Waqas SARWAR ABBASI,Shams UL ISLAM,Luna FAIZ,Hamid RAHMAN

aMathematics Department,Air University,Islamabad 44000,Pakistan

bMathematics Department,COMSATS Institute of Information Technology,Islamabad 44000,Pakistan

cMathematics and Statistics Department,Bacha Khan University,Charsadda 44000,Pakistan

KEYWORDS Drag reduction;Flow states;Gap spacing;Inline cylinders;Reynolds number

Abstract This study focuses on the transitions in flow states around two-,three-and four-inline square cylinders under the effect of Reynolds numbers at two different gap spacing values using the lattice Boltzmann method.For this purpose,Reynolds number is varied in the range 1–130 while two different values of spacing taken into account are gap spacing=2 and 5.Before going to actual problem,the code is tested for flow around a single square cylinder by comparing the results with experimental and numerical results of other researchers,and good agreement is found.The current numerical computations yield that for both spacing values and all combinations of cylinders there exist three different sates of flow depending on Reynolds numbers:steady state,transitional state and unsteady state.It is found that the range of Reynolds numbers for these flow states is different for both spacing values.At gap spacing=2 the range of Reynolds numbers for each flow state decreases by increasing the number of cylinders while at gap spacing=5 opposite trend is observed.The results also show that at gap spacing=2 the reduction in drag force is greater than the corresponding reduction at gap spacing=5.The maximum reduction in drag force is observed at Reynolds numbers=1 at both spacing values.Similarly,at both spacing values and all Reynolds numbers,the maximum reduction in drag force is observed for the case of four-inline square cylinders.

1.Introduction

Flow around cylindrical structures of square cross section has various applications in different areas of architectural,mechanical and civil engineering.For instance,such type of structures can be seen in most of the architectural designs like buildings,beams,fences,bridges and many external and internal support components.In gas turbine blades and heat exchangers,the cylinders of square cross section are used as potential vortex generators for cooling purpose.The chips used in electronic devices mostly have square/rectangular cross sections.Another example of flow around a cylinder of square cross section is the flame holder of the combustor.By searching the literature related to the fluid interaction with solid structures,it can be found that numerous experimental investigations have been carried out to study the characteristics of flow around circular/square cylindrical structures.In the work of Norberg,1it was found that the vortex shedding around a circular cylinder starts at Reynolds number(Re)≈47.The flow remains two-dimensional till Re≈160 and after that three-dimensional effects start appearing in the range 160＜Re＜190.After Re≈ 190 the flow becomes completely three-dimensional.Price et al.2experimentally visualized the flow around circular cylinder and divided it into four different regions regarding the behavior of shear layers and vortex generation mechanism.Wang and Tan3found in their experimental work that the vortex shedding and momentum exchange for the flow around square cylinders are slower as compared to circular cylinder.The vortex induced vibrations and oscillation mechanism of the flow around a square cylinder were also analyzed by Manzoor et al.4Some experimental studies about characteristics of single cylindrical obstacle can also be found in Refs.5–7.

In the case of two or more cylinders,the gap spacing(g)significantly affects flow characteristics.This fact was also observed in many experimental studies.Deng et al.8found that the placement of a second cylinder in the wake of a single cylinder,to a certain level of spacing,suppresses the three-dimensional instabilities.Sakamoto et al.9observed significant changes in time-averaged forces of tandem(inline)square cylinders depending on g at Re=27600.Kim et al.10termed g=2.5 as critical value of spacing and categorized the flow characteristics in two modes separated by critical spacing value.For the flow around three inline circular cylinders,Igarashi and Suzuki11divided the shear layers’behavior,depending on Re and g,in three categories:without reattachment,with reattachment and roll up.Liu12examined the dependence of flow induced forces on g and found that the drag coefficient(CD)of cylinders changes abruptly in the range between g=3.5 and 4 due to existence of critical spacing value in this range.According to the study made by Hetz et al.13,the Strouhal number(St)of five inline circular cylinders increases with increment in g.Some further experimental studies related to the fluid flow characteristics of two or more inline cylinders can be found in Refs.14–16.

