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Acoustic multipole sources for the lattice Boltzmann method

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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 q particle velocities, the source term can be decomposed into q orthogonal 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.
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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
iis 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
0u·u
2c2
0(2)
with a relaxation time τ;
i=1
τfif(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
i=(1 1
τ)wi
2c4
0X
j
(ξξc2
0δαβ)ξ ξ fneq
jfneq
i,
(4)
where fneq
i=fif(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+2t2, ∂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+αξ)f(0)
i=1
τf(1)
i+si(5)
at O(), and
t2f(0)
i+ (t1+αξ)11
2τf(1)
i
=1
τf(2)
i1
2(t1+αξ)si
(6)
at O(2). Then we define the monopole, dipole, and
quadrupole moments of sias
S0=X
i
si, Sα=X
i
ξsi, Sαβ =X
i
ξξ 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
=S01
2∂S0
∂t +Sα
∂xα,(8)
and a modified momentum conservation equation
∂ρuα
∂t +ρuαuβ
∂xβ
+∂p
∂xα
∂xβ
ν∂uβ
∂xα
+∂uα
∂xβ
=11
2
∂t Sατ∂Sαβ
∂xβ
(9)
+τ1
2
∂xβc2
sδαβS0+uαSβ+uβSαuαuβS0.
Here we have a pressure p=c2
0ρ, a kinematic shear vis-
cosity ν= (τ1
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 τ1
2limit,
corresponding to the low viscosities commonly chosen for
LB acoustics [3, 5], this equation is
1
c2
0
2p0
∂t2− ∇2p0=11
2
∂t ∂S0
∂t Sα
∂xα
+τ2Sαβ
∂xαxβ
,
(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) = ReZ()2
2S0(y)G(xy, t)
Sα(y)∂G(xy, t)
∂xα
(11)
+τSαβ (y)2G(xy, t)
∂xαxβdy,
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
eiˆ
k(31)eiˆ
k/2τeiˆ
k
2312
eiˆ
k/τ eiˆ
k/2τ eiˆ
k(31)
eI
= 0,
an analytical LB wavenumber including the aforemen-
tioned effects can be found to be
ˆ
k=ilnh3τ(ζ2ζ+ 1 ζ1) + ζ2+3ζ1
+3ζ1Ξi.h4+6τ(ζ1) 2ζi,
(13)
where the shorthand ζ=ehas been used, and
Ξ=(ζ+ 1)(ζ1)2(τζ + 1 τ)(3τ ζ2ζ+ 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 (q1)/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[i1
2π],cos[i1
2π]for i= 14,
2sin[2i1
4π],cos[2i1
4π]for i= 58,
wi=
4/9for i= 0,
1/9for i= 14,
1/36 for i= 58,
(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 11 0 0 1
21
2
w1
1
201
20 0 0 0 0
w201
201
20 0 0 0
w31
201
20 0 0 0 0
w401
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 x0y0coordinate system, ro-
tated π/4to the xyone, 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
21
2
Sy11
2
1
2
Sxx c2
011
2
1
2
Syy c2
011
2
1
2
Sxy
1
21
2
Syx
1
21
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 t0,
1
21
2cos(ωt/2) for 0t2π/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
0δαβS02G(x, t)
∂xαxβ
,
(17)
because from Eq. (12),
()2G(x, t) = ω2
4iH(2)
0(k|x|)eiωt =ω
k22G(x, t)
∂xαxα
.
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.
102101
102
101
100
101
1
kekq
O(12)
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= 2πc0is 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 1in 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 θsin 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. si0and 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, (q1)/2dipoles,
and (q1)/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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... These coefficients being related to diffusive phenomena, they were supposed to be negligible for high Reynolds number flows. In the end, all the aforementioned regularized models were also employed to simulate a large panel of distinct phenomena [69,71,[73][74][75][76][77][78]. ...
