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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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