Research Article High Performance Computation of a...

Research ArticleHigh Performance Computation of a Jet in Crossflow byLattice Boltzmann Based Parallel Direct Numerical Simulation

Jiang Lei1 Xian Wang1 and Gongnan Xie2

1State Key Laboratory for Strength and Vibration of Mechanical Structures Xirsquoan Jiaotong University Xirsquoan 710049 China2School of Mechanical Engineering Northwestern Polytechnical University Xirsquoan 710049 China

Correspondence should be addressed to Xian Wang wangxianmailxjtueducn

Received 19 September 2014 Accepted 26 January 2015

Academic Editor Reyolando M Brasil

Copyright copy 2015 Jiang Lei et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Direct numerical simulation (DNS) of a round jet in crossflow based on lattice Boltzmann method (LBM) is carried out on multi-GPU cluster Data parallel SIMT (single instruction multiple thread) characteristic of GPU matches the parallelism of LBM wellwhich leads to the high efficiency of GPU on the LBM solver With present GPU settings (6 Nvidia Tesla K20M) the present DNSsimulation can be completed in several hours A grid system of 15 times 108 is adopted and largest jet Reynolds number reaches 3000The jet-to-free-stream velocity ratio is set as 33 The jet is orthogonal to the mainstream flow direction The validated code showsgood agreement with experiments Vortical structures of CRVP shear-layer vortices and horseshoe vortices are presented andanalyzed based on velocity fields and vorticity distributions Turbulent statistical quantities of Reynolds stress are also displayedCoherent structures are revealed in a very fine resolution based on the second invariant of the velocity gradients

1 Introduction

A jet in crossflow (JICF) also known as transverse jetnormally describes a jet of fluid which enters and interactswith the crossflow and the resulting flow field JICF has wideapplications in many engineering fields such as gas turbineimpingement cooling and film cooling fuel injection incombustors thrust vectoring system in turbojet propulsionand reaction control for missiles For a traditional JICFproblem near the flow field of the transverse jet four types ofvortical structures can be observed [1] due to the interactionbetween mainstream and jet as shown in Figure 1 (1) thecounter-rotating vortex pair (CRVP) also known as kidneyvortices (2) the horseshoe vortices system (3) the jet shear-layer vortices due to Kelvin-Helmholtz instabilities (4) thewake vortices The CRVP and the horseshoe vortices arenormally defined by mean flow even though they may haveunsteady part The shear-layer vortices and the wake vorticesare naturally unsteady

Those vortical structures are very important to fluid flowand heat transfer behaviors of those fields where JICF isapplied If gas turbine film cooling is taken as an example film

coverage and the cooling effectiveness are closely relevant tothose large structures no matter whether they are definedas steady or naturally unsteady The ldquosteadyrdquo CRVP liftsthe coolant up and mixes with the mainstream weakeningthe film coverage and cooling effectiveness The ldquounsteadyrdquoshear-layer vortices have different directions determinedby the shape of the ejection hole and consequently eitherstrengthen or suppress the CRVP [2] To accurately predictthose large structures in which unsteadiness and anisotropyare inherent a time-resolved scheme is required in the flowfield calculation with fine spatial discretization

The flow field of JICF is characterized by both anisotropiclarge-scale structures which break down to smaller sizesThese resultant small scale structures are isotropic anddissipation is dominant The typical energy spectrum forthe JICF inherently requires capturing the flow structuresacross the spectrum of all scales and thus a time and spaceaccurate calculation is needed Direct numerical simulation(DNS) and large eddy simulation (LES) are commonly usedto resolve the turbulent characteristics of JICF in large-scale parallel computation devices However inherent fea-tures of Navier-Stokes equation or transport equation make

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 956827 11 pageshttpdxdoiorg1011552015956827

2 Mathematical Problems in Engineering

Counter-rotating vortex pair

Wake vorticesHorseshoe vortices

Jet shear-layervortices

Wall

Crossflow

Figure 1 Four vortical structures associated with the JICF as in [1]

the parallel processing with low efficiency Thus a highlyparallel scheme is extremely helpful to those large-scalesimulations of turbulent flow field

Lattice Boltzmannmethod (LBM) has been regarded as apromising candidate for years due to its major merits of fullyparallel algorithm Unlike conventional numerical schemesbased on assumption and discretizations of macroscopiccontinuum the LBM adopts microscopic models [3] Otheradvantages of LBM include (1) easy treatment of boundaryconditions and (2) easy programming As a result the LBMhas its wide application in scientific and engineering researchsuch as Biofluid and porous medium [4ndash6] However toresolve the turbulent flow LBM-based DNS and LES requirea very high grid resolution and thus huge computationresource is needed

Graphic Processing Unit (GPU) brings excellent hard-ware conditions for these simulations With the develop-ment of computing platforms for example CUDA [7] andOpenCL the use of GPU to accelerate nongraphic computa-tions has drawn much more attention [8 9] Due to its highperformance of floating-point arithmetic operation togetherwith wide memory bandwidth and good programmabilitygeneral-purpose GPU (GPGPU) has its huge advantage overCPU in turbulent flow simulation [10ndash12] and thus hasbeen applied in fields like weather prediction crystal growthprocess and air flow in the city [13ndash15]

The feasibility for LBM-based DNS has some preliminaryvalidation for turbulent flows between parallel plates [16]further verification for JICF is still empty In this paper thelattice Boltzmann based DNS on JICF model is performedusing D3Q19model in multi-GPUs With further validationof this code this paper aims to resolve the JICF flow fieldincluding the time-averaged and instantaneous velocitiesvorticities and turbulent quantities of Reynolds stress Thecharacteristic coherent hairpin structures are also revealedand discussed

