Asymptotic theory of nonlinear acoustic waves in a thermoacoustic prime-mover

摘要

A nonlinear evolution equation describing stabilization of oscillations in a thermoacoustic prime mover is derived. It predicts that, in the case of quasiadiabatic interaction of sound and thermal waves inside an inhomogeneously heated acoustically thin thermoacoustic stack, sound amplification can be proportional to the square root of frequency and can be frequency-independent depending on the value of Kramer's constant. It takes into account wave velocity dispersion (additional to that caused by thermal and viscous boundary layers) and also describes the shifts of resonance frequencies induced in the acoustic resonator by installation of the stack. The analytical description of the spectral characteristics of the thermoacoustic stack (valid for an arbitrary temperature distribution inside the stack) was found by transforming the differential equation for the sound propagation and backscattering in inhomogeneous media into an equivalent Volterra integral equation of the second kind. The latter has a solution in the form of an iterative sequence that converges absolutely and uniformly to its exact solution. The derived evolution equation also includes the usual quasi-linear differential term responsible for the nonlinear acoustic process in the gas. Analytical solution of this equation (in the case of weak dispersion) for the thermoacoustic prime mover filled with the gas, characterized by a small value of Kramer's constant, is found. Characteristic times of shock front formation, of wave amplitude stabilization and the characteristic amplitude of the stationary wave (and also their dependence on the stack heating) are determined. Conditions, when a shock front formation caused by nonlinear acoustic processes is the dominant mechanism for the saturation of wave amplitude are established. [References: 43]