Extended Lattice Boltzmann Model

Abstract

Conventional lattice Boltzmann models for the simulation of fluid dynamics are restricted by an error in the stress tensor that is negligible only for small flow velocity and at a singular value of the temperature. To that end, we propose a unified formulation that restores Galilean invariance and the isotropy of the stress tensor by introducing an extended equilibrium. This modification extends lattice Boltzmann models to simulations with higher values of the flow velocity and can be used at temperatures that are higher than the lattice reference temperature, which enhances computational efficiency by decreasing the number of required time steps. Furthermore, the extended model also remains valid for stretched lattices, which are useful when flow gradients are predominant in one direction. The model is validated by simulations of two- and three-dimensional benchmark problems, including the double shear layer flow, the decay of homogeneous isotropic turbulence, the laminar boundary layer over a flat plate and the turbulent channel flow.

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Article
Extended Lattice Boltzmann Model
Mohammad Hossein Saadat, Benedikt Dorschner and Ilya Karlin *


Citation: Saadat, M.H.; Dorschner,
B.; Karlin, I.V. Extended Lattice
Boltzmann Model. Entropy 2021,23,
475. https://doi.org/10.3390/
e23040475
Academic Editor: Carlos
Mejía-Monasterio
Received: 21 February 2021
Accepted: 13 April 2021
Published: 17 April 2021
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Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
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Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland;
msaadat@ethz.ch (M.H.S.); bdorschn@ethz.ch (B.D.)
*Correspondence: ikarlin@ethz.ch
Abstract:
Conventional lattice Boltzmann models for the simulation of fluid dynamics are restricted
by an error in the stress tensor that is negligible only for small flow velocity and at a singular value of
the temperature. To that end, we propose a unified formulation that restores Galilean invariance and
the isotropy of the stress tensor by introducing an extended equilibrium. This modification extends
lattice Boltzmann models to simulations with higher values of the flow velocity and can be used at
temperatures that are higher than the lattice reference temperature, which enhances computational
efficiency by decreasing the number of required time steps. Furthermore, the extended model also
remains valid for stretched lattices, which are useful when flow gradients are predominant in one
direction. The model is validated by simulations of two- and three-dimensional benchmark problems,
including the double shear layer flow, the decay of homogeneous isotropic turbulence, the laminar
boundary layer over a flat plate and the turbulent channel flow.
Keywords: lattice Boltzmann method; Galilean invariance; extended equilibrium
1. Introduction
The lattice Boltzmann method (LBM) solves a Boltzmann-type kinetic equation on
a discrete velocity set, forming the links of a space-filling lattice. Efficiency of the LBM
makes it attractive for the simulation of a wide range of problems in fluid dynamics, see,
for example, [1,2].
In this paper, we revisit the restrictions of LBM due to the geometry of the discrete
velocities. It is well known that standard LBM velocities yield a persistent error in the
fluid stress tensor, which breaks Galilean invariance and limits the operation range of
LBM to small flow velocities and a singular value of the lattice reference temperature; only
under these conditions can the error be ignored. While one can cope with this error in
most incompressible flow applications [
1
,
2
], it certainly affects high-speed compressible
flows [3–12]
and sometimes even low-speed isothermal cases [
13
]. Moreover, the same
error is amplified when using stretched (rectangular) lattices instead of the conventional
(cubic) lattice, where in addition to the corrupted Galilean invariance, the stress tensor
becomes anisotropic [14,15].
The extension of LBM beyond its classical operation domain has so far been addressed
with different techniques, depending on the desired outcome. Introducing lattices with
more velocities (multi-speed lattices) [
16
] is one technique, which comes with a significant
increase of computational cost and has been used mostly for high-speed compressible
flow applications [
16
]. In the standard LB setting, on the other hand, one approach is to
alter the relaxation rates and use a multi-relaxation time collision operator [
9
]. Another
approach to extend the flow velocity and temperature range of the standard cubic lattices
is to add correction terms to the original LBM [
4
–
8
,
10
–
12
,
17
–
21
]. The realization varies
among different authors but none address the general case of rectangular lattices. On the
other hand, rectangular lattices may improve the computational efficiency of the LBM
by using a coarser mesh in the direction of smaller gradients in the flow. Unlike other
approaches of handling non-uniform grids (e.g., Eulerian [
22
,
23
] and semi-Lagrangian off-
lattice
LBM [24–27]
or grid refinement techniques [
28
,
29
]), stretched lattices do not require
Entropy 2021,23, 475. https://doi.org/10.3390/e23040475 https://www.mdpi.com/journal/entropy

