Forced convection for flow across two tandem cylinders with rounded corners in a channel

International Journal of Heat and Mass Transfer 130 (2019) 1053–1069

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Forced convection for flow across two tandem cylinders with rounded corners in a channel Wei Zhang a,b, Xiaoping Chen a,b, Hui Yang a,b, Hong Liang c, Yikun Wei a,b,⇑ a

State-Province Joint Engineering Lab of Fluid Transmission System Technology, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China c Department of Physics, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China b

a r t i c l e

i n f o

Article history: Received 24 July 2018 Received in revised form 25 October 2018 Accepted 27 October 2018

Keywords: Forced convection Tandem cylinders Rounded corner Unsteady Gap ratio

a b s t r a c t This work presents the first numerical investigation on the forced convection of flow across two tandem cylinders with rounded corners in a channel at Re = 100. Both cylinders have the geometry of a square rounded at all corners with a radius of curvature R, which is non-dimensionalized as R+ = R/D where D is the cylinder diameter, thus the cylinder geometry can be square (R+ = 0.0), partially rounded (R+ = 0.1–0.4) or circular (R+ = 0.5). The two cylinders are separated at a distance in the streamwise direction as characterized by the parameter of gap ratio (GR) chosen at GR = 1(1)8. The objective of this work is to explore the effects of two significant parameters, i.e., gap ratio and corner radius, on the flow unsteadiness and heat transfer characteristics of the tandem arrangement that has not been studied before. The effects of the two parameters are exhibited and analyzed by the instantaneous temperature and vorticity fields, variation of representative aerodynamic and heat transfer quantities, spatial distributions of local heat transfer rate, flow behaviors in the gap and the near-wake regions, and temperature distribution and variation on the channel wall. The results are presented by time-averaged and fluctuating quantities to reflect both mean and pulsating behaviors. We observed that the cylinder geometry determines the unsteadiness of the near-wake flow after the downstream cylinder; the flow is always unsteady for square-like cylinders where the corner radius is small, while the flow can be stabilized by the circularlike cylinders with larger corner radii that the flow fluctuation is greatly weakened or even fully suppressed at small GRs. Numerical results also reveal that the gap flow is steady at small GRs and unsteady at large GRs, as categorized as steady gap flow regime and unsteady gap flow regime. There are drastic variations for the representative characteristic quantities at the critical GR where the gap flow transits from steady to unsteady. The different flow regimes categorized by GR and R+ also substantially determine the flow patterns in the gap and near-wake regions, the mean and fluctuating of heat transfer rate on the cylinder surface and the temperature variation on the channel wall. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The low-Re flow across isolated cylinders is a classical physical problem and is widely encountered in engineering applications in aero, civil and ocean circumstances. For incoming steady flow at Reynolds number above a critical value, the attached flow separates at the cylinder surface due to the adverse pressure gradient, and then the separated shear layer flow and wake flow are disturbed by the environmental perturbation, thus the flow transits from steady to unsteady. The unsteady flow is characterized by ⇑ Corresponding author at: State-Province Joint Engineering Lab of Fluid Transmission System Technology, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China. E-mail address: [email protected] (Y. Wei). https://doi.org/10.1016/j.ijheatmasstransfer.2018.10.125 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

the time-periodic fluctuation of global aerodynamic quantities, and the cylinders are also subjected to fluctuating forces which may lead to certain undesirable structural failures. The destabilization of the steady flow is inevitable as it is caused by the inherent instability of the flow. A number of researches have been conducted on low-Re flow across cylinders of various geometries for the thorough understandings of the physical mechanism and the development of potential flow control techniques. The flow across a single isolated cylinder is the fundamental for further researches on flow across multiple cylinders. Davis et al. [1] and Mukhopadhyay et al. [2] studied the two-dimensional flow across a square cylinder in a channel. They found that the channel confines the flow in the passages to the lateral sides of the cylinder; the gap flow is accelerated due to the reduction of cross-section area and significantly determines the evolution of wake vortices, resulting

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Nomenclature Symbols B CD CL D f GR H Nu P Pr R Re St t

blockage ratio drag coefficient lift coefficient cylinder diameter fluctuating frequency gap ratio channel height Nusselt number period Prandtl number corner radius Reynolds number Strouhal number time

in the increased drag coefficient and Strouhal number. For flow across a rectangular cylinder, Okajima [3] performed experimental study at Re = 70–20,000 and aspect ratio (AR, length-to-width ratio of the cylinder) AR = 1–4. The author observed the abrupt variation of Strouhal number at AR = 2–3 in the St-Re curve which is attributed to the flow pattern transition. The separated shear layer flow at the leading corners of the cylinder reattaches on either of the lateral surfaces for Reynolds number smaller than a critical value, while the flow fully detaches at higher Reynolds numbers. The associate numerical investigation is carried by Okajima [4] at AR = 0.6–8 and Re
105 where the drastic variation of Strouhal number is reproduced, and the conclusions are supported by Okajima et al. [5]. We reported in our earlier numerical work for flow across a triangular cylinder positioned at various inclinations in an infinite medium at Re = 100 [6], with emphasis on the flow unsteadiness and stability-sensitivity characteristics. The results revealed that the fluctuation of streamwise velocity is the most significant at the inclination angle of 40°, while the fluctuation of transverse velocity is the maximum as one of the corners or edges faces the incoming flow. The linear stability-sensitivity analysis demonstrates that the flow is the most unstable as one corner of the cylinder faces the incoming flow. Considering the temperature difference between the incoming flow and the cylinder in engineering applications, the heat transfer characteristics of forced convection for flow across a single cylinder are also studied. Abbassi et al. studied the forced convection for flow across a triangular cylinder in a channel at Reynolds number up to 250 [7] and 750 [8]; the bottom wall of the channel is kept at a higher temperature, while the temperature of the top wall, the cylinder and the incoming flow is relatively low. The flow transits to unsteady at Reynolds number in the vicinity of 45 which is slightly lower than the critical Reynolds number 47 for the circular cylinder and is equivalent to that of the square cylinder, and the StRe correction was proposed. The unsteady flow remarkably enhances the heat transfer with an augmentation of about 85% at Re = 250. For a square cylinder, Sharma & Eswaran [9] performed numerical simulation for both unconfined (B = 5%) and channelconfined (B = 10–50%) flows at Re = 50–150 in which the hot cylinder is prescribed with uniform heat flux or constant temperature thermal boundary conditions. The critical Reynolds number at which flow transits to unsteady first increases and then decreases with the blockage ratio, while the Strouhal number, mean drag coefficient and Nusselt number monotonically increases. There is criss-cross motion of vortices for the channel flow case. In the work of Moussaoui et al. [10], the flow across a 45° inclined square

u, v x, y Greeks h u

x

velocity components Cartesian coordinate

temperature circumferential position vorticity

Subscripts cr critical value in inflow loc local value avg time-averaged rms root-mean-square w wall

cylinder in a channel is studied at Re = 0–300 and B = 25%, where the bottom wall of the channel is maintained at high temperature and all other solid walls and the incoming flow are of low temperature. The flow transits to unsteady at Recr = 82 for this specific inclination angle which is much larger than that of the edgefacing orientation cylinder (Recr  45); the local heat transfer at the windward edges are almost temporarily constant, while it is unsteady on the leeward edges. The same physical problem was studied by Seyyedi et al. [11] at Re = 50–300 and B = 25% where a flat plate is placed downstream of the cylinder to stabilize the wake flow. The flat plate lowers the local heat transfer of the bottom wall by confining the wake vortex shedding, and there is an optimized streamwise position for the plate for the maximum averaged heat transfer rate of the system. Different from flow across a single isolated cylinder, in engineering applications there can be circumstances with multiple cylinders to meet the complicated design and technical requirements. The low-Re flow across tandem cylinders, mostly of square geometry, is the simplest configuration. The flow characteristics are more complex than the single cylinder case due to the interaction between the two cylinders thorough the gap flow. The wake flow of the upstream cylinder interacts with the downstream cylinder thus is greatly affected, which is crucial in determining the heat transfer performance of the twin cylinders system. The flow across tandem cylinders has been extensively studied and forced convection heat transfer is considered in some works. Valencia [12,13] numerically investigated the forced convection of fully developed flow across an array of rectangular cylinders at Re = 100–400 and GR = 1–4 by only considering one periodic unit. The introduction of multiple cylinders enhances the heat transfer between the channel walls and the fluid which is the most effective for GR  2. The philosophy for considering only one periodic unit was also employed by Bahaidarah et al. [14] to study flow across cylinders of circular, flat, oval and diamond shapes at Re = 25– 350. The factor of geometric shape was found to affect the heat transfer performance for Re > 50 more significantly. Compared with the circular geometry, the flat and oval cylinders exhibit greater flow resistance and heat transfer rate, while the flow is less resistant to the diamond cylinder for Re
250 and has the lowest heat transfer rate. Agarwal & Dhiman [15] studied flow across two tandem rectangular cylinders in a channel at Re = 1–40 and GR = 1– 4 to identify the effects of GR and Reynolds and Prandtl numbers. It was found that the flow transits to unsteady at Re = 30–40 and GR = 4; the averaged Nusselt number monotonically increases with GR because of the perturbed flow, and the heat transfer is more

