ICNMM200 · 2011. 3. 27. · th th International International ASME ASME Conference Conference on on Nanochannels, Nanochannels, Microchannels Microchannels and and Minichannels Minichannels

Copyright © 2009 by ASME1

ProceedingsProceedingsProceedings

Proceedings

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InternationalInternationalInternational

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ASMEASMEASME

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Nanochannels,Nanochannels,Nanochannels,

Nanochannels,

MicrochannelsMicrochannelsMicrochannels

Microchannels

andandand

and

MinichannelsMinichannelsMinichannels

Minichannels

ICNMM200ICNMM200ICNMM200

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FLOWFLOWFLOW

FLOW

SIMULATIONSSIMULATIONSSIMULATIONS

SIMULATIONS

INININ

IN

AAA

A

SUB-MICROSUB-MICROSUB-MICRO

SUB-MICRO

POROUSPOROUSPOROUS

POROUS

MEDIUMMEDIUMMEDIUM

MEDIUM

BYBYBY

BY

THETHETHE

THE

LATTICELATTICELATTICE

LATTICE

BOLTZMANNBOLTZMANNBOLTZMANN

BOLTZMANN

ANDANDAND

AND

THETHETHE

THE

MOLECULARMOLECULARMOLECULAR

MOLECULAR

DYNAMICSDYNAMICSDYNAMICS

DYNAMICS

METHODSMETHODSMETHODS

METHODS

S.S.S.

S.

TakenakaTakenakaTakenaka

Takenaka

Dept.Dept.Dept.

Dept.

ofofof

of

Mech.Mech.Mech.

Mech.

Eng.,Eng.,Eng.,

Eng.,

OsakaOsakaOsaka

Osaka

PrefecturePrefecturePrefecture

Prefecture

University,University,University,

University,

SakaiSakaiSakai

Sakai

599-8531,599-8531,599-8531,

599-8531,

JapanJapanJapan

Japan

[email protected]@[email protected]

[email protected]

K.K.K.

K.

SugaSugaSuga

Suga

Dept.Dept.Dept.

Dept.

ofofof

of

Mech.Mech.Mech.

Mech.

Eng.,Eng.,Eng.,

Eng.,

OsakaOsakaOsaka

Osaka

PrefecturePrefecturePrefecture

Prefecture

University,University,University,

University,

SakaiSakaiSakai

Sakai

599-8531,599-8531,599-8531,

599-8531,

JapanJapanJapan

Japan


[email protected]

T.T.T.

T.

KinjoKinjoKinjo

Kinjo

Comput.Comput.Comput.

Comput.

Phys.Phys.Phys.

Phys.

Lab.,Lab.,Lab.,

Lab.,

ToyotaToyotaToyota

Toyota

CentralCentralCentral

Central

R&DR&DR&D

R&D

Labs.,Labs.,Labs.,

Labs.,

Inc.,Inc.,Inc.,

Inc.,

AichiAichiAichi

Aichi

480-1192,480-1192,480-1192,

480-1192,

JapanJapanJapan

Japan

eee

e


[email protected]

S.S.S.

S.

HyodoHyodoHyodo

Hyodo

Comput.Comput.Comput.

Comput.

Phys.Phys.Phys.

Phys.

Lab.,Lab.,Lab.,

Lab.,

ToyotaToyotaToyota

Toyota

CentralCentralCentral

Central

R&DR&DR&D

R&D

Labs.,Labs.,Labs.,

Labs.,

Inc.,Inc.,Inc.,

Inc.,

AichiAichiAichi

Aichi

480-1192,480-1192,480-1192,

480-1192,

JapanJapanJapan

Japan

eee

e


[email protected]

ABSTRACTABSTRACTABSTRACT

ABSTRACT

In order to devise and establish a cost-effective strategy tosimulate flows in continuum to slip and transitional regimes,present study focuses on evaluating a lattice Boltzmannequation [Niu et al., Phys. Rev. E 76, 036711, 2007] formerlyproposed by the present authors’ group. The main test flow caseis a flow around a square cylinder situated in a sub-microchannel. Since a rather shorter streamwise domain size and aperiodic streamwise boundary condition are imposed, the flowregime is regarded as a part of an infinite cylinder array set in anarrow channel which can be considered as a kind of micro-porous media. For the assessment of the lattice Boltzmannsimulations, the molecular dynamics (MD) simulations usingLeonard-Jones potential are also performed. In the MDsimulations, novel boundary treatments are applied. The resultsof the square cylinder flow by both the simulations at Knudsennumber Kn=0.08 show reasonable agreement and confirm thereliability of the present lattice Boltzmann method (LBM). Thefact that the computational time of the present MD simulationto obtain the reliable statistics is 1800 times as much as that ofthe LBM clearly indicates the usability of the LBM forengineering applications even in the slip and transitional flowregimes.

