A Robust Control Approach to Understanding Nonlinear Mechanisms in Shear Flow Turbulence

A robust control framework is used to investigate a streamwise constant projection of the Navier Stokes equations for plane Couette flow. Study of this streamwise constant model is motivated by both numerical and experimental observations that suggest the prevalence and importance of streamwise and quasi-streamwise elongated structures. Small-amplitude Gaussian noise forcing is applied to a two-dimensional, three-velocity component (2D/3C) model to describe its response in the presence of disturbances, uncertainty and modeling errors. A comparison of the results with Direct Numerical Simulation (DNS) data demonstrates that the simulations capture salient features of fully developed turbulence. In particular, the change in mean velocity profile from the nominal laminar to the characteristic “S” shaped turbulent profile. The application of Taylor’s hypothesis shows that the model can also reproduce downstream information in the form of large-scale coherence resembling numerically and experimentally observed flow features. The 2D/3C model is able to generate “turbulent-like” behavior under small-amplitude stochastic noise. The laminar flow solution is globally stable, therefore transition to turbulence in this model is likely a consequence of the laminar flow solution’s lack of robustness in the presence of disturbances and uncertainty. In fact, large disturbance amplification is common in both this model and the linearized Navier Stokes equations. Periodic spanwise/wall-normal (z–y) plane stream functions are used as input to develop a forced 2D/3C streamwise velocity equation. The resulting steady-state solution is qualitatively similar to a fully turbulent spatial field of DNS data. Both numerical methods and a perturbation analysis confirm that the momentum transfer that produces a “turbulent-like” mean profile requires a nonlinear streamwise velocity equation. A system theoretic approach is used to study the amplification mechanisms that develop through the 2D/3C nonlinear coupling in the streamwise velocity equation. The spanwise/wall-normal plane forcing required to produce each stream function is computed vii and used to define an induced norm from this forcing input to the streamwise velocity. This input-output response is used to determine the energy optimal spanwise wavelength (i.e., the preferential spacing) over a range of Reynolds numbers and forcing amplitudes. This analysis serves to augment the well-developed linear theory through the addition of the nonlinear coupling in the streamwise velocity equation.