method (FVM) with 560×340 non-uniform grid and the Lattice Boltzmann automata (LBA) using 2000×320 uniform grid. Figure 3 displays the U and V velocity curves along the x and y axes respectively, for Reynolds number Re = 100. The results are very close to those of Breuer et al. with a difference of less than 1.69%.
From this figure, it can be seen that the Cdmean values compare favorably with those of Breuer et al. (Breuer et al., 2000) The exactness of our numerical code is also verified in the case of flow around a square cylinder controlled by a flat plate located downstream. Figure 5 illustrates the variation of the average drag coefficient for various gap spacings (0 ≤ g ≤ 11) at a fixed Reynolds number Re = 150. This figure shows that there is very good agreement between our results and those found experimentally by Okajima et al. (Okajima A, 1982) and those obtained numerically by Islam et al. (S. Ul Islam et al., 2015). Note that we found the almost same values as Turki et al. (Turki, 2008) for the same grid (500×80).

Results and discussion
Firstly, the distance between partitions, length, and position of partitions are fixed at a = 0.8d and Lp = g = 1d.
The effects of Re on flow patterns, drag coefficient, and Clrms are studied. Secondly, the effect of gap spacing between the square cylinder and the two partitions is studied for a fixed length of partitions (Lp = 1d), the distance separating the two control partitions (a = 0.8d), and Reynolds number (Re = 150). Thirdly, the length of the partitions is varied for the gap spacing (g = 4) and the Reynolds number (Re = 150). Finally, the effect of the distance separating the two control partitions is treated for the lengths (Lp = 1d), the spacing gap (g = 1), and the Renolds number (Re = 150).

Regime flow
In this part, the effect of the Reynolds number on the velocity contours is performed. For this purpose, the length of the two control partitions is fixed at Lp = 1d. The partitions are positioned behind the top and bottom edges of the cylinder at a distance of g = 1. As mentioned earlier, several studies exist in the literature that treat the flow of fluids around a square cylinder with and/or without control partition (Youssef Admi et al., 2022b;Ali et al., 2012;Breuer et al., 2000;Doolan, 2009;Hassanzadeh Saraei et al., 2021;S. Ul Islam et al., 2015;Shams Ul Islam et al., 2014;Mat Ali et al., 2011;Nidhul et al., 2015;Saha et al., 2003;Sohankar et al., 1998;Turki, 2008;Zhou et al., 2005). Among these studies, we find those that address the effect of the Reynolds number on several physical parameters (Nidhul et al., 2015;Sohankar et al., 1998). Sohankar et al. (Sohankar et al., 1998) have found that at Re ⩽ 150 the wake of a square cylinder is laminar, two-dimensional and characterized by the primary spanwise Kármán vortices. They have also concluded that the wake shows spanwise secondary instability and becomes three-dimensional in the range between Re = 150 and 200. Saha et al. present a numerical study of the spatial evolution of vortices and the transition to three-dimensionality in the wake of a square cylinder. The study is considered for a range of Reynolds numbers 150 ≤ Re ≤ 500. The authors find that the transition to three-dimensionality occurs at Reynolds numbers between 150 and 175. Likewise, they find that secondary vortices of the A-mode persist over the Reynolds number range of 175 to 240. On the other hand, at a Reynolds number of about 250, the secondary vortices of mode B are present. Also, Nidhul et al. (Nidhul et al., 2015)numerically investigated the flow past a square block with and without a detached flat plate by using CFD fluent. Their results show that for Re > 40, the flow past on the square cylinder becomes unstable and results in an oscillating flow whose oscillation amplitude increases downstream. This exerts a force on the cylinder in a lateral direction. These forces are reduced using the separated plate that suppresses the vortex shedding. Zhou et al. (Zhou et al., 2005) performed a numerical study on the reduction of fluid forces acting on a square cylinder in a two-dimensional channel using a control plate. It is well known that the vortex shedding phenomenon can be significantly modified for uniform flows passing a bluff body at different Reynolds numbers. The vortex structure behind the cylinder in a shear flow can also depend on the Reynolds number. In this part, the effect of the Reynolds number on the velocity contours is performed. For this purpose, the length of the two control partitions is fixed at Lp = 1d. The partitions are positioned behind the top and bottom edge of the cylinder at a distance of g = 1. Figure 6 displays the velocity contours for various Reynolds values (Re = 20, 65, 150, 300) characterizing different flow regimes. The results obtained show that the flow is stable and perfectly symmetrical about the horizontal axis of the channel for low values of Reynolds number (Re < 60). A detachment of a few small vortices appeared just downstream of the cylinder at a critical Reynolds number Re = 65. In this case, the previously obtained stability disappears and the flow is characterized by low amplitude undulations (Figure 6-b). When Re is in the interval (100 < Re < 250), a strong generation of vortices is observed along the channel and the regime becomes periodically variable in time. i.e., there is an alternation of positive and negative vortices generated by the lower and upper edges of the square cylinder. For high Reynolds numbers (Re > 270), the periodicity and symmetry, obtained before, disappear and the regime becomes intense. This instability is due to the great interaction between the shear layers produced by the ends of the cylinder and the control partitions. This causes oscillations of the flow behind the cylinder and behind the partitions. This gives rise to the formation of vortices of different sizes that propagate randomly in the flow direction.