Entropy 2021,23, 475 2 of 23
a substantial change in the standard LBM algorithm. Recent work on the stretched LBM
restores the isotropy of the stress tensor by using multi-relaxation time LBM models [
30
–
32
].
However, these approaches do not address the flow velocity and
temperature restrictions
.
In this paper, we propose a unified view on the three aspects of the problem, the
velocity range, the temperature range and grid stretching, which all stem from the same
error, induced by constraints of the discrete velocity set. In particular, we propose to use
an extended equilibrium, which restores the Galilean invariance and isotropy of the stress
tensor, enabling simulations at higher flow velocities, higher temperatures using both cubic
and stretched lattices, yielding increased accuracy and efficiency.
The paper is organized as follows—in Section 2, we start by presenting the discrete
kinetic equations, following the standard single-relaxation time lattice Bhatnagar–Gross–
Krook (LBGK) setting, as well as the equilibrium and extended equilibrium formulation,
followed by the derivation of the model’s hydrodynamic limit. In addition, the locally
corrected LBM of Reference [9] is compared to the extended LBGK in Appendix A.
Subsequently, in Section 3, we assess the validity, accuracy and performance of our
model using both two- and three-dimensional benchmark problems. As a first step, we
verify Galilean invariance, temperature independence and isotropy of the model on the
example of an advected decaying shear wave. It is shown that the theoretical viscosity
is recovered for both cubic as well as stretched lattices in a large range of temperatures
and advection velocities. This also indicates that the model can readily be extended to
high-speed compressible flows, provided that it is augmented with a suitable solver for the
total energy. Next, for the example of homogeneous isotropic turbulence, we demonstrate
that a speed-up can be achieved by using an operating temperature, which is larger than
the lattice reference temperature. The present model can also be viewed as an alternative to
preconditioned LBM [33] for accelerating the convergence rate but without the restriction
to steady flows. Finally, accuracy and performance are assessed for rectangular lattices
using the doubly periodic shear flow, laminar flow over a flat plate and turbulent channel
flow as examples. Conclusions are drawn in Section 4.
2. Discrete Kinetic Equations
2.1. LBGK
We consider the LBGK equation for the populations
fi
, associated to the discrete
velocities vifor i=0, . . . , Q−1,
fi(x+viδt,t+δt)−fi(x,t) = ω(fex
i−fi), (1)
where
x
denotes the location in space and
δt
is the time step. The extended equilibrium
fex
i
, which will be specified below, satisfies the local conservation laws for the density
ρ
and momentum ρu,
ρ=
Q−1
∑
i=0
fex
i=
Q−1
∑
i=0
fi, (2)
ρu=
Q−1
∑
i=0
vifex
i=
Q−1
∑
i=0
vifi. (3)
As we will show below, the relaxation parameter ωis related to the kinematic viscosity ν,
ν=1
ω−1
2RTδt, (4)
where
T
is the temperature and
R
is the gas constant. We now proceed with identifying the
extended equilibrium.

Entropy 2021,23, 475 3 of 23
2.2. Discrete Velocities and Factorization
We use the
D
3
Q
27 lattice, where
D=
3 denotes the spatial dimension and
Q=
27 is
the number of discrete speeds, which are given by
ci= (cix ,ciy,ciz),ciα∈ {−1, 0, 1}. (5)
With (5), we define the particles’ velocities viin a stretched cell as
vi= (λxcix ,λyciy,λzciz), (6)
where
λα
is the stretching factor in the direction
α
. To maintain on-lattice propagation, the
cell size is changed accordingly to δxα=λαδt.
The (normalized, M000 =1) moments Mlmn are defined using the convention
l→x,m→y,n→z;l,m,n=0, 1, 2, . . . , (7)
and thus
ρMlmn =
Q−1
∑
i=0
vl
ix vm
iy vn
iz fi. (8)
For convenience, we use a more specific notation for the first-order and the diagonal
second-order moments,
M100 =ux,M010 =uy,M001 =uz, (9)
M200 =Pxx ,M020 =Pyy,M002 =Pzz. (10)
We essentially follow [
34
] and consider a class of factorized populations. To that end, we
define a triplet of functions in the three variables, uα,Pαα and λα,
Ψ0(uα,Pαα,λα) = 1−Pαα
λ2
α
, (11)
Ψ1(uα,Pαα,λα) = 1
2uα
λα
+Pαα
λ2
α, (12)
Ψ−1(uα,Pαα,λα) = 1
2−uα
λα
+Pαα
λ2
α. (13)
For the vectors u,P, and λ,
u= (ux,uy,uz), (14)
P= (Pxx ,Pyy,Pzz), (15)
λ= (λx,λy,λz), (16)
we consider a product-form, associated with the discrete velocities vi(6),
Ψi(u,P,λ) = ∏
α=x,y,z
Ψciα(uα,Pαα,λα). (17)
The normalized moments of the product-form (17),
Mlmn =
Q−1
∑
i=0
vl
ix vm
iy vn
iz Ψi, (18)
are readily computed thanks to the factorization,
Mlmn =Ml00 M0m0M00n, (19)

Entropy 2021,23, 475 4 of 23
where
M000 =1, (20)
Ml00 =(λl−1
xux,lodd
λl−2
xPxx ,leven, (21)
M0m0=(λm−1
yuy,modd
λm−2
yPyy,meven, (22)
M00n=(λn−1
zuz,nodd
λn−2
zPzz,neven. (23)
For any stretching (16), the six-parametric family of normalized populations (17) is identi-
fied by the flow velocity (14) and the diagonal of the pressure tensor at unit
density (15)
,
and was termed the unidirectional quasi-equilibrium in Ref. [
34
]. We make use of the
product-form (17) to construct all pertinent populations, the equilibrium and the extended
equilibrium.
2.3. Equilibrium and Extended Equilibrium
The equilibrium
feq
i
is defined by setting
Pαα
(10) equal to the equilibrium diagonal
element of the pressure tensor at unit density,
Peq
αα =RT +u2
α. (24)
Substituting (24) into (17), we get
feq
i=ρΨi(u,Peq,λ). (25)
With (18), we find the pressure tensor and the third-order moment tensor at the equilibrium
(25) as follows,
Peq =
Q−1
∑
i=0
vi⊗vifeq
i=PMB, (26)
Qeq =
Q−1
∑
i=0
vi⊗vi⊗vifeq
i=QMB +˜
Q. (27)
The isotropic parts,
PMB
and
QMB
, are the Maxwell–Boltzmann (MB) pressure tensor and
the third-order moment tensor, respectively,
PMB =pI+ρu⊗u, (28)
QMB =sym(pI⊗u) + ρu⊗u⊗u, (29)
where
p=ρRT
is the pressure,
sym(. . . )
denotes symmetrization and
I
is the unit tensor.
The anisotropy of the equilibrium (25) manifests with the deviation
˜
Q=Qeq −QMB
in
(29), where only the diagonal elements are non-vanishing,
˜
Qαβγ =(ρuα(λ2
α−3RT)−ρu3
α, if α=β=γ,
0, otherwise. (30)
The origin of the diagonal anomaly (30) is the geometric constraint,
v3
iα=λ2
αviα
, which is
imposed by the choice of the discrete speeds (5), and is well known in the case of the
standard (unstretched) lattice with
λα=
1. A remedy in the latter case is to minimize
spurious effects of anisotropy by fixing the temperature T=TL,

