An extension of Proper Orthogonal Decomposition is applied to the wall layer of a turbulent channel flow (Re τ = 590), so that empirical eigenfunctions are defined in both space and time. Due to the statistical symmetries of the flow, the eigenfunctions are associated with individual wavenumbers and frequencies. Self-similarity of the dominant eigenfunctions, consistent with wall-attached structures transferring energy into the core region, is established. The most energetic modes are characterized by a fundamental time scale in the range 200-300 viscous wall units. The full spatio-temporal decomposition provides a natural measure of the convection velocity of structures, with a characteristic value of 12u τ in the wall layer. Finally, we show that the energy budget can be split into specific contributions for each mode, which provides a closed-form expression for nonlinear effects. Key words:where a n is stochastic and χ n are orthogonal, square integrable, functions. In all that follows the superscript refers to the mode index. It is important to note that U (t) represents a stochastic variable and not a sample. Realizations of U (t) will be noted u(t). The functions χ n (t) are the eigenfunctions of the covariance function K U (t, t ) = E[U t U t ], where the operator E refers to expectation with respect to the measure of U . In the Karhunen-Loève derivation, the variable t corresponds to time, but it could indicate any other variable -such as space.Lumley (1967) (see also Berkooz et al. (1993)) adapted the decomposition to Fluid Mechanics: the samples were constituted by flow realizations, and the ergodicity assumption was used to replace the covariance function corresponding to an ensemble average with † Email address for correspondence: Berengere.Podvin@limsi.fr arXiv:1805.01494v3 [physics.flu-dyn]

This paper concerns steady, high-Reynolds-number flow around a semi-infinite, rotating cylinder placed in an axial stream and uses boundary-layer type of equations which apply even when the boundary-layer thickness is comparable to the cylinder radius, as indeed it is at large enough downstream distances. At large rotation rates, it is found that a wall jet appears over a certain range of downstream locations. This jet strengthens with increasing rotation, but first strengthens then weakens as downstream distance increases, eventually disappearing, so the flow recovers a profile qualitatively similar to a classical boundary layer. The asymptotic solution at large streamwise distances is obtained as an expansion in inverse powers of the logarithm of the distance. It is found that the asymptotic radial and axial velocity components are the same as for a non-rotating cylinder, to all orders in this expansion.