Towards Control Oriented Models of Wall-bounded Turbulence

A wide range of reduced order models have proven beneficial in the study of transition and fully-developed wall-bounded turbulent flows, providing a vehicle for the analysis of the underlying flow physics, as well as a tractable means of identifying promising avenues of inquiry for experimental/numerical studies and flow control approaches. Restricted-nonlinear (RNL) models of wall-turbulence are examples of a physics-based approach comprised of first partitioning the Navier Stokes (NS) equations into coupled evolution equations for a streamwise constant mean flow and a streamwise-varying perturbation field (defined about that mean). Nonlinear interactions between perturbations are then neglected or parametrized. Modeling wall-turbulence via a streamwise constant flow coupled with a dynamically restricted streamwise-varying perturbation field is supported by experimental and analytical evidence of the prevalence and central role of streamwise elongated coherent structures in these flows. Simulation and analytical results demonstrate that the streamwise constant, but nonlinear, dynamics of the mean flow capture fundamental aspects of the flow physics (e.g. momentum transfer mechanisms), while the dynamical restriction eliminating streamwise varying nonlinear interactions in the perturbation field improves computational and analytical tractability over the NS equations. RNL simulations capture self-sustaining turbulence and accurately reproduce flow structures known to play a key role in the momentum transfer associated with higher skin-friction drag across a range of Reynolds numbers, even with as few as one streamwise-varying degree of freedom (non-zero Fourier coefficient). This vast order reduction motivates extensions to new applications, such as wind farms, where the lesser computational requirements have enabled parametric studies of vertical staggering of turbines.