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A Molecular Dynamics and LatticeBoltzmann Multiscale Simulation forDense Fluid FlowsW. J. Zhou a , H. B. Luan a , J. Sun b , Y. L. He a & W. Q. Tao aa Key Laboratory of Thermo-Fluid Science and Engineering of MOE,School of Energy & Power Engineering, Xi'an Jiaotong University,Xi'an, Shaanxi, People's Republic of Chinab School of Engineering and Materials Science, Queen Mary,University of London, London, United KingdomVersion of record first published: 22 Jun 2012.

To cite this article: W. J. Zhou , H. B. Luan , J. Sun , Y. L. He & W. Q. Tao (2012): A MolecularDynamics and Lattice Boltzmann Multiscale Simulation for Dense Fluid Flows, Numerical Heat Transfer,Part B: Fundamentals: An International Journal of Computation and Methodology, 61:5, 369-386

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A MOLECULAR DYNAMICS AND LATTICEBOLTZMANN MULTISCALE SIMULATION FOR DENSEFLUID FLOWS

W. J. Zhou1, H. B. Luan1, J. Sun2, Y. L. He1, and W. Q. Tao11Key Laboratory of Thermo-Fluid Science and Engineering of MOE,School of Energy & Power Engineering, Xi’an Jiaotong University, Xi’an,Shaanxi, People’s Republic of China2School of Engineering and Materials Science, Queen Mary, University ofLondon, London, United Kingdom

A molecular dynamics (MD)-lattice Boltzmann (LB) hybrid scheme has been adopted to

simulate dense fluid flows. Based on the domain decomposition method and the Schwarz

alternating scheme, the ‘‘Maxwell Demon’’ approach is used to impose boundary conditions

from the continuum to the atomistic region, while the ‘‘reconstruction operator’’ is implemen-

ted to construct the single-particle distribution function of the LB method from the results of

the MD simulation. Couette flows and the flow of a dense fluid argon around a carbon nano-

tube (CNT) are solved to validate the hybrid method. When the mesh of the LB domain is

refined and the size of corresponding sampling cells of the MD domain is reduced, the fluc-

tuations of the results between two successive iterations of the hybrid method become more

severe, although the results get closer to the MD reference solutions. To decrease the fluc-

tuation due to the mesh refinement, a new weighting function is proposed for the sampling

of MD simulation results. Numerical practice demonstrates its feasibility.

1. INTRODUCTION

The Navier-Stokes (NS) equations are based on the conventional continuumassumption and can be used to solve macroscopic flow problems. However, whenthe system becomes small and reaches the nanometer scale, the continuum assump-tion breaks down. Fortunately, molecular dynamics (MD) simulations can be usedto obtain details in small regions. Nevertheless, MD simulations are much moretime-consuming than continuum models and are limited to really small size in bothspace and time. Thus the multiscale simulation, or hybrid model, emerges, whichpossesses the advantages of both the MD simulation and the continuum methodand is becoming more and more popular. Microscopic details near the key positions

Received 22 April 2011; accepted 3 February 2012.

The authors wish to thank Dr. Hui Xu for very helpful technical discussion.

This work was supported by the Key Projects of National Natural Science Foundation of China

(No. 51136004, U0934005).

Address correspondence to W. Q. Tao, State Key Laboratory of Thermo-Fluid Science and

Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, 28 Xi’an Ning

Road, Xi’an, Shaanxi 710049, People’s Republic of China. E-mail: [email protected]

Numerical Heat Transfer, Part B, 61: 369–386, 2012

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790.2012.666144

369

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such as interfaces between fluid and solid, where the continuum assumption breaksdown, can be obtained by MD simulation, while the macro field of the remainingbulk region can be solved by continuum mechanics.

