# Higher-order Galilean-invariant lattice Boltzmann model for microflows: single-component gas.

**ABSTRACT** We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation (without any cubic error). In the first part of this work, we show that the present model can capture two important features of the microflow in a single component gas: Knudsen boundary layer and Knudsen Paradox. Finally, we present numerical results corresponding to Couette flow for two representative Knudsen numbers. We show that the off-lattice D3Q27 model exhibits better accuracy as compared to more widely used on-lattice D3Q19 or D3Q27 model. Finally, our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature-based procedure for the construction of the lattice Boltzmann model.

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**ABSTRACT:**In this paper, we compare two families of Lattice Boltzmann (LB) models derived by means of Gauss quadratures in the momentum space. The first one is the HLB(N;Qx,Qy,Qz) family, derived by using the Cartesian coordinate system and the Gauss-Hermite quadrature. The second one is the SLB(N;K,L,M) family, derived by using the spherical coordinate system and the Gauss-Laguerre, as well as the Gauss-Legendre quadratures. These models order themselves according to the maximum order N of the moments of the equilibrium distribution function that are exactly recovered. Microfluidics effects (slip velocity, temperature jump, as well as the longitudinal heat flux that is not driven by a temperature gradient) are accurately captured during the simulation of Couette flow for Knudsen number (kn) up to 0.25.International Journal of Modern Physics C 01/2014; 25(01):1340016. · 1.13 Impact Factor - SourceAvailable from: Gonçalo Silva[Show abstract] [Hide abstract]

**ABSTRACT:**The application of the lattice Boltzmann method (LBM) in three-dimensional isothermal hydrodynamic problems often adopts one of the following models: D3Q15, D3Q19, or D3Q27. Although all of them retrieve consistent Navier–Stokes dynamics in the continuum limit, they are expected to behave differently at discrete level. The present work addresses this issue by performing a LBM truncation error analysis. As a conclusion, it is theoretically demonstrated that differences among the aforementioned cubic lattices lie in the structure of their non-linear truncation errors. While reduced lattice schemes, such as D3Q15 and D3Q19, introduce spurious angular dependencies through non-linear truncation errors, the complete three-dimensional cubic lattice D3Q27 is absent from such features. This result justifies the superiority of the D3Q27 lattice scheme to cope with the rotational invariance principle in three-dimensional isothermal hydrodynamic problems, particularly when convection is not negligible. Such a theoretical conclusion also finds support in numerical tests presented in this work: a Poiseuille duct flow and a weakly-rotating duct flow.Journal of Computational Physics 07/2014; 269:259–279. · 2.49 Impact Factor - SourceAvailable from: Raúl Machado[Show abstract] [Hide abstract]

**ABSTRACT:**The influence of the use of the generalized Hermite polynomial on the Hermite-based lattice Boltzmann (LB) construction approach, lattice sets, the thermal weights, moments and the equilibrium distribution function (EDF) are addressed. A new moment system is proposed. The theoretical possibility to obtain a unique high-order Hermite-based singel relaxation time LB model capable to exactly match some first hydrodynamic moments thermally i) on-Cartesian lattice, ii) with thermal weights in the EDF, iii) whilst the highest possible hydrodynamic moments that are exactly matched are obtained with the shortest on-Cartesian lattice sets with some fixed real-valued temperatures, is also analyzed.Frontiers of Physics 01/2014; 9. · 1.36 Impact Factor

Page 1

Higher-order Galilean-invariant lattice Boltzmann model for microflows: Single-component gas

Wahyu Perdana Yudistiawan,1Sang Kyu Kwak,1D. V. Patil,2and Santosh Ansumali1,2

1Division of Chemical and Biomolecular Engineering, School of Chemical and Biomedical Engineering,

Nanyang Technological University, 637459 Singapore, Singapore

2Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India

?Received 20 May 2009; revised manuscript received 25 August 2010; published 5 October 2010?

We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete

velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with

correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation ?without any cubic error?. In

the first part of this work, we show that the present model can capture two important features of the microflow

in a single component gas: Knudsen boundary layer and Knudsen Paradox. Finally, we present numerical

results corresponding to Couette flow for two representative Knudsen numbers. We show that the off-lattice

D3Q27 model exhibits better accuracy as compared to more widely used on-lattice D3Q19 or D3Q27 model.

Finally, our construction of discrete velocity model shows that there is no contradiction between entropic

construction and quadrature-based procedure for the construction of the lattice Boltzmann model.

DOI: 10.1103/PhysRevE.82.046701PACS number?s?: 47.11.?j, 05.20.Dd

I. INTRODUCTION

The lattice Boltzmann ?hereafter LB? method has emerged

as an alternate viable tool to model a range of hydrodynamic

applications ?1–8?. By now, it is understood that the LB

model constitutes a well-defined hierarchy of approximation

to the Boltzmann equation based on discrete velocity sets

and is naturally equipped with relevant boundary conditions

derived from Maxwell-Boltzmann theory ?9–11?. A lot of

attention was given recently to the use of LB models for

simulation of gaseous flows in microdevices, where hydro-

dynamic approximation breaks down ?9,12–18?. Although,

so far lower-order LB model is massively used in practice,

recent works have indicated that the higher-order LB models

perform much better for resolving complex phenomena such

as Knudsen boundary layer ?18?, gaseous flow in small de-

vices ?19?, thermal flows ?20?, and even in the case of turbu-

lence ?21?. In the case of turbulence better performance

seems to be originating from the fact that the hydrodynamic

limit of the higher-order LB models is Galilean invariant

?21?. In order to recover the Galilean-invariant hydrodynam-

ics, it is crucial to have correct equilibrium third order mo-

ment at least up to the third order in the Mach number

?21–23? ?the term correct here means same as that obtained

from Maxwell-Boltzmann distribution?. The Galilean invari-

ance of a general class of LB models has been demonstrated

using numerical experiments in Refs. ?24,25?.

The basic idea that the LB method is an approximate, but

a systematic technique for solving the Boltzmann BGK

equation with increasing accuracy was proposed in Ref. ?26?.

Later, in Ref. ?10?, it was shown that the LB method approxi-

mates the Boltzmann BGK equation in terms of the Hermite

polynomials similar to the Grad’s moment method ?27?. This

idea was refined further in Refs. ?11,15?, which showed that

it is possible to formulate the LB method in a thermodynami-

cally consistent fashion ?8,28–32?, in a way similar to the

entropic formulation of the Grad’s moment method ?33?. In

these approaches, higher-order discrete velocity models are

constructed from roots of Hermite polynomials ?11?. How-

ever, the roots of the Hermite polynomials are irrational, and

the corresponding discrete velocities cannot be fitted into a

regular space-filling lattice. Recently, this problem was re-

solved by pointing out that a rational number approximations

of the Hermite quadrature is possible for constructing com-

putationally convenient on-lattice models ?22,23?. A few

other examples of higher-order on-lattice LB models were

also given in Refs. ?34,35?.

This route of working with the rational number approxi-

mation of the Hermite polynomial is quite convenient for the

turbulence modeling ?21–23?. However, it might just add ex-

tra computational cost with less appreciable gain in mixture

and/or microflow modeling. In the case of the mixture mod-

eling, this happens because even for the lower-order LB

model ?D2Q9 model? it is not always possible to match the

spatial discretization with the discrete velocity set for all the

components ?see for example ?36??. Similarly, for the micro-

flow modeling the accuracy of the discretization in the ve-

locity space is more crucial ?see for example ?37??. Thus

unlike turbulence, for microflows better accuracy and effi-

cient implementation for the space derivative is a secondary

issue. For example it is well known that the Knudsen layer

can be observed with the minimum of 16 discrete velocities

?which in 3D implies 64 velocities? when the LB method is

constructed via the route of Gauss-Hermite quadrature

?18,37?. It is interesting to note here that even this particular

higher-order LB model fails to reproduce Knudsen paradox

phenomena ?37?. In fact, numerical studies suggest that a

very high order LB model is needed to reproduce the Knud-

sen minima correctly ?37?. Thus, in order to model microf-

lows, it will be quite useful to have a higher-order LB model

with a reduced velocity set ?as compared to the Gauss-

Hermite quadrature route?.

