Numerical study on flow and aerodynamic characteristics: Square cylinder and eddy-promoting rectangular cylinder in tandem near wall

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Numerical study on flow and aerodynamic characteristics: Square cylinder and eddy-promoting rectangular cylinder in tandem near wall Dilip K. Maiti ∗ , Rajesh Bhatt Department of Mathematics, Birla Institute of Technology & Science, Pilani 333031, India

a r t i c l e

i n f o

Article history: Received 14 August 2013 Received in revised form 10 March 2014 Accepted 22 March 2014 Available online xxxx Keywords: Aerodynamic characteristics Rectangular eddy-promoter Square cylinder Isolated Tandem Wall

a b s t r a c t Numerically simulated results are presented for shear flow past a square cylinder (of height a) near a wall (at a gap height 0.5a) in presence of eddy promoting rectangular cylinders (of fixed height a with different widths b  a) to gain a better insight into the dependency of aerodynamic characteristics of both the cylinders on the parameters: spacing distance between the cylinders S (= D /a: 0.5  S  20) and aspect ratio r (= b/a: 0.1  r  1.0). The value of Reynolds number Re is kept as Re = 100 and 200. The governing unsteady Navier–Stokes equations are solved numerically based on the finite volume method on a staggered grid system using QUICK scheme. The resulting equations are then solved by an implicit, time-marching, pressure correction-based SIMPLE algorithm. The influence of numerical parameters on the validated code used in this study is demonstrated here. The strong dependency of vortex shedding (from both the cylinders) on aspect ratio r and spacing distance S are explored and, hence, a region of finite area in the Sr-plane is proposed in order to generate the unsteadiness in the steady flow of the downstream cylinder. An attempt is made to identify the different flow regimes depending on the flow patterns of the downstream cylinder, associated with the geometrical parameters (S and r). Owing to the differences in the basic shedding frequency of the square (downstream) cylinder from that of the rectangular cylinder (promoter) of different widths, the major issue of appearing multiple peaks in the spectrum of fluctuating lift coefficient of the downstream cylinder is addressed. The thrust force observed on the downstream cylinder in presence of the thinner promoter at closely spaced arrangement is justified presenting the surface pressure distribution. Finally, the present numerical results at large spacing distances are certified with some previous numerical and experimental findings. © 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction There have been numerous experimental and numerical studies of unsteady flow past bluff bodies. These studies have several technical applications such as compact heat exchangers, cooling of electronic components, drying of different materials (textiles, veneer, paper and film materials), cooling of glass, plastics and industrial devices, and so on. The objectives of most of the studies have been devoted to examine the unsteady nature of the flow behind the bluff bodies and the effects on heat transfer and flow-induced vibrations. It is known that the high dense ICs may generate high temperature during their operation. Therefore it is required to dissipate the excess heat generated in the ICs to make them work efficiently. As it can be found from the previous literatures (Yang and Fu [49] and Sharma and Eswaran [40]), the heat flux for the

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Corresponding author. Tel.: +91 1596 515641; fax: +91 1596 244183. E-mail address: [email protected] (D.K. Maiti).

http://dx.doi.org/10.1016/j.ast.2014.03.012 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.

case of flow with vortex shedding is higher than that without vortex shedding. An enhancement of heat transfer from heated cylinder/flate tube bank fin due to the interaction of vortices generated by a vortex generator was reported in some of the previous studies (Devarkonda and Humphery [14] and Zhu et al. [50]). An upstream cylinder of rectangular shape can generate the vortices and enhance the heat transfer from a heated square downstream cylinder in tandem arrangement. In the future, a systematic study will be conducted to maximize the heat transfer from the heated square cylinder, beginning with the problem of understanding the fundamental mechanism of unsteady interaction between tandem pair of cylinders without considering the energy equation. Studies on the problems of wake development and vortex shedding behind a rectangular cylinder in free-stream flows were investigated both numerically and experimentally by Davis and Moore [12] and Franke et al. [17], Patankar and Kelkar [36]. Davis and Moore [12,13] studied the vortex shedding from rectangular/square cylinder numerically in both uniform and channel flows. Flows around rectangular cylinder in unbounded domain were

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extensively studied experimentally and numerically by Okajima et al. [33,34]. Gennaro et al. [15] studied the effect of flow shear on the Strouhal number of cylinder vortex shedding. When the cylinder is placed in the proximity of a solid wall, the strengths of the upper and lower shear layers separated from the surfaces of the cylinder are not equal and the vortex shedding pattern is distorted. The form of the wake and the vortex shedding behind a cylinder in proximity of a wall were studied by several authors, namely, Bearman and Zdravkovich [4], Bosch and Rodi [10], and Bhattacharyya and Maiti [6–8]. Bhattacharyya and Maiti [6] observed that the vortex shedding frequency was higher for the laminar flow past a square cylinder near a wall with incident inlet shear flow than the unbounded flow. The dependence of flow characteristics of a rectangular cylinder near a wall on the incident velocity and on the gap height has been reported in the previous studies (Maiti [25,26]) under incident inlet shear flow. It has been reported there that the rectangular cylinder with r = 0.5 is the optimum size (among all the rectangular cylinders of r  1) to produce the stronger vortices in the wake irrespective of the position from the wall. It has also been reported there that the vortex shedding from a rectangular cylinder of lower aspect ratio occurs at comparatively lower Reynolds number. The studies on the several aspects of the unsteady flow past tandem circular cylinders arrangement were performed by Alam et al. [1], Sharman et al. [41], and the flow and heat transfer from other obstacles in tandem arrangements were conducted by Zhu et al. [50], Singha and Sinhamahapatra [42]. The flow over two circular cylinders, with the large diameter cylinder upstream of the smaller one, were experimentally studied by Baxendale and Grant [3] and Sayers and Saban [38] for different cylinder spacings, diameter ratios and stagger angles. Tatsutani et al. [47] investigated the unsteady flow and heat transfer for a pair of square cylinders (aligned on the centerline of the channel with uniform inlet velocity profile); they reported that Strouhal number St is larger for small-large tandem pair than the cylinder pair of the same size. In a similar study, Rosales et al. [37] numerically reported the pronounced differences in the unsteady behavior of cylinders in a fully developed parabolic velocity profile when compared with an initially uniform inlet velocity profile. Mixed convection through a horizontal channel with two isolated protruding blocks on the bottom wall was numerically studied by Wang and Jaluria [48]. They observed that the frequency and amplitude of perturbation are changed by adjusting the geometry of the promoter. A numerical study of the two- and three-dimensional unsteady flows over two square cylinders arranged in an inline configuration for Re ranged in [40, 1000] at S = 4 was performed by Sohankar [43]. The effect of the spacing distance, ranged in [0.3, 12], was also studied at selected Re. Three major regimes for the flow field are distinguished by Sohankar [43]. Malekzadeh and Sohankar [29] also reported three major regimes in the flow patterns, depending on the values of height and position of a control plate in unbounded region. Etminan et al. [16] numerically studied the unconfined flow characteristics around two square cylinders in both steady and unsteady laminar flow regimes at a fixed S = 5. Flow around an inline cylinder array consisting of six square cylinders subjected to unconfined uniform flow at a fixed Re = 100 is investigated numerically by Bao et al. [2]. They showed six different flow patterns, which appeared successively with the increase of spacing distance. For the flow of incompressible fluid past a pair of square cylinders in inline tandem arrangement, Lankadasu and Vengadesan [20] reported the negative drag force on the downstream cylinder at some shear rates. In a similar study (when both the cylinders placed near a wall at a fixed height 0.5 times the cylinder height), Bhattacharyya and Dhinakaran [5] observed that the vortex shed-

