Boundary-layer theory for Taconis oscillations in a helium-filled tube

Abstract

This paper develops a one-dimensional formulation to simulate Taconis oscillations in a helium-filled, quarter-wavelength tube in cryogenics within a framework of the boundary-layer theory. Dividing an acoustic field in the tube into a boundary layer on the wall and a main-flow region outside of it, the fluid-dynamical equations are averaged over the whole cross section of the tube, from which the equations averaged over the main-flow region are derived by using the boundary-layer solutions. Nonlinear theory is employed for the main-flow region, whereas the boundary layer is assumed to be described by the linear theory. Resultant equations for the main-flow region are posed in the form of integrodifferential equations due to memory effects by the boundary layer. An initial- and boundary-value problem is solved numerically for the evolution of a small disturbance. It is demonstrated that for temperature distribution of a smooth, step function, a transient behavior leading to emergence of self-excited Taconis oscillations can be simulated numerically. Although the first-order theory in the boundary-layer thickness has been regarded as being incapable of describing Taconis oscillations, it turns out to be applicable to a case with plausible temperature gradient.