An empirical torque relation for supercritical flow between rotating cylinders

Abstract

An empirical relationship for the torque transmitted by fluid friction to an outer cylinder as a function of the angular velocity of the inner cylinder has been obtained by analysis of experimental data published by Wendt, Taylor and Donnelly. With one exception it is found that the torque G has the functional form $G = a \Omega^{-1}_1 + b\Omega ^{1 \cdot 36}_1,$ where Ω₁ is the angular velocity of the inner cylinder and a and b are constants determined from the data. The formula applies to a range of values of Ω₁ above the onset of instability extending to about 10 times the critical angular velocity. The experiments also show that the finite amplitude analysis recently advanced by Stuart gives the correct variation of torque over a short range above the critical speed. At speeds well beyond critical it is found that G varies approximately as Ω^1.5₁, and that the variation of torque with gap width can be expressed as a simple power law with exponent about 0.31. In an appendix Dr G. K. Batchelor shows that these latter relations are consistent with the supposition that the flow is steady and consists of inviscid cores surrounded by boundary layers.