INTRODUCTIONINTRODUCTIONINTRODUCTION

INTRODUCTION

The recent rapid development of the fuel cell technology,flows in micro porous media have been focused on by manyresearchers (e.g. Pasaogullari and Wang, 2004; Niu et al.,2007a) since the catalytic layers and the electrolyte membranesof the fuel cells are usually made of micro-nano porous

materials. In such materials, the flows are usually distinguishedby large Knudsen numbers: Kn= , where is the/ 0.01H molecular mean free path of fluid and H is the characteristiclength of the flow domain. Accordingly, when one considersfluid flows in sub-micro flow geometries, it is necessary tounderstand the flows at the molecular level since the continuumNavier-Stokes equations are no longer applicable for such highKnudsen number flows. It is well known that processes in thesekinds of flows are described by the Boltzmann equation (BE) ofthe gas kinetic theory (Cercignani, 1975; Chapman andCowling, 1970). The numerical solution of the BE, eitherdirectly (Ohwada, 1976) or via the direct simulation MonteCarlo (DSMC) method (Bird, 1974), is very time expensive.This means that it is virtually impossible to perform suchsimulations for flows in complex large scale geometries evenby the modern computer environment. Thus, there are strongneeds for accurate models which allow engineering flowsimulations in microscale geometries at lower computationalcosts.

Recently, alternative strategies to simulate flows in thecontinuum to the slip and transitional regimes have been thusproposed. Nie et al. (2002) introduced Knudsen numberdependency into the relaxation parameter of the latticeBoltzmann equation and simulated micro-channel flows at0.01<Kn<0.4 using a slip velocity condition (e.g. Arkikic et al.,1997) on the channel walls. Shen et al. (2004) validated thisstrategy comparing the results of DSMC simulations ofmicrochannel flows. The authors’ group is also trying to devisean alternative strategy to simulate such flows in the continuumto the slip and transitional regimes (Niu et al., 2007). In our


lattice Boltzmann method (LBM), a diffuse scattering wallboundary condition, which represents the interactions betweengas molecules and the wall, as well as an effective relaxationtime associated with the Knudsen number are employed withthe D2Q21 velocity model. Based on this work, the presentstudy focuses on simulating a flow around a square cylinderarray situated in a sub-micro channel. The test flow fieldsconsidered are a plane Poiseuille channel flow and a flowaround a square cylinder in a channel.

In order to assess the LBM simulations of such sub-microflows, the molecular dynamics (MD) simulation method (Haile,1997), which is based on simpler Newtonian mechanics and isgetting popular in nanoscale flow simulations (e.g. Kinjo andMatsumoto, 1998), is also performed with novel boundarytreatments developed in this study. Although the current topicsdo not include phase changes yet, the MD simulation ispromising on treating phase changes and/or multiphase nano-micro flow phenomena which are important issues in the fuelcell technology. In fact, there is no special treatment is requiredfor the MD simulation to distinguish the phase difference (e.g.Matsumoto and Tanaka, 2007; Tsuda et al., 2007).

NOMENCLATURENOMENCLATURENOMENCLATURE

NOMENCLATURE

: force distribution functionF: channel heightH: nth-order Hermite polynomial nH: Knudsen numberKn: size of the simulation cellL: gass constantR: temperatureT: mainstream velocityU: bulk mainstream velocitybU: maximum mainstream velocitymaxU: force constant for LBEaaa

a

: sound speedcs: amount of elongation and contractiondL: single particle velocity distribution functionf: equilibrium distribution functioneqf: non-equilibrium distribution functionf

: regularization distribution functionf

: force constant for the MDg: molecular massm: number density of gas flown: fluid pressurep: intermolecular distancer: the closest distance when collision occursmr: distance between wall moleculeswr: timet: velocity of a moleculev: function of effective relaxation time: direction of velocity: time stept: strength of inter-atomic interaction

: potential energy: mean free path length: density: diameter of molecules: relaxation time: weighting function: velocity of the particleξξξ

ξ

LATTICELATTICELATTICE

LATTICE

BOLTZMANNBOLTZMANNBOLTZMANN

BOLTZMANN

METHODMETHODMETHOD

METHOD

(LBM)(LBM)(LBM)

(LBM)

Over the last decade, due to its conception, LBM(McNamara and Zanetti, 1988; Higuera and Jimenez, 1989) hasreceived a lot of attention because of its advantages in hydro-dynamics analyses. One of the advantages is applicability ofdescribing complex flow phenomena including multiphase flowin porous materials which are difficult to carry out bytraditional techniques such as those using Navier-Stokesequations (e.g. Martys and Chen, 1996; Keehm, et al., 2004;Pan et al., 2006). Another is computational efficiency becauseit is an intrinsically explicit method, and thus its parallelizationis straight forward. In addition, though it is explicit, itsnumerical stability and accuracy at Navier-Stokes level areguaranteed.

A brief description of the lattice Boltzmann equation (LBE)which is fundamental equation of LBM and its steps of theapplication to high Knudsen number flow are given insubsections.

