Robustness in Hydrodynamic Stability and Control of Transition

Bassam Bamieh, Dept. of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign

The prediction of when certain laminar flows become unstable and transition to turbulence is the central question in hydrodynamic stability theory. Traditional linear hydrodynamic stability theory is concerned with eigenvalue analysis of the dynamical operator of the linearized Navier-Stokes equations about a nominal laminar flow. It has been long recognized that the predictions of this theory do not always agree with experimental findings. However, in the past decade it has become recognized that the non-normality of the linear dynamical operator in strongly sheared flows plays a much more important role in stability than do its eigenvalues. Concepts such as the pseudo-spectrum, transient energy growth, and flow perturbation variance have been formulated to quantify the "degree of instability". It also turns out that certain coherent structures in turbulent near wall boundary layers are predictable as worst case growth modes of the linearized system.

There are striking similarities between this new theory and the central concepts in robust control theory. The latter is concerned with quantifying the "stability margin" of systems. The similarity is due to the fact that both theories are concerned with stability not as a yes/no question, rather with how "robust" is stability. This talk will give a tutorial account of the new linear hydrodynamic stability theory and the corresponding concepts of robust control. The above concepts will be interpreted in terms of the so-called H2 and H-infinity norms, and small gain stability criteria.

We will indicate how these robustness measures can guide the formulation of the important problem of control of transition to turbulence, and skin-friction drag reduction in wall bounded flows.