Contrary to experimental measurements,numerical computations can quickly provide the information about important flow characteristics such as wake structure mechanism,temporal histories of flow induced forces,and sensitivity of force coefficients to different flow parameters.These characteristics are very difficult to obtain experimentally.In numerical simulations,the fluid flow parameters and conditions can be easily changed by suitable modification in the input parameters without any time or labor cost.Several numerical studies have been reported regarding the flow mode transitions from steady to unsteady state around single cylindrical obstacle.The numerical investigations of Park et al.17yielded that the flow around a single circular cylinder remains steady at low Re and the unsteadiness starts at Re=47.Rajani et al.18found that the flow separation and formation of a steady closed near wake around circular cylinder occur at Re=6 while at Re=49 the steady wake becomes unsteady due to vortex generation mechanism.Similar observations were reported in the numerical work of Gera et al.19for the flow around a single square cylinder.They found that the flow remains steady up to Re=50 and instability occurs between Re=50 and 55.After this range, flow becomes completely unsteady.Also till Re≤250,the flow remains laminar and wake region is 2D.The transitions and chaos in the wake of a square cylinder were also investigated by Saha et al.20In addition to flow mode transitions,the dependence of flow induced forces on Re and g has also been a major concern of many numerical studies.According to the research conducted by Breuer et al.21,CDof a square cylinder strongly depends on small Re.Sohankar et al.22found a sharp drop in CDof square cylinder between Re=55 and 75.Vikram et al.23simulated the flow around two tandem square cylinders and reported that the vortex shedding frequency(fs)was suppressed due to introduction of second cylinder.It is reported by Etminan24that the root-mean-square values of drag and lift coefficients(CDrmsand CLrms)of two inline square cylinders increase by increasing the distance between cylinders.Vasel-Be-Hagh et al.25taken into account the flow around three cylinders and found that Re=42,63,150 are critical for pressure,viscous and total drag forces,respectively.Islam et al.26observed that CLrmsof three inline square cylinders was higher than the corresponding CDrmsvalues.In the work of Manzoor et al.,27it was reported that,at some spacing values,the downstream cylinders experience larger drag force as compared to upstream cylinders due to turbulence effects in flow even at low Re values.Some more numerical studies related to the fluid-solid interactions can also be found in Refs.28–31.

From above discussion,it can be deduced that the transitions in flow states around the structures of square cross section are not investigated thoroughly yet.From the perspective of applications,the bluff bodies of square cross section are very important.Flow around such bodies differs from those of circular cross section in many aspects like flow structure mechanism,variation of flow induced forces,etc.Therefore a detailed analysis of flow characteristics around bluff bodies of square cross section is very important in order to improve the design of structures and avoid losses.Keeping in view the above mentioned points,the current study will focus on:(A)the transition in flow states around inline array of cylinders under the effect of Re,(B)the effect of spacing on the transitional range of Re,(C)comparison of transitional range for different numbers of cylinders in array,and(D)effect of addition of cylinders,in array,on flow induced forces.

The rest of the paper is organized as follows:Section 2 consists of governing equations and numerical details,Section 3 contains problem description and boundary conditions,in Section 4 grid independence,domain independence and code validation study are presented,results are discussed in Section 5,and conclusions are drawn in Section 6.

2.Governing equations and numerical details

In this study,the Lattice Boltzmann Method(LBM)is used as numerical tool.Due to its simplicity and ease of applicability,LBM is widely used nowadays to simulate fluid flow problems.32–35Breuer et al.21compared the results obtained by Finite Volume Method(FVM)and LBM for flow around a square cylinder.They concluded that both methods captured fluid flow characteristics efficiently and the results are in good agreement.Despite of similarities,LBM also has several benefits in comparison to long-established CFD methods36.For example,continuity and momentum equation at macroscopic scale can be recovered using LBM,which are generally discretized and solved numerically by traditional CFD methods;we can get pressure fields directly by LBM,whereas using traditional CFD methods,for pressure fields,we have to solve Poisson equation which is computationally expensive;the third benefit of LBM is that the method can easily be parallelized due to locality of its steps(streaming and collision).

The governing continuity and momentum equations for the incompressible laminar flow,without external forces,considered in the current study are

where t is time,ν is viscosity,ρ is density.

The above equations can be achieved by applying the Chapmann Enskog expansion37to the discretized lattice Boltzmann equation:

where fiis density evolution function,x is position vector,eiis velocity directions,Δt is time step,τ is relaxation time,is equilibrium distribution function,i is index

Due to this fact,by LBM,Eq.(3)is used for fluid flow simulations instead of Eqs.(1)and(2).