W. Zhang et al. / International Journal of Heat and Mass Transfer 130 (2019) 1053–1069

significant for the upstream cylinder. Chatterjee & Biswas [16] performed systematic study on flow across two tandem square cylinders in the steady regime at Re
30, and observed the twin recirculation vortices downstream of the cylinder. For the unsteady flow, Chatterjee & Mondal [17] found that GRcr, i.e., the critical gap ratio at which the vortex starts to shed in the gap, decreases with the Reynolds number, and there is a jump of drag coefficient for the downstream cylinder close to GRcr; the Nusselt number increases with Re and GR. Farhadi et al. [18] simulated the cold flow across tandem hot square cylinders at Re = 100–300 and blockage ratio (B, ratio between cylinder width and channel height) from 12.5% to 50%. The results showed that the heat transfer performance is almost not affected by GR in the high-B regime. The heat transfer rate of the downstream cylinder increases with GR but decreases with B. At high GRs where vortex shedding emerges, the increases of drag coefficient and Nusselt number for the downstream cylinder are more significant than those of the upstream one. In the work of Rosales et al. [19,20], the flow across two tandem square cylinders of unequal size is investigated at Re = 500 and GR = 2 to identify the effect of channel confinement as the cylinders are placed closer to the channel wall. Both Nusselt number and drag coefficient decrease as the cylinders approach the channel wall because of the low-velocity flow in the boundary layer; the existence of the upstream small cylinder (‘‘eddy promoter”) substantially reduces the drag coefficient of the main cylinder compared with the single cylinder case. Sohankar & Etminan [21] studied flow across tandem square cylinder in an infinite medium at Re = 1–200 and GR = 5, and found the onset of vortex shedding in the range of Re = 35–40 which is smaller than that of the single cylinder. The fluctuating aerodynamic forces of the downstream cylinder are larger than those of the upstream one; there is flow pattern transition at Re = 55–60, resulting in the drastic variation of aerodynamic forces acting on the cylinders. Asif & Dhiman [22] numerically studied the forced convection of laminar flow across a triangular periodic array of heated cylinders at Re = 10– 100, and the volume fraction of the cylinder configuration ranges from 0.7 to 0.99. They found that as the Reynolds number increases, the size of the wake region increases continuously and the flow separates earlier on the cylinder surface. As the free volume fraction increases, the viscous friction and pressure drags in the array decrease. In our earlier work [23], we studied the forced convection of flow across two tandem rectangular cylinders in a channel at Re = 100 and B = 25% to investigate the effect of AR and GR. Two flow regimes are identified based on the gap flow pattern, i.e., the steady gap flow at GR
3 and unsteady gap flow at GR  4, regardless of the aspect ratio. The characteristic quantities experience a drastic variation as the gap flow transits to unsteady at GRcr, and the time-averaged and fluctuating flows in the gap and near-wake of downstream cylinder are also categorized into two families based on GR. Except for the introduction of aiding buoyancy, recent numerical and experimental works conclude that the flow across a cylinder can be stabilized by slightly modifying the cylinder geometry, mostly by altering the shape of cylinder corners such as rounding, cutting or chamfering. For rounded corner cylinders, the cylinder surface is smooth over the whole circumference without curvature discontinuity at the corners as that of the square geometry, and hence the attached flow is devoid of forced separation resulted from the abrupt geometrical variation. Rounded corner cylinders are frequently encountered in engineering applications because of the reduced stress concentration which helps improving mechanical strength and avoiding structural failure. The rounding radius R of the corners is normally non-dimensionalized by the cylinder diameter D as R+ = R/D, i.e. square cylinder with R+ = 0.0 and circular cylinder with R+ = 0.5, and partially rounded cylinder

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for 0.0 < R+ < 0.5. The effect of rounding corners on stabilizing the flow has been investigated mainly for high-Re flow across a single isolated cylinder. Tamura et al. [24] simulated the flow across a cylinder with rounded corners at high Reynolds number ReO (104–106). Compared with the square cylinder, the base pressure coefficient at the cylinder rear is smaller for the rounded geometry, and the Strouhal number is larger. Tamura & Miyagi [25] experimentally studied the turbulent flow across the cylinder at Re = 30,000, and found that the rounded corners promote the reattachment of separated shear layer on the lateral surface of the cylinder, resulting in a lower drag and an increased lift at all inclinations within the range 5° to 30°. Carassale et al. [26] experimentally studied the aerodynamic behavior of the cylinder inclined at 0–45°and Re = 17,000–230,000 with two nondimensional rounding radii R+ = 1/15 and 2/15. The rounded corners lead to the absence of fixed separation position and promote flow reattachment on the lateral surfaces of the cylinder. Hu et al. [27] experimentally investigated the effect of corner radius on the near-wake flow of an isolated cylinder at Re = 2600 and 6000. For the rounded corner cylinder, the wake vortices are significantly weakened with the first roll-up occurring relatively further away from the cylinder; the Strouhal number increases as represented by the decreased streamwise distances of neighboring vortices. The near-wake flow is comparably more sensitive to the leading corner radius; the width of the wake is reduced by 25% as R+ increases from 0.0 to 0.5. In the experimental work by Hu & Zhou [28], the cylinder is rounded at two corners and is positioned at various inclinations towards the incoming flow at Re = 2600– 8500. As R+ increases, the wake recirculation bubble expands in size with rising base pressure and results in the decreased drag and fluctuating lift and drag forces. The inclined cylinder causes the wake centerline to move toward the sharp corner and the shear layer flow is symmetric about the shifted centerline. We performed direct numerical simulation on flow across an isolated rounded corner cylinder at Re = 1000 to investigate the effect of corner radius on the development of separated and transitional flow [29]. The rounded corners significantly reduce the mean drag and force fluctuations. The statistical quantities do not monotonically vary with the corner radius, but exhibit drastic variations between the configurations of square cylinder and partially rounded cylinders, and between the latter and the circular cylinder. Recently, we performed two-dimensional global linear stability analysis and sensitivity analysis on low-Re flow across a rounded corner cylinder at Re
110 [30] to explore the physical mechanism for the stabilization of wake flow by rounding the corners. We found that the flow is stabilized as reflected by the smaller temporal growth rate of perturbation, and the stabilization is the most obvious when the corners are only partially rounded. The variation of stability is attributed to the different spatial variation trends of the backflow velocity in the near- and far-wake regions for various cylinder geometries. It was demonstrated in the above literatures that the low-Re flow across one or multiple isolated cylinders has been extensively studied. Most existing studies focus on the effect of Reynolds number, gap ratio and the spatial arrangement of multiple cylinders on the thermal and flow characteristics of the system. Although it has already been confirmed in a number of studies that the modification of cylinder geometry could notably (de)stabilize the flow. However, to the authors’ knowledge, so far there is no attempt to explore the effect of rounded corners on the flow across tandem cylinders especially when heat transfer between the cylinder and ambient fluid is taken into consideration. This physical problem is of engineering significance that the cylinders are normally rounded to reduce stress concentration and to satisfy the design requirement. In the present work, we perform the first exploration

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on the forced convection of low-Re flow across two tandem cylinders with rounded corners in a channel. The objective is to investigate the effect of rounded corners and corner radius on the flow unsteadiness, aerodynamic behaviors and heat transfer characteristics for the tandem cylinders configuration. Numerical simulation will be carried out for various corner radii including square, partially rounded and circular cylinders, and the gap ratio for the tandem cylinders is chosen to cover both steady and unsteady flow regimes identified in our earlier work. The effect of corner radius as well as the gap ratio on the thermal and flow characteristics will be presented by analyzing the thermal and flow fields, the aerodynamic and thermal quantities in both the time-averaged and temporal fluctuating perspectives. We also emphasize on the characteristics of flow in the gap and near-wake regions and discuss its relationship with the variations of thermal quantities.