Force statistics
The effect of the Reynolds number on the average drag coefficient (as well as on the temporal drag coefficient) and the mean square value of the lift coefficient are studied in this section. Some previous works that have studied the effect of the Reynolds number on the average drag coefficient around a square cylinder without and with a single control partition are cited here (Breuer et al., 2000;Feng et al., 2001;Shams Ul Islam et al., 2014). Breuer et al., 2000) used two methods (LBM and FVM) to study the confined flow around a square cylinder mounted inside a horizontal channel with a blocking ratio β =1/8 and for a range of Reynolds numbers between 0.5 and 300. For Re < 60, the authors find an excellent agreement between LBA and FVM results for the length of the recirculation region, while small deviations are detected for the drag coefficients in this range. Likewise, they calculated the drag coefficient for the other Reynolds number ranges and found that the drag coefficient of a confined cylinder also shows a local minimum at Re 150. Islam et al. (Shams Ul Islam et al., 2014) performed a numerical study of the flow along a square cylinder in a two-dimensional channel with a detached flat plate. They use a detached flat plate downstream to control the flow around and behind the square cylinder. The study of vortex generation, time trace analysis of drag and lift coefficients, and root mean square (rms) value of drag and lift coefficients are performed using the Boltzmann lattice method for a range of Reynolds numbers from 75 to 200. They found a reduction of more than 90% in the rms value of drag and lift coefficients for a reduced spacing, regardless of the Reynolds number. While wake development and vortex formation behind the square cylinder depend significantly on the Reynolds number. Figure 7-a displays the variation of the mean drag coefficient as a function of the Reynolds number. From this figure, it can be seen that the value of Cdmean is higher for low Reynolds numbers and it decreases with the increase of this number until it reaches a minimum at Re = 250. Then, it increases slightly until it obtains an almost stable value between Re = 270 and 300. It is well known that the drag force is the resultant of the viscous force due to the friction of a moving object flowing in the opposite direction of the fluid (i.e., it represents the resistance of an object in a fluid). Therefore, as viscosity increases, the Cdmean coefficient also increases. Thus, a viscous fluid means that the Reynolds number is low and therefore higher Cd values are obtained. More Re increases, more Cdmean decreases. This decrease reaches its minimum at Re = 250, where the flow is detached from the body by the two control partitions. After that, a strong vortex generation around the obstacle is observed, which causes a disturbance of the flow and increases the mean value of the drag coefficient.
The calculation of the average lift coefficient is not of interest since the latter admits a zero value for all the Reynolds numbers processed. It is, therefore, preferable to calculate its root mean square value Clrms. Figure 7-b illustrates the variation of this coefficient as a function of Re. We observe that Clrms gradually increase with the increase of Reynolds number.

