  We have carried out three-dimensional Navier-Stokes simulations, from
quiescent conditions to the limit cycle, of a traveling-wave thermoacoustic
heat engine (TAE) composed of a long variable-area resonator shrouding a
smaller annular tube, which encloses the hot (HHX) and ambient (AHX)
heat-exchangers, and the regenerator (REG). Simulations are wall-resolved, with
no-slip and adiabatic conditions enforced at all boundaries, while the heat
transfer and drag due to the REG and HXs are modeled. HHX temperatures have
been investigated in the range 440K - 500K with AHX temperature fixed at 300K.
The initial exponential growth of acoustic energy is due to a network of
traveling waves amplified by looping around the REG/HX unit in the direction of
the imposed temperature gradient. A simple analytical model demonstrates that
such thermoacoustic instability is a Lagrangian thermodynamic process
resembling a Stirling cycle. A system-wide linear stability model based on
Rott's theory is able to accurately predict the frequency and the growth rate,
exhibiting properties consistent with a supercritical Hopf bifurcation. The
limit cycle is governed by acoustic streaming -- a rectified flow resulting
from high-amplitude acoustics. Its key features are explained with an axially
symmetric incompressible model driven by the wave-induced stresses extracted
from the compressible calculations. These features include Gedeon streaming and
mean recirculations due to flow separation. The first drives the mean advection
of hot fluid from the HHX to a secondary heat exchanger (AHX2), in the thermal
buffer tube (TBT), necessary to achieve saturation of the acoustic energy
growth. The direct evaluation of the energy fluxes reveals that the efficiency
of the device deteriorates with the drive ratio and that the acoustic power in
the TBT is balanced primarily by the mean advection and thermoacoustic heat
transport.
