Environmental Engineering Reference
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Fig. 3 Case R a = 5,000. It begins forming a concentric circles pattern. a Time is t = 3 . 8. b Time
is t = 4 . 2
Fig. 4 a The time that it takes to the perturbation to overcome a certain threshold versus Rayleigh
number. b The time that it takes to the system to attain a stationary state versus Rayleigh number
steady state. Both quantities must be decreasing functions of the Rayleigh number.
Our simulation satisfies this criterion as we can see from Fig. 4 . The calculation of
the first quantity consists of obtaining the time for which the velocity at a certain
point is no longer zero (in a numerical sense, see Fig. 4 a). For the second time we
determine the time elapsed until the velocity variations at a certain point are less
than a value
(see Fig. 4 b). In both figures, we observe that the time is a decreasing
function of the Rayleigh number.
4 Conclusions
The numerical simulations reproduce some of the patterns observed in experiments
other than the annular cells. If the Rayleigh number is not much bigger than the
critical value, the steady state consists of a collection of concentric rolls. As the
Rayleigh number increases, the time to reach the steady state decreases, and the rolls
pattern becomes unstable, as we showed for R a
=
2,500 case. For the R a
=
5,000
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