3DSimulationofSelf-CompactingConcreteFlowBasedon MRT-LBMdownloads.hindawi.com/journals/amse/2018/5436020.pdf · Research Article 3DSimulationofSelf-CompactingConcreteFlowBasedon MRT-LBM

Research Article3D Simulation of Self-Compacting Concrete Flow Based onMRT-LBM

Liu-Chao Qiu and Yu Han

College of Water Resources and Civil Engineering China Agricultural University Beijing 100083 China

Correspondence should be addressed to Liu-Chao Qiu qiuliuchaocaueducn and Yu Han yh916uowmaileduau

Received 21 June 2017 Accepted 2 November 2017 Published 28 January 2018

Academic Editor Ying Li

Copyright copy 2018 Liu-Chao Qiu and Yu Han -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A three-dimensional multiple-relaxation-time lattice Boltzmann method (MRT-LBM) with a D3Q27 discrete velocity model isapplied for simulation of self-compacting concrete (SCC) flows In the present study the SCC is assumed as a non-Newtonianfluid and a modified HerschelndashBulkley model is used as constitutive mode -e mass tracking algorithm was used for modelingthe liquid-gas interface Two numerical examples of the slump test and enhanced L-box test were performed and the calculatedresults are compared with available experiments in literatures -e numerical results demonstrate the capability of the proposedMRT-LBM in modeling of self-compacting concrete flows

1 Introduction

As a highly flowable concrete the self-compacting concretedoes not require any vibration during casting processes andhas been considered as ldquothe most revolutionary developmentin concrete construction for several decadesrdquo [1] AlthoughSCC has been successful in plenty of applications manyproblems are encountered during construction because ofaggregate segregation air voids and the improper filling offormworks and most of these problems are closely relatedto the flowability of SCC -erefore it is important to wellunderstand and accurately predict the flow characteristics forthe success of SCC casting However the accurate predictionof the SCC flowing behavior is a big challenge especially inthe case of heavy reinforcement complex formwork shapesand large size of aggregates In this regard the numericalmodeling of SCC is an indispensable and inexpensive ap-proach not only as a tool for form filling prediction but also interms of determination of fresh concrete properties mixdesign and casting optimization Nowadays the numericalmodeling of fresh concrete flow has gained importance and itis becoming an important tool for the prediction and opti-mization of casting processes [2 3]

-e modeling of fresh SCC is not a trivial task because itinvolves a free-surface flow of a dense suspension witha wide range of particle sizes and shows non-Newtonian flow

behavior A complete description of SCC from the cementparticles to coarse aggregates is impossible with any com-puter model since accounting for broad particle size andshape distributions exceeds the computational limits of eventhe best supercomputers [4] -e fresh SCC can be treated asa one-phase fluid and simulated using CFD because it isflowable and the amount of coarse particles is lower thanthat in conventional concrete In terms of rheology thefresh SCC can be approximated as one of the known non-Newtonian materials such as the Bingham or HerschelndashBulkley fluid In case if the high shear rates are likely to occurin the casting processes the shear-thickening behavior canhappen and the HerschelndashBulkley model could be a moresuitable one At the end of the casting process the shear ratesare rather low and the yield stress practically dominates theflow -erefore the material can be modeled as a Binghamfluid to predict the final shape

At present many numerical techniques have been de-veloped to model the SCC flow by assuming it as a homo-geneous viscous fluid and using either the mesh-basedmethods such as the finite volume method (FVM) [5] andthe finite element method (FEM) [6] or the meshlessmethods like the smoothed particle hydrodynamics (SPH)and the lattice Boltzmann method (LBM) Due to the factthat the flow of SCC is a typical free-surface flow thetreatment of the interface and its position represents another

HindawiAdvances in Materials Science and EngineeringVolume 2018 Article ID 5436020 8 pageshttpsdoiorg10115520185436020

important numerical modeling issue In this regard themeshless methods have an advantage over the mesh-basedmethods As a meshless Lagrangian method the SPHmethodis good at modeling Newtonian and non-Newtonian flowswith a free surface [7] and has been chosen for simulating theSCC flow by Kulasegaram et al [8] Lashkarbolouk et al [9]Qiu [10] Abo Dhaheer et al [11] andWu et al [12] Howeverthe SPH method has difficulties in solving problems withcomplex solid boundary conditions For these reasons manyresearchers have made much effort to develop an alternativemethod called LBM for modeling the SCC flow due to the factthat it is easy for coding intrinsically parallelizable andapplicable to complex geometries straightforwardly For ex-ample Svec et al [13] and Leonardi et al [14] simulated thefresh SCC as the non-Newtonian fluid based on the single-relaxation-time (SRT) LBM Although the single-relaxation-time (SRT) model has been successful in many applicationsit is prone to numerical instability in complex flows [15]To overcome these difficulties the multiple-relaxation-time(MRT) model proposed by drsquoHumieres et al [16] is useful tostabilize the solution and to obtain satisfactory results becausethe MRT model allows the usage of an independently opti-mized relaxation time for each physical process [17] -ereforeChen et al [18] solved Bingham fluids by using theMRTmodelbut the free surface is not considered in their model In thepresent work we will develop a 3D multiple-relaxation-timeLBM with a mass tracking algorithm representing the freesurface to simulate the flow of fresh SCC

