The flow around three tandem circular cylinders

*Md. Mahbub Alam1, J.F. Derakhshandeh2, Q. Zheng3, S. Rehman4, Chunning Ji5 and F. Zafar6

1,3,6 Institute for Turbulence-Noise-Vibration Interaction and Control, Shenzhen

Graduate School, Harbin Institute of Technology, Shenzhen 518055, China 2School of Mechanical Engineering, University of Adelaide, Adelaide, Australia 4Center for Engineering Research, Research Institute, King Fahd University of

Petroleum and Minerals, Dhahran-31261, Saudi Arabia 5State Key Laboratory of Hydraulic Engineering Simulation & Safety, Tianjin University

Tianjin, 300072, China

*Email: alamm28@yahoo.com

ABSTRACT

This paper presents dependence of forces and flow structures on phase lags between vortex sheddings from three tandem cylinders. The flow around the three cylinders (cylinders 1, 2 and 3, respectively) of an identical diameter D is numerically

simulated at Re = 200 for spacing ratios *

1L = L1/D = 3.5-5.25 and *

2L = L2/D= 3.6 -5.5,

where L1 is the center-to-center spacing between cylinders 1 and 2, and L2 is that

between cylinders 2 and 3. The variations in *

1L and *

2L in these ranges correspond to

the phase lags 1 (between cylinders 1 and 2) and 2 (between cylinders 2 and 3) both changing from inphase to antiphase. The flow around the cylinders is more sensitive to

*

1L than to *

2L , while both 1 and 2 have more influences on cylinder 1 than on the

other two. An inphase condition (1 = 2 = inphase) corresponds to a high fluctuating lift and fluctuating shear-layer velocity, and a small drag, Strouhal number, and time-mean shear-layer velocity, for cylinder 1. When shear layers from cylinders 2 and 3

grow/accelerate on the same side at the same time (2 = inphase), they can pull the fluid at the same side of cylinder 1, adding to the flow/shear-layer velocity on the same side. In the next half cycle of the shedding, the same prevails on the opposite side of the cylinders, which leads to a high fluctuating lift and fluctuating shear-layer velocity.

On the other hand, an out-of-phase condition (1 = inphase/antiphase and 2 = antiphase/inphase) complements a small fluctuating lift and fluctuating shear-layer velocity. 1. INTRODUCTION

Because of its fundamental and practical importance, the flow around two circular cylinders in tandem arrangements has received a surge attention in the literature. Vortex formation, forces, vibration, noise, heat transfer, and energy

harvesting were the focus of the most investigations in the literature (e.g., Bearman 1984, Zdravkovich and Pridden 1977, Alam et al. 2003, 2007, Williamson and Govardhan 2004, Kitagawa and Ohta 2008, Sumner 2010, Kim et al. 2009, Alam and Meyer 2011, Bernitsas et al. 2008, and Derakhshandeh et al. 2016). Particularly, dynamics and formation of vortices between two tandem cylinders are very complex, and the centre-to-centre spacing L between the cylinders is one of the vital key parameters, governing the flow structure around and mutual interference between the cylinders (Alam and Meyer 2011, Derakhshandeh et al. 2014, 2015). The flow interference between two cylinders is non-linear (Alam 2016) and dependent on Reynolds number Re (Alam 2014).

There are several approaches classifying the flow structures around two

cylinders, based on spacing ratio *L = L/D, where D is the cylinder diameter.

Zdravkovich (1977) classified the flow as overshoot regime *L < 1.2-1.8 (depending on

Re), reattachment regime 1.2-1.8 < *L < 3.4-4.0, and, coshedding regime *L > 3.4-4.0,

where the shear layers shed vortices in the gap without reattaching to the downstream

cylinder. The *L separating the reattachment and coshedding regimes is known as the

critical spacing *

cL . The effect of *L on time-mean drag coefficient ( DC ) and Strouhal

number (St) was investigated in the literature whereas less attention has been devoted

to investigating fluctuating (rms) lift coefficient LfC . Experimental studies conducted by

Sakamoto et al. (1987) and Alam et al. (2006) revealed that for *L > *

cL the shedding

from the downstream cylinder is triggered by the oncoming vortices from the upstream

cylinder, as a result the phase lag () between the vortex sheddings from the two

cylinders varies linearly with *L . The has a significant effect on LfC of the

upstream cylinder (Alam et al. 2006); however, the study failed to dig out the physics behind the effect. Recently, Alam (2016) coped with the previous weakness and

developed an equation showing the relationship between LfC , *L , St, and . He

showed that is a non-linear function of *L , St and convective velocity of vortices.