In Eq.(3),fiis the density evolution function at position x and time t,and τ is relaxation time having values in the range 0.5＜τ＜2 for stability of method38.The equilibrium distribution functionin Eq.(3)is given by the relation:

where wiare the weighting coefficients.

These weighting coefficients are different in different models used in LBM38.For the two-dimensional nine-velocity particles(D2Q9)model used in current study,shown in Fig.1,the weighting coefficients are

Further details about LBM can be found in Refs.39,40.

Fig.1 Lattice structure for D2Q9model.

Different important flow parameters to be dealt with in this study are given below:drag coefficientis drga force,U∞is in flow velocity,d is size of cylinder),lift coefficientis lift force),Strouhal numbervortex she∑dding frequency)and average drag coefficientn is number of iterations).

3.Problem description and boundary conditions

The schematic diagram of the problem considered in current study is presented in Fig.2.Fig.2 contains a single-,two-,three-and four-square cylinder’s geometry to be analyzed in this work.In Fig.2(a),a single square cylinder of size d is placed inside a rectangular shaped computational domain of height H and length L.The distance from inlet position of domain to front surface of cylinder is Luwhile the distance from rear surface of cylinder to exit position of domain is Ld.At the inlet position,the stream wise and transverse velocities are set to U∞=0.04386 and 0 i.e.,u=U∞and v=026.At outlet position, convective boundary condition=0 is used.This boundary condition allows smooth passage of flow from outlet position without any significant disturbance in the inner domain41.No slip boundary condition u=v=0 with bounce back rule is applied to the solid surfaces like cylinder body,upper and lower walls of channel40.The single square cylinder geometry presented in Fig.2(a)serves as a reference for other geometries(Fig.2(b)–(d)).Values of Lu,Ld,H and boundary conditions are fixed for all geometries,whereas the values of L vary and depend on number of cylinders and gap spacing.For single square cylinder,the computational domain consists of L×H=32d×11d grid points while the numbers of grid points vary in the case of two-,three-and four-square cylinders depending on number of cylinders in domain and gap spacing between them.In Fig.2(b–d),c1,c2,c3and c4denote the first,second,third and fourth cylinder,respectively.

Fig.2 Geometry of problem.

4.Grid independence,domain independence and code validation

4.1.Grid independence

Before starting computations for actual problem,the grid independence study is performed in order to select such grid size which does not affect the results of computations.For this purpose,three different grid sizes i.e.,10-point grid,20-point grid and 40-point grid are selected along the surface of a single square cylinder,as test cases for computations,at Re=100.The resulting physical parameters like St,CDmeanand CLrmsobtained from these computations are presented in Table 1 along with percentage difference from 40 points grid results.It can be observed that the values of physical parameters are much affected by 10 points grid in terms of percentage difference compared to 20 points grid.The 20 points grid gives better results as compared to 10-point grid.On the other hand,40-point grid takes much time for convergence.So we have chosen 20 grid points on the cylinder surface for current study.Similar grid size was chosen in Refs.26,27.

4.2.Domain independence

In addition to grid size,the size of computational domain also has great influence on the characteristics of flow around cylindrical bodies.It can fasten or slow the vortex shedding process which affects the physical parameters significantly42.So in order to choose a suitable computational domain,CDmean,CLrmsand St are calculated at different values of Lu,Ldand H(Table 2).Table 2 shows that by fixing H and Ldat 11d and 25d respectively,the values of CDmean,CLrmsand St decrease by increasing Lufrom 4d to 8d.At Lu=6d and 8d,the difference in values is negligible.This indicates that after Lu=6d the effect of upstream distance almost vanishes.So we will take Lu=6d as upstream distance for rest of the computations.Similarly by fixing Luand H at 6d and 11d respectively,the values of CDmeanand St decrease by increasing Ldfrom 20d to 30d.Also there is minor difference between the results at Ld=25d and 30d.We will use Ld=25d as downstream distance for rest of the computations,because Ld=30d takes more computational time as it uses more number of grid points.Table 2 also indicates that H=9d is a suitable value for height of computational domain.This is because when H is increased from 9d to 13d,by fixing Luand Ldat 6d and 25d respectively,the values of CLrmsand St decrease.And there is minor difference in results at H=11d and 13d.So in order to ensure accuracy in less computational time,the moderate value H=11d is a suitable value of height of computational domain.These values for the size of computational domain agree well with the values chosen by other researchers26,43,44.