Entropy 2021,23, 475 5 of 23
RTL=1
3, (31)
In order to eliminate the linear term
∼uα
in (30). Thus, using the equilibrium (25) in the
LBGK Equation (1) imposes a two-fold restriction on the operation domain: the temper-
ature cannot be chosen differently from (31) and the flow velocity has to be maintained
asymptotically vanishing. Moreover, for stretched lattices, the anisotropy becomes even
more pronounced since it is not possible to eliminate the linear deviation in all directions
simultaneously by fixing any temperature.
Alternatively, the spurious anisotropy effects can be canceled out by extending the
equilibrium such that the third-order moment anomaly is compensated in the hydrody-
namic limit. Because the anomaly only concerns the diagonal (unidirectional) elements
of the third-order moments, the cancellation can be achieved by redefining the diagonal
elements of the second-order moments. As demonstrated below, in order to cancel the
errors, the diagonal elements Pex
αα for the extended equilibrium must be chosen as
Pex
αα =Peq
αα +δt2−ω
2ρω ∂α˜
Qααα, (32)
where spatial derivative is evaluated using a second-order central difference scheme.
Hence, the extended equilibrium fex
iis specified by using the product-form (17),
fex
i=ρΨi(u,Pex,λ). (33)
We shall now proceed with the derivation of the Navier–Stokes equations in the hydrody-
namic limit of the proposed extended LBGK model.
2.4. Hydrodynamic Limit with Extended Equilibrium
Taylor expansion of the shift operator in (1) to second order gives,
δtDi+δt2
2DiDifi=ω(fex
i−fi), (34)
where Diis the derivative along the characteristics,
Di=∂t+vi·∇. (35)
Introducing the multi-scale expansion,
fi=f(0)
i+δt f (1)
i+δt2f(2)
i+O(δt3), (36)
fex
i=fex(0)
i+δt f ex(1)
i+δt2fex(2)
i+O(δt3), (37)
∂t=∂(1)
t+δt∂(2)
t+O(δt2), (38)
substituting into (34) and using the notation,
D(1)
i=∂(1)
t+vi·∇, (39)
we obtain, from the zeroth to second order in the time step δt,
f(0)
i=fex(0)
i=feq
i, (40)
D(1)
if(0)
i=−ωf(1)
i+ωfex(1)
i, (41)
∂(2)
tf(0)
i+vi·∇ f(1)
i−ω
2D(1)
if(1)
i+ω
2D(1)
ifex(1)
i=−ωf(2)
i+ωfex(2)
i. (42)

Entropy 2021,23, 475 6 of 23
With (40), the mass and the momentum conservation (2) and (3) imply the solvability
conditions,
Q−1
∑
i=0
fex(k)
i=
Q−1
∑
i=0
f(k)
i=0, k=1, 2 . . . ; (43)
Q−1
∑
i=0
vifex(k)
i=
Q−1
∑
i=0
vif(k)
i=0, k=1, 2, . . . . (44)
With the equilibrium (25), taking into account the solvability conditions (43) and
(44)
, and also making use of the equilibrium pressure tensor (26) and (28), the first-order
Equation (41) implies the following relations for the density and the momentum,
∂(1)
tρ=−∇ · (ρu), (45)
∂(1)
t(ρu) = −∇ · (pI+ρu⊗u). (46)
Moreover, the first-order constitutive relation for the nonequilibrium pressure tensor
P(1)
is found from (41) as follows,
−ωP(1)+ωPex(1)=∂(1)
tPeq +∇ · Qeq, (47)
where
P(1)=
Q−1
∑
i=0
vi⊗vif(1)
i, (48)
Pex(1)=
Q−1
∑
i=0
vi⊗vifex(1)
i. (49)
With the help of Equations (28) and (29), the first-order constitutive relation (47) is trans-
formed to make explicit the contribution of the anomalous term (30),
−ωP(1)+ωPex(1)=∇ · ˜
Q+∂(1)
tPMB +∇ · QMB. (50)
The last term is evaluated using (45) and (46) to give,
∂(1)
tPMB +∇ · QMB =ρRT∇u+∇u†, (51)
where
(·)†
denotes transposition. Combining (51) and (50), the first-order constitutive
relation becomes,
−ωP(1)=∇· ˜
Q−ωPex(1)+ρRT∇u+∇u†. (52)
Note that, if we had used the equilibrium
feq
i
instead of the extended equilibrium
fex
i
in
(1), at this stage of the derivation, we get instead of (52),
−ωP(1)=∇· ˜
Q+ρRT∇u+∇u†. (53)
The anomalous term
∇· ˜
Q
cannot be canceled in the latter expression, rather, by choosing
T=TL
(31), its effect can be ignored but only under the assumption of an asymptotically
small flow velocity. Moreover, for a quasi-incompressible flow (
Ma →
0, density variation
∇ρ∼Ma2
, where
Ma
is a characteristic Mach number), it is possible to further reduce
the effect of the anomaly by rescaling the relaxation parameter [
9
], see a discussion in
Appendix A.