As far as the ways for exchanging information at the interface in the multiscalesimulation is concerned, generally it can be expressed as follows [1–4]. If theexchange of information at the interface (‘‘hand-shaking’’ region) is performed viaDirichlet type, then mathematically it can be expressed by

U ¼ CDu u ¼ RDU ð1aÞ

where U and u are the macroscopic parameter, and microscopic=mesoscopic para-meter, respectively. CD and RD are the Dirichlet compression and reconstructionoperators, respectively. The information at mesoscale or microscale level may betransferred to the macroscale level via Neumann type, that is, by supplying flux atthe interface; then we have [1–4]

q ¼ CNu u ¼ RDU ð1bÞ

where q is the interface flux of the continuum region.As indicated in [1–4], at the interface between different regions there will be a

mismatch in the kind and number of variables used by the different regions. TheDirichlet compression operator CD, which extracts the macroscopic parameters froma large amount of data at micro-or mesoscale level by some averaging or integrating

NOMENCLATURE

C lattice speed

cs lattice sound speed

CD Dirichlet compression operator

CN Neumann compression operator

d distance, velocity change rate

D2Q9 2-dimension, 9-velocity lattice

e error

f lattice distribution function, force

H height

l length

m mass of atom

N number of atoms or grid points

p pressure

q property of atom

Q macroscopic quantity, interface flux

r position vector, radius

rc cutoff radius

r0 radius of circle in MCIC scheme

RD Dirichlet reconstruction operator

t time

u velocity vector

u1 freestream velocity

w, W weighting factor

x, y, z Cartesian coordinates

X grid node

Dx space step

Dt time step

e characteristic energy

a, b coordinate direction indices

C boundary

dt time step

n kinematic viscosity

q density

r characteristic length

s nondimensional relaxation time

/ potential function

Superscripts

ArC related to an argon–carbon pair

eq equilibrium

Subscripts

c cutoff

i ith

j jth

k kth

370 W. J. ZHOU ET AL.

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principles, is easy to define, but the reconstruction operator RD, which should extenda small amount of macroscopic parameters into a large amount of parameters atmesoscale or microscale, is quite difficult to construct. Here the problem of one-to-many is encountered, since the macroscopic (mesoscopic) variables have to bemapped to more MD variables. The design of the compression and reconstructionoperators should abide by some basic physical laws or principles, such as mass,momentum, and energy conservation. In short, the exchange of information shouldbe conducted in a way that is physically meaningful, mathematically stable, compu-tationally efficient, and easy implement. It should be noted that the terminology‘‘operator’’ means: (1) It is an actual mathematical formula for transferring (convert-ing) results of different regions at the interface; or (2) it is a set of numerical treat-ments for transferring information which are developed from some fundamentalconsiderations. At present, the second is the most frequently encountered.

The pioneering work based on Eq. (1a), i.e., based on the state variable coup-ling, was done by O’Connell and Thompson [5]. An unsteady shear flow was inves-tigated, and the results from the hybrid method agreed well with the analyticalsolutions. Two main limitations of their coupling scheme are that the mass flux con-tinuation between the atomistic and continuum regions was not dealt with, and thetime scales of two different methods (micro versus macro) were not decoupled. Had-jiconstantinou and Patera [6] proposed a domain decomposition coupling schemebased on the Schwarz alternating method. They proposed a coupling scheme, calledthe Maxwell Demon method, to impose the continuum solutions to the boundary ofthe atomistic region. The implementation procedure of the Maxwell Demon will bepresented later. The use of a particle reservoir ensures that there is no net mass fluxinto the region of interest, so that the mass conservation condition at the interface isguaranteed. Using the Schwarz alternating method, the time scales of the MD simu-lation and the continuum method were successfully decoupled. The results from thehybrid method agreed well with the solutions from the pure continuum and MD ref-erence computations. Werder et al. [7] developed a hybrid scheme which can handlethe nonperiodic velocity boundary conditions of the atomistic domain, with the aidof a particle insertion algorithm named USHER [8] and specularly reflecting walls.An effective boundary force based on the radial distribution function was consideredto compensate for the lack of surrounding structure of the fluid. The simulationresults of the liquid argon flow around a carbon nanotube (CNT) from the hybridmethod were in good agreement with the fully MD reference solutions. Based onthe work by Werder et al. [7], Dupuis et al. [9] introduced a novel coupling schemeto solve the same problem. In this method, the lattice Boltzmann (LB) method wascoupled with the MD simulation to take advantage of the inherent mesoscopic char-acteristics of the LB method. The LB method was used for the entire domain, whilethe local details in the overlap region were provided by the MD simulation. The velo-city boundary condition around the CNT was presumed nonslipping to solve theNavier-Stokes (N-S) equations. A local forcing term g was added to the lattice Boltz-mann equation in order to enforce Dirichlet boundary conditions. During the infor-mation exchange procedure between the continuum and the atomistic region, thevelocities and the velocity gradients were both considered to improve the results.The results showed that the LB-MD hybrid model indeed had better performancethan the previous coupled macro method-MD model proposed by Werder et al.

