Influence of staggered position of a square cylinder of different height on the wake of downstream square cylinder near a plan wall

*Rajesh Bhatt1), Dilip K. Maiti2) and Md. Mahbub Alam3)#

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

Graduate School, Harbin Institute of Technology, Shenzhen, China 2) Department of Applied Mathematics with Oceanology and Computer Programming,

Vidyasagar University, Midnapur 721102, India #alamm28@yahoo.com

ABSTRACT A numerical study is conducted on two-dimensional flow over two staggered cylinders at a Reynolds number of 100. The governing equations are solved numerically through a pressure-correction-based iterative algorithm (SIMPLE) with the quadratic upwind interpolation for convective kinematics (QUICK) scheme for convective terms. The incident flow is a linear shear flow. The downstream cylinder is square, of height A, positioned near a wall with a gap height of 0.5A. The upstream cylinder is also square, but of different sizes, with height ratio a* = a/A = 1 and 0.5, respectively. In order to investigate the effect of position and a* of the upstream cylinder on the flow around and aerodynamic forces acting on the downstream cylinder, the gap height between the wall and upstream cylinder L*(= L/A = 0.1-3.0) and the streamwise spacing between the cylinders S*(=S/A= 0.5-7.0) are varied systematically. The flow structure around the downstream cylinder is highly dependent on L* and a*, experiencing several discontinuities. For the same L*, the interaction between vortices from the two cylinders is weaker for a* = 0.5 than for a* = 1.0, as such very small peaks are observed in the spectra of the lift coefficient of downstream cylinder for a* = 0.5. NOMENCLATURE a :height of the upstream square cylinder (m) A :height of the downstream square cylinder (m) L* :nondimensional gap height from upstream cylinder to plane wall S* :nondimensional gap spacing between the cylinders Re :Reynolds number (UA/ν) CD :normalized time averaged drag coefficient CL :normalized time averaged lift coefficient CL :rms of lift coefficient St :Strouhal number U :velocity at height A from the wall (m/s) V : kinematic viscosity of fluid

PC : time averaged pressure coefficient

1. INTRODUCTION Flow past cylinders is found in many practical engineering applications such as, compact heat exchangers, cooling of electronic components, drying of different materials (textiles, veneer, paper and film materials), cooling of glass, plastics and industrial devices, and so on. In order to accurately design the devices and structures, forces and interference effect of the neighbouring structures should be well understood. Both position and shape of neighbouring structure affect the flow pattern around a structure. For two tandem cylinders, the downstream cylinder is submerged in the wake of the upstream one. However, when the upstream cylinder is offset with respect to the downstream cylinder, vortex shedding mechanism and the force characteristics differ significantly from the inline case. There is a significant volume of articles on the flow around a single and tandem square cylinders. To our knowledge, there is no article available on the flow around a square cylinder in presence of the upstream cylinder of different heights. In the present study, square cylinders are chosen because they can easily be given a Cartesian grid system with higher accuracy. This study is important for many applications mentioned above, providing also some fundamental information of the flow around cylinders in the staggered arrangement. Studies on the problems of wake development and vortex shedding behind a rectangular cylinder in free-stream flows were investigated both numerically and experimentally by Davis and Moore (1982), Franke et al. (1990) and Okajima et al. (1990). When the cylinder is placed in the proximity of a solid wall, the strength of the upper and lower shear layers separated from the surfaces of the cylinder is not identical and the vortex shedding pattern is therefore distorted. The effect of wall proximity on the flow around a single cylinder was investigated by Martinuzzi et al. (2003), Bhattacharyya and Maiti (2004), Mahir (2009), Harichandan and Roy (2012) and Maiti (2012). Bhattacharyya and Maiti (2004) observed that the vortex shedding frequency from a square cylinder near a wall is higher for a shear flow than the uniform flow. The dependence of flow characteristics of a rectangular cylinder near a wall on the incident shear flow velocity and gap height has been reported in the previous studies (Maiti, 2011, 2012). The flow around two staggered circular cylinders has been extensively studied experimentally (Sumner (2000), Alam et al. (2005), Zhou et al. (2009)) and numerically (Lee et al. (2009), Akbari and Price (2005)). Nine flow patterns are identified, and processes of shear layer reattachment, induced separation, vortex pairing and synchronization, and vortex impingement, are elucidated by Sumner (2000). Akbari and Price (2005) identified five flow patterns, depending on the geometrical arrangement of the cylinders at a subcritical Reynolds number (Re) of 800. Lee et al. (2009) numerically identified ten distinct flow patterns for two circular cylinders at Re≤160. Alam et al. (2017) investigated flow around four side-by-side circular cylinders at Re =100. They identified four flow regimes (single bluff-body, flip-flopping, quasi-interlocked, and interlocked flow) depending on the gap spacing between the cylinders.

