Convection in Glycerol: Thermal Convection with Broken Symmetry

Jun Zhang, Stephen Childress, and Albert Libchaber

In an experiment set up and run at Rockefeller University, Zhang and Libchaber studied the these non-Boussinesq features of convection in a 20cm cube of glycerol, at Rayleigh numbers 10⁶-10⁹ and Prandtl numbers 10²-10³. In connection with this work we have developed a nonlinear thermal boundary layer model to explain the scaling observed in this and other experiments, and to predict the bulk temperature of the fluid. The letter deviates from the arithmetic mean of the wall temperatures, because of the differing structure of the top and bottom thermal layers.

We have also examined the mean velocity near within the thermal layers, and compared this with the theoretical model. In all respects the model agrees qualitatively with the experimental findings, thus explaining the non-Boussinesq effects as principally a result of the nonlinear effect of variable viscosity on the thermal boundary layer equation.

In an ongoing extension of this work, we have set up in the WetLab a second generation of the glycerol experiment, and developed a theoretical model of the large-scale flow developed in the experiment. In our model, the mean flow is viewed as an instability of well-established upward and downward thermal plumes.

[1] J. Zhang, S. Childress, and A. Libchaber, Non-Boussinesq effect: thermal convection with broken symmetry, Phys. Fluids {\bf 9}, (4), 1034 (1997).

[2] J. Zhang, S. Childress, and A. Libchaber, Non-Boussinesq effect: asymmetric velocity profiles in thermal convection. Submitted to Physics of Fluids as a brief communication (1997).

Abstract for ref [1]:
We investigate large Rayleigh number (10⁶-10⁹) and large Prandtl number (10²-10³) thermal convection in glycerol in as aspect ratio one cubic cell. The kinematic viscosity of the fluid strongly depends upon the temperature. The symmetry between the top and bottom boundary layers is thus broken, the so-called non-Boussinesq regime. In a previous paper Wu and Libchaber have proposed that in such a state the two thermal boundary layers adjust their length scales so that the mean hot and cold temperature fluctuations are equal in the center of the cell. We confirm this equality. A simplified two-dimensional model for the mean center temperature based on an equation for the thermal boundary layer is presented and compared with the experimental results. The conclusion is that the central temperature adjusts itself so that the heat fluxes from the boundary layers are equal, temperature fluctuations at the center symmetrical, at a cost of very different temperature drops and Rayleigh number for each boundary.

Abstract for ref [2]:
In thermal convection at high Rayleigh numbers, in the hard turbulent regime, a large scale flow is present. When the viscosity of the fluid strongly depends on temperature, the top-bottom symmetry is broken. In addition to the asymmetric temperature profile across the convection cell, the velocity profiles near the plate boundaries show dramatic differences from the symmetric case. We report here that the second derivative of the velocity profiles are of opposite signs in the thermal sub-layers, through measurements derived from the power spectrum of temperature time-series. As a result, the stress rate applied at the plates is maintained constant within a factor of 3, while the viscosity changes by a factor of 53i, in qualitative agreement with previous theory.