The maximum and minimum LfC imposing on the upstream cylinder are linked to the

in-phase ( = 2nπ) and anti-phase ( = (2n+1)π), respectively. Here, n stands for

integer numbers (n = 1, 2, etc.). Pertaining to the influence of on LfC , a few questions

may arise. (i) Will placing a third cylinder inphase or antiphase impact on LfC of the

upstream cylinder (Fig. 1a)? (ii) What is its impact on mean and fluctuating flow fields around the upstream cylinder? When the third cylinder is placed, two additional

parameters ( *

2L and 2) get involved, where *

2L is the center-to-center spacing ratio

between the second and third cylinders, and 2 is the phase between the vortex sheddings from the same cylinders. Assume here the spacing ratio between the first

and second cylinders is defined by *

1L and the corresponding shedding phase lag is 1.

Following the previous investigation, the objective of this study is to extend our

knowledge on the relationship of LfC , DC and St with *

1L , *

2L , 1 and 2 for the flow

around three cylinders. A set of numerical simulations at Re = 200 is conducted for

three cylinders with *

1L and *

2L both varying from 3.6 to 5.5 which leads 1 and 2

changing from inphase to antiphase. In addition to DC , LfC , St, 1 and 2, analyzed

and studied in details are flow structures, including vorticities, and time-mean and fluctuating pressure and flow fields. 2. COMPUTATIONAL TECHNIQUE The governing equations for an unsteady, viscous, laminar and incompressible fluid flow with constant properties are the continuity and momentum equations expressed in Cartesian coordinate as

0 u* . , and ***

*

uu.uu 2*

* Re

1)(

P

t,

where the freestream velocity U and the cylinder diameter D are used as the

reference speed and reference length to normalize the different parameters. The u* = (u*, v*) is the normalized velocity field, t* is the nondimensional time, and P* is the normalized static pressure. The differential equations are coupled and solved for the unknown P*, u* and v*. The gravity force is excluded. The computations are performed based on the finite volume method. While a standard scheme and a second-order upwind scheme are used to discretize pressure and velocity, respectively, the first-order implicit formulation is employed for time discretization. On the other hand, the coupling between the pressure and velocity fields is achieved using the SIMPLE technique. Re = 200 is adopted in the simulation (Alam 2016).

The schematic diagram of the flow model is given in Fig. 1(a). Three circular cylinders of the same diameter D are placed in tandem in a uniform flow, with spacing L1 between cylinders 1 and 2 and with spacing L2 between cylinders 2 and 3. The center of the upstream cylinder is the origin of the (x, y) coordinate system. The computational domain was 65D in the streamwise direction and 30D in the cross-stream direction, which gives a blockage ratio of 3.3% only, small enough to avoid blockage effect. The inlet boundary was 15D away from the center of the upstream cylinder. An O-xy grid system near the cylinders and a rectangular grid system away from the cylinders were used as shown in Fig. 1(b). The grid system was generated using Gambit. The number of grids for an O-grid system around a cylinder was 200 in the transverse direction and 60 in the radial direction. Therefore, a total of 200 points were on the cylinder surface. The grid in the radial direction was denser near the cylinder surface with the nearest grid being 0.005D away from the cylinder surface. A detailed mesh sensitivity and convergence test is done for single and two cylinders in Alam (2016). The same mesh system is employed in the present study; hence the detail is not repeated here.

While the no-slip boundary conditions are employed on the surfaces of the cylinders, symmetry boundary condition is adopted for the upper and lower boundary walls. In the domain, the initial flow velocities (at t* = 0) are given as u* = 1, v* = 0. For two cylinders, the critical spacing corresponding to the inphase mode was L* = 3.65, and that to the antiphase was L* = 5.25 (Alam, 2016). When the third cylinder is placed

behind the two, *

1L = 3.5 is identified as the critical spacing for the inphase between

the first two cylinders. After a series of simulations with varying *

1L and *

2L , inphage

and antiphase modes over all three cylinders were obtained at ( *

1L , *

2L ) = (3.5, 3.6) and

(5.25, 5.5), respectively, *

2L being slightly greater than *

1L for either mode.

