Galiullina E., Botteldooren D.

This paper is focused on efficient numerical simulation of secondary streaming arising inside an acoustic-resonance tube refrigerator. Secondary streaming is described by the second order time-average velocity, therefore field quantities to the second order need to be considered. The model that is derived in this work is based on the boundary-layer approximation of modified Navier-Stokes equations. The numerical problem is split up in two parts: first order, acoustical, and second order (streaming) approximations - by virtue of the method of successive approximations. Viscous losses near the walls are taken into account within the framework of the FDTD (finite-difference time-domain) discretisation of the acoustic equations using a subgrid-scale analytical approximation for the boundary layers. Therefore the grid parameters can be optimally tuned to solving the acoustical problem. The obtained first order quantities enter as nonlinear source terms in the equations for the second order approximation and thus trigger second harmonics and static fields. Numerical simulation of the second order equation results in acoustic streaming in the standing-wave thermoacoustic engine. The presented approach can however easily be extended to include other resonator based thermoacoustic engine configurations. The second order numerical model is free of boundary conditions at the edges of stack plates.