2 Mathematical Formulations

21 Multiple-Relaxation-Time LBM In this paper the SCCflow is solved based on the lattice Boltzmann method whichis considered as a very attractive alternative to the traditionalCFD especially in problems with complex boundary con-ditions In the LBM a finite number of velocity vectors eα areused to discretize the velocity space and the fluid motion isdescribed by a particle distribution function f(x eα t) which

is the probability density of finding particles with velocity eαat a location x and at a given time t -e LB equations canrecover the continuumNavierndashStokes equations bymeans ofthe ChapmanndashEnskog expansion if a proper set of discretevelocities was employed [17 19] -e D3Q27 (3 dimensionsand 27 velocities) discrete velocity model illustrated inFigure 1 was used in this study -e particle velocity vectorseα for this lattice model are given by

eα

(0 0 0)c α 0

(plusmn1 0 0)c (0 plusmn1 0)c (0 0 plusmn1)c α 1 6

(plusmn1 plusmn1 0)c (plusmn1 0 plusmn1)c (0 plusmn1 plusmn1)c α 7 18

(plusmn1 plusmn1 plusmn1)c α 19 26

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(1)

where c δxδt with δx and δt being the lattice spacing andthe time step respectively

A discretization of the Boltzmann equation in time andspace leads to the lattice Boltzmann equation [20 21]

fα x + eαδt t + δt( 1113857minusfα(x t) Λαj1113876fj(x t)minusfeq

j (x t)1113877

+ Fαδt

(2)

where fα is the distribution function of particles moving withvelocity eα Λαj is the collision matrix f eq is the equilibriumdistribution function and Fα is the external force

-e equilibrium distribution function f eq is obtainedusing the Taylor series expansion of theMaxwellndashBoltzmanndistribution function with velocity u up to second order Itcan be written as

feqα ρwα 1 +

eα middot uc2s

+eα middot u( 1113857

2 minus cs u| |( 11138572

2c4s1113890 1113891 (3)

where ρ is the fluid density u is the fluid velocity and thesound speed is cs c

3

radic -e weight coefficients for D3Q27

are given by

wα

827

α 0

227

α 1 6

154

α 7 18

1216

α 19 26

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

-e components of F are given as

Fα wαρeα middot a

c2s (5)

where a is the acceleration-e relaxation process has major influence on the

physical fidelity as well as numerical stability For the single-

1

23

4

5

6

7

11

1513

17

10

9

12

8

16

18

14

19

20

22

26

24

25

23

21

0

Figure 1 -ree-dimensional twenty-seven particle velocity(D3Q27) lattice

2 Advances in Materials Science and Engineering

relaxation-time (SRT) model the collision matrix is Λαj

minus1τδαj where δαj is the Kronecker symbol and τ is therelaxation time which is related to the kinetic viscosity by] c2s (τ minus 12)δt For the multiple-relaxation-time LBM thecollision matrix Λ can be written as [15]

Λ minusMminus1SM (6)

where the linear transform matrixM is a 27times 27 matrix -ediagonal matrix Smay be written as S diag (0 0 0 0 s4 s5s5 s7 s7 s7 s10 s10 s10 s13 s13 s13 s16 s17 s18 s18 s20s20 s20 s23 s23 s23 s26) with s4154 s5 s71τs1015 s13183 s1614 s17161 s18 s20198 ands23 s26174

-e macroscopic density ρ and velocity u are computed by

ρ sum26

α0fα

u sum26

α0eαfα +

12aδt

(7)

-e pressure p is related to the density by

p ρc2s (8)

22 Free Surface Modeling For the modeling of the liquid-gas interface the most straightforward way is to track all thephases for example liquid and gas Such a method has thehighest accuracy at the expense of high computational costs-e mass tracking algorithm [22] without considering thegas phase is employed in the present study due to the factthat it is simple fast and accurate In this algorithm thefluid domain is divided into liquid interface and gas nodes(Figure 2) -e liquid and interface nodes are active andsolved by the LBM and the remaining gas nodes are inactivewithout evolution equation Liquid and gas nodes are neverdirectly connected but through an interface node -eadopted mass tracking algorithm is applied directly at thelevel of the LBM so the algorithm mimics the free surfaceby modifying the particle distributions An additional mac-roscopic variable for the mass m (x t) stored in a node isrequired and defined as

m(x t) ρ(x t) for liquid nodes

0ltm(x t)lt ρ(x t) for interface nodes

m(x t) 0 for gas nodes

(9)