Balachandar and Parker (2002) considered the onset of vortex shedding in a periodic array of rectangular cylinders in both inline and staggered arrangement. They observed that the interaction between the cylinders becomes negligible when the transverse spacing between the cylinders is large enough and the critical Re approaches that of an isolated cylinder. Niu and Zhu (2006) in a numerical study of three-dimensional flow around two identical square cylinder in staggered arrangements evaluated the dependence of aerodynamic characteristics on the streamwise spacing. Sewatkar et al. (2009) performed a numerical study of the flow around a row of nine square cylinders placed normal to the oncoming flow. They identified different flow regimes based on the gap spacing of cylinders and proposed that wake interaction is strongly influenced by the jets in the gap region. Chatterjee et al. (2010) performed a numerical study for the flow around five square cylinders placed side-by-side and normal to the oncoming flow at Re = 150. Burattini and Agarwal (2013) with change in spacing between two side-by-side square cylinders observed various flow structures. Chern et al. (2010) observed a jet-like structure when oscillatory flows are given through the gap of two side-by-side cylinders. Rosales et al. (2001) compared results of the inline and offset square cylinder pairs at a fixed spacing ratio of 2. Their study revealed pronounced differences in the unsteady flow behaviour between the inline and offset pairs. Devarakonda (1994) examined the case of offset square cylinders exposed to a uniform flow at a fixed spacing ratio of 3 and observed that the Strouhal number is larger than that for an inline cylinders. Malekzadeh and Sohankar (2012) reported three major flow regimes depending on the height and position of a control plate upstream of a square cylinder. Bao et al. (2012) showed six different flow patterns, which appeared successively with the increase of spacing between six inline square cylinders. For a pair of inline square cylinders, Bhattacharyya and Dhinakaran (2008) observed that the onset of vortex shedding occurs for Re beyond 125 for all spacing (0.5- 5) examined. Maiti and Bhatt (2014a, 2014b) extended the above study considering the upstream cylinder as rectangular shape of different heights and widths in a tandem arrangement. It is reported that the transition (-from unsteady/steady to steady/unsteady-) of the flow over the upstream/downstream cylinder strongly depends on the shape and the position of the upstream cylinder. The above literature review indicates that shedding frequencies, aerodynamic forces and wakes of two or a group of cylinders are dependent on the inter-cylinder spacing, flow incidence angle and the cylinder shape. Although there are a number of studies on single cylinder and pair of cylinders, the flow past a square cylinder in presence of another square cylinder in staggered or tandem arrangement has not received much attention, especially for the case of incident shear flow. Here the objective of the work is to examine the wake flow structure, aerodynamic forces, and shedding frequency of a square cylinder lying in the wake an upstream square cylinder where the height and position of the upstream cylinder are varied. The effect of inter-cylinder spacing, shape of upstream cylinder, and gap height of upstream cylinder from a plan wall on the flow characteristics of the downstream square cylinder are investigated for uniform shear flow. Simulations have been performed for two different heights of the upstream

