On the Transition to Tuvhulent Convection 



The Second Transition 



The only theoretical study of stability of two-dimensional con- 

 vection In this Raylelgh number range Is that of Busse [ 1968] . He 

 shows that for Infinite Prandtl number, two-dimensional rolls having 

 wave-number p within a finite band (see Fig. 4) are stable to a 

 restricted class of Infinites Innal disturbances provided that R < 22,600. 

 If R > 22,600 rolls are unstable for all p. Busse shows further that 

 the roll plan form Is then unstable to a disturbance of rectangular 

 form with one side along the original roll axis. It Is not known from 

 this theory whether the resulting flow above 22,600 Is steady. It Is 

 also not known how the selection of (3 from this- band of possible 

 wave-numbers occurs. 



Laboratory studies [ Krlshnamurtl 1970a] show that two- 

 dimensional rolls do Indeed become unstable near this Raylelgh 

 number, which will be labelled Rjj . The "plan forms" (obtained 

 from the side) are shown In Fig. Da where that on the left shows rolls 

 below Rjj , that on the right shows the flow pattern above Rjj . The 

 three-dimensional disturbance that forms on the rolls above R^, 

 is consistent with Busse 's Instability to a rectangular disturbance. 

 Since the method of photography displays regions of strong shear, 

 the hypotenuse of the rectangle should appear bright. Thus, the 

 nature of the growing mode (which Is found experimentally to 

 attain a steady state) Is In agreement with Busse's result. It may be 

 noted that the rectangular disturbance of his theory Is one with sym- 

 mietry In the vertical. The point of transition Is also In good agree- 

 ment with that computed by Busse, for that wave-number P which 

 occurs In the experiment, although the selection mechanism of that 

 P Is not understood. Figure 5b shows the same transition when a 

 circular boundary of plexiglass has been Inserted within the rectangu- 

 lar region. Both Davis [ 1967] and Segel [ 1969] show that spatially 

 modulated rolls will line up with their axes parallel to the short side 

 of a rectangular container. In the almost square container, there 

 appeared to be little preference of orientation of the rolls; rolls 

 were seen along the line of sight as well as perpendicular to the line 

 of sight in two different repetitions of the same experiment. The 

 preference of rolls to line up with their axes parallel to the short 

 side may be re-expressed as a preference of the rolls to meet the 

 boundaries rather than lie along the boundaries. This effect Is dis- 

 played In Fig, 5b, Presumably circular rolls did not develop 

 because the plexiglass has thermal conductivity so close to that of 

 the fluid that there was negligible distortion of the conduction tem- 

 perature field and no fringing of the Isotherms since there was fluid 

 outside of the ring. 



Associated with this change from steady two-dimensional to 

 steady three-dimensional flow, there Is observed a discrete change 

 In slope of the heat flux curve (Fig, 5). This corresponds to the 

 second change of slope observed by Malkus [ 1954] . Rj, showed no 



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