Research Article Large-Scale Computations of Flow around...

Research ArticleLarge-Scale Computations of Flow around Two Cylinders bya Domain Decomposition Method

Hongkun Zhu1 Qinghe Yao1 and Hiroshi Kanayama2

1School of Engineering Sun Yat-sen University Guangzhou 510275 China2Department of Mathematical and Physical Sciences Japan Womenrsquos University Tokyo 112-8681 Japan

Correspondence should be addressed to Qinghe Yao yaoqhemailsysueducn

Received 25 September 2015 Revised 31 December 2015 Accepted 28 February 2016

Academic Editor Uchechukwu E Vincent

Copyright copy 2016 Hongkun Zhu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A parallel computation is applied to study the flow past a pair of cylinders in tandem at Reynolds numbers of 1000 by DomainDecomposition Method The computations were carried out for different sets of arrangements at large scale The modeling bydomain decomposition was validated by comparing available well-recognized results Two cylinders with different diameters werefurther investigated for different diameter ratios the wake width ratio and some properties of the critical space ratio that dominatesthe flow regime were discoveredThis result has important implications on future industrial application efforts as well as codes andstandards related to the two-cylinder structure

1 Introduction

Cylinder is one of the most common structures in engi-neering such as piers and chimney the struts of offshoreplatform the pipe of condenser and more When fluidflows through the cylinders the shedding vortex may causeinterference effect This effect may lead to the vibration ofthe cylinder structures or fatigue of the materials and as aresult the structure may be destroyed Due to its importancein engineering applications flow past two cylinders has beenstudied by experimental and numerical investigations for sev-eral decades [1] As early as 1977 Zdravkovich classified thecharacteristic of flow past two cylinders in tandem into threeregimes at low Reynolds number (Re) For more complicatedapplications Wu et al [2] studied two cylinders in tandemwith wind tunnel and water tunnel at the Re of 1000 Theflow visualization in the water tunnel showed the existence ofstreamwise vortices in spanwise direction To make out therelationship between different arrangements of two cylindersand Re Mittal et al [3] studied incompressible flows past twocylinders in tandem and staggered arrangements (Re = 100and Re = 1000) by a stabilized finite element method (FEM)Jester and Kallinderis [4] investigated the incompressibleflow about fixed cylinder pairs numerically and cylinderarrangements include tandem side-by-side and staggered

at Reynolds numbers of 80 and 1000 Hysteresis effectsand bistable biased gap flow in tandem arrangements werereproduced Different combinations of Re and arrangementsof cylinders have been reported by many [5ndash7] as is reportedin [8ndash17] and summarized by Sumner [1] most of the earlyresearches on this problem contribute to the flow structuresinduced by different spacing ratios (119871119863) Re and variantcylinder shapes

Recently there has been a lot of interest on the diameterratio (119889119863) of two cylinders in tandem Zhao et al studiedturbulent flow past two cylinders with different diametersnumerically The hydrodynamic force and vortex sheddingcharacteristics were proved to depend on the relative positionof small cylinders around the main cylinder [18] MahbubAlam and Zhou investigated the Strouhal number hydro-dynamic forces and flow structures and vortex sheddingfrequency of flow past two cylinders in tandemwith differentdiameters in wind tunnel [19] The diameter of upstreamcylinder varied from 024 to 1 of the downstream cylinderdiameter and the distance between two cylinders remains55 times of the diameter of the upstream cylinder Ding-Yong et al conducted the simulation of different spacingratios and different diameter ratios at Re = 200 usingFluent [20] Besides two cylinders Zhang et al determinedthe influence of the diameter ratio on the flow past three

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4126123 8 pageshttpdxdoiorg10115520164126123

2 Mathematical Problems in Engineering

cylinders in two dimensions by applied finite elementmethod[21]

As far as we know the flow structure of two cylindersin tandem affected by the changing of diameter ratio andspacing ratio is still uncertain and the computation scale ofnumerical experiments published research is very limited Toinvestigate the flow regime more concretely and more subtlylarge-scale simulations by Domain Decomposition Method(DDM) which is considered to have better accuracy and lesstime cost when comparing with conventional methods areimplemented [22 23]The large-scalemodelingwas validatedby comparing with othersrsquo reports for two cylinders of thesame diameter with different spacing ratios Moreover tostudy its influence to the flow past two cylinders in tandemmore comprehensively the flow structures at specified diam-eter ratios (119889119863 = 05 and 119889119863 = 1) with different spacingratios (119871119863 3ndash6) are investigated The critical spacing ratiothat affects the flow regime is expected to be determined fordifferent diameter ratios

This paper is organized as follows Section 2 introducesthe governing equations to be solved as well as the domaindecompositionmethod (DDM) In Section 3 the models andthe boundary conditions are described in detail Numericalresults for different spacing ratio and different diameter ratios(119889119863 = 1 and 119889119863 = 05) are present and comparedwith othersrsquo work in Section 4 At last Section 5 draws theconclusions of this work

