A comprehensive numerical study on the linear stability of mixed-convection flow
in a vertical pipe with constant heat flux is presented with particular emphasis on
the instability mechanism and the Prandtl number effect. Three Prandtl numbers
representative of different regimes in the Prandtl number spectrum are employed
to simulate the stability characteristics of liquid mercury, water and oil. The results
suggest that mixed-convection flow in a vertical pipe can become unstable at low
Reynolds number and Rayleigh numbers irrespective of the Prandtl number, in
contrast to the isothermal case. For water, the calculation predicts critical Rayleigh
numbers of 80 and −120 for assisted and opposed flows, which agree very well
with experimental values of Rac = 76 and −118 (Scheele & Hanratty 1962). It
is found that the first azimuthal mode is always the most unstable, which also
agrees with the experimental observation that the unstable pattern is a double spiral
flow. Scheele & Hanratty's speculation that the instability in assisted and opposed
flows can be attributed to the appearance of inflection points and separation is
true only for fluids with O(1) Prandtl number. Our study on the effect of the
Prandtl number discloses that it plays an active role in buoyancy-assisted flow and
is an indication of the viability of kinematic or thermal disturbances. It profoundly
affects the stability of assisted flow and changes the instability mechanism as well.
For assisted flow with Prandtl numbers less than 0.3, the thermal–shear instability
is dominant. With Prandtl numbers higher than 0.3, the assisted-thermal–buoyant
instability becomes responsible. In buoyancy-opposed flow, the effect of the Prandtl
number is less significant since the flow is unstably stratified. There are three distinct
instability mechanisms at work independent of the Prandtl number. The Rayleigh–Taylor
instability is operative when the Reynolds number is extremely low. The
opposed-thermal–buoyant instability takes over when the Reynolds number becomes
higher. A still higher Reynolds number eventually leads the thermal–shear instability
to dominate. While the thermal–buoyant instability is present in both assisted and
opposed flows, the mechanism by which it destabilizes the flow is completely different.