In the one-dimensional case, the issue of minimal discrete

velocity is well settled. It is understood that Galilean-

invariant hydrodynamics for the one-dimensional LB model

is possible with the minimum of 4 velocities, provided they

are chosen using the Gauss-Hermite quadrature route

?15,22?. Recently, it was shown that a rational number ap-

proximation with 5 discrete velocities allows an on-lattice

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model with the same accuracy ?22?. This agrees with the

usual understanding that the Gauss-Hermite quadrature is op-

timal in one-dimension ?10,15?. A remarkable result was ob-

tained in Refs. ?21,23?, where it was shown that in three

dimension it is possible to construct an on-lattice model with

the Galilean-invariant hydrodynamics with a velocity set of

just 41 members. On the other hand, the tensor-product

based Gauss-Hermite quadrature route requires a velocity set

of 64 members ?15?. A more compact 39-speed LB model

was given in ?35?. Here, we remind that in multidimensional

case, discrete velocity vectors are obtained by taking tensor

products of the one-dimensional velocity set. This result

demonstrated that in the multidimensional case tensor-

product based Gauss-Hermite quadrature is suboptimal. In-

deed, a similar result is known for lower-order LB models

too. In the case of lower-order models ?accurate up to the

third order in Mach number?, while the tensor-product based

Gauss-Hermite route requires 27 discrete velocities in three

dimension ?15?, two subsets of set with either 15 or 19 dis-

crete velocities are sufficient to construct models with the

same accuracy in the hydrodynamic limit. This important

observation that the tensor-product based Gauss-Hermite

quadrature route is suboptimal for the construction of the LB

models, is the starting point for the current work.

Indeed suboptimality of the tensor-product based Hermite

Polynomial in multidimensional case is discussed in detail

by ?38?. In Ref. ?38?, it was shown that quadratures with

predefined nodes can be constructed by solving appropriate

orthogonality conditions. In the context of lattice Boltzmann

such a possibility that existing higher-order LBM are not

optimal is discussed by ?35? and recently explored in great

details for all possible on–lattice cases by ?21,23,39?. In fact,

in Ref. ?38?, a detailed list of two- and three-dimensional

grids which are compact compared to Hermite representation

is reported. Theoretical possibilities of using these compact

grids for LB were discussed in Ref. ?35?. However, such

compact grids were never used in the lattice Boltzmann con-

text mostly because they are off-lattice and stability on such

off-lattice alternate is not well tested. Thus, recently all pos-

sible alternate which are on lattice is explored in Refs.

?21,23,39?. Furthermore, such choices of lattices in LBM is

on trial and error basis, where grids are chosen from consid-

eration of Gaussian quadrature and is thrown away if it is not

stable enough. For higher-order on-lattice models, it is

shown in great details that a lattice considered from quadra-

ture prospective is trustworthy only if appropriate H-function

?relevant for hydrodynamics? exist on that lattice ?21–23?. In

Ref. ?35? using quadrature route, remarkable result was

found that in contrast to result of Ref. ?23? ?which uses prun-

ing of an entropy equipped lattice?, it is possible to construct

an on-lattice model with sixth order accuracy by using just

39-discrete velocity set. Although, the link between 39 ve-

locity model and entropic models remains an open question.

To summarize, so far, in the lattice Boltzmann literature

three different approaches to construct higher-order lattices

were used. These approaches are tensor-product based

Gauss-Hermite quadrature route, projection of Gauss-

Hermite quadrature on a predefined lattice with appropriate

orthogonality condition ?as defined in Ref. ?38??, and pruning

of an entropy equipped tensor-product lattice. Recent works

?21–23,35,39? have shown that the first approach of con-

structing multidimensional Gauss-Hermite quadrature via

tensor product is clearly suboptimal. It seems that second

approach of projected Gauss-Hermite quadrature on appro-

priate lattice and third approach of pruning of an entropy

equipped tensor-product lattice are two unrelated indepen-

dent route. The reason being in principle it might be possible

to find a lattice which is consistent with quadrature but not

consistent with the entropy principle. As stated in Ref. ?39?:

“Although a number of high-order lattices are obtained using

the different approaches and found to be very effective in

extending the application domain of the LB method, the

comprehensiveness and minimality of those lattices have not

been established in general, neither are the connections

among the different approaches identified.”

In this work we show that as conjectured in Ref. ?39?,

there is no contradiction between the entropic lattice Boltz-

mann route and the quadrature route. In order to do so, in the

present work, an alternate framework to create discrete ve-

locity set is suggested. In this framework, unlike Ref.

?35,38,39?, the construction of quadrature is given a thermo-

dynamic interpretation. Furthermore, in contrast to Refs.

?21–23? a new way of grid construction, which does not rely

on pruning of tensor-product based grid, in framework of the

entropic lattice Boltzmann is proposed. It is shown that in

this framework the entropic formulation of the LB method

can be naturally extended to obtain a discrete velocity set

with a given accuracy. As an example, a 27-velocity LB

model with the Galilean-invariant hydrodynamic limit is de-

rived. The result shows that the new 27-velocity LB model

uses same grid as proposed in Ref. ?38?. In that respect, our

results can also be interpreted as thermodynamic justification

of higher-order quadrature as proposed by ?38?.

Furthermore, in the present work we have extended the

set of known analytical solution for the microflow as started

in Ref. ?18?. Our theoretical and numerical work clearly in-

dicate that off-lattice D3Q27 model is far more superior to

D3Q27 model used in the lattice Boltzmann setting.

The present work is organized as follows: in Sec. II a

brief review of the LB method is presented. In Sec. III, a new

construction framework for deriving the entropic LB models

with arbitrary accuracy and the relevant H-function is pre-

sented. In Sec. IV, a 27-velocity LB model with the sixth

order accuracy is derived using the new framework. In Secs.

V and VI, an appropriate isothermal and thermal equilibrium

distribution for the discrete velocity model is derived respec-

tively. In Sec. VII, the moment representation of the kinetic

equation is presented. In Sec. VIII, the hydrodynamic limit

of the discrete velocity model is derived to show that the

Galilean-invariant hydrodynamics is recovered. In Sec. IX,

the formal solution for the case of the unidirectional station-

ary flow is presented and the diffusive boundary condition is

used to obtain an explicit solution for the pressure driven and

the Couette flow. These results are analyzed further in the

Secs. X and XI. An illustrative numerical example has been

presented in Sec. XII. Finally, in Sec. XIII a brief discussion

on the conclusions and outlook of the present work is pro-

vided.

YUDISTIAWAN et al.

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II. LATTICE BOLTZMANN METHOD

Discrete velocity models are often used in the kinetic

theory of gases to describe the propagation of shock waves

?40?. Motivated by the search for the computationally effec-

tive microscopic schemes for the hydrodynamics, the con-

cept of discrete kinetic modeling was revived in Ref. ?41?. In

this pioneer work, it was shown that indeed a simple discrete

kinetic model on lattice can describe the Navier-Stokes hy-

drodynamics in appropriate limits. The key new idea was to

provide a reduced description of the molecular motion, suf-

ficient to describe the hydrodynamics at desired length

scales, by considering pseudoparticle dynamics, where par-

ticles are constrained to move along some fixed discrete di-

rection only. This concept was refined further in Refs. ?1–3?

to obtain the LB model, a viable hydrodynamic simulation

tool for the Navier-Stokes equations. In its typical formula-

tion, one works with a set of discrete populations f=?fi?

corresponding to the predefined discrete velocities ci?i

=1,¯,N? to represent the system. For this set of discrete

populations, the evolution equation is often written in the

BGK-form ?42? as

fi?x + c?t,t + ?t? = fi?x,t? + 2??fi

eq?f?x,t?? − fi?x,t??, ?1?

where, ? denotes the discrete relaxation time and fi

tional of f, is chosen in such a way that the correct hydro-

dynamic limit is recovered.

In the last few years, a lot of attention was paid on the

construction of the appropriate equilibrium distribution in the

discrete case. It was shown that it is possible to construct

discrete analog of the Maxwell-Boltzmann distribution by

proper choice of the H-function ?a necessity to ensure ther-

modynamic consistency? ?11,15,28–32?. As this extension,

broadly known as the entropic LB method, is a generaliza-

tion of the usual LB method, we will not distinguish between

the two formulations in the present discussion but present the

result for entropic formulations only.