... In parallel of the derivation of CM-LBMs, another type of collision models has been proposed to increase the numerical stability of the BGK-LBM without accounting for collisions in the moment space. Instead, these regularized collision models aim at filtering out nonhydrodynamic contributions that appear during the streaming step [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78]. This is done projecting the nonequilibrium VDF on Hermite polynomials up to a given order N. Hence, this regularized collision model can be summarized as ...
... In particular, this is true for models based on orthogonal bases (MRT and TRT), the regularized LBMs, and the K-LBM. For regularized LBMs, it is explained by the fact that in the one-dimensional case, either solving Eq. (74) or using the Gauss-Hermite quadrature formulation (64) leads to the very same formalism. Tensor product rules compiled in Eqs. ...
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Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. This article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows us to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant, or regularized form. In parallel with that, several bases (nonorthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post-collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. This work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of this article, general instructions are provided to help the reader with the implementation of the most complicated collision models.
... Based on the work performed by previous scholars [4][5][6], the lattice Boltzmann method gradually improved to be a trustable numerical methodology. It turned out that the lattice Boltzmann method is numerically capable of solving many mathematical and physical problems, including all sorts of partial differential equations [7][8][9][10], applications such as acoustic wave phenomena [11][12], multi-phase flows [13][14], combustion [15][16], fluid mechanics [17][18], and many others. ...
... According to different authors, the critical Reynolds number of the transitional flow from laminar steady to laminar unsteady in case S1 are expected to have the following values, 7500 [30,34], 6000-8000 [29,37,38], 8000 [31,32,33,36], 8000-8300 [11] and 8000-8051 [33,35,[39][40][41]. Koseff and Street [42,43] experimentally investigated the lid-driven cavity flow and determined the critical Reynolds number for the Hopf bifurcation was between 6000 and 8000. ...
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Wall-driven flow in square cavities has been studied extensively, yet it appears some main flow characteristics have not been fully investigated. Previous research on the classic lid-driven cavity (S1) flow has produced the critical Reynolds numbers separating the laminar steady and unsteady flows. Wall-driven cavities with two opposing walls moving at the same speed and the same (S2p) or opposite (S2a) directions have seldom been studied in the literature and no critical Reynolds numbers characterizing transitional flows have ever been investigated. After validating the LBM code for the three configurations studied, extensive numerical simulations have been undertaken to provide approximate ranges for the critical Hopf and Neimark-Sacker bifurcations for the classic and two two-sided cavity configurations. The threshold for transition to chaotic motion is also reported. The symmetries of the solutions are monitored across the various bifurcations for the two-sided wall driven cavities. The mirror-symmetry of the base solution for case S2p is lost at the Hopf bifurcation. The exact same scenario occurs with the pi-rotational symmetry of the base state for case S2a.
... 碰撞算子。它因形式简单、物理意义明朗而被广泛 采用。多种数学分析 [32][33] 显示Boltzmann方程是与 [33] 将公式(1)在 Hermit基上进行投影,推导获得任意阶平衡态分布函 数,本文采用四阶精度形式 [34][35] ...
... The azimuthal averaged acoustic emission is then reduced for more than 6dB compared to the I − 1 one. As a recall, a reduction of 3dB corresponds in dividing the sound intensity by a factor of 2. The I − 1 method generates a very strong dipole radiated noise [243,244], especially in the direction of the fine mesh. A very large reduction of this noise is achieved in most propagation directions, and particularly in the fine grid, as can also be seen on Fig. 6. 7. ...