MULTISCALE SCALING FOR DENSE FLUID FLOWS 371

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[7] when simulating the liquid argon flow around a CNT. All the above-mentionedmethods belong to the state variable exchange expressed by Eq. (1a).

A primary hybrid method based on direct flux exchange was proposed byFlekkoy et al. [10]. In their method, the coupling was directly realized by theexchange of mass and momentum fluxes across the particle–continuum interfacebetween the continuum and atomistic regions. Thus the conservation laws were nat-urally satisfied. Steady isothermal Couette and Poiseuille flows were simulated tovalidate their coupling scheme. Then Delgado-Buscalioni and Conveney [11]developed a similar scheme based on the one proposed by Flekkoy et al. [10]. In theirwork, unsteady fluid flows were investigated and the energy flux across the particle–continuum interface was also considered. The hybrid method based on direct fluxexchange was further developed by Wagner and Flekkoy [12], Delgado-Buscalioniet al. [13], De Fabritiis et al. [14], and Kalweit and Drikakis [15].

In this article, the hybrid scheme coupling the MD and the LBM is adopted.The LB method is adopted for resolving the continuum formulation because of itsinherent mesoscopic characteristics and its geometric flexibility [16]. Moreover, theunderlying kinetic nature of the LB equation is valuable for the simulation of micro-fluidics [17]. The Maxwell Demon method [6] is used to impose the boundary con-ditions of the atomistic region from the continuum region. What are differentfrom reference [6] includes following two aspects. First, the boundary conditionsof the continuum region are obtained by reconstructing the single-particle distri-bution function of the LB method from the results of the MD simulation throughthe ‘‘reconstruction operator’’ proposed in [1, 2]. With the ‘‘reconstruction operator’’approach, the selection of the continuum part out of the whole computationaldomain is more flexible because of the good geometric adaptive feature of the LBmethod. Second, a mesh refinement of the LB domain is needed when one wantsto acquire better results. Correspondingly, the size of sampling cells of the MDdomain is reduced. However, traditional weighting function used in MD to averagethe atomic properties cannot guarantee stability, due to the decrease of the size ofsampling cells in the MD simulation. A novel weighting function is proposed andis proven to be efficient.

The outline of the article is as follows. In Section 2, the basic principles of theMD simulation and the LB method and their coupling scheme are presented, andthen a new weighting function used to average the atomic properties is introduced.In Section 3, the hybrid scheme is demonstrated through simulations of Couette flowand flow of liquid argon around a CNT. Finally, some concluding remarks are given.

2. HYBRID SCHEME

2.1. Molecular Dynamics Simulation

In the atomistic region, the MD simulation is applied [18, 19]. The shiftedLennard-Jones (L-J) potential is used to describe the molecular interactions,

/ rð Þ ¼ 4err

� �12

� rr

� �6

� rrc

� �12

þ rrc

� �6" #

ð2Þ


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where e¼ 1.67� 10�21 J and r¼ 0.341 nm are the energy and length characteristicparameters, respectively. rc is a cutoff length beyond which the molecular interac-tions are not considered. The equation of motion used to update the accelerationof each molecular is

mid2ridt2

¼ �Xj 6¼i

q/ji

qrið3Þ

These Newtonian equations of motion are integrated using the leapfrop algor-ithm with a time step dt of 0.005s (s¼m1=2re�1=2, m is the molecular mass).

The other parameters are different for different problems and will be given indetail individually.