Another crucial ingredient in LB modeling is the choice

of the set of discrete velocities itself. An important progress

was made in Refs. ?10,43,44?, where it was shown that the

LB method is an approximate technique for solving the Bolt-

zmann BGK equation,

eq, a func-

?tfi+ ci???fi= −1

??fi− fi

eq?f??,

?2?

in the low Mach number limit. Here, ? is the relaxation time,

the set of discrete velocities are typically chosen as the root

of Hermite polynomials, and a low Mach number expansion

of the Maxwell-Boltzmann distribution evaluated at the node

of quadrature is used as discrete equilibrium fi

the problem with this approach was that one cannot ensure

positivity of the fi

work was later generalized to get the entropic LB method

?11,15?, where it was shown that it is sufficient to discretize

the continuous H-function using the Gauss-Hermite quadra-

ture as

eq. However,

eq. In order to fix this deficiency, the frame-

H =?

i=1

N

fi?ln?fi

wi?− 1?,

wi? 0,

?3?

with wias weights associated with quadrature and fi

minimum of this H-function under the constraint of the local

conservation. For example, in the case of isothermal hydro-

dynamics, we have the conservation law for the mass den-

sity, ?, and the momentum density, J?, defined as

eqas

? =?

i=1

N

fi,

J?=?

i=1

N

fici?.

?4?

So, in this case the equilibrium can be obtained as mini-

mizer of the H-function ?Eq. ?3?? under the constraint of the

fixed mass and the momentum density ?Eq. ?4??. An explicit

solution of this minimization problem for the commonly

used lattices of the LB method is presented in Ref. ?11?. This

approach was generalized further in Ref. ?22?, where it was

shown that the rational number approximation of the model

allows an on-lattice model with the same accuracy albeit

with increased number of discrete velocities. Later, in Refs.

?21,23?, it was shown that in the multi-dimensional case

number of discrete velocities can be drastically reduced by

considering only a subset of the set of discrete velocities

generated via the tensor product of the desired one-

dimensional set. These results suggest that the route of three-

dimensional lattices as a tensor product of one-dimensional

lattices is far from being optimal. Although it is possible to

construct a reduced set by pruning of tensor-product lattice

?21,23?, it is not obvious that this route is optimal. In the

subsequent sections, we will demonstrate that it is possible to

create a desired velocity set entirely from multi-dimensional

considerations and such a route leads to a discrete velocity

set with much reduced number of discrete velocities.

III. ENTROPIC QUADRATURE METHOD

In this section, we propose a set of ansatz needed to con-

struct a discrete velocity set equipped with H-function di-

rectly in multidimensional case. These ansatz should be un-

derstood as culmination of the set of the rules developed to

derive the entropic LB method ?11,15,21–23,28–32,45,46?.

Before discussing these ansatz, it is important to define a few

higher-order moments. In particular, typical to the Grad type

moment system ?27?, We define relevant second, third, and

fourth order moments, respectively, as follows:

P??=?

i=1

N

fi?ci?ci?−kBT0

m

????,

Q???=?

i=1

N

fi?ci?ci?ci?−kBT0

m

????ci?+ ???ci?+ ???ci???,

HIGHER-ORDER GALILEAN-INVARIANT LATTICE …

PHYSICAL REVIEW E 82, 046701 ?2010?

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R????=?

i=1

N

fi?ci?ci?ci?ci?+?kBT0

m?

2

???????+ ??????

+ ??????? −kBT0

+ ci?ci????+ ci?ci????+ ci?ci??????,

m

?ci?ci????+ ci?ci????+ ci?ci????

?5?

where, T0is some reference temperature, kBis the Boltz-

mann constant and m is the mass of the particle. It is often

convenient to work with reduced fourth order moment de-

fined as,

fi?ci

ci?ci??.

R??=?

i=1

N

2?ci?ci?−kBT0

m

????+ 5?kBT0

m?

2

???

−7kBT0

m

?6?

Here, we present the necessary set of ansatz as:

?1? Condition on Equilibrium Moments. Ideally, we would

like that the equilibrium values of the second order moment

P??, third order moment Q???and contracted fourth order

moment R?? are the same as those obtained from the

Maxwell-Boltzmann distribution, i.e.,

P??

MB=1

?J?J?,

Q???

MB=1

?2J?J?J?,

R??

MB=1

?2J?J?J2.

?7?

However, typically in a discrete velocity model, the equilib-

rium distributions will satisfy such conditions in an

asymptotic sense only. So, we would like that these condi-

tions are satisfied at least up to the fourth order in Mach

number, i.e., O?u4?. This is sufficient to recover the Galilean-

invariant hydrodynamics ?23?.

?2? Discrete H-function. It is sufficient to consider the

discrete H-function of the Kullback form as given by, Eq.

?3?. Here, the weights wiare unknown positive definite num-

bers. The formal expression for the equilibrium distribution

?in isothermal setting, where energy conservation is not con-

sidered? is

fi

eq= wiexp?? + ??ci?? ? wiA? exp???ci??,

?8?

where, ? and ??are the Lagrange multipliers associated with

the mass and momentum conservation and A=?−1exp ?

with A?0. We need to determine these weights such that the

equilibrium distribution has desired higher-order moments

?See Eq. ?7??. Indeed these two ansatz were used earlier in

Refs. ?22,23? to construct on-lattice higher-order discrete

Boltzmann equation.

?3? Constraints on weights. We claim that in order to sat-

isfy first two ansatz ?Eqs. ?7? and ?3??, it is sufficient that

apart from positivity constraint ?wi?0?, weights also obey

following set of constraints on the even moments,

?

i=1

N

wi= 1, ?

i=1

N

wici?ci?=?kBT0

m????,

?

i=1

N

wici?ci?ci?ci?=?kBT0

m?

2

?????,

?

i=1

N

wici?ci?ci?ci?ci?ci?=?kBT0

m?

3

???????,

?9?

where, symbol ? is used to denote symmetrized tensor gen-

erated from the Kronecker-delta. In particular,

?????= ??????+ ??????+ ??????,

???????= ????????+ ????????+ ????????+ ????????

+ ????????.

?10?

The set of conditions on the odd moments are

?

i=1

N

wici?= 0, ?

i=1

N

wici?ci?ci?= 0,

?

i=1

N

wici?ci?ci?ci?ci?= 0,

?

i=1

N

wici?ci?ci?ci?ci?ci?ci?= 0.

?11?

As stated earlier, the condition on the equilibrium moments

are satisfied up to the accuracy of O?u4? only. In fact, it can

be easily proven that this ansatz is just a direct consequence

of the previous two ansatz. The equivalence of the first two

constraints with the third one is one of the central results of

the present work. The practical consequence of this ansatz is

that the problem of finding reliable entropic LB model is

simplified to solving a set of algebraic equations coupled

with positivity constraints. We defer the proof of this equiva-

lence to later sections and propose few more ansatz, which

will allow an analytically solvable set of algebraic equations.

?4? Energy Dependent Weights.Any meaningful set of dis-

crete velocities is composed by choosing discrete velocities

with different energy E?cx

tation of energy shell was introduced in the Ref. ?30?. We

assume that the weights, wi, are just a function of energy, E.

In fact, all existing LB models satisfy this criteria.

?5? Symmetry Group of the Lattice. For any discrete ve-

locity set C, we must have following,

?i? Closure under Inversion. if a discrete velocity ci

??cix,ciy,ciz? is an element of the set i.e., ci?C, then −ci

?C. This closure, coupled with the ansatz 4, trivially ensures

that Eq. ?11? is satisfied.

?ii? Closure under Reflection. If a discrete velocity ci

??cix,ciy,ciz? is an element of the set i.e., ci?C, then all

possible reflection of it are also a member of the set ?i.e.,

??cix,?ciy,?ciz??C?. The first condition is just a special

case of the second one. Thus, any discrete velocity set con-

2+cy

2+cz

2. Thus, a convenient no-

YUDISTIAWAN et al.

PHYSICAL REVIEW E 82, 046701 ?2010?

