Bifurcation of internally heated flow in a vertical pipe

Abstract

Paper presented at the 9th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Malta, 16-18 July, 2012.

The development of fluid motion in an infinitely long circular pipe with homogeneously distributed internal heat source is examined numerically. The pipe is placed vertically in the gravity field with the pipe wall temperature being kept constant. The motion of the fluid is driven upward by the buoyancy force as well as downward by an applied pressure gradient along the pipe axis. Thus, the basic velocity profile can become inflectional and we may anticipate that the flow may become unstable in contrast to the isothermal pipe flow which is known to be linearly stable for any Reynolds number. We find that the linear instabilities always occur within the region where the basic velocity profile is inflectional but not totally reverse. Our nonlinear analysis indicates that there are two types of nonlinear solutions, referred to as spirals and ribbons. They bifurcate simultaneously from the same point on the neutral curve. Furthermore, the branch of the ribbon extends far inside the region where the basic state is linearly stable and reaches the isothermal limit, creating a nonlinear solution in ’pure’ pipe flow for the case with Pr = 0. For the case with Pr = 7 nonlinear interactions between spirals with different azimuthal wavenumbers are observed.