ding starts for Re beyond 125 for all values of spacing distance, and the wake of the downstream cylinder consisting of a series of negative vortices. Maiti and Bhatt [27] extended the above study considering the upstream cylinder of rectangular shape of different heights and widths. The upstream cylinder was placed towards the wall at gap heights 0.1 and 0.25 times the downstream cylinder height. The suppression of the vortex shedding from the downstream cylinder was reported due to diversion of the downstream cylinder’s gap flow depending on size of the upstream cylinder and its position with respect to the downstream cylinder and the wall at a fixed Re. From the above literature discussion, it is plausible that the eddy promoter, placed at the upstream side of an obstacle, can generate the developing boundary layers, swirl, and flow destabilization, and that depends on the shape of the promoter and other parameters such as angles and speed of attack, and the position with respect to the obstacle and wall. From the previous studies (Maiti [25,26]), it can be deduced that the rectangular shape for the promoter would be interesting in generating the vortices (passively) under the incidence of shear flow to dominate the state of the wake of a downstream square cylinder. To the knowledge of the authors, not a single published paper is available in the literature on the shear flow around a square cylinder near a wall in presence of an upstream cylinder of rectangular shape in an inline tandem arrangement. The main objective of this investigation can be divided into three folds: (i) depending of the flow over the upstream/downstream cylinder on the shape (rectangular/square) of the upstream cylinder, (ii) the dependency of the aerodynamic characteristics (namely Strouhal number, time-averaged and rootmean-square (RMS) values of the fluctuating forces) of an upstream rectangular cylinder (promoter) of different widths on the presence of a downstream square cylinder, and (iii) the dependency of the aerodynamic characteristics of a square cylinder on the presence and width of an upstream rectangular promoter, under the situation of different spacing distances between the cylinders. An attempt is made to propose a region for the critical spacing distance, for which the unsteadiness can be generated in the steady flow of the downstream cylinder, for different widths of the promoter at a fixed Re. The value of critical spacing distance for which the promoter starts to shed the vortices is searched for different sizes of the promoter. The investigation is completed documenting the comparison of the present results with those of some previous experimental and numerical results at higher spacing distances. 2. Problem formulation and numerical method 2.1. Problem formulation A wall lying along the x∗ -axis and a long cylinder of square cross-section of height a is placed parallel to the wall at a height 0.5a from the wall, as below this height a suppression of vortex shedding for Re < 125 was reported in the previous study (Bhattacharyya and Maiti [6]). Another cylinder of rectangular crosssection of aspect ratio r = b/a with width b and height a is placed in the upstream side of the square cylinder at a distance D and parallel to the wall at the same height (= 0.5a) from the wall (see Fig. 1). The upstream flow field is taken as a uniform shear flow U y∗ u ∗ = 0a where U 0 is the velocity at height a from the wall. The rationale for this choice of linear velocity profile near the wall has been discussed in the previous study (Maiti [25]). As there is no velocity scale U 0 directly, the prescribed slope λ of the incident velocity profile at the surface multiplied by a, leaving U 0 = λa is taken as the velocity scale based on obstacle height. The height of the cylinder a is considered as the characteristic length scale. It may be noted that there will not be any incidence flow if the velocity gradient λ approaches zero.

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achieved by advancing the flow field variables through a sequence of short time steps of duration t. 2.3. Definitions of aerodynamic characteristics The coefficients of the drag (C D ) and lift (C L ) on the cylinders are given by

C D = C Dp + C Dsh = and

FD 0.5ρ U 02 a

C L = C Lp + C Lsh =

Fig. 1. Schematics of the flow configuration.

2.2. Model equations with boundary and initial conditions The flow is considered to be two-dimensional and laminar. The non-dimensional Navier–Stokes equations are given by

∇·V=0 ∂V 1 + (V · ∇)V = −∇ p + ∇ 2 V ∂t Re

(1) (2)

The non-dimensional quantities V = (u , v ), p and t denote the velocity, pressure and time, respectively, and are defined as:

u= x=

u∗ U0 x∗ a

,

,

v= y=

v∗ U0 y∗ a

,

p= t=

,

p∗

ρ U 02

t∗U 0 a

(3)

Here ρ denotes the density of the fluid. The flow field is characU a terized by the parameters: (i) Reynolds number Re = ν0 , (ii) intercylinder spacing distance S = b . a

D , a

and (iii) aspect ratio of the rect-

Here ν denotes the kinematic viscosity of angular promoter r = the fluid. Leung et al. [23] reported that a very important range of Re lies under 500 in the cooling of PCB assemblies. The rationale for the assumptions of two-dimensional and laminar flow within the considered range [100, 200] of Re in the present study can be found in Bhattacharyya and Dhinakaran [5] and Maiti [25]. The mathematical form of the boundary conditions used in this study to solve Eqs. (1), (2) is:

u = y,

v =0

at the far upstream,

(4)

Further,

u = v = 0,

∂u = 0, ∂y

on the plane wall ( y = 0) and surfaces of both the cylinders,

(5)

v = 0 at the top lateral boundary.

(6)

At the outlet boundary, we considered the Sommerfeld radiation condition (7)

∂φ ∂φ + uc = 0, ∂t ∂x

(7)

where φ is any flow variable and u c is the local wave speed. The numerical treatment to the condition (7) has been discussed in the previous study (Maiti [25]). The flow is assumed to start impulsively at a particular value of Re. The converging solution (after the transient state) computed at this Re is considered as an initial solution for the case of other values (lower/higher) of Re. The transition time of a particular case is dependent on the initial conditions (impulsively started or starting from lower/higher Re). A time-dependent numerical solution is

(8)

where C Dp and C Dsh represent the drag coefficient due to pressure and viscous forces, respectively; similarly, C Lp and C Lsh represent the lift coefficient due to pressure and viscous forces, respectively. F D and F L are the integrated drag and lift forces, respectively, acting on the cylinders. The pressure coefficient C P is obtained as







C P = p ∗ − p ∗0 /

1 2



ρ U 02

(9)

where p ∗0 is the dimensional pressure at the inlet boundary. The fluctuating lift and drag components are presented using RMS (root-mean-square) values,

  N 1 C rms =  (C i − C )2

,

.

FL 0.5ρ U 02 a

N

(10)

i =1

Here C rms is the RMS coefficient for either lift or drag, C i is the instantaneous lift or the drag coefficient of a single point in time, C is the mean lift or drag coefficient while N is the number of samples used to calculate the RMS coefficients. The Strouhal number (St) based on the characteristic velocity U 0 and length a is expressed as follows

St = f a/U 0

(11)

where f is the frequency of vortex shedding. 2.4. Numerical method The computational domain is divided into Cartesian cells. The iterative algorithm SIMPLE (Patankar [35]), based on finite-volume method (FVM) with staggered grids, is applied. Leonard [22] concluded that finite volume formulations, in general, are considerably more accurate than the corresponding finite-difference formulation of the same formal order. A third order accurate Quadratic Upwind Interpolation for Convective Kinematics (QUICK) (Leonard [22]) is employed to discretize the convective terms and central differencing scheme for diffusion terms. Sengupta et al. [39] recommended that between two higher order schemes: QUICK and MUSCL (Monotone Upstream – Centered Scheme for conservation Laws), QUICK is the best one since the MUSCL scheme displayed a small overshoot over one at intermediate wave number, while QUICK scheme does not have this attribute and the QUICK scheme adds less numerical dissipation as compared to MUSCL scheme. Patankar [35] has shown that the use of upwind schemes always gives physically realistic solution, even for coarse grid. A fully implicit second order scheme is incorporated to discretize the time derivatives. As pointed out by Manson et al. [30], the main advantage of fully implicit temporal discretization scheme, i.e., unconditional stability and therefore larger time-steps, is of minor importance for truly unsteady flows, where the time-step t is strictly limited on accuracy grounds. At the initial stage of motion, t is taken to be 0.0001, which has been subsequently increased

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Fig. 2. The grid size distribution along x-direction (solid line) and y-direction (dotted line) for the case of promoter r = 0.5, S = 3. UP: upstream cylinder and DN: downstream cylinder.

to 0.001 after the transient state. A detailed discussion of the numerical methodology (staggered grid, FVM, QUICK and SIMPLE algorithms) used here has been made in the previous study (Bhattacharyya et al. [9]). 2.4.1. Influence of numerical parameters Karniadakis [19] has proposed a numerical error bar in the field of CFD. However, the identification and quantification of specific numerical error may be very difficult, especially for timedependence computations, non-uniform grids and complex geometries. Nevertheless, as in Ref. [19], the error bar in calculations of vortex shedding flows may be divided into four parts: errors due to computational domain size, boundary conditions, temporal and spatial errors. The inflow boundary lies at a distance L inlet = 10a from the front face of the upstream cylinder while the top lateral boundary lies at L top = 10a from the top face of the downstream cylinder. In this study, the convective boundary condition (7) at the outlet is implemented, which drastically reduces the downstream length of the computational domain and hence the convergence process becomes faster. This condition (7) is commonly known as the most computationally efficient mathematical condition compatible with the physics as the wave reflection at the outlet boundary can be avoided for unsteady flows. Sohankar et al. [46] recommended L outlet = 10a, the distance of the outflow boundary from the downstream cylinder rear face, for the case with condition (7) in the range of Re considered here. In the present study L outlet ranged in [15a, 17a] for Re ranged in [100, 200], Fig. 1. The upstream influence from the outlet is effectively damped out with this sufficiently large L outlet . The errors due to computational domain size and boundary conditions can be regarded as very small (within a few percent). Franke et al. [17] observed that the distance of the first grid point away from the body (δ1 ) has a particularly strong influence on the results and Sohankar et al. [45] suggested that the far-field resolution and the number of nodes distributed along the unit length of the square cylinder are the most crucial parameters for producing grid-independent results. Sohankar et al. [29,46,45,44] recommended that the grid size, if it is less than 0.20, far away from the cylinder has negligible effect on the final results. In this study, a non-uniform grid distribution is considered distributing