2 Formulations

21 Governing Equations Let 120597Ω be the boundary of a three-dimensional polyhedral domain Ω 1198671(Ω) is the first orderSobolev space and 1198712(Ω) is the space of 2nd power summablefunctions on Ω Under the assumption that the flow field isincompressible viscous and laminar the solving of themodelcan be summarized as finding (119906 119901) isin 1198671(Ω)3 times 1198712(Ω) suchthat for any 119905 isin (0 119879) the following set of equations hold [24]

120597119905119906 + (119906 sdot nabla) 119906 minus

1

120588nabla sdot 120590 (119906 119901) =

1

120588119891

nabla sdot 119906 = 0

in Ω times (0 119879)

(1)

where 119906 is the velocity [ms] 119901 is the pressure [Pa] 120588 is thedensity (const) [kgm3] 119891 is the body force [Nm3] 120590(119906 119901)is the stress tensor [Nm2] defined by

120590119894119895(119906 119901) equiv minus119901120575

119894119895+ 2120583119863

119894119895(119906)

119863119894119895(119906) equiv

1

2(120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

)

119894 119895 = 1 2 3

(2)

with the Kronecker delta 120575119894119895and the viscosity 120583 [kg(msdots)]

An initial velocity 1199060is applied in Ω at 119905 = 0 Dirichlet

boundary conditions

119906 = on Γ times (0 119879) (3)

R3

u(X t)

X(tnminus1) asymp X1(unminus1 Δt) X(tn) = x

Figure 1 A characteristic finite element scheme

and Neumann boundary conditions

3

sum

119895=0

120590119894119895119899119895= 0 on 120597Ω Γ times (0 119879) (4)

are also applied where Γ sub 120597Ω and 119899 is the outward normaldirection to 120597Ω

22 Characteristic Finite Element Scheme Using the defini-tion

119905119899equiv 119899Δ119905

119873119879equiv [

119879

Δ119905]

(5)

a characteristic finite element scheme approximates thematerial derivative in (1) at 119905119899 as follows [23]

120597119905119906 + (119906 sdot nabla) 119906 asymp

119906119899minus 119906119899minus1

(1198831(119906119899minus1

Δ119905))

Δ119905 (6)

where1198831(sdot sdot) is a position function see Figure 1

Let Iℎequiv 119870 denote a triangulation of Ω consisting

of tetrahedral elements and the subscript ℎ denotes therepresentative length of the triangulation The finite elementspaces are defined as follows

119883ℎequiv Vℎisin 1198620(Ω)3 Vℎ

1003816100381610038161003816119870isin 1198751(119870)3 forall119870 isin I

ℎ

119872ℎequiv 119902ℎisin 1198620(Ω) 119902

ℎ

1003816100381610038161003816119870isin 1198751(119870) forall119870 isin I

ℎ

119881ℎ() equiv V

ℎisin 119883ℎ Vℎ(119875) = (119875) forall119875 isin Γ

119881ℎequiv 119881ℎ(0)

119876ℎ= 119872ℎ

(7)

where the 1198620(Ω)3 in (7) denotes the continuous functionon Ω in three dimensions 119875

1(119870)3 denotes the first order

polynomial defined by 119870 in three dimensionsNote that piecewise linear interpolations are employed

for velocity and pressure (see (7) and Figure 2) which doesnot provide a sufficient condition to link the velocity andpressure space therefore the so-called inf-sup condition [25]should be satisfied In previous work [26] a penalty Galerkinleast-squares (GLS) stabilization method for pressure [27]

Mathematical Problems in Engineering 3

0

1

2

3

VelocityPressure

Figure 2 A tetrahedron element

was employed and it was found difficult to be applied for thesimulation of complex flows especially when the flow is veryturbulent the schemebecomes easy to divergeA stabilizationtechnique for the saddle point problem is employed and thefinite element scheme for (1) read as follows [22]

Find (119906119899ℎ 119901119899

ℎ) isin 119881ℎ(119892) times 119876

ℎ119873119879

119899=1such that

(

119906119899

ℎminus 119906119899minus1

ℎ(1198831(119906119899minus1

ℎ Δ119905))

Δ119905 Vℎ) + 119886 (119906

119899

ℎ Vℎ)

+ 119887 (Vℎ 119901119899

ℎ) + 119887 (119906

119899

ℎ 119902ℎ)

+1

1205882sum

119870isinIℎ

120591119870(nabla119901119899

ℎ minusnabla119902ℎ)119870

=1

120588(119891 Vℎ) +

1

1205882sum

119870isinIℎ

120591119870(119891 minusnabla119902

ℎ)119870

(8)

