Predicting transition ranges to turbulent viscous boundary layers in low Prandtl number convection flows

TL;DR

This paper uses high-resolution simulations to analyze turbulent convection at very low Prandtl numbers, predicting the Rayleigh number ranges where boundary layers become turbulent and the flow enters the ultimate convection regime.

Contribution

It provides the first detailed comparison of statistical turbulence quantities at very low Prandtl numbers and predicts transition ranges to turbulence using extrapolation methods.

Findings

Transition Rayleigh numbers decrease with lower Prandtl numbers.

Inertial effects dominate momentum transport at low Prandtl numbers.

Simulations suggest boundary layer turbulence onset at specific Rayleigh ranges.

Abstract

We discuss two aspects of turbulent Rayleigh-B\'{e}nard convection (RBC) on the basis of high-resolution direct numerical simulations in a unique setting; a closed cylindrical cell of aspect ratio of one. First, we present a comprehensive comparison of statistical quantities such as energy dissipation rates and boundary layer thickness scales. Data are used from three simulation run series at Prandtl numbers $Pr$ that cover two orders of magnitude. In contrast to most previous studies in RBC the focus of the present work is on convective turbulence at very low Prandtl numbers including $Pr=0.021$ for liquid mercury or gallium and $Pr=0.005$ for liquid sodium. In this parameter range of RBC, inertial effects cause a dominating turbulent momentum transport that is in line with highly intermittent fluid turbulence both in the bulk and in the boundary layers and thus should be able to trigger a transition to the fully turbulent boundary layers of the ultimate regime of convection for higher Rayleigh number. Secondly, we predict the ranges of Rayleigh numbers for which the viscous boundary layer will transition to turbulence and the flow as a whole will cross over into the ultimate regime. These transition ranges are obtained by extrapolation from our simulation data. The extrapolation methods are based on the large-scale properties of the velocity profile. Two of the three methods predict similar ranges for the transition to ultimate convection when their uncertainties are taken into account. All three extrapolation methods indicate that the range of critical Rayleigh numbers $Ra_c$ is shifted to smaller magnitudes as the Prandtl number becomes smaller.