uniform grids along the surfaces of both the cylinders with expansion factors for the far-fields (away from the surfaces) starting with δ1 . The grid distribution in x- and y-directions is shown in Fig. 2. Depending on the size of distributed grids, the horizontal and vertical lines of the computational domain can be divided into seven and three, respectively, distinct segments. A symmetry grid (along the x-direction about the mid-line in the spacing between the cylinders) starting with δ1 from any one of the interfacing faces is incorporated (segment-I V x in Fig. 2). The numerical grids taken in the gap spacing here is double of the grids used by Nikfarjam and Sohankar [32]. Along y-direction, a uniform grid with size 0.005 is considered between wall and top of the cylinder (within the segment I y in Fig. 2). Both Franke et al. [17] and Sohankar et al. [45] reported in their calculations that δ1 = 0.004 is sufficiently fine. Therefore, the value of δ1 is kept constant as δ1 = 0.004 for the present computation also. As depicted in Fig. 2, the use of a finer grid in the present study reduces the truncation error and effects of numerical diffusion. At constant Re, the t ratio  δ1 is a measure of the maximum Courant number, which for higher accuracy and physical realization in convective regions should be kept as small as possible i.e., temporal and spatial errors are coupled (Manson et al. [30]). In the present study, the t value of  δ1 is 0.025, much smaller than the values used by Sohankar et al. [29,46,45,44]. In fact, the maximum CFL is 0.6 over the whole computational domain for all the simulations. In particular, the CFL value is about 0.022 around the cylinders, spacing between the cylinders, and in the gap height between the wall and cylinders. It was reported in the previous studies (Maiti [25,26]) that the refinement on the total number of grids by a factor 4 changed St, C D , C L and C Lrms by less than 6% (Table 1 of Maiti [25] for different Re and Table 1 of Maiti [26] for different gap heights). The effect of grid refinement on St, C D , C L , C Lrms and C Drms for the case of upstream cylinder of r = 0.5 placed at S = 3 is summarized in Table 1 for Re = 100 and 200 varying grids between 400 × 300 and 1600 × 1200 with the first and second number being the number of mesh points in the x-direction and in the y-direction, respectively. Table 1 clearly reveals that refinement on the total number of grids by factors of 4 and 16 changed St, C D , C L , C Lrms and C Drms by less than 6.2% and 7.9%, respectively. The maximum percentage

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Table 1 Grid refinement study on St, C D , C L , C Lrms and C Drms for the case of upstream rectangular cylinder of aspect ratio r = 0.5 at S = 3, Re = 100 and 200 for different grid sizes. UP: upstream cylinder and DN: downstream cylinder. The value within the bracket is the percentage deviation of aerodynamic characteristics at a grid from that of the grid 625 × 400. Re 100

200

Grid

St

CD

CL

C Lrms

C Drms

UP

625 × 400 400 × 300 800 × 300 800 × 600 1600 × 1200

– – – – –

3.1922 3.0901(3.2) 3.0454(4.6) 3.0933(3.1) 3.1586(1.05)

0.4318 0.4111(4.8) 0.4184(3.1) 0.4240(1.8) 0.4486(3.9)

– – – – –

– – – – –

DN

625 × 400 400 × 300 800 × 300 800 × 600 1600 × 1200

– – – – –

1.2945 1.3436(3.8) 1.2414(4.1) 1.2647(2.3) 1.2924(0.16)

0.0847 0.0874(3.2) 0.0819(3.3) 0.0865(2.1) 0.0879(3.8)

– – – – –

– – – – –

UP

625 × 400 400 × 300 800 × 300 800 × 600 1600 × 1200

0.297 0.2993(0.78) 0.3002(1.10) 0.3043(2.47) 0.3015(1.52)

3.2011 3.1115(2.8) 3.3548(4.8) 3.2844(2.6) 3.2874(2.7)

0.2171 0.2104(3.1) 0.2102(3.2) 0.2236(3) 0.2249(3.6)

0.6131 0.6315(3) 0.5935(3.2) 0.6272(2.3) 0.5941(3.1)

0.1824 0.1871(2.4) 0.1771(3.0) 0.1877(2.7) 0.1763(3.4)

DN

625 × 400 400 × 300 800 × 300 800 × 600 1600 × 1200

0.3098 0.3252(4.9) 0.3172(2.4) 0.32125(3.6) 0.31004(0.07)

0.5444 0.5275(3.1) 0.5733(5.3) 0.5580(2.5) 0.5601(2.6)

0.0145 0.0151(3.8) 0.0152(4.6) 0.0149(2.6) 0.0139(4.5)

1.2666 1.3173(4) 1.2096(4.5) 1.2362(2.7) 1.2122(4.4)

1.2085 1.2512(3.4) 1.1555(4.5) 1.1827(2.1) 1.1587(4.2)

differences of St, C D , C L , C Lrms and C Drms at grid 800 × 600 from those at grid 800 × 300 are 1.3% (occurs at Re = 200, DN), 2.6% (occurs at Re = 200, DN), 6.1% (occurs at Re = 200, UP), 5.3% (occurs at Re = 200, UP) and 5.9% (occurs at Re = 200, UP) respectively. As it is seen from Table 1, the major differences of these aerodynamic characteristics occur on double the number of grids along the streamwise direction with fixed coarse grid along the transverse direction (i.e., from 400 × 300 to 800 × 300) while the differences of these characteristics are found minor varying between two finer grids of 800 × 600 and 1600 × 1200. The percentage deviations of St, C D , C L , C Lrms and C Drms computed at 625 × 400 from other girds are noted in Table 1. Hence it is quite clear from the ongoing discussion and Table 1 that the numerical results based on 625 × 400 (between two medium grids: 625 × 400 and 800 × 300) are closer to the results computed at two finer grids: 800 × 600 and 1600 × 1200. It is noted that the computation based on finer grid takes much more time than that on medium grids. Breuer et al. [11] recommended that a higher resolution is required only at higher Re. Therefore the grid 625 × 400 can be taken as reasonably fine for the calculation domain with r = 0.5 and S = 3. For other combinations of r and S, the number of grid points are taken accordingly by increasing/decreasing the number of grids distributed on the width of the upstream cylinder and in the spacing distance only. It may be noted that the effect of grid size on St and C D at various values of Re for uniform flow past a square cylinder placed in an unbounded region has been discussed in our earlier paper (Fig. 3 of Bhattacharyya and Maiti [7]) varying the grid sizes over the whole computational domain including δ1 . It was reported that the changes in solution due to halving the grid size occur on the third decimal place. 2.4.2. Validation of numerical code The numerical code used in the previous studies (Bhattacharyya and Maiti [6–8] for single square cylinder and Maiti [25,26] for single rectangular cylinder) has been used here incorporating the square cylinder on the downstream side of the rectangular cylinder in the computational domain. This code was validated for the case of (i) a square cylinder without plane wall for different Re (Fig. 3, Bhattacharyya and Maiti [7]), (ii) a square cylinder placed in a boundary layer at a gap height from a wall (Table 1 of Bhattacharyya et al. [9]), (iii) a square cylinder confined in a channel

(Fig. 4 of Bhattacharyya and Maiti [6]), and (iv) heat/mass transfer in a cavity in the presence/absence of lid motion (Table 1 of Maiti et al. [28]). The numerical methodology used here is the same as used in our previous study [27]. Thus, the previous published results of the first author [6–8,25,26,9,28] and the present authors [27] also show the validity of the used code for this work. The numerical code for the present case of tandem cylinders has also been validated with Rosales et al. [37] (for single/double square cylinder/s in channel under the parabolic approaching flow) and Bhattacharyya and Dhinakaran [5] (for inline tandem square cylinders under the present flow conditions), and presented in Fig. 3. Comparing the curves between ‘ P ’ and ‘R’ (for both single cylinder and double cylinders cases) in Figs. 3(a)–(c) and ‘ P ’ and ‘B’ (for both upstream and downstream cylinders) in Figs. 3(d)–(f), it may be noted that the maximum percentage differences of St, C D and C L of the present calculation from those of Rosales et al. [37] are 6.7% (occurs at d/h = 0.431 for single), 5.6% (occurs at d/h = 0.431 for double) and 7% (occurs at d/h = 0.431 for double), respectively, while these values are 5% (occurs at S = 2, downstream), 3.34% (occurs at S = 5, upstream) and 6.7% (occurs at S = 4, downstream), respectively, for the comparison with Bhattacharyya and Dhinakaran [5]. The comparison of the present results for the case of two square cylinders in in-line tandem arrangement placed in unbounded domain under uniform flow is made with Bao et al. [2] and Lankadasu and Vengadesan [20] in Fig. 4. The excellent agreement of the present results with those of previous results is found in St over the whole range of S. Although, the maximum differences are found in both the RMS (C Drms and C Lrms ) of the downstream cylinder at some S. The differences in these aerodynamic characteristics between the results of Bao et al. [2] and Lankadasu and Vengadesan [20] are also larger in Figs. 4(c)–(d). It may be noted that Bao et al. [2] and Lankadasu and Vengadesan [20] have used the numerical methodology different from that used in the present study, Bhattacharyya and Dhinakaran [5] and Rosales et al. [37]. The behavior of aerodynamic characteristics of both the cylinders at large spacing for the cases of unbounded and near wall are validated with the previous studies and discussed in Section 3.3. The grid-independent results and comparisons reveal a satisfactory agreement and would, hence, boost confidence in the generated results.

