## 1 Introduction and Literature Review  
Thermoacoustic systems are usually divided into engines and refrigerators depending on the direction of energy conversion between the acoustic and thermal energy. The working principle of the thermoacoustic refrigerator is illustrated in Figure 1. The acoustic driver at the end of the resonator supplies acoustic power, \(W_{ac}\), to the working gas inside the resonator. The standing wave acoustic field induces oscillatory motion of \"gas parcels\" coupled with their cyclic compression and expansion. Internal structures such as heat exchangers and stacks are placed at a location where the oscillating pressure and velocity are non-zero. A fragment of a single stack plate has been magnified at the bottom of Figure 1 together with a gas parcel undergoing an acoustic oscillation. The gas parcel is compressed as it moves to the right and the parcel temperature increases to be slightly higher than the plate temperature. Consequently, heat is being transferred from the parcel to the plate. As the flow reverses, the gas parcel expands and the temperature drops slightly lower than that of the plate. As a result,heat is being absorbed from the plate and the parcel returns to its original thermodynamic state. This completes the thermodynamic cycle. In essence, the acoustic wave provides power that allows the heat to be pumped up the temperature gradient. Conversely, if a high enough temperature gradient is imposed across the stack/regenerator, an acoustic power will be self-excited and useful energy will be produced [1]. This in turn could be extracted by a linear alternator to produce electricity, or used to drive a coupled thermoacoustic cooler. The challenges in achieving efficient thermoacoustic devices were discussed comprehensively by [PERSON] [1]. The difficulties with complex flows across the porous structures (stack, regenerator, and heat exchangers) that form the core of the thermoacoustic device have been raised as important research issues. Understanding phenomena such as vortex flow at the end of stacks [2], entrance effects [3], joining conditions [1], and streaming [4], is important as these could hold the key to an efficient device operation. The structure of the vortex pattern, as observed for example by [PERSON] et al. [5], is expected to influence the flow and possibly induce losses that may affect the efficiency of thermoacoustic systems. This has been illustrated through the observations of streamlines and viscous dissipation presented in the numerical investigation of [PERSON] and [PERSON] [6]. The viscous dissipation represents mechanical energy losses within the model. The choice of location for the stack/heat exchanger and blockage ratio (gap between plates) were reported as factors that influence mechanical energy losses. The same group has made several improvements to their model [7; 8; 9]. These involve adiabatic stacks and heat exchangers, all working in a thermoacoustic standing wave environment. In all models, a simplified periodic configuration was used covering the area between the plates and some open area next to the plates to include the possible contribution from the vortex shedding phenomenon. The natural convection was assumed to be very small compared to the oscillatory flow magnitude, and was hence neglected. The effect of natural convection in an oscillatory flow has been previously tested in an experiment involving a heated wire located in an empty resonator [10]. The natural convection was found to dominate the flow under a small velocity regime. This result, however, is not very conclusive because the temperature difference investigated was not widely varied. Furthermore, it is speculated that the presence of additional structures such as the stack and heat exchanger may alter the influence of the natural convection effect on the flow even when the velocity is relatively small [11].