Fig. 1 (a) Schematic of the computational domain and boundary conditions, (b)

structured grid distribution around three cylinders.

3. RELATIONSHIP OF 1 AND 2 WITH *

1L AND *

2L

Table 1 summarizes the relationship of phase lags 1 and 2 with *

1L and *

2L .

The 1 and 2 were estimated from cross-correlation between fluctuating lift of the

corresponding cylinders. With *

1L = 3.5 corresponding to an inphase shedding from

cylinders 1 and 2, the shedding from cylinders 2 and 3 becomes inphase at *

2L = 3.6,

antiphase at *

2L = 5.5 and neither inphase or antiphase at *

2L = 4.5. Hereafter, *

2L =

4.5 will be referred to as an intermediate mode between the inphase and antiphase.

Similarly, at *

1L = 4.5, an intermediate mode prevails for cylinders 1 and 2, while for

cylinders 2 and 3 there are inphase, intermediate and antiphase modes at *

2L = 3.6,

4.5 and 5.5, respectively. With *

1L = 5.25 corresponding to an antiphase mode for

cylinders 1 and 2, *

2L = 3.6, 4.5 and 5.5 complement inphase, intermediate and

antiphase modes for cylinders 2 and 3.

x

y

(a)

LdLu

2L

t

L1

D

u* = 1

v* = 0

P* = 0

Inlet

τxx = 0

τyx = 0

Outlet

v* = 0, τxy = 0

v* = 0, τxy = 0

L2

(b)

Cyl. 1 Cyl. 2 Cyl. 3

Fig. 2 Time-histories of lift forces, and vortex shedding for (a, b) L1

* = 3.5, L2* = 3.6, (c, d)

L1* = 3.5, L2

* = 5.5, (e, f) L1* = 5.25, L2

* = 3.6, and (g, h) L1* = 5.25, L2

* = 5.5.

100 105 110 115 120-2

-1

0

1

2

(b)

tU/D

CL

CL

(a)

y*

100 105 110 115 120-2

-1

0

1

2

(a)

CL

CL

(c)

50 55 60 65 70-2

-1

0

1

2

(c)

CL

(e)

CL

100 105 110 115 120-2

-1

0

1

2

(d)

CL

tU/D

600 720660630 690

(g)

CL

t*

x*

Cylinder 1; Cylinder 2; Cylinder 3

x*

(d)

(f)

(b)

(h)

x*

x*

y*

y*

y*

Table 1. Shedding modes as functions of *

1L and *

2L .

*

1L *

2L 1 2

3.5 3.6 0.12π (inphase) 0.06π (inphase)

3.5 4.5 0.12π (inphase) 0.40π

3.5 5.5 0.12π (inphase) 0.98π (antiphase)

4.5 3.6 0.62π 0.07π (inphase) 4.5 4.5 0.62π 0.39π 4.5 5.5 0.62π 0.96π (antiphase)

5.25 3.6 0.97π (antiphase) 0.08π (inphase) 5.25 4.5 0.97π (antiphase) 0.38π 5.25 5.5 0.97π (antiphase) 0.95π (antiphase)

In order to demonstrate the phase relationship discussed above, typical lift histories

and vortex shedding patterns are presented in Fig. 2 for the cases where the inphase and antiphase modes succeed. The vorticity patterns given are at the same phase at which cylinder 1 corresponds to the minimum lift. As such the vortex from the lower side of the cylinder 1 is in growing phase while that from the other side is saturated for

all patterns presented. For *

1L = 3.5, *

2L = 3.6 (Fig. 2b), growing and saturated

vortices appear on the lower and upper sides, respectively, for all cylinders, confirming

1 2 = inphase (see also Table 1). Obviously, the gap vortices impinging on cylinders

2 and 3 trigger the vortex sheddings from them. For *

1L = 3.5, *

2L = 5.5 (Fig. 3c, d), the

growing/impinging vortices appear on the lower side for cylinders 1 and 2, and on the

upper side for cylinder 3, leading to 1 inphase and 2 = antiphase. Similarly, a

condition of 1 = antiphase and 2 = inphase meets at *

1L = 5.25, *

2L = 3.6 (Fig. 2e, f)

while 1 = 2 = antiphase is achieved at *

1L = 5.25, *

2L = 5.5 (Fig. 2g, h).

