Kolmogorov’s spectral energy cascade in nonlinear acoustic and thermoacoustic wave turbulence

Gupta, Lodato, and Scalo (J. Fluid Mech. (2017), vol. 831, pp. 358-393) have demonstrated the existence of an equilibrium spectral energy cascade in shock waves formed as a result of continued thermoacoustic amplification consistent with Kolmogorov’s theory for high-Reynolds-number hydrodynamic turbulence. Recently, Gupta and Scalo (21^st International Symposium on Nonlinear Acoustics, 2018) have developed a rigorous theory of spectral energy cascade in an ensemble of nonlinear acoustic waves, which fully develop into randomly distributed shock waves resulting in acoustic wave turbulence. The dynamics are shown very similar to the homogeneous isotropic turbulence in a box. In this work, we elucidate the energy dynamics utilizing the mathematically exact energy corollary for second order nonlinear acoustics thus identifying the second-order energy norm for acoustics. For randomly initialized nonlinear waves, the mean energy in the domain decays with a -2/3 law in time due to coalescence of shock waves. In the spectral space, the energy corollary yields analytical expressions of energy, energy flux, and energy dissipation in spectral space. Analogous to the Kolmogorov's theory, we derive dimensionless scaling laws for energy spectra of acoustic wave turbulence.