  We have carried out wall-resolved unstructured fully-compressible
Navier--Stokes simulations of a complete standing-wave thermoacoustic
piezoelectric (TAP) engine model inspired by the experimental work of Smoker et
al. (2012). The model is axisymmetric and comprises a 51 cm long resonator
divided into two sections: a small diameter section enclosing a thermoacoustic
stack, and a larger diameter section capped by a piezoelectric diaphragm tuned
to the thermoacoustically amplified mode (388 Hz). The diaphragm is modelled
with multi-oscillator broadband time-domain impedance boundary conditions
(TDIBCs), providing higher fidelity over single-oscillator approximations.
Simulations are first carried out to the limit cycle without energy extraction.
The observed growth rates are shown to be grid-convergent and are verified
against a numerical dynamical model based on Rott's theory. The latter is based
on a staggered grid approach and allows jump conditions in the derivatives of
pressure and velocity in sections of abrupt area change and the inclusion of
linearized minor losses. The stack geometry maximizing the growth rate is also
found. At the limit cycle, thermoacoustic heat leakage and frequency shifts are
observed, consistent with experiments. Upon activation of the piezoelectric
diaphragm, steady acoustic energy extraction and a reduced pressure amplitude
limit cycle are obtained. A heuristic closure of the limit cycle acoustic
energy budget is presented, supported by the linear dynamical model and the
nonlinear simulations. The developed high-fidelity simulation framework
provides accurate predictions of thermal-to-acoustic and acoustic-to-mechanical
energy conversion (via TDIBCs), enabling a new paradigm for the design and
optimization of advanced thermoacoustic engines.
