Knudsen number effects on the nonlinear acoustic spectral energy cascade

We present a numerical investigation of the effects of gas rarefaction on the energy dynamics of resonating planar nonlinear acoustic waves. The problem setup is a gas-filled, adiabatic tube, excited from one end by a piston oscillating at the fundamental resonant frequency of the tube and closed at the other end; nonlinear wave steepening occurs until a limit cycle is reached, resulting in shock formation for sufficiently high densities. The Knudsen number, defined here as the ratio of the characteristic molecular collision timescale to the resonance period, is varied in the range $\mathrm{Kn}={10}^{-1}–{10}^{-5}$, from rarefied to dense regime, by changing the base density of the gas. The working fluid is Argon. A numerical solution of the Boltzmann equation, closed with the Bhatnagar-Gross-Krook model, is used to simulate cases for $\mathrm{Kn}\ge 0.01$. The fully compressible one-dimensional Navier-Stokes equations are used for $\mathrm{Kn}<0.01$ with adaptive mesh refinement to resolve the resonating weak shocks, reaching wave Mach numbers up to 1.01. Nonlinear wave steepening and shock formation are associated with spectral broadening of the acoustic energy in the wavenumber-frequency domain; the latter is defined based on the exact energy corollary for second-order nonlinear acoustics derived by Gupta and Scalo [Phys. Rev. E 98, 033117 (2018)], representing the Lyapunov function of the system. At the limit cycle, the acoustic energy spectra exhibit an equilibrium energy cascade with a $-2$ slope in the inertial range, also observed in freely decaying nonlinear acoustic waves by the same authors. In the present system, energy is introduced externally via a piston at low wavenumbers or frequencies and balanced by thermoviscous dissipation at high wavenumbers or frequencies, responsible for the base temperature increase in the system. The thermoviscous dissipation rate is shown to scale as ${\mathrm{Kn}}^2$ for fixed Reynolds number based on the maximum velocity amplitude, i.e., increasing with the degree of flow rarefaction; consistently, the smallest length scale of the steepened waves at the limit cycle, corresponding to the thickness of the shock (when present) also increases with $\mathrm{Kn}$. For a given fixed piston velocity amplitude, the bandwidth of the inertial range of the spectral energy cascade decreases with increasing Knudsen numbers, resulting in a reduced resonant response of the system. By exploiting dimensionless scaling laws borrowed by Kolmogorov's theory of hydrodynamic turbulence, it is shown that an inertial range for spectral energy transfer can be expected for acoustic Reynolds numbers ${\mathrm{Re}}_{{\mathit{U}}_{\max }}>100$, based on the maximum acoustic velocity amplitude in the domain.

Figure 1

Velocity perturbation in a high amplitude acoustic traveling wave (a), evolution of normalized spatial average of $u^2$ (b), and velocity spectra $|{\widehat{u}}_k{|}^2$ (c) at times $t_0,t_1,t_2,t_3$. The spectral broadening occurs due to the nonlinear terms in the governing equations resulting in the energy cascade from larger to smaller length scales.

Figure 2

Illustration of the investigated one-dimensional piston-tube setup during maximum compression of the gas at the adibatic wall ($t_1$) and approximately ${180}^{\circ }$ later in the shock resonance cycle ($t_2$).

Figure 3

Schematic of (a) the polynomial reconstruction of a quantity $\varphi \left(x\right)$ over a one-dimensional domain consisting of three elements, and (b) the staggered spectral difference spatial discretization approach used with solution ($•$) and flux points ($|$) within a single element. $N_{cv}$ is the number of elements and $p$ is the number of solution points within an element.

Figure 4

Schematic of the staggered flux conservative spatial discretization used for the BGK solver depicting the flux faces ($\mathrm{i}+1/2$) and cell centers (i).

Figure 5

Illustration of three grid sizes for the discretization of the molecular velocity space in $c_x$ and $c_r$.

Figure 6

Time series for (a) pressure fluctuations $p^{\prime }$ at $x=L$ and (b) velocity $u$ at $x=L/2$ for ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}=0.2$ and $\mathrm{Kn}={10}^{-5}$ as obtained from the NS-AMR solver. The figure insets—$\left(i\right),\left(ii\right)$ and $\left(iii\right)$—show the different stages in steepening of the planar acoustic wave until a limit cycle is achieved.

