Spectral Element Numerical Investigation of Flow...

Research ArticleSpectral Element Numerical Investigation of Flow betweenThree Cylinders in an Equilateral-Triangular Arrangement withDifferent Spacing Distances

Zhenzhong Bao Guoliang Qin Wenqiang He and YazhouWang

School of Energy and Power Engineering Xirsquoan Jiaotong University Xirsquoan Shanxi 710049 China

Correspondence should be addressed to Guoliang Qin glqinxjtueducn

Received 21 February 2018 Revised 11 April 2018 Accepted 22 April 2018 Published 23 May 2018

Academic Editor Adam Glowacz

Copyright copy 2018 Zhenzhong Bao et alThis 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

Two-dimensional incompressible Navier-Stokes equations are numerically solved using the high resolution spectral elementmethod at Reynolds number 200 The flow between three cylinders in an equilateral-triangular arrangement is investigated Thecenter-to-center spacing distance ratio between two circular cylinders is varied from 15 to 12 Present numerical results show thatthe flow patterns and force characteristics are the result of the combined effects of Reynolds number spacing distance configurationarrangement and incident angle For the small spacing distance ratio of 15 the well-known biased flow phenomenon in the gapof downstream cylinders is found And the biased flow is bistable in our study but not monostable A small spacing distancemeans lower Strouhal number drag and root-mean-square lift coefficients In the medium spacing distance ratio of 40 thesuppressed effect of vortex shedding for the presence of the side-by-side downstream cylinders disappeared Mean drag coefficientsof downstream cylinders are basically identical to the value of flow past around a single circular cylinder For the large spacingdistance ratio of 80 the effects between three cylinders basically disappearedThemean drag and lift coefficients root-mean-squarelift coefficients and Strouhal number of three cylinders are essentially equivalent to those values of a single circular cylinder

1 Introduction

Flows around circular cylinders are widespread among themodern industrial production and engineering practice suchas landing gear systems heat exchanger tubes offshoreplatforms pillar groups and nuclear reactors Thus duringthe past century numerous experiments and numericalsimulations have been carried out to study the flow regime offlows around circular cylinders with different arrangementNevertheless many works are focused on the single cylinderand a pair of cylinders [1ndash3] few take consideration ofthree or more cylinders arrangement configuration [4] Butflow between the three cylinders is frequently involved inindustrial applications such examples are shown in Figures1 and 2

Yan et al [5] Yang et al [6] investigated three cylindersin a staggered arrangement at Re = 200 with laser-inducedfluorescence flow visualization and lattice Boltzmann numer-ical method Two diverse flow characteristic steady (1 le

119879119863 le 12 and 25 le 119879119863 le 31) and unsteady flow(13 le 119879119863 le 24 and 32 le 119879119863 le 10) were found Zheng etal [7] investigated three cylinders in an equilateral-triangularT shaped and inverted-T shaped configuration with the finitevolume method Three flow pattern characteristics and forcecharacteristics were analyzed in detail Barros et al [8] pre-sented the convective laminar flow and heat transfer of threecylinders in a triangular arrangement at Re = 100 Prandtlnumber Pr = 071 using finite volume method Shaaban andMohany [9] numerically investigated the effect of distancebetween the upstream cylinder and middle cylinder onthe force characteristic with three uneven spacing cylindersconfiguration at Re = 200 Wu [10] using the second-order immersed boundary method carried out the numericalsimulation of three equilateral-triangular array cylinderswith different incident angle Bansal and Yarusevych [11]experimentally investigated flow around a cluster of threeequally spaced cylinders at Re = 21 times 103 POD analysissuggested that cluster orientation is a typical impact factor

HindawiShock and VibrationVolume 2018 Article ID 6358949 11 pageshttpsdoiorg10115520186358949

2 Shock and Vibration

Figure 1 Offshore platforms pillar groups

Figure 2 Oriental pearl radio and TV tower Shanghai

on the near wake development Qiu et al [12] experimentallymeasured the pressure coefficients of three cylinders in anequilateral-triangular arrangement Significant complicatedaerodynamic interference effects were observed betweenthree cylinders Several other literatures related to three ormore cylinders are also published [13ndash19] To conclude allthe above researches showed that the flow patterns and forcecharacteristics are the results of the combined effect of Respacing distance configuration arrangement and incidentangle

With the development of hardware technology numericalsolutions can be easily obtained compared to experimentresults Numerous numerical algorithms have been usedto solve the flow around cylinders such as finite vol-ume method finite element method immersed boundarymethod spectral method and meshless method Each algo-rithm has pros and cons However in order to capture moreflow details it has been a trend to apply high resolutionmethod into the solving of incompressible N-S equationsSpectral element method (SEM) mixes the exponential con-vergence property of spectral schemes and good geomet-ric flexibility feature of finite element method Therefore

exponential (p-type) and algebraic (h-type) convergence canbe respectively achieved by increasing polynomial order andelement numbers In addition the spectral element methodhas been applied tomany research fields such as flowandheattransfer [20 21] wave propagation [22] structure vibration[23] and aeroacoustics [24]

The paper is organized as follows In Section 2 a briefdescription of the governing equation is given In Section 3the accuracy of the spectral element numerical code isverified by the Kovasznay flow and flow past around asingle circular cylinder Then flows between three cylindersin an equilateral-triangular arrangement are investigated indetail Finally Section 5 contains the conclusions of ourwork

2 Governing Equationsand Numerical Methods

The governing equations used for unsteady and incompress-ible viscous flow are incompressible Navier-Stokes equationand continuity equation which are expressed as below

120597k120597119905 + k sdot nablak = minusnabla119901 + ]nabla2knabla sdot k = 0

(1)

where v is the velocity vector 119901 represents the pressure] is the kinematic viscosity ] = 1Re Re is Reynoldrsquosnumber with respect to the circular cylinder diameter119863 andcharacteristic velocity 119880 Re = 119880 sdot 119863]