LatticeLatticeLattice

Lattice

BoltzmannBoltzmannBoltzmann

Boltzmann

equationequationequation

equation

withwithwith

with

discretediscretediscrete

discrete

velocitiesvelocitiesvelocities

velocities

The LBE can be obtained by discretizing the velocity spaceof Boltzmann equation (BE) into a finite number of discretevelocities

{=1,...,d}. Although many techniques todiscretize the velocity space have been proposed, the presentstudy employs so called D2Q9 and D2Q21 discrete velocitymodels. Table 1 lists the sound speeds cs , the discrete velocities

and the weights parameter in both the models. Thediscretization is applied to uniform, regular spatial lattices asshown in Fig.1. Let xxx

x

be the Cartesian coordinates of theconfiguration space and

that of velocity space. The LBEdescribes evolutions of a single particle distribution functionf(xxx

x

,

, t) which is defined such that f(xxx

x

,

, t)d

dxxx

x

represents thenumber of particles in the phase space element d

dxxx

x

at time t,

TableTableTable

Table

1:1:1:

1:

MainMainMain

Main

parametersparametersparameters

parameters

ofofof

of

thethethe

the

discretediscretediscrete

discrete

velocityvelocityvelocity

velocity

models.models.models.

models.

Models cs2

(0,0)(±1,0),D2Q9

D2Q21

(±1,±1)(0,±1)

(0,0)(±1,0), (0,±1)

(±1,±1)(±2,0), (0,±2)

(±2,±2)(±3,0), (0,±3)

4/9 (α=0)1/9 (α=1-4)1/36 (α=5-8)91/324 (α=0)1/12 (α=1-4)2/27 (α=5-8)7/360 (α=9-12)1/432 (α=13-16)1/1620(α=17-20)

1/3

2/3

ξξξ

ξ


and can be written in the following BGK (Bhatnagar et al.,1954) form:

, , , ,0.5

,0.5

eqtf t t t f t f t f tt

t F tt

xxx

x

ξ x x xξ x x xξ x x x

ξ x x x

xxx

x

　　　　　　　　　 ,

(1)

where the equilibrium distribution function f eq is written as

(2)

2 2

21, 12

,

eqf tRT RTRT

ξ uξ uξ u

ξ u

ξ uξ uξ u

ξ u

uuu

u

xxx

x

(3)

2 2

2

2 2

2

1, 12

1 36

.

eqf tRT RTRT

RT RTRT

ξ uξ uξ u

ξ u

ξ uξ uξ u

ξ u

uuu

u

xxx

x

ξ uξ uξ u

ξ u

ξ uξ uξ u

ξ u

uuu

u

　　　　　　　

Eq.(2) retains up to the second-order term in the Hermiteexpansion and used in the D2Q9 velocity model whilst Eq.(3)retains further up to the third-order and used in the D2Q21velocity model. The ideal gas constant is R, and uuu

u

and T arerespectively the fluid macro density, velocity and temperature.The sound speed cs=(RT)1/2 is equal to 1/3 and 2/3 respectivelyin the D2Q9 and the D2Q21 models. The contribution of theforce term, F, is introduced as

22

, 1

1 : 22 .

ij

F tRT RT RT

HRT RT RT RT

ξ a ξ uξ a ξ uξ a ξ u

ξ a ξ u

a ua ua u

a u

xxx

x

ξ aξ aξ a

ξ a

ξ aξξ aξξ aξ

ξ aξ

uuu

u

　　　　　

(4)

where, H(n) is the Hermite polynomial. The variables uuu

u

andthe pressure p are respectively obtained by applying the integralof microscopic velocity moment as

(5),fd

0

0 2 ,

d tf

uuu

u

ξ aξ aξ a

ξ a

2.scp

ExtensionExtensionExtension

Extension

tototo

to

HighHighHigh

High

KnudsenKnudsenKnudsen

Knudsen

numbernumbernumber

number

flowsflowsflows

flows

So far, the equations which govern the macroscopicvariables of continuum flows are presented. Only when thenon-equilibrium part of distribution is very small and thedistribution function can be approximated by the equilibriumdistribution function, the equations can describe the fluidphenomena. However, when the Kn becomes larger and thenon-equilibrium part is no longer ignorable, the aforementionedLBE becomes invalid. According to our previous analysis (Niu

(a) D2Q9

(b) D2Q21FigureFigureFigure

Figure

1:1:1:

1:

DiscreteDiscreteDiscrete

Discrete


velocity

models;models;models;

models;

(a)(a)(a)

(a)

D2Q9D2Q9D2Q9

D2Q9

model,model,model,

model,

(b)(b)(b)

(b)

D2Q21D2Q21D2Q21

D2Q21

model.model.model.

model.

FigureFigureFigure

Figure

2:2:2:

2:

VariationVariationVariation

Variation

ofofof

of

thethethe

the

functionfunctionfunction

function

withwithwith

with

thethethe

the


Knudsen

number.number.number.

number.


et al., 2007b), the difficulty is cleared by using the threeprocedures described below.

A.A.A.

A.

