Identifying Optimal Mixing Strategies and Structures through Sobolev Norms
The mixing of fluids is a hugely important physical phenomenon for understanding processes such as, but not limited to, climate change and improving industrial efficiency. A particular question of interest is how one may optimise the underlying mixing underpinning these processes. Recent advances have shown that optimisation of certain mixing measures yield a higher quality mixing strategy while using less energy than those triggering turbulent flows. These multiscale measures, equivalent to Sobolev norms of negative index, have been employed successfully for this purpose. This thesis investigates these norms and their use in nonlinear optimal mixing problems. In Chapter 3, we compare the optimisation of the classical variance measure of mixing with that of the multiscale norms. These measures are found to be computationally efficient as they identify optimal perturbations which continue mixing of the passive scalar very well compared with variance-based strategies that were optimised over a longer horizon. These multiscale measures also lead to the identification of vertical coherent structures which correspond to effective mixing within the system. In Chapter 4, we study the effect of the underlying geometry of the passive scalar on mixing optimisation. This is investigated through two different problems of interest. In the first problem, we perform a comparative study between a rectilinear stripe geometry and a circular disc geometry. In this study, we find that the stirring of the scalar varies considerably depending on which geometry is prescribed. In the second problem, we restrict our attention to the disc geometry but add more discs to the domain to investigate how the mixing dynamics scale up. We find that while the qualitative behaviour of the solutions remains stable, but through non-local interactions the vortex structure exhibits some variability as the Péclet number is increased. Finally, in Chapter 5, we study a mixing problem with a prescribed stream function chosen according to observations from the studies performed in this thesis. We find that a closed form solution emerges for a special choice of a parameter of particular interest in the literature. We then derive through perturbation methods a system which describes mixing of a passive scalar based on free parameters which set the shape of the initial vortices.
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