The lattice Boltzmann method in acoustics

Abstract

The lattice Boltzmann method, a method based in kinetic theory and used for simulating fluid behaviour, is presented with particular regard to usage in acoustics. A point source method of generating acoustic waves in the computational domain is presented, and simple simulation results with this method are analysed. The simulated waves' transient wavefronts in one dimension are shown to agree with analytical solutions from acoustic theory. The phase velocity and absorption coefficients of the waves and their deviations from theory are analysed. Finally, the physical time and space steps relating simulation units with physical units are discussed and shown to limit acoustic usage of the method to small scales in time and space.

Full text

The lattice Boltzmann method in acoustics
Erlend Magnus Viggen
Acoustics Group, Department of Electronics and Telecommunications, NTNU, O.S. Bragstads Plass 2,
7034 Trondheim, Norway
(Dated: Presented at SSPA10 9 February 2010)
The lattice Boltzmann method, a method based in kinetic theory and used for simulating fluid
behaviour, is presented with particular regard to usage in acoustics. A point source method of
generating acoustic waves in the computational domain is presented, and simple simulation results
with this method are analysed. The simulated waves’ transient wavefronts in one dimension are
shown to agree with analytical solutions from acoustic theory. The phase velocity and absorption
coefficients of the waves and their deviations from theory are analysed. Finally, the physical time and
space steps relating simulation units with physical units are discussed and shown to limit acoustic
usage of the method to small scales in time and space.
I. INTRODUCTION
We can subdivide numerical methods in acoustics into
two main types:
Top-down methods which attempt to find a solution
to a set of differential equations by different meth-
ods of discretisation (e.g. finite difference and finite
element methods).
Bottom-up methods which compute microscale algo-
rithms which give rise to macroscale physical be-
haviour (e.g. cellular automata, the TLM method,
lattice gas automata, lattice Boltzmann method).
The lattice Boltzmann method is of the latter type,
being based in mesoscopic kinetic theory and used for
simulations of macroscopic behaviour. It can be seen
as either an evolution of the older method of lattice gas
automata, or as a discretisation of the Boltzmann equa-
tion, which is a fundament of kinetic theory. The method
works by tracking the movement and interaction of par-
ticle distributions inside the computational domain.
Since its beginning in 1988,
1
the method has been used
for a number of acoustically related purposes, includ-
ing simulations of acoustic streaming,
2,3
shock fronts,
4
particle motion in ultrasound fields,
5
flow simulations in
reed instrument mouthpieces,
6
and investigation of glot-
tal flow.
7
II. LATTICE BOLTZMANN BACKGROUND
A d-dimensional domain is discretised by an evenly
spaced lattice, defined by q different lattice vectors c
i
,
which also represent possible particle velocities. Each
node contains q different particle distributions f
i
(x, t),
each representing the density of particles at position x
and time t with velocity c
i
.
From these particle distributions, we can find each
node’s macroscopic quantities of density and momentum.
The density ρ is found by summing the particle distribu-
tions at a node,
ρ(x, t) =
X
i
f
i
(x, t), (1)
(a)D1Q3 (b)D2Q9
(c)D3Q19
FIG. 1. Velocity vectors in different lattices. Rest vectors
c
0
= 0 are not indicated.
while momentum is found by summing the individual mo-
menta of each particle distribution,
ρ(x, t)u(x, t) =
X
i
c
i
f
i
(x, t), (2)
where u is particle velocity.
A lattice is characterised by its number of dimensions
d and its number of velocity vectors q, and is denoted
by DdQq. Fig. 1 and Table I show the different velocity
vectors in the common D1Q3, D2Q9, and D3Q19 lattices.
The method’s time step update is described by the
lattice Boltzmann evolution equation,
f
i
(x + c
i
, t + 1) = f
i
(x, t) + Ω
i
(x, t). (3)
This equation states that incoming particle distributions
f
i
(x, t) are streamed to neighbouring nodes in their di-
rection of velocity, after being modified by the collision
33rd Scandinavian Symposium on Physical Acoustics The lattice Boltzmann method in acoustics 1