4. EFFECT OF 1 AND 2 ON DC AND LfC

Figures 3 and 4 show dependence on *

1L and *

2L of DC and LfC of the three

cylinders. The first, second and third columns are for cylinders 1, 2, and 3, respectively,

while the first, second and third rows are for *

1L = 3.5 (1= inphase), 4.5 and 5.25

(1=antiphase), respectively. At *

1L = 3.5, with increasing *

2L from 3.6 (2= inphase) to

5.5 (2= antiphase) DC grows for cylinders 1 and 2, but declines for cylinder 3 (Fig.

3a). DC behaves oppositely for cylinders 1 and 2 at *

1L = 5.25, decreasing with *

2L

(Fig. 3c1, c2). The opposite behaviour is connected to their relationship with 1 and 2.

That is, DC of cylinder 1 gets a minimum when 1 = 2 = inphase or antiphase and a

maximum when 1= inphase (antiphase) and 2= antiphase (inphase). For a given *

2L ,

DC for cylinders 1 and 2 swells with increasing *

1L while that for cylinder 3 decreases

between *

1L = 3.5 and 4.5, and increases between *

1L = 4.5 and 5.5. In the ranges of

*

1L and *

2L examined, cylinder 1 experiences more than three and fifteen times higher

DC than cylinders 2 and 3, respectively.

Fig. 3 Dependence of DC on

at different . (a)

= 3.5, (b) = 4.5, and (c)

= 5.25. The first, second and third columns are for the cylinders 1, 2 and 3, respectively.

On the other hand, LfC on cylinder 2 is the largest, more than two times that

on cylinder 1 or cylinder 3 (Fig. 4). With increasing *

2L , LfC wanes for cylinders 2 and

3 regardless of *

1L , while decaying and rising for cylinder 1 at *

1L = 3.5 and 5.25.

respectively. At *

1L = 4.5, the variation in LfC of cylinder 1 is not straight forward,

having a mixed behaviour. In summary, the dependence of LfC on *

2L for cylinder 1

at *

1L = 3.5 is opposite to that at *

1L = 5.25, as is for the case of DC . In other words,

when 1 = inphase ( *

1L = 3.5), the change in 2 from inphase ( *

2L = 3.6) to antiphase

( *

2L = 5.5) leads to a drop in LfC of cylinder 1 and an increase in DC . On the contrary,

when 1 = antiphase ( *

1L = 5.25), the change in 2 from inphase ( *

2L = 3.6) to antiphase

3.5 4.0 4.5 5.0 5.50.31

0.32

0.33

0.34

0.35

3.5 4.0 4.5 5.0 5.51.24

1.25

1.26

1.27

3.5 4.0 4.5 5.0 5.51.23

1.24

1.25

1.26

3.5 4.0 4.5 5.0 5.51.21

1.22

1.23

1.24

3.5 4.0 4.5 5.0 5.50.36

0.37

0.38

0.39

0.4

3.5 4.0 4.5 5.0 5.50.38

0.39

0.4

0.41

0.42

3.5 4.0 4.5 5.0 5.50.02

0.04

0.06

0.08

0.1

3.5 4.0 4.5 5.0 5.50.02

0.04

0.06

0.08

0.1

*

2L *

2L *

2L

1 32 1 32 1 32

3.5 4.0 4.5 5.0 5.50.02

0.04

0.06

0.08

0.1

(a2)(a1) (a3)

(b1) (b2) (b3)

(c1) (c2) (c3)

DC

DC

DC

(a)

(b)

(c)

( *

2L = 5.5) boosts LfC and weakens DC of cylinder 1. That is,

LfC of cylinder 1

reaches a maximum for a condition 1 = 2 = inphase or antiphase and a minimum for a

condition at which 1 and 2 are opposite to each other. On the other hand, the two

conditions are oppositely connected to DC . The observation explores that 2 (i.e., *

2L )

has a great influence on cylinder 1, though in the coshedding regime are cylinders 1

and 2. The 1 and 2 effects on LfC of cylinders 2 and 3 are perhaps small, hence

buried in the spacing effect.

Fig. 4. Dependence of CLf on

at different . (a)

= 3.5, (b) = 4.5, and (c)

= 5.25. The first, second and third columns are for the cylinders 1, 2 and 3, respectively.

