Overview
Research questions the long-held belief that statistically stationary states of Rayleigh-Bénard convection (RBC) are accurately described by statistically stationary solutions derived from the Boussinesq approximation. An analysis of the mechanical energy equation indicates that the implicit expression within the Boussinesq approximation for the mean conversion of internal energy into kinetic energy, through compressions and expansions, does not qualify as a valid approximation. This finding consequently challenges the validity of the exact Boussinesq expression which relates mean kinetic energy dissipation in a stationary state to the Rayleigh and Nusselt numbers.
Research Context
Rayleigh-Bénard convection (RBC) involves fluid motion driven by buoyancy forces arising from temperature differences. The Boussinesq approximation is a common simplification applied to the Navier-Stokes equations for such flows, assuming density variations are only significant in the buoyancy term. A fundamental assumption in the study of RBC has been that its statistically stationary states can be adequately characterized by stationary solutions derived using this approximation. This study directly investigates the justification for this assumption, particularly concerning the energy transformations within the system.
Approach
The research employed an analysis of the mechanical energy equation. This analysis focused on the expression within the Boussinesq approximation pertaining to the conversion of internal energy into kinetic energy, specifically through processes involving compressions and expansions of the fluid. Following this, the researchers derived new scaling relations for the mean kinetic energy dissipation in a stationary state. These derivations utilized assumptions that, in part, extended beyond the strict confines of the Boussinesq approximation. The analysis was conducted for both three-dimensional and two-dimensional systems.
Findings
- The expression for mean conversion of internal energy into kinetic energy by compressions and expansions, as implicitly given by the Boussinesq approximation, was found not to qualify as an approximation. This suggests a fundamental inaccuracy in the Boussinesq treatment of this energy conversion mechanism.
- The validity of the exact Boussinesq expression, which correlates the mean kinetic energy dissipation in a stationary state with the Rayleigh and Nusselt numbers, is therefore rendered questionable.
- Scaling relations for the mean kinetic energy dissipation in a stationary state were derived for three-dimensional systems and two-dimensional systems using assumptions partly outside the Boussinesq approximation.
- For the three-dimensional system, the derived scaling relation is similar to the Boussinesq expression, but it includes an unknown prefactor.
- For the two-dimensional system, the derived scaling relation is completely different from the Boussinesq expression.
- Statistically stationary solutions to the Boussinesq approximation in three dimensions are suggested to reproduce scaling relations between various statistical quantities quite well, despite not being approximations in a strict sense.
- Solutions to the Boussinesq approximation in two dimensions are suggested to completely fail in reproducing scaling relations between various statistical quantities.
Why This Matters
The findings challenge a foundational assumption in the study of Rayleigh-Bénard convection and the broader application of the Boussinesq approximation to fluid dynamics. The derived differences in scaling relations between two and three dimensions, and the observed limitations of the approximation in correctly representing energy conversions, highlight areas where current theoretical models may be insufficient. This suggests a need for re-evaluation in specific contexts of fluid dynamics research.
Potential Applications
The study suggests a future investigative step: performing Direct Numerical Simulations (DNS) of weakly compressible RBC and comparing these with DNS of Boussinesq RBC. This comparative analysis would serve to further investigate the identified discrepancies and the applicability of the Boussinesq approximation across different dimensionality and compressibility regimes.