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\title{Adjoint-based optimization with non-linear constraints applied to turbulent mixing}

\author{
{\bfseries  Sara Delport,\footnote{Ph.D. grant financed by the Institute for the Promotion of Innovation through
Science and Technology in Flanders (IWT-Vlaanderen). \texttt{sara.delport@mech.kuleuven.be}}~
  Martine Baelmans,\footnote{\texttt{martine.baelmans@mech.kuleuven.be}}~
  Johan Meyers\footnote{Also: Science Foundation -- Flanders (FWO -- Vlaanderen).\protect\\ \texttt{johan.meyers@mech.kuleuven.be} } }\\
Department of Mechanical Engineering, Katholieke Universiteit Leuven, \\
Celestijnenlaan 300A, B3001 Leuven, Belgium.}

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\maketitle



Thanks to the continuing increase of computational resources, it
is at present possible to apply adjoint-based optimization of
large parameter sets to the optimization of turbulent flow
phenomena. This study focusses on the handling of linear and
non-linear constraints in case of optimization of turbulent
mixing. We consider the optimization of initial perturbations of
the velocity field, such that the turbulent mixing at the end of
the flow simulation is maximum. Two constraints apply to these
initial perturbations: a continuity constraint, being linear, and
an energy constraint, which is nonlinear. The first is enforced
using parameter elimination. For the nonlinear constraint a
gradient-projection method is compared to the augmented Lagrangian
method. We find that the gradient-projection method is more robust
than the augmented Lagrangian. Further, the latter method is not
always yielding satisfactory optima and the method appears very
sensitive to the computational resolution. Moreover the gradient
projection method converges towards a reference solution.



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