Adjoint-based sensitivity and ignition threshold mapping in a turbulent mixing layer

Successful ignition in non-premixed turbulent flows remains a fundamental challenge in combustion systems. Current design strategies typically rely on iterative testing to map the spatial distribution of ignition probability. We propose to accelerate this by formulating the adjoint of the perturbed and linearised governing equations in such a way that sensitivity of an ignition indicator can be obtained with a cost comparable to the flow solution. A space-time discrete adjoint method for multi-component chemically reacting flows is developed, and the gradient formed via the corresponding adjoint solution is used to identify regions favourable to ignition in a direct numerical simulation of non-premixed turbulent free shear flow. This approach requires a specific definition of an ignition metric, although this can be problematic because ignition either succeeds or fails after some period and thus gradients for some metrics become ill-defined near the ignition threshold. To this end, a quantity of interest is designed to provide short-time sensitivity in conjunction with an indicator function over a long-time period that informs whether successful ignition occurred. The gradients are used in a line-search algorithm to map the ignition boundary under specific constraints. Finally, parametric sensitivity is evaluated at different flow realisations to analyse factors governing local sensitivity in unsteady chemically reacting flows.