Consistency study of Lattice-Boltzmann schemes macroscopic limit

TLDR: In this paper, a Taylor-expansion of lattice-Boltzmann models is proposed, in which the collision kernel is reinterpreted as a closure for the stress-tensor equation.

Abstract: Owing to the lack of consensus about the way Chapman–Enskog should be performed, a new Taylor-expansion of lattice-Boltzmann models is proposed. In contrast to the Chapman–Enskog expansion, recalled in this manuscript, the method only assumes a sufficiently small time step. Based on the Taylor expansion, the collision kernel is reinterpreted as a closure for the stress-tensor equation. Numerical coupling of lattice-Boltzmann models with other numerical schemes, also encompassed by the method, is shown to create error terms whose scalings are more complex than those obtained via Chapman–Enskog. An athermal model and two compressible models are carefully analyzed through this new scope, casting a new light on each model's consistency with the Navier–Stokes equations.

Figures

Table III: Third-order isotropy defects of standard lattices.

Table II: Lattice-Boltzmann notations

Figure 1: Streamlines of the 2D circular ame simulation colored by velocity magnitude (in m/s). Left column: pressure-based model18, right column: density-based model17. Note the very di erent ranges of velocity magnitude from the two methods. The yellow contour is the heat release rate peak indicating the ame front.

Table I: Chapman-Enskog expansions in the lattice-Boltzmann literature.

Figure 2: Visualization of the temperature, heat release rate and velocity magnitude elds of the 2D circular ame as obtained with the pressure-based formulation18, at C = 9.24s.

Table IV: Generic LBM structure.

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Consistency study of Lattice-Boltzmann schemes
macroscopic limit
G. Farag, S. Zhao, G. Chiavassa, Pierre Boivin
To cite this version:
G. Farag, S. Zhao, G. Chiavassa, Pierre Boivin. Consistency study of Lattice-Boltzmann schemes
macroscopic limit. Physics of Fluids, American Institute of Physics, 2021, 33 (3), pp.037101.
�10.1063/5.0039490�. �hal-03160898�

Consistency study of Laice-Boltzmann schemes macroscopic limit
G. Farag,
1
S. Zhao (赵崧),
1, a)
G. Chiavassa,
1
and P. Boivin
1, b)
Aix Marseille Univ, CNRS, Centrale Marseille, M2P2, Marseille,
France
(Dated: January 18, 2021)
Owing to the lack of consensus about the way Chapman-Enskog should be performed,
a new Taylor-Expansion of Lattice-Boltzmann models is proposed. Contrarily to the
Chapman-Enskog expansion, recalled in this manuscript, the method only assumes an
suciently small time step. Based on the Taylor expansion, the collision kernel is rein-
terpreted as a closure for the stress-tensor equation. Numerical coupling of Lattice-
Boltzmann models with other numerical schemes, also encompassed by the method, are
shown to create error terms whose scalings are more complex than those obtained via
Chapman-Enskog. An athermal model and two compressible models are carefully ana-
lyzed through this new scope, casting a new light on each model’s consistency with the
Navier-Stokes equations.
a)
Also atCNES Launchers Directorate, Paris, France
b)
Electronic mail: pierre.boivin@univ-amu.fr
1