The numerical calculation was carried out using theopen-source code Nektar++ [25] The incompressible N-S equation solver in Nektar++ library is based on thespectral element method in space discretization and high-order time splitting method in time discretization In thespectral element method the computational domain Ω isdecomposed into several nonoverlapping subdomains Eachspectral element is mapped into a standard element Finallythe variational problem discretized by the means of SEMis equivalent to solving Differential Algebraic EquationsAnd the block diagram of the spectral element method forincompressible flow problems is shown in Figure 3 Moredetails can be obtained from the literatures [20 24 25]

3 Validation Study

31 Kovasznay Flow In order to validate the exponentialconvergence rate of the numerical scheme we consider thetwo-dimensional Kovasznay flow in a rectangular area Ω =[minus05 1] times [minus05 15] This flow problem is a good test for thealgorithm because an analytical closed form solution existsfor the incompressible flow field At the same time the flowpattern is similar to the wake flow of a circular cylinderSo Kovasznay flow can be used to judge the convergenceand numerical accuracy of the method for solving incom-pressible N-S equations The analytical solution [29] is givenby

119906 = 1 minus 119890120582119909 cos 2120587119910

Shock and Vibration 3

Meshgeneration

(Gmsh)

Nektar++solver

Advectionterm

Poisson term

Helmholtzterm

Initialconditions

Final solution

Preprocessing

Postprocessing

Time integration

Figure 3 The block diagram of the incompressible Navier-Stokessolution algorithm

V = 1205822120587119890120582119909 sin 2120587119910119901 = 1199010 minus 121198902120582119909

120582 = Re2 minus radicRe2

4 + 41205872(2)

where 1199010 is a reference pressure and 120582 is an intermediatevariable related to Re The computational domain Ω isdecomposed into 12 nonoverlapping subdomains with 3 ele-ments in the x direction and 4 elements in the y directionThetime step chosen is very small so that the numerical accuracyis only connected with polynomial order Figure 4 shows thecontours of velocity component in the 119909 direction at Re =40 Figure 5 shows the convergence rate of pressure and 119906velocity We can find that 1198712 norm error has an exponentialconvergence rate with increasing polynomial orderThe errorlevel approaches the value of 10ndash12 at 119875 = 14 However itdoes not decrease again at the higher polynomial order whichmeans that the calculations are reached machine error levelsTherefore the exponential convergence of numerical schemefor solving incompressible N-S equations was verified

32 Unsteady Flow Past a Single Circular Cylinder Flowaround a circular cylinder is another benchmark problemto test the accuracy of the SEM We consider the two-dimensional laminar flow past around a circular cylinderThe schematic of the numerical computation is presentedin Figure 6 The center of the circular cylinder is locatedat the origin of the coordinate plane The diameter of thecircular cylinder (119863) denotes unit characteristic length The

x

y

minus05 0 05 1minus05

0

05

1

15u

240212184156128100072044016minus012minus040

Figure 4 Contours of velocity component in the 119909 direction andstreamlines at Re = 40

minus1

minus3

minus5

minus7

minus9

minus11

minus13

Log(

L 2er

ror)

up

3 5 7 9 11 13 15 17

Polynomial order P

Figure 5 1198712 norm error convergence curves of pressure and 119906velocity for different polynomial order at Re = 40

entire computational domain dimensions are 50119863 times 40119863with a downstream distance of 30119863 a distance of 20119863 onboth upstream and either side of the cylinder The boundaryconditions adopted are as follows at the inflow boundarythe velocity inlet is set to be a characteristic velocity 119880 = 1at the outer boundary zero velocity gradient and pressure119901 = 0 are specified the upper and lower boundaries areboth symmetry conditions and no-slip boundary conditionis applied on the wall of circular cylinder We run numericalcomputations with 1603 unstructured triangle elements and140 structured square elements near the wall of the circularcylinder as shown in Figures 7 and 8 In order to capturemore


20D 30D

20D

D=1x

y

u = 0 v = 0

20D

p = 0u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y= 0 = 0

휕u 휕y= 0 = 0

Figure 6The computation domain and boundary conditions of theflow around a circular cylinder

xD

yD

minus20 minus10 0 10 20 30minus20

minus15

minus10

minus5

0

5

10

15

20

Figure 7 Computational grid for flow around a circular cylinder

details of the flow near the cylinder surface the mesh gridsare imposed as narrow as 004119863 At the same time the wakeregion is also refined The grid systems in the present paperare all generated by Gmsh [30] open-source finite elementgrid software The numerical simulation is started with zeroinitial conditions at 119905 = 0

In this case we fixed the grid size and a relatively smalltime step Δ119905 = 00008 by varying the polynomial orderin each element a grid independence study is investigatedThe comparison of present numerical results with differentpolynomial orders and other existing results are given inTable 1 It is shown that using a polynomial order 119875 = 5time-mean drag coefficient (119862119889) and Strouhal number (St)provides as good a result as those obtained from previousliteratures

Some nondimensional parameters used in the presentpaper are defined as follows

119862119889 = 21198651198891205881198802119863

xD

yD

minus1 minus05 0 05 1minus1

minus05

0

05

1

Figure 8 Local refined mesh around the circular cylinder

Table 1 Comparison of the mean drag coefficients and Strouhalnumber

119875 119862119889 St3 13473 019994 13463 019995 13465 019646 13463 01964Han et al [26] 1346 01953Ding et al [27] 1327 0196Cai et al [28] 1345 0200Lam et al [18] 132 0196

119862119897 = 21198651198971205881198802119863St = 119891119904119863119880 119862119901ave = 2 (119901ave minus 119901infin)1205881198802 119862119889 = 1119879 int

119879

0

119862119889119889119905119862119897 = 1119879 int

119879

0

119862119897119889119905

1198621015840119897= radic 1119879 int

119879

0

1198621198972119889119905(3)

where 119865119889 119862119889 are the drag force and drag coefficient 119865119897 119862119897are the lift force and lift coefficient 119891119904 is the vortex sheddingfrequencywhich can be obtained from the FFT analysis of thelift coefficient curve 119862119901ave is mean pressure coefficient 119901ave


