Application of dynamical system theory to coherent structures in the wall region

In an attempt to apply dynamical system approaches to turbulence, we derive closed sets of nonlinear ordinary differential equations for the wall region. This is achieved by Galerkin projection onto statistical "coherent structures", using the proper orthogonal decomposition (Lumley [1]) which converges optimally fast in the quadratic mean sense. Application of this definition to the instantaneous velocity field of the wall region shows the presence of streamwise rolls. A severe truncation of the dynamical system (10 dimensions) is studied to investigate the time behavior of these rolls. The energy transfer to unresolved modes is adjusted by a Heisenberg parameter. For large values of the parameter (large loss), the solution is steady and represents streamwise rolls having the experimentally observed cross-stream spacing. For lower values of the parameter, intermittency appears due to the presence of a heteroclinic attracting orbit in the phase space. The behavior of the streamwise rolls consists in a long time quasi-steady state followed by a growing oscillation and a sudden burst during which the rolls lose their identify. Subsequently the rolls reform and the process is repeated. These dynamics appear to minic the bursting behavior observed in the wall region. For lower parameter values, the regime is much more complex, apparently chaotic. The fluctuating pressure term appears as a forcing term at the upper boundary of the integration domain. Although its amplitude is very small, it acts as a trigger for bursts and equilibrates the mean time between the events which would otherwise increase with time as the trajectory is attracted closer and closer to the heteroclinic cycle. The whole mechanism is qualitatively the same when the wall region is artificially thickened (which is done by applying stretching transformations). This is in agreement with experimental results concerning drag-reduced flows.