Oscillating flow of a heat-conducting fluid in a narrow tube

Abstract

Thermoacoustic refrigeration occurs in periodic flow in a duct with heat transfer within the fluid and to the tube. This study considers the periodic limit cycle with large pressure oscillations that is obtained in a tube when prescribed, phase-shifted, periodic velocities at the tube ends, at frequencies lower than acoustic eigenmodes, sweep a length comparable to the tube length. The temperature differences between the two ends are of arbitrary magnitude, heat transfer in the transverse direction within the fluid is assumed to be very effective and the thermal mass of the wall is large. The geometry is two-dimensional, axisymmetric, and conduction is accounted for, not only in the fluid, but also with and within the tube wall. A perturbation solution valid in a local near-isothermal limit determines the equilibrium longitudinal temperature profile that is reached at the periodic regime, the pressure field including longitudinal gradients, and the longitudinal enthalpy flux. Results are presented for tubes open at both ends and also with one end closed. In the latter case, a singularity occurs in the temperature at the closed end, with behaviour identical to Rott's result for acoustic flow with small pressure amplitude. Other new results obtained for tubes open at both ends show that when velocities at both ends are in opposite phase, internal singularities in the temperature profiles may occur.