Spectral energy cascade and decay in nonlinear acoustic waves

TL;DR

This paper investigates the nonlinear spectral energy cascade and decay in acoustic waves using numerical simulations and theoretical analysis, revealing how energy broadens and dissipates across scales in ideal gases.

Contribution

It introduces a new set of nonlinear acoustics equations and an energy norm that captures spectral broadening and decay in nonlinear acoustic wave turbulence.

Findings

Spectral energy broadening occurs during initial evolution stages.

Viscous losses lead to monotonic decay of the energy norm after saturation.

The new energy norm acts as a Lyapunov function for the system.

Abstract

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.