2 Lattice Boltzmann Equations

Lattice Boltzmann equation is the special form of theBoltzmann-BGK equation [17] in which discretizationapplies to velocity time and space The Boltzmann equation

for the discrete velocity distribution on a discrete lattice is asfollows

120597119891119894

120597119905

+ e119894sdot nabla119891119894= Ω119894 (1)

where119891119894is the particle velocity distribution function and e

119894is

the particle velocity in the 119894th direction Here the scheme ofLBM is described as D119898Q119899 [18] ldquoD119898rdquomeans ldquo119898 dimensionrdquoand ldquoQ119899rdquo stands for ldquo119899 lattice speedsrdquo For two-dimensionalproblems D2Q9 model is most commonly used and means2119863 and 9 speeds For three-dimensional problems severalcubic models are used such as D3Q13 D3Q15 D3Q19 andD3Q27 (119899 = 13 15 19 or 27) Ω

119894is the collision operator

With Boltzmann-BGK approximation

Ω119894= minus

1

120591

(119891119894minus 119891

eq119894) (2)

Combine (1) and (2) then it changes to

120597119891119894

120597119905

+ e119894sdot nabla119891119894= minus

1

120591

(119891119894minus 119891

eq119894) (3)

where119891eq119894

is the local equilibrium and 120591 is the relaxation timeThe equilibriumdistribution formhas to be correctly adoptedto restore the Navier-Stokes equations For 3119863 models withthe 13- 15- 19- and 27-speed lattice an appropriate distribu-tion function is written as

119891eq119894= 120588120603119894[1 + 3e

119894sdot u + 9

2

(e119894sdot u)2 minus 3

2

u sdot u] (4)

In the present paper D3Q19 scheme is used with par-ticle velocity as shown in Figure 2 119890

0(0 0 0) 119890

1(1 0 0)

1198902(minus1 0 0) 119890

3(0 1 0) 119890

4(0 minus1 0) 119890

5(0 0 1) 119890

6(0 0 minus1)

1198907(1 1 0) 119890

8(minus1 1 0) 119890

9(1 minus1 0) 119890

10(minus1 minus1 0) 119890

11(1 0 1)

11989012(minus1 0 1) 119890

13(1 0 minus1) 119890

14(minus1 0 minus1) 119890

15(0 1 1) 119890

16(0 minus1

1) 11989017(0 1 minus1) and 119890

18(0 minus1 minus1) Corresponding weighting

factors are 1206030= 13 120603

1sim 1206036= 118 and 120603

7sim 12060318= 136

If (3) is further discretized in both space 119909 and time 119905 thediscretized form can be written as

119891119894(x + e

119894120575119905 119905 + 120575119905) minus 119891

119894 (x 119905) = minus1

120591

(119891119894 (x 119905) minus 119891eq

119894(x 119905))

(5)

Mathematical Problems in Engineering 3

z

x

y

12

3

4

5

6

11

12

13

14

78

910

15

16

17

18

Figure 2 D3Q19 LBMmodel

The above equation is the LBGK model in which 120591 = 120582120575119905

is the nondimensional relaxation time The viscosity in themacroscopic Navier-Stokes equation can be expressed as ] =(120591 minus 12)119862

119904

2120575119905 Equation (5) is usually solved in a standard

way assuming 120575119905 = 1 and divided into two steps as followsCollision step is as follows

119891119894 (x 119905) = 119891 (x 119905) minus 1

120591

(119891 (x 119905) minus 119891eq119894(x 119905)) (6)

streaming step is as follows

119891119894(x + e

119894 119905 + 1) =

119891119894 (x 119905) (7)

where 119891119894and

119891119894denote the pre- and postcollision states of

the distribution function Please note that in the collision step119891119894is calculated absolutely locally and in streaming step 119891

119894is

only relevant to its neighboring nodes Thus the LBM codeitself is highly suitable for parallel processing and capable ofobtaining high efficiency Macroscopic quantities such as 120588and can be calculated as follows

120588 =

119873

sum

119894=0

119891119894=

119873

sum

119894=0

119891eq119894

120588u =119873

sum

119894=0

119891119894e119894=

119873

sum

119894=0

119891eq119894e119894

(8)

The lattice speed of sound is 119862119904= 1radic3 and pressure can be

obtained from the state equation of the ideal gas

119901 = 120588119862119904

2=

120588

3

(9)

Considering boundaries of 119891 in transverse direction (front

and back boundaries) periodic boundary conditions are

used in this paper For solid walls the following boundaryconditions are applied [19]

119891119894= 119891

eq119894+ 119891

neq119894

= 119891eq119894+ 119891

neq119894-neighbor

= 119891eq119894+ (

119891119894-nieghbor minus 119891

eq119894-neighbor)

(10)

in which 119891eq119894

is calculated by macroscopic u and 120588 on bound-aries by (4) The nonequilibrium distribution on boundaries119891neq119894

is assumed to be of the same value as that in theneighboring inner node 119891neq

119894-neighbor 119891119894-neighbor and 119891eq119894-neighbor on

the inner node are computed by (6) and (4) respectively

3 GPU Settings

A well-known inherent advantage of lattice Boltzmannmethod is its parallelism Previous study [20] compared theacceleration performance between Navier-Stokes solver andcurrent LBM solver in which incompressible flow arounda cylinder was simulated on GPU (GeForce GTX280) andCPU (XeonE5420 25GHz) separately For theNavier-Stokessolver in which Red-Black scheme and multi-grids wereapplied the GPU acceleration over CPU was 137 Compar-atively for LBM on the same grid scale the acceleration was874 In addition to the acceleration performance of LBM onGPU as mentioned the difficulty of coding is much smallerfor LBM than Navier-Stokes on GPU