2.2. Lattice Boltzmann Method

In the continuum region, the two-dimensional (2-D) LB method is used due tothe absence of variation along the z direction. The main fluid flow in the bulk regionaway from the fluid–solid interface is described by the incompressible N-S equationsas follows:

r � u ¼ 0 ð4Þ

quqt

þ u � rð Þu ¼ � 1

qrpþ n Du; ð5Þ

where u is the velocity vector, p is the pressure, and q is the density. To solve Eqs. (4)and (5), the two-dimensional nine-velocity square lattice (D2Q9) model [20] is used.The nine velocities ci are c0¼ 0

ci ¼ c cos i � 1ð Þp 2=½ �; sin i � 1ð Þp 2=½ �f g for i ¼ 1; 2; 3; 4

ci ¼ c cos 2i � 1ð Þp 4=½ �; sin 2i � 1ð Þp 4=½ �f g for i ¼ 5; 6; 7; 8 ð6Þ

where c¼Dx=Dt, Dx is the spatial separation of the lattice, and Dt is the time step.The lattice Boltzmann equation with the BGK model [21, 22] is given by

fi xþ ci Dt; tþ Dtð Þ ¼ fi x; tð Þ � 1

sfi � f

ðeqÞi

h ið7Þ

where fðeqÞi is the ith equilibrium distribution function and s is the relaxation time.

The equilibrium distribution function is expressed as [20]

fðeqÞi ¼ tiq 1þ ciaua

c2sþ uaub

2c4sciacib � c2sdab� ��

ð8Þ

where a and b are the Cartesian coordinates, cs ¼ c=ffiffiffi3

pis the speed of sound, ti is a

weighting factor with t0¼ 4=9, t1¼ t2¼ t3¼ t4¼ 1=9, and t5¼ t6¼ t7¼ t8¼ 1=36.


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The density q and the fluid velocity u are given as

q ¼X8i¼0

fi qu ¼X8i¼0

cifi ð9Þ

2.3. The Hybrid Scheme

The domain decomposition method based on the alternating Schwarz method[23] is represented in Figure 1. The LB method is used to describe the continuumregion away from the fluid–solid interface, while the MD simulation is adopted todescribe the region near the interface, and both methods are assumed valid andapplied in the overlap region between C1 and C4. The alternating Schwarz methodis described as follows: In every iteration, given the boundary conditions on C4 bythe atomistic solution of the previous iteration and an outer boundary conditionby the specific problem, the N-S equations for the continuum region is solved bythe LB method. Then the continuum solution in turn offers a boundary conditionfor the MD simulation in the atomistic region between C2 and C3. This procedureis repeated until convergence toward a steady-state solution is achieved.

Obviously, the main difficulty in the atomistic–continuum coupling schemes isthe appropriate imposition of the boundary conditions between the continuum andthe atomistic region. The region between C1 and C2 is the reservoir region, which willbe treated in different ways for different problems. In the C ! P region between C2

and C3, the imposition of boundary condition from the LB to the MD method isaccomplished by the ‘‘Maxwell Demon’’ approach [6]. It is implemented as follows.At every time step, the velocities of molecules in the C! P region are reset through aMaxwellian distribution with mean and variance consistent with the local continuumfluid velocity and temperature resulted from LB solution. The boundary conditionfor the continuum region from the atomistic region is obtained by the ‘‘reconstruc-tion operator’’ method [1, 2]. In Figure 1, C4 is the computational boundary of the

Figure 1. Domain decomposition of the hybrid scheme.


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continuum region in the atomistic region. The single-particle distribution function ofthe grid point D can be given by following reconstruction operator:

fi ¼ f eqi 1� s Dt Uibc�2s Uiaqxaub þ nq2xaub þ nq�1Sabqxaq� � �

ð10Þ

where Uia¼ cia� ua, Sab¼ qxbuaþ qxaub, and n is the kinematic viscosity. The gradi-ents are calculated for the macroscopic quantities of the grid point E on the P ! Cboundary, which can be extracted through spatial and temporal averaging from themolecule configurations. In addition, the distance between the P ! C boundary andthe computational boundary of the continuum region C4 is always the same as thegrid size of the continuum region in the related direction.

Usually, thermal fluctuation should be considered in the LB equation whendealing with nanoscopic flows [24, 25], especially for fluid flow problems whereBrownian motion is important. However, in this article the LB method is used todescribe the continuum region away from the fluid–solid interface where the fluidis still in the hydrodynamic region, and the thermal fluctuation can be ignoredaccording to the derivation process shown in [1]. Therefore the coupling betweenMD and LB involves only mean flow fields, as done in [7, 9].