5.2
Effect of the gap spacing

Regime flow
The effect of the distance between the principal cylinder and the control partition (gap spacing) is widely studied in the literature (S. Ul Islam et al., 2015;Mat Ali et al., 2011;Rashidi et al., 2016). Islam et al. (S. Ul Islam et al., 2015) performed a numerical study on the impact of a downstream control plate on different flow regimes around a square rod for Re = 150 and different spacings (0 ≤ g ≤ 11). In this study, the authors classified the flow into three different regimes: the extended body flow (0 ≤ g ≤ 1.53), the reattachment flow (1.9 < g < 4) and the completely developed flow (4.8 ≤ g ≤ 11) flow regimes. Ali et al. (Mat Ali et al., 2011) used a detached flat plate downstream of the square cylinder to study a wake alternation for Re = 150. They observed two distinct types of flow regimes within the gap spacing between the cylinder and splitter plate and after this plate. An investigation review was conducted by Rashidi et al. (Rashidi et al., 2016) In this study, the authors discuss existing studies on different methods of controlling the destructive behaviour of the wake and suppression of vortex ejection behind bluff bodies. Likewise, this study presents a discussion on the advantages, limitations, energy efficiency, and specific applications of two existing control methods. Figure 8 illustrates the velocity contours in the wake of the cylinder in the presence of double detached plates at various gap spacings. For a narrow spacing (0.25 ≤ g ≤ 3.9), the extended body regime is clearly observed. The separate free shear layers of the extremity of the square block are attached to the two control plates. Consequently, the alternating vortex shedding is only observed after the plates (as for a single bluff body case). In this case, the positive and negative vortices produced by the upper and lower edges of the cylinder are quickly reattached to the control plates. Therefore, an alternating generation of vortices behind the two detached flat plates is clearly observed without any fusion or distortion. It is noted that for g < 3.9, all simulation cases have almost similar characteristics. Another flow regime is presented in figures 10 a-d where the spacing g varies from 4 to 5.5. This is the attachment flow where the distance between the block and the plates exceeds a crucial gap value (g = 3.9). The shear stress layers detached by the square cylinder are developed and then rapidly reattached to the control partitions. Consequently, the vortex shedding is only observed behind the detached partitions. For this range of g, a strong interaction of the shear layers with the control partitions is observed. Thus, it affects the length and size of the wake. Figure 9 a-d shows that the number and size of vortices shedding behind the control plates change completely compared to the previous case. Generally, in this flow regime, the form of the wake is enlarged just behind the plates, which reinforces the vortices shedding of the square cylinder. Consequently, the mean value of the drag coefficient is slightly elevated than the corresponding value for a unique cylinder (i.e. without control partitions). Figure 10 illustrates the third flow regime, named a completely developed flow regime. In this case, the vortex evacuated in the intermediary distance is totally developed. Also, one can clearly observe an alternation of positive and negative vortices produced from the top and bottom edge of the block almost as the case was using the square cylinder without control. However, the wake behind the detached partitions is characterized by the undulating behaviour and the formation of the vortex of different sizes.

Force statistics
Likewise, the fluidic forces exerted on the square cylinder are influenced by the variation of the gap spacing. Indeed, Islam et al. (S. Ul Islam et al., 2015) also indicate that the optimal gap between the square cylinder and the control plate is g = 0 (the plate is attached to the square cylinder) where the drag force coefficient admits a maximum reduction. Zhou et al. (Zhou et al., 2005) present a numerical study on the reduction of fluid forces acting on a square cylinder (prism) in a two-dimensional channel using a control plate. They find that not only is the drag on the square cylinder significantly reduced by the control plate, but also the fluctuation of the lift is suppressed. The optimal position of the control plate to minimise the drag on the square cylinder is found for each control plate height. They also find that there is an optimal position and size for the upstream plate to effectively suppress lift. Likewise, Doolan (Doolan, 2009) presents a numerical study of the flow around a single square cylinder (prism) controlled by an infinitely thin plate at Re = 150. Doolan (Doolan, 2009) finds that the addition of an infinitely thin plate in the near-wake has resulted in fundamental changes in the flow field. Indeed, the main characteristics of the flow between the cylinder and the plate are strong secondary vortices at the leading and trailing edges of the plate and an inverted flow environment above the plate. Also, Doolan (Doolan, 2009) finds that the force applied on the square cylinder was significantly reduced. However, the magnitude of the lift coefficient on the downstream plate was found to be the same as for the simple square cylinder. Figures 11, 12, and 13 illustrate the time plot of the drag coefficient (Cd) for various gap spacing ranges. For 0.25 ≤ g ≤ 3.9, the temporal variation of Cd of the square cylinder has an irregular profile, resulting from the short distance between the square and the plates. The shear layers produced by the block are partially or totally attached to the detached partitions. Note that for the cases where g < gcr show almost identical characteristics. Also, a reduction in the amplitude of the drag coefficient is observed with increasing spacing. From the graphs in Figure 11, it can be observed that the average value of the drag coefficient decreases from 1.296 for g = 0.25 to 1.256 for g = 3.9. This is justified by the variation in the length and width of the detached vortices behind the control plates. While there is a remarkable increase in the value of the drag for g between 4 and 5.5. This is clearly explained by the behavior of the velocity wake structure behind the plates. In this case, the shear layers crash with the control plates, which disturbs the intermediate zone between the square cylinder and the partitions. This increases the forces applied to the back surface of the block. Similarly, this range represents a transition from an irregular regime (g ≤ 3.9) to a regular regime (g ≤ 5.5). For elevated values of g (6 < g < 13), regular variations in the drag coefficient are observed; this is due to the large spacing existing between the square cylinder and the detached control plates allowing a regular development of the shear layers delivered by the upper and lower face of the square cylinder.
The comparison of graphs showing the temporal variation of the drag coefficient in this case with those obtained previously (tie flow) shows that they have the same profile but the amplitude of the fluctuation for all spacings in the tie flow regime is greater than in the fully developed flow regime. The amplitude of the drag coefficient is slightly higher at g = 5. Then it decreases until g = 6; thereafter, an almost constant behaviour is observed from g = 7 to g =13. It should be noted that some cases are not presented in this article.