Let (sdot sdot) defines the 1198712inner product the continuous

bilinear forms 119886 and 119887 in (8) are introduced by

119886 (119906 V) equiv2120583

120588(119863 (119906) 119863 (V)) (9)

119887 (V 119902) equiv minus1

1205882(nabla sdot V 119902) (10)

respectively The following stabilization parameter is em-ployed

120591119870= minΔ119905

2

ℎ119870

21003817100381710038171003817119906119899minus1

ℎ

1003817100381710038171003817infin

120588ℎ2

119896

24120583 (11)

where ℎ119870denotes the maximum diameter of an element 119870

and sdot infin

is the maximum norm

Uinfin = 06ms

x3x1

5D L

d D

+ +

11D

26D

Figure 3 Two cylinders in tandem

A weak coupling of finite element scheme in (8) isapplied and the element searching algorithm for Lagrange-Galerkin method only needs to be implemented once in eachnonsteady loop [23]

23 Domain DecompositionMethod The calculating area is acuboidThe left side of themodel is the entrance and the rightside is the free outlet The inlet is placed 5119863 to the upstreamcylinder and total length is 26119863 The nonslip boundaries areplaced 55119863 above and below the cylinders The height is 4119863see Figure 3 for a two-dimensional projection of the three-dimensional model

In order to investigate the relationship between themby large-scale simulation a parallel computation by DDM[22 23 28 29] which is considered to have better accuracyand less time cost is implemented in this work In the domaindecomposition system the whole computation domain issplit into119873 nonoverlapping parts where119873 is the number ofthreads in each smaller part an FEM process is precededFor the comparison of efficiency of DDM and FEM pleasesee [23]

A Linux cluster (Intel Xeon E5606213GHz times 44 LVRDIMM 12GB1333MHz times 44) in Sun Yat-sen Universitywas used for this computation For each computation model176 threads were created as can be seen in Figure 4

For each computational model nonsteady time step is setto 001 s and the number of total simulation time is 10 s Themaximum number of Degrees Of Freedom (DOF) is up to166 million and it takes the cluster about 58 hours Details ofthe computations for each model can be found in Table 1

3 Modeling and Boundary Conditions

Air is assumed to be the fluid in this work 119871 is the distancebetween the two centers of the cylinders The diameter ofupstream cylinder is d and the diameter of downstreamcylinder is 119863 (119863 = 0025m) The time step is 001 s and totalcomputation time for each model is 10 s

The kinematic viscosity of the fluid is 0000015m2s atconstant temperature and Re = 1000 The velocity of thecoming flow is 06ms The air flows into the model with aconstant horizontal velocity from the left to the right Thevertical velocity is prescribed to zero


Table 1 Computation information

Name Spacing ratio(119871119863)

Diameter ratio(119889119863) Number of elements Number of DOF Nonsteady loops Computation time (h)

Model 1ndash1 2 1 9420393 14130590 1000 515Model 1ndash2 215 1 9466412 14211123 1000 500Model 1ndash3 25 1 9566559 14390891 1000 435Model 1ndash4 3 1 9710157 14649874 1000 485Model 2ndash1 3 05 6908469 9309739 1000 310Model 2ndash2 3 1 9710157 14855135 1000 468Model 2ndash3 4 05 10146357 15654964 1000 532Model 2ndash4 4 1 9997736 15312582 1000 474Model 2ndash5 5 05 10428101 16114294 1000 497Model 2ndash6 5 1 10283886 15785861 1000 545Model 2ndash7 6 05 10715636 16597006 1000 575Model 2ndash8 6 1 10565696 16259744 1000 553

x3

x1

x2

Figure 4 The domain decomposition of a model the 176 parts and local details

The model of two cylinders in tandem with differentdiameter is shown in Figure 5 The diameter of the upstreamcylinder decreases to 05119863 The diameter of the downstreamcylinder remains the same 119889119863 is 05

4 Result and Discussion

This section presents the results of numerical simulationsfor two cylinders in tandem with different spacing ratios(119871119863) and diameter ratios (119889119863) To confirm the results thevisualization of computational results was conducted and itshowed the details of flow field Micro AVS by CYBERNETSYSTEMS was used to plot Figures 5ndash14

41 Validation In order to validate the current modelingnumerical results of two cylinders of the same diameterwith different spacing ratios (119871119863) were compared to well-recognized results Figures 6 and 7 show the comparison ofcurrent results with the numerical results reported by Jesterand Kallinderis [4]

As is shown in Figure 6 there is a recirculation betweenthe two cylinders at 119871119863= 2The vortex shedding occurs onlybehind the downstream cylinder