Entropy 2021,23, 475 7 of 23
In a contrast, using the present formulation, the cancellation is possible by finding
the corresponding expression for the correction term Pex(1), to which end we need to con-
sider the second-order contribution to the momentum equation. Applying the solvability
condition (43) and (44) to the second-order Equation (42), we obtain,
∂(2)
tρ=0, (54)
∂(2)
t(ρu) = −∇ · h1−ω
2P(1)+ω
2Pex(1)i. (55)
The latter is transformed by virtue of (52),
∂(2)
t(ρu) = −∇·−1
ω−1
2ρRT(∇u+∇u†)+∇ · 1
ω−1
2∇· ˜
Q−Pex(1). (56)
The last (anomalous) term is canceled out by choosing,
Pex(1)=2−ω
2ω∇· ˜
Q. (57)
Combining the result (57) with the zeroth-order (equilibrium) value, we arrive at the
extended pressure tensor
Pex =Peq +δtPex(1)
=pI+ρu⊗u+δt2−ω
2ω∇· ˜
Q. (58)
Since the anomalous contribution is a diagonal tensor, cf. Equation (30), the result (58) is im-
plemented with the extended equilibrium in the product-form by choosing the normalized
(at unit density) diagonal elements of the pressure tensor as follows,
Pex
αα =RT +u2
α+δt2−ω
2ρω ∂αρuαλ2
α−3RT −u2
α, (59)
which is equivalent to (32). Finally, combining the first- and second-order contributions
to the density and the momentum equation, (45), (46), (54) and (56), using a notation,
∂t=∂(1)
t+δt∂(2)
t
, and also taking into account the cancellation of the anomalous term in
(56), we arrive at the continuity and the flow equations as follows,
∂tρ+∇·(ρu) = 0, (60)
∂tu+u·∇u+1
ρ∇p+1
ρ∇·Π=0, (61)
where pis the pressure of ideal gas at constant temperature T,
p=ρRT, (62)
Πis the viscous pressure tensor,
Π=−µS, (63)
with Sthe rate of strain,
S=∇u+∇u†, (64)
and µthe dynamic viscosity,

Entropy 2021,23, 475 8 of 23
µ=1
ω−1
2pδt. (65)
The above considerations can be summarized as follows—because of the third-order
moment anomaly (30), the LBGK Equation (1) with the product-form equilibrium (25) is
restricted in several ways, namely:
(i)
The temperature is restricted to a single value, the lattice reference temperature
TL
(31);
(ii)
The flow velocity has to be asymptotically vanishing;
(iii)
Stretched velocities amplify these restrictions by making it impossible to cancel even
the linear (in velocity) anomaly in all the directions simultaneously.
Note that, in addition to all of the restrictions above, when using the conventional
second-order equilibrium obtained by retaining the terms up to the order of
∼uαuβ
in (25),
the anomaly becomes not only confined to the diagonal elements
Qeq
ααα
but also contaminates
the off-diagonal elements
Qeq
αββ
. While the diagonal anomaly (30) is genuine, that is, it is
caused by the geometry of the discrete velocities, this additional off-diagonal deviation is due
to an unsolicited second-order truncation of the product-form equilibrium (25).
The proposed revision of the LBGK model is based on extending the product-form
equilibrium such that the anomaly of the diagonal third-order moment is compensated
in the hydrodynamic limit by counter terms, which are added to the diagonal of the
equilibrium pressure tensor. With this, all three restrictions mentioned above are addressed
at once, without making a special distinction between the temperature, flow velocity or
stretching as separate causes for the anisotropy.
3. Numerical Results
In this section, we shall access the accuracy and performance of the proposed LB
model in a variety of scenarios of activating spurious anisotropy. First, we test Galilean
invariance, isotropy and temperature independence of the model with both regular and
rectangular lattices in the simulation of a decaying shear wave. Second, we validate the
model for the more complex case of decaying homogeneous isotropic turbulence and
show the effectiveness of using higher temperatures in saving compute time. Third, we
investigate the applicability of the proposed model with stretched lattices in a periodic
double shear layer flow, in a laminar flow over a flat plate, and finally in the case of the
turbulent channel flow. In the simulations below, the gas constant was set to
R=
1, the
time step is
δt=
1 and Grad’s approximation, as proposed in [
35
], was used for wall
boundary conditions.
3.1. Galilean Invariance, Isotropy and Temperature Independence Test
To probe the Galilean invariance and temperature independence of the model, the
kinematic viscosity
ν=µ/ρ
(4) is measured for the decay of a plane shear wave. First, we
consider the axis-aligned setup, with the initial condition,
ρ=ρ0,ux=a0sin(2πy/Ly),uy=Ma√T, (66)
where
Ma =u0/√T
is the advection Mach number,
a0=
0.001 is the amplitude,
Ly=
200
is number of grid nodes in the
y
direction,
ρ0=
1. The nominal kinematic viscosity is
set to
ν=
0.01. Periodic boundary conditions are imposed in both
x
- and
y
- directions.
The numerical viscosity (
νnum
) is measured by fitting an exponential to the time decay of
maximum flow velocity
ux,max ∼e−νt(2π/Ly)2
. In this special case, the diagonal anomaly
(30)
is dormant and does not trigger any spurious effects because the derivatives
∂x˜
Qxxx
and
∂y˜
Qyyy
both vanish. Consequently, the extended equilibrium (33) becomes equivalent
to the product-form equilibrium (25) in this case.
In order to compare with the standard LBGK, the standard velocities
λα=
1 were
used in this simulation. Figures 1and 2show the importance of using the product-form
equilibrium (25) as opposed to the conventional LBGK model with the second-order equi-

Entropy 2021,23, 475 9 of 23
librium. A strong dependence of the viscosity on the reference frame for the second-order
equilibrium can be seen in Figure 1, where the viscosity drops with increasing advection
Mach number. This well-known artifact of the second-order equilibrium is due to the non-
vanishing anomaly in the off-diagonal moments
Qeq
αββ
and, unlike the diagonal anomaly, is
caused only by the approximate treatment of the product-form equilibrium. Moreover, as
shown in Figure 2, even at a small enough velocity this spurious feature improves only
at the lattice reference temperature
TL
. In contrast, as is shown in
Figures 1and 2
, the
product-form equilibrium of the present model is able to accurately predict the viscosity in
this setup for a wide range of temperatures and reference frame velocities.
Ma
νnum / ν
0.1 0.2 0.3 0.4 0.5 0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Second-order equilibrium
Product-form equilibrium
T = 1/3
Figure 1.
Numerical measurement of viscosity for axis-aligned setup at temperature
T=
1
/
3 for
different velocities. The exact solution corresponds to νnum/ν=1.
Next, in order to trigger the anisotropy of the deviation terms (30) and to show
the necessity of using the extended equilibrium, the shear wave is rotated by
π/
4. The
anisotropy is further increased by also conducting simulations on a stretched grid with
λx=
2. The temperature is kept at
T=
1
/
3. The viscosity measurement is shown in
Figure 3for different advection Mach numbers and stretching factors. It can be observed
that the model lacks Galilean invariance for larger velocities when using the product-form
equilibrium without correction (25). Furthermore, the stretching factor
λx=
2 results in a
significant hyper-viscosity since the deviation (30) in this case amounts to a large positive
number. However, once the correction term is included and the extended equilibrium
(33)
is used, the present model recovers the imposed viscosity, independent of the frame
velocity and stretching factor.