a square cylinder in a channel [23], in which the results of Nusselt number, Strouhal number, and mean and fluctuating forces are in good agreement with those in Ref. [9]. For the distance from the leeward side of the downstream cylinder to the outlet boundary, Sohankar et al. [31] studied the flow across an inclined rectangular cylinder in a channel at Re
200 and found a distance of Ld = 26D is large enough for this type of geometrical configuration, as reflected by the negligible variation of aerodynamic and thermal quantities as the distance further increases. The value employed in our work Ld = 50D is believed to be sufficient to produce domain independent solutions. The comparisons of the data are further presented in the following discussions. 2.2. Numerical methods The flow is incompressible at the present Reynolds number and is governed by the continuity, momentum and energy equations of non-dimensional form:

2. Numerical setup 2.1. Physical model The geometrical configuration of the physical problem is schematically shown in Fig. 1. Two identical cylinders are placed in tandem at the centerline of a horizontal channel. The corners of both cylinders are rounded at a radius of curvature R, which is scaled as R+ = R/D with D the cylinder diameter; the corner radii under investigation ranges from R+ = 0.0 for square cylinder to R = 0.5 for circular cylinder with an interval of DR+ = 0.1. The width of the channel is fixed at H = 4D, i.e. blockage ratio B = 25%, to permit the development of unsteady wake in the transverse direction. The parameter of gap ratio is defined as the ratio between the faceto-face distance of the two cylinders and the cylinder diameter, which ranges from GR = 1 to GR = 8 with DGR = 1 to cover both the possibly steady and unsteady flow regimes. The inflow and outflow boundaries are placed at Lu = 10D and Ld = 50D away from the nearest cylinder surface to minimize the influence of artificially imposed boundary conditions on the near-wake flow. Both cylinders are assumed isothermal with surface temperature hw, while the channel walls are kept adiabatic. A parabolic inflow velocity profile for the working fluid (air, Pr = 0.7) is prescribed at the inflow boundary with averaged streamwise velocity uin and temperature hin, while fully developed flow is assumed at the outflow boundary. The Reynolds number of the inflow is fixed at 100 as non-dimensionalized by D and uin under which the flow is twodimensional. The sizes of the computational domain upstream and downstream of the tandem cylinders are determined based on the conclusions in our earlier studies and those of others. The distance Lu = 10D from the inlet boundary to the windward side of the upstream cylinder is determined in our earlier work for flow past

@uk ¼ 0; @xk

ð1Þ

@ui @uk ui @p 1 @ 2 ui ¼ þ ; þ @xi Re @xk @xk @t @xk

ð2Þ

@h @uk h 1 @2h þ ¼ : @t @xk RePr @xk @xk

ð3Þ

where the primitive variables are non-dimensionalized by the reference length D, velocity uin, pressure qu2in and time D/uin. The temperature is scaled as h = (h⁄  hin)/(hw  hin) where h⁄ is the dimensional temperature. The boundary conditions of the nondimensional variables are mathematically written as:

u ¼ 1:5½1  ð2y=HÞ2 ;

v ¼ h ¼ 0;

@u=@x ¼ @ v =@x ¼ @h=@x ¼ 0;

at the inlet;

at the outlet;

u ¼ v ¼ @h=@y ¼ 0;

at the channel walls;

u ¼ v ¼ 0;

at the cylinder surface:

h ¼ 1;

The governing Eqs. (1)–(3) are spatially discretized in the fluid domain on a multi-block structured grid as exemplified in Fig. 2. The grid is of H-type for the square cylinder geometry, and hybrid H-O type for the rounded cylinder cases where the O-grid is employed around the cylinder to adapt to the curved surface. The spatial resolution is determined based on the geometric configuration (single or tandem, R+ and GR) and ranges from 768  256 to 1536  256 in the present study, with at least 128

Fig. 1. Configuration of the physical problem.

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Fig. 2. Schematic of the multi-block structured grid at R+ = 0.2 and GR = 3. The grid is shown at every fourth gridline in each direction for clarity. The thick lines denote the boundaries of adjacent blocks in parallel computation.

grids within each unit length on the cylinder surface. The wallnormal size of the first layer grid at the cylinder surface and channel walls is about 0.005D to well resolve the boundary layer flow and accurately predict the local heat transfer rate. The governing equations are solved using our in-house second-order finite difference code. The discretized equations are solved by a semi-implicit fractional step method [32]. The time marching in one physical step consists of four sub-steps including the predictor step for velocity, the solution of pressure, the corrector step for velocity and the updating of temperature. The differential equations for the four sub-steps are formulated as:


n1
n2 n1=2  un1 3 @uk ui 1 @uk ui 1 @ 2 ui i  ¼ ; þ 2 @xk 2 @xk Re @xk @xk Dt

ð4Þ

n1=2 @ 2 pn 1 @uk ¼ ; @xk @xk Dt @xk

ð5Þ

@pn ; @xi

ð6Þ

uni ¼ uin1=2  Dt


n1
n2 hn  hn1 3 @uk h 1 @uk h 1 @ 2 hn  ¼ : þ 2 @xk 2 @xk RePr @xk @xk Dt

3. Results and discussion 3.1. Instantaneous thermal and flow patterns

n1=2

ui

coefficients and surface-integrated Nusselt number for both cylinders. We found that by doubling the number of grid in each direction, the maximum relative difference of the above quantities is around 1.0%. Since R+ = 0.0(0.1)0.5 and GR = 1(1)8 are considered here along with simulations for single cylinder, the computational cost will be much expensive with refined resolution for the tens of cases. The resolutions demonstrated above (from 768  256 to 1536  256) will be used in all computations in this work.

ð7Þ

The non-dimensional time step size is about 0.003D/uin as constrained by the CFL constraint which is sufficient to capture the unsteady thermal and flow behaviors. The time-integration continues for at least 500D/uin to guarantee that the time-periodicity of the flow is achieved and time-averaged solution is obtained with a reasonable accuracy. The code used in the present simulations has been extensively used and well validated in our earlier studies for flow across a single cylinder in unconfined domain (see Table 1 in Ref. [6]) and flow in enclosed space [33,34]. For geometrically and physically similar configurations, we analyzed the flow across two circular or square cylinders in tandem in an infinite medium [35], and presented detailed results for the mean and fluctuating forces for both cylinders. The results are quantitatively compared the benchmark solutions in literatures and good agreement is achieved (see Table 3 in Ref. [35]). Considering the forced convection heat transfer, the flow across tandem rectangular cylinders has been studied in our earlier work [23] in which we carefully validated the code by computing the mean Nusselt number, Strouhal number, mean drag coefficient and root-mean-square fluctuating lift coefficient. The quantitative results are compared with those of Sharma & Eswaran [9], and the results are consistent with each other which confirm the reliability of our code. In this work, we performed grid independence study for flow across two tandem cylinders for four cases at R+=0.0 and 0.5 and GR = 1 and 8, and computed the timeaveraged and root-mean-square values of fluctuating lift and drag

The effect of R+ and GR on the unsteady thermal and flow patterns is first presented and quantitatively analyzed by the instantaneous vorticity and temperature distributions for both the single and tandem cylinders configurations. Fig. 3 gives the instantaneous fields for the single cylinder case to provide direct benchmark comparisons for the following tandem cylinders configurations. For flow across a square cylinder, the incoming flow is forced to separate at the leading sharp corners and forms recirculating bubbles at the lateral edges of the cylinder. However, the geometrical discontinuity no longer exists as the corners are rounded and the adverse pressure gradient at the windward and lateral edges is greatly alleviated, thus separation occurs naturally which can even be observed at the leeward edge, and the recirculation bubbles are reduced in size [29]. It is seen in the figure that there is no notable difference regarding the spatial structure of the vorticity field, i.e., vorticity contours in the near- and far-wake regions, under the various R+. The position for the occurrence of initial vortex shedding slightly moves toward the cylinder as R+ increases; the vorticity with maximum positive magnitude is observed roughly at x = 3 for the square-like cylinders (R+ ? 0.0), while it moves to x = 2 for the circular-like cylinders (R+ ? 0.5). The instantaneous temperature field exhibits similar spatial structure. The isotherms present the typical downstream convection structure and R+ does not have noticeable influence. The fluid temperature rapidly decreases in the far-wake and approaches to that of the incoming flow (hin = 0). In conclusion, R+ has minor effect on flow across a single cylinder. The instantaneous vorticity and temperature fields for the tandem cylinders configurations are given in Figs. 4–6 at several representative gap ratios. At GR = 2, the gap flow between the cylinders is steady for all R+ as reflected by the symmetric distributions of vorticity about the centerline. This is attributed to the small separating distance of the twin cylinders which constrains the vortex detachment and shedding in the gap [23]. For the square-like cylinder at R+ = 0.0–0.3, the flow in the gap and to the lateral edges of cylinder-A is quasi-steady and imposes almost no perturbation to flow around cylinder-B, thus the unsteadiness of

