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Free keywords:
Bénard convection; turbulent convection
Abstract:
Rayleigh–Bénard convection, i.e. the flow of a fluid between two parallel plates
that is driven by a temperature gradient, is an idealised set-up to study thermal
convection. Of special interest are the statistics of the turbulent temperature field,
which we are investigating and comparing for three different geometries, namely
convection with periodic horizontal boundary conditions in three and two dimensions
as well as convection in a cylindrical vessel, in order to determine the similarities and
differences. To this end, we derive an exact evolution equation for the temperature
probability density function. Unclosed terms are expressed as conditional averages of
velocities and heat diffusion, which are estimated from direct numerical simulations.
This framework lets us identify the average behaviour of a fluid particle by revealing
the mean evolution of a fluid with different temperatures in different parts of the
convection cell. We connect the statistics to the dynamics of Rayleigh–Bénard
convection, giving deeper insights into the temperature statistics and transport
mechanisms. We find that the average behaviour is described by closed cycles in
phase space that reconstruct the typical Rayleigh–Bénard cycle of fluid heating up at
the bottom, rising up to the top plate, cooling down and falling again. The detailed
behaviour shows subtle differences between the three cases.