Thesis
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Predicting landing gear noise is a major concern for an aircraft manufacturer, since it contributes to about 40% of the total aircraft noise during the approach phases. Flight tests and those carried out in anechoic wind tunnels have enabled the understanding of noise generation mechanisms, as well as the design of low noise devices. However, these methods are time consuming and costly to set up. The use of computational fluid dynamics (CFD) is thus emerging as an essential complement to these experimental approaches. The flow around landing gears is complex and highly unsteady, and the noise generated is broadband by nature. Given these characteristics, it is therefore necessary to use unsteady methods with high-fidelity turbulence modeling such as Large Eddy Simulation (LES), to predict these acoustic sources. The lattice Boltzmann method (LBM) is a numerical approach that has recently shown a strong potential for this type of application, thanks to its accuracy, its low restitution time and its ability to handle complex geometries. It is consequently adopted for this thesis. Aeroacoustic simulations require a high level of accuracy since acoustic fluctuations, which are several orders of magnitude smaller than aerodynamic ones, must be properly captured and propagated. Nevertheless, the non-conforming grid interfaces used in LBM have the inconvenience of generating spurious vorticity and acoustics that propagate in the fluid core, which may affect the noise predictions. The PhD objective is to develop new grid coupling models in the "LaBS/ProLB" LBM solver, and to validate them in the context of landing gears aeroacoustics. Two main directions are addressed to overcome these phenomena: 1/ A study of the numerical scheme in the fluid core is performed, highlighting the involvement of non-hydrodynamic modes, specific to the LBM, in the generation of vorticity and of a portion of the spurious acoustics generated at mesh interfaces. After a thorough study of the implication of these modes, an appropriate collision model (H-RR) is chosen to filter them out during a simulation. The stability and accuracy of several LBM schemes including the H-RR one under typical aeroacoustic simulation conditions are also investigated. This study highlights stability issues, as well as questionable precision of many advanced LBM schemes available in the literature. 2/ A direct coupling algorithm between two grids of different resolution is proposed. This algorithm allows to greatly improve the accuracy of the non-conforming grid interfaces, and hence to reduce the spurious acoustic emission produced by the crossing of vortices composing the wakes. Finally, the LAGOON landing gear allows for the validation of these numerical ingredients. An aerodynamic study and then an aeroacoustic one via a coupling with an acoustic propagation code based on the Ffowcs Williams and Hawkings analogy (FW-H) are conducted. The limitations of this analogy in its solid formulation, mostly used to predict landing gear noise, are exposed. Lastly, the effect of extra components of increasing complexity on the noise generated is investigated.
... Thus, the method is routinely used for incompressible hydrodynamic simulations both in low Reynolds number creeping flow regime as well as for high Reynolds number turbulent flow regime [1][2][3][6][7][8][9]]. An extension of the lattice Boltzmann method (LBM) for acoustics is relatively recent [5,[10][11][12][13][14][15]. These works have established the capability of LBM to correctly reproduce fundamental acoustic phenomena and highlighted the low dissipative behavior of the LBM. ...
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We present an energy conserving lattice Boltzmann model based on a crystallographic lattice for simulation of weakly compressible flows. The theoretical requirements and the methodology to construct such a model are discussed. We demonstrate that the model recovers the isentropic sound speed in addition to the effects of viscous heating and heat flux dynamics. Several test cases for acoustics and thermal and thermoacoustic flows are simulated to show the accuracy of the proposed model.
... Thus, the method is routinely used for incompressible hydrodynamic simulations both in low Reynolds number creeping flow regime as well as for high Reynolds number turbulent flow regime [1][2][3][6][7][8][9]]. An extension of the lattice Boltzmann method (LBM) for acoustics is relatively recent [5,[10][11][12][13][14][15]. These works have established the capability of LBM to correctly reproduce fundamental acoustic phenomena and highlighted the low dissipative behaviour of the LBM. ...
Preprint
We present an energy conserving lattice Boltzmann model based on a crystallographic lattice for simulation of weakly compressible flows. The theoretical requirements and the methodology to construct such a model are discussed. We demonstrate that the model recovers the isentropic sound speed in addition to the effects of viscous heating and heat flux dynamics. Several test cases for acoustics, thermal and thermoacoustic flows are simulated to show the accuracy of the proposed model.