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structed in this way will satisfy Eq. ?11? trivially. Further-

more, as this condition ensures that there is no preference on

a specific direction, so for any natural number n and m,

?

i=1

N

cix

2n=?

i=1

N

ciy

2n=?

i=1

N

ciz

2n,

?

i=1

N

cix

2nciy

2m=?

i=1

N

cix

2nciz

2m=?

i=1

N

ciy

2nciz

2m.

?12?

This means in order to satisfy conditions on even moments

?Eq. ?9??, it is sufficient to satisfy 7 scalar equations,

?

i=1

N

wi= 1, ?

i=1

N

wicix

2=kBT0

m

,

?

i=1

N

wicix

4= 3?kBT0

m?

2

,

?

i=1

N

wicix

2ciy

2=?kBT0

m?

2

,

?

i=1

N

wicix

6= 15?kBT0

m?

3

,

?

i=1

N

wicix

2ciy

4= 3?kBT0

m?

3

,

?

i=1

N

wicix

2ciy

2ciz

2=?kBT0

m?

3

.

?13?

We therefore need at least seven degree of freedom in the

model to obtain the sixth order accuracy.

IV. CONSTRUCTION OF VELOCITY SET

The ansatz 5 mentioned in the previous section is trivially

satisfied if we sample the discrete velocities from the cubic

bravice lattice. Thus, apart from the zero energy vector ?c

=?0,0,0?? other simple choices to generate energy shell is to

sample discrete velocities from either of the three cubic lat-

tices, i.e., simple cubic ?SC?, face-centered cubic ?FCC?, or

body-centered cubic ?BCC? structures ?47?. Here, we remind

the reader that we need to satisfy seven nonlinear algebraic

equations along with inequalities wi?0. As any energy shell

has two degrees of freedom ?magnitude of the energy and

weight associated with the shell?, apart from zero energy

shell, we need to have at least three more energy shells.

The energy shells are chosen via a trial-and-error proce-

dure. Only available guideline is that one would like to have

as few as possible energy shells and within the energy shell

the number of the discrete velocities being as few as pos-

sible. The optimal choice is to choose three energy shells

from the SC structure. However, that set is inadmissible as

weights are negative for that choice. However, it is possible

to satisfy all equations with positive values of the weights if

we chose each of the three energy shells from different struc-

ture simultaneously ?one each from SC, FCC, and BCC

structure, see Fig. 1?. Thus, instead of assuming that the

magnitude of energy are in ratio of 1:2:3, we put it as

a2:2b2:3d2, where a, b, and d are the distortion parameters.

Denoting w0, wa, wb, and wdbeing weights corresponding to

shell with energy zero, a2, 2b2, and 3d2, respectively, Eq.

?13? may be simplified. Further, the last two conditions on

sixth moments ?Eq. ?13?? may be solved to obtain,

8d6?kBT0

wd=

1

m?

3

,

wb=

1

2b6?kBT0

m?

3

,

?14?

which when substituted in a condition for fourth moment

?Eq. ?13?? gives

a4?1 −1

wa=1

b2?kBT0

m???kBT0

m?

2

,

2

b2+1

d2=?kBT0

m?

−1

.

?15?

Furthermore, when substituted in second moment equation

gives,

d2+ b2= 2a2.

?16?

Finally, the first condition on sixth order moment gives,

a6wa= 5?kBT0

m?

3

.

?17?

It is this condition, which differentiate the current model

from standard D3Q27 model. If, we do not insist on this

condition, we may impose the condition a=b=c to obtain,

a = b = c =?3kBT0

m

,

?18?

which gives the standard D3Q27 model. Further, we note

that from Eqs. ?15? and ?16?, a2and d2may be written in

terms of b2as

a2=b2

2

b2−kBT0

m

b2− 2kBT0

m

,

d2=

b2kBT0

m

b2− 2kBT0

m

.

?19?

Insisting on Eq. ?17? gives a system of equations. So, finally

we obtain two valid solutions, referred as Basis 1 and Basis

2, ?which satisfy wi?0? corresponding to

FIG. 1. Admissible energy shells: notice that unlike typical

D3Q27 lattice Boltzmann model we are not assuming that magni-

tude of energy are in ratio 1:2:3.

HIGHER-ORDER GALILEAN-INVARIANT LATTICE …

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b =?6 ??15.

?20?

In this way, we can see that given the constraints, we are

left with only one parameter to tune, which is b. Conditional

upon b2?2

tive and thus always have solution?s?. At b2=3

cover the normal D3Q27 model, with a6wa=4?

posed to 5?

?2

pertinent to compare the current choice of 27-velocity LB

model with the usual Hermite based D3Q27 model ?see for

example ?11??. In the usual D3Q27 model, the three energy

shell are sampled from the same cube, so energy of the shells

are in the ratio 1:2:3. As fixation of the energy ratio will

reduce the available degrees of freedom, it is clear that usual

D3Q27 model fails to satisfy all the seven conditions. How-

ever, in the current case no such restriction is imposed, so we

have managed to satisfy all seven conditions on the moment

of the weights ?Eq. ?13??. However, the penalty is that the

discrete velocity set is no longer space-filling. Thus, com-

pared to usual D3Q27 model, the implementation of the ad-

vection will be nontrivial.

kBT0

m, we may see that a2and d2are always posi-

kBT0

m

kBT0

m?3as op-

we re-

kBT0

m?3given by Eq. ?17?. Also note that for b2

m, we have a6wa?4?

kBT0

kBT0

m?3. Hence, at this juncture, it is

V. ISOTHERMAL EQUILIBRIUM DISTRIBUTION

In this section, the explicit expression for the equilibrium

distribution ?Eq. ?8?? is presented. Our task here is to find the

Lagrange multipliers using Eq. ?4? in terms of conserved

moments ? and u?. For this nonlinear problem, the explicit

solution is not available. However, we know that at zero

velocity ?u?=0?, the Lagrange multipliers are A=1, ??=0.

Since, LB method works in subsonic region where Mach

number is considerably small, we can take the Mach number

?velocity? to be the smallness parameter, and work out a

perturbative scheme around the zero velocity. So we intro-

duce a formal smallness-parameter, ?, which may also be

termed as a book-keeping parameter ?this is because, at the

end of the perturbation analysis is set equal to 1? and write,

A = 1 + ?A?1?+ ?2A?2?+ ?3A?3?+ ?4A?4?+ ?5A?5?+ ?6A?6?+ ¯ ,

??= ?B?

?1?+ ?2B?

?2?+ ?3B?

?3?+ ?4B?

?4?+ ?5B?

?5?+ ?6B?

?6?+ ¯ .

?21?

Upon substituting above expansion in Eq. ?8? and coupled

with Eq. ?4?, we obtain a solution up to sixth order of ? ?at

?=1?,

A = 1 −

u2

m??kBT0

+

u4

m??kBT0

2−

u6

m??kBT0

3+ O?u8?,

??=

u?

m??kBT0

+ O?u7?.

?22?

Finally the higher-order moment can be computed to obtain,

P??

eq− P??

MB= O?u6?,

Q???

eq

− Q???

MB= O?u5?,

R??

eq− R??

MB= O?u4?.

?23?

Hence, the desired moments up to order O?u4? are recovered.

This shows, it is sufficient that the discrete velocity models

satisfy the ansatz 2 and 3.

VI. THERMAL EQUILIBRIUM DISTRIBUTION

In this section, we try to construct an energy conserving

model because the off-lattice is higher-order accurate model.

This requires the inclusion of energy conservation along with

the mass and momentum conservation constraint. This means

the function to be minimized is

? =??fi?ln?fi

wi?− 1?− ?fi− ??ci?fi− ?ci

2fi?dci,

?24?

where, ?, ??, and ? are the Lagrange multipliers associated

with the mass, momentum and energy conservation, respec-

tively. Here, we present an explicit expression for the ther-

mal equilibrium distribution. For an algebraic convenience

we rewrite the formal expression of the equilibrium distribu-

tion as

fi

eq= wi? exp?? + ??ci?+ ?ci

2?.

?25?

The way we have constructed the model ensures that at zero

velocity and T=T0, we have

fi

eq?u = 0,T = T0? = wi?.

?26?

Hence, we have to solve following the system of equations:

?

i

fi

eq= ?,

?

i

fi

eqci?= ??u?,

?

i

fi

eqci

2= ?2?u2+ 3p0+ 3??p − p0?.

