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Acoustic multipole sources for the lattice Boltzmann method

Erlend Magnus Viggen∗

Department of Electronics and Telecommunications, NTNU, 7034 Trondheim, Norway

By including an oscillating particle source term, acoustic multipole sources can be implemented

in the lattice Boltzmann method. The eﬀect of this source term on the macroscopic conservation

equations is found using a Chapman-Enskog expansion. In a lattice with qparticle velocities, the

source term can be decomposed into qorthogonal multipoles. More complex sources may be formed

by superposing these basic multipoles. Analytical solutions found from the macroscopic equations

and an analytical lattice Boltzmann wavenumber are compared with inviscid multipole simulations,

ﬁnding very good agreement except close to singularities in the analytical solutions. Unlike the

BGK operator, the regularized collision operator is proven capable of accurately simulating two-

dimensional acoustic generation and propagation at zero viscosity.

PACS numbers: 47.11.-j, 43.20.Rz

I. INTRODUCTION

The lattice Boltzmann (LB) method is a relatively re-

cent advance in computational ﬂuid dynamics which dif-

fers from traditional methods in that it solves the equa-

tions of ﬂuid dynamics indirectly using a straightforward

discretization of the Boltzmann equation [1]. It has re-

cently also been applied to practical cases in acoustics [2–

4] and aeroacoustics [5–7].

This article describes a new method to generate

acoustic multipole sources in LB simulations. Acous-

tic monopole point sources have previously been imple-

mented by completely replacing the particle distribution

in a node with an equilibrium distribution having a spec-

iﬁed oscillating density [8–13]. In this article we take

a diﬀerent approach by adding a particle source term

to the LB equation. Unlike the previous method, the

new method does not unphysically disturb the under-

lying ﬂow, and it also allows dipoles, quadrupoles, and

complex multipole superpositions.

II. LATTICE BOLTZMANN WITH

OSCILLATING SOURCE TERM

The LB method works by evolving the distribution

function fi(x, t)on a square numerical grid. This func-

tion represents the density of particles with position x

and velocity ξiat time t. The velocities ξiand their asso-

ciated distribution functions fiare restricted to a discrete

set. From the moments of fiwe can ﬁnd the macroscopic

quantities of density ρ(x, t) = Pifi(x, t)and momentum

density ρu(x, t) = Piξifi(x, t).

fiis evolved using the lattice Boltzmann equation,

fi(x+ξi, t + 1) = fi(x, t)+Ωi(x, t) + si(x, t),(1)

∗erlendmagnus.viggen@sintef.no;

This post-print was published in Physical Review E 87, 023306

(2013), DOI: 10.1103/PhysRevE.87.023306

where siis the aforementioned particle source term, and

Ωiis a collision operator. Most common is the BGK

operator, which relaxes fito an equilibrium

f(0)

i=ρwi1 + ξi·u

c2

0

+(ξi·u)2

2c4

0−u·u

2c2

0(2)

with a relaxation time τ;

Ωi=−1

τfi−f(0)

i.(3)

In Eq. (2), c0is the inviscid speed of sound and wiis a

set of weighting coeﬃcients; both depend on the choice

of velocity set ξi.

This article primarily uses the regularized collision op-

erator [14], which behaves similarly to BGK but sup-

presses nonhydrodynamic moments of fiby relaxing

them to equilibrium in each time step. It may thus

be seen as a conceptually simple, eﬃcient, and gener-

ally applicable multiple relaxation time operator which

improves on the accuracy and stability of the BGK op-

erator, in particular at low viscosities. The regularized

collision operator is given by

Ωi=(1 −1

τ)wi

2c4

0X

j

(ξiαξiβ −c2

0δαβ)ξjα ξjβ fneq

j−fneq

i,

(4)

where fneq

i=fi−f(0)

i. In this notation, Greek in-

dices indicate vector or tensor components, and repeated

Greek indices in a term imply a summation (i.e. aαbα=

Pαaαbα). δαβ is the Kronecker delta.

The source term’s eﬀect on the conservation equations

can be analyzed using a Taylor and Chapman-Enskog

expansion [15]. Because Eqs. (3) and (4) have the same

hydrodynamic moments [14], performing the analysis us-

ing Eq. (3) gives results which are valid for both. fiis

expanded around f(0)

iin orders of the Knudsen number,

and siand the derivatives are also similarly expanded;

fi=f(0)

i+f(1)

i+2f(2)

i+. . . , si=si,

∂t=∂t1+2∂t2, ∂α=∂α.