DiffuseDiffuseDiffuse

Diffuse

scatteringscatteringscattering

scattering

boundaryboundaryboundary

boundary

conditionconditioncondition

condition

(DSBC)(DSBC)(DSBC)

(DSBC)

The boundary condition used in the LBM is usuallyspecular reflection, so the velocity and the temperature of awall are not reflected into the distribution of the reflectedmolecules. However, from a microscopic viewpoint, the wallboundary condition should include the physics on the wallbecause the fluid and the wall molecules are interacted witheach other. Therefore, the incident particles are modeled to bereflected with the information of the Maxwell distributionfunction at the wall boundary. The modeled form is written inthe LBM frame as

,

,, , ,,

w eqweq

w

f tf t f t

f t

ξ u n xξ u n xξ u n x

ξ u n x

x xx xx x

x x

ξ u n xξ u n xξ u n x

ξ u n x

(6) 0; 0 ,w w ξ u n ξ u nξ u n ξ u nξ u n ξ u n

ξ u n ξ u n

where nnn

n

is the unit wall normal vector, is the wall,eqwf

equilibrium distribution function, and the subscripts w, ', respectively mean the wall and the directions of the incidentand reflected particles.

B.B.B.

B.

EffectiveEffectiveEffective

Effective

relaxationrelaxationrelaxation

relaxation

timetimetime

time

In continuum flow, the relaxation time can be defined interms of viscosity then flow is under control of the Reynoldsnumber: Re. In contrast, Kn is a fundamental dimensionlessnumber in non-continuum flow. Therefore, for applying to highKn flows, the relaxation time needs to be associated with Kn.Specifically, In microscale geometries, the relaxation time isinfluenced by the presence of walls, and the effective relaxationtime * can be modeled as

(7) * 2Kn Kn Kn ,sc L

where L is the characteristic length confined between walls in amicroscale flow geometry. The function can be expressed as

(8) 2Kn arctan Kn ,

ba

where the coefficient a and b are set a =2 and b =3/4 based onexperimental investigation (Guo et al., 2006). Fig.2 shows therelation between Kn and . The profile (decreases as Knincreases) indicates that some molecules will hit walls and theirflight time (effective relaxation time *) may be shorter thanthe mean free time (relaxation time ) defined in an unboundedsystem.

C.C.C.

C.

RegularizationRegularizationRegularization

Regularization

procedureprocedureprocedure

procedure

Generally speaking, the distribution function f has analiasing error because it cannot be entirely projected on theHermite space. Such an error is usually very small, but it can beno longer neglected when the system is far from equilibriumbecause of high Knudsen effect. To resolve this problem, theregularization procedure was previously introduced forimproving numerical stability (Chen et al., 2006). Theprocedure is implemented as the following. First, thedistribution function f is divided as

(9),eqf f f

where f' is the non-equilibrium part of the distribution. Second,it is necessary to convert f' to a new distribution which lieswithin the subspace spanned by the first three Hermitepolynomials. Using the Hermite polynomials, is given

(10)

22

0

33

0

12

6 .

d

i js s

d

i j ks s

f H fc c

B H fc c

ξξξ

ξ

ξ ξξ ξξ ξ

ξ ξ

ξξξ

ξ

ξ ξ ξξ ξ ξξ ξ ξ

ξ ξ ξ

　　　

Here, the coefficient B is given by B =1. By substituting fin Eq.(1) to Eq.(9) after converting f' of Eq.(9) to Eq.(10), onecan obtain the following form:

(11)

0.5, ,0.5

, .0.5

eq tf t t t f t ft

tF tt

xxx

x

ξ xξ xξ x

ξ x

xxx

x

　　　　　　　　　

MOLECULARMOLECULARMOLECULAR

MOLECULAR

DYNAMICSDYNAMICSDYNAMICS

DYNAMICS

METHODMETHODMETHOD

METHOD

Molecular dynamics simulation consists of the numericalsolution of classical equations of motion to calculate themotions of N molecules interacting via model potentials. Oneof the well-known model potential is the Lennard-Jones (6-12)potentialwhich is defined as

(12) 12 6

4,

rr r

where r is the intermolecular distance, and are the welldepth and the diameter of the molecules. This was used in theearliest study of the properties of argon and is illustrated inFig.3. In the system which is modeled by the Lennard-Jonesmolecules only, the classical equation of motion can berewritten in the nondimensional form:

f

f


(13)2 * 2 *

*2 2 * ,d rdt m r

where r*=r/, t*=t/ (=t/(/m2)1/2) and *=/ are thenondimensional length, time and energy, respectively. (It iscommon that the calculation values are expressed by thenondimensional parameters.) However, to make a physicalinterpretation, it is sometimes expressed as finite values interms of argon. In addition, the nondimensional number densityN* (=N3) and temperature T* (=kBT/) are used in theLennard-Jones fluid.

In actual calculations, to hold down the calculation effortof the interactions, the potential effects are truncated over cutoff distance rc which is defined as rc = . In the2.5 5.5 present study, rc = 3.0 is considered.

TheTheThe

The

interactionsinteractionsinteractions

interactions

betweenbetweenbetween

between

thethethe

the

fluidfluidfluid

fluid

andandand

and

wallwallwall

wall

moleculesmoleculesmolecules

molecules

When fluid flows in a finite space surrounded by walls, theflow is governed by the interaction between the fluid and thewall molecules. Thus, it is necessary to evaluate the flowphenomena in terms of the intermolecular potential energybetween the fluid and the wall which varies with the structuresand the properties of the system.

