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 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 Kn≥0.01. The fully compressible one-dimensional Navier-Stokes equations are used for 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)2470-004510.1103/PhysRevE.98.033117], 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 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 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 Re_{U_{max}}>100, based on the maximum acoustic velocity amplitude in the domain.