Direct-adjoint-looping methods for generalised stability problems

It is of great practical importance to understand how flows undergo the transition to turbulence, as turbulence typically hugely increases mixing, transport and dissipation within flows of environmental and industrial interest. It is commonly believed that `normal' mode flow instabilities play a central role in such transition processes, and the conventional argument is that the `most unstable' normal mode will dominate the nonlinear evolution of the flow, and hence lead the flow to transition. However, the underlying linearized operator is non-normal, and so it is possible for substantial transient growth of perturbations to occur. A particularly attractive method to consider such transient growth problems is the so-called `direct-adjoint looping' method, which can be generalised to consider fully nonlinear perturbations, where the developing perturbations can reach a sufficiently large amplitude to nontrivially modify the `base flow'. This method is particularly well-suited to consider generalised stability problems, where the measure which is being extremised is not necessarily the `energy' of the developing perturbation. Indeed, there are several interesting mathematical issues about the most appropriate measures to use, and this project will approach the general issue of perturbation `growth' in a range of environmentally and industrially important flows from a variety of mathematical and computational directions.