On low-dimensional modeling of channel turbulence

We investigate a sequence of low-dimensional models of turbulent channel flows. These models are based on the extraction of the Karhunen-Loève (KL) eigenfunctions from a large-scale simulation in a wide channel with R_* = 180 (based on the friction velocity). KL eigenfunctions provide an optimal coordinate system in which to represent the dynamics of the turbulent flow. The hierarchy of KL modes identifies the most energetic independent phenomena in the system. A series of Galerkin projections is then used to derive a dynamical approximation to flows. In order to capture essential aspects of the flow in a low-dimensional system, a careful selection of modes is carried out. The resulting models satisfy momentum and energy conservation as well as yielding the amount of viscous dissipation found in the exact system. Their dynamics includes modes which characterize the flux, rolls, and propagating waves. Unlike previous treatments the instantaneous streamwise flow is time dependent and represented by KL flux modes. The rolls correspond to the streaks observed in experiments in the viscous sublayer. Propagating waves which first appear in the model flow at low Reynolds number (R_* ∼ 60) persist through the chaotic regime that appears as the Reynolds number is increased. Statistical measures of the chaotic flows which have been generated by the models compare favorably with those found in full-scale simulations.