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Doktorsdisputation

Optimal Control of Boundary Layer Transition


Respondent	Huvudhandledare	Bihandledare	Datum
Markus Högberg	Dan Henningson	Martin Berggren	2001-09-11

Opponent
P. Luchini,

Betygsnämd
Håkan Gustavsson, LTU Bo Wahlberg, KTH, S3 Per Weinerfelt, SAAB Aerospace, Future Products, Linköping

Abstract

Methods for optimal control of transition in boundary layers are investigated and developed in this thesis. A model problem is studied in order to investigate an approximative method for objective function gradient computations. The approximation is to use the continuous formulation of the equations, instead of the discrete counterpart, to derive the optimization problem. The conclusion is that the approximative method is sufficiently accurate for the purpose of transition control. A nonlinear control approach using the Navier--Stokes equations and the associated adjoint equations to minimize an objective function measuring the energy of the perturbation to a laminar flow is developed and tested using direct numerical simulations. A similar optimization problem is posed, using the Orr--Sommerfeld--Squire equations, which can be solved directly to obtain localized physical space feedback control laws. The performance of these control laws is quantified in direct numerical simulations by computing transition thresholds. It is shown that the threshold values can be increased by about 500% for a random perturbation. By using a physically motivated modification of the objective function it is shown that these linear controllers are also able to relaminarize a low Reynolds number turbulent flow. In this linear framework an estimator in the form of an extended Kalman filter is developed and shown to have exponential convergence using the normal derivative of the normal vorticity as a wall measurement. The estimator and controller are combined into a compensator for which transition thresholds are computed. In this case the threshold value for the random perturbation is only increased by about 48%. The linear control approach is then applied in direct numerical simulations of spatially developing boundary layer flows with successful reduction of perturbation energies for Tollmien--Schlichting waves and optimal perturbations in the Blasius boundary layer. In a Falkner-Skan-Cooke flow the control strategy also reduces the energy of traveling and stationary, saturated cross-flow vortices.
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