TABLE I. Basic velocity vectors and their weighting factors
for the three lattices used in this article. The complete set of
vectors is all possible spatial permutations of the basic vectors
given here.
(a) D1Q3
c
i
t
i
(0) 2/3
(±1) 1/6
(b) D2Q9
c
i
t
i
(0, 0) 4/9
(±1, 0) 1/9
(±1, ±1) 1/36
(c) D3Q19
c
i
t
i
(0, 0, 0) 1/3
(±1, 0, 0) 1/18
(±1, ±1, 0) 1/36
operator Ω
i
(x, t). This collision operator represents col-
lisions by redistributing particles between distributions
in such a way that conservation of mass and momentum
is upheld.
The simplest and most common collision operator is
the lattice Bhatnagar-Gross-Krook (LBGK) operator,
Ω
i
= −
1
τ
[f
i
− f
eq
i
] , (4)
which is based on relaxation to an equilibrium distri-
bution f
eq
i
(x, t) with a relaxation time τ . We have here
simplified our notation by considering (x, t) implicit. The
equilibrium distribution is given by
f
eq
i
= ρt
i

1 +
u · c
i
c
2
s
+
(u · c
i
)
2
2c
4
s
−
u
2
2c
2
s

, (5)
which is the particle distribution which maximises en-
tropy, in analogy with the Maxwell-Boltzmann distribu-
tion in gases. The equilibrium distribution is constructed
from the node’s pre-collision values of ρ and u. c
s
is the
lattice speed of sound, which is equal to 1/
√
3 for the lat-
tices given in Table I and Figure 1, while t
i
is a weighting
factor which ensures that the lattices satisfy certain sym-
metry properties necessary for isotropic behaviour.
8
The
values of t
i
are given in Table I for each lattice.
The macroscopic behaviour of the lattice Boltzmann
method can be found by a procedure known as Chapman-
Enskog analysis, where a multiple-scale analysis of the
Taylor expansion of equation 3 is performed.
8,9
This anal-
ysis is performed under the assumptions of low excursions
from equilibrium (all terms of order Ma
3
or higher are ne-
glected), and the fluid being an isothermal perfect gas.
It results in the equation of continuity and the compress-
ible Navier-Stokes equation, with kinematic shear and
bulk viscosities ν and ν
0
given directly by the relaxation
time τ as
ν = c
2
s

τ −
1
2

, (6a)
ν
0
=
2
3
ν. (6b)
Because of the assumptions taken, these results are only
valid in the isothermal limit of low Mach number.
Since the method results in a macroscopic behaviour
consistent with the compressible Navier-Stokes equation
under these constraints it should be possible to use for
simulations of acoustics, since acoustic particle velocities
tend to be very small compared to the speed of sound,
and since the isothermal perfect gas approximation holds
for small acoustic compressions and rarefactions.
In fluid mechanics, the lattice Boltzmann method has
largely been used to simulate incompressible flow. It has
therefore been associated with incompressible flow simu-
lations, and there has unfortunately been relatively little
research into wave propagation with the method.
III. POINT SOURCE
The author’s method of researching acoustic wave
propagation with the lattice Boltzmann method has been
to perform benchmark comparisons with acoustic theory,
where acoustic behaviour with known theoretical solu-
tions is simulated with the lattice Boltzmann method.
Since wave propagation usually involves some sort of
wave source, it is necessary to use a point source to per-
form these kinds of simulation. Unfortunately, no arti-
cles describing point sources for the lattice Boltzmann
method with the BGK collision operator could be found.
Only related concepts have been developed so far.
3,10
The initial attempt by the author at creating a point
source for the lattice Boltzmann method works by defin-
ing one or more nodes as source nodes, where the particle
density ρ is forced to oscillate around an equilibrium den-
sity ρ
0
,
ρ = ρ
0
+ ρ
src
sin

