Acoustic wave propagation at nonadiabatic conditions: The continuum limit of a thin acoustic layer

Existing works on sound propagation in rarefied gases have focused on wave transmission at adiabatic conditions, where a reference uniform equilibrium state prevails. To extend these studies, we analyze the propagation of acoustic waves in a slightly rarefied gas at nonadiabatic conditions, where arbitrarily large reference temperature and density gradients are imposed. Considering a planar slab configuration, constant wall heating is applied at the confining walls to maintain the nonuniform reference thermodynamic distributions. Acoustic excitation is then enforced via small-amplitude harmonic wall oscillations and normal heat-flux perturbations. Focusing on continuum-limit conditions of small Knudsen numbers and high actuation frequencies (yet small compared with the mean collision frequency), the gas domain affected by wall excitation is confined to a thin layer (termed “acoustic layer”) in the vicinity of the excited boundary, and an approximate solution is derived based on asymptotic expansion of the acoustic fields. The application of thermoacoustic wall excitation necessitates the formation of an ever thinner “thermal layer” that governs the transmission of the wall's unsteady heat flux into sound waves. The results of the approximate analysis, supported by continuum-model finite differences and direct simulation Monte Carlo calculations, clarify the impacts of system nonadiabaticity and the gas kinetic model of interaction on sound propagation. Primarily, reference wall heating results in an extension of the acoustic layer and consequent sound-wave radiation over larger distances from the wall source. Considering the entire range of inverse power law (repulsion point center) interactions, it is also found that wave attenuation is affected by the kinetic model of gas collisions, yielding stronger decay rates in gases with softer molecular interactions. The results are used to generalize the counterpart adiabatic-system findings for the amount of boundary heat flux required for the silencing of vibroacoustic sound at nonadiabatic reference conditions.

Figure 1

Schematic of the double-layer configuration containing the gas thermal and acoustic layers. The parameters $\mathrm{Kn}\ll 1$ and $\omega \gg 1$ considered are chosen such that the layers are asymptotically thin compared with the unity slab width.

Figure 2

(a)–(c) Variation with $x$ of the acoustic pressure generated by vibroacoustic (${\overline{U}}_w^{\left(1\right)}=0.02, {\overline{Q}}_w^{\left(1\right)}=0$) wall excitation for (a), (c) hard-sphere ($n=0$) and (b) Maxwell-type ($n=1$) gas with $\mathrm{Kn}={10}^{-4},\omega =700$ and (a,b) $R_T=2$; (c) $R_T=10$. The solid blue lines, dashed red curves, and black crosses present finite-difference-based, asymptotic, and DSMC results, respectively, whereas the dash-dotted green curves depict the reference adiabatic ($R_T=1$) signal. All results are presented at period time, $t=2\pi /\omega$. (d) Variation with $R_T$ of the relative decay ratio $\mathcal{D}$ [see Eq. (36)] for a hard-sphere gas.

Figure 3

Validity of the asymptotic solution for a hard-sphere gas at $R_T=2$ and $\mathrm{Kn}={10}^{-4}$ subject to vibroacoustic (${\overline{U}}_w^{\left(1\right)}=0.02, {\overline{Q}}_w^{\left(1\right)}=0$) wall excitation: variations with $\omega$ of the extremal acoustic pressure adjacent to the $x=0$ wall at time $t=2\pi /\omega$, as predicted by the finite-difference (solid blue line) and asymptotic (dashed red curve) calculations.

Figure 4

Variation with $x$ of the acoustic pressure generated by vibroacoustic (${\overline{U}}_w^{\left(1\right)}=0.02, {\overline{Q}}_w^{\left(1\right)}=0$) wall excitation for a Maxwell-type ($n=1$) gas with $\mathrm{Kn}={10}^{-4},\omega =700$, and (a) $R_T=2$; (b) $R_T=10$. The solid blue lines, dashed red curves, and black crosses present finite-difference-based, asymptotic, and DSMC results, respectively, whereas the dash-dotted green curves depict the reference adiabatic ($R_T=1$) signal. All results are presented at period time, $t=2\pi /\omega$. Different from Figs. 2 and 3 and the next figures, the heated wall is located at $x=1$, whereas the actuated $x=0$ boundary is set at the reference temperature $T^{\left(0\right)}=1$.

Figure 5

Variation with $x$ of the (a) acoustic pressure and (b) velocity perturbation generated by thermoacoustic (${\overline{U}}_w^{\left(1\right)}=0,{\overline{Q}}_w^{\left(1\right)}=0.02\times 5p^{\left(0\right)}/2$) wall excitation for a hard-sphere gas with $R_T=2,\mathrm{Kn}={10}^{-4}$, and $\omega =700$. The solid blue lines, dashed red curves, and black crosses present finite-difference-based, asymptotic, and DSMC results, respectively, whereas the dash-dotted green curves present the reference adiabatic ($R_T=1$) signal. All results are presented at period time, $t=2\pi /\omega$.

Figure 6

Validity of the acoustic and thermal layer approximations in the case of thermoacoustic wall excitation: variations with $x$ of the velocity (blue lines) and heat-flux (red curves) perturbations for a hard-sphere gas at $R_T=2$ and $\mathrm{Kn}={10}^{-3}$ subject to wall excitation with ${\overline{Q}}_w^{\left(1\right)}=0.02\times 5p^{\left(0\right)}/2,{\overline{U}}_w^{\left(1\right)}=0$, and $\omega =100$. The solid, dashed, and dash-dotted lines present the finite-difference-based, acoustic layer, and thermal layer solutions, respectively. The vertical dotted line confines the $x=\sqrt{\mathrm{Kn}/\omega }$ characteristic size of the thermal layer. The results are presented at period time, $t=2\pi /\omega$.