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Fig. 3. Instantaneous vorticity (left) and temperature (right) fields for the single cylinder case obtained at the moment of maximum lift for cylinder-A: R+ = 0.0(0.1)0.5 from top to bottom. The isolines are plotted at x = 5.0(0.5)5.0 and h = 0.1(0.1)1.0.

near-wake flow of cylinder-B is minor and similar to flow across a streamwise rectangular cylinder [31], as represented by the weakly growing shedding of the elongated recirculating wake vortices. Consequently, the flow in the central region of the channel becomes unsteady as a result of its inherent instability and exhibits the pattern of a single vortex street due to the more stable separated shear layer [29], rather than the double vortex street type as observed for the single cylinder configuration (see Fig. 3). It is worth noticing that for the circular-like cylinder at R+ = 0.4 and 0.5, the wake flow of cylinder-B becomes completely steady as exhibited by the perfect twin symmetric vortices about the wake centerline (y = 0) without any shedding. The transition of flow from unsteady to steady with increasing R+ is also illustrated by the isotherms that for square-like cylinders, the thermal field is perturbed in the central channel by the transverse shedding vortices. The transverse non-uniformity of the isotherms is reduced as R+ increases, and finally results in the symmetric isolines for the circular-like cylinders at R+ = 0.4 and 0.5 corresponding to the steady wake flow. This observation is different from the flow across a streamwise elongated cylinder in which the wake flow always exhibits time-periodic shedding although the incoming flow is steady. The rounding of the corners could suppress the flow unsteadiness at Reynolds number far larger than the critical value not only in the gap where the flow is confined but also in the wake where the flow is permitted to fully develop. The gap flow for the GR = 4 arrangement is unsteady for all R+. For the square geometry, the vortex generated at the rear sharp corners of cylinder-A is permitted to relatively fully develop to detach and shed away from the cylinder surface, and forms the double vortex street downstream of cylinder-B because of the extensive incoming perturbed flow from the gap. As R+ increases, the streamwise position of the initial vortex shedding in the gap

is getting close to cylinder-A, the tendency of which is the same as that of the single cylinder although the confinement from cylinder-B also affects. The shedding vortices in the gap strongly interacts with flow around cylinder-B and results in the substantially perturbed wake flow compared with those of the single and GR = 2 configurations, as reflected by the strong transverse oscillation of wake vortices. The effect of gap perturbation in intensifying the unsteadiness of flow around cylinder-B is also observed from the instantaneous thermal field. The isotherms to the lateral edges of cylinder-B is notably affected that the local isotherms at the leeward edge greatly deviate from symmetric distribution pattern. The channel walls are heated by the transverse shedding vortices and the temperature is higher than the incoming flow which will be discussed in the following sections. The gap flow fully develops at the largest gap ratio GR = 8 that the initial vortex shedding is no longer constrained by cylinder-B. However, the velocity of the gap flow is still low as confined by both cylinders and channel walls, thus there is large velocity gradient at the separated shear layers in the gap as represented by the streamwise stretched vortices. The effect of gap perturbation on the wake is more obvious in two aspects. First, the flow pattern on the windward edge of cylinder-B is complex due to the impingement of incoming unsteady flow where the local pressure gradient field is affected. Second, compared with the above arrangements, the damping of vorticity magnitude in the streamwise direction is relatively slow especially in the far-wake region. As R+ increases, the position for the initial shedding vortex gradually approaches to cylinder-A. The transverse oscillation of the wake vortices is more significant in its affecting scope, as reflected by the vorticity field and the isotherms that the fluid temperature at the channel walls is heated in the region more close to cylinder-B.

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Fig. 4. Instantaneous vorticity (left) and temperature (right) fields for GR = 2 obtained at the moment of maximum lift for cylinder-A: R+ = 0.0(0.1)0.5 from top to bottom. The isolines are plotted at x = 5.0(0.5)5.0 and h = 0.1(0.1)1.0.

Fig. 5. Instantaneous vorticity (left) and temperature (right) fields for GR = 4 obtained at the moment of maximum lift for cylinder-A: R+ = 0.0(0.1)0.5 from top to bottom. The isolines are plotted at x = 5.0(0.5)5.0 and h = 0.1(0.1)1.0.

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Fig. 6. Instantaneous vorticity (left) and temperature (right) fields for GR = 8 obtained at the moment of maximum lift for cylinder-A: R+ = 0.0(0.1)0.5 from top to bottom. The isolines are plotted at x = 5.0(0.5)5.0 and h = 0.1(0.1)1.0.

It is concluded from the above analysis that the gap flow determines the unsteadiness of near-wake flow downstream of cylinder-B; the unsteady gap flow at larger GR perturbs the flow around cylinder-B and results in the significant oscillating wake flow which is of double vortex street type. The rounding of the corners could even fully suppress the transition to unsteady of the gap and near-wake flows at small GR. 3.2. Characteristic global quantities The dependency of flow unsteadiness on R+ and GR can be quantitatively revealed by the characteristic thermal and flow quantities. The pattern of near-wake flow downstream of cylinder-B, i.e., steady or unsteady, are presented by the Strouhal number St = fD/uin and is given in Fig. 7 at all corner radii and gap ratios; the results for the single cylinder case are also given for comparison to analyze the effect of the tandem arrangement. For the single cylinder, the Strouhal number is around 0.29 at R+ = 0.0–0.1 and increases to 0.30 for higher R+ at the present Reynolds number, which is consistent with the value of 0.30 obtained by Sharma & Eswaran [9,36] for flow past a square cylinder. The variation tendency is consistent with the observations of flow across a rounded corner cylinder in an infinite medium at higher Reynolds numbers Re  O(103) [27]. The consistent results reflect that the confinement from the channel walls at the present blockage ratio (25%) has almost negligible effect in damping the wake unsteadiness in terms of the fluctuating frequency. However, the Strouhal number does not monotonically vary with R+; it is the highest for the partially rounded cylinder at R+ = 0.2 but slightly decreases as R+ further increases. For the tandem cylinders arrangements, the flow downstream of cylinder-B is unsteady for R+ = 0.0–0.2, while the near-wake flow transits to steady at GR = 1 for the R+ = 0.3 cylinder

Fig. 7. Variation of Strouhal number with R+ and GR. The solid symbols and dashed horizontal lines at GR = 0 denote the results of single cylinder case.

and GR = 1–2 for cylinders at R+ = 0.4–0.5 where the Strouhal number is missing in the figure. The steady flow in the gap and nearwake regions is also exhibited in Fig. 4 and reflects that the flow across square-like cylinders with relatively sharper corners is more unstable. A noticeable variation is observed in the St-GR curve for the R+ = 0.0–0.3 cylinders that the Strouhal number is higher than that of the single cylinder case at GR = 3 but becomes lower at GR = 4. The drastic decreasing is attributed to the transition of gap flow from steady to unsteady at this specific GR because the flow across cylinder-A is less confined by cylinder-B and behaves

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as passing a single cylinder [23]. As GR further increases, the Strouhal number gradually approaches to the result of single cylinder case but is always slightly lower at all R+ because of the slight confinement acting on the gap flow. For the R+ = 0.4–0.5 cylinders at the small gap ratio GR = 3, the weakly unstable wake flow downstream of cylinder-B takes more time to fully develop and form the periodic shedding pattern, thus the Strouhal number is relatively low. The gap flow affects the aerodynamic performances of the two cylinders by its interactions with the cylinders and the pressure in the gap. Fig. 8 shows the time-averaged drag coefficient computed as:

C D;av g ¼

1 P

Z

t 0 þP

C D ðtÞdt;

