A quasi-one-dimensional model of thermoacoustics in the presence of mean flow

In thermoacoustic regenerators, the interaction of thermo-viscous boundary layers and axial temperature gradients causes a conversion from thermal energy to acoustic power or vice versa. In this paper, an improved analytical model for thermoacoustic boundary layer effects in the presence of mean flow is derived and analyzed. Previous formulations of the thermo-acoustic effect take into account effects of mean flow on acoustic propagation only implicitly, i.e. in as much as mean flow influences the mean temperature field. The new model, however, includes additional terms in the perturbation equations, which describe explicitly the interaction between steady mean flow and acoustics. For a parallel plate pore the three-dimensional thermoacoustic equations are derived and reduced to a transversally averaged system of differential equations by applying Green׳s function technique and suitable assumptions. The resulting one-dimensional perturbation equations are then solved numerically for two sets of boundary conditions to obtain the linear scattering matrix coefficients. The solutions, generated for a wide range of frequencies, can be applied in a low-order “network model” context to study the stability of thermoacoustic devices. The impact of mean flow on the thermoacoustic interaction is investigated and validated against full computational fluid dynamics simulations of laminar, compressible flow for one specific configuration. It is shown that at low frequencies (Womersley number <1) the new formulation predicts the acoustic behavior more accurately than the earlier formulations. Finally, the ideas and benefit of further improved and more complex models for higher Mach numbers are discussed.