Towards full-scale numerical simulations of a traveling-wave thermoacoustic Stirling heat engine

We have carried out the first three-dimensional numerical simulation of a thermoacoustic Stirling heatengine (TASHE). The goal is to lay the groundwork for full-scale, time-resolved Navier-Stokes simulations that allow for the direct description of the non-linear processes limiting the overall efficiency of such devices. The TASHE model adopted for the present study is a double-Helmholtz resonator with the engine module at one end. The latter is composed of an infinitely thin annular tube holding the heat-exchanger/regenerator (HX/REG) unit – the only component not directly resolved. All walls are modeled as adiabatic, no-slip boundaries with a local wall-normal resolution resolving the boundary layer thickness. An imposed temperature difference across the regenerator of ∆T = 200K is sufficient to initiate the thermoacoustic instability leading to the generation of acoustic energy in the system. After the application of the heat source, the acoustic fluctuations in the engine rapidly settle towards an exponentially amplified pure tone at 59.8 Hz. By tracking a fluid parcel in the regenerator it is shown how the thermoacoustic instability intensifies exclusively plane waves traveling in the direction of the imposed temperature gradient. The Lagrangian thermodynamic history of the fluid parcel resembles a Stirling heat-engine cycle, unveiling the conversion mechanisms of heat into mechanical work central to any TASHE. A well-defined acoustic network of traveling waves looping around the HX/REG unit is established and analyzed with the aid of the time-resolved data. A simple lumped-parameter model is used to investigate the linear stability properties of the initial acoustic perturbation revealing, in particular, that the acoustic energy growth is maximized at the lowest resonant frequencies of the system. Non-linear effects are detectable from the very beginning and become dynamically important only after the first two seconds of operation. It is shown that intense acoustic fluctuations in the feedback loop component drive complex system-wide streaming flow patterns responsible, in particular, for the mean advection of hot fluid away from the HX/REG (thermal leakage) in the direction of the imposed temperature gradient and around the annular tube. This unwanted effect is contained by the introduction of a second ambient heat-exchanger, commonly found in the design of real TASHEs, allowing for the establishment of a dynamical thermal equilibrium in the system. A limit cycle in fully non-linear conditions is finally obtained for an acoustic amplitude of 178 dB.