In this research, the wall is simulated by an argon atommonolayer, which is schematically shown in Fig.4. Wallmolecules form regular triangles whose side length of rw = 0.7.Each of the wall molecules is tethered to fixed lattice sitelocations by a harmonic spring with a large spring constant.

A fluid molecule is located near the wall molecules withthe distance of r0. Each of interactions between two moleculesis the Lennard-Jones potential itself. However, the total wall-fluid interaction should include not only the nearestneighboring molecule but also surrounding ones. Therefore, theeffects of the surrounding molecules are considered by thenumbering molecules as follows: the nearest neighboringmolecule is set n=1, and its surrounding six molecules aredefined as n=2. Then the next twelve molecules around n = 2are defined as n=3. From this numbering rule, the potentialenergy between the fluid molecule and the wall molecules up ton=k is written as the following form of :

12 612 6

0 2 2 2 21 0 0

12 6

2 22 20 0

4 4 *6

4 *6 11 1

k

nn w w

w w

r Ar r r r r r

kr k r r k r

(14)

where A is the parameter which determines the strength of thewall-fluid interaction. The present study sets A = 1 after a seriesof validity tests shown in the later section. Fig.5 shows theprofiles of Eq.(14) with the parameter A = 1. In the figure, it isobvious how far a fluid molecule should be affected by the wallmolecules and how much the total potential energy is incomparison with the simple Lennard-Jones potential. It can be

found that the potential energy almost converges by the stage of up to n=5. And the well depth of the actual potential upto n=5 is about 7.5 times as large as that of the Lennard-Jonespotential. The features of the flows differ by the variation ofthis well depth. It is thus important to set an appropriateparameter according to the flow conditions.

ElongationElongationElongation

Elongation

andandand

and

contractioncontractioncontraction

contraction

ofofof

of

periodicperiodicperiodic

periodic


boundary


condition

In conventional MD simulations, periodic boundaryconditions are employed to avoid the effects of domainsurfaces. Fig.6 shows a schematic illustration of the periodicboundary condition in a two-dimensional space. Periodic imagecells of the original cell are put around the original one. The

FigureFigureFigure

Figure

3:3:3:

3:

Lennard-JonesLennard-JonesLennard-Jones

Lennard-Jones

potentialpotentialpotential

potential

showingshowingshowing

showing

thethethe

the

rrr

r

-12-12-12

-12

andandand

and

rrr

r

-6-6-6

-6

contributions.contributions.contributions.

contributions.

FigureFigureFigure

Figure

4:4:4:

4:

SchematicSchematicSchematic

Schematic

drawingdrawingdrawing

drawing

ofofof

of

thethethe

the

wallwallwall

wall

structure.structure.structure.

structure.

TheTheThe

The

leftleftleft

left

sidesideside

side

isisis

is

ananan

an

edgeedgeedge

edge

view,view,view,

view,

andandand

and

thethethe

the

rightrightright

right

sidesideside

side

isisis

is

aaa

a

schematicschematicschematic

schematic

elevationelevationelevation

elevation

ofofof

of

thethethe

the

wall.wall.wall.

wall.

WallWallWall

Wall

moleculesmoleculesmolecules

molecules

areareare

are

presentpresentpresent

present

ononon

on

thethethe

the

outsideoutsideoutside

outside

(n(n(n

(n

3).3).3).

3).


molecules in the image cells move in the same way as in theoriginal cell. Thus the atom that leaves the original cell througha boundary simultaneously reenters the cell through theopposite boundary.

In the aforementioned periodic boundary conditions, thefluid consists of the molecules in the original cell does notexhibit fluidic motion expect thermal fluctuations of themolecules. In conventional method, a gravity-like constantforce is introduced to generate flows in the MD simulationmethod (e.g. Rapaport et al., 1995). Fig.7 shows flowgeneration by the gravity-like force. In this method, forceacting on the simulation cell is imitated by the body force .gIn actual fluid systems, a driving force of the flow generation isnot only gravity-like force. Another typical driving force in thefluid flow is a pressure gradient:

. (15)drive p fff

f

By using the equation of state of the fluid , the Tpp ,pressure difference can be converted to density difference. Inthis study, a new boundary condition for the flow generation isdeveloped by modifying the conventional periodic boundarycondition. The density gradient is generated by contraction ofthe image cell in upstream side and elongation in downstreamside as shown in Fig.8; the upstream side is contracted to

, and the downstream side is elongated to .L dL L dLIn order to validate this new boundary treatment, flows in

micro channel were simulated preliminary. The fluid consistedof the Lennard-Jones molecules, whose inter-atomic interactionwas expressed by Eq.(12). The number of the fluid moleculeswas 5632 and the number of the wall molecules was 1620. Theedge lengths of the simulation domain were Lx=18.9,Ly=18.1865 and Lz=41.0. The amount of the elongation and thecontraction was dL=0.1 along the x-direction. The results wereconfirmed to be reasonable as discussed in the later section.(The velocity profiles of the MD simulations are discussed withthe LBM and the DSMC results.) The density profile along x-direction is shown in Fig.9. As mentioned above, there is adensity gradient which can be the source of driving force forthe flow generation. In our method, very small elongation andcontraction, dL=0.1, can generate the small density gradient.This method seems to be more natural to generate flow inmolecular dynamics simulations than direct driving by applyingconstant forces.