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Fig. 3. Comparison among Strouhal number (St), time-average drag coefficient (C D ), and time-average lift coefficient (C L ) results for single/double square cylinder/s with Rosales et al. [37]: (a)–(c) at Re = 500, and double square cylinders with Bhattacharyya and Dhinakaran [5]: (d)–(f) at Re = 200. In (a)–(c): d is the distance between the top of the cylinder and the upper channel wall, h is the channel height, and the values for the double cylinders are of the downstream cylinder.

Fig. 4. Comparison among Strouhal number (St), time-average drag coefficient (C D ), and RMS of lift (C Lrms ) and drag (C Drms ) coefficients results for double square cylinders in uniform flow with Bao et al. [2] and Lankadasu and Vengadasan [20] at Re = 100.

3. Results and discussion

The numerically simulated results are presented for an upstream rectangular cylinder (also to be known as promoter) of r = 0.1, 0.25, 0.5, 0.75 and 1.0 and different values of S (0.5  S  20) at Re = 100 and 200. In order to delineate the behavior of the aerodynamic characteristics due to this tandem arrangement, the

computed results of a single rectangular cylinder (Maiti [25]) are often used for the comparison. 3.1. Fluid flow analysis at Re = 100 Under the incident of shear flow over cylinder/s placed at a gap height L = 0.5a from a wall: Maiti [25] reported the critical Reynolds number (Recr-iso-r ) for onset of alternating vortex shedding from an isolated cylinder of aspect ratio r = 1.0, 0.75,

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Fig. 5. Streamlines over the cylinders at Re = 100 for closely spaced-arrangement (S = 0.5) (left side) and different spacing distances (right side) for r = 0.75, 0.5, 0.25 and 0.1.

0.5, 0.25 and 0.1 at 125, 110, 100, 80 and 75, respectively, and Bhattacharyya and Dhinakaran [5] reported the steady flow over a downstream square cylinder in presence of an upstream square cylinder in an inline tandem arrangement for all S at Re = 100. Therefore it would be interesting to observe the flow patterns of the downstream square cylinder in the wake of an upstream cylinder of rectangular shape (r < 1) at Re = 100 in an inline tandem arrangement, and vice versa. 3.1.1. Dependency of flow on width and position of promoter The streamlines are presented in Fig. 5 for all the cases of r < 1.0 at different S. The numerical flow visualization for closelyspaced arrangement (S = 0.5), as displayed in Fig. 5 (left side), indicates that the form of steady wake of the downstream cylinder is similar to the wake due to a single rectangular cylinder (of r > 1) exposed to a shear flow in wall proximity. The flow around the cylinders at this spacing remains steady for all considered values of r. The fluid flowing over the cylinders and the flow separating from the sharp corners of the leading edge of the promoter (not from the downstream cylinder) form a stagnant pair of vortices attached to the rear face of the downstream cylinder. The wall induced negative shear layer diffuses part of the vorticity in the positive shear layer issuing from the wall-side face of the cylinders. The wake length of the downstream cylinder increases with the reduction of width of the promoter. The fluid within the spacing between the cylinders can be considered as stagnant for this closely-spaced arrangement since the closed wake cannot be formed behind the promoter due to insufficient spacing. With the increase of spacing distance S within their respective subcritical-spacing (< S cr : the spacing at which the promoter starts to sheds the vortices and consequently the unsteadiness is generated in the steady flow of a square downstream cylinder), the shear layers separated from the promoter are allowed to merge on a line and a closed wake with a stagnant pair of counterrotating vortices is formed between the cylinders (Fig. 5 (right side)). However, the unsteady interaction between the shear layers of the promoter is interrupted due to lack of sufficient spacing and the vortex shedding from the promoters is perfectly suppressed. Therefore, the flow over the rectangular cylinders of r  0.5 is

largely affected by the presence of a downstream cylinder placed within their subcritical spacing range (0 < S  3 for r = 0.5 and 0 < S  2.5 for r  0.25). As seen in Fig. 5, the flow of the downstream cylinder remains steady and the wake length of the downstream cylinder decreases with the increase of spacing distance for all r. The time histories of the lift coefficient (C L ) of both the cylinders, associated spectrum (by Fast Fourier Transformation (FFT)) along with the instantaneous vorticity and pressure contours for the promoters of r = 0.75, 0.5, 0.25 and 0.1 at their respective critical spacing (S cr ) are presented in Fig. 6. Upon increasing the spacing (to S = 3.5 for r = 0.5 and S = 3.0 for r = 0.25 and 0.1: Figs. 6 (b)–(d)), the rectangular cylinders of r  0.5 developed their own unsteady vortex shedding wake within the spacing. It is interesting to note from Fig. 6(a) that the rectangular cylinder of r = 0.75 also started to shed the vortices in front of the downstream cylinder at S = 3.5 at this lower Re (< Recr-iso-0.75 ). Consequently, the upper shear layer of the downstream cylinder does curl up in a periodic manner (vorticity contours in Figs. 6(a)–(d)); however, as for the small Re (< Recr-iso-1.0 ) the wake periodicity of the downstream cylinder observed in the fluctuating lift coefficients and their associated spectrum in Fig. 6 should not be interpreted as an alternating vortex shedding in the present study of shear flow near a wall. The wake of the downstream cylinder consists of a single row of negative vorticity (previously observed by Bhattacharyya and Maiti [8] and Maiti [26] for single cylinder). It is evident from the pressure contours in Fig. 6 that the upstream side of promoter is subjected to the higher pressure zone, therefore pressure experienced by the downstream cylinder is comparatively low, because the downstream cylinder enveloped in the wake of upstream cylinder. The same shedding frequency is found in Fig. 6 for both the cylinders at this Re. This confirms that the unsteady shedding mechanism of the downstream cylinder is completely controlled by the vortices shedding from the promoter. It is evident from Figs. 5 and 6 that the critical spacing S cr for the promoter of r = 0.75, 0.5, 0.25 and 0.1 is registered at 3.5, 3.5, 3 and 3, respectively. It may be noted that the computation for the unsteadiness

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Fig. 6. Time histories of the lift coefficient (C L ), associated spectrum along with the vorticity and pressure contours at Re = 100 for different promoters at their respective critical spacing (S cr ). In all the vorticity plots, vorticity is drawn at the instant corresponding to the maximum in the respective lift of promoter. In vorticity/pressure contours, solid lines: positive vorticity/pressure and dotted lines: negative vorticity/pressure. UP: upstream cylinder and DN: downstream cylinder.

Fig. 7. Gap flow of the downstream cylinder ((a)–(b): profile of mean horizontal velocity component (u) along the vertical direction ( y) at x = 1.0 for r = 0.75, 0.5, 0.25 and 0.1 and S = 0.5 and S cr ; (c): mean surface pressure distribution (C P ) along the bottom of the downstream cylinder and the wall for r = 0.5 and 0.1 at S = S cr ) and the upstream cylinder ((d): profile of u along y at x = − S for S = 2.5, 3.0 and 3.5 and r = 0.5 and 0.1) at Re = 100. The profile of isolated rectangular cylinder of r = 0.5 and 0.1 is also presented in (d). UP: upstream cylinder and DN: downstream cylinder.

for a particular case is carried out only at some discrete values of S with the increment S = S + 0.5 near S cr . Fig. 7 presents the mean horizontal velocity (u) profiles along the vertical direction ( y) at the exit position of the gap flow of

downstream cylinder (between the downstream cylinder’s lower surface and the wall, see plots a and b) and promoter (between the promoter’s lower surface and the wall, see plot d), and the mean surface pressure distribution (C P ) along the downstream cylinder’s

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Fig. 8. Vorticity contours for the case of promoter with r = 0.75 at Re = 100 for different S at t = 100. Solid lines: positive vorticity, dotted lines: negative vorticity.

