Thermoacoustic technologies rely on a direct power conversion between acoustic and thermal energies using well known thermoacoustic effects. The presence of the acoustic field leads to oscillatory heat transfer and fluid flow processes within the components of thermoacoustic devices, notably heat exchangers. This paper outlines a two-dimensional ANSYS FLUENT CFD (computational fluid dynamics) model of flow across a pair of hot and cold heat exchangers that aims to explain the physics of phenomena observed in earlier experimental work. Firstly, the governing equations, boundary conditions and preliminary model validation are explained in detail. The numerical results show that the velocity profiles within heat exchanger plates become distorted in the presence of temperature gradients, which indicates interesting changes in the flow structure. The fluid temperature profiles from the computational model have a similar trend with the experimental results, but with differences in magnitude particularly noticeable in the hot region. Possible reasons for the differences are discussed. Accordingly, the space averaged wall heat flux is discussed for different phases and locations across both the cold and hot heat exchangers. In addition, the effects of gravity and device orientation on the flow and heat transfer are also presented. Viscous dissipation was found to be the highest when the device was set at a horizontal position; its magnitude increases with the increase of temperature differentials. These indicate that possible losses of energy may depend on the device orientation and applied temperature field.