DeterminationDeterminationDetermination

Determination

ofofof

of


Knudsen

numbersnumbersnumbers

numbers

ofofof

of

thethethe

the

MDMDMD

MD

simulationssimulationssimulations

simulations

As noted in the earlier section, the Knudsen number is adimensionless number defined as the ratio of the molecularmean free path length to a representative length scale . HIn this research, is the width of a channel whilstH depends a great deal on the property of the system and the kindof the particles. The estimation method for used in thepresent study is as follows.

First, a one-dimensional model of inter-molecular collisionwhich is shown in Fig.10 is considered. Using the principle ofequipartition of energy, the kinetic energy of a molecule with equipartition of energy, the kinetic energy of a molecule with

FigureFigureFigure

Figure

5:5:5:

5:

ComparisonComparisonComparison

Comparison

ofofof

of

thethethe

the

fluid-wallfluid-wallfluid-wall

fluid-wall

potentialpotentialpotential

potential

atatat

at

differentdifferentdifferent

different

kkk

k

(1(1(1

(1

kkk

k

5).5).5).

5).

FigureFigureFigure

Figure

6:6:6:

6:

SchematicSchematicSchematic

Schematic

picturepicturepicture

picture

ofofof

of

thethethe

the

periodicperiodicperiodic

periodic


boundary


condition

forforfor

for

two-dimensiontwo-dimensiontwo-dimension

two-dimension

alalal

al

spacespacespace

space

...

.

FigureFigureFigure

Figure

7:7:7:

7:

FlowFlowFlow

Flow

generationgenerationgeneration

generation

bybyby

by

constantconstantconstant

constant

force.force.force.

force.

FigureFigureFigure

Figure

888

8

...

.

FlowFlowFlow

Flow

generationgenerationgeneration

generation

bybyby

by

elongationelongationelongation

elongation

andandand

and

contractioncontractioncontraction

contraction

ofofof

of

imageimageimage

image

cells.cells.cells.

cells.


the velocity v is equal to the mean energy:

. (16)21 3

2 2 Bmv k T

When the molecules collide, the kinetic energy of the moleculeis replaced with an equal volume of the potential energy. Thus,the following formula is obtained by the conservation law ofenergy:

, (17) 21 3

2 2LJ m Br mv k T

where, rm is the closest distance for the intermolecular collision.A repulsion force F(rm) is defined as a threshold value which isdetermined whether the collision is present.

(18)

.m

LJm

r

dF rdr

If the force acting on the molecule exceeds the threshold value,a collision occurs. To calculate the mean free path length,practically, the motion and the force of sample molecules aretraced. The intermolecular force acting on a sample moleculefor 500 time steps is shown in Fig.11. The time intervalbetween a collision point (the point when the force steps overthe threshold value) and next collision point is a mean freetime. Obviously, the migration distance in the mean free time isthe mean free path length. Finally, the mean free path isobtained by taking the reasonable statistics of the migrationdistance.

RESULTSRESULTSRESULTS

RESULTS

ANDANDAND

AND

DISCUSSIONSDISCUSSIONSDISCUSSIONS

DISCUSSIONS

First, in Poiseuille channel flows, the effects of differentwall-fluid interactions used in the MD method are discussed.By varying the coefficient A in Eq.(14), we discuss how muchthe channel flow profiles are affected. Second, to confirm thevalidity of the LBM and the MD simulations of sub-microscalegas flows, Poiseuille channel flows are discussed. Third,simulations of flows around a square cylinder in a channel arediscussed. In the MD simulations, Kn is a resultant value of thecomputation. In order to compare the results, such a Kn is usedas a computational condition of the LBM simulation.

EffectsEffectsEffects

Effects

ofofof

of


different

wall-fluidwall-fluidwall-fluid

wall-fluid

interactionsinteractionsinteractions

interactions

An atomic channel consisting of two parallel walls isconsidered for the MD simulations. The fluids consist of theLennard-Jones molecules, whose wall-fluid interactions areexpressed by Eq.(14) with A. The numbers of the fluid and thewall molecules are 4096 and 2100, respectively. The simulationdomain’s lengths are Lx=21.0, Ly=21.218 and Lz=40.0. Fig. 12compares the normalized velocity profiles at different values of

A. As the coefficient A decreases, which means the wall is gas-

FigureFigureFigure

Figure

9:9:9:

9:

DensityDensityDensity

Density

profileprofileprofile

profile

alongalongalong

along

x-direction.x-direction.x-direction.

x-direction.

FigureFigureFigure

Figure

10:10:10:

10:

SSS

S

chematicchematicchematic

chematic

viewviewview

view

ofofof

of

thethethe

the

collisioncollisioncollision

collision

model.model.model.

model.