Regime flow
In the literature, a limited number of studies have been performed to investigate the length of the control partition compared to the studies realized to investigate the effect of the gap spacing between the principal cylinder and the control partition. Ali et al. (Mat Ali et al., 2011) performed a numerical study on flows around a square cylinder of lateral length with and without a separator plate at a Reynolds number of 150. In this study, the length of the separator plate varies from = 0.5 to = 6 . The authors find that the splitter plate introduces a strong hydrodynamic interaction in the wake near the cylinder and that the length of the plate significantly affects the flow structure. Indeed, they observed three flow regimes by varying the length of the separator plate: For short lengths closer to the trailing edge. For longer plate lengths ( ≳5 ), a regime in which the free shear layers attach to the separator plate is observed. Another numerical study was performed by Admi et al. (Admi et al., 2022b) on fluid flow and thermal convection phenomena around a heated square cylinder controlled by three downstream partitions using the Boltzmann multiple relaxation time network method at a fixed Reynolds number (Re = 150). They find that maximum vortex suppression is observed at a critical length of the control partitions Lp = 4d. The effects of the position and length of a detached downstream plate on the wake of a square cylinder were studied numerically by Ali et al. (Ali et al., 2012) at a Reynolds number of 150. The authors identified two flow regimes: the first regime is characterized by the completion of vortex formation downstream of the gap and the second regime is characterized by the completion of vortex formation within the gap. They found that there are abrupt changes in the integral properties observed between regime I and II, with the transition occurring at the critical gap distance Gc = 2.3D. Also, they observed that the plate has no significant effect on the generation of the von Kármán vortex when the separation is beyond ∼5.6D.  Figure 14 illustrates the instantaneous velocity contours for six different lengths. These plots make it possible to interpret the topology of the flow reigning around the cylinder controlled by two detached partitions. We observe that there is a very strong generation of vortices behind the plates, particularly in the first case where the flow is disturbed in the gap between the block and the control partitions. This strong generation decreases with increasing partition length, where the size of the vortexes decreases and the regime becomes symmetrical and quasi-stable.

