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Wall-resolved large-eddy simulations (LES) of the incompressible Navier–Stokes equations together with empirical modelling for turbulent Taylor–Couette (TC) flow are presented. LES were performed with the inner cylinder rotating at angular velocity Ω_i and the outer cylinder stationary. With R_i, R_₀ the inner and outer radii respectively, the radius ratio is η = 0.909 . The subgrid-scale stresses are represented using the stretched-vortex subgrid-scale model while the flow is resolved close to the wall. LES is implemented in the range Re_i = 10⁵-10⁶ where Re_i = Ω_iR_id/v and d = R₀-R_i is the cylinder gap. It is shown that the LES can capture the salient features of the flow, including the quantitative behaviour of spanwise Taylor rolls, the log variation in the inner-cylinder mean-velocity profile and the angular momentum redistribution due to the presence of Taylor rolls. A simple empirical model is developed for the turbulent, TC flow for both a stationary outer cylinder and also for co-rotating cylinders. This consists of near-wall, log-like turbulent wall layers separated by an annulus of constant angular momentum. Model results include the Nusselt number Nu (torque required to maintain the flow) and measures of the wall-layer thickness as functions of both the Taylor number Ta and η. These are compared with results from measurement, direct numerical simulation and the LES. A model extension to rough-wall turbulent flow is described. This shows an asymptotic, fully rough-wall state where the torque is independent of Re_i/Ta, and where Nu ~ Ta^(1/2).

We note that the mirror-symmetric solution in plane Couette flow, found recently by Gibson, Halcrow & Cvitanović (J. Fluid Mech., vol. 611, 2009, pp. 107–130) and Itano & Generalis (Phys. Rev. Lett., vol. 102, 2009, p. 114501), belongs to the solution group classified as ‘ribbon' in rotating-plane Couette flow (RPCF). It represents a subcritical (in terms of the system rotation) solution at zero rotation rate on the three-dimensional tertiary flow branch which bifurcates from the second streamwise-independent flow in RPCF. The way of its appearance is similar to that of the Nagata solution (J. Fluid Mech., vol. 217, 1990, pp. 519–527), which lies on the subcritical three-dimensional tertiary flow branch bifurcating from the first streamwise-independent flow in RPCF., Published online: 14 June 2013

We numerically simulate turbulent Taylor-Couette flow for independently rotating inner and outer cylinders, focusing on the analogy with turbulent Rayleigh-B\'enard flow. Reynolds numbers of $Re_i=8\cdot10^3$ and $Re_o=\pm4\cdot10^3$ of the inner and outer cylinders, respectively, are reached, corresponding to Taylor numbers Ta up to $10^8$. Effective scaling laws for the torque and other system responses are found. Recent experiments with the Twente turbulent Taylor-Couette ($T^3C$) setup and with a similar facility in Maryland at very high Reynolds numbers have revealed an optimum transport at a certain non-zero rotation rate ratio $a = -\omega_o / \omega_i$ of about $a_{opt}=0.33-0.35$. For large enough $Ta$ in the numerically accessible range we also find such an optimum transport at non-zero counter-rotation. The position of this maximum is found to shift with the driving, reaching a maximum of $a_{opt}=0.15$ for $Ta=2.5\cdot10^7$. An explanation for this shift is elucidated, consistent with the experimental result that $a_{opt}$ becomes approximately independent of the driving strength for large enough Reynolds numbers. We furthermore numerically calculate the angular velocity profiles and visualize the different flow structures for the various regimes. By writing the equations in a frame co-rotating with the outer cylinder a link is found between the local angular velocity profiles and the global transport quantities., Comment: Under consideration for publication in JFM, 31 pages, 25 figures

Introduction The objective of the following lecture notes is to provide a consistent account of the development of hydrodynamic instabilities in open flows such as mixing layers, wakes and boundary layers. These prototype shear flows are commonly encountered in a variety of technological applications in aerodynamics, mechanical and chemical engineering, and in many geophysical processes in the oceans and atmospheres. A physical understanding of the transition to turbulence in “simple” shear flows gives rise to fundamental issues that constitute the core of this course. Flows are typically classified according to their open or closed nature and to their laminar or turbulent character. In open flows , fluid particles are not recycled within the physical domain of interest but leave it in a finite time (Huerre and Monkewitz, 1990), as in the mixing layer generated downstream of a splitter plate by the merging of two parallel streams in relative motion. By contrast, in closed flows , fluid particles always remain within the same physical region, as in Taylor–Couette flow between two counter-rotating cylinders (Andereck et al ., 1986). The distinction between laminar and turbulent flows is related to the degree of spatial and temporal coherence of a given flow (Tennekes and Lumley, 1972; Monin and Yaglom, 1975; Landau and Lifshitz, 1987a; Lesieur, 1991; Frisch, 1995). Typically, turbulent flows arise at high Reynolds numbers and are characterized by the presence of a wide range of spatial and temporal scales, three-dimensional vorticity fluctuations and a certain degree of unpredictability.

Measurements at moderately large Reynolds numbers in a finite laminar circular Couette flow show that the tangential motion near the axial plane of symmetry is two-dimensional, within experimental accuracy, but is nevertheless strongly modified by end conditions.