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uniform shear flow the flow field is similar to a single bluff body flow. For the inline tandem arrangement the flow remains steady from both the cylinders and the recirculation length behind the downstream cylinder is increased. With the increase of gap height (L* ≈ 1.0) for lower S*, the shear layers either wraps around or reattaches on the downstream cylinder. The positive shear layers from the upstream cylinder are enveloped by the negative shear layer of both the cylinders and then forced the negative shear layers pf the downstream cylinder interact with positive counterpart. If one compares the formation of vortices from the cylinders with the larger S*, it is clearly perceived that the upstream cylinder vortices shed in the gap between the cylinders. Further increase in L* results small positive vortices from the wake of the downstream cylinder for intermediate spacing. When the cylinder is located in the higher velocity zone a wider wake is formed behind the cylinder. The negative vortices from the upstream cylinder has less impact on the shear layers of the downstream cylinder. However, the positive vortices from the upstream cylinder hit the downstream cylinder and then roll. As results, the positive vortices are stretched towards the wake of the cylinder. The strength of the negative shear layers around the downstream cylinder is effectively reduces. The negative vortices found in the wake are mainly from the upstream cylinder. For L* ≥ 2, the vortices shed from the upstream cylinder evolves in the vertical direction. Higher the gap height of upstream cylinder from the wall, the more elongation of the positive shear layers from the upstream cylinder towards downstream. The shear layers do not roll-up completely and the effect of the vortices on the downstream cylinder reduces. The vortices behind the downstream cylinder are fairly close to each other. They interact with each other, forming a very complex wake. Such vortex interaction results in more complex wake interference around the downstream cylinder. The vortex shed from the upstream cylinder attacks the front face of the downstream cylinder at lower L*. However, at higher L* instead of impinging on the front face, vortices pass through the upper surface of the downstream cylinder. Fig. 5 shows the effect of reduction of height of the upstream cylinder on flow structure for different S* and L*. The effect of decreasing the height of the upstream cylinder is clearly visible from the vorticity contours. The flow pattern for the case of a*=1.0 is more complex in comparison to the case of a*=0.5. The positive and negative vortices shed alternately from the upstream cylinder and consequently move downstream. It is noticed that with reduction in height the elongated vortices are reduced. With the increase of L*, the positive vortices are more elongated and dominate the wake. The negative vortices from the upstream cylinder are fairly round whereas positive vortices being oblate in shape. With the increase in L* the interaction between the vortices decreases. The wake flow of the upstream cylinder of square shape is similar to the wake of a single square cylinder under the shear flow (Saha et al. (2003) and Cheng et al. (2005)).

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Fig. 8. C(b) a*=0 5.3 Time To undeof the daverage10 (CL). isolated shows qvariation(S*= 3)decreasthe valupattern, negativeand it invortices isolated downstr

Fig. 9. Ccylinder

Contours o0.5 at differ

e average

erstand thedownstreamed drag CD

The averacylinder v

qualitativeln of CD wi, and mines nominae of a singwhen the

e gauge prncreases w(ref. Fig. 4value for

eam cylind

Contours of: (a) a*=1.

of RMS of lrent L* and

ed forces

e effect of m cylinderand lift CL

aged forcevalues at ly similar th L* and nimum forally up to ggle isolatede vortex shressure is with gap h4 and 5). Fall a* and

der can be

f normalize.0 and (b) a

ift coefficied S*.

position ar, contour L coefficiene coefficienRe=100. Tbehavior a* is foun

r larger sap height

d cylinder. hedding frfound to a

height. It rFor offsettid S*. A ranoted from

ed time avea*=0.5 at d

ent of down

nd shape diagram

nts for diffents are noThe CD ofwith chan

nd maximupacings (L* = 0.5, thThis jump

rom the cyact on theresults to ng case n

ange of L*m Fig. (9).

eraged dradifferent L*

nstream cy

of the upsis present

erent a* at rmalized wf the down

nging the um upto thS* = 7). hen increa

p in CD corrylinder stae rear face

increase ear L* = 1at which

ag coefficie* and S*.