Forces statics
In this section, the effect of the length of the control baffles on the aerodynamic forces is investigated. In this sense, several researches have been carried out.  used LBM to numerically study the incompressible flow around a square cylinder placed downstream of a control bi-partition in a horizontal channel at a Reynolds number of 250. The authors find that the fluid flow exhibits a complex structure in the space between the cylinder and the bi-partition and that the fluid forces acting on the cylinder decrease slightly for the spacing range w/d = 1-4 except for a height h = 0.4 and 0.6. When the position of the control bi-partition w/d =5, the vortex shedding behind the cylinder is almost completely suppressed for small and large values of h, i.e. for h = 0.1-0.3 and 0.8-1.0. Similarly, the authors find that the timeaveraged drag tends to decrease gradually as the spacing of the control bi-partitions increases in the range of w/d, except for h = 0.4-0.6. Moreover, for h = 0.8-1.0, the timeaveraged drag is negative and it is significantly reduced. It should be noted that the amplitude of fluctuating lift on the square cylinder is successfully suppressed using the control bi-partition, and it can be completely suppressed by carefully choosing the height and position of the control bipartition. Sakamoto et al. (Sakamoto et al., 1997) evaluated the fluid forces acting on a square prism by changing the width of a flat plate and its position on the centre line. They found that the optimum width of the plate for suppression of fluid forces is approximately 10% of that of the prism inserted upstream. Admi et al. (Youssef Admi et al., 2022a) find that the maximum percentage reduction in drag coefficient reaches 17.33% when the length of the control plates reaches its critical value (Lp = 4d).  In this work, to study the effect of the length of the control walls only, we carried out a study on the gap spacing, to find out the optimal position of the two control plates. Also, the Reynolds number was fixed at 150. Six cases are treated when g varies from 0.25 to 9 to get the optimal case of control wall placement. For each g-spacing, we varied the length of the partitions from Lp = 1d to Lp = 5d and considered the case of Lp = 3d as a Cdmean reference. Table 1 shows the average values of the drag coefficient for Lp = 3d. The smallest Cdmean value is found for g = 0.25. However, the average drag coefficient admits high values in the cases where g = 0.25; 1; 2; 9 for all partition lengths except in the case where Lp = 3d. Likewise, Table 2 shows that in the case where g = 4, the average Cd value remains almost constant and takes values of about 1.25 for all lengths (except in the case where Lp = 1d). This is observed in Figure 15, which shows the temporal variation of the drag coefficients.
A difference of less than 0.03% is observed between the Cdmean value for Lp = 2d and for Lp = 5d, where the Cd presents its minimum. This gives the possibility of using partitions of 2d or 3d length instead of using partitions of Lp = 5d length. Since the values of Cdmean are very close, then the lift coefficient values can contribute to the choice of the length of the partitions. From the curve showing the variation of the lift coefficient (Figure 16), it can be seen that it has its maximum for the length Lp = 2d. Therefore, for beneficial flow control, i.e., having reduced values of Cd and high values of Cl, it is preferable to use partitions of length Lp = 2d placed parallel downstream of the cylinder at a position g = 4.

5.4
Effect of partitions position

Regime flow
In the last part, we studied the effect of the vertical position of the partitions (i.e. the distance between the two control partitions) for g =1d and Re=150. Figure 17 shows the velocity contour structures for different vertical positions of the partitions. The flow structures show that the presence of the two control partitions introduces a strong hydrodynamic interaction in the wake near the cylinder. This leads to a strong generation of vortices in the wake in all cases tested. In fact, the layers of airborne particles at the edges of the cylinder fall onto the two control plates, and this strong fluid-structure interaction increases the dynamics of the fluid particles in the wake near the cylinder.  Figure 18 presents the variation of the drag coefficient as a function of time for each position of the partitions. From this figure, we see that there is an inversely proportional relationship between the coefficient Cd and the vertical position of the partitions, i.e. the more the distance between the two partitions decreases, the more the value of Cd increases. In the case where the two partitions are associated, the drag coefficient has reached a higher value. This is because the fluid flow delivered by the shear layers strongly interacts with the partitions and causes undulations in the intermediate zone. This increases the fluidic forces acting on the square cylinder. Likewise, the lift coefficient exhibits its greatest value in the case where the detached partitions are placed behind the upper and lower edges of the square cylinder (see Figure 19).

Conclusions
In this paper, a numerical simulation of the laminar flow of a fluid around a square block arranged in a twodimensional horizontal channel controlled by two partitions was presented. The numerical approach used is the Lattice Boltzmann method with multiple relaxation times. The study of the effect of the Reynolds number shows that as the Reynolds number increases, the fluidic forces acting on the cylinder decrease until they reach a minimum value at Re = 250 and then start to increase again. Likewise, the root mean square coefficient increases with the Reynolds number. This increase becomes faster for 200 ≤ Re ≤ 300. The study of the gap spacing variation at a fixed Reynolds number (Re = 150) shows three different regimes: the extended body regime (0.25 ≤ g ≤ 3.9), the attachment flow regime (4 ≤ g ≤ 5.5), and the fully developed flow regime (7 ≤ g ≤ 13). This study shows that the drag coefficient reaches its minimum Cd = 1.25 at a critical spacing g = 3.9. This shows a reduction of 12.5% compared to the case without a control device. At this critical spacing, a suppression of vortices behind the square cylinder is only observed for the extended body regime. Also, a 12.95% reduction in drag coefficient is achieved for a critical length Lp = 3d. The last study shows that the variation of the distance between the two control partitions shows that the optimal position of the partitions is behind the upper and lower ends of the square cylinder (a = 0.8d). These results show that the control mechanism with simple devices has a significant advantage over the uncontrolled case. For beneficial application in further research in different fields of engineering science, it is recommended to place the two partitions of length Lp= 3d behind the ends of the bluff body at a distance g = 4. This results in a significant energy saving. Note that the extension to 3D calculations and higher Reynolds numbers is being further investigated in ongoing research.