Spectral energy cascade and decay in nonlinear acoustic waves

We present a numerical and theoretical investigation of nonlinear spectral energy cascade of decaying finite-amplitude planar acoustic waves in a single-component ideal gas at standard temperature and pressure (STP). We analyze various one-dimensional canonical flow configurations: a propagating traveling wave (TW), a standing wave (SW), and randomly initialized Acoustic Wave Turbulence (AWT). We use shock-resolved mesh-adaptive direct numerical simulations (DNS) of the fully compressible one-dimensional Navier-Stokes equations to simulate the spectral energy cascade in nonlinear acoustic waves. We also derive a new set of nonlinear acoustics equations truncated to second order and the corresponding perturbation energy corollary yielding the expression for a new perturbation energy norm $E^{(2)}$. Its spatial average, <$E^{(2)}$> 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 break down of three-dimensional coherent vorticity -- resulting in the formation of smaller scales. Upon saturation of the spectral energy cascade i.e. fully broadened energy spectrum, the onset of viscous losses causes a monotonic decay of <$E^{(2)}$> in time.