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Licentiate seminar

Towards optimal design of vehicles with low drag: Applications to sensitivity analysis and optimal control


Defendant Main Advisor Extra Advisor Date
Jan  Pralits Dan Henningson Ardeshir Hanifi 2001-06-06

Opponent
Peter Schmid, Department of Applied Mathematics, University of Washington

Evaluation committee

Abstract

The intention of the project is to develop a methodology for optimal design of vehicles with low drag, and the aim is to automatically incorporate a transition prediction method into the optimization process. The fundamental tool in this analysis are the Parabolized Stability Equations (PSE) which can predict the growth of disturbances in non-parallel boundary layers. The nonlocal growth rate is used to calculate the N-factor in the so called e^N-method which can then be correlated in experimental data (\eg wind tunnel or flight tests) to determine the value of N that corresponds to laminar-turbulent transition. An optimization procedure which results in changes of the geometry in turn causes changes in the external pressure distribution. The pressure distribution is obtained from inviscid equations \eg the Euler equations which can then be used in the solution of the corresponding boundary layer equations (BLE). Finally, the stability analysis is done on the computed boundary layer. The chosen optimization procedure is gradient based and is formulated as an optimal control problem where the aim is to minimize an objective function balancing a measure of the state and the control. Here, the gradients are identified from so called adjoint equations. An outline is first presented, on how to solve the optimal design problem. However, derivations and results regarding modifications in the geometry are not presented in this thesis. The problem has instead been divided into a number of smaller parts which serve both as an interesting application by itself and provide knowledge which is useful for the optimal design problem. In the first application, the gradients (sensitivities) of the disturbance kinetic energy at a given position in the flow field due to unsteady forcing on the wall and within the boundary layer are derived. The gradients are identified from the adjoint of the PSE (APSE). Further, an application to optimal disturbance control is outlined where the unsteady disturbance velocity is used as control variable (blowing/suction) on the surface of a given geometry. In the second application, an optimal control problem is presented in which the wall normal velocity of the steady mean-flow is optimized to control disturbance growth in the whole flow domain. Here, the gradient of the objective function with respect to the control is derived from a coupling between the APSE and the adjoint of the boundary layer equations (ABLE). Related problems have also been solved regarding the gradient accuracy and the treatment of cases when different types of disturbances are present simultaneously.
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