Spatio-temporal proper orthogonal decomposition of turbulent channel flow

Abstract

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]

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