Fig. 9. Flow characteristics at Re = 100 for r = 0.1: (a) equivorticity lines at S = 20, (b) instantaneous lift coefficient (C L ) of both the cylinders against time at S = 20, and (c) spectra of fluctuating lift coefficient of downstream cylinder for different S. In vorticity contours, solid lines: positive vorticity, dotted lines: negative vorticity. UP: upstream cylinder and DN: downstream cylinder.

lower surface and the plane wall (see plot c). As seen in Fig. 7(a), the velocity profiles overshoot for all the promoter cases of r < 1 and the gap flow of the downstream cylinder does not change appreciably by the presence of the promoter (notably for r = 0.1, Fig. 5 (left side)) for the closely-spaced arrangement. With the increase of spacing distance S within the critical spacing range, the diversion of the downstream cylinder’s gap flow is clearly exhibited in Fig. 7(b). The width of the promoter has marginal role on the downstream cylinder’s gap flow for S = 0.5, while nominal for S = S cr . Fig. 7(c) shows that the pressure distribution (C p ) in the gap flow between the downstream cylinder and wall in the tandem arrangement is similar to that in the isolated cylinder of steady case. The pressure distribution difference between the sides is almost zero in the core of the gap region for all the cases. In fact the range of x at which the pressure distribution difference is almost zero is larger for the case of tandem arrangement compared to the isolated case. This implies that the gap flow is unidirectional, and the core flow resembles that of channel flow. As a result, the interaction between the shear layers of the downstream cylinder gets lost. This mechanism for the suppression of vortex shedding from a square cylinder placed near a wall was experimentally proposed by Martinuzzi et al. [31] and numerically confirmed by Bhattacharyya and Maiti [8]. This validates the argument stated in Figs. 6(a)–(d) that vortex shedding observed from the downstream cylinder is due to rolling of shedding vortex (negative one) of the promoter. The mechanism of onset/suppression of vortex shedding from the promoter of r  0.5 as observed in Figs. 5, 6 is discussed based on velocity profile presented in Fig. 7(d). It is quite clear that the gap flow (which plays a major role in the shedding of vortex behind the cylinder near a wall) becomes weaker in the presence, compared to the absence, of the downstream cylinder. Lee et al. [21] reported that in the cases where the vortex shedding occurs, the value of maximum velocity is greater than in the cases without vortex shedding. Also, the position at which maximum velocity occurs is closer to the lower surface of the cylinder. As can be seen in this figure, when the spacing is increased, the gap flow becomes stronger and the velocity profile approaches that of an isolated cylinder case of vortex shedding, and hence the vortex shedding from the promoter is to be observed at the critical spacing (S cr = 3.5 for r = 0.5 and S cr = 3 for r = 0.1; Figs. 6(b) and 6(d), respectively). The justification for the vortex shedding from a promoter of lower aspect ratio at comparatively lower critical spacing can be found from Fig. 7(d).

3.1.2. Existence of unsteadiness of downstream cylinder at large spacing It is clear from the above discussion (based on Figs. 6(b)–(d)) that the upstream cylinder of rectangular shape with r  0.5 generates the developing boundary layers, which swirl, and hence the unsteadiness is generated in the steady flow of a square downstream cylinder at a lower Re (< Recr-iso-1.0 ). It is also seen in the Fig. 6(a) that the unsteadiness in the steady flow of a rectangular cylinder of r = 0.75 is generated at a Re (< Recr-iso-0.75 ) employing a square cylinder in the downstream side at some specific S. In order to examine the persistence of this generated flow behavior on the downstream/upstream cylinder at large S, the flow patterns are presented for two extreme values of r (< 1): r = 0.75 in Fig. 8 and r = 0.1 in Fig. 9. As seen in Fig. 8(a), the unsteadiness in the flow of the upstream cylinder of r = 0.75 is persisting for S = 5.0. The possible explanation for the shedding of vortices from the promoter of r = 0.75 in Figs. 6(a) and 8(a) at this lower Re (< Recr-iso-0.75 ) may be that in presence of the downstream cylinder at these spacings the positive shear layer issuing from the promoter is forced to interact with its negative counterpart. As a result, the upper shear layer of downstream cylinder does curl up in a periodic manner. At sufficiently large S = 6, as seen in Fig. 8(b), the two free shear layers from the promoter eventually convect downstream without interacting to each other and the Karman vortex street behind the promoter is completely absent, similarly to that of an isolated rectangular cylinder of r = 0.75. Hence the steady vortical structure of an isolated square cylinder is returned to the downstream cylinder for S  6. As is seen in Figs. 9(a)–(b), the unsteadiness is there in the flow over the downstream cylinder in the presence of the thinnest promoter (r = 0.1) notwithstanding at very large spacing S = 20. From Fig. 9(a) it is immediately evident that the shedding of vortices from the promoter is an inherent behavior of a cylinder of r = 0.1 placed at a gap height L = 0.5a at this Re (> Recr-iso-0.1 ) and S (> S cr ). The shedding vortex (mainly the negative one) from the promoter hits the front face (and finally rolls over the upper face) of the downstream cylinder. A constant Strouhal number is found from the fluctuating lift of the downstream cylinder at each of spacing S varying from S = 3 to S = 20 (Fig. 9(c)). As one may expect the magnitude of fluctuating lift of the downstream cylinder decreases upon increasing the spacing S (Fig. 9(c)).

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Table 2 Classification of flow regimes depending on geometrical parameters (S and r) and flow patterns observed on the downstream cylinder at Re = 100. UP: upstream cylinder and DN: downstream cylinder. Regime

r

UP’s flow

Gap spacing flow

DN’s flow

Figure

I (S  0.5)

∀r < 1

Steady of single cylinder of r > 1

Fluid is stagnant

Steady of single cylinder of r > 1

5 (left side) and 7(a)

II (0.5 < S < S cr )

∀r < 1

Steady: shedding suppressed for r  0.5

Closed wake with stagnant pair of counter rotating vortices

Steady: gap flow suppressed

5 (right side) and 7(b)–(d)

III (S cr  S  S cr )

IIIa (r = 0.75)

Forced shedding

Rolling of shedding vortices from UP

Generated unsteadiness

6(a) and 8(a)

IIIb (r  0.5)

Inherent shedding

Rolling of shedding vortices from UP

Generated unsteadiness

6(b)–(d), 9 and 7(d)

IVa (r = 0.75)

Steady of isolated cylinder of r = 0.75

Shear layers of UP convect downstream without interacting to each other

Steady of isolated square cylinder

8(b)

IVb (r  0.5)

Inherent shedding

Rolling of shedding vortices from promoter up to a certain spacing and finally diffused before the front face of DN

Steady of isolated square cylinder

IV (S > S cr )

Table 3 Variation of amplitude of spectrum of lift coefficient of the upstream cylinder with spacing S for different r at Re = 200. S

r = 0.75

r = 0.5

r = 0.25

r = 0.1

1.5 2.0 2.5 3.0

0.0309 0.0128 0.0021 1.0950

0.0369 0.0030 0.0029 0.0114

0.0143 0.0010 0.0012 1.2068

0.0048 0.0041 0.000057 0.6472

3.2. Influence of thinning and position of promoter at Re = 200 Fig. 10. Proposed zone in the Sr-plane for which unsteadiness in the steady flow of square downstream cylinder can be generated at Re = 100 (< Recr-iso-1.0 ) by selecting the values of S and r from the shaded zone.

3.1.3. Proposed zone for critical spacing for unsteadiness of downstream cylinder and classification of flow patterns It is plausible from the discussion in Section 3.1.1 that there exists a critical spacing S cr at which the unsteadiness in the steady flow of a square downstream cylinder is generated for each value of r (< 1) considered here. While plotting these values in Sr-plane in Fig. 10, it is observed that the critical spacing S cr is dependent on the aspect ratio r at Re = 100. At the same time, one can deduce from the discussion in Section 3.1.2 that the dynamic interaction of the shear layers of downstream cylinder with the shedding vortices of the promoter reduces with the increase of spacing. As a consequence, the original steady/unsteady flow of an isolated square cylinder is to be expected from the downstream cylinder in the physical domain after a certain large S (say, S cr : the upper bound for S cr ). Beyond this value of S cr , there should not be any effect of presence of the promoter on the behavior of flow over the downstream cylinder. For example, in Fig. 8, the value of S cr is ‘5’ for r = 0.75. However, as seen in Fig. 9, computationally it is very difficult to find the value of S cr for each value of r as it requires a lot of unsteady computations varying S based on the increment S = S + 1. Hence, a region of finite area could be proposed in the Sr-plane (as displayed in Fig. 10 based on computations: r = 0.1, 0.25, 0.5, 0.75 and 1.0) to generate the unsteadiness in the flow of a square downstream cylinder at a lower Re = 100 (< Recr-iso-1.0 ). According to the values of the geometrical parameter (S) and based on the flow patterns observed on the downstream cylinder due to consideration of rectangular promoter r < 1, four major regimes are identified at this Re = 100. Key features of the flow patterns are summarized in Table 2.