It is worth noting that in our hybrid scheme the LB method does not cover theentire domain as was done in [9]. Therefore, the boundary conditions needed to solvethe N-S equations need not be assumed in advance but can be provided by the MDsimulation at each iteration.

2.4. The Weighting Function

The methods used to extract macroscopic quantities from discrete atomicproperties are crucial. In a 2-D system, the macroscopic quantity Qij of a grid Xij,where i and j represent the locations in the x and y directions, respectively, is gener-ally obtained from the neighboring N atoms by

Qij ¼XNk¼1

qkWij;k ð11Þ

where qk is the property q of atom k. The weighting function Wij,k measures thecontribution of atom k to quantity Qij of grid node Xij and is given by

Wij;k ¼ wk xið Þwk yj� �

ð12Þ

The simplest type of weighting functions is called nearest-grid-point (NGP). Inthis scheme, the properties of an atom are totally assigned to the nearest grid point,i.e., W(NGP)ij, k¼ 1 if the distances between the atom and the grid point in the x and ydirections are less than Dx and Dy, respectively, otherwise W(NGP)ij, k¼ 0. In thisstudy a more accurate weighting function named cloud-in-cell (CIC) [26, 27] is alsoadopted, which assigns the properties of an atom to the nearest four grid points asdepicted in Figure 2a. It can be expressed as

wkðxiÞ ¼1þ xk�xi

Dx ðxi � Dx < xk < xiÞ1� xk�xi

Dx ðxi < xk < xi þ DxÞ

�ð13Þ


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where xk and xi are the atom and the grid point positions in the x direction, respect-ively, and wk(yj) has a similar expression as wk(xi). However, in our numerical prac-tice it is found that when the mesh becomes so fine that there is very few atoms ineach block sized by Dx�Dy, the fluctuations increase when averaging the atomproperties using the NGP or CIC scheme, which is due to the fact that the numberof atoms assigned to each grid point is reduced. To solve this problem, a novel typeof weighting function named modified cloud-in-cell (MCIC) is proposed. As shownin Figure 2b, the dots represent atoms and the triangles denote the grid points insidethe circle of radius r0. The properties of an atom are distributed to the grid pointslocated inside the circle whose center coincides with the atom position. Here wechoose the radius of the circle as r0¼ 1.0r. The MCIC weighting function can bedefined as

Wij;k ¼ ðdij;kÞ�1=PNm¼1

ðdij;kÞ�1 dij;k < r0

0 otherwise

8<: ð14Þ

where dij,k is the absolute distance between the atom k and the grid point Xij, and N isthe number of grid points inside the circle. Use of the MCIC scheme can guaranteethat the number of atoms assigned to each grid point remains unchanged when themesh of the LB method changes.

3. RESULTS AND DISCUSSION

In this section, the hybrid scheme is first validated through simulations of steadyCouette flows and is then used to investigate the flow of liquid argon around a CNT.Finally, the novel weighting function is used to adapt the refinement of the mesh.

3.1. Couette Flows

As a good benchmark test, Couette flows are first simulated to validate ourhybrid scheme. In our simulation, the fluid argon is confined between two parallel

Figure 2. Distribution schemes of atom properties to the surrounding grid points.


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walls at y¼ 0 and 68.12r. The top wall is moving at a velocity U¼r=s while the bot-tom wall is kept still. The LB method is implemented in the upper subregion of14.60r� y� 68.12r (i.e., the continuum region), the MD simulation is used in thelower subregion of 0� y� 34.06r (i.e., the atomistic region), and the overlap regionlies in the domain of 14.60r� y� 34.06r, as depicted in Figure 3. The flow in thecontinuum region is assumed to be fully developed, hence is one-dimensional(1-D) in the sense that it changes only in the y coordinate, while the particle motionin the atomistic region is three-dimensional (3-D).