ð8Þ

when subjecting to the incoming perturbations from unsteady shedding vortices, especially the impingement of vortices on the windward edge of the cylinder. The drag coefficient does not monotonically vary with R+; the partially rounded cylinder has the highest drag in the steady gap flow regime and the lowest drag when gap flow is unsteady, while the circular and square cylinders exhibit notably low and high drag in the steady and unsteady gap flow regimes, respectively. Since the cylinders are placed in the channel in a symmetric manner about the wake centerline, the time-averaged lift coefficient is zero for both cylinders and is not discussed here. The effect of R+ and GR on the flow unsteadiness can be reflected by the rootmean-square value of the fluctuating lift coefficient which is defined as:

t0

for both cylinders at various R+ and GR. The instantaneous drag coefficient is computed by considering the pressure difference between the windward and leeward edges and the shear stress at the lateral edges. The drag coefficient of the single cylinder first decreases and then increases with R+; for the square cylinder case, the mean drag coefficient is about 3.48 which is quite close to the benchmark data of 3.6 by Sharma & Eswaran [9,36]. For the tandem cylinders, the drag is generally higher for cylinder-A at small GR and for cylinder-B at large GR; no negative drag (‘‘drag inversion”) is observed for cylinder-B at all GRs which is different from the tandem cylinders in an infinite medium [37], reflecting the effect of channel confinement. For cylinder-A, the flow separation and vortex detachment in the gap is restricted at small GR, thus the pressure is not as low as that of the fully detached flow which results in the relatively lower drag. As GR increases, the gap flow is fully developed with massive separation and vortex shedding, consequently the drag coefficient increases and approaches to that of the single cylinder case at large GR. The corner radius has more substantial effect on the drag coefficient compared with GR. The drag generally decreases with R+ for both steady and unsteady gap flows except for the critical GR where the gap flow transits to unsteady, and the decreasing is less apparent at large R+ where the gap flow gets close to that of single cylinder case. For cylinder-B, the drag is quite low at small GR due to the steady gap flow and weakly unsteady wake flow that the pressure difference between the windward and leeward edges is small; it abruptly jumps as the gap flow transits to unsteady for GR beyond the critical value. Different from the result of cylinder-A which approaches to the result of single cylinder case, the drag for cylinder-B at large GR is at least 10% higher

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C L;rms

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 t0 þP ¼ ½C L ðtÞ  C L;av g 2 dt; P t0

ð9Þ

where CL,avg = 0 is the time-averaged lift coefficient, and is presented in Fig. 9. For the single cylinder case, the magnitude of RMS lift is 0.27 for the square cylinder and 0.55 for the circular cylinder, compared with the reference values of 0.27 [9,36] and 0.50 [38]. The lift fluctuation monotonically increases with R+ and the increasing is more pronounced for circular-like geometries, reflecting the strong unsteadiness for flow across circular-like cylinders. This variation trend with R+ is different from the threedimensional low-Re flow across a single isolated cylinder where the fluctuation is minor for the partially rounded cylinders but significant for the square and circular cylinders [29]. The difference is mainly attributed to the pattern for the destabilization of separated shear layer; for the three-dimensional flow, the destabilization of quasi-2D separated shear layer experiences the spanwise corrugation followed by the onset of vortex shedding, while for twodimensional flow case the destabilization is primarily resulted from the local adverse pressure gradient field which is greatly affected by R+. For tandem cylinders at GR = 1–2, since the gap size is small and the gap flow is steady, the flow across tandem cylinders is roughly equivalent to flow across a streamwise elongated cylinder [23], thus the flow to the lateral edges of the cylinder is quasi-steady and feed minor perturbation to flow around cylinder-B and results in the weakly unsteady near-wake flow. The corresponding fluctuating lift is negligible at these GRs. As the gap flow transits to unsteady with increasing GR, the lift fluctuation is significantly intensified for both cylinders for GR greater than the critical value, i.e., GRcr = 4 for the square-like cylinders at R+ = 0.0–0.2 and GRcr = 3 for the circular-

Fig. 8. Variations of mean drag coefficient of cylinder-A (left) and cylinder-B (right). The solid symbols and dashed horizontal lines at GR = 0 denote the results of single cylinder case.

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Fig. 9. Variations of root-mean-square value of lift coefficient of cylinder-A (left) and cylinder-B (right). The solid symbols and dashed horizontal lines at GR = 0 denote the results of single cylinder case.

like cylinders at R+ = 0.3–0.5. As GR further increases beyond GRcr, the lift fluctuation of cylinder-A decreases and approaches to that of the single cylinder with negligible difference from GR = 5, indicating the limited confinement effect of cylinder-B on the gap flow unsteadiness. However, due to the incoming shedding vortices from the gap, the lift fluctuation for cylinder-B decreases in the unsteady gap flow regime up to the largest gap ratio GR = 8, and is generally over five times that of the single cylinder configuration. The different flow patterns in the gap and near-wake regions affect the local heat transfer rate and determine the mean heat transfer of the tandem cylinders system. The mean Nusselt number is used to denote the total heat transfer rate of each cylinder which is defined as:

Nuav g ¼

1 P

Z

t 0 þP

t0

Z 2p 0

Nuðu; tÞdudt;

ð10Þ

where the temporal-spatial integration of the instantaneous local Nusselt number Nu(u,t) is performed all over the cylinder surface during a whole shedding cycle. The variation of mean Nusselt number is given in Fig. 10. The mean Nusselt number is 4.7 for the square cylinder and is consistent with the value of 4.8 obtained by Sharma & Eswaran [9]. It is noted that for both the single and

tandem cylinders configurations, the mean Nusselt number monotonically increases with R+ at all GRs although the surface area of the cylinder actually monotonically decreases, i.e., the surface area of the circular geometry (pD) is smaller than that of the square geometry (4D). The contrary variation trends of Nuavg and surface area indicate that the higher heat transfer rate is attributed to the intensified unsteady flow which enhances the local heat transfer. For cylinder-A, the mean Nusselt number is smaller than that of the single cylinder in the steady gap flow regime due to the weakened heat transfer at the lateral and leeward edges. The heat transfer is notably enhanced at GRcr where the gap flow first transits to unsteady, while the overshoot is minor since the contribution of heat transfer at the leeward edge is still limited. As GR further increases, the mean Nusselt number approaches to that of the single cylinder roughly at GR = 5 as the two cylinders are sufficiently separated, reflecting that the confinement has almost no effect on the heat transfer of gap flow. Compared with cylinder-A, the mean Nusselt number for cylinder-B is relatively smaller mainly for two reasons. First, the fluid around cylinder-B has already been heated by its upstream counterpart that the temperature difference between the solid and fluid is reduced. Second, the local heat transfer rate at the windward edge of the cylinder is lower which is supposed to contribute the most to the total heat transfer. The transition of

Fig. 10. Variations of mean Nusselt number of cylinder-A (left) and cylinder-B (right). The solid symbols and dashed horizontal lines at GR = 0 denote the results of single cylinder case.

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gap flow with increasing GR has more pronounced effect on the heat transfer of cylinder-B as represented by the abrupt jump at GRcr. The heat transfer is always affected by the confined gap flow even for the largest GR where the averaged Nusselt number is still lower than the single cylinder result for the partially rounded and circular cylinders. 3.3. Local heat transfer characteristics The above discussions clearly demonstrate the effect of tandem arrangement on the general aerodynamic and thermal characteristics of the two cylinders related with the flow patterns in the gap and near-wake of cylinder-B. In order to comprehensively quantify how the variations of GR and R+ affect the heat transfer between the solid and fluid, Fig. 11 gives the circumferential distribution of time-averaged local Nusselt number on the surfaces of both cylinders, in which the solid and dashed lines denote the curves for cylinder-A and cylinder-B, respectively. The curves are exhibited for half of the circumference considering the symmetry of the geometrical configuration and flow pattern: from u = 0° at the windward stagnation point to u = 180° at the leeward stagnation point as shown in Fig. 1. For the square geometry, there is a drastic increase of local Nusselt number at the sharp corners for both cylinders; the magnitude of the former one is more than five times that of the lateral edges, while the magnitude of the latter one is much smaller due to the reduced temperature difference between the solid and fluid. This peak Nusselt number still exists for the partially rounded cylinders especially at the leading corners, while the local Nusselt number at the rear corners is only negligibly higher than that of the neighboring lateral and leeward edges and monotonically decreases with R+; it completely diminishes for the circular geometry where there is no discontinuity of surface curvature. It is noted that the distribution pattern of Nusselt number for both cylinders of the present tandem configuration

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is quite similar to that observed for single cylinder or side-by-side cylinders configuration where the flow pattern is actually greatly different. Sanyal & Dhiman [39,40] studied the low-Re flow past two cylinders in side-by-side arrangement. They concluded that the shear layers near the lateral edges has almost the same temperature as the cylinder, thus reduces the thermal stratification and leads to a decreased wall heat flux (local Nusselt number) along the surface in the streamwise direction, which is also observed for the upstream cylinder in this study where the cold incoming fluid is first heated. Moreover, a local minimum of Nusselt number is seen at the rear edge around the corners where forced flow separation occurs, which is especially notable for the square-like cylinders; the similar trend is also reported by Sanyal & Dhiman [39,40] for the side-by-side cylinders configuration and Bhattacharyya & Mahapatra for the single cylinder [41]. The distribution characteristics are remarkably different for the two cylinders. For cylinder-A, the magnitude of local Nusselt number is generally the maximum at the windward edge and minimum at the leeward edge except for the circular-like cylinders where the heat transfer at the leeward edge is enhanced by pulsating flow. The gap ratio has almost no effect on the local heat transfer at the windward and lateral edges, and the result is almost perfectly identical with that of the single cylinder which reflects the negligible effect of confinement from cylinder-B. The local heat transfer at the leeward edge is notably affected by the gap flow, thus can be deviated from the single cylinder result and is particularly weakened at small GR because of the quasi-steady gap flow. As R+ increases, the magnitude of Nusselt number at the leading stagnation point monotonically increases from Nuloc,avg  7.5 to Nuloc,avg  12 which is primarily determined by the cylinder geometry. However, the maximum Nusselt number at the leading corners gradually decreases in magnitude and approaches to that of the leading stagnation point. The abrupt jump at the rear sharp corners is also rapidly weakened with increasing R+ which is still observ-

Fig. 11. Circumferential distribution of time-averaged local Nusselt number.