5. EFFECTS OF 1 AND 2 ON TIME-MEAN FLOW STRUCTURE

Figures 5 and 6 show the contours of normalized time-averaged streamwise

velocity *u (= Uu / ) around the three cylinders for *

1L = 3.5 and 5.5 corresponding to

1 = inphase and antiphase, respectively. The same increment in the contours is used

to facilitate a comparison. The enclosed wake bubble with *u = 0 is identified as the

recirculation bubble (Alam and Zhou 2007; Alam et al. 2011). Obviously, the wake

3.5 4.0 4.5 5.0 5.50.47

0.48

0.49

0.50

3.5 4.0 4.5 5.0 5.51.00

1.02

1.04

1.06

1.08

3.5 4.0 4.5 5.0 5.50.25

0.30

0.35

0.40

0.45

3.5 4.0 4.5 5.0 5.50.94

0.96

0.98

1.00

1.02

3.5 4.0 4.5 5.0 5.50.45

0.46

0.47

0.48

3.5 4.0 4.5 5.0 5.50.54

0.55

0.56

0.57

3.5 4.0 4.5 5.0 5.50.4

0.45

0.5

0.55

0.6

3.5 4.0 4.5 5.0 5.51.12

1.14

1.16

1.18

1.20

CLf

*

2L *

2L *

2L

CLf

1 32 1 32 1 32

3.5 4.0 4.5 5.0 5.50.30

0.35

0.40

0.45

0.50

CLf

(a2)(a1) (a3)

(b1) (b2) (b3)

(c1) (c2) (c3)

(c)

(b)

(a)

bubble size is sensitive to *

1L and *

2L , as are DC and LfC . A couple of observations

can be pointed out here. Firstly, at a given *

1L , from cylinders 1 to 3, (i) the lateral and

streamwise sizes of the wake bubble shrink and elongate, respectively, and (ii) *u on

both front and side surfaces of the cylinder gets smaller which can be seen from the

concentration of contour lines. The observation is consistent with the variation in DC

from cylinders 1 to 3. Secondly, as *

2L increases from 3.6 to 5.5, the maximum *u

marked on the side surface of cylinder 1 increases and dwindles for *

1L = 3.5 and 5.25,

respectively. It is the reason explaining why DC for the cylinder 1 grows for *

1L = 3.5

(Fig. 3a1) but drops for *

2L = 5.25 (Fig. 3c1).

Fig. 5 Contours of time-averaged streamwise velocity *u for L1

* = 3.5 with change in L2*

as (a) L2* = 3.6, (b) L2

* = 4.5, and (c) L2* = 5.5. The number in the first cylinder

represents the maximum value of streamwise velocity.

(a)

(b)

(c)

1.278

1.280

1.283

y*

y*

y*

x*x* x*

Fig. 6 Contours of time-averaged streamwise velocity *u for L1

* = 5.25 with change in

L2* as (a) L2

* = 3.6, (b) L2* = 4.5, and (c) L2

* = 5.5. The number in the first cylinder represents the maximum value of streamwise velocity.

Both *

1L and *

2L play significant roles in the intrinsic features of the flow.

However, the *

1L effect is more significant than *

2L . The difference of their roles can be

illustrated in Fig. 7 where maximum streamwise velocity *

maxu on the side surface of

cylinder 1 is plotted as a function of *

1L and *

2L . For a given *

2L , *

maxu grows with *

1L .

The growth is, however, smaller between *

1L = 4.5 and 5.25 than between *

1L = 3.5

and 4.5. On the other hand, *

maxu with *

2L varies oppositely for *

1L = 3.5 and 5.25,

swelling for the former *

1L but dwindling for the latter. The variation is conspicuously

(a)

(b)

1.311

1.310

1.310

(c)

y*

y*

y*

x*x* x*

connected to 1 and 2 as marked in the figure. As observed, for a given *

1L , a low *

maxu

corresponds to 1= 2= inphase ( *

1L = 3.5, *

2L = 3.6) or antiphase ( *

1L = 5.25, *

2L =

5.5), while a high *

maxu complements 1 = inphase, 2 = antiphase ( *

1L = 3.5, *

2L = 5.5)

or vice versa ( *

1L = 5.25, *

2L = 3.6). Interestingly, these conditions are the same for

low and high DC . The relationship between St and *

2L at different *

1L is shown in Fig.

8, reflecting the same view as observed in the *

maxu variations (Fig. 7).

