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VORTEX METHODS AND VORTEX MODELS FOR FLOW CONTROL PROBLEMS
Bartosz PROTAS
Department of Mathematics and Statistics
McMaster University,
Hamilton, ON, Canada
Email: bprotas@mcmaster.ca
URL: www.math.mcmaster.ca/~bprotas
In this presentation we will discuss two Lagrangian approaches
to the problem of flow control: a vortex method used as a tool
for solution of adjoint equations arising in flow optimization
problems, and a point vortex model used as an "ultimate reduced
order model" for feedback control design. Both these strategies
are derived using methods of Modern Control Theory and their
precise mathematical characterization will be presented together
with computational results. The two approaches represent two
opposite extremes as regards complexity of the underlying models
and to fix attention in the talk we will focus on rotational
control of laminar vortex shedding in the wake behind a circular
cylinder. In the first strategy we seek to determine the optimal
control for the full Navier-Stokes system by minimizing a cost
functional which represents the drag. The cost functional
gradient is determined using adjoint equations which are solved
in the Lagrangian (vorticity) formulation. This control strategy
leads to significant sustained drag reduction obtained with a
very small control effort. The method, however, is very costly
as regards the computational cost. In the second approach we
use the F\"oppl system (1913) as a reduced order model for
vortex shedding in the cylinder wake. This model will be
characterized from the control-theoretic perspective. It will
be shown that the cylinder wake can be stabilized using a
Linear-Quadratic-Gaussian (LQG) feedback control design based
on the F\"oppl model. The two approaches will be compared and
perspectives will be outlined as regards design of intermediate
approaches that could bridge vortex methods and vortex models
for flow control purposes.
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Professor Protas will also give a more specialized talk in the morning:
Department of Computer Science Numerical Analysis Seminar
10:10 AM Friday 12 November in Bahen Center (BA) Room 1210
"TOWARDS A MULTI-SCALE FRAMEWORK FOR COMPUTATIONAL
FLOW CONTROL AND ESTIMATION"
http://www.math.toronto.edu/i-aim/public/Abstracts/protas.10am