Marginally stable and turbulent boundary layers in low-curvature Taylor-Couette flow

TL;DR

This paper uses marginal stability theory to explain the rotation-number dependence of torque in high-radius-ratio Taylor-Couette flow, accurately predicting observed torque maxima and boundary layer behaviors.

Contribution

It introduces a marginal stability-based model that quantitatively explains the torque variation and boundary layer characteristics in Taylor-Couette flow at high radius ratios.

Findings

Model explains broad torque shoulder near R_Ω≈0.2

Predicts boundary layer shear Reynolds numbers for R_Ω<0.07

Matches numerical simulations for torque maxima

Abstract

Marginal stability arguments are used to describe the rotation-number dependence of torque in Taylor-Couette (TC) flow for radius ratios $\eta \geq 0.9$ and shear Reynolds number $Re_S=2\times 10^4$. With an approximate representation of the mean profile by piecewise linear functions, characterized by the boundary-layer thicknesses at the inner and outer cylinder and the angular momentum in the center, profiles and torques are extracted from the requirement that the boundary layers represent marginally stable TC subsystems and that the torque at the inner and outer cylinder coincide. This model then explains the broad shoulder in the torque as a function of rotation number near $R_\Omega\approx 0.2$. For rotation numbers $R_\Omega < 0.07$ the TC stability conditions predict boundary layers in which shear Reynolds numbers are very large. Assuming that the TC instability is bypassed by some shear instability, a second maximum in torque appears, in very good agreement with numerical simulations. The results show that, despite the shortcomings of marginal stability theory in other cases, it can explain quantitatively the non-monotonic torque variation with rotation number for both the broad maximum as well as the narrow maximum.