Entropy 2021,23, 475 10 of 23
T / TL
νnum / ν
0.4 0.8 1.2 1.6 2
1
2
3
4
5Second-order equilibrium
Product-form equilibrium
Ma = 0.1
Figure 2.
Numerical measurement of viscosity for axis-aligned setup at Mach number
Ma =
0.1 for
different temperatures. The exact solution corresponds to νnum/ν=1.
Ma
νnum / ν
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.5
1
1.5
2
2.5
3
3.5
Product-form equilibrium (λx=1)
Product-form equilibrium (λx=2)
Extended equilibrium (λx=1, 2)
T = 1/3
Figure 3.
Numerical measurement of viscosity for rotated setup at temperature
T=
1
/
3 for different
velocities and stretching ratios. The exact solution corresponds to νnum/ν=1.
3.2. Decaying Homogeneous Isotropic Turbulence
In order to further validate the model as a reliable method for the simulation of
complex flows and to show the application of using higher temperatures, decaying homo-
geneous isotropic turbulence was considered. The initial condition, in a box of the size
L×L×L
, was set at unit density and constant temperature along with a divergence-free
velocity field, which follows the specified energy spectrum,
E(κ) = Aκ4e−2(κ/κ0)2, (67)
where
κ
is the wave number,
κ0
is the wave number at which the spectrum peaks and
A
is the parameter that controls the initial kinetic energy [
36
]. The initial velocity field is

Entropy 2021,23, 475 11 of 23
generated using a kinematic simulation as proposed in [
37
]. The turbulent Mach number is
defined as
Mat=√u·u
cs, (68)
where
cs=√T
is the speed of sound. The Reynolds number is based on the Taylor
microscale,
Λ2=u2
rms
(∂xux)2, (69)
and is given by
ReΛ=ρurms Λ
µ, (70)
where
urms =√u·u/3
is the root mean square (rms) of the velocity and overbar denotes
the volume average over the entire computational domain.
Simulations were performed at
Mat=
0.1,
ReΛ=
72,
κ0=
16
π/L
, at two different
temperatures,
T=
1
/
3 and
T=
0.55, and with
L=
256 grid points. Figure 4shows a
snapshot of the velocity magnitude
√u·u
at time
t∗=t/τ=
1.0, where
τ=LI/urms,0
is
the eddy turnover time, which is defined based on the initial rms of the velocity and the
integral length scale LI=√2π/κ0.
Figure 4.
Velocity magnitude in lattice units for the decaying homogeneous isotropic turbulence at
Mat=0.1, ReΛ=72 and t∗=1.0 with temperature T=0.55.
To quantitatively assess the accuracy of the model at different temperatures, the time
evolution of the turbulent kinetic energy,
K=3
2u2
rms, (71)
normalized with its initial value (
K0
), and of the Taylor microscale Reynolds number are
compared in Figures 5and 6with results from direct numerical simulations (DNS) [
36
]. It
is apparent that the two working temperatures yields almost identical results that agree
well with the DNS simulation. This indicates that the correction terms do not degrade the
accuracy of the model at higher temperatures, even though the magnitude of error term
(30) is higher due to amplification of the linear term.

Entropy 2021,23, 475 12 of 23
t*
K / K0
12345
0.2
0.4
0.6
0.8
1
T = 1/3
T = 0.55
DNS
Figure 5. Time evolution of the turbulent kinetic energy for decaying isotropic turbulence at Mat=
0.1, ReΛ=72. Lines: present model; symbol: DNS [36].
t*
ReΛ
10-1 100
40
60
80
100
120
T = 1/3
T = 0.55
DNS
Figure 6.
Time evolution of the Taylor microscale Reynolds number for decaying isotropic turbulence
at Mat=0.1, ReΛ=72. Lines: present model; symbol: direct numerical simulations (DNS) [36].
The immediate advantage of using the present model at a temperature higher than the
lattice temperature
TL=
1
/
3 is that it effectively increases the characteristic velocity (here
urms,0
) and therefore the time step by a factor of
√T/TL
. A larger time step is equivalent
to fewer number of time steps. The present model, therefore, speeds up the simulation
by a factor of
√T/TL
compared to the conventional LBM, which can operate only at the
lattice temperature
TL
. Furthermore, this speedup strategy can be used for both steady
and unsteady flows. This is in contrast to the preconditioned LBM [
33
], which works
by altering the effective Mach number and therefore reduces the disparity between the
speeds of the acoustic wave propagation and the waves propagating with the fluid velocity,
cf. [33]
. This makes preconditioned LBM restricted to steady state applications. In contrast,