140 142 144 146 148 150 152 154 156 158 160

t

CdCl

130

132

134

136

138

140

C d

minus14

minus10

minus06

minus02

02

06

10

14

C l

Figure 9 Temporal evolution curve of lift coefficients and dragcoefficients

13

08

03

minus02

minus07

minus12

minus17

Cpav

e

0 20 40 60 80 100 120 140 160 180

휃

Rajani et alpresent

Figure 10 Distribution of time averaged pressure coefficients on thesurface of circular cylinder

is mean pressure 119901 119901infin are free stream pressure 120588 is fluiddensity 119879 is one stable period 119862119897 1198621015840119897 are the time-mean androot-mean-square lift coefficients

Figure 9 shows the temporal evolution curve of liftcoefficients and drag coefficients at Re = 200 As what can beseen from the graph shown in Figure 9 the vortex sheddinghas developed to be a stable periodic fashion The periodof lift coefficient is twice the period of drag coefficient Inaddition the time averaged pressure coefficient on the surfaceof circular cylinder is also an important parameter for char-actering the flow around a circular cylinder Figure 10 showsthe comparison of 119862119901ave between our numerical result andthe literature results from [31] Good agreement is observedand it shows the capabilities of the spectral element methodencapsulated inNektar++ for solving the incompressible flowproblems again

20D 30D

D=1x

y

20D

20D

p = 0s12

3u = 0 v = 0

u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y = 0 = 0

휕u 휕y = 0 = 0

Figure 11The computation domain and boundary conditions of theflow around three circular cylinders

minus20 minus10 0 10xD

yD

20 30minus20

minus15

minus10

minus5

0

5

10

15

20

Figure 12 Computational grid for flow around three circularcylinders with 119878119863 = 40

4 Unsteady Flow between ThreeCircular Cylinders

Now we consider the two-dimensional laminar flow pastbetween three circular cylindersThe schematic of the numer-ical computation is presented in Figure 11 Three circularcylinders are placed in the shape of an equilateral triangleThe center of the upstream circular cylinder (denoted withcylinder 1) is always located at the origin of the coordinatesAnother two cylinders are located at downstream in a side-by-side configuration (denoted with cylinder 2 and 3 respec-tively) 119878 is the center-to-center spacing between two circularcylinders The boundary conditions are the same as the onesadopted by flow past a single cylinder Computational gridfor flow around three circular cylinders with 119878119863 = 40 ispresented in Figures 12 and 13 3893 unstructured triangleelements and 330 structured square elements are used for thecase 119878119863 = 40 Just like the single circular cylinder we fixthe grid size and time step by varying the polynomial orderin each element a grid independence test is also examined


Table 2 Comparison of the mean drag coefficient and Strouhalnumber of flow around three cylinders

119875 119862119889 St

31 11814 0182 13470 0183 13465 018

41 11811 0182 13459 0183 13459 018

51 11810 0182 13459 0183 13460 018

xD

yD

minus2 0 2 4 6minus4

minus2

0

2

4

Figure 13 Local refined mesh around the three circular cylinders

Table 2 displays the comparison of 119862119889 and St betweendifferent polynomial orders Owing to the enough elementnumbers it is shown that grid independence is obtained usinga polynomial order 119875 = 3 However in order to present theflow patterns more clearly the polynomial order 119875 = 4 isadopted in the below numerical simulations

41 Flow Patterns Previous research had identified that thespacing distance has a vital impact on the flowpattern charac-teristicsTherefore in this paper we do a study of the spacingdistance 119878119863 from 15 to 12 at Re = 200 The flow patterncharacteristics of three typical spacing distances are discussedin below Figures 14ndash16 show presented computed resultsfor instantaneous vorticity contours pressure contours andstreamlines at 119878119863 = 15 40 and 80 with Re = 200

For the small spacing distance case of 119878119863 = 15 cross-flow around three cylinders with an equilateral-triangulararrangement is similar to a single bluff body flow As canbe seen from Figure 14(a) the single-row vortex street wasproduced in the downstreamofwake region Vortex sheddingof upstream cylinder is fully suppressed due to the presence of

the side-by-side downstream cylindersThe existence of well-known biased flow phenomenon in the gap of downstreamcylinders is proved And owing to this asymmetry flowthe flow pattern becomes irregularly and spontaneouslyMoreover it is of interest that we find the biased flow isbistable in our study but not monostable in literatures [716] This is also noticed by Sumner [1] Zhiwen et al [32]They explained that the bistable characteristic was relatedto the transient presentation and breakdown of large gapvortices behind the cylinder with the wider wake Gu andSun [33] also found this biased bistable flow at Re = 14000with 119878119863 = 17 One reason for the inconsistency betweendifferent researchers may be linked to the mixed effects ofchanges in Re and spacing distance Further comprehensiveresearch for this biased flow phenomenon should be carriedout

For the medium spacing distance case of 119878119863 = 40Figures 15(a) and 15(b) showed that cross-flow around threecylinders with an equilateral-triangular arrangement are allpartially developed The three-row vortex street was pro-duced in the downstream of wake region The suppressedeffect of vortex shedding for the presence of the side-by-sidedownstream cylinders is disappeared Gap flow is no longerdominated by the downstream cylinders And every cylinderproduces an obvious single vortex street in the wake regionThe wake of upstream cylinder is sandwiched by the wakesof the side-by-side downstream cylinders Owing to thisrestriction the wake extent of upstream cylinder becomesmore relatively narrow while those at the downstreambecome wider Moreover the vortex shedding behind threecylinders nearly presents an in-phase synchronized fashionThe results show that there is a strong interaction betweenthe vortex shedding of cylinder 1 with those from the innersides of downstream cylinders 2 and 3 However the vortexshedding of the outer sides of downstream cylinders 2 and3 does not take part in this merging process

For the large spacing distance case of 119878119863 = 80 asshown in Figures 16(a) and 16(b) cross-flow around threecylinders with an equilateral-triangular arrangement is allcompletely developed The three-row well-defined vortexstreet was produced in the downstream of wake regionThe flow pattern of any one of three cylinders only hasa slight difference with that of cross-flow around a singlecylinder And this slight difference can be seen from theinstantaneous streamlines in the wake region of Figure 16(b)Vortex shedding between cylinder 1 and cylinder 2 nearlypresents an in-phase synchronized fashion On the contraryVortex shedding between cylinder 2 and cylinder 3 nearlypresents an antiphase synchronized fashion

