  We have carried out boundary-layer-resolved, unstructured fully-compressible
Navier--Stokes simulations of an ultrasonic standing-wave thermoacoustic engine
(TAE) model. The model is constructed as a quarter-wavelength engine,
approximately 4 mm by 4 mm in size and operating at 25 kHz, and comprises a
thermoacoustic stack and a coin-shaped cavity, a design inspired by Flitcroft
and Symko (2013). Thermal and viscous boundary layers (order of 10
$\mathrm{\mu}$m) are resolved. Vibrational and rotational molecular relaxation
are modeled with an effective bulk viscosity coefficient modifying the viscous
stress tensor. The effective bulk viscosity coefficient is estimated from the
difference between theoretical and semi-empirical attenuation curves.
Contributions to the effective bulk viscosity coefficient can be identified as
from vibrational and rotational molecular relaxation. The inclusion of the
coefficient captures acoustic absorption from infrasonic ($\sim$10 Hz) to
ultrasonic ($\sim$100 kHz) frequencies. The value of bulk viscosity depends on
pressure, temperature, and frequency, as well as the relative humidity of the
working fluid. Simulations of the TAE are carried out to the limit cycle, with
growth rates and limit-cycle amplitudes varying non-monotonically with the
magnitude of bulk viscosity, reaching a maximum for a relative humidity level
of 5%. A corresponding linear model with minor losses was developed; the linear
model overpredicts transient growth rate but gives an accurate estimate of
limit cycle behavior. An improved understanding of thermoacoustic energy
conversion in the ultrasonic regime based on a high-fidelity computational
framework will help to further improve the power density advantages of
small-scale thermoacoustic engines.
