Direct Simulation of a Flow Produced by a Plane Wall
Oscillating in Its Normal Direction
Taku Ohwada and Masashi Kunihisa
Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University
Abstract. A nonlinear wave driven by a plane wall oscillating in its normal direction is analyzed numerically
on the basis of the Boltzmann equation for hard-sphere molecules and the Maxwell-type boundary condition
by using the direct simulation Monte-Carlo. The computation is carried out for the case where the wall
reciprocates with a constant speed. When the frequency of the oscillation is much lower than the collision
frequency, the occurrence of a sawtooth-like wave is observed as in the case of nondissipative ideal gas and
harmonic oscillation [Y. Inoue & T. Yano, J. Acoust. Soc. Am. 94, 1632-1642 (1993)]. As the frequency
of oscillation increases, the periodic disturbance decays rapidly but the acoustic streaming, a unidirectional
flow usually observed as the time average of mass flux, remains as the propagation of a shock wave. The
effect of the acoustic Mach number, the acoustic Reynolds number, and the accommodation coefficient of
the wall on the strength of the acoustic streaming is examined.
1
INTRODUCTION
A plane wall oscillating in its normal direction is one of the simplest models of sound generator. The
fluiddynamic problem of the propagation of a plane wave from this simple source into a uniform gas at rest
is characterized by the acoustic Mach number M and the acoustic Reynolds number Re. In the case of
1 Re and M 1, the flow field is described by the well-known wave equation. As the acoustic Mach
number increases, shock waves appear and the isentropic condition no longer holds. This case is studied
numerically on the basis of the Euler equation in Ref. [1], where the formation of a sawtooth-like wave,
the occurrence of the acoustic streaming, which is a unidirectional flow observed as the time average of the
mass flux and is one of the striking nonlinear phenomena in acoustics, and the density decrease near the
source due to the occurrence of the acoustic streaming, etc., are revealed. On the other hand, the NavierStokes equation seems to give the adequate description for the case of small or finite acoustic Reynolds
numbers. However, the analysis for this case should rely on kinetic theory, since the acoustic Reynolds
number is inversely proportional to the Knudsen number Kn defined by the ratio of the mean free path to
the displacement amplitude of the oscillation, i.e., Re∼ 1/Kn. The frequency of the oscillation may become
comparable to the collision frequency of the gas molecules even for M 1 and the propagation of a sound
wave at extremely high frequency has been investigated by several authors on the basis of linearized kinetic
equations (M 1). The attenuation rate and the phase speed are obtained as functions of the frequency
and satisfactory agreement with the experimental data is confirmed. The reviews of the existing theoretical
and experimental works and the references are found in Refs. [2, 3, 4, 5]. The case of M ∼ 1, where the
nonlinear effects, such as the occurrence of the acoustic streaming, are expected to appear, has not yet been
well explored, however.
In the present paper, we formulate the problem of a nonlinear wave driven by a plane wall oscillating
in its normal direction as a Stefan problem of the Boltzmann equation faithfully and carry out the DSMC
computation. It will be shown that the periodic disturbance decays rapidly but the acoustic streaming
remains as the propagation of a shock wave. The effect of the acoustic Mach number, the acoustic Reynolds
number, and the accommodation coefficient of the wall on the strength of the acoustic streaming will be
examined.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
202
2
FORMULATION OF PROBLEM
2.1
Problem and notation
A semi-infinite expanse of a gas (X1 > 0) is bounded by a plane wall (X1 = 0) with a uniform temperature
T0 , where Xi is the Cartesian coordinate system. The gas is at rest and uniform with the density ρ0 and
temperature T0 . At the time t̄ = 0 the wall beginstarts to oscillate. For the sake of simplicity of the later
computation, we consider the case where the wall reciprocates between 0 < X1 < 2D with a constant speed
V . We investigate the time evolution of the gas on the basis of the Boltzmann equation for hard-sphere
molecules and the Maxwell-type boundary condition.
The main notation employed in this paper is summarized as follows: l0 is the mean free path of the
gas
the initial equilibrium state at rest (for hard sphere molecules, l0 is given by l0 =
√ molecules in
−1
[ 2πσ 2 (ρ
/m)]
,
where σ and m are the diameter and mass of the molecule, respectively); X1 = l0 x;
√0
t̄ = (l0 / 2RT0 )t is the time, where R is the specific gas constant; (2RT0 )1/2 ζi is the molecular velocity;
ρ0 (2RT0 )3/2 f(x, t, ζi ) is the distribution function of gas molecules; ρ0 ρ, (2RT0 )1/2 u, and T0 T are the density,
X1 component of the flow velocity (the other components are zero in the present problem), and temperature
of the gas, respectively.
2.2
Basic equation
The nondimensional Boltzmann equation in the present spatially one-dimensional case is
∂f
∂f
+ ζ1
= J(f, f).