At 119871119863 = 215 both reattachment and two vortex streetsoccur between two cylinder which presents a bistable

state [4] When reattachment occurs the vortices shed fromthe upstream cylinder fluctuate and reattach to the down-stream cylinder see Figure 7 Recirculation is observedbetween the two cylinders and vortex shedding is observedbehind the downstream cylinder all the time at steady state

At 119871119863 = 25 the vortex shedding occurs behind bothupstream and downstream cylinders as is shown in Figure 8The vortex sheds from the upstream cylinder sharply andimpinges to the downstream cylinder

The computational results are also validated by compar-ing with the experiment conducted by Wu et al [2] and thecomparison of the vortex contour is shown in Figure 9

As can be seen from Figure 9 the numerical andexperimental results agree with each other very well thevortices separate from the upstream cylinder reattach to thedownstream cylinder The vortex shedding occurs only afterthe downstream cylinder There is no independent vortexstreet between two cylinders

The computational results present the same characteristicand tendency with recognized results of other researchersThe characteristics of the flow and the vortex shedding areboth performed as expectedThe numerical experiments alsoconvinced us that the parallel computation with DomainDecomposition Method has good accuracy and efficiency inlarge-scale simulation which encouraged us tomove forwardto investigate the effect of the diameter ratio


x3

x2

x1

(a)

x3

x2

x1

(b)

Figure 5 Two cylinders in tandem with different diameter ratios ((a) 119889119863 = 1 (b) 119889119863 = 05)

(a) (b)

Figure 6 Streamlines of Model 1ndash1 at 119905 = 800 s ((a) Current results (b) Jester and Kallinderis)

(a) (b)


(a) (b)


440 s

(a) (b)

Figure 9 Contour of Model 1ndash1 at 119905 = 440 s ((a) Current results (b) Wu et al)


1000 s

(a)

1000 s

(b)

Figure 10 Contour at 119871119863 = 3 119905 = 1000 s ((a)Model 2ndash1 (b)Model2ndash2)

42 The Effect of Diameter Ratio According to the researchconducted by Ding-Yong et al [20] the diameter ratio hasstrong effects on the characteristic of flow field aroundtwo cylinders in tandem In the research the downstreamcylinder had smaller diameter than the upstream one Withthe increasing of the diameter ratio (119889119863) the critical spacingratio increases Mahbub Alam and Zhou also investigatedflow past two cylinders in tandem with different diametersexperimentally in wind tunnel [19]The diameter of upstreamcylinder varied from 024 to 1 of the downstream cylinderdiameter However the spacing ratio kept 55 times ofupstream cylinder diameter

To determine the influence of the diameter ratio (119889119863)to the flow field more comprehensively this paper conductedthe study by decreasing the diameter of the upstream cylinderto half of the downstream cylinder The space between twocylinders remains the same as the corresponding simula-tion at 119889119863 = 1 The visualization of the flow field showsthe characteristic between the gap and vortices street Inall arrangements the vortex shedding occurs behind bothupstream and downstream cylinders However the volumeof the vortices and the complexity of wake flow are different

As can be seen from Figure 10 at 119871119863 = 3 the vorticesare smaller and but the wake behind downstream cylinderis wider at 119889119863 = 05 compared with 119889119863 = 1 With 119889119863 =1 the vortices shed from the upstream cylinder sharply andimpinged to the downstream cylinder At 119889119863 = 05 theinterference on the upstream cylinder is less than that at119889119863 = 1

In Figure 11 at 119871119863 = 4 the vortices are smaller at 119889119863 =05 comparedwith 119889119863= 1Thewake behind the downstreamcylinder shares about the same area for119889119863=05 and119889119863= 1The interference on the upstream cylinder is smaller at 119889119863 =05 than that at 119889119863 = 1

At 119871119863= 5 the vortices andwake behind the downstreamcylinder appears smaller and narrower at119889119863=05 comparedwith 119889119863 = 1 The interference on the upstream cylinder ismuch smaller at 119889119863 = 05 than that at 119889119863 = 1 see Figure 12

At119871119863= 6 the vortices andwake behind the downstreamcylinder are much smaller and narrower at 119889119863 = 05compared with 119889119863 = 1 The interference on the upstreamcylinder is negligible at 119889119863 = 05 than that at 119889119863 = 1 seeFigure 13

1000 s

(a)

1000 s

(b)


1000 s

(a)

1000 s

(b)

Figure 12 Contour at 119871119863= 5 119905= 1000 s ((a)Model 2ndash5 (b)Model2ndash6)

Overall the wake after the downstream cylinder at 119889119863 =05 is clearer than that at 119889119863 = 1The vortices that shed fromthe smaller upstream cylinder impinge to the downstreamcylinder and lead to a simpler behavior of wake It is supposedthat there exists a critical spacing ratio dominating the flowregime Additionally the different diameter ratio also gaverise to the change of critical spacing ratio FromFigures 10ndash13it is known that