The parameters of the atomistic region are described as follows. The density ofliquid argon is q¼ 0.81mr�3, the temperature of the fluid during the whole simula-tion is kept constant at T¼ 1.1ekB

�1, where kB is Boltzmann’s constant. Control ofthe temperature is achieved through a Langevin thermostat [28] with the dampingrate s�1, which is applied only in the z direction, normal to the bulk flow. Thusthe viscosity of the liquid argon at given density and temperature is m¼ 2.14esr�3

[5]. The lower wall consists of two layers of atoms in face-centered cubic (FCC) (11 1) structure. The interactions between fluid and solid atoms are also calculatedthrough Eq. (2), two groups of parameters (ewf=e, rwf=r, qw=q)¼ (0.6, 1.0, 1) and(0.6, 0.75, 4), where ewf, rwf, and qw are the fluid–solid energy scale, the fluid–solidlength scale, and the wall density, respectively. The former group of parametersyields a nonslip fluid–wall boundary condition, and the latter yields a slip one[29]. All the interactions are truncated at rc¼ 2.2r. As the sampling cell size in theatomistic region of this problem is relatively large, the simplest weighting functionNGP is adopted when averaging the atomic properties. The Reynolds number basedon the channel height is Re¼ 25.8.

In the atomistic region, periodic boundary conditions are imposed in the x andz directions, and the simulation sizes along these two directions are lx¼ lz¼ 13.62r.The continuum region is discretized by 11� 11 uniform grids with Dx¼Dy¼ 4.87r,and the periodic boundary condition is applied for the inlet and outlet boundaries.At the P ! C boundary, the macroscopic quantities such as velocities are presumedto be uniform along the x direction and are taken from the atomistic region by aver-aging the properties of all atoms within a volume of dimensions lx�Dy� lz whosecenter is located at the grid point concerned. In the overlap region, the thicknesses

Figure 3. Schematic diagram of Couette flow.


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of the reservoir layer and the C ! P region are Dy and 0.2r, respectively. To ensurethat the atoms always stay in the simulation domain, an external depress force isadded to the atoms in the reservoir region [30]:

fy ¼ �p0ry� y2

1� y� y2ð Þ y1 � y2ð Þ=ð15Þ

where p0 represents the average pressure of the atomistic region, and y1 and y2denote the coordinate values in the y direction of the upper and lower boundariesof the reservoir region, respectively.

In each iteration of the hybrid scheme, the LB method runs for 20,000 steps(totally 3.5 ns), and the MD simulation runs for 60,000 steps (totally 0.65 ns), ofwhich the last 40,000 steps are used to provide the next boundary condition forthe continuum region.

Figure 4 shows the steady solutions of the Couette flows from the hybridmethod and the MD reference simulation, where H represents that height of thechannel. The solid and the dash lines are the pure MD reference solutions of thetwo parameter groups and the symbols represent the solutions of the hybrid scheme.It can be seen that the hybrid results are in good agreement with the pure MD ref-erence solutions, under slip and no-slip fluid–wall boundary conditions.

3.2. Flow Past a CNT

In this section, the flow of liquid argon around a CNT is investigated using thehybrid scheme. All the configuration parameters are the same as [7, 9]. The CNT, ofchirality (16, 0) and radius r¼ 1.836r, is located at the center of the whole computa-tional domain of 88.11r� 88.11r� 13.19r (30 nm� 30 nm� 4.49 nm in real units).The interactions between argon and carbon atoms are also calculated throughEq. (2), with eArC¼ 0.572e, rArC¼ 1.0r. All the interactions are truncated at

Figure 4. Comparisons of the steady velocity profiles from the hybrid method and the pure MD simula-

tion results under slip and no-slip boundary conditions.


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rc¼ 2.94r. In the hybrid simulation, the size of the MD region is 29.37r�29.37r� 13.19r. As shown in Figure 5, the MD region is restricted by an outerboundary C1, while the LB region is bounded by an inner boundary C4. Uniform gridsare used for the LB region, with grid size Dx¼Dy¼ 1.47r. A flow velocity ofux¼ 0.634r=s and uy¼ 0 is specified at the inlet (x¼ 0) and outlet (x¼ 88.11r)boundaries. The density is chosen to be q¼ 0.6mr�3; the temperature is maintainedat T¼ 1.8ekB

�1 by using the Langevin thermostat [28] in the z direction with thedamping rate s�1. Thus the viscosity of the fluid argon needed by the LB method isn¼ 1.5 esm�1 [31], and the Reynolds number based on the CNT diameter is Re¼ 1.5.