Table 1 Spatial resolution effects at Re=100.

4.3.Code validation

4.3.1.Qualitative validation

Generally the flow around a single cylinder is used as benchmark for the analysis of flow characteristics around multiple cylinders23,27,29.In this study,we have also analyzed the flow around a single square cylinder in order to validate our code.

The resulting streamlines and lift coefficient variation at different values of Re are given in Figs.3 and 4,respectively(only some representative cases shown).These figures indicate three different states of flow:the first one is steady state,the second one is transitional state and the third one is unsteady state.In steady state,the flow,after interacting with cylinder,moves steadily without any vortex formation(Fig.3(a)),which can also be confirmed from the lift coefficient graph at Re=1(Fig.4(a)).The main reason for steady state is low value of Re.Transitional state starts from Re=47.Two recirculating eddies appear adjacent to the downstream position of cylinder(Fig.3(b)).These eddies show that there is some unsteadiness appearing in flow.The lift coefficient for this case also indicates that the flow is initially steady but after some time unsteadiness occurs in flow(Fig.4(b)).The increasing amplitude of lift cycles shows that the unsteadiness is dominating the steadiness of flow with passage of time.This transitional state indicates the weakening effect of viscous forces.This state ranges from Re=47 to 52 and after that the flow becomes completely unsteady.The variation of streamlines in the wake of cylinder indicates the generation of vortices(Fig.3(c)).Due to unsteadiness,the lift coefficient becomes periodic,having the same amplitude cycles,unlike the transitional state(Fig.4(c)).

From above,it can be concluded that the flow around a single square cylinder is steady in the range 1≤Re≤47,transitional in the range 48≤Re≤52 and unsteady for Re≥53.Our findings agree well with those of Gera et al.19and Sohankar et al.22These observations indicate that the results obtained from the present code have good qualitative agreement with those of other researchers.

4.3.2.Quantitative validation

To ensure quantitative validity of present code,we have compared the values of St and CDfor flow around a single square cylinder with those of other researchers at Re=100(Table 3).This table shows good agreement between present results and those of other researchers,which indicates that the present code can calculate the forces acting on cylinders in an efficient way.

Table 2 Effect of computational domain at Re=100.

Fig.3 Streamline representation of flow around a single cylinder at different Re.

5.Results

Flow past multiple cylindrical structures mostly depends on two parameters:Re and g.In this section,the variation of flow states around two-,three-and four-inline cylinders,under the effect of Reynolds numbers,will be discussed at two different gap spacings i.e.,g=2 and 5.As we have already seen from the above section that there are three main states of flow around a single cylinder:steady state,transitional state and unsteady state,and each state has its own range of Re,in this section we will investigate the range of Re,for these flow states,in the case of different numbers of cylinders in the flow field.Furthermore,from previous studies,it can be deduced that at g=2 there is a strong proximity effect of multiple cylinders25.The flow characteristics of each cylinder are influenced by other cylinders due to narrow space between these bodies.But at g=5 the proximity effect becomes weaker and each cylinder in the flow field is able to shed its own vortices13.That’s why we have selected these two spacing values in order to analyze the variation of flow states under the effect of Re at both small and high spacing between the cylinders.This selection allows us to analyze the flow characteristics in two aspects:(A)when there is strong influence of multiple bodies on each other and(B)when these multiple bodies have independent flow characteristics.It should also be noted that to analyze flow state variation around each geometry,the computations are started from Re=1 and stopped at such value of Re at which flow achieves unsteady state.That’s why the range of Re for computations around each geometry is different.The maximum range for the whole study is Re=1–130.Also it is important to mention here that to shorten the length of the paper only some important representative cases are discussed.The steady state flow is discussed for the case of two-inline cylinders,transitional state is discussed for the case of three-inline cylinders,while unsteady state is discussed for the case of fourinline cylinders for the sake of brevity.

