Transition to turbulence scaling in Rayleigh-B\'{e}nard convection

TL;DR

This paper investigates how the moments of kinetic energy dissipation rate depend on Reynolds number in Rayleigh-Bénard convection, revealing non-monotonic behavior at low Reynolds numbers before aligning with turbulence scaling laws.

Contribution

It provides the first detailed analysis of dissipation rate moments in thermal convection across a wide Reynolds number range, connecting initial complex flow states to turbulence scaling.

Findings

Normalized moments show non-monotonic behavior at low Re

Data partially agree with turbulence theory

Implications for transition to turbulence in convection

Abstract

If a fluid flow is driven by a weak Gaussian random force, the nonlinearity in the Navier-Stokes equations is negligibly small and the resulting velocity field obeys Gaussian statistics. Nonlinear effects become important as the driving becomes stronger and a transition occurs to turbulence with anomalous scaling of velocity increments and derivatives. This process has been described by V. Yakhot and D. A. Donzis, Phys. Rev. Lett. 119, 044501 (2017) for homogeneous and isotropic turbulence (HIT). In more realistic flows driven by complex physical phenomena, such as instabilities and nonlocal forces, the initial state itself, and the transition to turbulence from that initial state, are much more complex. In this paper, we discuss the Reynolds-number-dependence of moments of the kinetic energy dissipation rate of orders 2 and 3 obtained in the bulk of thermal convection in the Rayleigh-B\'{e}nard system. The data are obtained from three-dimensional spectral element direct numerical simulations in a cell with square cross section and aspect ratio 25 by A. Pandey et al., Nat. Commun. 9, 2118 (2018). Different Reynolds numbers $1 \lesssim {\rm Re}_{\ell} \lesssim 1000$ which are based on the thickness of the bulk region $\ell$ and the corresponding root-mean-square velocity are obtained by varying the Prandtl number Pr from 0.005 to 100 at a fixed Rayleigh number ${\rm Ra}=10^5$. A few specific features of the data agree with the theory but the normalized moments of the kinetic energy dissipation rate, ${\cal E}_n$, show a non-monotonic dependence for small Reynolds numbers before obeying the algebraic scaling prediction for the turbulent state. Implications and reasons for this behavior are discussed.