On the Transition to Turbulent Convection 



The Second Transition 



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

 vection in the Rayleigh number range of the second transition is that 

 of Busse [ 1968] . He shows that for infinite Prandtl number, two- 

 dimensional rolls having wave-nunnber p within a finite band (see 

 Fig. 4) are stable to a restricted class of infinitesimal disturbances 

 provided that R< 22,600. If R > 22,600 rolls are unstable for all 

 (3. 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 p from this band of possible wave-numbers occurs. 



Laboratory studies [ Krishnamurti 1970a] show that two- 

 dimensional rolls do indeed become unstable near this Rayleigh 

 number, which will be labelled R-jj . The "plan forms" (obtained 

 fronn the side) are shown in Fig. 5a where that on the left shows 

 rolls below Rjj , that on the right shows the flow pattern above R^ . 

 The three-dimensional disturbance that forms on the rolls above 

 Rj2 is consistent with Busse's instability to a rectangular distur- 

 bance. 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 

 symmetry in the vertical. The point of transition is also in good 

 agreement with that computed by Busse, for that wave-number p 

 which occurs in the experiment. Figure 5b shows the same transi- 

 tion when a circular boundary of plexiglass has been inserted into 

 the rectangular 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 perpen- 

 dicular 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 displayed 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 con- 

 duction temperature 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. 5a). This corresponds to the 

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



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