?27?

Further, using the perturbative procedure as per in Sec. V, we

get an expression for thermal equilibrium as

eq= wi??1 +u?ci?

T0

u2

2T0

2T0

2T0

+u?ci?

T0

T0

fi

T0?1 −?

T

− 1??+1

2?

1

T0

ci

2− 3??

T

T0

− 1?

−

+

1

2?u?ci??2−u?u2

− 1??

2??,

2ci?+

1

6T0

T

T0

3?u?ci??3

− 1?

2T0?

T

1

ci

2− 3?+?

2?

15

8

−

5

4T0

ci

2

+

1

8T0

2ci

2ci

?28?

and moments at equilibrium are

YUDISTIAWAN et al.

PHYSICAL REVIEW E 82, 046701 ?2010?

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???

eq= ?u?u?−1

3?u2???,

q?

eq= ?5

2Tu?+ ?u2u?

2.

?29?

The moments q?and ???are defined, respectively, as

q?=1

2?

i=1

N

fici

2ci?,

???=?

i=1

N

fici?ci?,

?30?

and

ci?ci?=?ci?ci?−1

Dci

2????,

ci?ci?ci?=?ci?ci?ci?−

1

D + 2ci

2????ci?+ ???ci?+ ???ci???,

?31?

where, D being the dimension of space. Also, for sixth order

accurate models,

2?1 + 2?T

T0

Rˆeq= 15?T0

− 1??+ 10?T0u2,

Rˆ??= 7T0u?u?−7

3?T0u2???,

?32?

where, the moments Rˆand Rˆ??are defined, respectively, as

Rˆ=?

i=1

N

fici

2ci

2,

Rˆ??=?

i=1

N

fici

2ci?ci?.

?33?

Thus, we can see that we do recover the desired moments up

to the order O?u4?, which also shows that it is sufficient that

discrete velocity models satisfy the ansatz 2 and 3.

VII. MOMENT CHAIN AND RESEMBLANCE

TO GRAD’S METHOD

In order to compare the present model with a typical

Grad’s moment system, it would be convenient to write the

moment chain for the present kinetic equation, Eq. ?2?. As,

we have 27 discrete velocities, we can have only 27 indepen-

dent moments. In this particular setup, we will write it in-

stead in a slightly different form, in such a way that all of

moments are still independent after the reduction to the nor-

mal D3Q27 model ?special case where b=?3kBT0

present model is different than the usual Grad representation,

wherein, higher-order moments are included only after the

inclusion of all the lower-order moments. The set of 27 in-

dependent moments that we choose are

m?. The

M = ??,J?,P,???,q?,Rˆ,Rˆ??,Nˆ???,??,

?34?

where

P =?

i=1

N

fici

2,

Nˆ???=?

i=1

N

fici

2ci?ci?ci?,

? =?

i=1

N

fici

2ci

2ci

2.

?35?

As we are dealing with a discrete velocity model system,

one can always write the closed form of the moment chain.

Once we have decided the choice of independent moments,

we can write the moment system. In the present case, they

give rise to a closed chain of 27 independent equations, as

we have 27 discrete velocities. Here, it needs to be reminded

that the energy conservation is absent in an isothermal dis-

crete velocity model. In thermal model, we have balance of

mass, momentum and energy as

?t? + ??J?= 0,

?tJ?+ ?????+1

D??P = ?g?,

?tP + 2??q?= 0.

?36?

The evolution equations for the second order moments ???

are,

?t???+ ??Q???+2

5???q?+ ??q?? −

4

15??q????

=1

?????

eq− ????.

?37?

We have equation of motion for the heat-flux, q?, from the

discrete kinetic Eq. ?3? as

2???Rˆ??+1

?tq?+1

3Rˆ????=1

??q?

eq− q?? +g?5p0

2

?38?

and we have closure relations for other third order tensorial

moments as

Qxyz=

1

3d2Nˆxyz,

Qxyy− Qxzz=

1

2b2?Nˆxyy− Nˆxzz?,

Qzxx− Qzyy=

1

2b2?Nˆzxx− Nˆzyy?,

m?

2?Jx

m

2?Nˆxxx−?10b6− 48b4?kBT0

Qxyy+ Qxzz=

1

3D1??15b6− 78b4?kBT0

+ 63b2?kBT0

+ 90?kBT0

m?

m?

kBT0

−?15b4− 75b2?kBT0

m?

m?

HIGHER-ORDER GALILEAN-INVARIANT LATTICE …

PHYSICAL REVIEW E 82, 046701 ?2010?

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+ 74b2?kBT0

m?

2

− 84?kBT0

m?

3?qx?,

Qyzz+ Qyxx=

1

3D1??15b6− 78b4?kBT0

+ 63b2?kBT0

+ 90?kBT0

+ 74b2?kBT0

m?

−?15b4− 75b2?kBT0

2?Nˆyyy−?10b6− 48b4?kBT0

m?

m?

m?

2?Jy

kBT0

m

m?

m?

2

− 84?kBT0

m?

3?qy?,

?39?

where

D1= 4b6− 33b4?kBT0

m?+ 116b2?kBT0

m?

2

− 147?kBT0

m?

3

.

?40?

A. Fourth order moments

The evolution equations for the fourth order moments are

?tRˆ??−

2

15??n????+ ??Nˆ???+1

5???n?+ ??n??

=1

??Rˆ??

eq− Rˆ???.

?41?

?tRˆ+ ??n?=1

??Rˆeq− Rˆ?,

?42?

where

n?=?

i=1

N

fici?ci

2ci

2.

?43?

B. Fifth order moments

Similarly, following are the evolution equations for the

fifth order moments:

2b2?kBT0

?b2− 2kBT0

?tNˆxyz+

m?

m??2b2− 7kBT0

m?

??x?2b2?yz− Rˆyz?

+ ?y?2b2?xz− Rˆxz? + ?z?2b2?xy− Rˆxy??

=1

??Nˆxyz

eq− Nˆxyz??44?

?tNˆxxx+ ?x?r4P + r5Rˆ+ r6? + r7?xx+ r8Rˆxx?

+ ?y?

2b2?b4− 4b2kBT0

m

5?2b2− 7kBT0

+ ?z?

2b2?b4− 4b2kBT0

m

5?2b2− 7kBT0

− 6b4?b2− 6kBT0

5?2b2− 7kBT0

m?kBT0

m

m??b2− 2kBT0

− 2?kBT0

m??b2− 2kBT0

− 6b4?b2− 6kBT0

5?2b2− 7kBT0

m?

2?

?xy

+

m?

m?

Rˆxy?

m?kBT0

m

m??b2− 2kBT0

− 2?kBT0

m??b2− 2kBT0

m?

2?

?xz

+

m?

m?

Rˆxz?

=1

??Nˆxxx

eq− Nˆxxx??45?

?tNˆyyy+ ?x?

− 6b4?b2− 6kBT0

5?2b2− 7kBT0

2b2?b4− 4b2kBT0

5?2b2− 7kBT0

m?kBT0

m

m??b2− 2kBT0

− 2?kBT0

m??b2− 2kBT0

m?

2?

?xy

+

mm?

m?

Rˆxy?

+ ?y?r4P + r5Rˆ+ r6? + r7?yy+ r8Rˆyy?

+ ?z?

2b2?b4− 4b2kBT0

m

5?2b2− 7kBT0

− 6b4?b2− 6kBT0

5?2b2− 7kBT0

m?kBT0

m

m??b2− 2kBT0

− 2?kBT0

m??b2− 2kBT0

m?

2?

?yz

+

m?

m?

Rˆyz?

=1

??Nˆyyy

eq− Nˆyyy??46?

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?tNˆzzz+ ?x?

− 6b4?b2− 6kBT0

5?2b2− 7kBT0

2b2?b4− 4b2kBT0

5?2b2− 7kBT0

+ ?y?

2b2?b4− 4b2kBT0

5?2b2− 7kBT0

m?kBT0

m

m??b2− 2kBT0

− 2?kBT0

m??b2− 2kBT0

− 6b4?b2− 6kBT0

5?2b2− 7kBT0

− 2?kBT0

m??b2− 2kBT0

m?

2?

?xz

+

mm?

m?