2π
T
t

. (7)
In the simplest kind of free-field simulation, a point
source is placed in the middle of a periodic 1D, 2D, or
3D system at equilibrium and is left to propagate waves
outwards until the first wavefront approaches the bound-
ary. The waves are then measured as a function of dis-
tance from the point source. The 2D system is sketched
in Figure 2, and results in one to three dimensions are
shown in Figure 3.
x
y
r
FIG. 2. Sketch for the two-dimensional case of the periodic
system used in free-field simulations.
In Figure 3, the numerical results have been matched
with modified (i.e. scaled and phase-shifted) steady-state
analytical solutions of the system.
11
As can be seen, there
is a very good visual match between the wave shape in
33rd Scandinavian Symposium on Physical Acoustics The lattice Boltzmann method in acoustics 2

r
∆ρ/ρ
0
0
20
40 60 80 100 120
140
−4
−2
0
2
4
×10
−7
(a)D1Q3 lattice
r
√
r
∆ρ/ρ
0
020406080
100
120 140
−1
0
1
×10
−7
(b)D2Q9 lattice
r
r∆ρ/ρ
0
020406080
100
120 140
−5
0
5
×10
−8
(c)D3Q19 lattice
FIG. 3. Non-steady-state lattice Boltzmann solution of the radial field from a free-field point source in a viscous medium at
t = 200 (circles), along with steady-state analytical solution of the same (lines), in one-, two- and three-dimensional lattices
with τ = 0.75. The point source used has a period T = 25 and an amplitude ρ
src
/ρ
0
= 10
−6
. Note that in the two- and
three-dimensional lattices, the solutions are scaled with
√
r and r respectively, for visibility.
theory and simulation, and the rate of viscous damping
seems correct.
Some problems with the point source were also found
from these simulations. First, the radiated waves from
the source do not have the expected amplitude compared
with the source’s amplitude. Also, in two and three di-
mensions (Figures 3(b) and 3(c)), there is some unex-
pected chaotic behaviour near the source — the neigh-
bouring points are out of phase with the source, resulting
in a propagated wave which has been phase-shifted com-
pared to the source. These factors are the reasons for
the aforementioned scaling and phase shift when match-
ing analytical and simulated results.
The peaks which are visible around the first wavefronts
around r = 116 in Figure 3 are not errors. When sound is
radiated into a viscous fluid, a slowly decaying transient
occurs around the first wavefront. This was described
analytically in 1D by Blackstock in 1967,
12
and the sim-
ulated wavefront is compared with his analytical solu-
tion in Figure 4. As we can see, the agreement is very
good. This illustrates the power of the lattice Boltzmann
method to naturally simulate acoustic behaviour beyond
33rd Scandinavian Symposium on Physical Acoustics The lattice Boltzmann method in acoustics 3

r
∆ρ/ρ
0
80 100 120
140
−4
−2
0
2
4
6
8
×10
−8
FIG. 4. First wavefront from Fig. 3(a) (circles), with ana-
lytical solution (Blackstock (1967)) of such a wavefront for a
viscous medium (line).
the standard wave equation.
IV. DEVIATIONS IN PHASE VELOCITY AND
ABSORPTION
The point source allows us to examine if spatially
damped waves (i.e. waves which originate at a point and
are affected by viscous absorption) in the lattice Boltz-
mann method behave as theoretically expected. By set-
ting up a one-dimensional simulation such as the one de-
scribed in the previous section we can measure the phase
velocity and absorption coefficient of the outgoing waves.
An acoustic wave where the absorption is only affected
by the viscous relaxation time
τ
S
=
1
c
2
s