5.1.Steady state

The first state of flow observed in this study is steady state.The corresponding vorticity contour for steady state flow,at Re=1 and g=2,is presented in Fig.5 for two-inline cylinders.In this flow state,the shear layers remain attached with cylinders in the form of bubbles without any movement to downstream wake region.The main reason for this steady state flow is that due to small Re,the viscous forces are stronger enough.These forces have great influence on flow and do not allow the movement of flow.Also flow is symmetric with respect to the centerline of cylinders.The corresponding temporal histories of drag and lift coefficients for steady state flow are presented in Fig.6.This graph shows that there is no variation in these forces.Their linearity confirms the steady state of flow.

With the increment in spacing to g=5,it can be seen that each cylinder generates a separate bubble(Fig.7).At g=2 due to narrow space between the cylinders,a single bubble was formed for both cylinders(Fig.5).Furthermore,at g=5,although spacing is increased,the flow is still in steady state without vortex formation similar to the case of g=2.This indicates that low Re has great influence on the flow as compared to g.Due to steady state flow,the drag and lift forces are linear(Fig.8).By comparing Figs.6 and 8,it can be observed that the drag forces acting on cylinders at g=2 are smaller in magnitude than the corresponding drag forces at g=5.This is due to the fact that with increment in spacing values, flow moves inside the gaps.As a result,higher drag forces act on the cylinders as compared to the corresponding drag forces at smaller spacing values.Similar characteristics of flow were observed for the cases of three-and four-inline cylinders( figures not shown).

Fig.4 Lift coefficient variation for flow around single cylinder at different Re.

5.2.Transitional state

A representative case of transitional state of flow around three inline cylinders,at Re=105 and g=2,is shown in Figs.9 and 10.In this state of flow,the shear layers emerging from the first cylinder move till the exit position of computational domain(Fig.9).This behavior of flow is different from the steady flow where we have seen only bubbles around the cylinders.Also the unsteadiness occurs in flow due to transverse oscillations in far wake region of cylinders.This unsteadiness can be witnessed from the temporal variation of lift coefficients.Unlike steady state flow,the lift coefficient is constantinitially but after some time it becomes nonlinear which indicates transition in flow from steady to unsteady state.According to Vasel-Be-Hagh25,the unsteadiness in flow around three inline circular cylinders starts at Re≥101.

Table 3 Comparison of physical parameters with those of other researchers at Re=100.

Fig.5 Vorticity contour for steady state flow around two-inline cylinders at Re=1,g=2.

Fig.6 Temporal histories of force coefficients for steady state flow around two-inline cylinders at Re=1,g=2.

Fig.7 Vorticity contour for steady state flow around two-inline cylinders at Re=1,g=5.

Fig.8 Temporal histories of force coefficients for steady state flow around two-inline cylinders at Re=1,g=5.

Fig.9 Vorticity contour for transitional state of flow around three-inline cylinders at Re=105,g=2.

Also it is important to highlight here that at g=2,the transitional flow state for three-inline cylinders appears at Re=105.This means that flow remains in steady state in the range 1≤Re≤104 for three-inline cylinders.Similarly at g=2,the range of Re for steady state flow around two-and four-inline cylinders is 1–109 and 1–103,respectively( figures not shown).

Fig.10 Temporal histories of force coefficients for transitional state of flow around three-inline cylinders at Re=105,g=2.

The corresponding vorticity contour and drag and lift variation graphs for transitional state of flow at g=5 are shown in Figs.11 and 12.From these figures,it can be observed that at g=5 the behavior of shear layers and force coefficients is similar to that observed at g=2.The only difference is the value of Re at which this flow state occurs.At this spacing value(g=5),transitional flow state starts at relatively low Re values as compared to those at g=2.Due to the fact that with increment in spacing between cylinders,the effect of cylinders on each other’s flow characteristics reduces.As a result,the shear layers coming from upstream cylinders are not stabilized in such a way as at g=2.Due to this fact,at high spacing value,the flow shows unsteadiness at relatively low Re than the corresponding unsteadiness at low spacing values.The appearance of transitional state at Re=68 indicates that at g=5 the flow remains steady in the range 1≤Re≤67 for three-inline cylinders.Similarly it was also observed in our investigations that for the cases of two-and four-inline cylinders,the flow remains steady in the ranges 1≤Re≤62 and 1≤Re≤67,respectively( figures not shown).

