On high Taylor number Taylor vortices

TL;DR

This paper investigates high Taylor number Taylor vortices through numerical and theoretical methods, revealing how the Nusselt number varies with axial period and providing asymptotic analysis that links to turbulence regimes.

Contribution

It introduces a detailed asymptotic analysis of Taylor vortices at high Taylor numbers, connecting steady solutions to turbulence characteristics and extending scaling laws to Rayleigh-Bénard convection.

Findings

Nusselt number exhibits two peaks as axial period shortens.

Theoretical Nusselt number scales with the quarter power of Taylor number.

Maximum Nusselt number aligns with experimental values near turbulence onset.

Abstract

Axisymmetric steady solutions of Taylor-Couette flow at high Taylor numbers are studied numerically and theoretically. As the axial period of the solution shortens from about one gap length, the Nusselt number goes through two peaks before returning to laminar flow. In this process, the asymptotic nature of the solution changes in four stages, as revealed by the asymptotic analysis. When the aspect ratio of the roll cell is about unity, the solution captures quantitatively the characteristics of the classical turbulence regime. Theoretically, the Nusselt number of the solution is proportional to the quarter power of the Taylor number. The maximised Nusselt number obtained by shortening the axial period can reach the experimental value around the onset of the ultimate turbulence regime, although at higher Taylor numbers the theoretical predictions eventually underestimate the experimental values. An important consequence of the asymptotic analyses is that the mean angular momentum should become uniform in the core region unless the axial wavelength is too short. The theoretical scaling laws deduced for the steady solutions can be carried over to Rayleigh-B\'enard convection.