42 Force Statistics Figures 17ndash19 show the temporal evolu-tion curve of lift and drag coefficients for three cylinders at119878119863 = 15 40 and 80 Variations of mean drag coefficientsmean lift coefficients root-mean-square lift coefficients andSt with different spacing distances for three cylinders areillustrated in Figures 20ndash23

Due to the existence of the biased flow phenomenonin the case of 119878119863 = 15 the first impression for curvesin Figure 17 is of very large irregular and disordered The


xD

yD


minus3

minus1

1

3

5

minus10 100806040200minus02minus04minus06minus08

(a)xD

yD


minus3

minus1

1

3

5

minus04 0403020100minus01minus02minus03

(b)

Figure 14 Snapshots of instantaneous flow fields at 119878119863 = 15with Re = 200 (a) vorticity contours and (b) pressure contours and streamlines

xD

yD


minus3

minus1

1

3

5

minus10 minus08 minus06 minus04 minus02 00 02 04 06 08 10

(a)xD

yD


minus3

minus1

1

3

5

minus04 0100minus01minus02minus03 02 03 04

(b)


xD

yD


minus5

minus3

minus1

1

3

5

7


(a)xD

yD


minus5

minus3

minus1

1

3

5

7


(b)


drag and lift coefficients of cylinder 1 are all obviously lessthan the values of other two cylinders The mean drag andRMS lift coefficients of cylinders 2 and 3 are basicallyidentical Lift coefficients have the nearly the samemagnitudebut opposite sign The St of three circular cylinders is smallenough in comparison with the value of flow past around asingle circular cylinder Therefore it can be concluded thata small spacing distance tends to have a lower St drag andRMS lift coefficients

In the medium spacing distance case of 119878119863 = 40 allcoefficients versus time history present a sinusoidal shapeexcept the drag coefficients of cylinders 2 and 3 Dragcoefficients of cylinders 2 and 3 have the same magnitudebut some difference in phase The phase of lift coefficients forthree cylinders is all identical and this also suggests the sameSt will be obtainedThis character can be attributed to the in-phase synchronized vortex shedding behind three cylindersThemean drag coefficients of cylinders 2 and 3 are basically


400 405 410 415 420 425 430 435 440

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

065

070

075

080

085

090

095

100

105

110

115

C d

minus04

minus03

minus02

minus01

00

01

02

03

04

C lFigure 17 Temporal evolution curve of lift and drag coefficients forthree cylinders at 119878119863 = 15

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

260 265 270 275 28011

12

13

14

15

16

17

C d

minus3

minus2

minus1

0

1

2

3

C l

Figure 18 Temporal evolution curve of lift and drag coefficients forthree cylinders at 119878119863 = 40

identical to the value of flow past around a single circularcylinder except for the fact that the mean drag coefficientsof cylinder 1 are a bit less than the value of a single circularcylinder The reason for this can be referred to the stronginteraction between the vortex shedding of cylinder 1 withthose from the inner sides of downstream cylinders 2 and3

For the large spacing distance case of 119878119863 = 80 dueto the antiphase synchronized vortex shedding temporalevolution curves of drag coefficients for cylinders 2 and 3are in-phase Moreover the maximum drag coefficients ofcylinder 2 correspond to the minimum drag coefficients ofcylinder 3 and vice versa The mean drag coefficients ofthree cylinders are basically identical to the value of the singlecircular cylinder The cylinder 1 is a bit less than it andthe other two cylinders are a bit bigger than it As is shownin Figure 23 the St of three cylinders maintains the same

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

180 185 190 195 200125

130

135

140

145

150

C d

minus20

minus15

minus10

minus05

00

05

10

15

20

C l


15

14

13

12

11

10

09

08

07

06

C d

Cd 1Cd 2

Cd 3Single cylinder

2 3 4 5 6 7 8 91 11 12 1310

SD

Figure 20 Variations of mean drag coefficients with differentspacing distances for three cylinders

value Also the St of three cylinders is slightly larger thanthe value of a single cylinder Therefore it can be concludedthat there is also a slight difference between the flow pastaround a single circular cylinder and three cylinders withan equilateral-triangular arrangement in the large spacingdistance case of 119878119863 = 80

With the increase of the spacing distance the mean dragcoefficients become increasingly obvious The growth rateof cylinder 1 is slower than the other two cylinders andthere is an abrupt jump in the mean drag coefficients ofcylinder 1This also can be seen fromFigures 21 and 22 Keepincreasing the spacing distance the mean drag coefficients ofthree cylinders maintain a constant number respectively

Additionally mean lift coefficients have the nearly samemagnitude but opposite sign as illustrated in Figure 21 Thisis caused by the negative pressure distributions around thecylinder 2 and 3Mean lift coefficients of upstream cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder

Figure 21 Variations of mean lift coefficients with different spacingdistances for three cylinders

2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l

Figure 22 Variations of RMS lift coefficients with different spacingdistances for three cylinders

always keep a constant value of 0 which is coincidingwith theresult of a single cylinder RMS lift coefficients of cylinders2 and 3 reach a maximum at 119878119863 = 40 and then declinegradually to 0 at 119878119863 = 80 and thereafter maintain thisvalue with the bigger spacing distance RMS lift coefficientsof cylinders 1 reach a maximum at 119878119863 = 40 and thenmaintain this value which is close to that obtained by asingle cylinderThe St of three cylinders become increasinglyobvious with small spacing distance In the medium spacingdistance the St are all less than the value of a single cylinderHowever in the large spacing distance the St are all greaterthan the value of a single cylinder We also found that the Stof three cylinders are almost equivalent to each other for allspacing distances

5 Conclusions

In this paper the spectral element method which combinesthe advantages of the spectral method and finite element

