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

Turbulent Taylor-Couette flow with arbitrary rotation frequencies omega(1), omega(2) of the two coaxial cylinders with radii r(1) < r(2) is analysed theoretically. The current J(omega), of the angular velocity omega(x, t) = u(phi)(r, phi, z, t)/r across the cylinder gap and and the excess energy dissipation rate E, 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-Benard 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 eta = r(1)/r(2) or the gap width d = r(2) - r(1) between the cylinders. A quantity or corresponding to the Prandtl number in Rayleigh-Benard flow can be introduced, sigma= ((1 + eta)/2)/root eta)(4). Taylor-Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh-Benard flow. The analogue of the Rayleigh number is the Taylor number, defined as Ta proportional to (omega(1) - omega(2))(2) times a specific geometrical factor. The experimental data show no pure power law, but the exponent ce of the torque versus the rotation frequency omega(1) depends on the driving frequency omega(1). An explanation for the physical origin of the omega(1)-dependence of the measured local power-law exponents alpha(omega(1)) is put forward. Also, the dependence of the torque on the gap width eta is discussed and, in particular its strong increase for eta -> 1.