(1) When the spacing ratio is less than the criticalvalue there are only few vortices shedding after thedownstream cylinder but no vortex shedding occursafter the upstream cylinder see Figure 10(b)

(2) When the spacing ratio reaches the critical valuethe vortex shedding occurs after both cylinders seeFigure 11(b)

(3) When the spacing ratio keeps increasing the vorticesshed from the upstream cylinder will not attachto the downstream cylinder and there will be twoindependent vortex streets see Figure 13(a)

It can be seen that the critical spacing ratio increaseswith the increasing of diameter ratio Moreover the critical


1000 s

(a)

1000 s

(b)


dD = 05

dD = 1

40 50 6030

Spacing ratio (LD)

00

10

20

30

40

50

60

70

Wak

e wid

th ra

tio (W

D)

Figure 14 Wake width ratio versus spacing ratio

spacing ratio at 119889119863 = 1 is between 3 and 4 and thecritical spacing ratio at 119889119863 = 05 is smaller than 3 Thisphenomenon coincides with what we know decreasing 119889119863leads to the narrower wake and smaller vortices after theupstream cylinder [19] before 119871119863 reaches the critical valuethe vortex street is suppressed between the two cylinders [30]To explain the mechanism behind this wake width (119882) ismeasured for Figures 10ndash13 and wake width ratio is definedby119882119863 see Figure 14

At 119889119863 = 05 the wake and vortices behind upstreamcylinder are narrower and smaller which reduces the interfer-ence between the two cylinders leading to the narrower wakebehind downstream cylinder as well At 119889119863 = 1 and 119871119863 = 3the vortex street is suppressed behind the upstream cylinderConsequently the wake width is narrower compared to thecase at 119889119863 = 05 119871119863 = 3 However the wake behindthe upstream cylinder is no longer suppressed when keepincreasing the spacing ratio Thus the interference increasesAt 119889119863 = 1 the wake behind the upstream cylinder is widerthan at 119889119863 = 05 The increasing of spacing ratio will evencause wider wake behind the downstream cylinder due tostronger interference until two independent vortex streetsoccur

5 Conclusions

This paper tested the parallel computation with DomainDecomposition Method in large-scale computation by sim-ulating flow past two cylinders in tandemThe arrangementsof spacing ratio 119871119863 between 2 and 6with diameter ratio 119889119863equal to 1 and 05 were studied

(a) By applying this method to simulate flow past twocylinders in tandem and comparing the results withothersrsquo work it showed that the results have betteraccuracy and the computation costs less time Thecredibility and the viability were verifiedThemethodwill have broad application in large-scale computa-tion in the future

(b) The diameter ratio plays an important role in flowpast two cylinders in tandem The upstream cylinderwith smaller diameter will decrease the vortex volumeand the interference between two cylinders With adecreasing of 119889119863 the vortices behind the upstreamcylinder appear more but smaller Meanwhile withan increasing spacing ratio 119871119863 the wake becomesnarrower at 119889119863 = 05 but becomes wider at 119889119863 = 1

(c) The critical spacing ratio increases with the diameterratio increasingMoreover the critical spacing ratio at119889119863 = 1 is between 3 and 4 and it is smaller than 3 at119889119863 = 05

In real word engineering the structure of cylinders ismore complicated which should be considered in futurestudy

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to thank Mr Cansen Zhen for someof numerical experiments at early stageTheNational ScienceFoundation ofChinaGrants 11202248 and 11572356 supportsthis work

References

[1] D Sumner ldquoTwo circular cylinders in cross-flow a reviewrdquoJournal of Fluids and Structures vol 26 no 6 pp 849ndash899 2010

[2] J Wu L W Welch M C Welsh J Sheridan and G JWalker ldquoSpanwise wake structures of a circular cylinder andtwo circular cylinders in tandemrdquo Experimental Thermal andFluid Science vol 9 no 3 pp 299ndash308 1994

[3] S Mittal V Kumar and A Raghuvanshi ldquoUnsteady incom-pressible flows past two cylinders in tandem and staggeredarrangementsrdquo International Journal for Numerical Methods inFluids vol 25 no 11 pp 1315ndash1344 1997

[4] W Jester andY Kallinderis ldquoNumerical study of incompressibleflow about fixed cylinder pairsrdquo Journal of Fluids and Structuresvol 17 no 4 pp 561ndash577 2003

[5] Y Bao D Zhou and C Huang ldquoNumerical simulation of flowover three circular cylinders in equilateral arrangements at low


Reynolds number by a second-order characteristic-based splitfinite element methodrdquo Computers and Fluids vol 39 no 5 pp882ndash899 2010

[6] F Xu and J-P Ou ldquoNumerical simulation of vortex-inducedvibration of three cylinders subjected to a cross flow in equi-lateral arrangementrdquo Acta Aerodynamica Sinica vol 28 no 5pp 582ndash590 2010

