We derive the velocity profiles in strongly turbulent Taylor-Couette flow for the general case of independently rotating cylinders. The theory is based on the Navier-Stokes equations in the appropriate (cylinder) geometry. In particular, we derive the axial and the angular velocity profiles as functions of distance from the cylinder walls and find that both follow a logarithmic profile, with downwards-bending curvature corrections, which are more pronounced for the angular velocity profile as compared to the axial velocity profile, and which strongly increase with decreasing ratio η between inner and outer cylinder radius. In contrast, the azimuthal velocity does not follow a log-law. We then compare the angular and azimuthal velocity profiles with the recently measured profiles in the ultimate state of (very) large Taylor numbers. Though the qualitative trends are the same – down-bending for large wall distances and the (properly shifted and non-dimensionalized) angular velocity profile ω+(r) being closer to a log-law than the (properly shifted and non-dimensionalized) azimuthal velocity profile

$u^+_{\varphi }(r)$
uφ+(r)quantitativedeviations are found for large wall distances. We attribute these differences to the nonlinear dependence of the turbulent ω-diffusivity on the wall distance and partly also to the Taylor rolls and the axial dependence of the profiles, neither of which are considered in the theoretical approach. Assuming that the first origin is the most relevant one, we calculate from the experimental profile data how the turbulent ω-diffusivity depends on the wall distance and find a linear behavior for small wall distances as assumed and a saturation behavior for very large distances, reflecting the finite gap width: But in between the ω-diffusivity increases stronger than linearly, reflecting that more eddies can contribute to the turbulent transport (or they contribute more efficiently) as compared to the plane wall case.

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The velocity fluctuation scale u* is also called wall velocity unit, velocity fluctuation, or velocity fluctuations amplitude, and in accordance with literature we will use all these synonyms.
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