Spectral energy cascade in thermoacoustic shock waves

We have investigated thermoacoustically amplified quasi-planar nonlinear waves driven to the limit of shock-wave formation in a variable-area looped resonator geometrically optimized to maximize the growth rate of the quasi-travelling-wave second harmonic. Optimal conditions result in velocity leading pressure by approximately $40^{\circ }$ in the thermoacoustic core and not in pure travelling-wave phasing. High-order unstructured fully compressible Navier–Stokes simulations reveal three regimes: (i) modal growth, governed by linear thermoacoustics; (ii) hierarchical spectral broadening, resulting in a nonlinear inertial energy cascade, (iii) shock-wave-dominated limit cycle, where energy production is balanced by dissipation occurring at the captured shock-thickness scale. The acoustic energy budgets in regime (i) have been analytically derived, yielding an expression of the Rayleigh index in closed form and elucidating the effect of geometry and hot-to-cold temperature ratio on growth rates. A time-domain nonlinear dynamical model is formulated for regime (ii), highlighting the role of second-order interactions between pressure and heat-release fluctuations, causing asymmetry in the thermoacoustic energy production cycle and growth rate saturation. Moreover, energy cascade is inviscid due to steepening in regime (ii), with the $k$ th harmonic growing at $k/2$ -times the modal growth rate of the thermoacoustically sustained second harmonic. The frequency energy spectrum in regime (iii) is shown to scale with a $-5/2$ power law in the inertial range, rolling off at the captured shock-thickness scale in the dissipation range. We have thus shown the existence of equilibrium thermoacoustic energy cascade analogous to hydrodynamic turbulence.