Rˆxz?

m?kBT0

m

m??b2− 2kBT0

m?

2?

?yz

+

mm?

m?

Rˆyz?

+ ?z?r4P + r5Rˆ+ r6? + r7?zz+ r8Rˆzz?

=1

??Nˆzzz

eq− Nˆzzz??47?

?t?Nˆxzz− Nˆxyy? +

2b4

?2b2− a2??x?− a2??zz− ?yy? + Rˆzz− Rˆyy?

−

2b4

?3d2− 2b2??y?3d2?xy− Rˆxy?

2b4

?3d2− 2b2??z?3d2?xz− Rˆxz?

+

=1

??Nˆxzz

eq− Nˆxyy

eq− Nˆxzz+ Nˆxyy?,

?t?Nˆyxx− Nˆyzz? +

2b4

?3d2− 2b2??x?3d2?yx− Rˆyx? +

2b4

?2b2− a2?

??y?− a2??xx− ?zz? + Rˆxx− Rˆzz?

2b4

?3d2− 2b2??z?3d2?yz− Rˆyz?

−

=1

??Nˆyxx

eq− Nˆyzz

eq− Nˆyxx+ Nˆyzz?,

?t?Nˆzxx− Nˆzyy? +

2b4

?3d2− 2b2??x?3d2?xz− Rˆxz?

−

2b4

?3d2− 2b2??y?3d2?yz− Rˆyz? +

2b4

?2b2− a2?

??z?− a2??xx− ?yy? + Rˆxx− Rˆyy?

=1

??Nˆzxx

eq− Nˆzyy

eq− Nˆzxx+ Nˆzyy?,

?48?

where

r4= −

2b6?b2−kBT0

m??b2− 7kBT0

m??b2− 3kBT0

m??2b2− 7kBT0

m?

m??3b2− 7kBT0

?b2− 2kBT0

m??kBT0

m?,

r5=

2b2?b8− 6b6kBT0

3?b2− 2kBT0

m

+ 22b4?kBT0

m??b2− 7kBT0

m?

m??2b2− 7kBT0

2

− 54b2?kBT0

m?

m??3b2− 7kBT0

3

+ 49?kBT0

m?

m?

4?

,

r6=

2?4b6− 33b4?kBT0

15?b2− 7kBT0

m?+ 116b2?kBT0

m??2b2− 7kBT0

m?

m??3b2− 7kBT0

2

− 147?kBT0

m?

m?

3?

,

HIGHER-ORDER GALILEAN-INVARIANT LATTICE …

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r7=

2b4?b2−kBT0

5?b2− 2kBT0

b2?5b4− 18b2kBT0

5?b2− 2kBT0

m??2b2− 3kBT0

m??3b2− 7kBT0

+ 17?kBT0

m??3b2− 7kBT0

m?

m?

m?

m?

,

r8=

m

2?

.

?49?

We also have closure

n?=

5

2?b2− 2?D1??4b10− 24b8?kBT0

− 216b4?kBT0

− 89b6?kBT0

+ 686?kBT0

m?+ 88b6?kBT0

m?

m?

m?

2

m?

m?+ 448b4?kBT0

m?

3

+ 196b2?kBT0

4?q?−?6b8

− 931b2?kBT0

2

m?

m?

3

4?Nˆ????????−?6b10− 24b8?kBT0

+ 18b6?kBT0

m?

2?J??kBT0

m??

?50?

C. Sixth order moment

Finally, the evolution equations for the sixth order mo-

ments may be written as

?t? + ???r1J?+ r2q?+ r3Nˆ????????? = −1

??? − ?eq?,

?51?

where

?eq=

3?b4+ 23b2kBT0

m

− 49?kBT0

m?

2?

b2− 2kBT0

m

??kBT0

m?

2

,

r1= −

3b6?b2−kBT0

2?b2− 2kBT0

m??17b6− 24b4?kBT0

m?

m?− 187b2?kBT0

m?+ 116b2?kBT0

m?

m?

2

+ 294?kBT0

− 147?kBT0

m?

m?

3?

2?4b6− 33b4?kBT0

23??kBT0

m?,

?52?

r2=

b4?17b10− 135b8?kBT0

m?+ 797b6?kBT0

?b2− 2kBT0

m?

2

− 3189b4?kBT0

m?+ 116b2?kBT0

m?

3

+ 6026b2?kBT0

m?

m?

m?

4

− 4116?kBT0

3?

m?

5?

m?

2?4b6− 33b4?kBT0

2

− 147?kBT0

,

r3= −

5b2?b2− 7kBT0

4?b2− 2kBT0

m??2b2− 7kBT0

m??4b6− 33b4?kBT0

m??3b2− 7kBT0

m?+ 116b2?kBT0

m??5b2− 3kBT0

m?

m?

m?

2

− 147?kBT0

3?

.

?53?

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The moment system described above ?Eqs. ?36?–?38?,

?41?, ?42?, ?44?–?48?, and ?51?? is equivalent to the original

kinetic equation. Indeed, a very similar set of equations will

be obtained if the Grad’s 26-moment system is extended with

the sixth order moment ? as an independent variable. What

is very different from the usual Grad’s moment system is the

choice of the variable itself. It is not really obvious why the

Grad’s 26-moment system should be extended by just one

higher-order moment. Typically, when Grad’s system is ex-

tended, one includes next higher-order moment ?or some

meaningful contraction of it? into the list of variables. In the

present context, the choice of the 27th variable as ? ?which

is a sixth order rather than a fifth order moment? emerged

automatically from the choice of the lattice itself. Such un-

usual extensions of the Grad’s moment system is a typical

feature of the LB type equations ?18?, although significance

of such extension is not yet clear. All one can say at this

juncture is that, such unusual extension of the Grad’s system

leads to boundary condition in a natural way via discrete

equivalence of diffusive boundary conditions ?9?.

VIII. HYDRODYNAMIC LIMIT

It is expected that discrete kinetic equation, in the hydro-

dynamic limit, which is limit of Knudsen number going to

zero, will lead to Navier-Stokes type equation. The usual

procedure to obtain the transport coefficients and the hydro-

dynamic equation is to do the Chapman-Enskog expansion

of the kinetic equation ?48?. In this procedure, one writes the

distribution function f and its time derivative in the powers

of Knudsen number, Kn.

The hydrodynamic variables are not expended and in or-

der to define time-derivatives consistency condition is used,

which means that the derivatives of all other variables are

evaluated using chain rule via time derivatives of the con-

served quantities. Thus, we have

???

?0?=J?J?

?

,

Qˆ???

?0?=J?J?J?

?2

.

?54?

Using Eq. ?36?, we define time derivatives as

?t

?0?? = − ??J?,

?t

?0?J?= − ???J?J?

?

+ ?kBT0

m

????,

?55?

the chain rule gives

?t

?0????

?0?= ??t

?0????????

?2??J?−?J?

????J?J?

?

eq

??? + ??t

?0?J???????

?????

????.

eq

?J??

= −J?J?

????+J?

+ ?kBT0

m

?56?

Also, using Eq. ?37?, we have

?t

?0????

?0?+ ??Qˆ???

?0?+kBT0

m

???J?+ ??J?? = − ???

?1?,

?57?

which can be simplified using Eq. ?56? to obtain

???

?1?= − ?kBT0

m???

J?

?+ ??

J?

??.

?58?

which means in the hydrodynamic limit, we do recover the

Navier-Stokes equation for an isothermal model as

?t? + ??J?= 0,

????= ???????

?tJ?+ ???J?J?

?

+ ?kBT0

m

J?

?+ ??

J?

???,

?59?

where the viscosity coefficient is ?=??kBT0/m. The impor-

tant thing to notice here is that only ???

the first-order expansion. So, it is sufficient that the equilib-

rium values of those match with those obtained from the

Maxwell-Boltzmann distribution. Thus, it confirms that the

ansatz 1 is correct.

A similar computation with the energy conserving model

recover the equation of motion as a Navier-Stokes-Fourier

system.

eqand Qˆ???

eq

appear in

?t? + ??J?= 0,

?tJ?+ ???J?J?

= ???????

?

+ ?kBT0

m

????

??−2

J?

?+ ??

J?

3??

J?

?????,

?tT +J?