INTRODUCTION
The Navier-Stokes-Fourier (NSF) system of conservation equations is widely accepted, to
study mass, momentum and energy conservation in uid systems. Yet, its derivation from a
more general and purely atomistic point of view is one of the challenges of the 6
𝑡ℎ
Hilbert
problem
1
. Formal solutions of the Boltzmann equation (BE)
2
were obtained following pertur-
bation theory
3–5
, but nding full thermo-hydrodynamic solutions of the BE remains an active
research topic in mathematics
6
.
Nonetheless, this lack of theoretical understanding on the link between the NSF and BE for-
malisms has not slowed down the rapid development of Lattice-Boltzmann methods (LBM), now
an invaluable simulation tool widely used in the engineering and scientic communities. LBM
emerged in the 1980𝑠 and consist of a specic BE discretization. (i) First a discrete set of veloci-
ties is used to represent the velocity space, leading to the discrete velocity Boltzmann equation
(DVBE). (ii) Second, time and space are discretized, as in most computational uid dynamics
(CFD) methods. Albeit initially limited to low-Mach athermal ows, the range of applicabil-
ity has been steadily growing, to encompass compressible ows
7–10
, multiphase ows
11–16
and
combustion
17–19
.
In understanding the link between the equations resolved by LBM and the macroscopic NSF
system, the so-called Chapman-Enskog (CE) expansion
20
is the most popular method, provided
as an Appendix to most LBM papers. Yet, the CE expansion can have limitations in understanding
specic aspects of modern LB methods. For instance, the aforementioned applications (beyond
athermal ows) often correspond to Knudsen numbers too high for the underlying LBM theory
to hold, but LBMs reportedly yields reasonable results nonetheless. The impact of the choice of
collision kernel, central in the method’s robustness
21
, is also hard to study with the CE expansion,
often carried out with a simplied Bhatnagar-Gross-Krook BGK collision model
22
. Last but not
least, the CE expansion can not be easily performed for the wide variety of models in which a
LB distribution is resolved coupled to another distribution or scalar (which can represent energy,
species, or any transported scalar).
The purpose of the present study is threefold. First, we provide a review of the methods tradi-
tionally used to derive the macroscopic equations from a given LBM. Second, the implicit assump-
tions underlying the CE expansion are discussed. Third, we propose a rigorous and systematic
method to analyze LBMs, based on a modied equation analysis
23–25
using a Taylor expansion
2

in time and space. Although use of Taylor expansion to that goal is already reported in the LB
literature
26–28
for athermal models, the presented method is the rst – to the authors knowledge
– to encompass arbitrary LB numerical schemes with multi-physics coupling, arbitrary collision
kernel, arbitrary force terms and arbitrary non-dimensional numbers. It will be shown that the
method allows to identify error terms beyond the CE expansion, necessary to fully understand
recent LB models.
The article is organized following these three goals. After a brief Section I introducing the NSF
system of equations along with the necessary notations, Section II focuses on the continuous BE.
We recall two popular methods used to analyze it and derive a NSF system from the BE, namely
the CE expansion
20
and the Grad moment system
29
. Section III discusses the application of the CE
formalism to LBMs. In particular, we will point out the lack of consensus found in the literature
around the CE expansion. Underlying assumptions and limitations are also discussed.
Section IV contains the principal novelty of the present work, following the arguments pre-
sented in Sections II-III, and proposes an alternative to the CE expansion formalism. The step-
by-step algorithm to build and understand a LB scheme is thoroughly explained. Resting on a
naive Taylor expansion of the numerical scheme this method is seen to be fully deductive and
ansatz-free in the sense that its derivation automatically and unequivocally gives the conditions
for the scheme to be consistent to an expected set of macroscopic equations in the small time-step
limit Δ𝑡 → 0, while keeping the so-called acoustic scaling coecient Δ𝑡/Δ𝑥 constant
30
.
As a rst textbook example, the classical athermal BGK
30
is analyzed through the scope of the
Taylor expansion in Section V. Then, in light of the proposed step-by-step algorithm, Section VI
proposes a new interpretation of the LB collision kernel strictly based on macroscopic equations
instead of the usual kinetic interpretation. Lastly, the scope of our new theoretical framework
is illustrated for two advanced LB thermal models recently published by our group, namely the
RR-𝜌
31,32
and RR-𝑝
7
models, respectfully in Sections VII and VIII.
I. THE NAVIER-STOKES-FOURIER SYSTEM
Before carrying out any comparison between the BE and NSF systems, it is useful to introduce
the NSF governing equations, along with appropriate denitions.
3

