This paper considers nonlinear acoustic waves propagating unidirectionally in a gas-filled
tube under an axial temperature gradient, and examines whether the energy
flux of the waves can be amplified by thermoacoustic effects. An array of Helmholtz
resonators is connected to the tube axially to avoid shock formation which would
otherwise give rise to nonlinear damping of the energy flux. The amplification is
expected to be caused by action of the boundary layer doing reverse work, in the
presence of the temperature gradient, on the acoustic main flow outside the boundary
layer. By the linear theory, the velocity at the edge of the boundary layer is given
in terms of the fractional derivatives of the axial velocity of the gas in the acoustic
main flow. It is clearly seen how the temperature gradient controls the velocity at the
edge. The velocity is almost in phase with the heat flux into the boundary layer from
the wall. With effects of both the boundary layer and the array of resonators taken
into account, nonlinear wave equations for unidirectional propagation in the tube
are derived. Assuming a constant temperature gradient along the tube, the evolution
of compression pulses is solved numerically by imposing the initial profiles of both
an acoustic solitary wave and of a square pulse. It is revealed that when a positive
gradient is imposed, the excess pressure decreases while the particle velocity increases
and that the total energy flux can indeed be amplified if the gradient is suitable.
Errata to ?The effect of an axial temperature gradient on the steady motion of a large droplet in a tube? by S. K. Wilson