-e mass is calculated by

m x t + δt( ) m x t( ) +sumα

kα1113876 fα x + eαδt t( 1113857

minusfα x t( )1113877

(10)

where fα and fα are the particle distributions with oppositedirections and

kα

m(x t) + m x + eαδt t( 11138571113858 1113859

2 for interface nodes

1 for liquid nodes

0 for gas nodes

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)-e interface node becomes a fluid node when the mass

reaches its density with m (x t) ρ (x t) and vice versa theinterface node becomes a gas node when the mass dropsdown to zero with m (x t) 0

23 Modified HerschelndashBulkley Model In this study thefresh SCC is assumed as viscoplastic fluids to consider itsnon-Newtonian behavior Among the constitutive relationsof viscoplastic fluids the HerschelndashBulkley model is prob-ably the most commonly used because of its simplicity andflexibility -e standard HerschelndashBulkley model is de-scribed by

_c 0 σ lt σy

σ σy + k _cn σ ge σy(12)

where _c is the shear rate (sminus1) σ is the stress (Pa) σy is theyield stress (Pa) and k is the consistency index (Pamiddotsn) n isa measure of the deviation of the fluid from Newtonian(the power-law index) For a fluid with n gt 1 the effectiveviscosity increases with shear rate and the fluid is calledshear-thickening or dilatant fluid For a fluid with0 lt n lt 1 the effective viscosity decreases with shear rateand the fluid is called shear-thinning or pseudoplasticfluid

-e standard HershelndashBulkley model becomes discon-tinuous at less shear rates and causes instability duringnumerical solution due to the fact that the non-Newtonianviscosity becomes unbounded at small shear rates In orderto overcome such discontinuity the standard HerschelndashBulkley model can be modified as [23]

Real interface

Gas-liquid interfaceInterface nodes

Liquid nodes

Gas nodes

Figure 2 Mass tracking algorithm scheme

Advances in Materials Science and Engineering 3

μ μy σ lt σy

μ σy

_c+

k

_c_c

n minusσy

μy

1113888 1113889

n

1113890 1113891 σ ge σy

(13)

where μ is the apparent viscosity of fluid and the yieldingviscosity μy can be estimated from experimental rheological

curves and can be assumed to be the slope of the line ofshear stress versus shear rate curve before yielding -emodified HerschelndashBulkley model combines the effects ofBingham and power-law behavior in a fluid For low strainrates the material acts like a very viscous fluid with yieldviscosity μy As the strain rate increases and the yield stressthreshold σy is passed the fluid behavior is described bya power law

(a)

100 mm

200 mm

300

mm

(b)

YX

Z

(c)

Figure 3 Slump test (a) Experiment setup [24] (b) model size and (c) LBM discretization

Velocity (mmiddotsminus1)

000 011 022 034 045 056 067 079

(a)


000 019 038 056 075 094 113 131

(b)


000 007 014 020 027 034 041 048

(c)


000 000 000 000 001 001 001 001

(d)


000 000 000 000 001 001 001 001

(e)


000 000 000 001 001 001 001 002

(f)

Figure 4 -e velocity distribution during the slump test (a) t 01 s (b) t 02 s (c) t 03 s (d) t 05 s (e) t 10 s and (f) t 20 s


3 Numerical Examples and Validation

In this section the developed MRT-LBM is applied to sim-ulate the slump flow of fresh SCC to validate the capability ofthe proposed model in modeling SCC flow And the model isalso used to simulate the passing ability and the filling abilityof SCC by using an enhanced L-box test

31 SlumpTest -e slump flow is the most commonly usedtest in SCC technology It measures flow spread and op-tionally the flow time t50 -e numerical simulation ofthe slump test in our study is based on a laboratory ex-periment by Huang [24] Figure 3(a)-e dimensions of thecone are 300mm height 200mm lower diameter and100mm upper diameter Figure 3(b) -e SCC used in thisstudy has the following properties yield stress 100 Paplastic viscosity 50 Pamiddots and density 2300 kgm3 In thestandard form of the slump test only the final spread valueof the slump is measured to evaluate the flowing behaviorof the SCC