FigureFigureFigure

Figure

11:11:11:

11:

VariationVariationVariation

Variation

ofofof

of

thethethe

the

intermolecularintermolecularintermolecular

intermolecular

forceforceforce

force

actingactingacting

acting

ononon

on

aaa

a

moleculemoleculemolecule

molecule

withwithwith

with

timetimetime

time

step.step.step.

step.

TheTheThe

The

thresholdthresholdthreshold

threshold

valuevaluevalue

value

isisis

is

estimatedestimatedestimated

estimated

atatat

at

aboutaboutabout

about

808080

80

atatat

at

TTT

T

***

*

=2.0.=2.0.=2.0.

=2.0.

FigureFigureFigure

Figure

12:12:12:

12:


Comparison

ofofof

of

thethethe

the

normalizednormalizednormalized

normalized


velocity

profilesprofilesprofiles

profiles

ofofof

of

PoiseuillePoiseuillePoiseuille

Poiseuille

flowsflowsflows

flows

atatat

at


different

AAA

A

bybyby

by

thethethe

the

MDMDMD

MD

simulations.simulations.simulations.

simulations.


A. As the coefficient A decreases, which means the wall isgasphobic, the slip velocity on the wall increases and thegeneral shape of the profile becomes flatter as if the Knudsennumber increases. This shows that indeed the flow issignificantly influenced by the interaction between the fluid andthe wall molecules. The following discussions thus use A=1 asa neutral wall condition.

PPP

P

lanelanelane

lane


Poiseuille

channelchannelchannel

channel

flowflowflow

flow

sss

s

Again, an atomic channel consisting of two parallel wallsis considered for the MD simulations whilst a 2D channel withheight H is considered for LBM simulations as illustrated inFig.13. As in our previous work (Niu et al., 2007b), thepresently applied LBE can capture Knudsen effects and this can

be also confirmed in Fig. 14 where the normalized velocityprofiles of plane Poiseuille channel flows at Kn=0.1 and 1 arecompared with the data of the DSMC (Beskok et al., 1999).The discrete velocity models of LBM are the D2Q9 and theD2Q21 models for the 2D uniform Cartesian lattice of 60 60.(Such a density of the lattice was confirmed to be fine enoughin Niu et al., 2007b). The number of iterations of the LBMsimulations is 5,000 and the results are fully converged. Theresults of the D2Q21 model well accord with the DSMCresults. Although the D2Q9 model can capture the generaltendency, its results always underestimate the slip velocities atthe plate walls compared to those of the DSMC and the D2Q21model. This is because in D2Q9 model, only up to the second-order terms are retained in the Hermite expansion of theMaxwellian distribution described by Eq.(2). The D2Q21model allows approximations to the Boltzmann equation to beconstructed by retaining terms of third-order in the Hermite

TableTableTable

Table

2:2:2:

2:

ComputationalComputationalComputational

Computational

conditionsconditionsconditions

conditions

ofofof

of

thethethe

the

MDMDMD

MD


simulations

ofofof

of


Poiseuille

flows.flows.flows.

flows.

(a)

(b)

FigureFigureFigure

Figure

13:13:13:

13:

FlowFlowFlow

Flow

fieldfieldfield

field

geometriesgeometriesgeometries

geometries

ofofof

of


Poiseuille

flowflowflow

flow

simulations;simulations;simulations;

simulations;

(a)(a)(a)

(a)

3D3D3D

3D


channel

forforfor

for

thethethe

the

MD,MD,MD,

MD,

(b)(b)(b)

(b)

2D2D2D

2D


channel

forforfor

for

thethethe

the

LBM.LBM.LBM.

LBM.

FigureFigureFigure

Figure

14:14:14:

14:


Comparison

ofofof

of

thethethe

the


normalized


velocity


profiles

ofofof

of


Poiseuille

flowsflowsflows

flows

atatat

at


different


Knudsen


numbers

bybyby

by

thethethe

the

D2Q9D2Q9D2Q9

D2Q9

andandand

and

thethethe

the

D2Q21D2Q21D2Q21

D2Q21

models.models.models.

models.

FigureFigureFigure

Figure

15:15:15:

15:


Comparison

ofofof

of

thethethe

the


normalized


velocity


profiles

ofofof

of


Poiseuille

flowsflowsflows

flows

atatat

at

threethreethree

three


different


Knudsen


numbers

bybyby

by

thethethe

the

MDMDMD

MD

andandand

and

thethethe

the

DSMC.DSMC.DSMC.

DSMC.


expansion and using a Gauss-Hermite quadrature of third-orderto obtain the discrete velocity values. For isothermal sub-microscale fluids, the Hermite expansion up to the third order isrequired to model the momentum equation at the Burnett level.

In Table 2, the computed cases for the MD simulations aresummarized. The nondimensional lengths of the domains are onthe basis of molecular diameter . The numbers of the fluid andthe wall molecules are also listed in Table 2. Variations of threeKnudsen number are the results of changing the channel heightwith keeping a constant number density in the flows. By usingthe Knudsen number calculated from the MD simulation, thecorresponding DSMC results (Beskok etc., 1999) are plotted.To obtain reasonable flow statistics, the MD simulations wereperformed for 1-6 million time steps. Fig.15 illustrates the

normalized velocity profiles of plane Poiseuille channel flowsof the MD and the DSMC solutions with Kn=0.066, 0.132, and0.265. The MD solutions are in good agreement with those ofthe DSMC profiles. It is then found that the fluid-wallinteraction based on Eq.(14) with A=1 is realistically reasonablefor the sub-microscale gas flows.