However the expense to LBMrsquos parallelism is also obviousdue to substantial variables and consequent huge memoryconsumption For the D3Q19 model in single precision cal-culation the theoretical GPU memory requirement on 15 times108 grids (1024times 256times 600) is over 30G Besides data transferbetween GPUs is realized byMPI (message passing interface)and CudaMemcpy() sentences in CUDA From the previousexperience [11] if the number of GPUs is less than 10 the1119863 domain partitioning is more efficient than 2119863 and 3119863 Inthe current research 6GPUs of Nvidia Tesla K20M are used


Six processors

1024 times 256 times 100



Data communication MPI + CudaMemcpy

Figure 3 1D partitioning in 119911-direction

Lx

Lx1 Lx2

Ly

Lz

X

Y

Z

D

Figure 4 Physical model of jet in crossflow

20

15

10

5

10minus1

100

101

102

z+

U+

Sublayer

Turbulentcore

Figure 5 Streamwise velocity boundary layer profile

and the current 1119863 partitioning of the GPUs is illustrated inFigure 3

4 Physical and Numerical Models

The physical model of the jet in cross flow problem is shownin Figure 4 The computational domain is hexahedral inshape The mainstream flow is in 119909-direction while the jetis orthogonal to the mainstream in 119911-direction and ejectedfrom the bottom plane uniformly with diameter 119863 Originof the coordinates is located at the center of the roundjet exit The flow domain is zero-gradient in pressure and

no-slip boundary conditions are applied on the bottom planeexcluding the jet exit On transverse planes (with normalin 119910-direction) periodic boundary conditions are appliedThe inlet turbulent profile of velocity is given through acalculation procedure similar to the so-called ldquorescalingprocessrdquo as described in [21] The results presented in thispaper are based on flow parameters and boundary conditionslisted as follows

(i) Ejecting hole diameter (119863) = 75(ii) Domain length (119871119909119863 = (1198711199091+1198711199092)119863 = 5+7) = 12(iii) Domain span (119871119910119863) = 34


UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16

LBM-DNSExperiment by Meyer et al

UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0


y = NY2

y = NY2

y = NY2

y = NY2

x = 05D

x = 3D

zD

x = 05D

x = 3D

zD

zD

zD

Figure 6 Streamwise and spanwise velocities profiles after the jet

(iv) Domain height (119871119911119863) = 64

(v) Free-stream turbulence intensity (TI) = 3

(vi) Boundary layer height (120575119863) = 02

(vii) Velocity ratio (VR = 119880119895119880infin) = 33

Reynolds number Re119895is defined based on jet velocity

119880119895and film hole diameter 119863 as Re

119895= 119880119895119863120592 The largest

Reynolds number Re119895reaches 3000 on grids (1024 times 256 times

600) and several calculations are performed with differentdimensions grids and jet locations trialed as shown inTable 1Thediscussion is based on the 1st case in the tableTheflow domain is kept in the ldquolow-speedrdquo range and velocitymagnitude is smaller than 03 times of the lattice sound speed

119862119904 Thus the LBGK model used in current study is suitable

[22]

5 Model Validation

Nondimensional time-averaged streamwise velocity 119880+

=

119880119906120591versus 119911+ = 119911119906

120591120592 is presented in Figure 5The location

is selected before the jet at 119909 = minus2119863 and 119910 = 1198731198842 toeliminate the influence from the jet A turbulent boundarylayer profile is obtained consisting of laminar sublayer andlog layer which is consistent with the Von Karman mixinglength theory

Further comparisons with experiments are made inregions where the mainstream is disturbed by the jet asshown in Figure 6 The experiments were done by Meyer


Table 1 Testing cases matrix

Jet diameter (119863) Domain (119871119909 times 119871119910 times 119871119911) Jet location (1198711199091) Mesh75 12D times 34D times 8D 26D 896 times 256 times 60075 136D times 34D times 8D 4D 1024 times 256 times 600100 10D times 5D times 8D 4D 1024 times 512 times 60075 85D times 34D times 64D 26D 640 times 256 times 48075 85D times 4D times 64D 26D 640 times 300 times 480

Etu

r10

0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

101

102

Frequency (Hz)

minus53

x = 6D z = 4D y = NY2

Figure 7 Spectral power density of velocity fluctuations

et al [23 24] using stereoscopic PIV in a JICF problemTime-averaged streamwise and spanwise velocities (119880119880

infin

and119882119880infin) are presented in spanwise direction (with respect

to 119911119863) at locations of 119909 = 05119863 119910 = 1198731198842 and 119909 = 3119863 119910 =1198731198842 Results show good agreement with experimental dataIn current simulation a uniformly distributed velocity profileis set at the jet exit while in experiment the jet is suppliedthrough a tube and a turbulent profile has been developedinside itThus an overestimation of the jetrsquos momentum nearits boundary of the jet hole can be expected and seen inFigure 6

6 Results and Analysis

61 Energy Spectra Fast Fourier transformation (FFT) isperformed to a time series of turbulent energy 119864tur at thesample frequency of 2000 at 119909 = 6119863 119911 = 4119863 and 119910 =

1198731198842 The turbulent energy is defined as 119864tur = (12)(11990610158402+

V10158402 + 11990810158402) The corresponding turbulent power spectrum

is presented versus frequency as shown in Figure 7 Thecalculated turbulence decay rate is close to the theoreticalcanonical value of minus53 and no obvious spectrum buildupis observed which indicate the current mesh resolution isadequate