2

Here, is an expansion parameter indicating the order

of the Knudsen number. After inserting Eq. (3) into

Eq. (1), Taylor expanding, collecting terms according to

their Knudsen number order, and performing some alge-

bra, we ﬁnd

(∂t1+∂αξiα)f(0)

i=−1

τf(1)

i+si(5)

at O(), and

∂t2f(0)

i+ (∂t1+∂αξiα)1−1

2τf(1)

i

=−1

τf(2)

i−1

2(∂t1+∂αξiα)si

(6)

at O(2). Then we deﬁne the monopole, dipole, and

quadrupole moments of sias

S0=X

i

si, Sα=X

i

ξiαsi, Sαβ =X

i

ξiαξiβ si.(7)

Finally, combining the diﬀerent moments of Eqs. (5)

and (6) [1, 15], we end up with a modiﬁed mass con-

servation equation

∂ρ

∂t +∂ρuα

∂xα

=S0−1

2∂S0

∂t +∂Sα

∂xα,(8)

and a modiﬁed momentum conservation equation

∂ρuα

∂t +∂ρuαuβ

∂xβ

+∂p

∂xα−∂

∂xβ

ν∂uβ

∂xα

+∂uα

∂xβ

=1−1

2

∂

∂t Sα−τ∂Sαβ

∂xβ

(9)

+τ−1

2∂

∂xβc2

sδαβS0+uαSβ+uβSα−uαuβS0.

Here we have a pressure p=c2

0ρ, a kinematic shear vis-

cosity ν= (τ−1

2)c2

0ρ, and a bulk viscosity of 2ν/3. The

O(u3)error term [1] has been neglected.

An inhomogeneous linear wave equation can be derived

from these conservation equations. In the τ→1

2limit,

corresponding to the low viscosities commonly chosen for

LB acoustics [3, 5], this equation is

1

c2

0

∂2p0

∂t2− ∇2p0=1−1

2

∂

∂t ∂S0

∂t −∂Sα

∂xα

+τ∂2Sαβ

∂xα∂xβ

,

(10)

where the acoustic pressure p0(x, t) = p(x, t)−p0is the

pressure’s deviation from a rest state. Unlike when this

equation is derived in the continuous Boltzmann equa-

tion case [16], Sαβ does not fully disappear with viscos-

ity. This is a fortuitous consequence of the discretization

error inherent in the lattice Boltzmann scheme.

In the simpliﬁed case of a time-harmonic source term

with angular frequency ω, the solution to this equation

can be found using standard methods [17] to be

p0(x, t) = ReZiω −(iω)2

2S0(y)G(x−y, t)

−Sα(y)∂G(x−y, t)

∂xα

(11)

+τSαβ (y)∂2G(x−y, t)

∂xα∂xβdy,

where G(x, t)is the time-harmonic Green’s function. The

three terms in the integral represent monopoles, dipoles,

and quadrupoles, respectively. For the two-dimensional

case used in the following sections,

G(x, t) = 1

4iH(2)

0(k|x|)eiωt,(12)

where H(2)

nis the nth order Hankel function of the second

kind and k=ω/c0is the wavenumber [18].

However, waves in LB simulations have non-ideal

wavenumbers, due to discretization errors and viscous

eﬀects [19, 20]. It has been shown that linear LB wave

propagation can be formulated as an eigenvalue prob-

lem [1, 19]. From the corresponding characteristic poly-

nomial,

1

3

e−iˆ

k(3−1/τ)e−iˆ

k/2τ−e−iˆ

k/τ

2/τ 3−1/τ 2/τ

−eiˆ

k/τ eiˆ

k/2τ eiˆ

k(3−1/τ)

−eiωI

= 0,

an analytical LB wavenumber including the aforemen-

tioned eﬀects can be found to be

ˆ

k=ilnh3τ(ζ2−ζ+ 1 −ζ−1) + ζ−2+3ζ−1

+√3ζ−1√Ξi.h4+6τ(ζ−1) −2ζi,

(13)

where the shorthand ζ=eiω has been used, and

Ξ=(ζ+ 1)(ζ−1)2(τζ + 1 −τ)(3τ ζ2−ζ+ 3 −3τ).