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able for the circular-like cylinders but vanishes for the circular cylinder. Another observation regarding the effect of R+ is the circumferential variation of Nusselt number on the flat edges. The variation is quite mild for square-like cylinders especially on the lateral and leeward edges, and drastic variation is only found in the region around the corners; however, the circumferential variation gets more pronounced for the partially rounded cylinders with noticeable Nusselt number variation all over the cylinder. The curves for cylinder-B are greatly deviated from the results of cylinder-A and the single cylinder case because of the heated and possibly perturbed gap flow. The magnitude of local Nusselt number is normally much smaller at the windward edge for two reasons. The first is the weakly recirculating gap flow that the convection at the windward edge is minor. The second is that the upstream fluid has already been heated by cylinder-A and the temperature difference between solid walls and ambient fluid is low. The gap ratio has a pronounced effect on the local heat transfer rate especially at the windward edge. As GR increases, the Nusselt number at the windward and leeward edges generally substantially increases; the former is attributed to the transition of gap flow to unsteady, while the latter is resulted from the incoming gap flow perturbation which intensifies the near-wake flow. The maximum Nusselt number around the leading corners is getting close to the leading stagnation point for the partially rounded and circular cylinders. It is noted that as R+ increases, the local heat transfer rate at the leading stagnation point gradually increases in the unsteady gap flow regime. In order to analyze the unsteady heat transfer characteristics, the circumferential distribution of fluctuating amplitude of local Nusselt number is presented in Fig. 12. Although the gap flow transits to unsteady at GRcr, the fluctuating Nusselt number on the surface of single cylinder and cylinder-A is negligible at the windward and lateral edges and is quite minor at the leeward edge. As R+ increases, the fluctuation at the downstream half of lateral edge and the whole leeward edge slightly increases and is nearly the

same in magnitude as that of the single cylinder. Considering the smaller magnitude of the Nusselt number at the leeward edge, the minor fluctuation would only slightly contribute to the fluctuation of total heat transfer which is consequently negligible. For gap ratio above GRcr, the fluctuation is much violent for cylinderB especially at the windward and lateral edges because of the perturbation from incoming shedding vortices. The fluctuating amplitude depends on R+ where unsteady gap flow emerges, and the fluctuation monotonically decreases all over the cylinder surface as R+ increases. Notable fluctuation is observed at GR = 4 and 8 for the square-like cylinders R+ = 0.0–0.2, while it can be observed at a smaller gap ratio GR = 3 for larger R+, indicating that the flow across circular-like cylinders could induce more unstable flow in the gap. It is worth noticing that the fluctuating heat transfer is the most significant all over the cylinder surface at GRcr where the gap flow first transits to unsteady; although the gap flow better develops as GR further increases, the fluctuating amplitude monotonically decreases attributed to the reduced unsteadiness of gap flow. 3.4. Characteristics of gap and near-wake flows It is concluded from the above section that the heat transfer characteristics are dependent on the flow unsteadiness especially in the gap. Fig. 13 gives the streamwise distribution of timeaveraged and fluctuating amplitude of u-velocity in the gap between the cylinders, in which the solid and dashed lines denote the time-averaged value and fluctuating amplitude, respectively. The curves are given at several representative GRs to cover both steady and unsteady gap flow regimes and around GRcr where flow transition occurs, and the results for single cylinder are provided for comparison. One major effect of R+ on the time-averaged gap flow is that for the square cylinder geometry, the flow in the gap behaves as it is in an open rectangular cavity confined by the vertical sidewalls, while the confinement is relieved by rounding the

Fig. 12. Circumferential distribution of fluctuating amplitude of local Nusselt number.

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Fig. 13. Time-averaged and fluctuating amplitude of u-velocity along the wake centerline (y = 0) in the gap. Dx is the distance measured from the leeward stagnation point of cylinder-A.

corners that the cavity flow is connected to the ambient flow by the two ramps, thus the flow unsteadiness is affected. Generally, the time-averaged velocity for the tandem cylinders arrangement is in good agreement with that of the single cylinder and deviation is only observed in the region just upstream of cylinder-B. For the square cylinder, there is reverse flow at the rear of cylinder-A with negative u-velocity because of the fluid recirculation; the reverse flow reaches a maximum negative velocity and then recovers to positive with increasing magnitude. For steady gap flow, the reverse flow occupies the whole gap and forms the steady recirculation bubbles, and the magnitude of maximum negative velocity is slightly intensified by increasing GR. For unsteady gap flow, forward flow with positive magnitude of u-velocity is observed as a result of the detachment and shedding of vortices. The curve is in good agreement with that of the single cylinder in the majority region of the gap and deviation only occurs close to cylinder-B where flow starts to stagnate. As the gap flow transits from fully reversal to partially forward, the reattachment point (uavg = 0) first moves upstream and then downstream with GR that the unsteady gap flow at GRcr has the smallest recirculation bubble in the streamwise direction. The curves obtained at large GR coincide with the result of single cylinder. The streamwise distribution of fluctuating amplitude of uvelocity is generally consistent with that of the single cylinder in the region close to cylinder-A. Very good consistency is even observed for flow within almost the whole gap at GR = 8 which reflects the relatively fully developed gap flow and minor effect of cylinder confinement on the flow unsteadiness. It is seen that the fluctuating amplitude of gap flow almost does not vary with increasing R+ both in magnitude and distribution pattern. The steady gap flow is identified by the zero fluctuation in the whole gap at GR = 2–3 for the square-like cylinders of R+ = 0.0–0.2, and at GR = 2 for the circular-like cylinders of R+ = 0.3–0.5, which is consistent with the observations made in Figs. 4–6. As the flow

starts to transit to unsteady at GRcr, the fluctuation is drastically intensified in the region at the rear of cylinder-A, and is much larger in magnitude than the results obtained at other GR since the shedding vortices are confined by cylinder-B and is squeezed toward cylinder-A. The small plateau of Du close to cylinder-B is resulted from the impingement of vortices at the windward edge that the local flow unsteadiness is substantial. The flow in the near-wake region of cylinder-B is affected by GR and R+ mainly because of the incoming perturbations of the gap flow. The streamwise distributions of time-averaged value and fluctuating amplitude of u-velocity in the near-wake unconfined region of cylinder-B are presented in Fig. 14. There is recirculating near-wake flow as shown by the negative u-velocity at the cylinder rear. Although the near-wake flow is isolated from the gap flow by cylinder-B, the curves are clearly categorized into two families corresponding to the steady and unsteady gap flows at small and large GR, respectively. Compared with flow in the unsteady gap flow regime, the flow in the steady gap flow regime has a larger recirculation zone downstream of cylinder-B with the reattachment point (uavg = 0) at around Dx = 2 for all R+; the recovery rate of u-velocity to the incoming flow is relatively low and the recovery is slower than the single cylinder case. However, the near-wake flow at large GR exhibits rapid recovery especially in the region close to the cylinder, thus the velocity magnitude is much larger. The corner radius has a noticeable effect on the near-wake flow behaviors including the size of recirculation zone, the maximum negative velocity of the reverse flow and the velocity recovery. For the circular-like cylinders, GR has minor effect on the u-velocity distribution for the unsteady flow in both the recirculation zone and the recovery zone as revealed by the overlapping curves. The size of recirculation zone and the maximum negative velocity monotonically increase with GR for the square-like cylinders, while both of them are larger in magnitude compared with the circular-like

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Fig. 14. Time-averaged and fluctuating amplitude of u-velocity along the wake centerline (y = 0) in the near-wake of cylinder-B. Dx is the distance measured from the leeward stagnation point of cylinder-B. For captions see Fig. 13.

