Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders

Abstract

Turbulent Taylor–Couette flow with arbitrary rotation frequencies ω1, ω2 of the two coaxial cylinders with radii r1 < r2 is analysed theoretically. The current Jω of the angular velocity ω(x,t) = uϕ(r,ϕ,z,t)/r across the cylinder gap and and the excess energy dissipation rate ϵw due to the turbulent, convective fluctuations (the ‘wind’) are derived and their dependence on the control parameters analysed. The very close correspondence of Taylor–Couette flow with thermal Rayleigh–Bénard convection is elaborated, using these basic quantities and the exact relations among them to calculate the torque as a function of the rotation frequencies and the radius ratio η = r1/r2 or the gap width d = r2 − r1 between the cylinders. A quantity σ corresponding to the Prandtl number in Rayleigh–Bénard flow can be introduced, $\sigma = ((1 + \eta)/2)/\sqrt{\etaacute;)^4$. In Taylor–Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh–Bénard flow. The analogue of the Rayleigh number is the Taylor number, defined as Ta ∝ (ω1 − ω2)2 times a specific geometrical factor. The experimental data show no pure power law, but the exponent α of the torque versus the rotation frequency ω1 depends on the driving frequency ω1. An explanation for the physical origin of the ω1-dependence of the measured local power-law exponents α(ω1) is put forward. Also, the dependence of the torque on the gap width η is discussed and, in particular its strong increase for η → 1.