Fig. 7 Dependence on L1* and L2

* of *

maxu on the side surface of cylinder 1.

Fig. 8. Dependence of St on L1

* and L2*.

3.5 4.0 4.5 5.0 5.5

1.280

1.285

1.290

1.295

1.300

1.305

1.310

*

maxu

*

2L

L1* = 3.5

= 4.5

= 5.25

1 = inphase

2 = inphase

1 = inphase

2 = antiphase

1 = antiphase

2 = inphase1 = antiphase

2 = antiphase

3.5 4.0 4.5 5.0 5.50.165

0.170

0.175

0.180

= 3.5

= 4.5

= 5.25

St

*

2L

*

1L

Figures 9 and 10 show the time-mean pressure coefficient PC contours for *

1L =

3.5 and 5.25, respectively, at different *

2L . PC on both front and side surfaces retreats

from cylinder 1 to cylinder 3, again consistent with *u contours and DC variations.

Paying attention to PC on the side surface of cylinder 1, we can see that the minimum

PC for *

1L = 3.5 decreases as *

2L increases from 3.6 to 5.5 (Fig. 9), while that for *

1L

= 5.25 augments (Fig. 10).

Fig. 9. Contours of CP around the three cylinders for L1

* = 3.5 with change in L2* as (a)

L2* = 3.6, (b) L2

* = 4.5, and (c) L2* = 5.5. The number in the first cylinder represents the

minimum value ofCP .

(a)

-1.236

(b)

-1.241

(c)

-1.247

x* x* x*

y*

y*

y*

Fig. 10. Contours of CP around the three cylinders for L1

* = 5.25 with change in L2* as

(a) L2* = 3.6, (b) L2

* = 4.5, and (c) L2* = 5.5. The number in the first cylinder represents

the minimum value ofCP.

The minimum pressure coefficient min,PC on the side surface is extracted and its

relationship with *

1L and *

2L are presented in Fig. 11. Evidently, a larger *

1L complements a negatively increased min,PC regardless of *

2L . As DC , *

maxu , St and

min,PC are interconnected, min,PC is also reliant on 1 and 2 accordingly. Therefore, for

a given *

1L , the condition 1= 2= inphase or antiphase corresponds to lower

magnitudes of DC , *

maxu , St and min,PC , while the other condition 1= inphase, 2=

antiphase or vice versa yields higher magnitudes of DC , *

maxu , St and min,PC .

(a)

(b)

(c)

x* x* x*

-1.313

-1.311

-1.310y*

y*

y*

Fig. 11 Dependence on L1

* and L2* of minimum time-averaged pressure coefficient on

the side surface of cylinder 1.

6. EFFECTS OF 1 AND 2 ON COHERENT FLOW FLUCTUATION

Both 1 and 2 have a substantial influence on the mean flow as discussed above. So have they on the fluctuating flow as identified in the CLf variations (Fig. 4).

The effect of 1 and 2 on the fluctuating flow field or a parameter can be understood from the instantaneous fluctuating field that can be obtained as the instantaneous field minus mean field, for instance, instantaneous fluctuating streamwise velocity

***~ uuu where u* is the instantaneous streamwise velocity. In other words, *~u is

the instantaneous streamwise fluctuating velocity where the datum is the mean velocity.

Since our focus is to understand how 1 and 2 affect the flow on the upper and lower

sides of the upstream cylinder, *~u contours at the instant corresponding to the

minimum (downward) lift force of cylinder 1 at different L* are compared in Fig. 12 for

cylinder 1 only. Negative and positive *~u on the upper and lower sides, respectively,

are the signatures of a smaller u* on the upper side and a higher u* on the lower side. Here one should not be confused with negative (reverse flow) velocity on the upper side. As the vortex shedding occurs from the two sides in an alternating fashion, the flow velocity with reference to the time-mean flow will be oscillating back and forth. The red and blue numbers in the cylinders represent the maximum and minimum values of