SquareSquareSquare

Square

cylindercylindercylinder

cylinder

flowflowflow

flow

As a preliminary step for applying the present approachesto more complex systems, a flow in a channel with a squarecylinder placed in the center of region is considered as shownin Fig.16. The flow regime is regarded as a part of an infinitecylinder array set in a narrow channel which can be consideredas a kind of micro-porous media. In Table 3, the computationalconditions for the MD simulation are summarized. The LBMsimulations are also carried out on the same grid size as thePoiseuille flows. The number of iterations of the LBMsimulations is 40,000 and the results are fully converged.Meanwhile, 7 million time steps are required to obtainreasonable flow statistics by the MD simulation.

Fig.17 compares the normalized velocity profiles of flowsat x/H=0.5 at Kn=0.084. Both the results of the LBMsimulations by the D2Q9 and the D2Q21 models agree wellwith that of the MD simulation. This indicates the effectivenessof the extended LBE for sub-microscale in such a complex flowregime. Since it can be seen that the profile of the D2Q9 modelis unexpectedly reasonable, the D2Q9 model may be applicableto simulate flows in more complex micro porous media. Thisimplies that retaining terms of third-order in the Hermiteexpansion and using a Gauss-Hermite quadrature of third-orderto obtain the discrete velocity values are not significantlyimportant for sub-microscale porous flows.

The computational time to perform 7 million time steps ofthe MD simulation was 1800 times as long as that of the LBMsimulation to get convergence. This confirms that our extendedLBE is engineeringly useful though well examined MDsimulations produce trustworthy results. Another advantage ofthe LBM may be flexibility for the system with a complex

FigureFigureFigure

Figure

17:17:17:

17:


Comparison

ofofof

of

thethethe

the


normalized


velocity


profiles

forforfor

for


Poiseuille

flowsflowsflows

flows

bybyby

by

D2Q21,D2Q21,D2Q21,

D2Q21,

D2Q9D2Q9D2Q9

D2Q9

andandand

and

thethethe

the

MDMDMD

MD

atatat

at

x/x/x/

x/

HHH

H

===

=

0.5.0.5.0.5.

0.5.

TableTableTable

Table

3:3:3:

3:

ComputationalComputationalComputational

Computational

conditionsconditionsconditions

conditions

ofofof

of

thethethe

the

MDMDMD

MD

simulationsimulationsimulation

simulation

forforfor

for

aaa

a

flowflowflow

flow

aroundaroundaround

around

aaa

a

squaresquaresquare

square

cylindercylindercylinder

cylinder

ininin

in

aaa

a

channel.channel.channel.

channel.

(a)

(b)

FigureFigureFigure

Figure

16:16:16:

16:

FlowFlowFlow

Flow

fieldfieldfield

field

geometriesgeometriesgeometries

geometries

ofofof

of

thethethe

the


simulations

ofofof

of

planeplaneplane

plane


Poiseuille


channel

flows;flows;flows;

flows;

(a)(a)(a)

(a)

atomicatomicatomic

atomic

3D3D3D

3D


channel

forforfor

for

thethethe

the

MD,MD,MD,

MD,

(b)(b)(b)

(b)

2D2D2D

2D


channel

forforfor

for

thethethe

the

LBM.LBM.LBM.

LBM.


boundary. For the MD simulations, it is difficult to make theboundary of complicated shapes because the walls are formedby molecules.

CONCLUSIONSCONCLUSIONSCONCLUSIONS

CONCLUSIONS

In the present study, simulations of the fluid flows at highKnudsen numbers are performed by both the LBM and the MDmethod. The following remarks are summarized.

1. For wall flow simulations by the MD method, it isnecessary to set the intermolecular potential energy between thefluid and the wall which varies with the structures andproperties of the system.2. A new inlet/outlet boundary condition for flow generationby applying elongation and contraction of the image cell areintroduced. This condition is natural for pressure-gradientdriven flows.3. Sub-microscale flow characteristics at high Knudsennumbers are correctly reproduced by the LBM and MDsimulations.4. The LBM solutions are in good agreement with those ofthe DSMC and the MD simulations and this means that theLBM is promising for engineering applications of sub-microscale flows.5. The simple D2Q9 model of the LBM shows comparableperformance to that of the D2Q21 model in the square cylinderflow. This implies that using the D2Q9 model may be enoughfor engineering simulation of micro-porous flows though it isnecessary to examine its performance in more complex flows.

ACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTS

ACKNOWLEDGMENTS

This work was partly supported by the Core Research forEvolutional Science and Technology (CREST) of JapanScience Technology (JST) Agency (No. 228205R), and theJapan Society for the Promotion of Science through a Grant-in-Aid for Scientific Research (B) (No. 18360050).

REFERENCESREFERENCESREFERENCES

REFERENCES

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