2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021

Figure 23 Variations of St with different spacing distances for threecylinders

methodwas adopted to solve the flowbetween three cylindersin an equilateral-triangular arrangement The simulationresults of Kovasznay flow and flow past around a single cir-cular cylinder showed a good consistency with the availableliteratures And this indicated that the algorithmwas efficientand high-order accuracy

For the small spacing distance case of 119878119863 = 15 theexistence of well-known biased flow phenomenon in the gapof downstream cylinders was provedHowever it is of interestthat we found the biased flow is bistable in our study butnot monostable A small spacing distance presented lower Stdrag and RMS lift coefficients

In the medium spacing distance case of 119878119863 = 40 thesuppressed effect of vortex shedding for the presence of theside-by-side downstream cylinders disappeared The resultsshowed that there was a strong interaction between the vortexshedding of cylinder 1 with those from the inner sides ofdownstream cylinders 2 and 3 The mean drag coefficientsof cylinders 2 and 3 were basically identical to the value offlow past around a single circular cylinder except for the factthat the mean drag coefficients of cylinder 1 were a bit lessthan the value of a single circular cylinder

For the large spacing distance case of 119878119863 = 80 theflow pattern of any one of three cylinders only has a slightdifference with that of cross-flow around a single cylinderThe mean drag and lift coefficients RMS lift coefficients andSt of three cylinders were essentially equal to the value of asingle circular cylinder This result illustrated that the effectsbetween three cylinders basically disappeared for 119878119863 ge 80Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest


Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 51776155)

References

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

[2] Y Zhou and M M Alam ldquoWake of two interacting circularcylinders a reviewrdquo International Journal of Heat and FluidFlow vol 62 pp 510ndash537 2016

[3] Z Li M A Prsic M C Ong and B C Khoo ldquoLarge EddySimulations of flow around two circular cylinders in tandem inthe vicinity of a plane wall at small gap ratiosrdquo Journal of Fluidsand Structures vol 76 pp 251ndash271 2018

[4] LMa YGao ZGuo andLWang ldquoExperimental investigationon flow past nine cylinders in a square configurationrdquo FluidDynamics Research vol 50 no 2 pp 1ndash47 2018

[5] W Yan J Wu S Yang and Y Wang ldquoNumerical investigationon characteristic flow regions for three staggered stationarycircular cylindersrdquo European Journal of MechanicsmdashBFluidsvol 60 pp 48ndash61 2016

[6] S Yang W Yan J Wu C Tu and D Luo ldquoNumerical investi-gation of vortex suppression regions for three staggered circularcylindersrdquoEuropean Journal ofMechanicsmdashBFluids vol 55 nopart 1 pp 207ndash214 2016

[7] S Zheng W Zhang and X Lv ldquoNumerical simulation of cross-flow around three equal diameter cylinders in an equilateral-triangular configuration at low Reynolds numbersrdquo Computersamp Fluids vol 130 pp 94ndash108 2016

[8] GM Barros G Lorenzini L A Isoldi L AO Rocha and EDdos Santos ldquoInfluence ofmixed convection laminar flows on thegeometrical evaluation of a triangular arrangement of circularcylindersrdquo International Journal of Heat and Mass Transfer vol114 pp 1188ndash1200 2017

[9] M Shaaban and A Mohany ldquoFlow-induced vibration of threeunevenly spaced in-line cylinders in cross-flowrdquo Journal ofFluids and Structures vol 76 pp 367ndash383 2018

[10] Y LWu ldquoNumerical simulation of flows pastmultiple cylindersusing the hybrid local domain free discretization and immersedboundary methodrdquo Ocean Engineering vol 141 pp 477ndash4922017

[11] M S Bansal and S Yarusevych ldquoExperimental study of flowthrough a cluster of three equally spaced cylindersrdquo Experimen-tal Thermal and Fluid Science vol 80 pp 203ndash217 2017

[12] X Qiu H Liu M He R He and J Dong ldquoExperimentalstudy for the cross-flow around three cylinders in an isoscelesright triangle configurationrdquo Journal of Wind Engineering ampIndustrial Aerodynamics vol 170 pp 185ndash196 2017

[13] Y Bao Q Wu and D Zhou ldquoNumerical investigation of flowaround an inline square cylinder array with different spacingratiosrdquo Computers amp Fluids vol 55 pp 118ndash131 2012

[14] A B Harichandan and A Roy ldquoNumerical investigation of lowReynolds number flow past two and three circular cylindersusing unstructured grid CFR schemerdquo International Journal ofHeat and Fluid Flow vol 31 no 2 pp 154ndash171 2010

[15] Z Han D Zhou T He et al ldquoFlow-induced vibrations of fourcircular cylinders with square arrangement at low Reynoldsnumbersrdquo Ocean Engineering vol 96 pp 21ndash33 2015

[16] M Tatsuno H Amamoto and K Ishi-i ldquoEffects of interferenceamong three equidistantly arranged cylinders in a uniformflowrdquo Fluid Dynamics Research vol 22 no 5 pp 297ndash315 1998

[17] M Zhao K Kaja Y Xiang and L Cheng ldquoVortex-inducedvibration of four cylinders in an in-line square configurationrdquoPhysics of Fluids vol 28 no 2 pp 1ndash31 2016

[18] K LamW Q Gong and R M C So ldquoNumerical simulation ofcross-flow around four cylinders in an in-line square configu-rationrdquo Journal of Fluids and Structures vol 24 no 1 pp 34ndash572008

[19] E Feldshtein J Jozwik and S Legutko ldquoThe influence of theconditions of emulsionmist formation on the surface roughnessof aisi 1045 steel after finish turningrdquo Advances in Science andTechnology Research Journal vol 10 no 30 pp 144ndash149 2016

[20] G Tang L Cheng F Tong L Lu and M Zhao ldquoModesof synchronisation in the wake of a streamwise oscillatorycylinderrdquo Journal of Fluid Mechanics vol 832 pp 146ndash169 2017

[21] D Liu Y-L Zheng A Moore and M Ferdows ldquoSpectral ele-ment simulations of three dimensional convective heat transferrdquoInternational Journal of Heat and Mass Transfer vol 111 pp1023ndash1038 2017

