Closing the Loop: Adjoint-Based Methods for Flow Stability, Control & Mixing
Russell Love Theatre
In very many circumstances, there is an interest in 'optimising' a fluid flow, in the sense of achieving some desired objective in the 'best' possible way. Examples of such circumstances include identifying the so-called 'minimal seed', i.e. the perturbation to a given (laminar) flow that can trigger transition to turbulence with minimal energy input, or identifying forcing strategies of given magnitude which can mix fluids of different compositions completely over a finite time horizon. Such problems are inherently nonlinear, and yet can be addressed efficiently and algorithmically when posed in (constrained) variational form. Specifically, imposing the constraint that the 'direct' flow fileds must always satisfy the underlying Navier-Stokes equations can be achieved using Lagrange multipliers that vary in space and time, which are known as 'adjoint variables'. It has recently been appreciated that these adjoint variables satisfy closely related partial differential equations, and that the optimal 'direct' and 'adjoint' variables can be determined algorithmically in an iterative 'loop' using time-stepping codes, where the two classes of variables satisfy consistency conditions at the start and end time of the loop. Here, I review the underlying mathematics of this 'direct-adjoint-looping' (DAL) method, and apply it to several different problems in turbulence transition, control and mixing optimisation. Of course, identification of the 'best' way to achieve such objectives requires the definition of an appropriate measure, and I also discuss approaches to identifying the most computationally efficient and robust measures to use for a specific problem of interest.
Professor Colm Caulfield, University of Cambridge