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A combined immersed boundary–lattice Boltzmann approach is used to simulate the dynamics of the fluid–structure interaction of a hollow sealing strip under the action of pressure difference. Firstly, the multiple relaxation times LBM model, hyper-elastic material model and immersed boundary method were deduced. According to the strain characteristics of hyper-elastic materials and the specific situation of friction between the elastic boundary and solid boundary, the internal force and the external force on the immersed boundary were discussed and deduced, respectively. Then, a 2D calculation model of the actual hollow sealing strip system was established, during which technical problems such as the equivalent wall thickness of the sealing strip and the correction of the stiffness of the contact corner were solved. The reliability of the model was verified by comparing results of FEM simulation of quasi-static deformation. Following this, the simulation results of three typical cases of sealing strips were presented. The results show that when the sealing strip fails, there will be a strong coupling phenomenon between the flow field and the sealing strip, resulting in the oscillation of the flow field and the sealing strip at the same frequency.
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The lattice-Boltzmann method, in its classical form, is a hyperbolic-leaning equation system which requires long term time-marching solutions to attain quasi-steady state, and much research has been done to improve the convergence performance of the algorithm. Nevertheless, previous approaches have seen limited use in the literature, either due to high complexity, a lack of integrability, and/or instability considerations. In this study, we propose a new acceleration scheme that utilizes information carried by pressure waves propagating in the simulated domain to achieve accelerated convergence to steady and quasi-steady state solutions. The formulated algorithm achieves accurate final flow fields and is in excellent agreement for tested benchmark problems. We show that this scheme is highly robust for a wide range of relaxation parameters in the single-relaxation time and the multiple-relaxation time formulations of the LBM, and effectively apply the algorithm to both obstacle-driven and shear-driven flows, with an observed time reduction to steady state behavior of more than half. Furthermore, the method is successfully tested on a complex, unsteady flow employing the KBC entropic multirelaxation operator – this exhibited a significant reduction of the flow transient stage of up to 63.8%, and proves the scheme to work with the full triad of major LB collision operators. In terms of numerical implementation, the relative cleanness and ‘bolt-on’ nature of the proposed algorithm allows for ease of application and increased universality, making it ideal for a previously unfilled role in current LBM development.
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Chapter
In this chapter, a novel computational method for flow, lattice Boltzmann method, is introduced. We first present the fundamentals and general implements of the method, followed by non-reflective boundary condition techniques, which is important for acoustic simulations. The von Neumann analysis shows lattice Boltzmann method is promising for acoustic simulations. In the latter part of this chapter, we present the applications of lattice Boltzmann method on sound phenomena, such as aeroacoustics, non-linear sound effect and acoustic levitation.
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As the numerical resolution is increased and the discretisation error decreases, the lattice Boltzmann method tends towards the discrete-velocity Boltzmann equation (DVBE). An expression for the propagation properties of plane sound waves is found for this equation. This expression is compared to similar ones from the Navier-Stokes and Burnett models, and is found to be closest to the latter. The anisotropy of sound propagation with the DVBE is examined using a two-dimensional velocity set. It is found that both the anisotropy and the deviation between the models is negligible if the Knudsen number is less than 1 by at least an order of magnitude.
Thesis
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An introduction is given to the lattice Boltzmann method and its background, with a view towards acoustic applications of the method. To make a larger range of acoustic applications possible, a point source method is proposed. This point source is applied to simulate cylindrical waves and plane waves, and is shown to give a very good numerical result compared with analytic solutions of viscously damped cylindrical and plane waves. Good results are found for simulations of Doppler effect, diffraction, and viscously damped standing waves. It is concluded that the lattice Boltzmann method could be suitable for simulating acoustics in complex flows, at ultrasound frequencies and very small spatial scales. The lattice Boltzmann method is shown to be unfeasible at lower frequencies or for larger systems.