[22] M R Machado and J M C Dos Santos ldquoReliability analysis ofdamaged beam spectral element with parameter uncertaintiesrdquoShock andVibration vol 2015 Article ID 574846 12 pages 2015

[23] Y-X Huang H Tian and Y Zhao ldquoEffects of cable on thedynamics of a cantilever beam with tip massrdquo Shock andVibration vol 2016 Article ID 7698729 11 pages 2016

[24] A Beck and C-D Munz ldquoDirect aeroacoustic simulationsbased on high order discontinuous Galerkin schemesrdquo inComputational Acoustics vol 579 of CISM Courses and Lectpp 159ndash204 Springer Cham 2018

[25] C D Cantwell D Moxey A Comerford et al ldquoNektar++an open-source spectralhp element frameworkrdquo ComputerPhysics Communications vol 192 pp 205ndash219 2015

[26] ZHanD Zhou XGui and J Tu ldquoNumerical study of flowpastfour square-arranged cylinders using spectral elementmethodrdquoComputers amp Fluids vol 84 pp 100ndash112 2013

[27] H Ding C Shu K S Yeo and D Xu ldquoSimulation ofincompressible viscous flows past a circular cylinder by hybridFD scheme and meshless least square-based finite differencemethodrdquo Computer Methods Applied Mechanics and Engineer-ing vol 193 no 9-11 pp 727ndash744 2004

[28] S-G Cai A Ouahsine J Favier and Y Hoarau ldquoMovingimmersed boundary methodrdquo International Journal for Numer-ical Methods in Fluids vol 85 no 5 pp 288ndash323 2017

[29] L I G Kovasznay ldquoLaminar flow behind a two-dimensionalgridrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 44 no 1 pp 58ndash62 1948

[30] CGeuzaine and J F Remacle ldquoGmsh a 3-Dfinite elementmeshgenerator with built-in pre- and post-processing facilitiesrdquoInternational Journal for Numerical Methods in Engineering vol79 no 11 pp 1309ndash1331 2009

[31] B N Rajani A Kandasamy and S Majumdar ldquoNumericalsimulation of laminar flow past a circular cylinderrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 33 no 3 pp 1228ndash12472009

[32] W Zhiwen Y Zhou and H Li ldquoFlow-visualization of a twoside-by-side cylinder wakerdquo Journal of Flow Visualization andImage Processing vol 9 no 2 pp 121ndash138 2002


[33] Z Gu and T Sun ldquoClassification of flow pattern on three circu-lar cylinders in equilateral-triangular arrangementsrdquo Journal ofWind Engineering amp Industrial Aerodynamics vol 89 no 6 pp553ndash568 2001

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom


Figure 1 Offshore platforms pillar groups

Figure 2 Oriental pearl radio and TV tower Shanghai

on the near wake development Qiu et al [12] experimentallymeasured the pressure coefficients of three cylinders in anequilateral-triangular arrangement Significant complicatedaerodynamic interference effects were observed betweenthree cylinders Several other literatures related to three ormore cylinders are also published [13ndash19] To conclude allthe above researches showed that the flow patterns and forcecharacteristics are the results of the combined effect of Respacing distance configuration arrangement and incidentangle

With the development of hardware technology numericalsolutions can be easily obtained compared to experimentresults Numerous numerical algorithms have been usedto solve the flow around cylinders such as finite vol-ume method finite element method immersed boundarymethod spectral method and meshless method Each algo-rithm has pros and cons However in order to capture moreflow details it has been a trend to apply high resolutionmethod into the solving of incompressible N-S equationsSpectral element method (SEM) mixes the exponential con-vergence property of spectral schemes and good geomet-ric flexibility feature of finite element method Therefore

exponential (p-type) and algebraic (h-type) convergence canbe respectively achieved by increasing polynomial order andelement numbers In addition the spectral element methodhas been applied tomany research fields such as flowandheattransfer [20 21] wave propagation [22] structure vibration[23] and aeroacoustics [24]

The paper is organized as follows In Section 2 a briefdescription of the governing equation is given In Section 3the accuracy of the spectral element numerical code isverified by the Kovasznay flow and flow past around asingle circular cylinder Then flows between three cylindersin an equilateral-triangular arrangement are investigated indetail Finally Section 5 contains the conclusions of ourwork

2 Governing Equationsand Numerical Methods

The governing equations used for unsteady and incompress-ible viscous flow are incompressible Navier-Stokes equationand continuity equation which are expressed as below

120597k120597119905 + k sdot nablak = minusnabla119901 + ]nabla2knabla sdot k = 0

(1)

where v is the velocity vector 119901 represents the pressure] is the kinematic viscosity ] = 1Re Re is Reynoldrsquosnumber with respect to the circular cylinder diameter119863 andcharacteristic velocity 119880 Re = 119880 sdot 119863]

The numerical calculation was carried out using theopen-source code Nektar++ [25] The incompressible N-S equation solver in Nektar++ library is based on thespectral element method in space discretization and high-order time splitting method in time discretization In thespectral element method the computational domain Ω isdecomposed into several nonoverlapping subdomains Eachspectral element is mapped into a standard element Finallythe variational problem discretized by the means of SEMis equivalent to solving Differential Algebraic EquationsAnd the block diagram of the spectral element method forincompressible flow problems is shown in Figure 3 Moredetails can be obtained from the literatures [20 24 25]

3 Validation Study

31 Kovasznay Flow In order to validate the exponentialconvergence rate of the numerical scheme we consider thetwo-dimensional Kovasznay flow in a rectangular area Ω =[minus05 1] times [minus05 15] This flow problem is a good test for thealgorithm because an analytical closed form solution existsfor the incompressible flow field At the same time the flowpattern is similar to the wake flow of a circular cylinderSo Kovasznay flow can be used to judge the convergenceand numerical accuracy of the method for solving incom-pressible N-S equations The analytical solution [29] is givenby

119906 = 1 minus 119890120582119909 cos 2120587119910


Meshgeneration

(Gmsh)

Nektar++solver

Advectionterm

Poisson term

Helmholtzterm

Initialconditions

Final solution

Preprocessing

Postprocessing

Time integration


V = 1205822120587119890120582119909 sin 2120587119910119901 = 1199010 minus 121198902120582119909


