Physica D 37 (1989) 1-10 North-Holland, Amsterdam SECTION I BOUNDARY LAYER TURBULENCE APPLICATION OF DYNAMICAL SYSTEM THEORY TO COHERENT STRUCTURES IN THE WALL REGION Nadine AUBRY t'*, Philip HOLMES 2'3, John L. LUMLEY a and Emily STONE 2 tSibley School of Mechc ;cal and Aerospace Engineering, Corndl Unioersity, Ithaca, NY 14853, USA "Department of Theore,,,, ,and Applied Mechanics, Corneil University, Ithaca, NY 14853, USA JDepartment of Mathematics and Center for Applied Mathematics, Corneli University, Ithaca, N Y 14853, US, I In an attempt to apply dynamical system a?proaches to tm'bulence, we derive closed sets of nonfinear 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 pa~ameter (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 stream"': ~e rolls consists in a long time quasi-steady state followed by a growing oscillation and a sudden burst during which the rolls lose their identity. Subsequently the rolls reiorm and the process is repeated. These dynamics appear to mimic 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 i~ in agreement with experimental results concerning drag-reduced flows. 1. Introduction Recent studies have shown that there may be some connection between dynamical systems the- ory and turbulence. The first link was estabfisbed by Lorenz [2]. RueUe and Takens [3] proposed a model in which turbulence would be a determinis- tic, chaotic regime reached after a small number of bifurcations. These ideas were totally opposite to Landau's theory in which turbulence involved mac,rarely many modes, xne question is to ~a~uw to what extent turbulent motion can be described by a low dimensional attractor similar to those found in low order dimensional systems. A corre- *New address: The Benjamin Levich Institute for Physico- Chemical Hydrodynamics, City College, New York, NY 100~1, USA. spondence has been shown in desed flow systems - i.e. systems constrained by boundaries- and close to transition such as Rayleigh-B&lard convection between two plates and Taylor-Couette flow be- tween rotating cylinders. However, the connection is much more difficult for open, fully developed turbulent flows since a large number of modes can easily be excited as :he biiurcation parameter varies. The main question is to determine whether the dynamics of a complex turbulent flow can be bly small, number of modes. The major difficulty is to extract some fe,v significant modes. As Monin [4] has pointed out, low dimensional models can- not hope to account for the detailed, high wave number, spatially chaotic aspects of fully devel- oped turbulence. Moreover, in open flow systems, arbitrary modal decompositions such as the 0167-2789/89/$03.50 © Elsevier Science Publishers B.V, (North-Holland •hvsics Publishing Division)