3.2.1. On vortex shedding, spectra and frequency of oscillation (Strouhal number) Table 3 presents the variation of amplitude (defined as the height of peak) of spectrum of fluctuating lift coefficient of the promoter with spacing S ( 3) for each of r < 1. Bhattacharyya and Dhinakaran [5] reported the vortex shedding from the upstream cylinder of square shape for S  3. Here it is observed that the downstream cylinder sheds vortices for all S and r at Re = 200. As a result, the separated shear layers from the promoter (attached to those from the downstream cylinder) oscillate, but the amplitude of oscillation decreases with the increase of spacing in their sub-critical spacing range (Table 3 for S = 1.5, 2 and 2.5). It is very difficult to determine the exact value of the critical spacing for the onset of vortex shedding from the promoter for each value of r as it requires many unsteady computations. However, it may be deduced from the variation of intensities of the peaks in the spectra with spacing shown in Table 3 that the vortex shedding from the promoter could be observed near S = 2.5 for all the cases of r < 1. As depicted in Fig. 11, the instantaneous lift signals are found sinusoidal (with a single peak in the associated spectrum) from all the promoters (Fig. 11(c) at S = 7.0) and the downstream cylinder at closely spaced arrangement (Fig. 11(b) for a particular case with r = 0.1 and S = 2.0). It is observed that the lift signals of the downstream cylinder in Fig. 11(b) are similar to those of an isolated square cylinder in Fig. 11(a). Unlike the cases of forced shedding from the downstream cylinder at Re = 100 (instantaneous C L of downstream cylinder in Figs. 6 and 9) and the absence of shedding from the promoter at Re = 200 (Fig. 11(b)), here in Figs. 11(d)–(e) at S = 7 and Re = 200 the flow over the square downstream cylinder becomes complicated. At this spacing S = 7, the promoter of aspect ratio r = 1.0, 0.75, 0.5, 0.25 and 0.1 sheds the vortices with respective shedding frequencies St = 0.3473, 0.3202, 0.3001, 0.2815 and 0.2782. As can be seen in Fig. 12, the vortices shedding from the

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Fig. 11. Time histories of the lift coefficient (C L ), associated spectrum at Re = 200: (a) Isolated square cylinder and (b) downstream cylinder before onset of vortex shedding from the promoter for a particular case of r = 0.1 and S = 2.0. Time histories of lift coefficient (C L ) at Re = 200 and S = 7.0 for different r: (c) Upstream cylinder, and (d) Downstream cylinder. (e) Spectra of respective fluctuating lift of the downstream cylinder, plotted in (d), at S = 7 for different r.

Fig. 12. Vorticity contours during one shedding cycle of the promoter of r = 0.5 with a period T (left side) and r = 0.1 with a period T  (right side) at S = 7 and Re = 200. Here t 0 is the time at which lift coefficient attains its minimum value. Solid lines: positive vorticity and dotted lines: negative vorticity.

promoter hit the downstream cylinder and at the same time the downstream cylinder develops its own vortices. These two interact with each other and result in a complicated multi-frequencies with different values and strengths in the spectrum of C L of the downstream cylinder (Fig. 11(e)). However, it is observed based on the study of the spectra of C L that the dominant frequencies of the flow for the downstream cylinder and promoter are similar. A little difference in the frequency of the downstream cylinder from that of the promoter is observed at spacing S near S cr due to consideration of shear flow near a wall (Bhattacharyya and Dhinakaran [5]). It is evident from Fig. 11(e) that the possibility of appear-

ing of the multiple peaks in the spectrum of lift coefficient of the downstream cylinder is more for a case of promoter of lower aspect ratio. The vortex shedding frequency is analyzed based on the values of the Strouhal number (St) calculated at the dominant peak in the spectra of C L . Fig. 13 shows the variation of the Strouhal number (St) of both the cylinders with spacing for each value of r. All the curves (of r < 1) plotted in Fig. 13(a) showed qualitatively similar behavior with changing the spacing S. The Strouhal numbers of all the promoters uniformly increase (with an abrupt jump) with S reaching their respective maximum value and then starting to de-

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Fig. 13. The non-dimensional shedding frequency (St): (a) of the promoter of different aspect ratios r in the presence of a square downstream cylinder, and (b) of the square cylinder in the presence of an upstream promoter of different aspect ratios r at Re = 200. Dotted lines refer to the values for their respective isolated case.

crease for further increase in S to reach their respective frequency of the case of absence of the downstream cylinder. It is evident from Fig. 13(a) that the multi-body interference has significant impact on the shedding frequency of all the promoters (especially of r  0.25) only in the critical range of spacing. Although all the five curves plotted in Fig. 13(b) (solid lines) scatter a little with a large sudden jump in the critical spacing range, they are similar to each other. The drastic change in the Strouhal number may be attributed to the transition from the vortex attachment of the promoter with downstream cylinder to the alternating vortex shedding from the promoter in the critical spacing range. Beyond this, the Strouhal number of the downstream cylinder increases very slowly with the increase of spacing and approaches that of the respective isolated rectangular cylinder. It is interesting to note that the shedding frequency of the downstream cylinder is reduced (from the value of its isolated case) due to the consideration of rectangular shape (r < 1) for the promoter. This indicates that introducing a rectangular cylinder against the approaching flow to a square cylinder (of the same height a as that of the rectangular cylinder) in an inline tandem arrangement decreases the vortex shedding velocity depending on the width b at which b  a. However, one may increase the shedding frequency of the downstream cylinder with the same set of promoters selecting other value of Re ( 500 where St of a rectangular cylinder of r < 1 is more than that of a square cylinder; Maiti [25]). Because, it is observed in Fig. 13(b) that the wake oscillation frequency for the flow past two tandem cylinders is overwhelmed by that of the respective promoter. It may be noted that the shedding frequency of a square cylinder was increased by Malekzadeh and Sohankar [29] applying a control plate of height less than that of the square cylinder.

3.2.2. On aerodynamic forces The magnitude of the fluctuating lift and drag coefficients of both the cylinders (in terms of their root mean square values: C Lrms and C Drms ) against the spacing (S) is plotted in Figs. 14(a) and (b) (for lift) and Figs. 14(c) and (d) (for drag) for different r. It is seen from Fig. 14(a) that the fluctuations on all the promoters remain extremely low in the sub-critical spacing range. There is a sudden rise in C Lrms of the promoter as the critical spacing is passed, which reveals the onset of vortex shedding from the promoter. Depending on the aspect ratio r, C Lrms of the promoter approaches to their respective isolated C Lrms with the increase of spacing. The discontinuity noted in Fig. 14(a) is also evident in Fig. 14(b) for all the aspect ratios in their respective critical spacing range. Inspection of Figs. 14(a) and (b) discloses the fact that C Lrms of the downstream cylinder is always higher than that of the promoter for all the cases considered here. This is because the flow interaction of the promoter with the downstream cylinder causes to amplify the fluctuations of forces on the downstream cylinder. It is observed from Figs. 14(a) and 14(b) that the magnitude of fluctuation of lift coefficient of the downstream cylinder at large S > 4 depends on the strength of rolling vortices originating from the promoter of a particular size. However, the change of C Lrms of the downstream cylinder with the increasing of spacing S is in the direction of its isolated value. It is worthy to note from Fig. 14 that the commencement of eddy shedding by the promoter (remarkably of aspect ratio r  0.5) induces increased fluctuations in the lift coefficients of both the cylinders (e.g., 32–67% for C Lrms of promoter and 61–565% for C Lrms of downstream cylinder depending on r and S). As seen in Figs. 14(c) and (d), the fluctuating drag coefficients of both the cylinders follow almost similar trends of their respective lift coefficients fluctuation (presented in Figs. 14(a) and (b)) as the spacing distance increases. A similar trend in C Drms was reported