62 Velocity Field and Vorticity The 2119863 time-averagedvelocity (119880119882) and instantaneous velocity (119906 119908) fields in

the middle plane of 119910-direction (119910 = 1198731198842) are plottedin Figures 8(a) 8(b) and 8(c) respectively The correspond-ing vorticity distributions based on the time-averaged andinstantaneous velocities are also shown in the figure In Figure8(a) the vertical jet is bended toward exit by the mainstreamflow and phenomenon like flow passing a solid obstacle isobserved in the wake region As shown in the enlarging plotof flow field the approaching wall boundary layer meets withadverse pressure gradient before the jet and consequentlyseparates and forms the horseshoe vortex Considering thevorticity field negative and positive values are observed inleading edge and trailing edge along the jet trajectory andtrivial values exist in the wake region Instantaneous velocityand vorticity fields at time steps 80000 and 120000 areshown in Figures 8(b) and 8(c) respectively Generation andshedding of Kelvin-Helmholtz vortices are introduced by theshear layers between the mainstream and the jet on bothleading edge and trailing edge The shed vortices are mixingwith the mainstream and dissipated in the downstreamregion quickly In the wake region of the jet wake vortices areobserved through those nontrivial vorticity values howeverno vortex tube is directly observed in this cross section viewTo compare the time-averaged fields with their instantaneouscounterparts the Kelvin-Helmholtz vortices are not availableafter averaging which proves the shear-layer induced vorticesare inherently unsteady and periodic


Horseshoevortex

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(a) Time-averaged

Shear-layervortices

Wakevortices

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(b) Time step = 80000

Shear-layervortices

vorticesWake

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(c) Time step = 120000

Figure 8 Time-averaged and instantaneous 2119863 velocity vector and vorticity at 119910 = 1198731198842 plane

The 2119863 instantaneous velocity (119906 V) and time-averagedvelocity (119880119881) fields are displayed at locations = 05119863 1119863and 2119863 respectively in Figure 9The corresponding vorticitydistributions are also displayed in the figure In Figure 9(a)the instantaneous velocity vectors and vorticity distributionsat time step 80000 are shown Separation of mainstreamoccurs near the mid-chord portion Further vortex sheddingand dissipation are observed in the downstream area Wakeregion is formed at the backside of the jet A characteristicflow pattern of the wake by the transverse jet is obtainedwhich is significantly different from that by a solid cylinder asdescribed in [1] In Figures 9(b) and 9(c) the time-averagedvelocity vectors and vorticity distributions are displayedSimilar to flow passing a solid cylinder along the stream-wise direction positive and negative values appear when

the mainstream starts to separate In downstream region thevorticity recovers to trivial values due to the dissipation ofthose periodically-shed vortices

The 2119863 time-averaged velocity (119881119882) fields are presentedin several streamwise planes at locations 119909 = 1119863 15119863and 3119863 in Figures 10(a) 10(b) and 10(c) respectively withvorticity distributions As shown in Figures 10(a) and 10(b)the CRVP system consists of two layers the upper one andthe lower one For a round hole as used in the current studythe rotating directions of the two pairs are the same Thisldquotwo-deckrdquo vortices structure is very similar to that observedin [2] by laser-induced fluorescence By comparing amongFigures 10(a) 10(b) and 10(c) it is found that when movingdownstream the vertical component (119882) of the 2119863 velocityreduces because of the jetrsquos bending At the same time


Wake vortices

Time step = 80000

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(a) 119911 = 05119863

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(b) 119911 = 1119863

Ref vector

01

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(c) 119911 = 2119863

Figure 9 2119863 velocity vector and vorticity at planes of 119911 = 05119863 1119863 and 2119863

the lower layer vortices lift in vertical direction and weakenin strength Finally in far downstream region (119909 = 3119863) the119882-component is at the same level of 119881-component since thejet is almost fully bended and those two layers fully emergewith strength further weakened

63 Turbulent Statistics Reynolds stress distributions of11990610158401199081015840 11990610158401199061015840 and 119908

10158401199081015840 are plotted in the midplane of 119910-

direction (119910 = 1198731198842) in Figures 11(a) 11(b) and 11(c)respectively The shear stress component of the Reynoldsstress tensor is shown in Figure 11(a) Strong anisotropy of theflow field can be observed in the flow field Negative values of11990610158401199081015840 appear in the leading edge of the jet with positive values

in the trailing portion In the wake region both positiveand negative values exist with big nonuniformity In Figures11(b) and 11(c) the normal Reynolds stress is displayedFor both 119906

10158401199061015840 and 119908

10158401199081015840 the highest values appear in the

leading and trailing portions of the jet due to large velocitygradients Right behind the jet in the recirculation zoneboth components maintain some nontrivial values especially

11990810158401199081015840 indicating the wrapping motion around the jet by the

fluid flow

64 Coherent Structure The isosurfaces of second invariantof velocity gradient 119876 at time steps 80000 and 120000with domain height of 1198731198852 are plotted in Figure 12 Thedefinition of 119876 is given by (11) in tensor form

119876 =

1

2

(119882119894119895119882119894119895minus 119878119894119895119878119894119895) = minus

1

2

(

120597119906119895

120597119909119894

)(

120597119906119894

120597119909119895

)

119882119894119895=

120597119906119894

120597119909119895

minus

120597119906119895

120597119909119894

119878119894119895=

120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

(11)

where119882119894119895and 119878119894119895denote the rate of rotation and rate of shear-

ing respectivelyThe hair-pin-shaped coherent structures areoriginated from the side and trailing edge of the ejectinghole and wrapping around the jet In the downstream regionthe hair-pin structure is weakened and dissipated into themainstream


CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863

Figure 10 Time-averaged 2119863 velocity vector and vorticity at planes of 119909 = 1119863 15119863 and 3119863


00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)

Figure 11 Reynolds stress distribution at 119910 = 1198731198842 plane

(a) Time step = 80000 (b) Time step = 120000

Figure 12 Isosurfaces of second invariant of velocity gradient

7 Conclusion

A direct numerical simulation of jet in crossflow based onlattice Boltzmann method is performed on multi-GPUs Thecurrent simulation takes about 10 hours based on largest gridof 15 times 108 Several points can be summarized and concludedas follows

(1) A turbulent boundary layer is observed before the jet(119909 = minus2119863 119910 = 1198731198842) consisting of laminar sublayerand turbulent core After the jet at 119909 = 05119863 119910 =

1198731198842 and 119909 = 3119863 119910 = 1198731198842 time-averaged stream-wise and spanwise velocity components (119880119880

infinand

119882119880infin) are compared with experiments and good

agreement is obtained


(2) The turbulent energy spectrum is plotted versusfrequency at 119909 = 6119863 119911 = 4119863 and 119910 = 1198731198842 A decayrate close to the theoretical value of minus53 is observed

(3) 2119863 velocity vectors and vorticity fields are plotted in119910- 119911- and 119909-direction planes Unsteady shear-layervortices are formed and shed along the leading andtrailing side of the jet trajectory Horseshoe vortex isgenerated in the plate boundary layer before the jetTwo-deck structure of the CRVP is retained and thelower pair is observed to rise along the streamwisedirection

(4) The normal components (11990610158401199061015840 and 11990810158401199081015840) and shearcomponent (11990610158401199081015840) of the Reynolds stress are revealedin the 119910 = 1198731198842 plane Strong anisotropy is observedfor the flow field due to the shear layer and wake

(5) Coherent structure represented by the second invari-ant of velocity gradient (119876) is plotted at differenttime steps The characteristic hair-pin vortices areobserved

Conflict of Interests

The authors declare that there is no any conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by 973 Project of China(2013CB035702) the National Natural Science Foundationof China (11302165 and 11402191) and Postdoctoral ScienceFoundation of China (2013M540741)

References

[1] T F Fric and A Roshko ldquoVortical structure in the wake of atransverse jetrdquo Journal of Fluid Mechanics vol 279 pp 1ndash471994

[2] B A Haven andMKurosaka ldquoKidney and anti-kidney vorticesin crossflow jetsrdquo Journal of FluidMechanics vol 352 pp 27ndash641997

[3] S Y Chen and G D Doolen ldquoLattice Boltzmann method forfluid flowsrdquoAnnual Review of Fluid Mechanics vol 30 pp 329ndash364 1998

[4] Y L He Y Wang and Q Li Lattice Boltzmann Method Theoryand Applications Science Press Beijing China 2009

[5] L Chen Q Kang Y Mu Y-L He and W-Q Tao ldquoA criticalreview of the pseudopotential multiphase lattice Boltzmannmodel methods and applicationsrdquo International Journal of Heatand Mass Transfer vol 76 pp 210ndash236 2014

[6] C K Aidun and J R Clausen ldquoLattice-Boltzmann method forcomplex flowsrdquo Annual Review of Fluid Mechanics vol 42 pp439ndash472 2010

[7] nVIDIA ldquoNVIDIA CUDA Compute Unified Device Architec-ture Programming Guide Version 20rdquo nVIDIA 2008

[8] I Buck T Foley and D Horn ldquoBrook for GPUs stream com-puting on graphics hardwarerdquo ACM Transactions on Graphicsvol 23 pp 777ndash786 2004

[9] J Kruger and R Westermann ldquoLinear algebra operators forGPU implementation of numerical algorithmsrdquo ACM Transac-tions on Graphics vol 22 no 3 pp 908ndash916 2003

[10] S Ogawa and T Aoki ldquoGPU Computing for 2-dimensionalincompressible-flow simulation based on multi-grid methodrdquoTransactions of JSCES Article ID 20090021 2009

[11] W Xian and A Takayuki ldquoMulti-GPU performance of incom-pressible flow computation by lattice Boltzmann method onGPUclusterrdquoParallel Computing vol 37 no 9 pp 521ndash535 2011

[12] D Rossinelli M Bergdorf G-H Cottet and P KoumoutsakosldquoGPU accelerated simulations of bluff body flows using vortexparticle methodsrdquo Journal of Computational Physics vol 229no 9 pp 3316ndash3333 2010

[13] T Shimokawabe T Aoki J Ishida K Kawano and C Muroildquo145 TFlops performance on 3990 GPUs of TSUBAME 20supercomputer for an operational weather predictionrdquo in Pro-ceedings of the 11th International Conference on ComputationalScience (ICCS rsquo11) vol 4 pp 1535ndash1544 June 2011

[14] X Wang and T Aoki ldquoHigh performance computation bymulti-node GPU cluster-TSUBAME 20 on the air flow inan urban city using lattice Boltzmann methodrdquo InternationalJournal of Aerospace and Lightweight Structures vol 2 no 1 pp77ndash86 2012

[15] T Miki X Wang T Aoki et al ldquoPatient-specific modellingof pulmonary airflow using GPU cluster for the applicationin medical practicerdquo Computer Methods in Biomechanics andBiomedical Engineering vol 15 no 7 pp 771ndash778 2012

[16] XWang Y ShangguanNOnoderaH Kobayashi andT AokildquoDirect numerical simulation and large eddy simulation on aturbulent wall-bounded flow using lattice boltzmann methodandmultiple GPUsrdquoMathematical Problems in Engineering vol2014 Article ID 742432 10 pages 2014