Fig.11 Vorticity contour for transitional state of flow around three-inline cylinders at Re=68,g=5.

Fig.12 Temporal histories of force coefficients for transitional state of flow around three-inline cylinders at Re=68,g=5.

5.3.Unsteady state

After transitional phase,the flow behavior changes to complete unsteady state.A representative case of four-inline cylinders at Re=118 and g=2 for this flow state is shown in Figs.13 and 14.In this state,the flow rolls up to form vortices after interacting with cylinders(Fig.13).These vortices move alternately in the flow field and form well known Karman vortex street.Also because of narrow space between cylinders,the vortices shed in downstream region only and the shear layers are slightly suppressed inside the gaps.Due to generation of vortices,the lift coefficients have periodic behavior with larger amplitude cycles of the most downstream cylinder.The drag coefficients still show constant behavior as observed for the cases of steady and transitional state.This is due to influence of smaller spacing between the cylinders.At such smaller spacing,the flow does not find enough space to roll up which prevents variation of drag force applied on each cylinder.This type of flow behavior is also termed as bluff body shedding due to the fact that all cylinders act like a single bluff body13.

Fig.13 Vorticity contour for unsteady state of flow around four-inline cylinders at Re=118,g=2.

Fig.14 Temporal histories of force coefficients for unsteady state of flow around four-inline cylinders at Re=118,g=2.

Furthermore,Fig.13 indicates that at g=2,the unsteady state of flow around four-inline cylinders occurs at Re=118.This means that the flow around four-inline cylinders remains in transitional state for 104≤Re≤117.Similarly for the cases of two-and three-inline cylinders,the flow remains transitional in the ranges 110≤Re≤124 and 105≤Re≤119.Lankadasu and Vengadesan43found that for two-inline square cylinders,the vortex shedding starts at Re=100 and g=3.Also according to Vasel-Be-Hagh et al.,25for g=2 the vortex shedding starts at Re=105 for three-inline cylinders.

In the unsteady state of flow,at g=5 the vortices can be seen within the gaps as well as in the downstream region of cylinders.This is due to the fact that with increment in spacing value to g=5,the flow finds enough space between the gaps to roll up in the form of vortices.Hetz et al.13termed such type of flow as gap shedding.By comparing Figs.13 and 15,it can be observed that the trend of vortices at g=5 is different from those observed at g=2.At g=2,the vortices travel in a single row fashion while at g=5,the vortices form two rows of packed vortices in downstream region of the last cylinder.Bao et al.29also observed such trend around six-inline square cylinders and termed it as double-row vortex street.The graph of force coefficients shows that,unlike g=2,the drag coefficient of all cylinders is no more constant and instead it shows nonlinearity(Fig.16).This is due to generation of vortices within the gaps between cylinders.The drag of the third and fourth cylinders initially decreases and then becomes steady periodic.It should be noted that the double row vortex also generates after the third cylinder(Fig.15).

Fig.15 Vorticity contour for unsteady state of flow around four-inline cylinders at Re=85,g=5.

Fig.15 also shows that at g=5 the unsteady state of flow occurs at Re=85 which means that the transitional state of flow around four-inline cylinders remains in the range 68≤Re≤84(see Section 5.2).Similarly,the range of Re for transitional flow state in the cases of two-and three-inline cylinders is 63≤Re≤72 and 68≤Re≤82,respectively.

5.4.Comparison of range of Reynolds numbers for different flow states

Fig.16 Temporal histories of force coefficients for unsteady state of flow around four-inline cylinders at Re=85,g=5.

Tables 4 and 5 represent the range of Re for different flow states around single,two-,three-and four-inline cylinders at g=2 and g=5,respectively.Table 4 shows that the range of Re for different flow states around a single cylinder is much less than that around more cylinders in the flow field at g=2.This is because of the fact that addition of cylinders in the flow field stabilizes the shear layers emerging from most upstream cylinder.This results in retaining a particular state of flow at relatively higher values of Re as compared to corresponding values of Re for single cylinder.Also this table indicates that for multi-body case the range of Re for each flow state decreases by increasing the number of cylinders in the flow field.Normally it should not happen because the more the number of bodies in flow filed is,the higher the range of Re for a particular flow state should be.This might be due to the effect of critical spacing i.e.,g=2.According to most researchers,the critical spacing between inline bodies remains in the range 2≤g≤4 depending on Re10,12,29.So it can be concluded from here that at g=2 the effect of spacing between cylinders is dominant as compared to addition of bodies.

