### Doctorandus/a PhD student

### Partners

### Promotie / Defence

Language: nl

Waar: aula van de Tweede Hoofdwet, 01.02, Kasteelpark Arenberg 41, 3001 Heverlee

### Promotor / Supervisor

- Prof. dr. ir. Martine Baelmans (promotor)
- Prof. dr. Johan Meyers (co-promotor)

### Samenvatting van het onderzoek / Summary of Research

Optimal Control of a Turbulent Mixing Layer

In turbulent flows relatively small perturbations can grow strongly and, for example, reduce drag or enhance mixing of two fluids. This enablesflow control with actuators. A general description, in time and space, of the flow perturbations induced by the actuators immediately involves many parameters. Optimisation of these parameters leads to the optimal perturbations. Numerical optimisation of flow control is within reach thanks to the recent advances in computational power of supercomputers. This thesis focuses on optimisation of open-loop control for turbulent flows. The used control is described by a large number of parameters. The optimisation minimises a cost functional that is formulated based on the solution of a Navier-Stokes simulation.

To limit the required computational power, the temporal mixing layer is selected as flow case for optimisation. The control consists of the perturbations on the initial mean-flow field. The perturbations are optimised subject to a linear and a non-linear constraint. The continuity equation imposes that the perturbations are divergence-free. This is a linear constraint on the parameters. The perturbations are limited to the ones with an imposed, low, energy-level. To this end, the optimisation takes into account a non-linear constraint on the parameters.

This thesis makes three contributionsto the domain of research. The first two contributions are on an optimisation method for turbulent flows that takes into account a non linear constraint on the parameters. The third contribution is the investigation of the impact of the control on mixing layers at long time horizons.

The first contribution is to optimisation methods for large parameter spaces, and cost functionals that require the solution of a Navier-Stokes simulation. Because the parameter space used has many dimensions, gradient-based optimisation is selected. Since the Navier-Stokes equations are computationally expensive partial differential equations, the continuous adjoint method is used to calculate the gradient efficiently. The adjoint boundary conditions are discussed for a general cost functional. This discussion provides insight into the restrictions, imposedby the adjoint method, on the choice of the cost functional. The convergence of the gradient, calculated using the adjoint method, is verified.

The second contribution is the comparison of two methods to impose the non-linear constraint: the augmented Lagrangian method, and the gradient projection method. Both satisfy the Karush-Kuhn-Tucker conditions. Comparison shows that the gradient projection method is the more robust technology in combination with the gradient calculated with the adjoint method.

The optimisation method is applied to optimise a mixing layer to five different cost functionals. These are based on the following properties at the time horizon: the momentum thickness, the turbulent kinetic energy, the mean-flow kinetic energy, the total kinetic energy, and the enstrophy. The first three cost functionals are shown to lead to the same optimum. That optimum results in large two-dimensional vortex structures with maximum impact on mean momentum, but with low dissipation. The remaining two cost functional, on the other hand, the kinetic energy and the enstrophy cost functionals, result in optimal solutions that havecomplex three-dimensional vortex structures at different scales.

Finally, the impact of control on mixing layers at long time horizons is investigated. Temporal mixing layers behave self-similar at long time horizons. In the self-similar state, the enstrophy is constant in time and the kinetic energy evolves linearly. This could imply that the control has no impact on the flow at long time horizons and as such, it would no longer be possible for the optimisation to improve the cost functional. Nevertheless, tests show that, up to the longest time horizon investigated, the optimisation leads to a significant decrease of the cost functional. The minimisation of the kinetic energy at the time horizon leads to a significant acceleration of the transition to self-similarity. The enstrophy maximisation on the other hand delays the onset of self-similarity.

Additional tests with noise on the optimal perturbations have shown that these are robust. The values of the cost functional increase onlyslightly even for high levels of noise.

### Volledige tekst van het doctoraat / full text

### Examencommissie / Board of examiners

- Prof. dr. ir. Martine Baelmans (promotor)
- Prof. dr. Johan Meyers (co-promotor)
- Prof. dr. ir. Yves Willems (voorzitter/chairman)
- Prof. dr. ir. Stefan Vandewalle (secretaris/secretary)
- Prof. dr. ir. Eric Van den Bulck
- Prof. dr. Moritz Diehl
- Prof. dr. Grégoire Winckelmans , Université Catholique de Louvain (U.C.L.), MECA
- Prof. dr. Michael A. Leschziner , Imperial College London