-e numerical simulation of the slump flow is per-formed based on the developed MRT-LBM In our sim-ulation the lattice spacing is set to 001m for the LBMdiscretization Figure 3(c) -e value of the yieldingviscosity was chosen to be 1000 times higher than thevalue of the yield stress with the power-law index beingn 1 Figure 4 presents some snapshots at different in-stants of the material shape with the velocity magnitudeIn Figure 5 experimental and numerical results for thematerial shape at the end of the flow are compared It canbe seen from this figure that the calculated slump flowspread diameter is about 624mm which shows a perfectmatch between the experimental and simulated flowdistance that can be observed in terms of the maximumspread distance -is proves the correctness of the pro-posed model and its ability to simulate the free surfaceflow of fresh SCC

32 Enhanced L-Box Test -e L-box test is generally used forassessing the passing ability and filling ability of fresh SCC in

(a)

60

50

40

30

20

10

0

z (cm

)

x (cm)6050403020100

(b)

Figure 5 Final spread of the slump test (a) Experiment [24] and (b) simulation

(a)

SCC

150 cm

600 cm

300

cm

150

cm

150

cm

150 cm

150

cm45

0 cm

(b)

SCC

Y

XZ

(c)

Figure 6 -e enhanced L-box test (a) Experiment setup [24] (b) model size and (c) LBM discretization


confined spaces In this section a laboratory experiment of anenhanced L-box test by Huang [24] was simulated using thedeveloped MRT-LBM -e enhanced L-box consists of the

concrete reservoir slide gate and horizontal test channel withfour ball obstacles (Figure 6(a)) -e vertical section is filledwith concrete and subsequently the gate is lifted to allow


0000 0000 0000 0000 0000 0000 0000 0000

(a)


0000 0002 0003 0005 0006 0008 0010 0011

(b)


0000 0001 0002 0003 0004 0005 0006 0007

(c)


0000 0001 0002 0003 0004 0004 0005 0006

(d)


0000 0001 0001 0002 0003 0004 0004 0005

(e)


0000 0001 0001 0002 0002 0003 0003 0004

(f)


0000 0001 0001 0002 0002 0003 0003 0004

(g)


0000 0000 0001 0001 0002 0002 0003 0003

(h)


0000 0001 0001 0002 0002 0003 0003 0004

(i)


0000 0000 0001 0001 0002 0002 0003 0003

(j)


0000 0000 0001 0001 0002 0002 0002 0003

(k)


0000 0004 0008 0012 0016 0020 0024 0028

(l)

Figure 7 -e velocity distribution of the enhanced L-box test (a) t 0 s (b) t 25 s (c) t 50 s (d) t 75 s (e) t 100 s (f) t 125 s (g)t 150 s (h) t 175 s (i) t 200 s (j) t 225 s (k) t 250 s and (l) final shape


concrete to flow into the horizontal section When the flowstops one measures the reached height of fresh SCC afterpassing through the specified gaps of balls and flowingwithin a defined flow distance With this reached heightthe passing or blocking behavior of SCC can be estimated-e dimensions of the enhanced L-box test are shown inFigure 6(b)

In our simulation the SCC is placed at the middle upperpart of the vertical container which is then suddenly re-leased and the concrete begins to spread under the gravityloading Figure 6(c) -e numerical simulation is performedbased on the developed MRT-LBM and the lattice spacing isset to 001m for the LBM discretization -e value of theyielding viscosity was chosen to be 1000 times higher thanthe value of the yield stress with the power-law index beingn 1 SCC used for this study was the same concrete as in theslump test with the following properties yield stress 50 Paplastic viscosity 50 Pamiddots and density 2300 kgm3 -etotal volume of the material is V 13 L Some snapshots ofthe SCC shape with the velocity magnitude at differentinstants are presented in Figure 7 Figure 8 compares theexperimental and numerical results for the final spread of theSCC flow -e simulated flow spread agrees well with theexperimental results -is in turn further validates theproposed model in modeling the passing ability and fillingability of fresh SCC in confined spaces A slight discrepancyin the shape of the SCC can be noted which may be due tothe fact that the balls are perfectly touched by the lateral wallin simulation but there are existing gaps between them in theexperiment

4 Conclusions

In this paper a multiple-relaxation-time LBM with a D3Q27discrete velocity model for modeling the flow behavior offresh SCC was proposed -e rheology of the fresh SCC wasapproximated as a non-Newtonian material using themodified HerschelndashBulkley fluid model -e free surface wasmodeled based on the mass tracking algorithm which isa simple and fast algorithm that conserves the mass precisely-e numerical simulation of slump flow of fresh SCCwas firstperformed to validate the capability of the proposed model inmodeling SCC flow And then the model was further used tosimulate the passing ability and the filling ability of fresh SCCin confined spaces based on an enhanced L-box test -esimulated results agree well with the corresponding experi-mental data in the published literature -is proves that theproposed MRT-LBM is suitable for numerical simulations ofthe fresh SCC flows It should be noted that the main problemfor the application of the present model to real engineeringproblems is its high computational cost -erefore furtherinvestigations might be needed to consider hardware accel-eration and parallel computing to make the proposed modelmore useful and versatile

