Spectral energy cascade and decay in nonlinear acoustic waves

We present a numerical and theoretical investigation of the nonlinear spectral energy cascade of decaying finite-amplitude planar acoustic waves in a single-component ideal gas at standard temperature and pressure. We analyze various one-dimensional canonical flow configurations: a propagating traveling wave (TW), a standing wave (SW), and randomly initialized acoustic wave turbulence (AWT). Due to nonlinear wave propagation, energy at the large scales cascades down to smaller scales dominated by viscous dissipation, analogous to hydrodynamic turbulence. We use shock-resolved mesh-adaptive direct numerical simulation (DNS) of the fully compressible one-dimensional Navier-Stokes equations to simulate the spectral energy cascade in nonlinear acoustic waves. The simulation parameter space for the TW, SW, and AWT cases spans three orders of magnitude in initial wave pressure amplitude and dynamic viscosity, thus covering a wide range of both the spectral energy cascade and the viscous dissipation rates. The shock waves formed as a result of the energy cascade are weak $(M\phantom{\rule{4pt}{0ex}}l1.4)$, and hence we neglect thermodynamic nonequilibrium effects such as molecular vibrational relaxation in the current study. We also derive a set of nonlinear acoustics equations truncated to second order and the corresponding perturbation energy corollary yielding the expression for a perturbation energy norm ${E}^{(2)}$. Its spatial average, $\ensuremath{\langle}{E}^{(2)}\ensuremath{\rangle}$, satisfies the definition of a Lyapunov function, correctly capturing the inviscid (or lossless) broadening of spectral energy in the initial stages of evolution---analogous to the evolution of kinetic energy during the hydrodynamic breakdown of three-dimensional coherent vorticity---resulting in the formation of smaller scales. Upon saturation of the spectral energy cascade, i.e., a fully broadened energy spectrum, the onset of viscous losses causes a monotonic decay of $\ensuremath{\langle}{E}^{(2)}\ensuremath{\rangle}$ in time. In this regime, the DNS results yield $\ensuremath{\langle}{E}^{(2)}\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{-}2}$ for TWs and SWs, and $\ensuremath{\langle}{E}^{(2)}\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{-}2/3}$ for AWT initialized with white noise. Using the perturbation energy corollary, we derive analytical expressions for the energy, energy flux, and dissipation rate in the wave number space. These yield the definitions of characteristic length scales such as the integral length scale $\ensuremath{\ell}$ (characteristic initial energy containing scale) and the Kolmogorov length scale $\ensuremath{\eta}$ (shock thickness scale), analogous to the K41 theory of hydrodynamic turbulence [A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 9 (1941)]. Finally, we show that the fully developed energy spectrum of the nonlinear acoustic waves scales as ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{k}{k}^{2}{\ensuremath{\epsilon}}^{\ensuremath{-}2/3}{\ensuremath{\ell}}^{1/3}\ensuremath{\sim}C\phantom{\rule{0.16em}{0ex}}f(k\ensuremath{\eta})$, with $C\ensuremath{\approx}0.075$ constant for TWs and SWs but decaying in time for AWT.