*~u in the wake. Interestingly, given *

1L = 3.5, while the maximum *~u on the lower side

decreases with *

2L , the minimum *~u on the upper side increases. The opposite trends

prevail at *

1L = 5.25, while a mixed behaviour exists at *

1L = 4.5. In order to clearly

explore the trends, the maximum and minimum *~u (i.e., *max

~u and *min

~u , respectively)

are plotted against *

2L for each *

1L as shown in Fig. 13(a-c). With 1 = inphase ( *

1L =

3.5 4.0 4.5 5.0 5.5-1.32

-1.31

-1.30

-1.29

-1.28

-1.27

-1.26

-1.25

-1.24

-1.23

= 3.5

= 4.5

= 5.25

1 = inphase

2 = inphase1 = inphase

2 = antiphase

1 = antiphase

2 = inphase

1 = antiphase

2 = antiphase

min,PC

*

2L

*

1L

3.5), as 2 changes from inphase ( *

2L = 3.6) to antiphase ( *

2L = 5.5), *max

~u and *min

~u

shrink and increase, respectively (Fig. 13a). On the contrary, given 2 antiphase ( *

1L

= 5.25), when 2 = inphase ( *

2L = 3.6) takes turns with 2 = antiphase ( *

2L = 5.5), they

show opposite trends (Fig. 13c). In a nutshell, a higher magnitude of *max

~u and *min

~u

results from a condition 1= 2= inphase or antiphase, and a lower magnitude of them is

accompanied by the other condition 1= inphase, 2= antiphase or vice versa. Similar

observation is made on maximum fluctuating cross-stream velocity *

max

~v and *

min

~v .

Fig. 12. Contours of

***~ uuu around and behind cylinder 1 corresponding to the

instant at which cylinder 1 is subjected to the minimum lift (downward). (a-c) L1* = 3.5;

L2* = 3.6, 4.5 and 5.5. (d-f) L1

* = 4.5; L2* = 3.6, 4.5 and 5.5. (g-i) L1

* = 5.25; L2* = 3.6, 4.5

and 5.5. The red and blue numbers in the cylinders represent the maximum and minimum values in the wake.

-0.45-0.35-0.15

-0.05

0.450.35

0.250.150.0

5

-0.35

-0.25-0.15-0

.05

0.4

0.3

0.20.1

0

-0.25

x*

y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(a)

0.450.35

-0.45-0.35-0.25

-0.15

-0.05

0.4

0.3

0.20.1

0

-0.35

-0.25-0.15-0.0

50.250.15

0.05

x*

y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(b)

-0.4-0.3-0.2-0.1

-0.35

0.450.35

0.250.150.05

-0.4-0

.3

-0.2-0.1

0

0.5

0.40.3

0.20.1

0

x*

y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(d)

0

-0.4-0.3-0.2-0.1

0.45

0.350.25

0.150.0

5

0.40.3

0.20.1

-0.3

5

-0.2

5

-0.1

5-0.0

5

x*

y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(e)

0.4

0.3

0.20.1

-0.4-0.3-0.2-0.1 0.5

0.40.3

0.2

0.1

0

-0.4-0

.3

-0.2

-0.1

0

x*

y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(f)

-0.3-0.2-0.1

0.350.25

0.150.05

-0.4

5-0.35

-0.25

-0.0

5

0.5

0.40.3

0.2

0.1

0

-0.15

x*

y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(g)

-0.35-0.15

-0.05

0.5

0.40.3

0.2

0.1

0

0.4

0.3

0.20.1

-0.4-0.3

-0.2-0.1

0

-0.25

x*

y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(h)

0

0.40.3

0.20.1

-0.35

-0.3

-0.25-0.15

-0.05

0.5

0.4

0.2

0.1

0

-0.4

-0.3

-0.2-0.1

0

0.3

x*

y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(i)

0.15

-0.45-0.35-0.25

-0.15

-0.05

0.450.35

0.250.150.05

-0.35

-0.25-0.15-0.0

5

0.45

0.35

0.25

0.0

5

x*y*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(c)

-0.446

0.451

-0.396

0.416

-0.438

0.448

-0.444

0.447

-0.475

0.467

-0.485

0.474

-0.395

0.413

-0.500

0.484

-0.399

0.420

L1

* =

3.5

L1

* =

4.5

L1

* =

5.2

5

L2* = 3.6 L2

* = 4.5 L2* = 5.5

(a) (b)(c)

(f)(e)(d)

(g) (h) (i)

Fig. 13. Dependence on L2* of maximum and minimum *~u (i.e.,

*max

~u and *min

~u ),

extracted from figure 12, on the upper and lower sides of cylinder 1. (a) L1* = 3.5, (b) L1

*

= 4.5 and (c) L1* = 5.25.