In the overlap region, the C!P region is the domain between C2 and C3 andhas a thickness of 2.94r. The reservoir region has a thickness of 3.67r, and no restric-tion is imposed to atoms in this region. Due to the existence of the reservoir region,periodic boundary conditions can be imposed in all three directions x, y, and z, andthere is no need to consider the insertion of atoms near the boundary as treated in [7].For determining the reconstruction operator containing velocity gradient (Eq. (10))on the boundary C4, the macro quantities at the lattice nodes along two adjacentboundaries, the boundary C4 and C5, are needed. The information on C4 can beobtained from the LBM solutions on that line, while the macro quantities at the lat-tice nodes along the boundary C5 (i.e., the P ! C boundary) are obtained by aver-aging the properties of all atoms within a volume of certain size whose center iscoincident with the lattice node. The numbers of steps of LB and MD methods ineach iteration of the hybrid scheme are the same as in the Couette flows: In each iter-ation, the LB method runs for 20,000 steps, and the MD simulation runs for 20,000steps for equilibration and a further 40,000 steps to collect the next boundary con-dition for the continuum region. The hybrid simulation starts from the LB regionwith an initial condition ux¼ 0.634r=s and uy¼ 0 at the inner boundary C4.

To evaluate the deviation of the hybrid results from the pure MD simulationresults, an error ei at iteration i is expressed as

ei ¼ 1

N

Xm2X

uim � um;pure

u1

ð16Þ

Figure 5. Schematic diagram of argon flow around a CNT (color figure available online).


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where N is the number of cells in the whole computational domain X, u1 representsthe free-stream velocity, uim is the hybrid velocity of cell m at iteration i, and um;pure isthe corresponding pure MD result. Meanwhile, the change rate of the velocity field di

is used to describe the stability of the hybrid scheme, and it is given by

di ¼ 1

N

Xm2X

uim � ui�1m

u1

ð17Þ

To study the influence of different mesh resolutions to the hybrid results, threecases are considered as follows.

Case 1. The grid size of the LB domain is Dx¼Dy¼ 1.47r, the correspondingsampling cells of the MD domain are 20� 20� 1.

Case 2. The mesh of the LB domain is doubly refined on the basis of case 1, thus thegrid size is Dx¼Dy¼ 0.735r, and the corresponding sampling cells of the MDdomain are 40� 40� 1.

Case 3. The mesh of the LB domain is doubly refined on the basis of case 2, thus thegrid size is Dx¼Dy¼ 0.3675r, and the corresponding sampling cells of the MDdomain are 80� 80� 1.

The above three cases are simulated using both CIC and MCIC schemes. Inorder to compare with the results of [7, 9], real units are used hereafter. The timeevolution of errors ei of three cases using the CIC scheme is depicted in Figure 6.As can be seen, the error decreases when the mesh is refined from case 1 to case 3.The average errors between iterations 50 and 100 of case 1 to case 3 are 1.54%,1.25%, and 1.00%, respectively. The corresponding time evolution of the rate of velo-city change di using the CIC scheme is shown in Figure 7. Unlike the trend of change

Figure 6. Time evolution of error ei between the hybrid results using the CIC scheme and the pure MD

reference solutions for three different cases.


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of error ei, the rate di increases with the mesh refinement, and the average rates ofvelocity change between iterations 50 and 100 for case 1 to case 3 are 0.27%, 0.33%,and 0.40%, respectively. Figure 6 and 7 imply that although hybrid results get betterwhen the mesh is refined, fluctuations between successive iterations increase at thesame time. The reason is that when the mesh gets finer, the number of atoms ass-igned to each grid point decreases when averaging the atomic properties as the size ofsampling cells becomes small. Fortunately, this problem can be avoided when usingthe MCIC scheme. In the MCIC scheme, the number of atoms assigned to each gridpoint stays fixed when the mesh is refined, as long as the radius r0 remains unchanged.

