Direct numerical simulations of Taylor–Couette flow for vanishingly small inner cylinders

We investigate turbulent Taylor–Couette flow between two concentric cylinders, where the inner cylinder of radius $r_i$ rotates while the outer one of radius $r_o$ remains stationary. Using direct numerical simulations, we examine how varying the radius ratio $\eta = r_i / r_o$ from $\eta = 0.714$ down to $0.0244$ affects the flow characteristics at low to moderate Reynolds numbers. Our results show significant changes in the flow structures and statistics in the limit of a vanishingly small inner radius. The turbulent kinetic energy, scaled with the friction velocity at the inner cylinder, does not exhibit a self-similar scaling; instead, it decreases with decreasing $\eta$ . The turbulent kinetic energy budgets reveal that the locations of peak production and total dissipation are independent of $\eta$ , whereas their amplitudes decrease as $\eta$ increases. The pressure–velocity correlation near the inner cylinder is large for small $\eta$ and its amplitude decreases with increasing $\eta$ , while the turbulent transport term exhibits the opposite trend. Numerical simulations for $\eta \leqslant 0.5$ show that, for our specific set-up, a rather good collapse of the distribution of the normalised torque versus the Taylor number ( $ \textit{Ta}$ ) is obtained when the latter is defined according to Chandrasekhar ( Hydrodynamic and Hydromagnetic Stability , Oxford Univ. Press, 1961), with a tendency towards a $ \textit{Ta}^{1/3}$ regime at sufficiently large $ \textit{Ta}$ .