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Fig. 14. The RMS: (a) and (c) for the promoter of different aspect ratios r in the presence of a square downstream cylinder, and (b) and (d) for the square cylinder in the presence of an upstream promoter of different aspect ratios r at Re = 200. Dotted lines in (a) and (c) refer the values for their respective isolated cases.

by Tatsutani et al. [47] in a channel flow when the spacing was increased. As seen in Fig. 14, the drag coefficient of all the promoters fluctuates much less than their respective lift coefficient. The presence of the promoter is more pronounced on C Drms , in comparison to that on C Lrms , of the downstream cylinder since the increment in C Drms of the downstream cylinder is found around 437.18% to 8532% (depending on r and S  3) of that of the isolated cylinder. The drag force on both the cylinders is discussed presenting the time-averaged drag coefficients in Fig. 15. The instantaneous values are averaged over approximately 10 shedding cycles in the saturated state where the starting processes are negligible to obtain the time average value. As plotted in Fig. 15(a), with the increase of spacing and depending on the aspect ratio r, the drag coefficient of all the promoters decreases initially followed by a dramatic increase and then finally approaching their respective isolated value. This shows that interference from the downstream cylinder reduces to be negligible. These results are analogous to those reported for tandem square cylinders in channel by Tatsutani et al. [47]. The drag coefficient of promoter is found inversely proportional to the width of that promoter (also reported in Maiti [25,26] for single rectangular cylinder). The shape of all the curves in Fig. 15(b) is essentially the same. It is plausible that the drag force on the downstream cylinder is highly affected when this cylinder lies directly in the unsteady wake of an upstream cylinder. The negative drag force on the downstream cylinder (i.e., the downstream cylinder experiences a force of attraction toward the upstream cylinder) is found to exist for the cases of promoter of r  0.5. The thrust force on the downstream cylinder was also observed previously by Lankadasu and Vengadesan [20] (numerically under shear flow) and Liu and Chen [24] (experimentally under uniform flow). A specific range of S (depending on r < 1) at which the C D of downstream cylinder registered a value very close to zero can be noted from Fig. 15(b). The C D of downstream cylinder becomes almost independent of r for S  4 and rises gradually with the increase in spacing, similarly to that of the second cylin-

Fig. 15. Time-averaged drag coefficient (C D ): (a) of the promoter of different aspect ratios r in presence of a square downstream cylinder, and (b) of the square cylinder in presence of an upstream promoter of different aspect ratios r at Re = 200. Dotted lines in (a) refer the values for their respective isolated case.

der of Bao et al. [2]. However, the C D is nearly 50.69% of that of a single square cylinder even at the largest S = 15, due to the shel-

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Fig. 16. Time-averaged surface pressure distribution (C P ) on front (solid lines) and rear (dashed lines) faces of: (a) upstream cylinder (UP) and (b) downstream cylinders (DN) for different r = 1.0, 0.5 and 0.1 at a fixed S = 0.5, while (c) downstream cylinder for a fixed r = 0.1 at three different spacings S = 0.5, 2.0 and 5.0 and comparison with isolated square cylinder at Re = 200. Table 4 Effect of aspect ratio of the promoter on time-averaged drag coefficient due to pressure (C Dp ) and shear (C Dsh ) for upstream (UP) and downstream (DN) cylinders for S = 0.5 and 5.0 at Re = 200. Source

S

C Dp , C Dsh

r = 1.0

r = 0.75

UP

0 .5

C Dp C Dsh

2.9775 −0.0762

3.0705 −0.0198

3.2316 0.0115

3.6705 0.0131

4.0998 0.0048

5.0

C Dp C Dsh

3.0710 −0.0569

3.1964 −0.0165

3.5824 0.0144

4.1444 0.0160

4.5071 0.0104

0.5

C Dp C Dsh

0.2738 0.2897

0.2405 −0.2569

0.0049 −0.2561

−0.3666 −0.2435

−0.7257 −0.2367

5.0

C Dp C Dsh

1.6337 −0.0013

1.7051 −0.0790

1.5961 −0.0658

1.4732 0.0519

1.5039 0.0097

DN

tering effect from the promoter of the same height, regardless of what the width is. Sharman et al. [41] (for circular cylinders) and Bao et al. [2] (for square cylinder) reported the C D of downstream cylinder around 50% (at S = 10) and 56% (at S = 15) of that of the single cylinder, respectively. 3.2.3. On time-averaged surface pressure distribution The pressure contribution C Dp (= C Dp,front − C Dp,rear ) to the overall drag coefficient of both the cylinders is exemplified in details presenting the time-averaged surface pressure distribution C P along the front and rear faces of the cylinders in Fig. 16. As is seen in Figs. 16(a)–(b) for S = 0.5, the negative gauge pressure is found to act on the rear face of the promoter and on the downstream cylinder from both the faces. Fig. 16(a) for S = 0.5 shows negligible change in C P (> 0) of front face but considerable change in C P of rear face (more negative) with the decrease of aspect ratio r. This results in an increase in C Dp for the promoter, which can be verified from Table 4: UP at S = 0.5. On the contrary, as can be seen in Fig. 16(b) for the downstream cylinder, the pressure distribution (C P ) along the front face is more sensitive (compared to that along the rear face) to the aspect ratio r. Notably, the strongest suction in the space between the cylinders, due to the largest magnitude

r = 0.5

r = 0.25

r = 0.1

of negative pressure coefficients on the rear face of the promoter and front face of the downstream cylinder, is found in presence of the thinnest promoter r = 0.1. This results in a largest thrust force on the downstream cylinder reported in Fig. 15(b) at S = 0.5. Furthermore, the maximum negative contribution of C Dp (along with the negative contribution of C Dsh noted in Table 4: DN at S = 0.5) to the total C D of the downstream cylinder is found for the case of r = 0.1 in Fig. 16(b), which may be validated from Table 4: C Dp for DN at S = 0.5. One possible explanation for the above suction may be that the fluid exchange between the gap spacing and the ambient fluid reduces by thinning the promoter. The distance between the corners of generating the separated shear layers (for the promoter) and the front face of the downstream cylinder decreases with r at a fixed S (Fig. 5). Moreover the gap flow between the wall and promoter’s lower face plays a role as a jet flow along the wall and becomes stronger upon decreasing r (Maiti [26]) at lower S. It is observed from 16(c) that the stagnation point on the front face of the downstream cylinder moves upward with the increase of spacing for a particular size of the promoter (r = 0.1). The stagnation pressure for the downstream cylinder remains negative even at S = 5, similarly to that of Bhattacharyya and Dhinakaran [5].

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Table 5 The absolute percentage deviation of the results of a single square cylinder from those of upstream (UP) and downstream (DN) cylinders by Bao et al. [2], Lankadasu and Vengadesan [20], Bhattacharyya and Dhinakaran [5] and the present study in an inline tandem square cylinders of the same size in a uniform unbounded flow at Re = 100. S

Source

St

CD

C Drms

C Lrms

UP, DN

UP

DN

UP

DN

UP

DN

5

Bao [2] Lankadasu [20] Bhattacharyya [5] Present

11.03 – 8.61 15.10

4.26 6.22 12.86 10.87

26.80 38.15 24.32 30.08

1690 973.00 – 1615

16 730 7665.48 – 28 392

51.89 39.18 – 77.42

535.78 466.98 – 401.35

7

Bao [2] Lankadasu [20] Present

10.75 – 10.98

6.00 4.55 10.22

53.91 41.16 51.43

929 95.13 886.25

3077 652.21 2923

13.15 3.39 26.93

491.5 599.27 425.50

8

Bao [2] Present

10.62 8.31

5.13 9.58

52.20 54.60

622 1842

3537 3378

7.78 15.94

462.10 414.82

622 – 587

4760 4590

5.10 – 11.37

417.36 – 374.91

5220 5034

2.26 10.34

399.47 373.74

10

11

Bao [2] Bhattacharyya [5] Present

5.241 3.28 6.77

2.53 0.55 7.1

46.73 47.47 48.95

Bao [2] Present

5.24 6.02

2.26 6.75

45.83 48.08

468 439.7

3.3. Validation of the present results at large spacing distance Table 5 provides the absolute percentage deviation of the results of a single square cylinder from those of first (upstream) and second (downstream) cylinders by Bao et al. [2], Lankadasu and Vengadesan [20], Bhattacharyya and Dhinakaran [5] and the present study in an inline tandem two square cylinders of the same size in a uniform unbounded flow at Re = 100. Lankadasu and Vengadesan [20] reported that the variation of the aerodynamic characteristics (St, C D , C Lrms and C Drms ) of both the cylinders with the spacing S ranged in [12,36] is not constant for different shear parameters. Bao et al. [2] observed that first and second cylinders (in six-cylinder arrangement) behave similarly with two-cylinder tandem configuration in terms of aerodynamic force statistics for S  15. As seen in Table 5, the Strouhal numbers of Bao et al. [2] are found constant when the spacing S ranged in [12,13] and [33,15], which are around 10.5% and 5.24%, respectively, lower than that the single cylinder. The experimental data of Huhe et al. [18] for two cylinders in tandem arrangement also showed that the Strouhal number stays constant for 5  S  10. As observed in Figs. 13–15, the overall variation of the aerodynamic characteristics of both the cylinders (promoter and downstream) with spacing S exhibits the same qualitative trend as that of both the cylinders (first and second, respectively) of Bao et al. [2] for both the cases: two and six cylinders. The present results of upstream (of square shape) and downstream cylinders for the flow configuration considered here agree well with those of Bhattacharyya and Dhinakaran [5] for spacing S  5 (Fig. 3). The absolute percentage deviations of St, C D and C L of an isolated square cylinder from those of upstream (and downstream) cylinder of Bhattacharyya and Dhinakaran [5] at S = 5 are 1.5%, 8.46%, and 5.94%, respectively (and 0.84%, 55.32% and 2.26%, respectively) may be noted here.

