Piacsek 



Recently, Chen and Whitehead (1968) have performed a careful laboratory 

 study of Benard convection by introducing small controlled perturbations prior 

 to the onset of motion, producing two-dimensional rolls of arbitrary width-to- 

 depth ratio. The conditions employed corresponded to the finite -amplitude 

 stability problem for a constant viscosity, large Prandtl number, Boussinesq 

 fluid with rigid, conducting boundaries. They found that two-dimensional cells 

 with width-to- depth ratios close to unity are stable at ail Ra investigated 

 (Ra^ < Ra < 2.5 Ra^), whereas for moderately too large or too small values 

 they tend to undergo size adjustments toward a preferred value of 1.1. Snyder 

 (1967) has performed experimental studies on wave-number selection of finite 

 amplitude between differentially rotating cylinders (the "Taylor" problem). He 

 showed that a finite-amplitude secondary flow may have any wavelength within 

 a range which depends on the amplitude. The reason is that the problem is in- 

 herently time-dependent, and that the actual wavelength selected is determined 

 by the initial conditions of the problem. He has experimentally found hysteresis 

 effects similar to the ones reported here. Meyer (1967) has performed a some- 

 what similar numerical experiment on preferred wavelengths in the nonlinear 

 region of Taylor flow. He made a big box of two complete cells and determined 

 the preferred length by varying the box length until the ratio of energy con- 

 tained in the even harmonics to that of the odd harmonics was a maximum. 



Numerical experiments have been performed on Benard convection by 

 Deardorff (1964, 1965) and Fromm (1965), but only for perturbations of \' =2 

 and horizontal (nondimensional) width y = 2,4, 8, and 20. 



The present investigation consists of two approaches. One purpose is to 

 determine the wavelength that transports the maximum heat, and then find if 

 that is the preferred wavelength. To this end, a single roll was considered, for 

 Ra = 20000 and y - \'/2 = 1.00. The latter value was chosen because it lies 

 very near the analytically predicted value of 1.002 by Roberts (1965), for Ra = 

 4000. The circulation in each roll in two-dimensional convection is in the op- 

 posite sense from that of its neighbors, and such that within each unidirectional 

 vortex there is symmetry about a diagonal. Furthermore, each is symmetrical 

 with respect to its neighbors; hence the cell boundaries can be defined as 

 "surfaces of symmetry." Such a definition was used by Chandrasekhar (1961) 

 and Stuart (1964) in their treatment of the Benard problem. It is sufficient, 

 therefore, to find the flow fields in only a half cell or single vortex, and from 

 that one may construct the whole circulation (see Fig. 8). The boundary condi- 

 tions on these surfaces become the following: no mass or heat flow may cross 

 the cell wall, and no stress may act upon it, i.e., they become frictionless, in- 

 sulating lids. It is of interest to note that heretofore these boundary conditions 

 have not been used in numerical work on Benard convection. Both Fromm (1965) 

 and Deardorff (1964) have assumed either rigid lateral boundaries or periodic 

 conditions on them. The present conditions enable us to deal with half -cells 

 only. 



The flow was started by assuming a linear temperature profile due to con- 

 duction only, with the temperature contrast already at its final value, and apply- 

 ing a temperature perturbation of the form A • cos nx • sin m. The finite- 

 difference versions of Eqs. (24) to (27) were iterated in time until a fully developed 

 steady- state convective flow was obtained. The aspect ratio 7 was then incre- 

 mented (or decremented) in steps of A > = .2, always using the previous steady 



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