ylinder (CL’

stream cylited for thedifferent L

with the constream cygap heighhe intermeThe mea

ases very rresponds trts. As se

e of the doin the form.0, the CD negligible

ent (CD) of

’) : (a) a*=1

inder on the normaliz

L* in Fig.9 orrespondinylinder forht. Appareediate gapn drag co

rapidly andto a chang

een in Fig.ownstreammation of is very nedrag force

downstrea

1.0 and

he forces zed time (CD) and ng single r both a* ently, the

spacing oefficient

d crosses ge in flow . 11, the cylinder stronger

ear to the e on the

am

Similar cylinder = 7. HowdownstrreductioS* = 0.5values o

Fig. 10. cylinder

The timcylinder intermedFig. 11 faces ofundersuaffectedobserveedge. Aresults spressurethe increpresenccylinder upstreamapproacpressuresingle cyof presepressurelength a

to the CD,is mostly

wever, the eam cylindn in CL of

5. The dowof L*.

Contours : (a) a*=1.

e-averageat selecte

diate spac(left side :f the squa

urface. The by the pd that vari

At lower L* shift in the e distributiease in L*

ce of the uis reduc

m cylindeches to the distributiylinder pre

ence of thee distributiond strengt

, the behasame at represence

der in comthe downs

wnstream

of normaliz.0 and (b) a

d surface ed values oing (S* = 3 a* = 1.0 a

are cylindee PC distrpresence oation in L*≤ 0.25, thlocation ofon (betwe

*, indicatingupstream cced from r from ine isolatedion on the

essure dist upstreamon for diffeth of vortice

avior of CL

espective sof the ups

mparison tostream cylicylinder is

zed time aa*=0.5 at d

pressure of L* (cons3.0), and laand right ser with a sribution onof the ups*, the distrie PC distf the maximen the frog that the cylinder, ththe isolate

nline arrand value. Fe downstretribution. Itcylinder is

erent a* is es behind t

L coefficienspacing, fostream cylio its drag inder is fou

s found to

averaged lifdifferent L*

distributiosidering tharge spacside : a* =suction (−

n the downstream cybution of tribution is mum pressnt and readrag force

he pressureed single ngement a

For a* = 0eam cylindt is noted fs reduced associate

the cylinde

nt with a* or exampleinder has lcounterpa

und aroundbe free fro

ft coefficie* and S*.

n PC arouhree casesing (S*=7.

= 0.5). The− PC ) peak nstream cyylinder, rem

PC are biasame for

sure on thear faces ofe increasese on the f

cylinder and the w0.5, at larder is simfrom Fig. 1compared

ed with the er.

and L* oe, the effecess effect art for all d L* = 1 foom lift forc

nt (CL) of d

und the do : closely 0)) at vario

e pressure develop a

ylinder fromarkably fased towaboth a*. R

e front facef the cylinds with incrront face ovalue. Wwall the ge gap h

milar and a11 that for to a* = 1.0variation

n the dowct is minim

on lift forcS*. The m

or all S*, nce for som

downstream

ownstreamspaced (Sous L* is pis negativ

at the fronnt face is for a* = 1rds the top

Reduction e. The diffeder) increarease in L*of the dow

While offsetaverage eight the approachea* = 0.5 t

0. The diffein vortex f

wnstream um at S*

ce on the maximum otably at

me higher

m

m square S* = 0.5), plotted in ve on the nt of the strongly

1.0. It is p leading of height erence in ases with *. Due to wnstream tting the pressure average

es to the he effect erence in formation

Fig. 11. in absenS*. 6. CONC This stupresencproximityobserva ⊙ Th

poremunhigthe

⊙ Th

cylthemaspainc

Time-avernce and pre

CLUSIONS

udy presence of upstrey to a wtions are r

e wake flosition (L* amains steasteady flow

gher L*. Ree large stre

e pressurlinder incree average aximum atacing (S* ≈

creases ra

raged surfaesence of

S

nts the nueam squarwall underreported fro

ow of the dand S*) anady similarw from theeduction oetched vort

e differenceases withdrag force

t the inter≈ 7.0). Thepidly and

ace pressuupstream

merical inre cylinderr the inciom this stu

downstread height o

r to the sine downstref height oftices into re

ce betweeh increase e on the domediate s

e CD decreacrossed th

ure distribusquare cyl

nvestigationr of differenidence of

udy :

m square of the upstrngle isolateeam squarf upstreamegular vort

en the fronin the gap

ownstreamspacing (Sases for thhe isolated

ution arounlinder of di

n of flow ant heights f uniform

cylinder isream squaed cylinderre cylinder

m cylinder (tices.