A. Navier-Stokes denitions
Mass and momentum conservation read
𝜕𝜌
𝜕𝑡
+
𝜕𝜌𝑢
𝛽
𝜕𝑥
𝛽
= ¤𝑚 , (1)
𝜕𝜌𝑢
𝛼
𝜕𝑡
+
𝜕

𝜌𝑢
𝛼
𝑢
𝛽
+𝑝𝛿
𝛼𝛽
− T
𝛼𝛽

𝜕𝑥
𝛽
= 𝜌F
𝛼
, (2)
where 𝜌 is the volume mass, 𝑢
𝛼
is the local velocity vector and 𝑝 is the pressure. In addition, ¤𝑚
and 𝜌F
𝛼
are respectively any forcing term in the mass an momentum equations. These forces
can model physical phenomena e.g. gravity or mass source, but they can also correspond to
numerical terms such as sponge-zones
33
. Lastly, T
𝛼𝛽
is the stress tensor,
T
𝛼𝛽
= 𝜇

𝜕𝑢
𝛼
𝜕𝑥
𝛽
+
𝜕𝑢
𝛽
𝜕𝑥
𝛼
−𝛿
𝛼𝛽
2
3
𝜕𝑢
𝛾
𝜕𝑥
𝛾

, (3)
with 𝜇 the shear viscosity. The bulk viscosity is neglected in the framework of this paper, but
can readily be included in the analysis.
Recombining Eqs. (1, 2) we obtain the kinetic tensor 𝜌𝑢
𝛼
𝑢
𝛽
equation
𝜕𝜌𝑢
𝛼
𝑢
𝛽
𝜕𝑡
+
𝜕𝜌𝑢
𝛼
𝑢
𝛽
𝑢
𝛾
𝜕𝑥
𝛾
+𝑢
𝛼
𝜕

𝑝𝛿
𝛾𝛽
− T
𝛾𝛽

𝜕𝑥
𝛾
+𝑢
𝛽
𝜕

𝑝𝛿
𝛼𝛾
− T
𝛼𝛾

𝜕𝑥
𝛾
= 𝜌F
𝛼
𝑢
𝛽
+ 𝜌F
𝛽
𝑢
𝛼
− ¤𝑚𝑢
𝛼
𝑢
𝛽
, (4)
not to be confused with the kinetic energy evolution equation, corresponding to half the trace of
the tensor evolution Eq. (4). When the ow is assumed to be athermal, the system is fully closed
by assuming, e.g.
𝑝 = 𝜌𝑐
2
𝑠
, (5)
where 𝑐
𝑠
is the constant sound speed.
B. Fourier system denitions
When thermal eects cannot be neglected, one needs to consider additionally the total energy
density 𝜌𝐸 equation
𝜕𝜌𝐸
𝜕𝑡
+
𝜕

(𝜌𝐸 +𝑝)𝑢
𝛽
+𝑞
𝛽
−𝑢
𝛼
T
𝛼𝛽

𝜕𝑥
𝛽
= 𝜌F
𝛾
𝑢
𝛾
+ 𝜌 ¤𝑞 , (6)
with the total energy 𝐸 dened as the sum of internal and kinetic energies, 𝐸 = 𝑒 +𝑢
𝛼
𝑢
𝛼
/2. In
Eq. (6), 𝑞
𝛼
corresponds to the heat ux, and ¤𝑞 is an energy source.
4

Frequently asked questions

Q1. What are the contributions in "Consistency study of lattice-boltzmann schemes macroscopic limit" ?

Farag et al. this paper presented a systematic method to analyze Lattice-Boltzmann models based on a Taylor-Expansion, which allows to derive the set of macroscopic equations consistent with the model. 

Q2. What are the future works in "Consistency study of lattice-boltzmann schemes macroscopic limit" ?

A major advantage of the method over the traditional Chapman-Enskog framework is that it allows to study more carefully numerical errors. Numerical coupling of Lattice-Boltzmann models with other numerical schemes are shown to create error terms whose scalings are more complicated than what the ChapmanEnskog expansion suggests. Note that although the authors focused on standard, nearest-neighbors lattices such as D2Q9, D3Q19 and D3Q27, the presented method can be extended to larger lattices.