## 1 Introduction  
Oscillatory wall-bounded flow emerges in a variety of engineering applications, one of the good examples being thermoacoustic devices. These are known as a technology that offers greener alternatives for energy conversion applications in areas such as waste heat recovery, solar power or environmentally-friendly cooling technologies. One of the keys to a better design of thermoacoustic systems lies in the understanding of the flow and heat transfer within the internal structures in oscillatory conditions. Experimental studies often reveal signs of nonlinearities that are insufficiently addressed by the linear models typically used for design predictions [1; 2]. These may be a result of phenomena such as acoustic streaming, vortex shedding or turbulence in the flow [3; 4; 5]. Previous studies of oscillatory flows identified many interesting features of the transition process [6; 7; 8; 9]. The resulting turbulent flow is characterised by the sudden, explosive appearance of turbulence towards the end of the acceleration phase of the cycle. Turbulent flow is sustained throughout the deceleration phase, while during the early stages of the acceleration phase, the production of turbulence essentially stops, and velocity profiles agree with laminar theory [8]. Nevertheless, during this period, the disturbances retain a small, but finite energy level. Theoretical [1; 10] and numerical [11; 12; 13; 14; 15] works related to thermoacoustic processes typically assume a laminar flow (limited to low drive ratios). The critical Reynolds numbers for the transition to turbulence in an oscillatory flow are reviewed by [PERSON] et al. [6]. Their experimental results agree with the critical value of 400 of the Reynolds number defined by [PERSON] and [PERSON] [7] based on the Stokes layer thickness. The flow regions are categorised into laminar, transitional and turbulent. However, oscillatory flows of a Reynolds number higher than 400 are shown experimentally to experience a stage of relaminarization where the velocity profiles match the laminar prediction during the initial stage of the acceleration phase and change to turbulent-like profiles at the later phase, before turning back to laminar [8; 9]. An additional complication in the transition/relaminarization processes lies in the appearance of temperature-dependent fluid properties, compressibility effects or additional forces, such as gravity [8]. Therefore, necessary precautions need to be taken in the modelling of such flow. As for the size of the computational domain, [PERSON] and [PERSON] [13] suggested that the length of the domain should not be too short or too long. The former alters turbulence characteristics while the latter can result in relaminarization. These observations were made for a turbulent oscillatory flow inside a pipe. They may not be directly applicable to oscillatory flow across parallel-plate structures, but these ideas need to be borne in mind. Experimental work reported by [PERSON] et al. [16; 17] has shown that the introduction of a temperature distribution along the fins (compared to isothermal situation) results in flow asymmetry within the channels and in the immediate vicinity outside and temperature-driven buoyant flow, giving rise to convective currents some distance away from the plates. The flow asymmetry observed through the velocity profiles shown in [17] may be related to the temperature-dependent properties of the fluid and additional flow forcing due to natural convection. [PERSON] [12] performed numerical studies of the heat exchanger arrangement of [PERSON] et al. [17] and further comparisons between experiment and CFD were made in [18]. The laminar model was limited only to the area between the heat exchanger plates and neglected the gravity effects. The natural convection observed in the experiment was modelled as \"heat loss\" by introducing a fictitious heat sink and heat source next to the heat exchangers. This approach provides a general idea about the magnitude of heat losses occurring in the experiment.