Explanation of constant mean angular momentum in high-Reynolds-number Taylor--Couette turbulence in terms of history effects

TL;DR

This paper explains the nearly constant mean angular momentum in high-Reynolds-number Taylor--Couette turbulence using Reynolds-averaged Navier--Stokes equations and history effects of Reynolds stress, validated by experiments.

Contribution

It introduces a model incorporating the Jaumann derivative to account for history effects, successfully predicting angular momentum profiles in turbulent TC flows.

Findings

Reynolds stress convection is key to angular momentum distribution.

History effects of Reynolds stress influence turbulence in curved flows.

The model accurately predicts constant angular momentum in co-rotating cases.

Abstract

This study discusses the mechanism of the emergence of nearly constant mean angular momentum profiles, which are widely observed in curved turbulent flows including the bulk region of Taylor--Couette (TC) flows. For high-Reynolds-number TC flows where the inner and outer cylinders are weakly counter-rotating and co-rotating, both the bulk and boundary layers become turbulent without Taylor rolls, referred to as the featureless ultimate regime (UR). Thus, we utilize the Reynolds-averaged Navier--Stokes (RANS) equations to explain the mechanism of the nearly constant mean angular momentum. High-Reynolds-number experiments of TC turbulence are performed for reference, where the angular velocity ratio $a = -\omega_\mathrm{out}/\omega_\mathrm{in}$ is in the range $-0.5 \le a \le 0.1$. Verification of the RANS based on the conventional algebraic Reynolds stress model suggests that convection of the Reynolds stress is essential for predicting the angular momentum profile. This indicates that the physical origin of the nearly constant angular momentum is the history effect of the Reynolds stress. To rigorously incorporate the convection effect into the Reynolds stress, we employ the Jaumann derivative as a covariant time derivative. The model that takes into account the history effect involving the normal stress difference successfully predicts the nearly constant mean angular momentum in the co-rotating cases. This study suggests the significance of the history effects for understanding curved or rotating turbulent flows in terms of the statistical analysis.