Transient high-Rayleigh-number thermal convection with large viscosity variations

Abstract

The characteristics of thermal convection in a fluid whose viscosity varies strongly with temperature are studied in the laboratory. At the start of an experiment, the upper boundary of an isothermal layer of Golden Syrup is cooled rapidly and maintained at a fixed temperature. The fluid layer is insulated at the bottom and cools continuously. Rayleigh numbers calculated with the viscosity of the well-mixed interior are between 106 and 108 and viscosity contrasts are up to 106. Thermal convection develops only in the lower part of the thermal boundary layer, and the upper part remains stagnant. Vertical profiles of temperature are measured with precision, allowing deduction of the thickness of the stagnant lid and the convective heat flux. At the onset of convection, the viscosity contrast across the unstable boundary layer has a value of about 3. In fully developed convection, this viscosity contrast is higher, with a typical value of 10. The heat flux through the top of the layer depends solely on local conditions in the unstable boundary layer and may be written \[Q_{\rm s} = - CK_{\rm m} (\alpha g/\kappa \nu_{\rm m})^{\frac{1}{3}} \Delta T^{\frac{4}{3}}_{\rm v}\], where km and νm are thermal conductivity and kinematic viscosity at the temperature of the well-mixed interior, κ thermal diffusivity, α the coefficient of thermal expansion, g the acceleration due to gravity. ΔTv, is the ‘viscous’ temperature scale defined by \[\Delta T_{\rm v} = - \frac{\mu (T_{\rm m})}{({\rm d}\mu /{\rm d}T)(T_{\rm m})}\] where μ(T) is the fluid viscosity and Tm the temperature of the well-mixed interior. Constant C takes a value of 0.47 ± 0.03. Using these relations, the magnitude of temperature fluctuations and the thickness of the stagnant lid are calculated to be in excellent agreement with the experimental data. One condition for the existence of a stagnant lid is that the applied temperature difference exceeds a threshold value equal to (2ΔTv).