[7] A Zhang and L Zhang ldquoNumerical simulation of three equis-paced circular cylindersrdquo Chinese Journal of Applied Mechanicsvol 20 no 1 pp 31ndash36 2003

[8] K K Chen J H Tu and C W Rowley ldquoVariants of dynamicmode decomposition boundary condition Koopman andFourier analysesrdquo Journal of Nonlinear Science vol 22 no 6 pp887ndash915 2012

[9] H Ding C Shu K S Yeo and D Xu ldquoNumerical simulationof flows around two circular cylinders by mesh-free least-square-based finite difference methodsrdquo International JournalforNumericalMethods in Fluids vol 53 no 2 pp 305ndash332 2007

[10] Z Han D Zhou Y Chen X Gui and J Li ldquoNumericalsimulation of cross-flow around multiple circular cylinders byspectral element methodrdquo Acta Aerodynamica Sinica vol 32no 1 pp 21ndash37 2014

[11] P Hao G Li L Yang and G Chen ldquoLarge eddy simulation ofthe circular cylinder flow in different regimesrdquo Chinese Journalof Applied Mechanics vol 29 no 4 pp 437ndash443 2012

[12] A G Kravchenko and P Moin ldquoNumerical studies of flow overa circular cylinder at ReD=3900rdquo Physics of Fluids vol 12 no 2pp 403ndash417 2000

[13] G V Papaioannou D K P Yue M S Triantafyllou and G EKarniadakis ldquoThree-dimensionality effects in flow around twotandem cylindersrdquo Journal of FluidMechanics vol 558 pp 387ndash413 2006

[14] A Kumar and R K Ray ldquoNumerical study of shear flowpast a square cylinder at reynolds numbers 100 200rdquo ProcediaEngineering vol 127 pp 102ndash109 2015

[15] J O Pralits F Giannetti and L Brandt ldquoThree-dimensionalinstability of the flow around a rotating circular cylinderrdquoJournal of Fluid Mechanics vol 730 pp 5ndash18 2013

[16] G Schewe ldquoReynolds-number-effects in flow around a rect-angular cylinder with aspect ratio 15rdquo Journal of Fluids andStructures vol 39 pp 15ndash26 2013

[17] F X Trias A Gorobets and A Oliva ldquoTurbulent flow arounda square cylinder at Reynolds number 22000 a DNS studyrdquoComputers and Fluids vol 123 pp 87ndash98 2015

[18] M Zhao L Cheng B Teng and G Dong ldquoHydrodynamicforces on dual cylinders of different diameters in steady cur-rentsrdquo Journal of Fluids and Structures vol 23 no 1 pp 59ndash832007

[19] M Mahbub Alam and Y Zhou ldquoStrouhal numbers forcesand flow structures around two tandem cylinders of differentdiametersrdquo Journal of Fluids and Structures vol 24 no 4 pp505ndash526 2008

[20] Y Ding-Yong L Hong-Chao and W Chang-Hai ldquoNumericalsimulation of viscous flow past two tandem circular cylindersof different diametersrdquo Periodical of Ocean University of Chinano 2 pp 160ndash165 2012

[21] P Zhang Z-D Su L-D Guang and Y-L Li ldquoEffect of thediameter change on the flow past three cylindersrdquo Journal ofHydrodynomics vol 27 no 5 pp 554ndash560 2012

[22] Q Yao and H Kanayama ldquoA coupling analysis of thermalconvection problems based on a characteristic curve methodrdquo

Theoretical and Applied Mechanics Japan vol 59 pp 257ndash2642011

[23] Q Yao and Q Zhu ldquoA pressure-stabilized Lagrange-Galerkinmethod in a parallel domain decomposition systemrdquo Abstractand Applied Analysis vol 2013 Article ID 161873 13 pages 2013

[24] Q Yao and H Zhu ldquoNumerical simulation of hydrogen dis-persion behaviour in a partially open space by a stabilizedbalancing domain decomposition methodrdquo Computers andMathematics with Applications vol 69 no 10 pp 1068ndash10792015

[25] F Brezzi and M Fortin Mixed and Hybrid Finite ElementMethods Springer New York NY USA 1991

[26] Q-H Yao and Q-Y Zhu ldquoInvestigation of the contaminationcontrol in a cleaning room with a moving AGV by 3D large-scale simulationrdquo Journal of Applied Mathematics vol 2013Article ID 570237 10 pages 2013

[27] F Brezzi and J Douglas ldquoStabilized mixed methods for theStokes problemrdquo Numerische Mathematik vol 53 no 1-2 pp225ndash235 1988

[28] J Mandel ldquoBalancing domain decompositionrdquo Communica-tions in Numerical Methods in Engineering vol 9 no 3 pp 233ndash241 1993