[17] P L Bhatnagar E P Gross andMKrook ldquoAmodel for collisionprocesses in gasesrdquo Physical Review vol 94 no 3 pp 511ndash5251954

[18] Y H Qian D DrsquoHumieres and P Lallemand ldquoLattice BGKmodels for navier-stokes equationrdquo EuroPhysics Letters vol 17no 6 pp 479ndash484 1992

[19] Z-L Guo C-G Zheng and B-C Shi ldquoNon-equilibriumextrapolation method for velocity and pressure boundary con-ditions in the lattice boltzmann methodrdquo Chinese Physics vol11 no 4 pp 366ndash374 2002

[20] D Xu G Chen X Wang and Y Li ldquoDirect numericalsimulation of the wall-bounded turbulent flow by Lattice Boltz-mannmethod based onmulti-GPUsrdquoAppliedMathematics andMechanics vol 34 no 9 pp 1ndash9 2013

[21] F Muldoon and S Acharya ldquoDirect numerical simulation ofpulsed jets-in-crossflowrdquoComputers amp Fluids vol 39 no 10 pp1745ndash1773 2010

[22] Z L Guo and C G Zheng Theory and Applications of LatticeBoltzmann Method Science Press Beijing China 2010

[23] K E Meyer O Ozcan P S Larsen and C H WestergaardldquoSteroscopic PIV measurements in a jet in crossflowrdquo in Pro-ceedings of the 2nd International Symposium on Turbulence andShear Flow Phenomena Stockholm Sweden June 2001

[24] K E Meyer O Ozcan P S Larsen and C H WestergaardldquoFlow mapping of a jet in crossflow with stereoscopic PIVrdquoJournal of Visualization vol 5 no 3 pp 225ndash231 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Counter-rotating vortex pair

Wake vorticesHorseshoe vortices

Jet shear-layervortices

Wall

Crossflow

Figure 1 Four vortical structures associated with the JICF as in [1]

the parallel processing with low efficiency Thus a highlyparallel scheme is extremely helpful to those large-scalesimulations of turbulent flow field

Lattice Boltzmannmethod (LBM) has been regarded as apromising candidate for years due to its major merits of fullyparallel algorithm Unlike conventional numerical schemesbased on assumption and discretizations of macroscopiccontinuum the LBM adopts microscopic models [3] Otheradvantages of LBM include (1) easy treatment of boundaryconditions and (2) easy programming As a result the LBMhas its wide application in scientific and engineering researchsuch as Biofluid and porous medium [4ndash6] However toresolve the turbulent flow LBM-based DNS and LES requirea very high grid resolution and thus huge computationresource is needed

Graphic Processing Unit (GPU) brings excellent hard-ware conditions for these simulations With the develop-ment of computing platforms for example CUDA [7] andOpenCL the use of GPU to accelerate nongraphic computa-tions has drawn much more attention [8 9] Due to its highperformance of floating-point arithmetic operation togetherwith wide memory bandwidth and good programmabilitygeneral-purpose GPU (GPGPU) has its huge advantage overCPU in turbulent flow simulation [10ndash12] and thus hasbeen applied in fields like weather prediction crystal growthprocess and air flow in the city [13ndash15]

The feasibility for LBM-based DNS has some preliminaryvalidation for turbulent flows between parallel plates [16]further verification for JICF is still empty In this paper thelattice Boltzmann based DNS on JICF model is performedusing D3Q19model in multi-GPUs With further validationof this code this paper aims to resolve the JICF flow fieldincluding the time-averaged and instantaneous velocitiesvorticities and turbulent quantities of Reynolds stress Thecharacteristic coherent hairpin structures are also revealedand discussed

2 Lattice Boltzmann Equations

Lattice Boltzmann equation is the special form of theBoltzmann-BGK equation [17] in which discretizationapplies to velocity time and space The Boltzmann equation

for the discrete velocity distribution on a discrete lattice is asfollows

120597119891119894

120597119905

+ e119894sdot nabla119891119894= Ω119894 (1)

where119891119894is the particle velocity distribution function and e

119894is

the particle velocity in the 119894th direction Here the scheme ofLBM is described as D119898Q119899 [18] ldquoD119898rdquomeans ldquo119898 dimensionrdquoand ldquoQ119899rdquo stands for ldquo119899 lattice speedsrdquo For two-dimensionalproblems D2Q9 model is most commonly used and means2119863 and 9 speeds For three-dimensional problems severalcubic models are used such as D3Q13 D3Q15 D3Q19 andD3Q27 (119899 = 13 15 19 or 27) Ω

119894is the collision operator

With Boltzmann-BGK approximation

Ω119894= minus

1

120591

(119891119894minus 119891

eq119894) (2)

Combine (1) and (2) then it changes to

120597119891119894

120597119905

+ e119894sdot nabla119891119894= minus

1

120591

(119891119894minus 119891

eq119894) (3)

where119891eq119894

is the local equilibrium and 120591 is the relaxation timeThe equilibriumdistribution formhas to be correctly adoptedto restore the Navier-Stokes equations For 3119863 models withthe 13- 15- 19- and 27-speed lattice an appropriate distribu-tion function is written as

119891eq119894= 120588120603119894[1 + 3e

119894sdot u + 9

2

(e119894sdot u)2 minus 3

2

u sdot u] (4)

In the present paper D3Q19 scheme is used with par-ticle velocity as shown in Figure 2 119890

0(0 0 0) 119890

1(1 0 0)

