On the Use of a Complex Frequency for the Description of Thermoacoustic Engines

In this paper, a formulation is proposed to describe the process of thermoacoustic amplification in thermoacoustic engines. This formulation is based on the introduction of a complex frequency which is calculated from the transfer matrices of the thermoacoustic core and its surrounding components. The real part of this complex frequency represents the frequency of self-sustained acoustic oscillations, while its imaginary part characterizes the amplification/attenuation of the wave due to the thermoacoustic process. This formalism can be applied to any type of thermoacoustic engine including stack-based or regenerator-based systems as well as straight, closed loop or coaxial duct geometries. It can be applied to the calculation of the threshold of thermoacoustic instability, but it is also well-suited for the description of the transient regime of wave amplitude growth and saturation due to non linear processes. All of the above mentioned aspects are described in this paper.