[29] A Toselli and O B Widlund Domain DecompositionMethodsmdashAlgorithms and Theory Springer Berlin Germany2005

[30] M M Zdravkovich ldquoREVIEWmdashreview of flow interferencebetween two circular cylinders in various arrangementsrdquo Jour-nal of Fluids Engineering vol 99 no 4 pp 618ndash633 1977

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


cylinders in two dimensions by applied finite elementmethod[21]

As far as we know the flow structure of two cylindersin tandem affected by the changing of diameter ratio andspacing ratio is still uncertain and the computation scale ofnumerical experiments published research is very limited Toinvestigate the flow regime more concretely and more subtlylarge-scale simulations by Domain Decomposition Method(DDM) which is considered to have better accuracy and lesstime cost when comparing with conventional methods areimplemented [22 23]The large-scalemodelingwas validatedby comparing with othersrsquo reports for two cylinders of thesame diameter with different spacing ratios Moreover tostudy its influence to the flow past two cylinders in tandemmore comprehensively the flow structures at specified diam-eter ratios (119889119863 = 05 and 119889119863 = 1) with different spacingratios (119871119863 3ndash6) are investigated The critical spacing ratiothat affects the flow regime is expected to be determined fordifferent diameter ratios

This paper is organized as follows Section 2 introducesthe governing equations to be solved as well as the domaindecompositionmethod (DDM) In Section 3 the models andthe boundary conditions are described in detail Numericalresults for different spacing ratio and different diameter ratios(119889119863 = 1 and 119889119863 = 05) are present and comparedwith othersrsquo work in Section 4 At last Section 5 draws theconclusions of this work

2 Formulations

21 Governing Equations Let 120597Ω be the boundary of a three-dimensional polyhedral domain Ω 1198671(Ω) is the first orderSobolev space and 1198712(Ω) is the space of 2nd power summablefunctions on Ω Under the assumption that the flow field isincompressible viscous and laminar the solving of themodelcan be summarized as finding (119906 119901) isin 1198671(Ω)3 times 1198712(Ω) suchthat for any 119905 isin (0 119879) the following set of equations hold [24]

120597119905119906 + (119906 sdot nabla) 119906 minus

1

120588nabla sdot 120590 (119906 119901) =

1

120588119891

nabla sdot 119906 = 0

in Ω times (0 119879)

(1)

where 119906 is the velocity [ms] 119901 is the pressure [Pa] 120588 is thedensity (const) [kgm3] 119891 is the body force [Nm3] 120590(119906 119901)is the stress tensor [Nm2] defined by

120590119894119895(119906 119901) equiv minus119901120575

119894119895+ 2120583119863

119894119895(119906)

119863119894119895(119906) equiv

1

2(120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

)

119894 119895 = 1 2 3

(2)

with the Kronecker delta 120575119894119895and the viscosity 120583 [kg(msdots)]

An initial velocity 1199060is applied in Ω at 119905 = 0 Dirichlet

boundary conditions

119906 = on Γ times (0 119879) (3)

R3

u(X t)

X(tnminus1) asymp X1(unminus1 Δt) X(tn) = x

Figure 1 A characteristic finite element scheme

and Neumann boundary conditions

3

sum

119895=0

120590119894119895119899119895= 0 on 120597Ω Γ times (0 119879) (4)

are also applied where Γ sub 120597Ω and 119899 is the outward normaldirection to 120597Ω

22 Characteristic Finite Element Scheme Using the defini-tion

119905119899equiv 119899Δ119905

119873119879equiv [

119879

Δ119905]

(5)

a characteristic finite element scheme approximates thematerial derivative in (1) at 119905119899 as follows [23]

120597119905119906 + (119906 sdot nabla) 119906 asymp

119906119899minus 119906119899minus1

(1198831(119906119899minus1

Δ119905))

Δ119905 (6)

where1198831(sdot sdot) is a position function see Figure 1

Let Iℎequiv 119870 denote a triangulation of Ω consisting

of tetrahedral elements and the subscript ℎ denotes therepresentative length of the triangulation The finite elementspaces are defined as follows

119883ℎequiv Vℎisin 1198620(Ω)3 Vℎ

1003816100381610038161003816119870isin 1198751(119870)3 forall119870 isin I

ℎ

119872ℎequiv 119902ℎisin 1198620(Ω) 119902

ℎ

1003816100381610038161003816119870isin 1198751(119870) forall119870 isin I

ℎ

119881ℎ() equiv V

ℎisin 119883ℎ Vℎ(119875) = (119875) forall119875 isin Γ

119881ℎequiv 119881ℎ(0)

119876ℎ= 119872ℎ

(7)

where the 1198620(Ω)3 in (7) denotes the continuous functionon Ω in three dimensions 119875

