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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.
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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???,
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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-
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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.
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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 …
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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?
YUDISTIAWAN et al.
PHYSICAL REVIEW E 82, 046701 ?2010?
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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.
HIGHER-ORDER GALILEAN-INVARIANT LATTICE …
PHYSICAL REVIEW E 82, 046701 ?2010?
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Page 12
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
YUDISTIAWAN et al.
PHYSICAL REVIEW E 82, 046701 ?2010?
046701-12
Page 13
?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 …
PHYSICAL REVIEW E 82, 046701 ?2010?
046701-13
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?
0000.2 0.20.20.40.4 0.4 0.60.6 0.6 0.8 0.80.8111 1.21.21.2 1.4 1.41.4 1.6 1.61.6 1.8 1.81.8222
000
0.1 0.10.1
0.20.20.2
0.3 0.30.3
0.4 0.40.4
0.50.5 0.5
0.6 0.60.6
0.7 0.7 0.7
0.8 0.8 0.8
Kn KnKn
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
YUDISTIAWAN et al.
PHYSICAL REVIEW E 82, 046701 ?2010?
046701-14
Page 15
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
D2Q9 Basis 1 Basis 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
0.0848528 0.6 1.060650.9316950.527046
HIGHER-ORDER GALILEAN-INVARIANT LATTICE …
PHYSICAL REVIEW E 82, 046701 ?2010?
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Page 16
of R13. Further, it behaves much better than D2Q16 model
used in literature as the present model can capture boundary
layer as well as Knudsen minimum.
XII. NUMERICAL ILLUSTRATION
In this section, numerical result corresponding to the Cou-
ette flow simulations has been illustrated for on-lattice and
off-lattice representations with D3Q27 velocity model. The
numerical setup has been explained earlier in Sec. IX. In the
computational domain the moving walls are considered to be
located at z=0 and z=L. Here, two representative Knudsen
numbers ?Kn=0.5 and 1.0? are considered and an isothermal
equilibrium distribution function given by Eq. ?8? is used.
The flow Mach number is kept constant ?Ma=0.075? in these
simulation runs. A diffusive wall boundary condition given
by Eq. ?69? is used for walls in z direction and a periodic
boundary condition is employed in x direction. The compu-
tational procedure is described as follows:
?i? A formal short-time solution for Eq. ?2? may be written
by evaluating the usual BGK collision kernel ??i=1/??fi
−fi??, using the trapezoidal rule ?assuming linear interpola-
tion?
eq
fi?x + c?t,c,t + ?t? = fi?x,c,t? +?t
2??i?fi?x,c,t??
+ ?i?fi?x + c?t,c,t + ?t??? + O??t3?.
?81?
?ii? First, in order to create an efficient explicit numerical
scheme we transform the distribution functions, fi, in terms
of giusing the following:
gi?x,c,t? = fi?x,c,t? −?t
2??fi
eq?x,c,t? − fi?x,c,t???82?
and hence the corresponding LB evolution equation using
the above transformation is
gi?x + c?t,t + ?t? = gi?x,t? + 2??gi
eq?g?x,t?? − gi?x,t??,
?83?
where, ?=?t/?2?+?t?.
?iii? The Eq. ?83? may be written as consisting of two-
steps, namely, local-collision and streaming. The postcolli-
sion distribution functions ?gi
?? may be found as
gi
??x,t? = gi?x,t? + 2??gi
eq?x,t? − gi?x,t??,
?84?
and thus computed gi
on-lattice representation as
?’s may then be streamed for the usual
gi?x,t + ?t? = gi
??x − c?t,t?.
?85?
?iv? Further, the grid-spacing, ?x=?y=?z and the time-step,
?t=?x/b is chosen.