From Table 5,it can be observed that the range of Re for different flow states around inline cylinders,at g=5,is relatively closer to that for the flow states around single cylinder.This is due to the fact that at g=5 the distance between cylinders increases.As a result,the effect of rear bodies reduces and each cylinder behaves like single body.Also this table shows that the range of Re for each flow state increases by increasing the number of cylinders in the flow field.This trend is opposite to that observed at g=2.From this observation,it can be clinched that along with Re the spacing between cylinders also has great influence on the change in flow states around multiple bodies.

5.5.Force statistics

5.5.1.Reduction of drag

According to previous studies,the addition of one or more cylinders in the wake of single body significantly affects the fluid forces.9,12,23We have also examined this phenomenon in current study.For this purpose,the percentage reduction,with respect to single cylinder,in the average drag force(CD-mean)of the first cylinder(c1)is calculated for each combination of multiple cylinders considered in this study.These values are presented in Tables 6 and 7 for g=2 and 5,respectively.

Table 6 shows that maximum reduction occurs in the case of four-inline cylinders for all values of Re.So it can be concluded that the more the number of bodies is,in the wake of single body,the more the reduction in drag force will be.Furthermore,for all combinations of cylinders,the maximum reduction can be seen at Re=1.In the range 1≤Re≤50,the values of%reduction decrease for all combinations approaching to their respective minimum values.And for Re＞50,the values of%reduction increase for all combinations of cylinders.It should also be noted from Table 6 that as Re increases,CDmeandecreases for single cylinder as well as for all combinations of multiple cylinders.According to Breuer41,Re has a great effect on the drag force of square cylinder.

At g=5,the highest reduction in CDmeanoccurs at Re=1 and minimum reduction occurs at Re=50(Table 7).Also like g=2 the maximum reduction can be seen for the case of fourinline cylinders at all values of Re.Furthermore,by comparing Tables 6 and 7,it can be observed that at g=2,CDmeanhas high reduction as compared to that at g=5 for low Re range

i.e.,1≤Re≤40 as well as at high values of Re i.e.,90≤Re≤110,for all combinations.However at moderate values of Re i.e.,60≤Re≤80,the drag force reduces more at g=5 as compared to that at g=2.This indicates that at low and high values of Re,the spacing value g=2 has greater influence on reduction in drag forces as compared to Re.But at g=5,the moderate values of Re have greater influence on the reduction of drag force as compared to g.Also it is important to highlight here that at g=5 the flow changes its states in moderate Re range i.e.,60≤Re≤85,while at g=2 the flow states change at high values of Re for all combinations of cylinders.

Table 4 Range of Re for transitions in flow states at g=2 along with single cylinder values.

Table 5 Range of Re for transitions in flow states at g=5 along with single cylinder values.

Table 6 Percentage reduction in CDmeanof c1for two-,three-and four-inline cylinders at g=2.

Table 7 Percentage reduction in CDmeanof c1for two-,three-and four-inline cylinders at g=5.

5.5.2.Variation of CDmeanof four-inline cylinders with Reynolds numbers

Fig.17 Variation of average drag force with Reynolds number for flow around four-inline cylinders.

Fig.18 Variation of Strouhal number with Reynolds number for flow around three-inline cylinders.

The variation of CDmeanfor flow around four-inline cylinders with Re is shown in Fig.17,at g=2 and g=5,respectively.At g=2,CDmeanof all cylinders show decreasing trend with increment in Re(Fig.17(a)).The reason is that as Re increases,the viscous forces become weaker.Due to this fact,the thickness of shear layers reduces,which results in weaker drag forces.Also by comparing this graph with Table 4,it can be seen that CDmeanis higher in steady state of flow than that in transitional and unsteady states of flow.But in unsteady state,CDmeanis less than that in the other two states of flow for both the spacing values.Fig.17(a)also shows that CDmeanof the second cylinder jumps from positive to negative values at Re=50 and remains negative till Re=120.This is due to the fact that at these values of Re(50＜Re≤120),the second cylinder produces strong back flow and the drag force acts like thrust force which results in negative values of CDmean.On the other hand,at g=5,CDmeanof all cylinders is positive for all studied values of Re(Fig.17(b)).According to Igarashi and Suzuki,11the drag coefficient of middle cylinder remains negative for g＜3.53 for flow around three circular cylinders.The negative values of CDmeanwere also found in the work of Bao et al.29By comparing Fig.17(b)with Table 5,it can be observed that in steady and transitional states CDmeanof all cylinders decreases as Re increaes,while in unsteady state it shows increasing trend with increment in Re.