Figure 8 shows the time evolution of errors ei of three cases using the MCICscheme. The error decreases when the mesh is refined, which is similar to the resultof the CIC scheme. The average errors between iterations 50 and 100 of cases 1 to 3when using the MCIC scheme are 1.59%, 1.18%, and 1.03%, respectively. The trendof rate of velocity change with the MCIC method, however, is different from the onewith the CIC method, which can be seen from Figure 9. The average rates betweeniterations 50 and 100 of case 1 to case 3 when using the MCIC scheme are 0.31%,0.28%, and 0.31%, respectively, which maintain almost at the same level. Obviously,it can be seen that the results from the MCIC scheme are much more stable thanthose from the CIC scheme.

In Figure 10, the converged hybrid simulation results of case 3 using the MCICmethod are compared with the pure MD reference solutions. Figure 10c indicatesthat the ux velocity contour of the hybrid results is highly consistent with the pureMD solutions, especially in the wake region of the CNT. Figure 11 shows the timeevolution of the velocity ux along the horizontal centerline y¼ 15 nm and the verticalcenterline x¼ 15 nm. After about 20 iterations, the velocity profiles of the hybridresults agree well with the MD reference solutions.

Figure 7. Time evolution of the rate of velocity change of the hybrid results using the CIC scheme for three

different cases.


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Moreover, the pure LB method is also adopted to see whether the fluid flowaround the CNT can be appropriately captured. In the pure LB method, the CNTis treated as a cylinder, and zero velocities are imposed at its surface. The finallyadopted lattice spacing Dx¼ 0.25r and a further decrease in the lattice spacing does

Figure 8. Time evolution of error ei between the hybrid results using the MCIC scheme and the pure MD

reference solutions for three different cases.

Figure 9. Time evolution of the rate of velocity change of the hybrid results using the MCIC scheme for

three different cases.


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not affect the results. The ux distributions along x and y are presented in Figure 11.From the figure it can be seen that even though as a whole the results from the pureLB method agree with the MD reference solutions quite well, deviations much largerthan 1% are observable. This comparison further proves the necessity of the hybridscheme.

It is also interesting to note that in [32] the LBM was used to simulate thewhole flow domain around a nano-size obstacle, and its was indicated that in orderto capture the recirculation just behind the obstacle the lattice spacing of the LBmethod, Dx¼ 0.25r should be adopted. However, in this article, the flow featuresnear the CNT is obtained by MD simulation, and the other computational domainis resolved by LB. Our results show that in our hybrid scheme, when the LB gridspacing Dx¼ 0.3675r, the deviation between results of the hybrid scheme and thepure MD method is only 1.00%.

Figure 10. Comparisons of the velocity contours between the converged hybrid solutions and the pure

MD reference results. (a) Pure MD results. (b) Hybrid solutions. (c) Comparison of two results (dashed

lines, hybrid solutions; solid lines, MD reference results).


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As far as the computational time is concerned, the time consumption of thehybrid scheme is only about one-fifth of that of the pure MD simulation.

4. CONCLUSIONS

In this article, a hybrid scheme has been adopted for dense fluid flow. MDsimulations are used in the region near the fluid–solid interface, while LB methodsare used in the bulk region away from the fluid–solid interface, where the continuumassumption is still valid. During the coupling procedure, the ‘‘Maxwell Demon’’ [6]approach is used to impose the boundary conditions for the atomistic region fromthe continuum region, and the boundary conditions of the continuum region isobtained by the ‘‘reconstruction operator’’ [1, 2].

The steady Couette flow is simulated to validate the hybrid scheme, and theresults agree well with the pure MD reference solutions. Then the flow of fluid argonaround a CNT is investigated. The mesh refinement of the LB method accompaniedby the reduction in size of sampling cells of the atomistic region lead to better resultsbut severe fluctuations when using the CIC weighting function. A novel weightingfunction named MCIC is proposed. The results show that the MCIC methodperforms better than the CIC method when the mesh is refined.

Moreover, it is not necessary to use the LB method for the entire computa-tional domain as was done in [9] when using our hybrid scheme, because the distri-bution function at the boundary of the LB method can be easily reconstructed due tothe ‘‘reconstruction operator’’ method. Thus the characteristics of the fluid–solidinterface, such as slip or no-slip boundary conditions, do not need to be known inadvance, which facilities the use of our hybrid scheme for complex geometries andboundary conditions.

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Figure 11. Time evolution of the ux velocity component: (a) along the horizontal centerline y¼ 15 nm;

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