Low dimensional models of the wall region in a turbulent boundary layer: new results

Abstract Using an optimally convergent representation, a low dimensional model is constructed, which embodies in a streamwise-invariant form the effects of streamwise structure. Results of Stone show that the model is capable of mimicking the stability change due to favorable and unfavorable pressure gradients. Results of Aubry et al. suggest that polymer drag reduction is associated with stabilization of the secondary instabilities, as has been speculated. Results of Bloch and Marsden indicate that drag can be reduced by feedback, and that this is mathematically equivalent to polymer drag reduction. The authors showed that dynamical systems based on the Proper Orthogonal Decomposition have, on the average, the best short term tracking time (the time that a model tracks the true system accurately; essential for control) for a given number of modes. In recent work, the authors have shown that several assumptions made on an intuitive basis in the work of Aubry et al. may be justified formally. Berkooz has made rigorous estimates using the proper orthogonal decomposition showing that a structured turbulent flow, such as the wall layer, has a phase space representation that remains within a thin slab centered on the most energetic modes for most of the time. Campbell and Holmes have shown several results in connection with symmetry breaking in systems with structurally stable heteroclinic cycles. This work is relevant to our models of interacting coherent structures in boundary layers with discrete spanwise symmetry, such as that caused by riblets, which are known to produce drag reduction.