Geometry, Ordering, and Efficiency in the Turbulent Cascade

Nicholas T. Ouellette, Department of Civil and Environmental Engineering, Stanford University

Hopeman 224

Turbulent flows are inherently multi-scale. The mechanism that drives motion on many scales arises from the nonlinearity in the Navier-Stokes equations, which expresses the interaction of wavenumber triads that couple dynamics on different length scales. In turbulence, these triads self-organize to produce a net transfer of energy from the scales at which it is injected into the flow to the scales at which it is dissipated. Typically, we think about this transfer of energy in a Fourier description; but in doing so, we obscure its mechanistic origins and lose any connection to the spatial structure of the flow field. I will discuss an alternative way to study the spatiotemporally localized exchange of energy between scales that is based on a filtering technique. Using this methodology, I will describe our recent work in both two-dimensional and three-dimensional  turbulence aimed at characterizing the geometric structure of the turbulent cascade, which is encoded in the relative alignment of the turbulent strain rate and a turbulent stress that is the manifestation of the nonlinearity in the Navier-Stokes equations. This alignment in turn allows us to define a notion of an efficiency for the turbulent cascade, which brings us to the puzzling observation that, in a certain sense, turbulent flows are surprisingly bad at being turbulent.