Adjoint based analysis for swirling and reacting flows

Abstract

Adjoint techniques are a useful tool in applied mathematics, enabling the efficient evaluation of gradient information. This thesis showcases the use of adjoints in a reacting and swirling flow setting. Additionally, algorithmic techniques are developed to facilitate the use of adjoints for the complicated governing equations that arise from reacting flow.

A resolvent study of an M-flame is conducted to determine the optimal forcing that provides the greatest amplification in flow dynamics. This study extends the concept of a flame transfer function to provide structural mechanisms through which amplification is obtained. Adjoint information is used to efficiently conduct the study, and to extend its validity to different parameter values. A blend of the Orr mechanism and other mechanisms that make use of the azimuthal shear is found to provide the optimal gain, which is achieved for the $m=2$ azimuthal mode in the non-swirling case, with a change to the $m=0$ mode when swirl is introduced.

A weakly non-linear analysis of an incompressible swirling jet is also performed to determine how harmonic forcing modifies the precessing instability caused by high swirl. This study reveals that forcing against the mean-flow swirl causes stabilisation of the instability, with axisymmetric and co-rotating forcing leading to destabilisation. In both cases the base-flow modification induced by the driven response provides the key mechanism through which the unstable mode is modified. A generalised, alternative formulation to this problem is also developed in which the weakly non-linear equations are found directly from the numerical code via finite differences, allowing for future analyses to be carried out for complex governing equations.

Lastly, an algorithmic study is conducted, showing how direct-adjoint looping can be accelerated using a parallel-in-time approach. This technique is found to be beneficial for both linear and non-linear governing equations, leading to more efficient optimisation studies.