∂t
∂x
(1)
For hard-sphere molecules the collision term J(f, f) is given by
1
J(f, f) = √
(f f∗ − ff∗ )|ni (ξi − ni )|dΩ(ni )dξ,
2 2π
f = f(x, t, ζi ),
f = f(x, t, ζi ),
(2)
f∗ = f(x, t, ξi ),
f∗ = f(x, t, ξi ),
ζi = ζi + ni nj (ξj − ζj ),
(3)
ξi = ξi − ni nj (ξj − ζj ),
(4)
where ni is the unit vector, dΩ(ni ) is the solid angle element in the direction of ni , the domain of integration
is the whole velocity space for ξi and all direction for ni .
We define
√ the nondimensional speed and nondimensional displacement amplitude of the oscillating plate
by v = V / 2RT0 and d = D/l0 , respectively. Then the Maxwell-type boundary condition on the oscillating
wall is written as
f(xb (t), t, ζi ) = f(xb (t), t, −ζ1 + 2vb (t), ζ2 , ζ3 ) + α
√
ρw = −2 π
ρw
exp[−(ζ1 − vb (t))2 − ζ22 − ζ32 ]
π 3/2
ζ1 <vb (t)
(ζ1 − vb (t))fdζ,
(ζ1 > vb (t)),
(5)
(6)
where α is the accommodation coefficient of the wall and xb (t) and vb (t) are the nondimensional displacement
and velocity of the wall, respectively. In the simple case where the wall reciprocates between 0 < x < 2d
with the constant speed v, xb (t) and vb (t) are given by
for (t − mod(t, ts ))/ts is odd
vmod(t, ts )
xb (t) =
(7)
2d − vmod(t, ts ) for (t − mod(t, ts ))/ts is even,
v
vb (t) =
−v
for (t − mod(t, ts ))/ts is odd
for (t − mod(t, ts ))/ts is even,
203
(8)
where
ts = 2d/v.
(9)
The initial condition is
1
exp(−ζi2 ).
π 3/2
The macroscopic variables are given by the moment of f:
ρ = fdζ, u = (1/ρ) ζ1 fdζ, T = (2/3ρ) (ζj − uj )2 fdζ.
f(x, t = 0, ζi ) =
2.3
(10)
(11)
Parameters of problem
The solution of the above Stefan problem is characterized by the parameters v, d, and α. The first two
parameters are related to the acoustic Mach number M and acoustic Reynolds number Re, which are defined
by
Dω
(γ + 1)c0 D
M =
,
Re =
,
(12)
c0
δ
where γ is the ratio of specific heats, ω is the angular frequency of the oscillation and corresponds to
√
ω = πV /(2D) in the present problem, c0 = γRT0 is the sound speed, and δ is the diffusivity of sound,[1]
defined by
δ = µ[µB /µ + (γ − 1)/Pr + 4/3]/ρ0 ,
(13)
bulk viscosity, and Prandtl number, respectively). For hard-sphere
(µ, µB , and Pr are the viscosity,
√
molecules, γ = 5/3, µ = ( π/4)γ1 ρ0 (2RT0 )1/2 l0 (γ1 = 1.270042), µB = 0, and Pr= 0.660694. Then
M ∼ v and Re∼ d.
3
RESULTS AND DISCUSSIONS
The DSMC computation was carried out for typical values of the parameters (v = 0.25, 0.5, 1, and 2;
d = 0.15625, 0.625, 2.5, 5, and 10; α = 0, 0.5, and 1). The computational parameters are as follows. The
cell size ∆x is 0.05 (near the wall)∼0.25 (far from the wall); the time step ∆t is 0.01∼0.1 depending on the
frequency of the oscillation; the number of particles in a large cell is 100 for the initial uniform state; the
final result of DSMC is obtained as the average of 500∼1000 cases for different seeds of random number.
Figure 1 shows the time evolution of ρ, u, and T in the case of (v, d, α)=(2,20,0). During 0 < t < Ts /2
the gas is compressed by the wall and the first shock wave is formed, where Ts = 4d/v is the oscillation
period and Ts = 40 in the present case. During Ts /2 < t < Ts the wall is pulled back and the rarefaction
region is formed behind the shock. At the beginning of the second compression, the flow velocity u around
the wall is negative (u = −v at the wall) and the relative speed between the gas and the wall during the
second compression Ts < t < 1.5Ts is larger than that during the first compression. This causes a large
temperature rise after the second compression. Figure 2 shows the time evolution of these macroscopic
variables after t = 1.5Ts . The head of the wave is a shock wave and the flow velocity is positive even in the
rarefaction phases behind it. This region spreads as the time passes. That is, the acoustic streaming, which
is a unidirectional flow usually observed as the time average of the mass flux, occures as the propagation of
a shock wave. Because of the occurrence of the acoustic streaming, the density near the wall decreases. At
the same time, the temperature of the gas near the wall increases. Figure 3-6 shows the distributions of ρ,
u, and T at t = 10Ts for different values of v; (v, d, α) = (0.25, 20, 0), (0.5, 20, 0), (1, 20, 0), and (2, 20, 0).