Entropy 2021,23, 475 13 of 23
the present model enables us to increase the speed of sound without changing the Mach
number. This increases the effective time step of the solver. Therefore, the present model
increases the computational efficiency by decreasing the number of required time steps.
Note that the theoretical temperature range of the model (like any other models based
on the D1Q3 lattice) is 0
≤T≤
1, beyond that the populations become negative and the
model is unstable. Therefore, while small temperature is possible but not beneficial, large
temperature greater than 1 is out of the stability domain
3.3. Periodic Double Shear Layer
The next validation case to test the accuracy of the proposed model with the stretched
lattice is the periodic double shear layer flow with the initial condition,
ux=u0tanh(α(y/L−0.25)),y≤L/2,
u0tanh(α(0.75 −y/L)),y>L/2, (72)
uy=δu0sin(2π(x/L+0.25)), (73)
where
L
is the domain length in both
x
and
y
directions,
u0=
0.1 is characteristic velocity,
δ=
0.05 is a perturbation of the
y
-velocity and
α=
80 controls the width of the shear layer.
The Reynolds number is set to Re =u0L/ν=104and the temperature is T=1/3.
Figure 7shows the vorticity field at non-dimensional time
t∗=tu0/L=
1 using the
conventional square lattice
λx=λy=
1 and the rectangular lattice with
λx=
2,
λy=
1.
Both lattice models perform qualitatively the same.
-0
−
0.5
−
l
Figure 7.
Vorticity field for double shear layer flow at
t∗=
1 with regular lattice (
left
) and stretched
lattice (right). Vorticity magnitude is normalized by its maximum value.
To quantify the effect of stretching on the accuracy, the time evolution of the mean
kinetic energy and of the mean enstrophy
Ω=ω2/u2
0
L2
, with
ω
the vorticity magnitude,
are compared in Figure 8. The results show only minor discrepancies, which indicates the
validity of the model also on stretched meshes.

Entropy 2021,23, 475 14 of 23
t*
K / 0.5 u0
2
012345
0.86
0.88
0.9
0.92
0.94 256×512 (λx = 2)
512×512 (λx = 1)
t*
Ω
012345
100
150
200 256×512 (λx = 2)
512×512 (λx = 1)
Figure 8.
Evolution of kinetic energy (
left
) and enstrophy (
right
) for double shear layer flow at
Re =104.
3.4. Laminar Boundary Layer over a Flat Plate
The next test case validates our model for wall-bounded flows. We consider the
laminar flow over a flat plate with an incoming Mach number
Ma∞=u∞/√T∞=
0.1,
temperature
T∞=
1
/
3 and Reynolds number
Re =ρ∞u∞L/µ=
4000, where
L
is the
length of flat plate. Since the flow gradients in the transverse
y
-direction are much larger
compared to the gradients in the streamwise
x
-direction, the mesh can be stretched in
x
-direction without significantly affecting the accuracy of the results. The computational
domain was set to
[Lx×Ly]=[
200
×
200
]
and a rectangular lattice with
λx=
2 was used.
The flat plate starts at a distance of
Lx/
4 from the inlet and symmetry boundary conditions
were imposed at 0
≤x≤Lx/
4. In Figure 9, the horizontal velocity profile at the end of
the plate is compared with the results of a regular lattice and with the Blasius similarity
solution, where ηis the dimensionless coordinate [38],
η=yru∞
νx. (74)
It can be seen that results for the regular and the rectangular lattice nearly coincide
and agree well with the Blasius solution. Thus, the model achieves accurate results with
half of grid points compared to the regular lattice. Furthermore, the distribution of skin
friction coefficient over the plate,
Cf=τwall
1
2ρ∞u2
∞
, (75)
with the wall shear stress
τwall =µ(∂u
∂y)y=0
, is shown in Figure 10 in comparison with the
analytical solution
Cf=
0.664
/√Rex
, where
Rex=u∞x/ν
[
38
]. Also here, the results of
the model with the regular and the stretched velocities are almost identical and in good
agreement with the analytical solution.

Entropy 2021,23, 475 15 of 23
u / U∝
η
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
(400×200) (λx=1)
(200×200) (λx=2)
Figure 9.
Comparison of the velocity profile at
x=Lx
for flow over a flat plate at different stretching
ratios. Lines: present model; symbols: Blasius solution.
x / L
Cf
0.2 0.4 0.6 0.8 1
0.02
0.04
0.06
0.08
0.1
0.12
0.14
(400×200) (λx=1)
(200×200) (λx=2)
Figure 10.
Comparison of the skin friction coefficient for flow over a flat plate at different stretching
ratio. Lines: present model; symbols: analytical solution.
3.5. Turbulent Channel Flow
In the final test case, we assess the accuracy and performance of the extended LBM
for the turbulent flow in a rectangular channel, for which many numerical [
39
–
41
] and
experimental [
42
,
43
] results are available. The channel geometry was chosen as
[
5.6
H×
2H×2H], where His the channel half-width. The friction Reynolds number,
Reτ=uτH
ν, (76)

Entropy 2021,23, 475 16 of 23
based on the friction velocity
uτ=pτw/ρ
, was set to
Reτ=
180. The initial friction
velocity was estimated by
uτ=u0
1
KlnReτ+5.5 , (77)
where
K=
0.41 is the von Kármán constant and
u0=
0.1 is the mean center-line velocity.
Periodic boundary conditions were imposed in the streamwise
x
-direction and the spanwise
z-direction. The flow was driven by a constant body force in the x-direction,
g=Re2
τν2/H3. (78)
In order to accelerate the transition to turbulence, a non-uniform divergence-free forcing field
as proposed in [44] was added to the flow for some period of time, until t∗=tH/uτ=5.
Similar to the previous test case, grid stretching in
x
-direction with
λx=
1.4 was used
in order to reduce the number of grid points in that direction while the temperature was
set to T=0.55, same as in Section 3.2
A snapshot of the velocity magnitude √u·uis shown in Figure 11.
Figure 11.
Snapshot of the velocity magnitude in lattice units for turbulent channel flow at
Reτ=
180
with λx=1.4.
Quantitatively, we compare the mean velocity profile with the DNS results of [
40
] in
Figure 12. In wall units, the mean velocity is given by
u+=¯
u/uτ
and the spatial coordinate
is
y+=yuτ/ν
. The statistics are collected after 30 eddy turnover times, i.e., after
t∗=
30.
It is apparent that the viscous sublayer (
y+<
5), the buffer layer (5
<y+<
30) and the
log-law region (
y+>
30) are captured well with our model and the mean velocity profile
agrees well with that of the DNS.

