Numerical simulation of a thermoacoustic refrigerator I. Unsteady adiabatic flow around the stack

A low Mach-number compressible flow model for the simulation of acoustically driven flow in a thermoacoustic stack is constructed. The model is based on the assumption that the acoustic wavelength is much larger than the characteristic hydrodynamic lengthscale. Thus, a simplified description of the flow is obtained which still retains the essential features of acoustically induced velocity oscillations near solid boundaries. A vorticity-based formulation of the governing equation is derived which relies on the Helmholtz decomposition of the velocity vector into irrotational and divergence-free components. Irrotational motion is used to represent the action of acoustic waves. Meanwhile the divergence-free velocity component is used to capture the nonlinear vortical perturbations due to noslip boundaries. A simplified version of the model is applied to analyze unsteady flow in the neighborhood of an idealized thermoacoustic stack which consists of a periodic array of thin plates placed in an acoustic standing wave. Computed results are used to analyze, for different stack configurations, the nonlinear response of the flow to different acoustic driving amplitudes and frequencies. In particular, it is shown that the flow is dominated by the motion of vortices which result from the shedding of boundary layers from the edges of the stack. The dependence of energy losses on stack configuration and operating conditions is also examined.