Finite-amplitude instability of mixed-convection in a heated vertical pipe

The instability of flow in a heated vertical pipe is studied using weakly nonlinear instability theory for both stably and unstably stratified cases. It is found that the dominant instability for stably stratified flow is a thermal-buoyant instability, while that of the unstably stratified case is a Rayleigh-Taylor instability. The weakly nonlinear theory predicts supercritical instability for the stably stratified case, in agreement with experimental observations. In this case, it is found that a wide band of wave numbers are linearly unstable soon after the onset of the initial instability. This limits the range for which the weakly nonlinear results are accurate in this case since the theory considers the growth of a single dominant wave. The results of the weakly nonlinear calculations for unstably stratified flow indicate that the flow is potentially subcritically unstable, again in agreement with the experimental observations. On the other hand, the theory predicts that a large amplitude disturbance will be necessary to initiate subcritical instability, while the amplitude of a supercritical disturbance will grow quickly as the magnitude of Ra increases. Therefore, another possible flow transition that is consistent with experimental observations involves rapid growth of the first azimuthal mode of a supercritical Rayleigh Taylor instability, followed by secondary instabilities that lead quickly to turbulence. Analysis of energy transfer in the fundamental wave demonstrates that the thermal-buoyant instability is supercritical because increases in the viscous dissipation rate and the rate of transfer of energy from the fundamental wave back into the mean flow overcome the destabilizing effect of an increase in the rate of buoyant production. Subcritical instability occurs with the Rayleigh-Taylor mode when the disturbance amplitude increases to the point that the combined destabilizing effects caused by a change in the shape of the fundamental wave induced by nonlinear effects become larger than the stabilizing effects due to the production of the harmonic wave and the distortion of the mean-flow. The increase in heat transfer rates due to instability predicted by the weakly nonlinear theory is smaller than the experimental observations. However, it is demonstrated that experimentally observed increases in Nu are predicted if the effects of additional waves are included in an approximate manner.