Generalized Maxwell state and H theorem for computing fluid flows
using the lattice Boltzmann method
Pietro Asinari
1
and Ilya V. Karlin
2,3
1
Department of Energetics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
2
Institute of Energy Technology, ETH Zurich, 8092 Zurich, Switzerland
3
School of Engineering Sciences, University of Southampton, SO17 1BJ Southampton, United Kingdom
Received 9 September 2008; revised manuscript received 17 February 2009; published 17 March 2009
Generalized Maxwell distribution function is derived analytically for the lattice Boltzmann LB method. All
the previously introduced equilibria for LB are found as special cases of the generalized Maxwellian. The
generalized Maxwellian is used to derive a different class of multiple relaxation-time LB models and prove the
H theorem for them.
DOI: 10.1103/PhysRevE.79.036703 PACS numbers: 47.11.-j, 05.20.Dd
A branch of kinetic theory—the lattice Boltzmann LB
method—has recently met with a remarkable success as a
powerful alternative for solving the hydrodynamic Navier-
Stokes equations, with applications ranging from large Rey-
nolds number flows to flows at a micron scale, porous media,
and multiphase flows, see, e.g., 1–3 and references therein.
The LB method solves a fully discrete kinetic equation for
populations f
x , t, designed in a way that it reproduces the
Navier-Stokes equations in the hydrodynamic limit in D di-
mensions. Populations correspond to discrete velocities v
for =0,1,..., Q - 1, which fit into a regular spatial lattice
with the nodes x. This enables a simple and highly efficient
algorithm based on a nodal relaxation and b streaming
along the links of the regular spatial lattice. On the other
hand, numerical stability of the LB method remains a critical
issue 2. Recalling the role played by the Boltzmann H theo-
rem in enforcing macroscopic evolutionary constraints the
second law of thermodynamics, pertinent entropy functions
have been proposed 4 –8. The full connection of LB to
kinetic theory was established by the discrete-velocity analog
of the Maxwellian see Eq. 2 below.
Admittedly, however, other heuristic methods were pro-
posed recently to enhance stability of LB. The rationale be-
hind one of them, the multiple relaxation time MRT9–11,
is sound. Since the incompressible flow is the only concern,
the bulk viscosity arising in the quasicompressible LB
scheme can be viewed as a free parameter and tuned in order
to enhance stability. However, in spite of popularity of the
MRT method, to date, it cannot be considered as a consistent
kinetic theory but rather a numerical trick where tuning of
parameters is based on experience rather than on physics.
In this paper, we present a different consideration of the
LB models and derive a crucial result: the closed-form gen-
eralized equilibrium see Eq. 3 below. The generalized
equilibrium is the analog of the anisotropic Gaussian and is a
long-needed relevant distribution in the LB method. This
finding further allows us to introduce an innovative class of
entropy-based MRT LB models which enjoy both the H
theorem and the additional free-tunable parameter for con-
trolling the bulk viscosity, where the range is dictated by the
entropy.
For the sake of presentation and without any loss of gen-
erality, we consider the popular nine-velocity model, the so-
called D2Q9 lattice, of which the discrete velocities are v
0
= 0,0 and v
i
= c ,0 and 0, c for i =1-4, and v
i
= c , c for i =5-8 12 where c is the lattice spacing.
Recall that the D2Q9 lattice derives from the three-point
Gauss-Hermite formula 13 with the following weights of
the quadrature w-1 =1 / 6, w0 =2 / 3, and w+1 =1 / 6. Let
us arrange in the list v
x
all the components of the lattice
velocities along the x axis and similarly in the list v
y
. Analo-
gously let us arrange in the list f all the populations f
.
Algebraic operations for the lists are always assumed com-
ponent wise. The sum of all the elements of the list p is
denoted by p =
i=0
Q-1
p
i
. The dimensionless density , the
flow velocity u, and the second-order moment pressure ten-
sor are defined by = f , u
i
= v
i
f , and
ij
= v
i
v
j
f ,
respectively.
On the lattice under consideration, the convex entropy
function H function is defined as 5
H f = f ln f /W , 1
where W = wv
x
wv
y
. The H-function minimization prob-
lem is considered in the sequel. It is well known 5 that the
equilibrium population list f
M
is defined as the solution of
the minimization problem f
M
= min
f P
M
H f , where P
M
is
the set of functions such that P
M
= f 0: f = , v f = u.
In other words, minimization of the H function Eq. 1 under
the constraints of mass and momentum conservation yields
6
f
M
=
=x,y
wv
2- u
/c
2u
/c + u
/c
1- u
/c
v
/c
,
2
where z =
3z
2
+ 1. A remarkable feature of equilibrium
2 which it shares with the ordinary Maxwellian is that it is
a product of one-dimensional equilibria. In order to ensure
the positivity of f
M
, the low Mach number limit must be
considered, i.e., |u
| c.
In this paper, we derive a different constrained equilib-
rium, or quasiequilibrium 14, by requiring, in addition, that
the diagonal components of the pressure tensor have some
prescribed values. Hence let us introduce a different minimi-
zation problem. The quasiequilibrium population list f
G
is
PHYSICAL REVIEW E 79, 036703 2009
1539-3755/2009/793/0367035 ©2009 The American Physical Society 036703-1