This wavenumber is used in this article when comparing

analytical and numerical solutions.

III. MULTIPOLE BASIS

For a velocity set with qvelocities, sican be seen

as a q-dimensional vector which can be found from

aq-dimensional orthogonal basis Mjas si(x, t) =

Aij Mj(x, t).Aij can be chosen so that each compo-

nent of Mjrepresents the strength of a particular multi-

pole. As all LB velocity sets are symmetric and have

an odd number of velocities, one reasonable choice is

to have one monopole in addition to (q−1)/2pairs of

oddly symmetric dipoles and evenly symmetric longitudi-

nal quadrupoles; one such pair for each pair of opposing

velocities ξi.

3

For the two-dimensional D2Q9 lattice, where

ξi=

0,0for i= 0,

sin[i−1

2π],cos[i−1

2π]for i= 1–4,

√2sin[2i−1

4π],cos[2i−1

4π]for i= 5–8,

wi=

4/9for i= 0,

1/9for i= 1–4,

1/36 for i= 5–8,

(14)

and c0= 1/√3, the source term can be decomposed in

this way as

s0

s1

s2

s3

s4

s5

s6

s7

s8

=

w00 0 −1−1 0 0 −1

2−1

2

w1

1

201

20 0 0 0 0

w201

201

20 0 0 0

w3−1

201

20 0 0 0 0

w40−1

201

20 0 0 0

w50 0 0 0 1

√801

40

w60 0 0 0 0 1

√801

4

w70 0 0 0 −1

√801

40

w80 0 0 0 0 −1

√801

4

M0

Mx

My

Mxx

Myy

Mx0

My0

Mx0x0

My0y0

.

(15)

The monopole has been chosen so that particles are

added at equilibrium. An x0–y0coordinate system, ro-

tated π/4to the x–yone, has been deﬁned for the diag-

onal dipoles and quadrupoles. Table I shows how these

nine multipoles map onto the moments deﬁned in Eq. (7).

It indicates that lateral quadrupoles can be made by

superposition of the diagonal longitudinal quadrupoles:

subtracting My0y0from Mx0x0and normalizing.

TABLE I. Nonzero moments of the D2Q9 basis multipoles Mj

M0MxMyMxx Myy Mx0My0Mx0x0My0y0

S01

Sx11

√2−1

√2

Sy11

√2

1

√2

Sxx c2

011

2

1

2

Syy c2

011

2

1

2

Sxy

1

2−1

2

Syx

1

2−1

2

IV. NUMERICAL VERIFICATION

To determine the correctness of the radiated ﬁelds of

these multipoles, a multipole point source was placed at

x= 0 in a system originally at rest with a density ρ0.

The simulated radiated ﬁeld was compared with the cor-

responding analytical solution for several representative

multipoles. The simulations presented in this article were

performed at zero viscosity, i.e. τ=1

2. To avoid ripple

caused by sudden onset of the source, the source’s am-

plitude was multiplied with an envelope function

E(t) =

0for t≤0,

1

2−1

2cos(ωt/2) for 0≤t≤2π/ω,

1for 2π/ω ≤t.

(16)

The point source was left to radiate waves until the ﬁrst

wavefront neared the edge of the simulated system, at

which point the simulation was stopped and the results

were compared with the analytical solution.

To avoid nonlinearities aﬀecting the results, the LB

method was linearized by removing the O(u2)terms in

Eq. (2). As the resulting dynamics and macroscopic

equations are linear, this allows the use of a complex

phasor source, which in turn allows a more simple and

accurate analysis. The real part of the complex radiated

ﬁeld represents its physical value, and the magnitude rep-

resents its amplitude.

For a two-dimensional complex time-harmonic mul-

tipole point source, si(x, t) = siδ(x)eiωt , at τ=1

2,

Eq. (11) becomes

p0(x, t) = iωS0G(x, t)−Sα

∂G(x, t)

∂xα

+1

2Sαβ −c2

0δαβS0∂2G(x, t)

∂xα∂xβ

,

(17)

because from Eq. (12),

(iω)2G(x, t) = −ω2

4iH(2)

0(k|x|)eiωt =ω

k2∂2G(x, t)

∂xα∂xα

.

The three right-hand terms in Eq. (17) are only af-

fected by monopole, dipole, and quadrupole multipole

strengths, respectively; Table I indicates that the two

components of the third term cancel for M0.