4
3
ν + ν
0

= 2τ − 1 (8)
can be shown to have a phase velocity of
c
p
= c
s

1 +
3
8
(ωτ
S
)
2

(9)
and an absorption coefficient of
α =
ω
2c
s
(ωτ
S
) (10)
in the limit of ωτ
S
 1.
11
If we define a characteristic number
K =
ωτ
S
2π
(11)
we see that the phase velocity c
p
should be constant if
K is held constant. We also see that the quantity αT /K
always should be constant.
Figure 5 shows the measured phase velocity as a func-
tion of λ
−2
in four cases where K was held constant while
T was varied, while Figure 6 shows αT/K as a function
of λ
−2
in the same cases. With both the phase velocity
and absorption, there is a deviation from theory of order
O(k
2
+ K
2
), so that the simulations agree with theory in
the limit k → 0, K → 0.
The results of a regression analysis in λ
−2
and K
2
on
the results can be used to predict the values of c
p
and
λ
−2
c
p
01
2
3
×10
−3
0.575
0.576
0.577
0.578
0.579
FIG. 5. Measured values of c
p
(gray circles), with predictions
based on a polynomial regression (black lines) for four dif-
ferent characteristic numbers. In ascending order, these lines
represent K = 1 × 10
−3
, 5 × 10
−3
, 8 × 10
−3
, 1 × 10
−2
.
λ
−2
αT /K
0
1
23
×10
−3
34
34.2
34.4
34.6
34.8
FIG. 6. Measured values of αT /K (gray circles), with predic-
tions based on a polynomial regression (black lines). In de-
scending order, these lines represent K = 1 ×10
−3
, 5 ×10
−3
,
8 × 10
−3
, 1 × 10
−2
.
α a priori with an accuracy of respectively 0.001 55 %
and 0.0176 % compared to the measurements performed
here. The lines in Figures 5 and 6 are predictions based
on these regressions.
V. UNIT CONSIDERATIONS
We have hitherto used lattice units, where the space
and time steps have been normalised to 1. The lattice
units can be related to a physical system through the
physical time step ∆t and physical space step ∆x. The
speed of sound and viscosity are related in lattice and
physical units through
c
s,phy
= c
s,lat
∆x
∆t
, (12a)
ν
phy
= ν
lat
∆x
2
∆t
. (12b)
33rd Scandinavian Symposium on Physical Acoustics The lattice Boltzmann method in acoustics 4

Inserting for the known lattice speed of sound and vis-
cosity and solving for ∆t and ∆x, we find
∆t =
ν
phy
c
2
s,phy
(τ − 1/2)
, (13a)
∆x =
ν
phy
√
3
c
s,phy
(τ − 1/2)
. (13b)
Inserting physical values for air at 20
◦
C, this becomes
∆t =
1.30 × 10
−10
τ − 1/2
s, (14a)
∆x =
7.73 × 10
−8
τ − 1/2
m, (14b)
which shows that the time and space steps are restricted
to orders of respectively ns and µm, since τ − 1/2 can-
not be arbitrarily close to zero for accuracy and stability
reasons
13
.
This problem does not occur when using the lattice
Boltzmann method to simulate incompressible flow, since
in that case the speed of sound is not a physically rele-
vant quantity. This affords considerably more freedom in
choosing units.
14
VI. CONCLUSION
The lattice Boltzmann method has several desirable
properties, in particular an inherent simplicity. The al-
gorithm is easy to implement and use, and complex ge-
ometries of no-slip hard walls can be implemented simply
by letting wall nodes bounce incoming particles back to
their point of origin. Since each time step update is local
to each node, the computational domain can be divided
into subdomains handled by different processors, and it
is possible to achieve a near-linear speedup in number of
processors.
15
It has been shown here that waves simulated with the
lattice Boltzmann method suffer from deviations from
theory of order O(k
2
+ (ωτ
S
)
2
) in phase velocity and
absorption, at least with the BGK collision operator. It
may be possible to solve this problem by changing to a
more advanced collision operator or by using an extended
lattice.
Unfortunately, acoustic use of the lattice Boltzmann
method seems to be limited to very high frequencies and
very small geometries, limiting the area of usage to ul-
trasound microacoustics in fluids. Hopefully this can be
rectified, but the manner in which this might be done is
not yet clear to the author.
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33rd Scandinavian Symposium on Physical Acoustics The lattice Boltzmann method in acoustics 5