1(119870)3 denotes the first order

polynomial defined by 119870 in three dimensionsNote that piecewise linear interpolations are employed

for velocity and pressure (see (7) and Figure 2) which doesnot provide a sufficient condition to link the velocity andpressure space therefore the so-called inf-sup condition [25]should be satisfied In previous work [26] a penalty Galerkinleast-squares (GLS) stabilization method for pressure [27]


0

1

2

3

VelocityPressure



Find (119906119899ℎ 119901119899

ℎ) isin 119881ℎ(119892) times 119876

ℎ119873119879

119899=1such that

(

119906119899


ℎ(1198831(119906119899minus1

ℎ Δ119905))

Δ119905 Vℎ) + 119886 (119906

119899

ℎ Vℎ)

+ 119887 (Vℎ 119901119899

ℎ) + 119887 (119906

119899

ℎ 119902ℎ)

+1

1205882sum

119870isinIℎ

120591119870(nabla119901119899


=1

120588(119891 Vℎ) +

1

1205882sum

119870isinIℎ

120591119870(119891 minusnabla119902

ℎ)119870

(8)



119886 (119906 V) equiv2120583

120588(119863 (119906) 119863 (V)) (9)

119887 (V 119902) equiv minus1

1205882(nabla sdot V 119902) (10)


120591119870= minΔ119905

2

ℎ119870

21003817100381710038171003817119906119899minus1

ℎ

1003817100381710038171003817infin

120588ℎ2

119896

24120583 (11)


and sdot infin

is the maximum norm

Uinfin = 06ms

x3x1

5D L

d D

+ +

11D

26D















x3

x1

x2














x3

x2

x1

(a)

x3

x2

x1

(b)


(a) (b)


(a) (b)


(a) (b)


440 s

(a) (b)



1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


dD = 05

dD = 1

40 50 6030

Spacing ratio (LD)

00

10

20

30

40

50

60

70

Wak

e wid

th ra

tio (W

D)




5 Conclusions






Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

2

3

VelocityPressure



Find (119906119899ℎ 119901119899

ℎ) isin 119881ℎ(119892) times 119876

ℎ119873119879

119899=1such that

(

119906119899


ℎ(1198831(119906119899minus1

ℎ Δ119905))

Δ119905 Vℎ) + 119886 (119906

119899

ℎ Vℎ)

+ 119887 (Vℎ 119901119899

ℎ) + 119887 (119906

119899

ℎ 119902ℎ)

+1

1205882sum

119870isinIℎ

120591119870(nabla119901119899


=1

120588(119891 Vℎ) +

1

1205882sum

119870isinIℎ

120591119870(119891 minusnabla119902

ℎ)119870

(8)



119886 (119906 V) equiv2120583

120588(119863 (119906) 119863 (V)) (9)

119887 (V 119902) equiv minus1

1205882(nabla sdot V 119902) (10)


120591119870= minΔ119905

2

ℎ119870

21003817100381710038171003817119906119899minus1

ℎ

1003817100381710038171003817infin

120588ℎ2

119896

24120583 (11)


and sdot infin

is the maximum norm

Uinfin = 06ms

x3x1

5D L

d D

+ +

11D

26D















x3

x1

x2














x3

x2

x1

(a)

x3

x2

x1

(b)


(a) (b)


(a) (b)


(a) (b)


440 s

(a) (b)



1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


dD = 05

dD = 1

40 50 6030

Spacing ratio (LD)

00

10

20

30

40

50

60

70

Wak

e wid

th ra

tio (W

D)




5 Conclusions






Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














x3

x1

x2














x3

x2

x1

(a)

x3

x2

x1

(b)


(a) (b)


(a) (b)


(a) (b)


440 s

(a) (b)



1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


dD = 05

dD = 1

40 50 6030

Spacing ratio (LD)

00

10

20

30

40

50

60

70

Wak

e wid

th ra

tio (W

D)




5 Conclusions






Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x3

x2

x1

(a)

x3

x2

x1

(b)


(a) (b)


(a) (b)


(a) (b)


440 s

(a) (b)



1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


dD = 05

dD = 1

40 50 6030

Spacing ratio (LD)

00

10

20

30

40

50

60

70

Wak

e wid

th ra

tio (W

D)




5 Conclusions






Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


1000 s

(a)

1000 s

(b)








1000 s

(a)

1000 s

(b)


dD = 05

dD = 1

40 50 6030

Spacing ratio (LD)

00

10

20

30

40

50

60

70

Wak

e wid

th ra

tio (W

D)




5 Conclusions






Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1000 s

(a)

1000 s

(b)


dD = 05

dD = 1

40 50 6030

Spacing ratio (LD)

00

10

20

30

40

50

60

70

Wak

e wid

th ra

tio (W

D)




5 Conclusions






Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Large-Scale Computations of Flow around...

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