???T +2T

3??????5

−2

3??

3??

J?

?

= −

2

2??T?+ ??

??????.

J?

??????

J?

?+ ??

J?

??

J?

?60?

IX. UNIDIRECTIONAL FLOW: STATIONARY

SOLUTION

In order to verify the usefulness of the modified D3Q27,

presented in this work, we chose to illustrate the stationary

solutions of the models for pressure driven and Couette

flows. We wish to compare the solution obtained with the

current model with that obtained from the Boltzmann BGK

solution. We have chosen this set-up as the Couette flow was

earlier analyzed in detail using the LB equation ?see for ex-

ample ?18??. It is known from there that Knudsen layer is

predicted only by D2Q16 model, which in three dimension

means a model with 64 discrete velocities. Thus, it is a good

set up to compare the effectiveness of the current model.

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A. Setup description and outline of solution

We consider the fluid to be enclosed by two parallel plates

normal to z direction and separated by a distance of L. The

bottom plate at z=−L/2 moves unidirectionally with the ve-

locity ?U1,0,0? while the top place at z=L/2 moves with

velocity ?U2,0,0?. It is assumed that tube is infinitely long in

x direction and constant density gradient is imposed in the x

direction. We aim to find the steady state solution to the

kinetic equation in this particular setup.

Integration of the steady state moment system is done

under following three assumptions:

?i? The flow is unidirectional, where all the fields except

density depend only on the z coordinate.

?ii? Mass does not flow through the walls.

?iii? For low Mach number flows, the nonlinearities can be

ignored and it is sufficient to consider the linearized moment

system.

As the result, we find the inner solution for all the mo-

ments. This inner solution is a parametric family that de-

pends on yet undetermined constants of integrations.

In order to find the constants of integrations, we need to

include the boundary condition, which is available in popu-

lation representation. Thus, we either should transform the

inner solution obtained earlier ?in terms of moments? to

population, or to find the moment representation at the

boundary ?where the relevant populations are taken from the

boundary, and the rest from the inner population?. Note that

this solution is still dependent on the same constants of inte-

grations. Matching this solution with the inner solution will

give us the value of integration constant, and thus solving the

entire problem. Here, solving for Jxwill be prioritized. The

same strategy is already used and elaborated in details in

?49?.

B. Inner solution to the unidirectional

stationary moment system

In this part, we start with the assumption that the flow is

in a steady state and is unidirectional ?all the fields depend

only on the z coordinate due to the nature of the setup, which

is infinite in x and y directions? with the exception of ?,

which only depends on x direction. The continuity equation

in this limit simplifies to ?zJz=0, and the boundary condition

?no mass leak from the wall? implies Jz=0. Further, the mo-

mentum conservation equation in the x-direction simplifies

as

?xz= −dp0

dxz + k1,

?61?

where p0=??kBT0/m?. Here, we assume that the pressure

drop, dp0/dx, is a constant. Equations ?37? and ?41? may be

reformulated in the following form:

?z?h1Jx+ h2qx+ h3Nˆxyy+ h4Nˆxzz? = −1

??xz,

?z?h5Jx+ h6qx+ h7Nˆxyy+ h8Nˆxzz? = −1

?Rˆxz,

?62?

where h1,h2,...,h8are constants. Equation ?62? may be

solved by eliminating Nxyyand Nxzzusing the fact that

+ 22b2?kBT0

10b4− 55b2kBT0

m

??

+ 18b2?kBT0

10b4− 55b2kBT0

m

??

−

4b6− 16b4kBT0

mm?

m?

2

+ 70?kBT0

2?z

2Rˆxz

= −1

?zNˆxyy−

b4− 12b2kBT0

5?b2− 2kBT0

m?

+ 70?kBT0

m

+ 21?kBT0

m?

m?

2

kBT0

m

?zJx?

6b6− 24b4kBT0

m

2

m?

2?z

2Rˆxz

= −1

?zNˆxzz−

b4− 12b2kBT0

5?b2− 2kBT0

m

+ 21?kBT0

m?

m?

2

kBT0

m

?zJx?

,

?63?

and eliminating qxusing Eq. ?38?,

?z

2Rˆxz= −1

??z?qx−5

2Jx

kBT0

m?.

?64?

We note that here we already incorporate the fact that

?z

2?xz=0, thus we have

− ?

b2− 3kBT0

m

2b2− 7kBT0

m

?z

2Rˆxz+kBT0

m

?zJx= −1

??xz,

− ?b2?

= −1

2b4− 8b2kBT0

m

+ 5?kBT0

+ 14?kBT0

m?

m?

2

2b4− 11b2kBT0

m

2?

?z

2Rˆxz+ 7?kBT0

m?

2

?zJx

?Rˆxz,

?65?

which has a solution for Rˆxzas

Rˆxz= 7?xz

kBT0

m

+ A1sinh?

z

?3???+ A2cosh?

z

?3???,

?66?

where, A1and A2are just integration constants determined by

the boundary condition ?refer next section? and

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?3? =?

b4− 4b2kBT0

m

+ 6?kBT0

m?

2

b2− 2kBT0

m

.

?67?

Substituting Eq. ?66? into Eq. ?65?, we solve for Jx,

Jx=dp0

dx

z2

2?−k1z

?

+ k2+

1

?3?

b2− 3kBT0

m

2b2− 7kBT0

m?A2sinh?

z

?3???

+ A1cosh?

z

?3????.

?68?

Here, we obtained a family of moments that is dependent on

four integration constants ?k1, k2, A1, and A2?. To determine

these, we need to specify boundary conditions at the walls.

Note that this is an advantage of LB hierarchy, since it is

well known that it is not possible to provide self-consistent

boundary conditions for the other moments methods ?such as

the Grad’s systems? ?27?. In the present case, this is possible

because the boundary conditions for the LB kinetic equations

are formulated in terms of populations rather than moments

?9?.

C. Diffusive wall boundary condition

Boundary conditions for discrete velocity models are for-

mulated in terms of distribution function. Thus, in order to

apply boundary conditions, it is more convenient to come

back from the moment representation to the distribution rep-

resentation. For the present system, we apply the classical

Maxwell’s diffusive wall boundary condition. In this condi-

tion, particles that reach the wall are redistributed in a way

consistent with the mass-balance and normal-flux condition,

fi?c·n?0=

?

cj·n?0

??cj· n??fj

??cj· n??fj

?

cj·n?0

eq??,Uwall?

fi

eq??,Uwall?,

?69?

where n is the inner normal at the wall, and Uwallis the wall

velocity. This boundary condition redistributes the popula-

tions that reach the wall according to the equilibrium distri-

bution of the population that leaves the wall.

Since the solution for the moments must be continuous,

we should have the inner solution same as the result obtained

from the boundary condition, where fiis taken from Eq. ?69?

whenever c·n?0 and taken from the Grad representation fi

?see Ref. ?49??, if otherwise. With such definition of fi, we

can have boundary conditions by taking the moment of fi

with respect to cixand cixciz

respectively.

Since we have 4 equations and 4 unknowns, the system

can be solved unambiguously. Substituting the inner solution

and solving for the unknowns give rise to the following

solutions:

2, on the top and bottom wall,

k1=

???U?b??b2− 2kBT0

m?

3/2

+ 2?kBT0

m?

3/2?cosh?

z

?3???+ b2?b4− 4b2kBT0

D

m???b2− 2kBT0

D

m

+ 6?kBT0

m?

2

sinh?

z

?3????

A1=

???U2b?2b2− 7kBT0

m

−?kBT0

m?

?70?

k2= −dp0

dx?L2

8?

m

kBT0

+

1

2b?2 +?b2m

dx???kBT0

m

m??kBT0

−?b2− 2kBT0

kBT0

− 2?L + 2??

+U1+ U2

2

1

DP

dp0

− b2??b2− 2kBT0

m?L + 2b?b2− 3kBT0

m???sinh?

m

+ 2?b2− 2kBT0

???kBT0

m?

m

L

2?3???

?71?

A2=

2b2− 7kBT0

m

DP

dp0

dx?

b??b2m

kBT0

− 2 − 1?L

+ 2b2?

b2m

kBT0

?b2m

− 3

kBT0

− 2?

??