5.5.3.Variation of Strouhal number of three-inline cylinders with Reynolds numbers

Fig.18 presents the variation of St with Re at g=2 and g=5,respectively.It is worth mentioning here that St indicates shedding frequency of vortices and it is calculated by applying the Fast Fourier Transform(FFT)to lift coefficients.We have seen from Section 5.1 that in steady flow state the vortices do not shed and lift coefficient has con-stant behavior.That’s why for steady state flow the St values do not exist.Fig.18 shows the St values for transitional and unsteady states of flow around three-inline cylinders.From Fig.18(a),it can be seen that St is generated at Re=105 which is the starting point of the transitional flow state around three-inline cylinders(Table 4).It exhibits increasing behavior with increment in Re.This is due to the fact that as Re increases,the shear layers become thin,which results in increment in the values of St.According to Vasel-Be-Hagh et al.,25St of three-inline circular cylinders also increases with increment in Re at g=2.Also St of all cylinders is similar at all Re except Re=115.This similarity in the values of St indicates the dominancy of vortex shedding frequency over the cylinder interaction frequency.Also by comparing Table 4 and Fig.18(a),it can be deduced that the maximum value of St occurs in the unsteady state of flow i.e.,at Re=130,while its minimum value occurs in the quasi unsteady state of flow i.e.,at Re=105.Similarly at g=5,St first decreases up to Re=80 and then it shows increasing trend with increment in Re(Fig.18(b)).Its maximum value occurs at Re=110(unsteady state)and the minimum value occurs at Re=80(quasi unsteady state).

6.Conclusions

Numerical computations were performed to study the flow around different combinations of inline cylinders using the lattice Boltzmann method.The main purpose of this study was to analyze the transitions in flow states around two-,three-and four-inline square cylinders under the effect of Reynolds numbers at two different spacing values g=2 and 5.The results were presented in the form of vorticity contours,streamlines,temporal variation of fluid forces and variation of physical parameters with Reynolds numbers.The main findings of this study are given below:

(1)For both spacing values and all combinations of cylinders studied,there exist three different states of flow depending on Reynolds numbers:steady state,transitional state and unsteady state.

(2)For two-inline cylinders,it was found that at g=2,the flow remains in steady state for 1≤Re≤109,in transitional state for 110≤Re≤124 and becomes completely unsteady for Re≥125;at g=5,the flow remains steady for 1≤Re≤62,transitional for 63≤Re≤72 and becomes completely unsteady for Re≥73.

(3)For three-inline cylinders,it was found that at g=2,the flow remains in steady state for 1≤Re≤104,in transitional state for 105≤Re≤119 and becomes completely unsteady for Re≥120;at g=5,it was found that the lf ow remains steady for 1≤Re≤67,transitional for 68≤Re≤82 and becomes completely unsteady for Re≥83.

(4)For four-inline cylinders,it was found that at g=2,the flow remains in steady state for 1≤Re≤103,in transitional state for 104≤Re≤117 and becomes completely unsteady for Re≥118;at g=5,it was found that the flow remains steady for 1≤Re≤67,transitional for 68≤Re≤84 and becomes completely unsteady for Re≥85.

(5)It was found that at g=2,the range of Re for each flow state decreases by increasing the number of cylinders in the array,while at g=5,the opposite trend was observed.

(6)The maximum reduction in average drag force,of the first cylinder for each combination,was observed for the case of four-inline cylinders at both spacing values.Also the results show that at g=2 the reduction in drag is greater than the corresponding reduction in drag at g=5.The maximum reduction was observed at Re=1 for all combinations of cylinders.

(7)It was found that,for four-inline cylinders,the steady state flow has maximum average drag force,while in unsteady state the average drag force was found to be minimum at both spacing values.