cylinders; the velocity magnitude monotonically decreases with GR. The streamwise distribution of fluctuating amplitude of uvelocity shows that there are two local maxima correspond to the detachment of vortices at the cylinder leeward edge and the initial shedding of vortices at around Dx = 4–6, respectively. The flow unsteadiness is viscously damped during its downstream convection because the destabilization of wake flow is through the mechanism of convective instability. The fluctuation is greatly attenuated whose magnitude approaches to zero in the far-wake, which is an indication of the absence of absolute instability in the computational domain [42,43]. The fluctuation is more significant for the circular-like cylinders which can be up to 0.75uin. For the square cylinder, the maximum fluctuation appears at different streamwise positions downstream of cylinder-B depending on GR, while the positions are quite consistent for other geometries. It can be anticipated from the figure that the steady gap flow feeds minor perturbation to flow around cylinder-B, while the unsteady gap flow exerts additional perturbation to the near-wake flow through the shedding vortices, thus amplifies the flow unsteadiness. The distribution of magnitude of maximum velocity fluctuation along the streamwise direction in the gap region and in the wake of cylinder-B could partially reveal the stability of the flow. Conventionally, the (in)stability characteristics of the flow is only dependent on the steady-state base flow and is not directly related to the flow fluctuation. However, it has been confirmed in our earlier work [44] that for low-Re flow past an isolated cylinder, the flow unsteadiness is more violent (as measured by larger RMS values of fluctuating lift, drag and Nusselt number) when the flow is more unstable (as quantified by the larger growth rate of perturbation obtained by global linear stability analysis). The stability of flow past two tandem circular or square cylinders in an infinite medium has been investigated in our earlier work [35]; since the forced convection heat transfer has no effect on the flow dynamics, the

results could partially demonstrate the flow stability for cylinders in a channel. It is found that the stability of flow is dominated by the destabilization of flow (detachment of vortices and roll-up of shear layers) in the gap for large gap ratios, while the wake flow downstream of cylinder-B is more significant for small gap ratios. The observations on the distribution of velocity fluctuation are quite in consistent with the findings in this work. 3.5. Temperature of channel walls For the present physical problem, the temperature of incoming flow upstream of the cylinders is constant at h = 0 as that of inflow. The cold fluid is heated by the hot cylinder(s) and its temperature gradually increases and approaches a constant value in the farwake. Since the channel walls are assumed thermally adiabatic, its temperature is determined by the local fluid motion as a result of the transversely oscillating shedding vortices and exhibits an increasing manner, as shown in Fig. 15. The two channel walls have the same time-averaged temperature distribution due to the symmetry of the configuration. It is noted that the wall temperature keeps at h = 0 as that of the inflow in the region approximately at Dx = [0,5] since it takes time for the transversely oscillating shedding vortices to heat the channel wall. The temperature rapidly increases for Dx > 5 and finally saturates around a constant value. The curves are distinctly categorized into two families depending on the gap flow pattern, i.e., steady or unsteady. The curves corresponding to the unsteady gap flow are quite close to each other in terms of growth rate of initial stage and the far-wake temperature, both of which are higher than those of the steady gap flow regime. The channel wall temperature grows at a relatively slower rate in the steady gap flow regime at small GR; the growth rate increases with GR and is the largest at GRcr where the temperature growth is the most rapid especially at the initial stage, while it gradually decreases with GR in the unsteady gap flow regime. It is noted that

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Fig. 15. Streamwise variation of time-averaged temperature of the channel wall. Dx is the distance measured from the leeward stagnation point of cylinder-A.

Fig. 16. Streamwise variation of fluctuating amplitude of temperature on the channel wall. Dx is the distance measured from the leeward stagnation point of cylinder-A.

the heating of the channel walls is generally more effective as R+ increases; the temperature of channel wall in the far-wake region at Dx=50 varies in the range 0.110–0.125 for the square cylinder,

while it keeps at around 0.120 from Dx = 15 to Dx = 50 with slight attenuation. This is mainly attributed to the more unstable wake flow across the circular-like cylinders that the transverse oscilla-

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tion of shedding vortices is more significant, as revealed by the isotherms in Figs. 3–6. Fig. 16 gives the streamwise distribution of fluctuating amplitude of temperature on the channel wall to reflect how the shedding vortices affect the local near-wall flow. The fluctuation is negligible in the region approximately at Dx = [0, 4] since the downstream convecting vortices have not heated the local wall, and in the far-wake region because the vortices are viscously damped and the flow recovers to be quasi-steady. The fluctuation is mainly observed at the streamwise positions where the cylinders locate and the near-wake region. For the single cylinder case, the fluctuation generally monotonically increases with R+. For the tandem cylinders arrangements, there are multiple peaks observed for the square cylinder case corresponding to the shedding of vortices in the near-wake region, in which the second is less notable especially at small GR. The pattern of variation of fluctuating temperature in the streamwise direction is also revealed by Sanyal & Dhiman [40] for the configuration of side-by-side cylinders although the measurement is on the temperature in the wake region. The RMS temperature reaches maximum at a certain position in the wake because of the roll-up of thermal layers (see Fig. 21c in [40]). It is noticed that GR does not affect the temperature fluctuation in a monotonic way; the fluctuation is minor at small GR since the two cylinders act as a single body and the wake unsteadiness is confined, while it is the most pronounced at GRcr for the square-like cylinders and gradually increases at the largest GR for the circular-like cylinders. 4. Conclusions This work presented the first numerical investigation on flow across two tandem cylinders with rounded corners in a channel at Re = 100. The cylinder geometry varies from square to circular by rounding all corners, and the partially rounded cylinders are considered. The effects of gap ratio (GR) characterizing the separating distance of two cylinders and the corner radius (R+) are analyzed at dimensionless parameters R+ = 0.1(0.1)0.5 and GR = 1(1) 8. Numerical results derive the following main conclusions:

(4) The rounding of the corners reduces the magnitude of local peak of time-averaged Nusselt number, but enhances local heat transfer at the leading stagnation point. The pattern of distribution of local Nusselt number is quite similar to that of a single cylinder or side-by-side cylinders configuration. The gap ratio has minor effect on the distribution at leading and lateral edges of cylinder-A, while the effect is substantial for cylinder-B that the magnitude monotonically increases with R+. The local Nusselt number only notably fluctuates for cylinder-B in the unsteady gap flow regime, and the fluctuation decreases with GR over the whole cylinder surface. (5) The time-averaged flow in the gap is generally consistent with the result of single cylinder case with minor difference on the size of recirculation zone, and the flow unsteadiness is the most pronounced at GRcr where the flow first transits to unsteady. The time-averaged flow in the near-wake of cylinder-B is dependent on the pattern of gap flow; it has a small recirculation zone and rapid flow recovery for unsteady gap flow, and the velocity fluctuation is pronounced at the cylinder rear; however, the recovery rate is much lower for steady gap flow and the flow unsteadiness is mild. (6) The streamwise distribution of temperature on the thermally adiabatic channel walls is determined by GR and is distinctly categorized into two families corresponding to the steady and unsteady gap flows. The magnitude of temperature for the unsteady gap flow is remarkably higher than that of the steady gap regime due to the heating of the channel walls by the transversely oscillating shedding vortices. The fluctuation of channel wall temperature is only pronounced in the unsteady gap flow regime. Conflict of interest The author declare that there is no conflict of interest. Acknowledgement

(1) Two regimes are identified for the gap flow between the two cylinders, i.e., steady gap flow at small GR and unsteady gap flow at large GR. The gap flow transits from steady to unsteady at the critical gap ratio GRcr = 4 for the squarelike cylinders at R+ = 0.0–0.2, while the transition occurs at GRcr = 3 for the circular-like cylinders at R+ = 0.3–0.5. (2) The flow across tandem cylinders can be fully stabilized by rounding the corners in the steady gap flow regime. The near-wake flow downstream of cylinder-B is weakly unstable compared with the single cylinder, as represented by the weakly shedding vortices in the streamwise direction. The stabilization is more significant as R+ increases and the wake flow could even transit to steady, i.e., at GR = 2 and R+ = 0.4–0.5. (3) There are drastic variations of characteristic aerodyanmic and thermal quantities at GRcr where the gap flow transits to unsteady. The Strouhal number decreases; the timeaveraged drag coefficient, fluctuating lift coefficient and total heat transfer rate increase for both cylinders and the increasing is much significant for cylinder-B. As R+ increases, the unsteadiness of gap flow is more pronounced, and the total heat transfer rate monotonically increases at all GRs although the surfaces area actually reduces.