?72?

where, the denominators, D and DPare, respectively, defined

as

HIGHER-ORDER GALILEAN-INVARIANT LATTICE …

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Page 14

D =?b??b2− 2kBT0

+ 2b4??cosh?

+?b4− 4b2kBT0

+ 2b?2?kBT0

m?

?3???

3/2

+ 2?

kBT0

m?

3/2??

m

kBT0

L

z

m

+ 6?

kBT0

m?

2?b2?

m???sinh?

m

kBT0

L

m

+?b2− 2kBT0

z

?3???

?73?

DP= b2?b4− 4b2kBT0

+ b??b2− 2kBT0

m

+ 6?

kBT0

m?

2

cosh?

3/2?sinh?

L

2?3???

m?

3/2

+ 2?

kBT0

m?

L

2?3???.

?74?

We note that up to here, both on-lattice scheme and off-

lattice scheme are valid. It is easy to check that for b=?3, we

recover back the characteristic result of D3Q27. In the next

section, we will substitute the parameter of off-lattice

scheme directly to get some numerics to compare with the

available data. We also define Knudsen number as:

Kn=?

L?3kBT0

m

.

?75?

X. COUETTE FLOW

The shear stress is the quantity of interest in the Couette

flow. We define dimensionless shear stress consistent with

Ref. ?50? as

??U2− U1??2?m

?xz

?= −

?xz

kBT0

.

?76?

This dimensionless shear stress for the first and second basis

is given as per the following equation:

??1coth??2

??4+ ?5Kn?coth??2

?xz

?=

Kn?+ ?3?Kn

Kn?+ ??6+ ?7Kn?

.

?77?

In Table I, we have listed the constants ?1,...,?7appearing

in the dimensionless shear stress equation. This dimension-

less shear stress is plotted in Fig. 2.

The convergence of the discrete Boltzmann equation to-

ward its continuous counterpart can also be analyzed via so-

lution at Kn→?. For example, we know that the infinite

Knudsen limit of the dimensionless shear stress ?eff

Boltzmann-BGK equation is equal to unity. The comparison

is tabulated in Table II. From the table, it is evident that the

current model with the first basis is converging much faster

to the Boltzmann equation compared to any other approxi-

mations. Furthermore, the quality of result also suggests that

the model with the first basis performs much better compared

to the second one. Yet another useful quantity which shows

nontrivial behavior in high Knudsen number cases is nondi-

mensional centerline velocity gradient Y, defined as

U2− U1?dux

kBT0?

?for the

Y = 1 −

1

dz/L?

z=0

= 1 +

m

A1

?Kn−k1

Kn?3kBT0

m?.

?78?

000 0.20.20.20.4 0.40.4 0.6 0.60.6 0.8 0.80.81111.2 1.21.2 1.4 1.41.41.6 1.61.61.8 1.81.8222

000

0.10.1 0.1

0.20.2 0.2

0.30.3 0.3

0.4 0.4 0.4

0.5 0.50.5

0.60.6 0.6

0.70.7 0.7

0.80.80.8

KnKn Kn

σ*xz

Boltzmann−BGK

D2Q9

D2Q16

Basis 1

Basis 2 Basis 2Basis 2

σ*xz

Boltzmann−BGK

D2Q9

D2Q16

Basis 1Basis 1

σ*xz

Boltzmann−BGK

D2Q9

D2Q16

FIG. 2. Shear stress profile for Couette flow.

TABLE I. Constants appearing in a generic expression for dimensional shear stress ?Eq. ?77??.

?1

?2

?3

?4

?5

?6

?7

Basis 1

Basis 2

6.93489

18.1766

0.303784

0.217371

7.23518

6.95667

4.79193

12.5598

7.1253

21.9968

4.99943

4.80697

8.82956

8.96816

TABLE II. Comparison of effective shear viscosity at Kn→?

between the Boltzmann-BGK model and various LB models.

Model

?eff

?

Boltzmann-BGK

D2Q9

D2Q16

Current Basis 1

Current Basis 2

1

0.723

1.113

0.973

0.826

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The result and its relative error are tabulated in Table III.

From the table, it is evident that the current model with the

first basis is converging much faster to the Boltzmann equa-

tion compared to any other approximations. Furthermore, the

quality of result also suggest that the model with first basis

performs much better compared to the second one.

XI. KNUDSEN PARADOX

It is well known that for the pressure driven flow, so

called “Knudsen paradox” behavior where the flow rate

shows a minimum as a function of the Knudsen number. In

the present section, we compare the result from the present

discrete velocity model with the continuous Boltzmann-BGK

equation as well as existing Hermite based lattice Boltzmann

models. For pressure driven flows, we can set U1=U2=0 in

the general solution Eq. ?68?. In this setup the quantity of

interest is the dimensionless flow rate Q defined as

L2?dp0

Q = −

1

dx?

−1?2kBT0

m?

z=−L/2

z=L/2

Jxdz.

?79?

In order to compare present result with the existing result in

the literature, we follow the convention in the literature and

redefined Knudsen number into Kn

From the Boltzmann-BGK equation, we know that in this

particular setup, the dimensionless flow rate defined in Eq.

?79? theoretically should have a minimum at Kn

Here, we write the expression for dimensionless flow rate

in a generic way and construct Table IV as a comparison

?, such that Kn

?=Kn?2/3.

??1 ?51?.

with different findings.

Q =

1

?+ ?1+ ?2Kn

6Kn

?−?3+ ?4Kn

?+ ?5Kn

Kn

?2

?6coth?

?7

??+ 1

?80?

It can be shown that the profile given by first Basis indeed

exhibits Knudsen paradox with Knudsen minimum occurring

at Kn?0.5886. For the second basis, with the flat profile,

which has no minimum and distinct value in the limit of

Knudsen number going to infinity. Thus, similar to D2Q16

model, the second basis in the present case do not exhibit

Knudsen paradox behavior. We can see that the first term of

the flow rate ?order of Kn

Stokes limit. In Table IV and using present notation, we have

compared the result with the Cercignani quadratic approxi-

mation using the slip flow model. This has previously used in

comparing D2Q9 model in ?49?.

Finally, we would like to comment on the other possible

models for microflow. Recently, the Grad’s moment method

was modified to obtain the R13 equation ?53,54?, for which

boundary condition was developed in Ref. ?52?. In recent

year, it has been shown that R13 with proper boundary con-

dition gives as good result as the LB method for microflows

?52?. The constants appearing in the expression of the flow

rate using the R13 approach as in ?52? have been tabulated in

Table IV.

From Fig. 3, it is visible that our current approach with

the first basis gives an almost exact agreement with the result

?−1? basically represents the Navier-

TABLE III. Deviation of nondimensional velocity gradient from Navier-Stokes value for the current

models, D2Q9 ?49?, and the Boltzmann-BGK kinetic equations ?50?. Percentage error of the value of devia-

tion is relative to Boltzmann-BGK value.

Kn

Values

Error

?%?

Boltzmann-BGK

D2Q9 Basis 1Basis 2

D2Q9Basis 1Basis 2

0.06124

0.12247

0.17496

0.24495

0.30619

0.61237

0.81650

1.22474

0.09134

0.1648

0.2136

0.2664

0.3041

0.4290

0.4821

0.5556

0.10911

0.1968

0.2592

0.3288

0.3798

0.5509

0.6202

0.7101

0.09152

0.1749

0.2384

0.3096

0.3611

0.5301

0.5986

0.6880

0.09973

0.1860

0.2480

0.3165

0.3663

0.5332

0.6024

0.6930

19.450

19.387

21.358

23.427

24.890

28.408

28.646

27.808

0.1971

5.775

10.403

13.953

15.785

19.071

19.462

19.244

9.185

12.864

16.104

18.806

20.454

24.289

24.953

24.730

TABLE IV. Constants appearing in a generic expression for dimensionless flow rate ?Eq. ?80??.

Ref.

?1

?2

?3

?4

?5

?6

?7

Basis 1

Basis 2

Ref. ?49?

Ref. ?52?

1.08152

1.14248

1.01617

0.97108

2.0

2.0

1.5324

1.1333

0.17096

0.06999

2.06084

2.00291

6.21071

11.40948

1.04330

1.01249

0.248039

0.177483

0.08485280.6 1.060650.931695 0.527046

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