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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.

1

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Equation to Simulate Lattice-Gas Automata”, Physical

Review Letters 61, 2332–2335 (1988).

2

D. Haydock and J. M. Yeomans, “Lattice Boltzmann sim-

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(2001).

3

D. Haydock and J. M. Yeomans, “Lattice Boltzmann simu-

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A 36, 5683–5694 (2003).

4

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Physics A 33, 3917–3928 (2000).

5

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Greated, “Numerical simulation of particle motion in an

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6

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(2007).

7

B. R. Kucinschi, A. A. Afjeh, and R. C. Scherer, “On the

application of the lattice Boltzmann method to the inves-

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(2008).

8

J. Latt, “Hydrodynamic limit of lattice Boltzmann equa-

tions”, Ph.D. thesis, University of Geneva (2007).

9

E. M. Viggen, “The lattice Boltzmann method with appli-

cations in acoustics”, Master’s thesis, Norwegian Univer-

sity of Science and Technology (2009).

10

B. Chopard, P. O. Luthi, and J.-F. Wagen, “Lattice

Boltzmann method for wave propagation in urban micro-

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11

L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders,

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12

D. T. Blackstock, “Transient solution for sound radiated

into a viscous ﬂuid”, J. Acous. Soc. Am. 41, 1312–1319

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13

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to Model and Simulate Complex Systems”, Advances in

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14

J. Latt, “Choice of units in lattice Boltzmann simulations”,

Freely available online at http://lbmethod.org/_media/

howtos:lbunits.pdf.

15

“Palabos benchmarks”, URL http://www.lbmethod.org/

palabos/benchmarks.html.

33rd Scandinavian Symposium on Physical Acoustics The lattice Boltzmann method in acoustics 5