Covariant algebraic Reynolds stress modelling of curvature effects in high-Reynolds-number Taylor--Couette turbulence

Nearly constant mean angular momentum profiles are widely observed in curved turbulent flows, including the bulk region of Taylor--Couette (TC) flows, where the inner and outer cylinders have weakly counter-rotating and co-rotating conditions. For high-Reynolds-number TC flows under these conditions, both the bulk and boundary layers become turbulent without Taylor rolls, referred to as the featureless ultimate regime (UR). In this study, we examine Reynolds-averaged Navier--Stokes (RANS) models to predict the nearly constant mean angular velocity as a one-dimensional problem in the featureless UR of TC turbulence. High-Reynolds-number experiments of TC turbulence are performed for reference, where the radius ratio is $\eta = r_\mathrm{in}/r_\mathrm{out} = 0.732$ and 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 model using the algebraic Reynolds stress model (ARSM) suggests that convection of the Reynolds stress is essential for predicting the angular momentum profile. We introduce the Jaumann derivative as a covariant time derivative to develop ARSMs that incorporate the convection effect in a covariant manner. The proposed ARSM using the Jaumann derivative of the term composed of the strain and vorticity tensors successfully predicts the nearly constant mean angular momentum for a wide range of angular velocity ratios in the co-rotating case. The modelling approach incorporating time-derivative terms is a candidate for expressing curvature effects while satisfying the covariance of the Reynolds stress tensor.