The waves radiated from three representative multi-

poles, simulated with both the BGK and regularized col-

lision operators at a source frequency ω= 2π/25, are

compared with the analytical solution from Eq. (17) in

Fig. 1. While neither collision operator is unstable for

this simulation, the BGK LB results are heavily aﬀected

by spurious oscillations, particularly for the higher-order

multipoles. On the other hand, the regularized LB re-

sults show no such oscillations, and are only signiﬁcantly

in error very close to the point source. At this point there

is a singularity in the analytical solution which cannot be

captured with any similar discrete simulation methods,

such as ﬁnite diﬀerence time domain methods. Similar

errors were also reported in two and three dimensions

for the previous LB monopole point source method [9].

Also, since we are comparing a non-steady state simu-

lated result and a steady-state analytical solution, there

is naturally a discrepancy near the ﬁrst wavefront; this

area is therefore not shown in Fig. 1.

The fundamental diﬀerence between the two collision

operators is that the BGK operator has a relaxation time

τfor all moments, while the regularized operator relaxes

4

−0.1

0

0.1

0.2

ρ0/M0

BGK LB Reg. LB Analytical

−0.2

−0.1

0

0.1

ρ0/Mx

0 10 20 30 40 50 60 70 80

−0.1

0

0.1

0.2

|x|

ρ0/Mxx

FIG. 1. (Color online) Physical acoustic density ρ0=

Re(p0/c2

0)normalized by by multipole strength Mjagainst

distance |x|to point source. Results were measured along the

x–axis. Top to bottom: Monopole, x–dipole, xx–quadrupole.

10−210−1

10−2

10−1

100

101

1/λ

kekq

O(1/λ2)

q= 1

q= 2

FIG. 2. 1- and 2-norm of the relative error of the monopole

pressure amplitude, found by Eq. (18), compared with second

order convergence.

nonhydrodynamic moments to equilibrium in each time

step [14]. At τ=1

2, the BGK operator fully overrelaxes

nonhydrodynamic moments in such a way that their am-

plitude does not decrease. By increasing τin these sim-

ulations, the spurious oscillations in the BGK results are

reduced and localized to the region around the source.

This indicates that the diﬀerence shown in Fig. 1 be-

tween the two operators is caused by nonhydrodynamic

moments generated in the source node. The regularized

collision operator at τ=1

2is used exclusively in the re-

mainder of this article.

In a region well away from the source and the ﬁrst

wavefront, the q-norm of the relative error [21] of the

monopole pressure amplitude is

kekq=

1

λ2

3λ

X

|x|=λ

|p0∗|−|p0|

|p0|

q

1/q

,(18)

where λ= 2πc0/ω is the wavelength, p0is the analyt-

ical solution, and p0∗ is the simulated solution. The 1-

90

120

150

180

210

240 270 300

330

360

30

60

0

0.5

1

+−

(a)

90

120

150

180

210

240 270 300

330

360

30

60

0

0.5

1

++

(b)

90

120

150

180

210

240 270 300

330

360

30

60

0

0.5

1

+−

+−

(c)

210

240 270 300

330

360

30

60

90

120

150

180

0

0.5

1

+

−

−

(d)

FIG. 3. (Color online) Directivity at k|x|= 25, normalized by

the analytical solution. Lobe phase is indicated by plus and

minus sign. Circles and line indicate numerical and analytical

solutions, respectively. (a) x–dipole. (b) xx–quadrupole. (c)

xy–quadrupole. (d) Rotated supercardioid.

and 2-norms were found for a number of diﬀerent resolu-

tions, and the results are shown as function of the spatial

resolution 1/λ in Fig. 2. The overall convergence of the

radiated wave is clearly seen to be second order; the same

order as the LB method itself.

The directivity at k|x|= 25 of the dipole and the longi-

tudinal and lateral quadrupoles, simulated at ω= 2π/50,

is shown in Fig. 3(a)–(c). In all three cases, there is an

excellent agreement between the numerical and analyti-

cal solutions.

The basic multipoles may be superposed to form more

complex ones. Rotation by an angle θmay be per-

formed by applying a rotation matrix αij =cos θ−sin θ

sin θcos θ

to a dipole vector Di=Sx

Sylike Drot

i=Pjαij Dj,

or to a quadrupole tensor Qij =Sxx Sxy

Syx Syy like Qrot

ij =

Pm,n αimαj nQmn . Figure 3(d) shows a rotated super-

cardioid formed by superposing a dipole and a longitu-

dinal quadrupole [22], both normalized to the same am-

plitude and rotated an angle θ=π/6. This composite

multipole is highly directive.