4 + 41205872(2)



x

y


0

05

1

15u

240212184156128100072044016minus012minus040


minus1

minus3

minus5

minus7

minus9

minus11

minus13

Log(

L 2er

ror)

up

3 5 7 9 11 13 15 17

Polynomial order P




20D 30D

20D

D=1x

y

u = 0 v = 0

20D

p = 0u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y= 0 = 0

휕u 휕y= 0 = 0


xD

yD


minus15

minus10

minus5

0

5

10

15

20





119862119889 = 21198651198891205881198802119863

xD

yD


minus05

0

05

1





119879

0

119862119889119889119905119862119897 = 1119879 int

119879

0

119862119897119889119905

1198621015840119897= radic 1119879 int

119879

0

1198621198972119889119905(3)



140 142 144 146 148 150 152 154 156 158 160

t

CdCl

130

132

134

136

138

140

C d

minus14

minus10

minus06

minus02

02

06

10

14

C l


13

08

03

minus02

minus07

minus12

minus17

Cpav

e

0 20 40 60 80 100 120 140 160 180

휃

Rajani et alpresent




20D 30D

D=1x

y

20D

20D

p = 0s12

3u = 0 v = 0

u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y = 0 = 0

휕u 휕y = 0 = 0



yD

20 30minus20

minus15

minus10

minus5

0

5

10

15

20






119875 119862119889 St

31 11814 0182 13470 0183 13465 018

41 11811 0182 13459 0183 13459 018

51 11810 0182 13459 0183 13460 018

xD

yD


minus2

0

2

4











xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus5

minus3

minus1

1

3

5

7


(a)xD

yD


minus5

minus3

minus1

1

3

5

7


(b)





400 405 410 415 420 425 430 435 440

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

065

070

075

080

085

090

095

100

105

110

115

C d

minus04

minus03

minus02

minus01

00

01

02

03

04


t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

260 265 270 275 28011

12

13

14

15

16

17

C d

minus3

minus2

minus1

0

1

2

3

C l




t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

180 185 190 195 200125

130

135

140

145

150

C d

minus20

minus15

minus10

minus05

00

05

10

15

20

C l


15

14

13

12

11

10

09

08

07

06

C d

Cd 1Cd 2

Cd 3Single cylinder

2 3 4 5 6 7 8 91 11 12 1310

SD






2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l



5 Conclusions


2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021










Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Meshgeneration

(Gmsh)

Nektar++solver

Advectionterm

Poisson term

Helmholtzterm

Initialconditions

Final solution

Preprocessing

Postprocessing

Time integration


V = 1205822120587119890120582119909 sin 2120587119910119901 = 1199010 minus 121198902120582119909


4 + 41205872(2)



x

y


0

05

1

15u

240212184156128100072044016minus012minus040


minus1

minus3

minus5

minus7

minus9

minus11

minus13

Log(

L 2er

ror)

up

3 5 7 9 11 13 15 17

Polynomial order P




20D 30D

20D

D=1x

y

u = 0 v = 0

20D

p = 0u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y= 0 = 0

휕u 휕y= 0 = 0


xD

yD


minus15

minus10

minus5

0

5

10

15

20





119862119889 = 21198651198891205881198802119863

xD

yD


minus05

0

05

1





119879

0

119862119889119889119905119862119897 = 1119879 int

119879

0

119862119897119889119905

1198621015840119897= radic 1119879 int

119879

0

1198621198972119889119905(3)



140 142 144 146 148 150 152 154 156 158 160

t

CdCl

130

132

134

136

138

140

C d

minus14

minus10

minus06

minus02

02

06

10

14

C l


13

08

03

minus02

minus07

minus12

minus17

Cpav

e

0 20 40 60 80 100 120 140 160 180

휃

Rajani et alpresent




20D 30D

D=1x

y

20D

20D

p = 0s12

3u = 0 v = 0

u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y = 0 = 0

휕u 휕y = 0 = 0



yD

20 30minus20

minus15

minus10

minus5

0

5

10

15

20






119875 119862119889 St

31 11814 0182 13470 0183 13465 018

41 11811 0182 13459 0183 13459 018

51 11810 0182 13459 0183 13460 018

xD

yD


minus2

0

2

4











xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus5

minus3

minus1

1

3

5

7


(a)xD

yD


minus5

minus3

minus1

1

3

5

7


(b)





400 405 410 415 420 425 430 435 440

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

065

070

075

080

085

090

095

100

105

110

115

C d

minus04

minus03

minus02

minus01

00

01

02

03

04


t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

260 265 270 275 28011

12

13

14

15

16

17

C d

minus3

minus2

minus1

0

1

2

3

C l




t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

180 185 190 195 200125

130

135

140

145

150

C d

minus20

minus15

minus10

minus05

00

05

10

15

20

C l


15

14

13

12

11

10

09

08

07

06

C d

Cd 1Cd 2

Cd 3Single cylinder

2 3 4 5 6 7 8 91 11 12 1310

SD






2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l



5 Conclusions


2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021










Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



20D 30D

20D

D=1x

y

u = 0 v = 0

20D

p = 0u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y= 0 = 0

휕u 휕y= 0 = 0


xD

yD


minus15

minus10

minus5

0

5

10

15

20





119862119889 = 21198651198891205881198802119863

xD

yD


minus05

0

05

1





119879

0

119862119889119889119905119862119897 = 1119879 int

119879

0

119862119897119889119905

1198621015840119897= radic 1119879 int

119879

0

1198621198972119889119905(3)



140 142 144 146 148 150 152 154 156 158 160

t

CdCl

130

132

134

136

138

140

C d

minus14

minus10

minus06

minus02

02

06

10

14

C l


13

08

03

minus02

minus07

minus12

minus17

Cpav

e

0 20 40 60 80 100 120 140 160 180

휃

Rajani et alpresent




20D 30D

D=1x

y

20D

20D

p = 0s12

3u = 0 v = 0

u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y = 0 = 0

휕u 휕y = 0 = 0



yD

20 30minus20

minus15

minus10

minus5

0

5

10

15

20






119875 119862119889 St

31 11814 0182 13470 0183 13465 018

41 11811 0182 13459 0183 13459 018

51 11810 0182 13459 0183 13460 018

xD

yD


minus2

0

2

4











xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus5

minus3

minus1

1

3

5

7


(a)xD

yD


minus5

minus3

minus1

1

3

5

7


(b)





