Applicability of Boussinesq approximation in a turbulent fluid with constant properties

TL;DR

This paper derives criteria to assess when the Boussinesq approximation remains valid in turbulent buoyant flows, showing it generally holds in laboratory conditions but may fail at extremely high Rayleigh numbers or in atmospheric convection.

Contribution

It introduces four criteria for evaluating the applicability of the Boussinesq approximation in turbulent convection with constant fluid properties, linking validity to stratification and temperature differences.

Findings

Most laboratory experiments satisfy the Boussinesq criteria.

Ultra high Rayleigh number helium experiments may violate the approximation.

Potential failure of Boussinesq approximation in atmospheric convection at high Rayleigh numbers.

Abstract

The equations of motion describing buoyant fluids are often simplified using a set of approximations proposed by J. Boussinesq one century ago. To resume, they consist in assuming constant fluid properties, incompressibility and conservation of calories during heat transport. Assuming fulfilment of the first requirement (constant fluid properties), we derive a set of 4 criteria for assessing the validity of the two other requirements in turbulent Rayleigh-B\'enard convection. The first criterion $\alpha \Delta \ll 1 $ simply results from the incompressibility condition in the thermal boundary layer ($\alpha$ and $\Delta$ are the thermal expansion coefficient and the temperature difference driving the flow). The 3 other criteria are proportional or quadratic with the density stratification or, equivalently with the temperature difference resulting from the adiabatic gradient across the cell $\Delta_{h}$. Numerical evaluations with air, water and cryogenic helium show that most laboratory experiments are free from such Boussinesq violation as long as the first criterion is fulfilled. In ultra high Rayleigh numbers ($Ra>10^{16}$) experiments in He, one of the stratification criteria, scaling with $\alpha \Delta_{h}$, could be violated. This criterion garanties that pressure fluctuations have a negligible influence both on the density variation and on the heat transfer equation through compression/expansion cycles. Extrapolation to higher $Ra$ suggests that strong violation of Boussinesq approximation could occur in atmospheric convection.