Entropy 2021,23, 475 17 of 23
y+
u+
100101102
0
5
10
15
20
Present
DNS
Figure 12.
Comparison of the mean velocity profile in a turbulent channel flow at
Reτ=
180 with
λx=1.4.
For a more thorough analysis, we compare the root mean square of the velocity
fluctuations with the DNS data in Figure 13. Here,
ux,rms =qu0
xu0
x
and
uy,rms
and
uz,rms
are defined in a similar way. It can be seen that the results are in excellent agreement
with the DNS results [
40
]. This demonstrates that the LBGK model, also in the presence
of a severe anisotropy triggered by stretched velocities, can be used for the simulation of
high Reynolds number wall-bounded flows once the corrections are incorporated with the
extended equilibrium.
y+
ux, rms, uy, rms, uz, rms
0 50 100 150
0
0.5
1
1.5
2
2.5
3
ux, rms
uy, rms
uz, rms
Figure 13.
Comparison of the rms of the velocity fluctuations in a turbulent channel flow at
Reτ=
180
with λx=1.4. Symbols: present model; lines: DNS [40].
4. Conclusions
While even with the standard discrete speeds (5) it is possible to develop an error-free,
fully Galilean invariant kinetic model in the co-moving reference frame, it does require

Entropy 2021,23, 475 18 of 23
off-lattice particles’ velocities [
45
,
46
]. Sticking with the fixed, lattice-conform velocities
(6)
, one is faced with an inevitable and persistent error, which spoils the hydrodynamic
equations whenever the flow velocity is increased or the temperature deviates from the
lattice reference value, or the discrete speeds are stretched differently in different directions.
We proposed an upgrade of the LBGK model to enlarge its operation domain in terms
of velocity, temperature and grid stretching by suggesting an extended equilibrium. The
extended equilibrium is realized through a product-form, which allows us to compensate
the diagonal third-order moment anomaly in the hydrodynamic limit by adding consistent
correction terms to the diagonal elements of the second-order moment. As a result, the
extended LBGK model restores Galilean invariance and temperature independence in a
sufficiently wide range, and can also be used with rectangular lattices. Similar to previous
proposals [
4
–
6
,
10
,
12
], the relaxation term of the present model remains almost local as
it uses only nearest-neighbor information for computation of the first-order derivatives
in the extended equilibrium populations. The extended LBGK model was validated in a
range of benchmark problems, probing different aspects of anomaly triggered either by
increased velocity or temperature deviation from the lattice reference temperature, or by
grid stretching. In all cases, the extended LBGK model featured excellent performance and
accuracy in both two and three dimensions. In particular, the simulation of homogeneous
isotropic turbulence demonstrated the expected speed-up when a higher temperature was
used, while simulations of the laminar boundary layer and of the turbulent channel flow
using stretched grids demonstrated good accuracy with a reduced number of grid points.
Furthermore, the present model can be extended to other applications including but
not limited to high-speed compressible flows, which can be achieved by incorporating
another solver for the total energy (see, e.g., the models proposed in [
10
,
12
]). Advanced
collision models, such as multiple relaxation times (MRT) schemes, can also readily be
employed in the present approach, which can be beneficial when running under-resolved
simulations. These two avenues shall be subject of further development and application of
the extended LBM.
•
Note added in proof: After the paper was submitted to peer-review, a later preprint
“Central Moment MRT Lattice Boltzmann Method on a Rectangular Lattice” by E.
Yahia and K. N. Premnath, arXiv:2103.02119 [physics.flu-dyn], became known to us
which adresses a correction for the two-dimensional rectangular lattice in a multiple
relaxation time setting.
Author Contributions:
Conceptualization, I.K.; methodology, M.H.S., B.D. and I.K.; analysis, M.H.S.
and I.K.; implementation, M.H.S.; simulation, M.H.S.; validation, M.H.S. and B.D.; visualization,
M.H.S.; simulation—supervision, B.D.; writing—original-draft preparation, M.H.S.; writing—review
and editing, I.K., M.H.S. and B.D.; supervision, I.K. All authors contributed to writing the paper. All
authors have read and agreed to the published version of the manuscript.
Funding:
This work was supported by the European Research Council (ERC) Advanced Grant
834763-PonD and the ETH Research Grant ETH-13 17-1. Computational resources at the Swiss
National Super Computing Center CSCS were provided under the Grant s897.
Data Availability Statement: Not applicable.
Acknowledgments:
We thank anonymous referee who suggested comparison with the local correc-
tion method of Ref. [
9
]. Authors are grateful to Seyed Ali Hosseini at ETHZ for computing spectral
dissipation presented in Figure A1.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
LBM Lattice Boltzmann method
LBGK Lattice Bhatnagar–Gross–Krook

