Mode interaction models for near-wall turbulence

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Abstract
Intermittent bursting events, similar to those characterizing the dynamics of near-wall turbulence, have been observed in a low-dimensional dynamical model (Aubry et al. 1988) built from eigenfunctions of the proper orthogonal decomposition (Lumley 1967). In the present work, we investigate the persistency of the intermittent behaviour in higher - but still of relatively low-dimensional dynamical systems. In particular, streamwise variations which were not accounted for in an explicit way in Aubry et al.'s model are now considered. Intermittent behaviour persists but can be of a different nature. Specifically, the non-zero streamwise modes become excited during the eruptive events so that rolls burst downstream into smaller scales. When structures have a finite length, they travel at a convection speed approximately equal to the mean velocity at the top of the layer (y+ ≈ 40). In all cases, intermittency seems to be due to homoclinic cycles connecting hyperbolic fixed points or more complex (apparently chaotic) limit sets. While these sets lie in the zero streamwise modes invariant subspace, the connecting orbits consist of nonzero streamwise modes travelling downstream. Chaotic limit sets connected by quasi-travelling waves have also been observed in a spatio-temporal chaotic regime of the Kuramoto–Sivashinsky equation (Aubry & Lian 1992a). When the limit sets lose their steadiness, the elongated rolls become randomly active, as they probably are in the real flow. A coherent structure study in our resulting flow fields is performed in order to relate our findings to experimental observations. It is shown that streaks, streamwise rolls, horseshoe vortical structures and shear layers, present in our models, are all connected to each other. Finally, criteria to determine a realistic value of the eddy viscosity parameter are developed.

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