1198902(minus1 0 0) 119890

3(0 1 0) 119890

4(0 minus1 0) 119890

5(0 0 1) 119890

6(0 0 minus1)

1198907(1 1 0) 119890

8(minus1 1 0) 119890

9(1 minus1 0) 119890

10(minus1 minus1 0) 119890

11(1 0 1)

11989012(minus1 0 1) 119890

13(1 0 minus1) 119890

14(minus1 0 minus1) 119890

15(0 1 1) 119890

16(0 minus1

1) 11989017(0 1 minus1) and 119890

18(0 minus1 minus1) Corresponding weighting

factors are 1206030= 13 120603

1sim 1206036= 118 and 120603

7sim 12060318= 136

If (3) is further discretized in both space 119909 and time 119905 thediscretized form can be written as

119891119894(x + e

119894120575119905 119905 + 120575119905) minus 119891

119894 (x 119905) = minus1

120591

(119891119894 (x 119905) minus 119891eq

119894(x 119905))

(5)


z

x

y

12

3

4

5

6

11

12

13

14

78

910

15

16

17

18




119904



119891119894 (x 119905) = 119891 (x 119905) minus 1

120591

(119891 (x 119905) minus 119891eq119894(x 119905)) (6)


119891119894(x + e

119894 119905 + 1) =

119891119894 (x 119905) (7)

where 119891119894and



119894is


120588 =

119873

sum

119894=0

119891119894=

119873

sum

119894=0

119891eq119894

120588u =119873

sum

119894=0

119891119894e119894=

119873

sum

119894=0

119891eq119894e119894

(8)



119901 = 120588119862119904

2=

120588

3

(9)




119891119894= 119891

eq119894+ 119891

neq119894

= 119891eq119894+ 119891

neq119894-neighbor

= 119891eq119894+ (

119891119894-nieghbor minus 119891

eq119894-neighbor)

(10)

in which 119891eq119894





3 GPU Settings




Six processors






Lx

Lx1 Lx2

Ly

Lz

X

Y

Z

D


20

15

10

5

10minus1

100

101

102

z+

U+

Sublayer

Turbulentcore








UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16


UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0


y = NY2

y = NY2

y = NY2

y = NY2

x = 05D

x = 3D

zD

x = 05D

x = 3D

zD

zD

zD


(iv) Domain height (119871119911119863) = 64






119895= 119880119895119863120592 The largest




[22]

5 Model Validation


=

119880119906120591versus 119911+ = 119911119906







Etu

r10

0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

101

102

Frequency (Hz)

minus53

x = 6D z = 4D y = NY2



infin











Horseshoevortex

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(a) Time-averaged

Shear-layervortices

Wakevortices

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004


Shear-layervortices

vorticesWake

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004







Wake vortices

Time step = 80000

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(a) 119911 = 05119863

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(b) 119911 = 1119863

Ref vector

01

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(c) 119911 = 2119863







10158401199061015840 and 119908




fluid flow


119876 =

1

2

(119882119894119895119882119894119895minus 119878119894119895119878119894119895) = minus

1

2

(

120597119906119895

120597119909119894

)(

120597119906119894

120597119909119895

)

119882119894119895=

120597119906119894

120597119909119895

minus

120597119906119895

120597119909119894

119878119894119895=

120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

(11)




CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863



00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

x

y

12

3

4

5

6

11

12

13

14

78

910

15

16

17

18




119904



119891119894 (x 119905) = 119891 (x 119905) minus 1

120591

(119891 (x 119905) minus 119891eq119894(x 119905)) (6)


119891119894(x + e

119894 119905 + 1) =

119891119894 (x 119905) (7)

where 119891119894and



119894is


120588 =

119873

sum

119894=0

119891119894=

119873

sum

119894=0

119891eq119894

120588u =119873

sum

119894=0

119891119894e119894=

119873

sum

119894=0

119891eq119894e119894

(8)



119901 = 120588119862119904

2=

120588

3

(9)




119891119894= 119891

eq119894+ 119891

neq119894

= 119891eq119894+ 119891

neq119894-neighbor

= 119891eq119894+ (

119891119894-nieghbor minus 119891

eq119894-neighbor)

(10)

in which 119891eq119894





3 GPU Settings




Six processors






Lx

Lx1 Lx2

Ly

Lz

X

Y

Z

D


20

15

10

5

10minus1

100

101

102

z+

U+

Sublayer

Turbulentcore








UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16


UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0


y = NY2

y = NY2

y = NY2

y = NY2

x = 05D

x = 3D

zD

x = 05D

x = 3D

zD

zD

zD


(iv) Domain height (119871119911119863) = 64






119895= 119880119895119863120592 The largest




[22]

5 Model Validation


=

119880119906120591versus 119911+ = 119911119906







Etu

r10

0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

101

102

Frequency (Hz)

minus53

x = 6D z = 4D y = NY2



infin











Horseshoevortex

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(a) Time-averaged

Shear-layervortices

Wakevortices

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004


Shear-layervortices

vorticesWake

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004







Wake vortices

Time step = 80000

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(a) 119911 = 05119863

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(b) 119911 = 1119863

Ref vector

01

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(c) 119911 = 2119863







10158401199061015840 and 119908




fluid flow


119876 =

1

2

(119882119894119895119882119894119895minus 119878119894119895119878119894119895) = minus

1

2

(

120597119906119895

120597119909119894

)(

120597119906119894

120597119909119895

)

119882119894119895=

120597119906119894

120597119909119895

minus

120597119906119895

120597119909119894

119878119894119895=

120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

(11)




CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863



00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Six processors






Lx

Lx1 Lx2

Ly

Lz

X

Y

Z

D


20

15

10

5

10minus1

100

101

102

z+

U+

Sublayer

Turbulentcore








UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16


UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0


y = NY2

y = NY2

y = NY2

y = NY2

x = 05D

x = 3D

zD

x = 05D

x = 3D

zD

zD

zD


(iv) Domain height (119871119911119863) = 64






119895= 119880119895119863120592 The largest




[22]

5 Model Validation


=

119880119906120591versus 119911+ = 119911119906







Etu

r10

0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

101

102

Frequency (Hz)

minus53

x = 6D z = 4D y = NY2



infin











Horseshoevortex

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(a) Time-averaged

Shear-layervortices

Wakevortices

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004


Shear-layervortices

vorticesWake

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004







Wake vortices

Time step = 80000

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(a) 119911 = 05119863

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(b) 119911 = 1119863

Ref vector

01

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(c) 119911 = 2119863







10158401199061015840 and 119908




fluid flow


119876 =

1

2

(119882119894119895119882119894119895minus 119878119894119895119878119894119895) = minus

1

2

(

120597119906119895

120597119909119894

)(

120597119906119894

120597119909119895

)

119882119894119895=

120597119906119894

120597119909119895

minus

120597119906119895

120597119909119894

119878119894119895=

120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

(11)




CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863



00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16


UUinfin

8

7

6

5

4

3

2

1

0minus04 0 04 08 12 16

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0

WUinfin

0 08 16 24 32

8

7

6

5

4

3

2

1

0


y = NY2

y = NY2

y = NY2

y = NY2

x = 05D

x = 3D

zD

x = 05D

x = 3D

zD

zD

zD


(iv) Domain height (119871119911119863) = 64






119895= 119880119895119863120592 The largest




[22]

5 Model Validation


=

119880119906120591versus 119911+ = 119911119906







Etu

r10

0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

101

102

Frequency (Hz)

minus53

x = 6D z = 4D y = NY2



infin











Horseshoevortex

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(a) Time-averaged

Shear-layervortices

Wakevortices

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004


Shear-layervortices

vorticesWake

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004







Wake vortices

Time step = 80000

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(a) 119911 = 05119863

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(b) 119911 = 1119863

Ref vector

01

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(c) 119911 = 2119863







10158401199061015840 and 119908




fluid flow


119876 =

1

2

(119882119894119895119882119894119895minus 119878119894119895119878119894119895) = minus

1

2

(

120597119906119895

120597119909119894

)(

120597119906119894

120597119909119895

)

119882119894119895=

120597119906119894

120597119909119895

minus

120597119906119895

120597119909119894

119878119894119895=

120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

(11)




CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863



00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Etu

r10

0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

100

101

102

Frequency (Hz)

minus53

x = 6D z = 4D y = NY2



infin











Horseshoevortex

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(a) Time-averaged

Shear-layervortices

Wakevortices

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004


Shear-layervortices

vorticesWake

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004







Wake vortices

Time step = 80000

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(a) 119911 = 05119863

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(b) 119911 = 1119863

Ref vector

01

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(c) 119911 = 2119863







10158401199061015840 and 119908




fluid flow


119876 =

1

2

(119882119894119895119882119894119895minus 119878119894119895119878119894119895) = minus

1

2

(

120597119906119895

120597119909119894

)(

120597119906119894

120597119909119895

)

119882119894119895=

120597119906119894

120597119909119895

minus

120597119906119895

120597119909119894

119878119894119895=

120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

(11)




CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863



00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Horseshoevortex

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004

(a) Time-averaged

Shear-layervortices

Wakevortices

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004


Shear-layervortices

vorticesWake

Ref vector

01

Vorticity_Y

004

003

002

0010

minus001

minus002

minus003

minus004







Wake vortices

Time step = 80000

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(a) 119911 = 05119863

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(b) 119911 = 1119863

Ref vector

01

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(c) 119911 = 2119863







10158401199061015840 and 119908




fluid flow


119876 =

1

2

(119882119894119895119882119894119895minus 119878119894119895119878119894119895) = minus

1

2

(

120597119906119895

120597119909119894

)(

120597119906119894

120597119909119895

)

119882119894119895=

120597119906119894

120597119909119895

minus

120597119906119895

120597119909119894

119878119894119895=

120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

(11)




CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863



00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Wake vortices

Time step = 80000

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(a) 119911 = 05119863

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(b) 119911 = 1119863

Ref vector

01

Time-averaged

Vorticity_Z

0008

0006

0004

0002

0010

minus0002

minus001

minus0004

minus0006

minus0008

(c) 119911 = 2119863







10158401199061015840 and 119908




fluid flow


119876 =

1

2

(119882119894119895119882119894119895minus 119878119894119895119878119894119895) = minus

1

2

(

120597119906119895

120597119909119894

)(

120597119906119894

120597119909119895

)

119882119894119895=

120597119906119894

120597119909119895

minus

120597119906119895

120597119909119894

119878119894119895=

120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

(11)




CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863



00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(a) 119909 = 1119863

CRVP

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(b) 119909 = 15119863

CRVP

Ref vector

005

Vorticity_X

0008

0006

0004

00020

minus0002

minus0004

minus0006

minus0008

(c) 119909 = 3119863



00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00015

00013

00011

00009

00007

00005

00003

00001

minus00001

minus00003

minus00005

u998400w998400

(a)

0004

00035

0003

00025

0002

00015

0001

00005

u998400u998400

(b)

00025

00023

00021

00019

00017

00015

00013

00011

00009

00007

00005

w998400w998400

(c)




7 Conclusion




infinand










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article High Performance Computation of a...

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