V. COMPARISON WITH PREVIOUS METHOD

A previous method for acoustic monopole point sources

within the LB domain [8–13] works by replacing the dis-

tribution function fiin the source node in each time step.

It is replaced by an equilibrium distribution f(0)

ideter-

mined by the velocity u, found from fi, and a speciﬁed

5

oscillating density

ρ=ρ0+ρsrc sin(ωt).(19)

Thus, the original density and all information contained

in fneq

iis overwritten and lost in the source node in each

time step. Extending the previous method for dipoles

is possible by making uan oscillating function of time.

However, extending it further would not be possible, as

the equilibrium distribution it depends on is fully deﬁned

by ρand u.

Unlike the current method described in this article,

there have never been found any expressions for the pre-

vious method, either regression- or theory-based, to re-

late the amplitudes and phases of the source node and

the radiated wave. Previous comparisons of the previous

method with theory have been done by ad hoc scaling

and phase shifting of the analytical solution [8, 9]. Thus,

we cannot make fair direct quantitative comparisons with

theory like in the previous section.

However, the previous method systematically gener-

ates errors in the source node. This is most clearly seen in

a limiting case where the monopole source strength goes

to zero (i.e. si→0and ρsrc →0) and there is a back-

ground ﬂow ﬁeld with a density ρ(x, t) = ρ0+ρ0(x, t).

For the current method, Eq. (1) shows that the eﬀect of

the source vanishes. For the previous method, however,

Eq. (19) shows that the ﬁxed source node density ρ0will

result in a relative error |ρ0/(ρ0+ρ0)|. This error will

propagate outward, aﬀecting the rest of the ﬂow ﬁeld.

VI. CONCLUSION

By adding an oscillating particle source term to the

LB equation, acoustic multipole sources may be imple-

mented. These may be either spatially distributed or

point sources. Comparing simulations of the ﬁelds ra-

diated by monofrequency point multipole sources with

the corresponding ﬁelds predicted by theory, very good

agreement is found except in the vicinity of the source

node, where there is a singularity in the analytical solu-

tion. Similar errors were also reported for a previous LB

monopole point source method [9].

In a lattice with qvelocities, we suggest using a orthog-

onal multipole basis of one monopole, (q−1)/2dipoles,

and (q−1)/2longitudinal quadrupoles. These funda-

mental multipoles may be superposed to form more com-

plex sources, such as the highly directive source shown in

Fig. 3(d). This could be useful for simulating cases with

directed emission of sound in a complex ﬂuid ﬂow.

Because acoustic LB simulations are commonly per-

formed at very low viscosities, i.e. τclose to 1

2[3, 5], the

simulations in this article were performed using the reg-

ularized collision operator at the inviscid limit τ=1

2.

The simulations prove that the regularized operator al-

lows stable and accurate zero-viscosity LB simulation of

some phenomena, at minimum acoustic generation and

propagation. For the same case, the BGK operator gives

results with very large spurious oscillations unless the

viscosity is increased. This is because it does not at all

when τ=1

2suppress the nonhydrodynamic moments

generated by the source node, unlike the regularized op-

erator which fully suppresses them.

It is worth noting that while monopoles, dipoles, and

quadrupoles appear as source terms in the wave equa-

tion, higher-order multipoles, such as octupoles, do not.

This is also the case when the wave equation is similarly

derived from the continuous Boltzmann equation [16].

However, it is likely that such higher-order multipoles

would appear in the momentum equation if its deriva-

tion were carried to the Burnett level, where the equa-

tion contains additional terms with derivatives of higher

order. Thus, octupoles etc. are likely also possible in LB

simulations, if the symmetries of the chosen velocity set

permit.

Comparison with the derivation based on the continu-

ous Boltzmann equation also shows that the quadrupole

strength here is nonzero in the inviscid limit due to a

fortuitous discretization error in the lattice Boltzmann

scheme. In the continuous case, quadrupoles disappear

in this limit.

ACKNOWLEDGMENTS

The author wishes to thank Paul J. Dellar for suggest-

ing how to ﬁnd Eq. (13).

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