nt and reap height (L

m cylinder. S* ≈ 3.0), he inline tad cylinder

nd the dowfferent a* f

around a in staggershear flo

s significanre cylinder

r, while incr and it bea* = 0.5) r

ar faces oL*) and it The effectwhile minindem posivalue as

wnstream cfor differen

square cyred arrangow. The

ntly affecter. For lowecrease in Lecomes coresults red

of the dowresults to t of L*, a* imum at tition (L* = 0L* increas

ylinder

nt L* and

ylinder in ement in following

ed by the er L* flow L* results mplex at

duction in

wnstream increase on CD is

the large 0.5) then ses. The

effect of size and position have negligible effect on the average forces of the downstream cylinder at large spacing.

⊙ For closely spaced the intensity of peak of spectra decreases with increase in L*

for both the values of a*. The shedding frequency increases with increase in L* and this value is more for the case of a*= 0.5. However, the CLrms value is more for a* = 1.0. Beyond certain L* these values either remain constant or decrease with increase in L*. It is therefore noticed that at large L* the presence of upstream cylinder is negligible.

ACKNOWLEDGMENTS The authors acknowledge financial assistance of DST (INDIA) (San No. 100/IFD/3613/2008-09) and UGC-BSR (INDIA) (letter no.F.4-1/2006(BSR)/7-203/2009) for carrying out this work. Alam wishes to acknowledge the support given to them from National Natural Science Foundation of China through Grants 11672096 and from Research Grant Council of Shenzhen Government through grant JCYJ20160531191442288. REFERENCES Akbari, M.H. and Price, S.J., (2005), “Numerical investigation of flow patterns for staggered cylinder pairs in cross-flow”, Journal of Fluids and Structures 20, 533–554. Alam, M.M., Sakamoto, H., and Zhou, Y., (2005), “Determination of flow configurations and fluid forces acting on two staggered circular cylinders of equal diameter in cross-flow”, Journal of Fluids and Structures 21, 363–394. Alam, M.M., Zheng, Q., and Hourigan, K., (2017), “The wake and thrust by four side-by-side cylinders at a low Re”, Journal of fluids and Structures, 70, 131- 144. Burattini, P., and Agrawal, A., (2013), “Wake interaction between two side-by-side square cylinders in channel flow”, Computers and fluids, 77, 134–142. Balachandar, S., and Parker, S.J., (2002), “Onset of vortex shedding in an inline and staggered array of rectangular cylinders”, Physics of fluids 1070- 6631/2002/14(10)/3714/19/19.00. Bhattacharyya, S. and Maiti, D.K. (2004), “Shear flow past a square cylinder near a wall”, In. Journal of Engineering Science 42, 2119-2134. Bhattacharyya, S., and Maiti, D.K., (2005), “Vortex shedding from a square cylinder in presence of a moving ground”, Int. J. Numer. Meth. Fluids 48, 985-1000. Bhattacharyya, S., Maiti D.K, and Dhinakaran S., (2006), “Influence of buoyancy on vortex shedding and heat transfer from a square cylinder in wall proximity”, Numer heat transfer: Part A, 50, 585–606. Bhattacharyya, S. and Dhinakaran, S. (2008), “Vortex shedding in shear flow past tandem square cylinders in the vicinity of a plane wall”, Journal of Fluids and Structures 24, 400-417. Bao, Y., Qier, W., and Zhou, D., (2012), “Numerical investigation of flow around

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