3.5 4.0 4.5 5.0 5.50.41

0.42

0.43

-0.41

-0.40

-0.39

3.5 4.0 4.5 5.0 5.50.44

0.45

0.46

0.47

-0.45

-0.44

-0.43

-0.423.5 4.0 4.5 5.0 5.5

0.46

0.47

0.48

0.49

-0.51

-0.50

-0.49

-0.48

-0.47

*

min

~u*

max

~u

, lower side

, upper side

(b)

*

max

~u*

min

~u

*

max~u

*

min~u

(c)

*

max

~u *

min

~u

(a)

*

2L

In nature, fish schooling or birds flying in a group may provide some advantages in their cruise, improving the efficiency, where the rear ones may get hydrodynamic advantages of the wakes of the leading ones while the leading ones may also utilize the hydrodynamic advantages through phase lags between their sheddings. It would be interesting to investigate a series of fish-like structures to see whether fishes or birds exploit this technique as most creatures are naturally optimized. 7. CONCLUSIONS

A numerical simulation of the flow around three circular cylinders at Re = 200 is

conducted. The *

1L , and *

2L are varied systematically in the ranges 3.5-5.25 and 3.6-

5.5, respectively, so that four possible extreme conditions (1 = inphase, 2 = inphase),

(1 = inphase, 2 = antiphase), (1 = antiphase, 2 = inphase) and (1 = antiphase, 2 =

antiphase) are achieved. The focus is given on how 1 and/or 2 influence on DC , LfC ,

St, and flow structures, including *

maxu , min,PC , *

max~u , *

min~u , *

max~v , and *

min~v . The

detailed investigation supports the following conclusions.

Forces on and St of the cylinders are more sensitive to *

1L than to *

2L , while

both 1 and 2 have a great influence on cylinder 1 than on the other two. In the ranges

of *

1L and *

2L examined, cylinder 1 experiences more than three and fifteen times

higher DC than cylinders 2 and 3, respectively. This is because, from cylinders 1 to 3, *u on the front surface of cylinder gets smaller and the wake bubble shrinks and

elongates in the lateral and streamwise directions, respectively. On the other hand,

LfC on cylinder 2 is the largest, more than two times that on cylinder 1 or cylinder 3,

which is attributed to the strong impingement of the first gap vortices on cylinder 2.

Given 1 = inphase ( *

1L = 3.5), a modification of 2 from inphase ( *

2L = 3.6) to

antiphase ( *

2L = 3.6) leads to (i) DC growing, (ii) LfC decaying, and (iii) St dwindling

for cylinder 1. Opposite behaviours of DC , LfC , and St are observed when 2 transmutes

from inphase ( *

2L = 3.6) to antiphase ( *

2L = 5.5) with 1 = antiphase ( *

1L = 5.25). The

opposite behaviour is linked to the relationship between 1 and 2. An identical phase

condition (1 = 2 = inphase) corresponds to a higher LfC and a lower DC , St, *

maxu

and min,PC magnitudes, while an opposite phase condition (1= inphase/antiphase and

2= antiphase/inphase) complements a smaller LfC and a higher DC , St, *

maxu and

min,PC magnitudes.

At 1= 2= inphase, when shear layers from cylinders 2 and 3 grow/accelerate, they, because of the low pressure, pull the fluid at the same side of cylinder 1, and add

to the flow/shear-layer velocity on the same side, leading to larger magnitudes of *max

~u ,

*min

~u , *max

~v and *min

~v . This is the reason explaining why LfC is enhanced for this

condition. When 1 = inphase and 2 changes from inphase to antiphase, magnitudes

of *max

~u , *min

~u , *max

~v and *min

~v all shrink because the contribution of cylinder 3 shifts

from the same side to the opposite side. As such, LfC follows suit. On the other hand,

given 1 = antiphase and 2 changes from inphase to antiphase, magnitudes of *max

~u ,

*min

~u , *max

~v and *min

~v all grow because the contribution of cylinder 3 shifts from the

opposite side to the same side, contributing to the growth of LfC .

ACKNOWLEDGMENT

Support from Deanship of Scientific research of KFUPM through grant IN151026 is acknowledged. The contribution of Ma Zhe to the simulation is also gratefully acknowledged. REFERENCES Alam, M.M. (2014), “The aerodynamics of a cylinder submerged in the wake of another”,

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