2.

3.

4.

5. 4. Conclusions This numerical study provides qualitative insight into the basic structures of the flow over a square cylinder (of height a) in presence of an upstream rectangular promoter (of the same height a with different width b  a) as well as the quantitative information for the dependency of the aerodynamic characteristics of both the cylinders on spacing distance S and aspect ratio r = b/a. The major observations of the present study can be highlighted as follows: 6. 1. The transition (from unsteady/steady to steady/unsteady) of flow over the promoter/downstream cylinder strongly depends

–

on the shape (square/rectangular) and position of the promoter at a fixed Re. Hence a bounded region in the Sr-plane, for values of S cr for each value of r < 1, in order to generate the unsteadiness in the steady flow of the downstream cylinder employing a promoter against the approaching flow at a Re (= 100), is proposed. Numerical results showed four major flow patterns on the downstream cylinder, which appeared successively with the increase of gap spacing, namely, steady of single cylinder of r > 1, steady with suppressed gap flow, generated unsteadiness and steady of isolated square cylinder at Re = 100. The variation of intensities of the peaks in the spectra of fluctuating lift coefficient of the promoter with spacing showed that the vortex shedding from the promoter could be observed near S = 2.5 ∀r < 1 at Re = 200. The instantaneous lift signals of the downstream cylinder are reported sinusoidal with a single peak in the spectra at lower Re = 100 for the values of r and S from the above proposed zone. On the contrary, at higher Re = 200 when both the cylinders shed vortices for S  3, the multiple peaks are observed in the above spectra due to the difference in the basic shedding frequency of a square (downstream) from that of a rectangular (promoter) cylinder. The possibility of increasing/decreasing the vortex shedding frequency of the downstream cylinder (overwhelmed by that of the respective promoter) is explored here selecting the value of Re from the previous study (Maiti [25]). The elevated amplitude of eddy-shedding oscillation of the downstream cylinder in the considered tandem arrangement is dependent on the strength of rolling vortices originating from the promoter at Re = 200. It is worth to note that the strength of vortices of the downstream cylinder is enhanced remarkably by the promoter of r  0.5. The multi-body interference has impact on the drag coefficient of the promoter up to a certain spacing. However, the C D of the downstream cylinder at Re = 200 is reported nearly 50.69% of that of a single square cylinder even at S = 15 due to greater sheltering of the downstream cylinder by the promoter of the same height a, regardless of what the width b is. Notably, a strong suction in the gap spacing between the cylinders, resulting in a thrust force on the downstream cylinder, is observed for the case of promoter of r  0.5 at closely spaced arrangement. The deviation of the aerodynamic characteristics (of both the cylinders from their respective isolated cylinder values) for higher spacings are certified with those of some previous pub-

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lished works for the cases of uniform unbounded flow (at Re = 100) and shear flow near a wall (at Re = 200). Sohankar [43] reported the hysteresis for all Re larger than 100 for two cylinders in uniform flow. The hysteresis is likely to be observed in the present study at Re = 200 where both the cylinders shed the vortices after certain spacing. Also the abrupt change in the aerodynamic characteristics of both the cylinders, associated with the transition from the vortex attachment of the promoter with downstream cylinder to the alternating vortex shedding from the promoter in the critical spacing range, is reported here. However, the hysteresis is not addressed in this study. It is obvious that a more detailed investigation (taking the very small spacing increments and identifying different flow patterns) will have to be carried out. This is one of the possible future works, which will be taken care of in our next study on staggered tandem arrangement of a square cylinder along with a rectangular cylinder near a wall. Acknowledgements The authors are highly indebted to DST (India) (San. No. 100/ IFD/3613/2008-09 dated: 10.09.2008) for financial assistance and also UGC (India) under SAP & BSR for helping in completion of this study. We also express our deep appreciation to Dr. N. Singh (Dept. of Physics) for his computational support through a UGC Grant and Dr. G.S. Chauhan (Dept. of Humanities and Social Sc.) for proofreading the research paper. References [1] M. Alam, M. Moriya, K. Takai, H. Sakamoto, Fluctuating fluid forces acting on two circular cylinders in a tandem arrangement at a subcritical Reynolds number, J. Wind. Eng. Ind. Aerodyn. 91 (2003) 139–154. [2] Y. Bao, W. Qier, D. Zhou, Numerical investigation of flow around an inline square cylinder array with different spacing ratios, Comput. Fluids 55 (2012) 118–131. [3] A.J. Baxendale, I. Grant, Vortex shedding from two cylinders of different diameters, J. Wind. Eng. Ind. Aerodyn. 23 (1986) 427–435. [4] P.W. Bearmen, M.M. Zdravkovich, Flow around a circular cylinder near a plane boundary, J. Fluid Mech. 89 (1978) 33–47. [5] S. Bhattacharyya, S. Dhinakaran, Vortex shedding in shear flow past tandem square cylinders in the vicinity of a plane wall, J. Fluids Struct. 24 (2008) 400–417. [6] S. Bhattacharyya, D.K. Maiti, Shear flow past a square cylinder near a wall, Int. J. Eng. Sci. 42 (2004) 2119–2134. [7] S. Bhattacharyya, D.K. Maiti, Vortex shedding from a square cylinder in presence of a moving ground, Int. J. Numer. Methods Fluids 48 (2005) 985–1000. [8] S. Bhattacharyya, D.K. Maiti, Vortex shedding suppression for shear flow past a square cylinder near a plane wall, Acta Mech. 184 (2006) 15–31. [9] S. Bhattacharyya, D.K. Maiti, S. Dhinakaran, Influence of buoyancy on vortex shedding and heat transfer from a square cylinder in wall proximity, Numer. Heat Transf., Part A 50 (2006) 585–606. [10] G. Bosch, W. Rodi, Simulation of vortex shedding past a square cylinder near a wall, Int. J. Heat Fluid Flow. 17 (1996) 267–275. [11] M. Breuer, J. Bernsdorf, Y. Zeiser, F. Durst, Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume, Int. J. Heat Fluid Flow 21 (2000) 186–197. [12] R.W. Davis, E.F. Moore, A numerical study of vortex shedding from rectangles, J. Fluid Mech. 116 (1982) 475–506. [13] R.W. Davis, E.F. Moore, L.P. Purtell, A numerical-experimental study of confined flow around rectangular cylinders, Phys. Fluids 27 (1984) 46–59. [14] R. Devarkonda, J.A.C. Humphery, Interactive computational-experimental methodologies in cooling of electronic components, Report CML92-008, Computer Mechanics Laboratory, University of California at Berkeley, 1992. [15] M. Elmer Gennaro, Alysson K. Colaciti, Marcello A.F. Medeiros, On the controversy regarding the effect of flow shear on the Strouhal number of cylinder vortex shedding, Aerosp. Sci. Technol. 29 (2013) 313–320. [16] A. Etminan, M. Moosavi, N. Ghaedsharafi, Determination of flow configurations and fluid forces acting on two tandem square cylinders in cross-flow and its wake patterns, Int. J. Mech. 5 (2011) 63–74. [17] R. Franke, W. Rodi, B. Schonung, Numerical calculation of laminar vortex shedding flow past cylinders, J. Wind Eng. Ind. Aerodyn. 35 (1990) 237–257. [18] A. Huhe, M. Tatsuno, S. Taneda, Visual studies on wake structure behind two cylinders in tandem arrangement, Rep. Res. Inst. Appl. Mech. (1985) XXXII(99).

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