Krishnamurti 



The studies that will be described here were performed as 

 externally steady, fixed heat flux experiments. Rayleigh number 

 and heat flux were measured for fluids having Prandtl number from 

 10"2to 10". The Rayleigh number ranged from 10^ to 10^ Except 

 in the cases of air and mercury, the "plan form" of the convection 

 was obtained by viewing from the side. The time dependence was 

 determined by both the (x,t) photographs and thermocouples 

 internal to the fluid. In the cases of air and mercury time dependence 

 was determined only by the internal thermocouples. 



The First Transition 



In the order of increasing R, the first transition occurs at 

 the well-known critical Rayleigh number R,-. This is a transition 

 from the conduction state to one of steady cellular convection. It 

 occurs independently of the Prandtl number Pr where Pr = v//c. 

 The nature of the flow and the change in slope of the heat flux curve 

 have been predicted and experinaentally verified. For the vertically 

 symmetric problen^ the only stable finite amplitude solution of the 

 infinite number of possible steady solutions is the two-dimensional 

 roll [Schluter, Lortz and Busse 1965] . With a vertical asymmetry, 

 such as that produced by changing mean temperature or by variation 

 of material properties {'^,K,0!) with temperature , the conduction 

 state is subcritically unstable to finite amplitude disturbance, and 

 the flow near the critical point is hexagonal [ Busse 1962; Segel and 

 Stuart 1962; Krishnamurti 1968a, b] . In this discussion we restrict 

 our attention to the case in which rolls are the realized flow just 

 above Re. 



As the heat flux, and hence the Rayleigh number, are increased 

 above Re, steady two-dimensional rolls continue to be the observed 

 flow up to approximately iZ R^, for 10< Pr < 10^. The size of the 

 rolls becomes larger in this range, as shown in Fig. 4, where the 

 wave-number (3 is plotted against Raleigh number. This increased 

 size of the cell might be rationalized by an argument such as the follow- 

 ing. By averaging over the entire fluid the non-dimensionalized tem- 

 perature equation in the Boussinesq approximation one finds 



H = R(r„ + <w9; 



where H is the dimensionles s heat flux, 0-^ is the vertical tem- 

 perature gradient averaged over the entire fluid, w is the verticail 

 velocity, 6 is the departure of the temperature from a horizontal 

 average, and brackets indicate averaging over the entire fluid. 

 Thus Rcr^ is the heat flux due to conduction, ( w9) is the convective 

 heat flux. As the externally imposed heat flux is increased such that 

 R exceeds R(., the fluid transfers this larger flux through the cor- 

 relation (w9) . Consider a fluid parcel near the lower boundary. 

 Its temperature 9 is limited by the thermal diffusivity of the fluid 

 material. As H is continually increased, the fluid is forced to 



294 



