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

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 $a = -ω_\mathrm{out}/ω_\mathrm{in}$ 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.

💡 Research Summary

The paper investigates why high‑Reynolds‑number Taylor‑Couette (TC) turbulence often exhibits a nearly constant mean angular momentum profile (r Uθ ≈ const) in the bulk, a feature observed in both laboratory and astrophysical rotating flows. The authors focus on the “featureless ultimate regime” (UR), where both inner and outer cylinders rotate weakly counter‑ or co‑rotating, Taylor rolls disappear, and turbulence fills the entire gap. Using Reynolds‑averaged Navier‑Stokes (RANS) equations, they first show that conventional algebraic Reynolds‑stress models, which neglect the convection term of the Reynolds stress, fail to reproduce the flat angular‑momentum profile. By analysing the transport equations for the Reynolds‑stress tensor, they identify the convection of Reynolds stress as a “history effect”: the stress at a given location carries information about upstream turbulent motions, especially important in curved, rotating geometries where curvature terms appear in the production terms.

To incorporate this history effect rigorously, the authors adopt the Jaumann derivative—a covariant time derivative that removes spurious rotation of tensors when changing reference frames. This derivative allows the Reynolds‑stress model to retain the normal‑stress difference (Rrr − Rθθ) that arises from curvature‑induced production, while remaining frame‑indifferent. The resulting algebraic model includes terms proportional to the mean strain rate and to the rotation‑rate tensor, with coefficients calibrated against high‑Re experiments.

Experimental validation is performed for angular‑velocity ratios a = −0.5 … 0.1 at Taylor numbers exceeding 10⁹. Measurements of the azimuthal velocity profile and the Reynolds shear stress Rrθ confirm that in co‑rotating cases (a ≥ 0) the bulk angular momentum is essentially constant, whereas weak counter‑rotation introduces a slight radial gradient. The Jaumann‑based model reproduces these trends with high fidelity, markedly improving upon the conventional model (error reduction >70 %).

A key theoretical insight is the equivalence between a constant angular momentum and zero mean absolute vorticity. By transforming to a rotating frame, the authors show that the condition ⟨WAij⟩ = 0 (zero absolute vorticity) leads directly to r Uθ = const, linking the observed profile to neutral stability. Standard eddy‑viscosity models cannot capture this because they ignore frame‑rotation effects. The history‑dependent Reynolds‑stress formulation thus provides a physical mechanism for the neutral‑stability state.

In summary, the study (1) identifies Reynolds‑stress convection as the essential mechanism behind the flat angular‑momentum profile in high‑Re TC turbulence, (2) formulates a covariant algebraic Reynolds‑stress model using the Jaumann derivative that incorporates this history effect, and (3) validates the model against extensive experiments, demonstrating its ability to predict the nearly constant angular momentum in co‑rotating TC flows. The work offers a new perspective on rotating and curved turbulent flows, with implications for modeling astrophysical accretion disks, geophysical flows, and other systems where curvature and rotation play dominant roles.