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-e authors are grateful for funding from the NationalNatural Science Foundation of China (Grant nos 11772351and 51509248) and the Scientific Research and Experimentof Regulation Engineering for the Songhua River Main-stream in Heilongjiang Province China (Grant noSGZLKY-12)

References

[1] EFNARC Specification and Guidelines for Self-CompactingConcrete Scientific Research Publishing Surrey UK2002

[2] N Roussel M R Geiker F Dufour L N -rane andP Szabo ldquoComputational modeling of concrete flow generaloverviewrdquo Cement and Concrete Research vol 37 no 9pp 1298ndash1307 2007

[3] N Roussel and A Gram Simulation of Fresh Concrete FlowState-of-the Art Report of the RILEM Technical Committee222-SCF RILEM State-of-the-Art Reports vol 15 SpringerNetherlands 2014

[4] K Vasilic M Geiker J Hattel et al ldquoAdvanced Methods andFuture Perspectivesrdquo in Simulation of Fresh Concrete FlowN Roussel and A Gram Eds vol 15pp 125ndash146 SpringerNetherlands 2014

[5] N Roussel ldquoCorrelation between yield stress and slumpcomparison between numerical simulations and concreterheometers resultsrdquo Materials and Structures vol 39 no 4pp 501ndash509 2006

[6] B Patzak and Z Bittnar ldquoModeling of fresh concrete flowrdquoComputers and Structures vol 87 no 15 pp 962ndash9692009

(a)

(b)

Figure 8 Final shape of SCC in the enhanced L-box test (a)Experiment [24] and (b) simulation


[7] S Shao and E Y M Lo ldquoIncompressible SPH method forsimulating Newtonian and non-Newtonian flows with a freesurfacerdquo Advances in Water Resources vol 26 no 7pp 787ndash800 2003

[8] S Kulasegaram B L Karihaloo and A Ghanbari ldquoModellingthe flow of self-compacting concreterdquo International Journalfor Numerical and Analytical Methods in Geomechanicsvol 35 no 6 pp 713ndash723 2011

[9] H Lashkarbolouk M R Chamani A M Halabian andA R Pishehvar ldquoViscosity evaluation of SCC based on flowsimulation in L-box testrdquo Magazine of Concrete Researchvol 65 no 6 pp 365ndash376 2013

[10] L-C Qiu ldquo-ree dimensional GPU-based SPH modelling ofself-compacting concrete flowsrdquo in Proceedings of the thirdinternational symposium on Design Performance and Use ofSelf-Consolidating Concrete pp 151ndash155 RILEM PublicationsSARL Xiamen China June 2014

[11] M S Abo Dhaheer S Kulasegaram and B L KarihalooldquoSimulation of self-compacting concrete flow in the J-ring testusing smoothed particle hydrodynamics (SPH)rdquo Cement andConcrete Research vol 89 pp 27ndash34 2016

[12] J Wu X Liu H Xu and H Du ldquoSimulation on the self-compacting concrete by an enhanced Lagrangian particlemethodrdquo Advances in Materials Science and Engineeringvol 2016 Article ID 8070748 11 pages 2016

[13] O Svec J Skocek H Stang M R Geiker and N RousselldquoFree surface flow of a suspension of rigid particles in a non-Newtonian fluid a lattice Boltzmann approachrdquo Journal ofNon-Newtonian Fluid Mechanics vol 179ndash180 pp 32ndash422012

[14] A Leonardi F K Wittel M Mendoza and H J HerrmannldquoCoupled DEM-LBM method for the free-surface simulationof heterogeneous suspensionsrdquo Computational ParticleMechanics vol 1 no 1 pp 3ndash13 2014

[15] K N Premnath and S Banerjee ldquoOn the three-dimensionalcentral moment Lattice Boltzmann methodrdquo Journal ofStatistical Physics vol 143 no 4 pp 747ndash794 2011

[16] D drsquoHumieres I Ginzburg M Krafczyk P Lallemand andL-S Luo ldquoMultiple-relaxation-time lattice Boltzmannmodels in three dimensionsrdquo Philosophical Transactions of theRoyal Society A Mathematical Physical and EngineeringSciences vol 360 no 1792 pp 437ndash451 2002

[17] P Lallemand and L-S Luo ldquo-eory of the lattice Boltzmannmethod dispersion isotropy Galilean invariance and sta-bilityrdquo Physical Review E vol 61 no 6 pp 6546ndash6562 2000