Entropy 2021,23, 475 19 of 23
Appendix A. Comparison of Extended LBGK to Locally Corrected LBM
Below, we compare the locally corrected lattice Boltzmann model (LC LBM) [
9
] with
both the standard and the present extended LBGK. To that end, it suffices to consider
the one-dimensional
D
1
Q
3 lattice. In order to introduce the LC LBM, we begin with the
standard LBGK (δt=1, R=1),
fi(x+vi,t+1)−fi(x,t) = ω(feq
i−fi). (A1)
The equilibrium populations in (A1) are given by (24) and (25),
feq
i=ρΨiux,Peq
xx , 1,Peq
xx =T+u2
x,i∈ {−1, 0, 1}. (A2)
Thanks to the diagonal anomaly, the second-order asymptotic analysis of Section 2.4
results in the following viscous stress in the one-dimensional version of the Navier–Stokes
Equation (61),
Πxx =−21
ω−1
2ρT∂xux+˜
Πxx . (A3)
For the LBGK model, the anomalous (second) term in (A3) reads,
˜
ΠLBGK
xx =−1
ω−1
21−3T
2T−3u2
x
2T2ρT∂xux+ux(1−3T)−u3
x∂xρ. (A4)
Upon realizing that the first term of the anomalous contribution (A4) is similar in its
structure to the relevant (first) term in the LBGK stress (A3), the locally corrected (LC)
LBGK groups these two terms together and replaces the relaxation parameter
ω
in (A1)
with a new relaxation
ωLC
, which depends on the flow velocity. While the original
work [9]
addressed the case of the lattice temperature,
T=TL=
1
/
3 (31), we first consider a slightly
more general formulation for a flexible temperature parameter. Consequently, the locally
corrected relaxation parameter ωLC in the LBGK Equation (A1) is defined as,
1
ωLC −1
2=1
ω−1
2X, (A5)
where the renormalization factor Xreads,
X=1+1−3T
2T−3u2
x
2T−1
. (A6)
The LBGK model with the locally corrected relaxation parameter
ωLC
(A5) results in the
viscous stress of the form (A3), with the remaining error term,
˜
ΠLC
xx =−1
ω−1
2Xux(1−3T)−u3
x∂xρ. (A7)
For the sake of a discussion, let us introduce the local Mach number,
Max=ux/√T
. For
a quasi-incompressible (slow) flow, the density variation scales as
∂xρ∼Ma2
x
. Thus, for
T6=TL
, the error (A7) can be estimated as,
˜
ΠLC
xx ∼Ma3
x
. This is two orders of magnitude
lower than the error of the original LBGK at
T6=TL
, cf. Equation (A4), at small Mach
number. Moreover, by setting the temperature
T=TL=
1
/
3, it was first realized in
Ref. [9]
that the error (A7) reduces to,
˜
ΠLC
xx =1
ω−1
2 T3/2
LMa3
x
1−(3/2)Ma2
x!∂xρ. (A8)

Entropy 2021,23, 475 20 of 23
In this case, the scaling at
Max→
0 becomes,
˜
ΠLC
xx ∼Ma5
x
. In other words, the local
correction at
T=TL
provides a gain of two orders of magnitude in accuracy with respect to
the standard LBGK under the quasi-incompressible flow conditions [
9
]. This consideration
extends straightforwardly to the
D
2
Q
9 and
D
3
Q
27 lattices by constructing a multiple
relaxation time LBM that corrects the relaxation of each diagonal component of the pressure
tensor [9].
However, for a generic isothermal flow, the error (A7) becomes amplified through
the renormalization factor (A6) as the velocity increases and eventually diverges when
u2
x→(1−T)/3
. This error persists also for the special case
T=TL
(A8). On the other hand,
the second-order analysis of Section 2.4 reveals that the present LBGK with the extended
equilibrium (33) removes the entire anomalous term,
˜
Πex
xx =
0. Thus, the difference between
the extended LBGK and the LC LBM [9] is expected beyond the asymptotic Max→0.
In order to demonstrate this point, a spectral analysis was performed for the two-
dimensional
D
2
Q
9 lattice (see [
21
] for details of the spectral analysis in the LBM context).
The normalized spectral dissipation of acoustic modes
=(ωκ)/νκ2
x
, is shown in Figure A1,
for
T=TL
and the background flow velocity
(ux
,
uy) = (
0.3, 0
)
, for the three models, the
standard LBGK, the present extended LBGK and the LC LBM of Ref. [9].
It can be seen that, the extended LBGK recovers the correct dissipation rate in the
continuum limit (vanishing wave number
κx
), confirming its Galilean invariance. However,
both the standard LBGK and the LC LBM show deviations in the form of under-dissipation
at low wave numbers, while the deviation for the LC LBM is indeed smaller. This non-
vanishing deviation is amplified in cases with different working temperature and/or
non-unit stretching factor for both the standard LBGK and the LC LBM, which makes their
applications limited to the quasi-incompressible flow regime at the lattice temperature.
κx
ℑ(ωκ) / ν κ x
2
−2
−1.5
−1
−0.5
0
0π / 2 π
Figure A1.
Spectral dissipation of acoustic modes for different models. Red symbols: LBGK; black
symbols: extended LBM (33); blue symbols: LC LBM [
9
]; dashed line: Navier–Stokes. The velocity
and temperature are set to ux=0.3 and T=TL.

Entropy 2021,23, 475 21 of 23
x
ρ
0 200 400 600 800
0.4
0.6
0.8
1
1.2
1.4
1.6
LBGK
Extended LBM
LC LBM
Figure A2.
Comparison of density profile for shock tube problem at density ratio
ρl/ρr=
3, after
500 iterations. Solid line: LBGK; dashed line: extended LBM (33); symbols: LC LBM [9].
Finally, it is interesting to note that, in the case of shock capturing, all the three models
are expected to behave similarly, given that their respective dissipation rates at the wave
number
κx=π
are close in value, see Figure A1. This observation is confirmed by the
simulation of a shock tube with the following initial condition,
(ρ,ux,T) = (ρl, 0, 1/3),x≤L/2,
(ρr, 0, 1/3),x>L/2, (A9)
with
L=
800 grid points and viscosity
ν=
0.04. Results are presented in
Figures A2 and A3
for
ρl=
1.5,
ρr=
0.5, corresponding to the initial density ratio
ρl/ρr=
3.
Figures A2 and A3
demonstrate that all models produce almost indistinguishable results, with a similar oscil-
lation pattern at the shock front.
x
Ma
0 200 400 600 800
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure A3.
Mach number profile,
Ma =u/√TL
, for the shock tube problem at density ratio
ρl/ρr=
3,
after 500 iterations. Solid line: LBGK; dashed line: extended LBM (33); symbols: LC LBM [9].

Entropy 2021,23, 475 22 of 23
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