Figure 7

Power spectral densities of pressure at $x=L$ (a, b) and velocity at $x=L/2$ (c, d) at the limit cycle for $\mathrm{Kn}={10}^{-1}$ and $\mathrm{Kn}={10}^{-2}$ from NS-AMR (black) and BGK (gray) solvers. Only the odd harmonics have been plotted for $\widehat{u}$. Spurious high-frequency or wavenumber content due to round-off errors has been omitted.

Figure 8

Instantaneous profiles of (a) pressure fluctuation, $p^{\prime }$, and (b) velocity, $u$, normalized by their maximum values at the limit cycle for ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}=0.2$. The pressure field is plotted at the instance when the piston is at its mean position moving towards left, whereas the velocity is taken at the time instance when the piston is at its right extremum.

Figure 9

Variation of ${\mathrm{Re}}_{{\mathit{U}}_{\max }}$ against $\mathrm{Kn}$ and ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}$: (a) depicts the behavior of ${\mathrm{Re}}_{{\mathit{U}}_{\max }}$ versus Knudsen number for all the cases considered and (b) illustrates ${\mathrm{Re}}_{{\mathit{U}}_{\max }}$ scaled by ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}$ versus the Knudsen number. The degree of wave amplification due to resonance observed is the highest for the cases with the lowest ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}$.

Figure 10

Spatial variation of (a) pressure fluctuation intensity, $p_{\mathrm{rms}}^{\prime }$, and (b) velocity fluctuation intensity, $u_{\mathrm{rms}}$, at the limit cycle versus $x$ for ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}=0.02$. For increasing Knudsen number, the $p_{\mathrm{rms}}^{\prime }$ profile becomes sharper at the center indicating decreased movement of the pressure node at the center.

Figure 11

Comparison of the temporal spectra at the limit cycle for ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}=0.2$ (a) for pressure at $x=L$ and (b) velocity at $x=L/2$, for which only the odd harmonics are shown. Spurious high-frequency or wavenumber content due to round-off errors has been omitted.

Figure 12

Comparison of the temporal spectra of ${\widehat{E}}_k$ at the limit cycle for ${\mathrm{Re}}_{{\mathit{U}}_{\max }}=16.218$ (left); 2.889 (mid); 0.333 (right) at $x=L/2$. Spurious high-frequency or wavenumber content due to round-off errors has been omitted.

Figure 13

Comparison of the cycle-averaged spatial spectra of (a) $p^{\prime }\left(x\right)$ and (b) $u\left(x\right)$, and cycle-averaged (c) spectral flux, ${\widehat{\mathrm{\Pi }}}_k$, scaled by $A_{\mathrm{rms}}$, at the limit cycle for ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}=0.2$. Spurious high-frequency or wavenumber content due to round-off errors has been omitted. The ${\mathrm{Re}}_{{\mathit{U}}_{\max }}$ values for $\mathrm{Kn}={10}^{-1},{10}^{-2},{10}^{-3},{10}^{-4}$, and ${10}^{-5}$ are $0.396,3.153,16.252,54.518,$ and 174.571, respectively, at the limit cycle for ${\mathrm{Re}}_{{\mathit{u}}_{\mathit{p}}}=0.2$.

Figure 14

Comparison of the cycle-averaged spatial spectra of (a, b) ${\widehat{E}}_{\text{sp},k}$ and cycle-averaged (c) spectral flux, ${\widehat{\mathrm{\Pi }}}_k$, at the limit cycle for ${\mathrm{Re}}_{{\mathit{U}}_{\max }}=16.218$ (left); 2.889 (mid); 0.333 (right). Spurious high-frequency or wavenumber content due to round-off errors has been omitted.

Figure 15

Spatial variation of scaled mean acoustic power ${\dot{W}}_{\mathrm{ac}}$ for the cases with ${\mathrm{Re}}_{{\mathit{U}}_{\max }}=16.218$. The decreasing trend of the acoustic power with $x$ indicates that the acoustic power is directed towards the right of Fig. 2 with influx of acoustic energy at the left end from the piston.

Figure 16

Spatial variation of (a) scaled mean acoustic power derivative $\partial {\dot{W}}_{\mathrm{ac}}/\partial x$, (b) dissipation $\mathrm{\Phi }$, and (c) acoustic mechanical work $\mathrm{\Gamma }$ versus $x$ for the set of cases with ${\mathrm{Re}}_{{\mathit{U}}_{\max }}=16.218$.