Sound propagation through a rarefied gas in rectangular channels

Sound propagation through a rarefied gas inside a two-dimensional cavity is investigated on the basis of the linearized Boltzmann equation, where one of the cavity walls oscillates harmonically in the direction normal to its own surface and is considered as a sound source. An analytical solution at high oscillation frequencies is obtained and detailed numerical results for a wide range of gas rarefaction are presented. The influence of both the aspect ratio of the cavity and the oscillation frequency on the average gas pressure exerted on the oscillating plate is studied. It is found that, at large values of the aspect ratio, the average pressure oscillates when the sound frequency varies, due to the sound resonance and antiresonance along the oscillation direction of the plate. However, at small values of the aspect ratio, the average pressure is a monotonically decreasing function of the sound frequency, which cannot be observed in the corresponding one-dimensional counterpart. This is explained by the sound interference in the direction parallel to the oscillating plate. The influence of both the cavity aspect ratio and oscillation frequency on the sound speed is also investigated: Again it is found that a different aspect ratio leads to the different behavior of the sound speed as a function of the oscillation frequency.

Figure 1

Cavity geometry and two types of interference. The left plate oscillating harmonically in the $x$ direction is considered as the sound source. Type I and type II interference occur in the directions parallel and perpendicular to the motion of the oscillating plate, respectively.

Figure 2

(a) Amplitude of the average pressure at the oscillating plate when $A=\infty$: comparison between the results of the LBE for hard-sphere molecules (lines) and the Shakhov kinetic model (symbols) (adopted from Fig. 1 in Ref. [12]). Note that here $\delta =\sqrt{\pi }/2\text{Kn}S$ and $S$ are equivalent to the $\theta$ and $L$ defined in Ref. [12], respectively. (b) Absolute value of the relative difference in $|\overline{p}\left(0\right)|$ between the LBE for the hard-sphere and Maxwellian molecules.

Figure 3

Marginal perturbed distribution function at the oscillating plate $\int \int h(x=0,\mathbf{v})d{v}_{y}d{v}_z$ when $A=\infty ,\text{Kn}=5\sqrt{\pi }$, and the Strouhal number is (a) $S=0.1$, (b) $S=1$, and (c) $S=10$. Large discontinuities and rapid oscillations near $v_x=0$ can be clearly seen. Solid lines (or circles) and dashed lines (or triangles) show the real and imaginary parts of the distribution function, respectively, when $N_v=192$ (or 96). Note that following the rescaling of the perturbed distribution function in Eq. (11), $|h/f_{eq}|$ is not necessary far less than one.

Figure 4

Amplitude and phase of the average pressure in the free-molecular flow regime with $1/\text{Kn}=0$, when the Strouhal number is (a) $S=1.5$, (b) $S=3$, and (c) $S=9$. Along the direction of the arrow, the cavity aspect ratios are 0.125, 0.25, 0.5, 1, 2, and infinity, respectively. In the inset of (c), the phases of the average pressure for different aspect ratios are nearly indistinguishable, so only the phase for $A=\infty$ is shown.

Figure 5

(a) Amplitude and (b) phase of the average pressure at the oscillating plate in the free-molecular flow regime with $1/\text{Kn}=0$. Along the direction of the arrow, the cavity aspect ratios are 0.125, 0.25, 0.5, 1, 2, and infinity, respectively.

Figure 6

(a) Amplitude and (b) phase of the average pressure at the oscillating plate when $\text{Kn}=0.1$. Along the direction of the arrow, the cavity aspect ratios are 0.125, 0.25, 0.5, 1, 2, and infinity, respectively.

Figure 7

Variation of the sound phase speed near the sound source with the Strouhal number, at (a) large and (b) small values of the cavity aspect ratio.