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 traveling-wave second harmonic. Optimal conditions result in velocity leading pressure by 40 degrees in the thermoacoustic core and not in pure traveling-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 harmonic k growing at k/2-times the modal growth rate of the thermoacoustically sustained second harmonic. The frequency energy spectra 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 wave turbulence with strong analogies to Kolmogorov's theory for hydrodynamic turbulence.

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). Due to nonlinear wave propagation, energy at the large scales cascades down to smaller scales dominated by viscous dissipation, analogous to hydrodynamic turbulence. 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. The simulation parameter space for the TW, SW, and AWT cases spans three orders of magnitude in initial wave pressure amplitude and dynamic viscosity, thus covering a wide range of both spectral energy cascade and the viscous dissipation rates. The shock waves formed as a result of energy cascade are weak, and hence we neglect thermodynamic non-equilibrium effects such as molecular vibrational relaxation in the current study. 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. Its spatial average 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 spatial average of energy in time. In this regime, the DNS results show -2 power law decay regime for TW and SW, and -2/3 for AWT initialized with white noise. Using the perturbation energy corollary, we derive analytical expressions for the energy, energy flux, and dissipation rate in the wavenumber space. These yield the definitions of characteristic length scales such as the integral length scale (characteristic initial energy containing scale) and the Kolmogorov length scale (shock thickness scale), analogous to K41 theory of hydrodynamic turbulence (A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30 , 9 (1941)). Finally, we show the universal scaling of fully developed energy spectrum of the nonlinear acoustic waves, specifically for TW, SW, and AWT.

The fascinating mechanism of thermoacoustic oscillation was believed to occur only in fluids while our research shows theoretical and numerical evidence of the existence of thermoacoustic effects also in solid media. Two types of solid-state thermoacoustic engines (SSTAEs) are under our investigation: standing- and traveling-wave SSTAE. It has been proved numerically that the traveling wave can be thermally-induced in a metal loop; and the traveling-wave SSTAE performs better thermoacoustic performance than its standing-wave counterpart for the same wavelength. Our group has also investigated flexural-wave solid-state thermoacoustic instability.

We have carried out boundary-layer-resolved, unstructured fully-compressible Navier-Stokes simulations of several complete thermoacoustic engine models. A thermoacoustic-piezoelectric engine model, operating at 388 Hz and comprising a thermoacoustic core, a resonator, and a piezoelectric diaphragm, was simulated through limit cycle, with growth rates consistent with a numerical dynamical model based on Rott's theory. The piezoelectric diaphragm was modeled with multi-oscillator broadband time-domain impedance conditions (TDIBCs), which demonstrated higher fidelity over single-oscillator approximations. Limit cycle acoustic energy budgets were closed using the linear dynamical model.

An ultrasonic standing-wave thermoacoustic engine model, operating at 25 kHz and comprising a thermoacoustic core and coin-shaped cavity, was simulated through limit cycle, with the inclusion of bulk viscosity effects. Bulk viscosity is the result of vibrational and rotational molecular relaxation, and can be a significant attenuative term at high frequencies. Because bulk viscosity is a function of pressure, temperature, frequency, and relative humidity of air, maximal growth rate can become a complex function of base fluid properties.