400 405 410 415 420 425 430 435 440

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

065

070

075

080

085

090

095

100

105

110

115

C d

minus04

minus03

minus02

minus01

00

01

02

03

04


t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

260 265 270 275 28011

12

13

14

15

16

17

C d

minus3

minus2

minus1

0

1

2

3

C l




t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

180 185 190 195 200125

130

135

140

145

150

C d

minus20

minus15

minus10

minus05

00

05

10

15

20

C l


15

14

13

12

11

10

09

08

07

06

C d

Cd 1Cd 2

Cd 3Single cylinder

2 3 4 5 6 7 8 91 11 12 1310

SD






2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l



5 Conclusions


2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021










Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



140 142 144 146 148 150 152 154 156 158 160

t

CdCl

130

132

134

136

138

140

C d

minus14

minus10

minus06

minus02

02

06

10

14

C l


13

08

03

minus02

minus07

minus12

minus17

Cpav

e

0 20 40 60 80 100 120 140 160 180

휃

Rajani et alpresent




20D 30D

D=1x

y

20D

20D

p = 0s12

3u = 0 v = 0

u = 1v = 0

휕u 휕x = 0

휕v 휕x = 0

휕u 휕y = 0 = 0

휕u 휕y = 0 = 0



yD

20 30minus20

minus15

minus10

minus5

0

5

10

15

20






119875 119862119889 St

31 11814 0182 13470 0183 13465 018

41 11811 0182 13459 0183 13459 018

51 11810 0182 13459 0183 13460 018

xD

yD


minus2

0

2

4











xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus5

minus3

minus1

1

3

5

7


(a)xD

yD


minus5

minus3

minus1

1

3

5

7


(b)





400 405 410 415 420 425 430 435 440

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

065

070

075

080

085

090

095

100

105

110

115

C d

minus04

minus03

minus02

minus01

00

01

02

03

04


t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

260 265 270 275 28011

12

13

14

15

16

17

C d

minus3

minus2

minus1

0

1

2

3

C l




t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

180 185 190 195 200125

130

135

140

145

150

C d

minus20

minus15

minus10

minus05

00

05

10

15

20

C l


15

14

13

12

11

10

09

08

07

06

C d

Cd 1Cd 2

Cd 3Single cylinder

2 3 4 5 6 7 8 91 11 12 1310

SD






2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l



5 Conclusions


2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021










Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119875 119862119889 St

31 11814 0182 13470 0183 13465 018

41 11811 0182 13459 0183 13459 018

51 11810 0182 13459 0183 13460 018

xD

yD


minus2

0

2

4











xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus5

minus3

minus1

1

3

5

7


(a)xD

yD


minus5

minus3

minus1

1

3

5

7


(b)





400 405 410 415 420 425 430 435 440

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

065

070

075

080

085

090

095

100

105

110

115

C d

minus04

minus03

minus02

minus01

00

01

02

03

04


t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

260 265 270 275 28011

12

13

14

15

16

17

C d

minus3

minus2

minus1

0

1

2

3

C l




t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

180 185 190 195 200125

130

135

140

145

150

C d

minus20

minus15

minus10

minus05

00

05

10

15

20

C l


15

14

13

12

11

10

09

08

07

06

C d

Cd 1Cd 2

Cd 3Single cylinder

2 3 4 5 6 7 8 91 11 12 1310

SD






2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l



5 Conclusions


2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021










Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus3

minus1

1

3

5


(a)xD

yD


minus3

minus1

1

3

5


(b)


xD

yD


minus5

minus3

minus1

1

3

5

7


(a)xD

yD


minus5

minus3

minus1

1

3

5

7


(b)





400 405 410 415 420 425 430 435 440

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

065

070

075

080

085

090

095

100

105

110

115

C d

minus04

minus03

minus02

minus01

00

01

02

03

04


t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

260 265 270 275 28011

12

13

14

15

16

17

C d

minus3

minus2

minus1

0

1

2

3

C l




t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

180 185 190 195 200125

130

135

140

145

150

C d

minus20

minus15

minus10

minus05

00

05

10

15

20

C l


15

14

13

12

11

10

09

08

07

06

C d

Cd 1Cd 2

Cd 3Single cylinder

2 3 4 5 6 7 8 91 11 12 1310

SD






2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l



5 Conclusions


2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021










Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



400 405 410 415 420 425 430 435 440

t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

065

070

075

080

085

090

095

100

105

110

115

C d

minus04

minus03

minus02

minus01

00

01

02

03

04


t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

260 265 270 275 28011

12

13

14

15

16

17

C d

minus3

minus2

minus1

0

1

2

3

C l




t

Cd 1Cd 2Cd 3

Cl 1Cl 2Cl 3

180 185 190 195 200125

130

135

140

145

150

C d

minus20

minus15

minus10

minus05

00

05

10

15

20

C l


15

14

13

12

11

10

09

08

07

06

C d

Cd 1Cd 2

Cd 3Single cylinder

2 3 4 5 6 7 8 91 11 12 1310

SD






2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l



5 Conclusions


2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021










Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



2 3 4 5 6 7 8 9 10 11 12 131

SD

015

010

005

000

minus005

minus010

minus015

C l

Cl 1Cl 2

Cl 3Single cylinder


2 3 4 5 6 7 8 9 10 11 12 131

SD

C㰀l 1

C㰀l 2

C㰀l 3

Single cylinder

minus02

00

02

04

06

08

10

C㰀 l



5 Conclusions


2 3 4 5 6 7 8 9 10 11 12 131

SD

St 1St 2

St 3Single cylinder

013

014

015

016

St

017

018

019

020

021










Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Acknowledgments


References





































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Spectral Element Numerical Investigation of Flow...

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