[18] S G Chen C H Zhang Y T Feng Q C Sun and F Jinldquo-ree-dimensional simulations of Bingham plastic flowswith the multiple-relaxation-time lattice Boltzmann modelrdquoScience China Physics Mechanics and Astronomy vol 57no 3 pp 532ndash540 2014

[19] S Chen and G D Doolen ldquoLattice Boltzmann method forfluid flowsrdquo Annual Review of Fluid Mechanics vol 30 no 1pp 329ndash364 1998

[20] S Succi H Chen and S Orszag ldquoRelaxation approxima-tions and kinetic models of fluid turbulencerdquo Physica AStatistical Mechanics and its Applications vol 362 no 1pp 1ndash5 2006

[21] Z Guo C Zheng and B Shi ldquoDiscrete lattice effects on theforcing term in the lattice Boltzmann methodrdquo PhysicalReview E vol 65 no 4 p 046308 2002

[22] C Korner M -ies T Hofmann N -urey and U RudeldquoLattice Boltzmann model for free surface flow for modelingfoamingrdquo Journal of Statistical Physics vol 121 no 1-2pp 179ndash196 2005

[23] P Prajapati and F Ein-Mozaffari ldquoCFD investigation of themixing of yield pseudo plastic fluids with anchor impellersrdquoChemical Engineering amp Technology vol 32 no 8 pp 1211ndash1218 2009

[24] M S Huang e Application of Discrete Element Method onFilling Performance Simulation of Self-Compacting Concrete inRFC PhD thesis Tsinghua University Beijing China 2010in Chinese


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important numerical modeling issue In this regard themeshless methods have an advantage over the mesh-basedmethods As a meshless Lagrangian method the SPHmethodis good at modeling Newtonian and non-Newtonian flowswith a free surface [7] and has been chosen for simulating theSCC flow by Kulasegaram et al [8] Lashkarbolouk et al [9]Qiu [10] Abo Dhaheer et al [11] andWu et al [12] Howeverthe SPH method has difficulties in solving problems withcomplex solid boundary conditions For these reasons manyresearchers have made much effort to develop an alternativemethod called LBM for modeling the SCC flow due to the factthat it is easy for coding intrinsically parallelizable andapplicable to complex geometries straightforwardly For ex-ample Svec et al [13] and Leonardi et al [14] simulated thefresh SCC as the non-Newtonian fluid based on the single-relaxation-time (SRT) LBM Although the single-relaxation-time (SRT) model has been successful in many applicationsit is prone to numerical instability in complex flows [15]To overcome these difficulties the multiple-relaxation-time(MRT) model proposed by drsquoHumieres et al [16] is useful tostabilize the solution and to obtain satisfactory results becausethe MRT model allows the usage of an independently opti-mized relaxation time for each physical process [17] -ereforeChen et al [18] solved Bingham fluids by using theMRTmodelbut the free surface is not considered in their model In thepresent work we will develop a 3D multiple-relaxation-timeLBM with a mass tracking algorithm representing the freesurface to simulate the flow of fresh SCC

2 Mathematical Formulations

21 Multiple-Relaxation-Time LBM In this paper the SCCflow is solved based on the lattice Boltzmann method whichis considered as a very attractive alternative to the traditionalCFD especially in problems with complex boundary con-ditions In the LBM a finite number of velocity vectors eα areused to discretize the velocity space and the fluid motion isdescribed by a particle distribution function f(x eα t) which

is the probability density of finding particles with velocity eαat a location x and at a given time t -e LB equations canrecover the continuumNavierndashStokes equations bymeans ofthe ChapmanndashEnskog expansion if a proper set of discretevelocities was employed [17 19] -e D3Q27 (3 dimensionsand 27 velocities) discrete velocity model illustrated inFigure 1 was used in this study -e particle velocity vectorseα for this lattice model are given by

eα

(0 0 0)c α 0

(plusmn1 0 0)c (0 plusmn1 0)c (0 0 plusmn1)c α 1 6

(plusmn1 plusmn1 0)c (plusmn1 0 plusmn1)c (0 plusmn1 plusmn1)c α 7 18

(plusmn1 plusmn1 plusmn1)c α 19 26

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(1)

where c δxδt with δx and δt being the lattice spacing andthe time step respectively

A discretization of the Boltzmann equation in time andspace leads to the lattice Boltzmann equation [20 21]

fα x + eαδt t + δt( 1113857minusfα(x t) Λαj1113876fj(x t)minusfeq

j (x t)1113877

+ Fαδt

(2)

where fα is the distribution function of particles moving withvelocity eα Λαj is the collision matrix f eq is the equilibriumdistribution function and Fα is the external force

