Spectral energy cascade in nonlinear acoustic and thermoacoustic waves

Abstract

Gupta, Lodato, and Scalo (JFM, 2017) have demonstrated the existence of an equilibrium spectral energy cascade in shock waves formed as a result of continued modal thermoacoustic amplification consistent with Kolmogorov’s theory for high-Reynolds-number hydrodynamic turbulence. In this work, we develop 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. In analogy to hydrodynamic turbulence, the dynamics are shown very similar to the homogeneous isotropic turbulence in a box. To elucidate the energy dynamics, we derive 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 spectral energy, spectral energy flux, and spectral energy dissipation. We derive the spectral energy scaling laws based on the Kolmogorov length scale which corresponds to the shock thickness in acoustic wave turbulence.Gupta, Lodato, and Scalo (JFM, 2017) have demonstrated the existence of an equilibrium spectral energy cascade in shock waves formed as a result of continued modal thermoacoustic amplification consistent with Kolmogorov’s theory for high-Reynolds-number hydrodynamic turbulence. In this work, we develop 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. In analogy to hydrodynamic turbulence, the dynamics are shown very similar to the homogeneous isotropic turbulence in a box. To elucidate the energy dynamics, we derive 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 spectral e...