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1992 A thermal convection loop

Christophe LETELLIER
09/05/2009
Yuzhou Wang, Jonathan Singer & Haim Bau
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Haim Bau

When the Rayleigh-Bénard experiments were carried out, the behaviors observed as the temperature difference \Delta T is increased did not correspond at all to those computed with the model — too simplified — of Lorenz. The basic reason is that the approximations made to obtain the Lorenz system are not applicable unless the fluid is at rest (there is no convection in the fluid). In the midst of the 1970s, as the Lorenz system began to be adopted as a description for some types of irregular behavior, an experiment [1] was conceived that corresponded to the dynamics described by the Lorenz equations : it was the inverse of the usual procedure. Usually the model is modified to describe the experiment ; this time the experiment was designed to suit the model.

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Fig. 1 : Schematic description of the convection experiment.

The experiment consisted of an annular ring, with internal diameter d=0.03 m, in a vertical orientation (Fig. 1). The ring has diameter D=0.76 m. A fluid is allowed to circulate within the tube. The liquid is heated by a heating ribbon in the lower half of the tube and cooled by a water cooling jacket in the upper half of the tube. The fluid temperature was measured at two diametrically opposite points in the ring, in the cooled part of the ring.

- The system

Following the procedure used by Lorenz Haim Bau’s group reduced the Navier-Stokes equations to three ordinary differential equations [2]


  \left\{
    \begin{array}{l}
    \dot{x} = \sigma (y-x) \\[0.1cm]
    \dot{y} = -y-xz \\[0.1cm]
    \dot{z} = -z + xy -R
    \end{array}
  \right.

where R is the Rayleigh number and  \sigma the Prandtl number. These equations are not exactly similar to the those obtained by Lorenz, but the departure does not imply notable changes in the nature of its solution. For instance, with R=18.5 and \sigma=6, this system produces a chaotic attractor (Fig. 2) topologically equivalent to the Lorenz attractor. This attractor is governed by a tearing mechanism that takes place in the neighborhood of the saddle fixed point located at the origin of the phase space.

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Fig. 2 : Chaotic "Lorenz" attractor solution to the Wang-Singer-Bau system.

For other parameter values - R=128 and \sigma=21.7 - the chaotic attractor is topologically equivalent to a "Burke and Shaw" attractor (Fig. 3).

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Fig. 3 : Chaotic "Burke and Shaw" attractor solution to the WAng-Singer-Bau system.

[1] H. F. Creveling, J. F. de Paz, J. Y. Baladi & R. H. Schoenhals, Stability Characteristics of a single thermal convection loop, Journal of Fluid Mechanics, 67, 65-84, 1975.

[2] Y.-Z. Wang, J. Singer & H. H. Bau, Controlling Chaos in a thermal convectionloop, Journal of Fluid Mechanics, 237, 479-498, 1992.

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