-e equilibrium distribution function f eq is obtainedusing the Taylor series expansion of theMaxwellndashBoltzmanndistribution function with velocity u up to second order Itcan be written as

feqα ρwα 1 +

eα middot uc2s

+eα middot u( 1113857

2 minus cs u| |( 11138572

2c4s1113890 1113891 (3)

where ρ is the fluid density u is the fluid velocity and thesound speed is cs c

3

radic -e weight coefficients for D3Q27

are given by

wα

827

α 0

227

α 1 6

154

α 7 18

1216

α 19 26

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

-e components of F are given as

Fα wαρeα middot a

c2s (5)

where a is the acceleration-e relaxation process has major influence on the

physical fidelity as well as numerical stability For the single-

1

23

4

5

6

7

11

1513

17

10

9

12

8

16

18

14

19

20

22

26

24

25

23

21

0

Figure 1 -ree-dimensional twenty-seven particle velocity(D3Q27) lattice







ρ sum26

α0fα

u sum26

α0eαfα +

12aδt

(7)


p ρc2s (8)





(9)



kα1113876 fα x + eαδt t( 1113857


(10)


kα

m(x t) + m x + eαδt t( 11138571113858 1113859


1 for liquid nodes

0 for gas nodes

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩




_c 0 σ lt σy




Real interface


Liquid nodes

Gas nodes



μ μy σ lt σy

μ σy

_c+

k

_c_c

n minusσy

μy

1113888 1113889

n

1113890 1113891 σ ge σy

(13)



(a)

100 mm

200 mm

300

mm

(b)

YX

Z

(c)



000 011 022 034 045 056 067 079

(a)


000 019 038 056 075 094 113 131

(b)


000 007 014 020 027 034 041 048

(c)


000 000 000 000 001 001 001 001

(d)


000 000 000 000 001 001 001 001

(e)


000 000 000 001 001 001 001 002

(f)








(a)

60

50

40

30

20

10

0

z (cm

)

x (cm)6050403020100

(b)


(a)

SCC

150 cm

600 cm

300

cm

150

cm

150

cm

150 cm

150

cm45

0 cm

(b)

SCC

Y

XZ

(c)






0000 0000 0000 0000 0000 0000 0000 0000

(a)


0000 0002 0003 0005 0006 0008 0010 0011

(b)


0000 0001 0002 0003 0004 0005 0006 0007

(c)


0000 0001 0002 0003 0004 0004 0005 0006

(d)


0000 0001 0001 0002 0003 0004 0004 0005

(e)


0000 0001 0001 0002 0002 0003 0003 0004

(f)


0000 0001 0001 0002 0002 0003 0003 0004

(g)


0000 0000 0001 0001 0002 0002 0003 0003

(h)


0000 0001 0001 0002 0002 0003 0003 0004

(i)


0000 0000 0001 0001 0002 0002 0003 0003

(j)


0000 0000 0001 0001 0002 0002 0002 0003

(k)


0000 0004 0008 0012 0016 0020 0024 0028

(l)





4 Conclusions




Acknowledgments


References







(a)

(b)
























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls









ρ sum26

α0fα

u sum26

α0eαfα +

12aδt

(7)


p ρc2s (8)





(9)



kα1113876 fα x + eαδt t( 1113857


(10)


kα

m(x t) + m x + eαδt t( 11138571113858 1113859


1 for liquid nodes

0 for gas nodes

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩




_c 0 σ lt σy




Real interface


Liquid nodes

Gas nodes



μ μy σ lt σy

μ σy

_c+

k

_c_c

n minusσy

μy

1113888 1113889

n

1113890 1113891 σ ge σy

(13)



(a)

100 mm

200 mm

300

mm

(b)

YX

Z

(c)



000 011 022 034 045 056 067 079

(a)


000 019 038 056 075 094 113 131

(b)


000 007 014 020 027 034 041 048

(c)


000 000 000 000 001 001 001 001

(d)


000 000 000 000 001 001 001 001

(e)


000 000 000 001 001 001 001 002

(f)








(a)

60

50

40

30

20

10

0

z (cm

)

x (cm)6050403020100

(b)


(a)

SCC

150 cm

600 cm

300

cm

150

cm

150

cm

150 cm

150

cm45

0 cm

(b)

SCC

Y

XZ

(c)






0000 0000 0000 0000 0000 0000 0000 0000

(a)


0000 0002 0003 0005 0006 0008 0010 0011

(b)


0000 0001 0002 0003 0004 0005 0006 0007

(c)


0000 0001 0002 0003 0004 0004 0005 0006

(d)


0000 0001 0001 0002 0003 0004 0004 0005

(e)


0000 0001 0001 0002 0002 0003 0003 0004

(f)


