Date of Award
Master of Engineering Science
Mechanical and Materials Engineering
Dr. J.M. Floryan
Natural convection in an infinite horizontal slot subject to spatially periodic heating has been investigated. The analysis is focused on the heating with the wavelength that is large when compared with the slot opening (small wave number limit). It has been shown that convection has a simple topology consisting of one pair of counter-rotating rolls per heating period when the heating intensity, as measured by a suitably defined Rayleigh number Ra, does not exceed the critical value of Ra=427. An explicit solution describing such flow structures has been developed. Secondary motions in the form of rolls aligned in the direction of primary rolls and concentrated around the hot spots occur for more intense heating. When 427 < Ra <~ 470 the secondary motions correspond to supercritical pitchfork bifurcations and can occur only if the heating wave number a is reduced below - 0.14. One of the branches of such bifurcations is associated with an odd number of secondary rolls per half period, with rolls at the hot spots rotating in the direction opposite to the primary rolls. The other branch is associated with an even number of secondary rolls per half period, with the rolls at the hot spots co-rotating with the primary rolls. The number of rolls increases without limit as a decreases with new rolls being pinched off in pairs. The pinching process for each branch occurs at different d s and as a result the branches alternate in producing larger net heat flow. Increase of heating intensity to Ra >~ 470 results in secondary motions occurring at larger values of a, i.e., a > ~ 0 .1 4 , and bifurcation changing character into "bifurcations from infinity". The critical points for such bifurcations move towards higher d s with increase of Ra. The main branch of such type of bifurcation is associated with one pair of rolls per heating period when a > 0.25 in the range of Ra studied. Decrease of a results in the formation of secondary rolls with the rolls at the hot spots co-rotating with the primary rolls. The new rolls are always pinched off in pairs with this branch having always an even number of rolls per half period. The lower part of the other branch is associated with one pair of rolls per heating period when a is very small. Increase of a results in pinching off of a
single roll counter-rotating with the primary rolls at the hot spots. Additional rolls are pinched off in pairs as one moves along this branch resulting in an odd number of rolls per half period. The above processes are insensitive to variations of Prandtl number for Pr = 0(1) as only small differences in the bifurcation diagrams have been observed when Pr changes from Pr = 0.7 to Pr = 7. It is shown that the observed phenomena are strictly associated with the small wave number limit of the external heating.
Asgarian, Ali, "Natural Convection in a Slot Subject to a Long Wavelength Heating" (2011). Digitized Theses. 3439.