The work was supported by NSFC (51706205, 11502236 and 11872337). W. Z. appreciates the support from Qianjiang Talent Program of Zhejiang Province. The valuable comments from the reviewers are highly appreciated. References [1] 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. [2] A. Mukhopadhyay, G. Biswas, T. Sundararajan, Numerical investigation of confined wakes behind a square cylinder in a channel, Int. J. Numer. Meth. Fl. 14 (1992) 1473–1484. [3] A. Okajima, Strouhal numbers of rectangular cylinders, J. Fluid Mech. 123 (1982) 379–398. [4] A. Okajima, Numerical simulation of flow around rectangular cylinders, J. Wind Eng. Ind. Aerod. 33 (1990) 171–180. [5] A. Okajima, H. Ueno, H. Sakai, Numerical simulation of laminar and turbulent flows around rectangular cylinders, Int. J. Numer. Meth. Fl. 15 (1992) 999– 1012. [6] W. Zhang, H. Yang, H.-S. Dou, Z. Zhu, Flow unsteadiness and stability characteristics of low-Re flow past an inclined triangular cylinder, ASME J. Fluid Eng. 139 (2017) 121203. [7] H. Abbassi, S. Turki, S.B. Nasrallah, Numerical investigation of forced convection in a plane channel with a built-in triangular prism, Int. J. Therm. Sci. 40 (2001) 649–658. [8] H. Abbassi, S. Turki, S.B. Nasrallah, Numerical investigation of forced convection in a horizontal channel with a built-in triangular prism, ASME J. Heat Transf. 124 (2002) 571–573.

W. Zhang et al. / International Journal of Heat and Mass Transfer 130 (2019) 1053–1069 [9] A. Sharma, V. Eswaran, Effect of channel confinement on the two-dimensional laminar flow and heat transfer across a square cylinder, Numer. Heat Tr. AAppl. 47 (2005) 79–107. [10] M.A. Moussaoui, M. Jami, A. Mezrhab, H. Naji, MRT-lattice Boltzmann simulation of forced convection in a plane channel with an inclined square cylinder, Int. J. Therm. Sci. 49 (2010) 131–142. [11] S.M. Seyyedi, H. Bararnia, D.D. Ganji, M. Gorji-Bandpy, S. Soleimani, Numerical investigation of the effect of a splitter plate on forced convection in a two dimensional channel with an inclined square cylinder, Int. J. Therm. Sci. 61 (2012) 1–14. [12] A. Valencia, Numerical study of self-sustained oscillatory flows and heat transfer in channels with a tandem of transverse vortex generators, Heat Mass Transfer 33 (1998) 465–470. [13] A. Valencia, Heat transfer enhancement due to self-sustained oscillating transverse vortices in channels with periodically mounted rectangular bars, Int. J. Heat Mass Trans. 42 (1999) 2053–2062. [14] H.M.S. Bahaidarah, M. Ijaz, N.K. Anand, Numerical study of fluid flow and heat transfer over a series of in-line noncircular tubes confined in a parallel-plate channel, Numer. Heat Tr. B-Fund. 50 (2006) 97–119. [15] R. Agarwal, A. Dhiman, Flow and heat transfer phenomena across two confined tandem heated triangular bluff bodies, Numer. Heat Tr. A-Appl. 66 (2014) 1020–1047. [16] D. Chatterjee, G. Biswas, The effects of Reynolds and Prandtl numbers on flow and heat transfer across tandem square cylinders in the steady flow regime, Numer. Heat Tr. A-Appl. 59 (2011) 421–437. [17] D. Chatterjee, B. Mondal, Forced convection heat transfer from tandem square cylinders for various spacing ratios, Numer. Heat Tr. A-Appl. 61 (2012) 381– 400. [18] M. Farhadi, K. Sedighi, M.M. Madani, Convective cooling of tandem heated squares in a channel, P. I. Mech. Eng. C-J. Mec. 223 (2009) 965–978. [19] J.L. Rosales, A. Ortega, J.A.C. Humphrey, A numerical investigation of the convective heat transfer in unsteady laminar flow past a single and tandem pair of square cylinders in a channel, Numer. Heat Tr. A-Appl. 38 (2000) 443– 465. [20] J.L. Rosales, A. Ortega, J.A.C. Humphrey, A numerical simulation of the convective heat transfer in confined channel flow past square cylinders: comparison of inline and offset tandem pairs, Int. J. Heat Mass Trans. 44 (2001) 587–603. [21] A. Sohankar, A. Etminan, Forced convection heat transfer from tandem square cylinders in cross flow at low Reynolds numbers, Int. J. Numer. Meth. Fl. 60 (2009) 733–751. [22] M. Asif, A. Dhiman, Analysis of laminar flow across a triangular periodic array of heated cylinders, J. Braz. Soc. Mech. Sci. 40 (2018) 350. [23] W. Zhang, H. Yang, H.-S. Dou, Z. Zhu, Forced convection of flow past two tandem rectangular cylinders in a channel, Numer. Heat Tr. A-Appl. 72 (2017) 89–106. [24] T. Tamura, T. Miyagi, T. Kitagishi, Numerical prediction of unsteady pressures on a square cylinder with various corner shapes, J. Wind Eng. Ind. Aerod. 74 (1998) 531–542. [25] T. Tamura, T. Miyagi, The effect of turbulence on aerodynamic forces on a square cylinder with various corner shapes, J. Wind Eng. Ind. Aerod. 83 (1999) 135–145.

1069

[26] L. Carassale, A. Freda, M. Marrè-Brunenghi, Experimental investigation on the aerodynamic behavior of square cylinders with rounded corners, J Fluid Struct. 44 (2014) 195–204. [27] J.C. Hu, Y. Zhou, C. Dalton, Effects of the corner radius on the near wake of a square prism, Exp. Fluids 40 (2006) 106–118. [28] J.C. Hu, Y. Zhou, Aerodynamic characteristics of asymmetric bluff bodies, ASME J. Fluid Eng. 131 (2009) 011206. [29] W. Zhang, R. Samtaney, Low-Re flow past an isolated cylinder with rounded corners, Comput. Fluids 136 (2016) 384–401. [30] W. Zhang, R. Samtaney, Effect of corner radius in stabilizing the low-Re flow past a cylinder, ASME J. Fluid Eng. 139 (2017) 121202. [31] A. Sohankar, C. Norberg, L. Davidson, Numerical simulation of unsteady lowReynolds number flow around rectangular cylinders at incidence, J. Wind Eng. Ind. Aerod. 69 (1997) 189–201. [32] Y. Zang, R.L. Street, J.R. Koseff, A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates, J. Comput. Phys. 114 (1994) 18–33. [33] W. Zhang, Y. Wei, X. Chen, H.-S. Dou, Z. Zhu, Partitioning effect on natural convection in a circular enclosure with an asymmetrically placed inclined plate, Int. Commun. Heat Mass Trans. 90 (2018) 11–22. [34] W. Zhang, Y. Wei, H.-S. Dou, Z. Zhu, Transient behaviors of mixed convection in a square enclosure with an inner impulsively rotating circular cylinder, Int. Commun. Heat Mass Trans. 98 (2018) 143–154. [35] W. Zhang, H.-S. Dou, Z. Zhu, Y. Li, Unsteady characteristics of low-Re flow past two tandem cylinders, Theor. Comp. Fluid Dyn. 32 (2018) 475–493. [36] A. Sharma, V. Eswaran, Effect of channel-confinement and aiding opposing buoyancy on the two-dimensional laminar flow and heat transfer across a square cylinder, Int. J. Heat Mass Trans. 48 (2005) 5310–5322. [37] M.M. Zdravkovich, Review of interference-induced oscillations in flow past two parallel circular cylinders in various arrangements, J. Wind Eng. Ind. Aerod. 28 (1988) 183–199. [38] K. Prasad, S.B. Paramane, A. Agrawal, A. Sharma, Effect of channel-confinement and rotation on the two-dimensional laminar flow and heat transfer across a cylinder, Numer. Heat Tr. A-Appl. 60 (2011) 699–726. [39] A. Sanyal, A. Dhiman, Wake interactions in a fluid flow past a pair of side-byside square cylinders in presence of mixed convection, Phys. Fluids 29 (2017) 103602. [40] A. Sanyal, A. Dhiman, Effect of thermal buoyancy on a fluid flowing past a pair of side-by-side square bluff-bodies in a low-Reynolds number flow regime, Phys. Fluids 30 (2018) 063603. [41] S. Bhattacharyya, S. Mahapatra, Vortex shedding around a heated square cylinder under the influence of buoyancy, Heat Mass Transfer 41 (2005) 824– 833. [42] X. Yang, A. Zebib, Absolute and convective instability of a cylinder wake, Phys. Fluids A 1 (1989) 689–696. [43] P. Huerre, P.A. Monkewitz, Local and global instabilities in spatially developing flows, Annu. Rev. Fluid Mech. 22 (1990) 473–537. [44] W. Zhang, R. Samtaney, Numerical simulation and global linear stability analysis on low-Re flow past a heated circular cylinder, Int. J. Heat Mass Trans. 98 (2016) 584–595.