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By adding a particle source term in the Boltzmann equation of kinetic theory, it is possible to represent particles appearing and disappearing throughout the fluid with a specified distribution of particle velocities. By deriving the wave equation from this modified Boltzmann equation via the conservation equations of fluid mechanics, multipole source terms in the wave equation are found. These multipole source terms are given by the particle source term in the Boltzmann equation. To the Euler level in the momentum equation, a monopole and a dipole source term appear in the wave equation. To the Navier-Stokes level, a quadrupole term with negligible magnitude also appears.
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An extended numerical scheme for the simulation of fluid flows by means of a lattice Boltzmann (LB) method is introduced. It is conceptually related to the lattice BGK scheme, which it enhances by a regularization step. The result is a numerical scheme that is both more accurate and more stable in the hydrodynamic regime.
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This is an introduction to the branch of fluid mechanics concerned with the production of sound by hydrodynamic flows. It is designed for a one semester introductory course at the advanced undergraduate or graduate level. Great care is taken to explain underlying fluid mechanical and acoustic concepts, and to describe fully the steps in a complicated derivation. The discussion deals specifically with low Mach number flows, which enables the sound produced by `vortex-surface' interactions to be analyzed using the `compact Green's function'. This provides a routine procedure for estimating the sound, and an easy identification of those parts of a structure that are likely to be important sources of sound.
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In this paper we address the problem of the time evolution of a perturbation around a steady base flow with the use of the lattice Boltzmann method (LBM). This approach, named base flow lattice Boltzmann method, is of great interest in particular for aeroacoustic fields where the acoustic perturbation, on the one hand, is almost exclusively influenced by the large scale average structures of the underlying flow, and on the other hand, has a low effect on the large structures. The method is implemented for weakly compressible flows and the results of the base flow lattice Boltzmann are compared with the standard single relaxation time LBM. The boundary conditions for the base flow lattice Boltzmann method are discussed, as well as the implementation of outflow conditions for acoustic waves.
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This paper investigates the reflection of sound waves as they propagate toward the open end of ducts issuing a cold subsonic mean flow in a stagnant fluid. The investigations are conducted by using a relatively new one-step numerical technique known as the lattice Boltzmann method. The results obtained in terms of end correction, l, and magnitude of the pressure reflection coefficient, |R|, for an unflanged pipe model are in very good agreement with theoretical predictions and experimental data for the range of Mach numbers M
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It is well-known that there exist several free relaxation parameters in the MRT-LBM. Although these parameters have been tuned via linear analysis, the sensitivity analysis of these parameters and other related parameters are still not sufficient for detecting the behaviors of the dispersion and dissipation relations of the MRT-LBM. Previous researches have shown that the bulk dissipation in the MRT-LBM induces a significant over-damping of acoustic disturbances. This indicates that MRT-LBM cannot be used to obtain the correct behavior of pressure fluctuations because of the fixed bulk relaxation parameter. In order to cure this problem, an effective algorithm has been proposed for recovering the linearized Navier-Stokes equations from the linearized MRT-LBM. The recovered L-NSE appear as in matrix form with arbitrary order of the truncation errors with respect to ${\delta}t$. Then, in wave-number space, the first/second-order sensitivity analyses of matrix eigenvalues are used to address the sensitivity of the wavenumber magnitudes to the dispersion-dissipation relations. By the first-order sensitivity analysis, the numerical behaviors of the group velocity of the MRT-LBM are first obtained. Afterwards, the distribution sensitivities of the matrix eigenvalues corresponding to the linearized form of the MRT-LBM are investigated in the complex plane. Based on the sensitivity analysis and the recovered L-NSE, we propose some simplified optimization strategies to determine the free relaxation parameters in the MRT-LBM. Meanwhile, the dispersion and dissipation relations of the optimal MRT-LBM are quantitatively compared with the exact dispersion and dissipation relations. At last, some numerical validations on classical acoustic benchmark problems are shown to assess the new optimal MRT-LBM.