For low frequency (v = 0.25), the profiles of the macroscopic variables are sawtooth-like as in the case of the
nondissipative ideal gas and harmonic oscillation.[1] As v increases, the strength of the acoustic streaming
increases and the temperature rise becomes more significant. Figures 7-9 show the results for different values
of d; (v, d, α) = (1, 10, 0), (1, 5, 0), (1, 2.5, 0). The periodic disturbance behind the head of the wave shrink
rapidly for small d (high frequency). The rapid decay of the periodic disturbance is inferred from the existing
theoretical results based on the linearized kinetic equation and the experimental data, which show that the
attenuation rate increases as the frequency does in the range 1/Ts 1. As a measure of the strength of the
acoustic streaming, the Mach number of the front shock wave Ms = Vs /(5RT0 /3)1/2 (Vs : the speed of head
204
of the wave) at t = 800 is tabulated in Table 1 as a function of v and in Table 2 as that of d; the macroscopic
variables in front of the head of the wave and those behind it satisfy the Rankin-Hugoniot condition within
the error of 1% for all cases of these tables. The strength of the acoustic streaming increases as v or d does
but its dependence on d is very small. The density of the gas near the wall decreases and the low density
region spreads as the time passes. For example, in the case of (v = 2, d = 2.5, α = 0), ρ(x = 0.125, t) is
0.070 at t = 10Ts , 0.031 at t = 100Ts , and 0.024 at t = 200Ts . Neither the formation of a vacuum region
nor that of a steady region is observed in the present computation, however.
Lastly, we show the influence of the accommodation coefficient α. In the case of α = 0, the impinging
molecules with high speed are emitted from the wall with high speed. These molecules contribute to the rise
of temperature near the wall and enhance the momentum transfer to the gas. As α increases, the number of
such molecules decreases and the temperature rise near the wall and the acoustic streaming are suppressed.
This is clearly seen in Fig. 10, where the density, flow velocity, and temperature distributions at t = 800 are
shown for different values of α; (v, d, α) = (1, 2.5, 0), (1, 2.5, 0.5), and (1, 2.5, 1). As d increases, the effect
of the boundary condition is localized; the effect of α on the strength of the acoustic streaming becomes
weaker. For example, in the case of (v, d) = (1, 20) [see Fig. 5 for α = 0], the profiles of ρ and u1 at t = 800
do not vary remarkably as α increases, while the temperature rise near the wall is suppressed [see Fig. 11
for α = 1].
4
t=20
3.5
ρ
4
4
3.5
3.5
ρ
3
2.5
t=40
ρ
3
3
2.5
2.5
2
2
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
50
100
x
0
0
200
u
1
0
-1
50
100
12
150
x
150
x
u
T
0
-1
-1
100
150
x
200
0
8
T
4
4
2
2
2
x
200
0
0
50
100
150
x
200
50
100
150
t=60
8
4
150
200
t=60
10
6
100
x
12
t=40
6
50
150
-2
50
6
0
0
100
1
0
10
8
50
2
t=40
12
t=20
0
0
200
1
0
200
10
T
100
-2
-2
0
50
2
t=20
2
u
150
t=60
x
200
0
0
50
100
150
x
200
Fig. 1: The distributions of the density ρ, flow velocity u, and temperature T at t = 20, 40, and 60 for (v, d, α) =
(2, 20, 0).
205
3
ρ
3
3
t=80
2.5
ρ
2
1.5
t=240
t=160
2.5
2.5
ρ
2
2
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
200
400
x
0
0
600
x
0
0
600
u
1
0
-1
400
20
x
0
0
-1
-1
-2
200
400
x
600
0
T
10
10
5
5
5
600
600
t=240
T
x
x
15
10
400
400
20
15
200
200
t=160
T
600
1
u
20
t=80
x
t=240
1
0
600
15
0
0
400
2
-2
-2
200
200
t=160
t=80
0
400
2
2
u
200
0
0
200
400
x
0
0
600
200
400
x
600
Fig. 2: The distributions of the density ρ, flow velocity u, and temperature T at t = 80, 160, and 240 for (v, d, α) =
(2, 20, 0).
1.2
0.3
1.6
0.2
ρ
1.4
1
T
u 0.1
0
0.8
1.2
-0.1
1
-0.2
0.6
0
1000
2000
x
3000
0
1000
2000
x
3000
0.8
0
1000
2000
x 3000
Fig. 3: The distributions of the density ρ, flow velocity u, and temperature T at t = 3200 for (v, d, α) = (0.25, 20, 0).