?v? In an off-lattice simulation, the usual streaming proce-
dure ?Eq. ?85?? is performed for the distribution functions
corresponding to the energy shell, b. Hence, a convection
term contribution needs to be computed for the energy shells
apart from b. This contribution is evaluated after calculating
the postcollision distribution functions using Eq. ?84? as
gi?x,t + ?t? ? gi
??x,t? −?gi
?
?x?
ci??t ? gi
??x,t?
+?gi
??x ? ci??t,t? ? gi
?x
??x,t?
?ci???t.
?86?
Here, a quasi-two-dimensional approximation has been
made. This is because, the illustrative example under consid-
eration is an unidirectional. The approximation implies
?gi/?y=0 and hence no convection in y direction. The gra-
dients of the distribution functions are evaluated using a
first-order upwind scheme. The scheme is stable and suffi-
ciently accurate. Further, we found that the use of central-
difference scheme is unstable. Here, we do acknowledge the
fact that for a full three-dimensional simulations, estimation
of the gradients of gi’s is an involved task.
A computational mesh comprises a total of ?16?1
?128? number of lattice points. In Fig. 4, the x-directional
velocity ?u? normalized with respect to the positive
x-directional wall velocity ?Uwall? has been plotted versus the
z coordinate. From Fig. 4, a very good comparison of the
numerical results ?shown using lines? with the analytical so-
lution ?shown using symbols? obtained from Sec. X? may be
clearly seen. Further, a slope of the curve predicted using the
off-lattice representation is observed to be larger than that of
on-lattice model. From this numerical exercise, we conclude
that an off-lattice LB ?D3Q27? is easily implementable with
a good accuracy for finite Knudsen flows.
XIII. OUTLOOK
In the present work, an alternate framework to create dis-
crete velocity set is suggested. It is shown that in this frame-
work the entropic formulation of the LB method can be natu-
rally extended to obtain a discrete velocity set with a given
accuracy. As an example, a 27-velocity LB model with the
10
−1
10
0
10
0
10
1
Kn
^
Q
Basis 1
Basis 2
Cercignani quadratic approx.
Boltzmann−BGK
D2Q16
R13
FIG. 3. Flow rate Q as a function of the resized Knudsen num-
ber Kn
?for Poiseuille flow.
YUDISTIAWAN et al.
PHYSICAL REVIEW E 82, 046701 ?2010?
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Page 17
Galilean-invariant hydrodynamic limit is derived. As com-
pared to existing model, the advantage of the current model
is that it not only captures the Knudsen layer effect which
does not exist in D2Q9 model, it also has a Knudsen mini-
mum which does not exist in D2Q16 model. Therefore, it
combines the advantages of D2Q9 and D2Q16 at the same
time. Thus to conclude, we have presented a novel model
which can predict both Knudsen layer effect and Knudsen
paradox, with rather minimal number of discrete velocities
for three-dimensional system. The presented scheme has
high efficiency for realistic applications due to the minimal
number of discrete velocities. It gives an edge for quantita-
tive computation in engineering application, given some
modification for computational purpose ?due to the fact that
it is off-lattice?. In part II of this work, we shall demonstrate
the usefulness of the current model in the case of two com-
ponent mixture.
ACKNOWLEDGMENTS
We would like to thank Dr. Xiaowen Shan for insightful
comments. SA and D.V. Patil are thankful to Department of
Science and Technology ?DST?, India for providing compu-
tational resources.
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0 0.2 0.40.6 0.81
−0.5
−0.25
0
0.25
0.5
Kn = 0.5
z
u/Uwall
(a)
0 0.20.4 0.6 0.81
−0.5
−0.25
0
0.25
0.5
Kn = 1.0
z
u/Uwall
(b)
FIG. 4. Comparison of the
analytical solution ?symbols; ?:
on-lattice, ?: off-lattice represen-
tation? with the numerical data
?dotted line: on-lattice, solid line:
off-lattice?. ?a? Kn=0.5; ?b? Kn
=1.0.
HIGHER-ORDER GALILEAN-INVARIANT LATTICE …
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