0000 0001 0001 0002 0002 0003 0003 0004

(g)


0000 0000 0001 0001 0002 0002 0003 0003

(h)


0000 0001 0001 0002 0002 0003 0003 0004

(i)


0000 0000 0001 0001 0002 0002 0003 0003

(j)


0000 0000 0001 0001 0002 0002 0002 0003

(k)


0000 0004 0008 0012 0016 0020 0024 0028

(l)





4 Conclusions




Acknowledgments


References







(a)

(b)
























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




μ μy σ lt σy

μ σy

_c+

k

_c_c

n minusσy

μy

1113888 1113889

n

1113890 1113891 σ ge σy

(13)



(a)

100 mm

200 mm

300

mm

(b)

YX

Z

(c)



000 011 022 034 045 056 067 079

(a)


000 019 038 056 075 094 113 131

(b)


000 007 014 020 027 034 041 048

(c)


000 000 000 000 001 001 001 001

(d)


000 000 000 000 001 001 001 001

(e)


000 000 000 001 001 001 001 002

(f)








(a)

60

50

40

30

20

10

0

z (cm

)

x (cm)6050403020100

(b)


(a)

SCC

150 cm

600 cm

300

cm

150

cm

150

cm

150 cm

150

cm45

0 cm

(b)

SCC

Y

XZ

(c)






0000 0000 0000 0000 0000 0000 0000 0000

(a)


0000 0002 0003 0005 0006 0008 0010 0011

(b)


0000 0001 0002 0003 0004 0005 0006 0007

(c)


0000 0001 0002 0003 0004 0004 0005 0006

(d)


0000 0001 0001 0002 0003 0004 0004 0005

(e)


0000 0001 0001 0002 0002 0003 0003 0004

(f)


0000 0001 0001 0002 0002 0003 0003 0004

(g)


0000 0000 0001 0001 0002 0002 0003 0003

(h)


0000 0001 0001 0002 0002 0003 0003 0004

(i)


0000 0000 0001 0001 0002 0002 0003 0003

(j)


0000 0000 0001 0001 0002 0002 0002 0003

(k)


0000 0004 0008 0012 0016 0020 0024 0028

(l)





4 Conclusions




Acknowledgments


References







(a)

(b)
























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls









(a)

60

50

40

30

20

10

0

z (cm

)

x (cm)6050403020100

(b)


(a)

SCC

150 cm

600 cm

300

cm

150

cm

150

cm

150 cm

150

cm45

0 cm

(b)

SCC

Y

XZ

(c)






0000 0000 0000 0000 0000 0000 0000 0000

(a)


0000 0002 0003 0005 0006 0008 0010 0011

(b)


0000 0001 0002 0003 0004 0005 0006 0007

(c)


0000 0001 0002 0003 0004 0004 0005 0006

(d)


0000 0001 0001 0002 0003 0004 0004 0005

(e)


0000 0001 0001 0002 0002 0003 0003 0004

(f)


0000 0001 0001 0002 0002 0003 0003 0004

(g)


0000 0000 0001 0001 0002 0002 0003 0003

(h)


0000 0001 0001 0002 0002 0003 0003 0004

(i)


0000 0000 0001 0001 0002 0002 0003 0003

(j)


0000 0000 0001 0001 0002 0002 0002 0003

(k)


0000 0004 0008 0012 0016 0020 0024 0028

(l)





4 Conclusions




Acknowledgments


References







(a)

(b)
























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls







0000 0000 0000 0000 0000 0000 0000 0000

(a)


0000 0002 0003 0005 0006 0008 0010 0011

(b)


0000 0001 0002 0003 0004 0005 0006 0007

(c)


0000 0001 0002 0003 0004 0004 0005 0006

(d)


0000 0001 0001 0002 0003 0004 0004 0005

(e)


0000 0001 0001 0002 0002 0003 0003 0004

(f)


0000 0001 0001 0002 0002 0003 0003 0004

(g)


0000 0000 0001 0001 0002 0002 0003 0003

(h)


0000 0001 0001 0002 0002 0003 0003 0004

(i)


0000 0000 0001 0001 0002 0002 0003 0003

(j)


0000 0000 0001 0001 0002 0002 0002 0003

(k)


0000 0004 0008 0012 0016 0020 0024 0028

(l)





4 Conclusions




Acknowledgments


References







(a)

(b)
























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






4 Conclusions




Acknowledgments


References







(a)

(b)
























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls

























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




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3DSimulationofSelf-CompactingConcreteFlowBasedon MRT-LBMdownloads.hindawi.com/journals/amse/2018/5436020.pdf · Research Article 3DSimulationofSelf-CompactingConcreteFlowBasedon MRT-LBM