206
1.4
2.4
0.4
1.2
ρ
2.2
1
u
0.8
0.2
T
2
1.8
0
0.6
1.6
0.4
-0.2
1.4
0.2
-0.4
1.2
0
0
500
1000
x1500
2000
0
500
1000
x
1500
1
0
2000
500
1000
x
1500
2000
Fig. 4: The distributions of the density ρ, flow velocity u, and temperature T at t = 1600 for (v, d, α) = (0.5, 20, 0).
1.5
8
7
0.5
ρ
1
u
T
0
6
5
4
3
-0.5
0.5
2
1
-1
0
0
500
1000
x
0
500
0
0
x 1000
500
x1000
Fig. 5: The distributions of the density ρ, flow velocity u, and temperature T at t = 800 for (v, d, α) = (1, 20, 0).
2
1
25
0.5
1.5
ρ
u
1
20
0
T
-0.5
15
-1
10
-1.5
0.5
5
-2
0
0
200
400
600
x
-2.5
0
800
200
400
600
x
0
0
800
200
400
600
x
800
Fig. 6: The distributions of the density ρ, flow velocity u, and temperature T at t = 400 for (v, d, α) = (2, 20, 0).
1.5
ρ
0.5
1
u
8
T
0
6
4
0.5
-0.5
2
0
0
200
400
600
x
-1
0
200
400
x
600
0
0
200
400
x
600
Fig. 7: The distributions of the density ρ, flow velocity u, and temperature T at t = 400 for (v, d, α) = (1, 10, 0).
Table 1: Ms versus v (d = 5, α = 0).
v
Ms
0.25
1.080
0.5
1.086
207
1
1.238
2
1.540
1.5
ρ
0.5
8
0
1
T
u
6
-0.5
4
-1
2
0.5
0
0
100
200
x
-1.5
0
300
100
200
x
0
0
300
100
200
x 300
Fig. 8: The distributions of the density ρ, flow velocity u, and temperature T at t = 200 for (v, d, α) = (1, 5, 0).
1.5
0.5
8
u
ρ
T
0
6
1
-0.5
4
-1
2
0.5
0
0
50
100
x 150
200
-1.5
0
50
100
x
150
0
0
200
50
100
x 150
200
Fig. 9: The distributions of the density ρ, flow velocity u, and temperature T at t = 100 for (v, d, α) = (1, 2.5, 0).
Table 2: Ms versus d (v = 1, α = 0).
Ms
0.15625
1.232
d
5
1.238
2.5
1.235
208
10
1.243
20
1.268
0.4
ρ
0.5
0.5
0
T
0.3
1
0
α =1
200
400
600
α =0
20
α =0
u
0.2
1
0.5
10
0.5
1
0.1
800
x
0
1000
200
400
600
800
x
1000
50
100
150
x
200
Fig. 10: The distributions of the density ρ, flow velocity u, and temperature T at t = 800 for α = 0, 0.5, and 1;
(v, d) = (1, 2.5).
1.5
8
0.5
ρ
T
u
1
6
0
4
-0.5
0.5
2
-1
0
0
500
x
1000
0
500
x
1000
0
0
500
x
1000
Fig. 11: The distributions of the density ρ, flow velocity u, and temperature T at t = 800 for (v, d, α) = (1, 20, 1).
4
SUMMARY
A nonlinear wave driven by a plane wall oscillating in its normal direction has been investigated on the
basis of the Boltzmann equation for hard sphere molecules and the Maxwell-type boundary condition. For
comparatively large values of Re, a sawtooth-like waveform is observed as in the case of nondissipative ideal
gas. For Re∼ O(1), the periodic disturbance decays rapidly but the acoustic streaming remains as the
propagation of a shock wave. The strength of the acoustic streaming increases as Re or M does but its
dependence on Re is small. The density of the gas near the wall continues to decrease due to the occurrence
of the acoustic streaming. The vibration of the wall also causes the increase of temperature of the gas near
the wall and it becomes larger as Re or M increases. The strength of the acoustic streaming decreases as α
increases. The effect of the boundary condition is appreciable even far from the wall for Re∼ O(1).
REFERENCES
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2. Cercignani, C., The Boltzmann equation and its applications, (Springer-Verlag, New York, 1988).
3. Thomas Jr., J.R. and Siewert C.E., “Sound-wave in a rarefied gas,” Transp. Theo. Stat. Phys., Vol. 8, 219 (1979).
4. Aoki, K. and Cercignani, C., “A technique for time-dependent boundary value problems in the kinetic theory of
gases Part II. Application to sound propagation,” ZAMP., Vol.35., 345 (1984).
5. Hadjiconstantinou, N.G. and Garcia, A.L., “Molecular Simulations of Sound Wave Propagation in Simple Gases,”
Phys. Fluids, Vol. 13, 1040 (2001).
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