  We propose a framework of Lagrangian Coherent Structures (LCS) to elucidate
the shear layers and coherent structures to understand the mechanism of
generation of tonal sound during thermoacoustic instability. Experiments were
performed on a bluff-body stabilized turbulent combustor in the state of
thermoacoustic instability. We use dynamic mode decomposition (DMD) on the
flow-field to identify dynamical regions where the acoustic frequency is
dominant. We find that the separating shear layer from the backward-facing step
of the combustor envelops a cylindrical vortex in the outer recirculation zone
(ORZ), eventually impinging on the top wall of the combustor during
thermoacoustic instability. The production of tonal sound is due to the
synchronous motion of this shear layer with the shear layer emerging from the
leading edge of the bluff-body causing the fluid to squeeze and expand between
them periodically. To de-synchronize the oscillations in an optimal manner, we
track the Lagrangian saddle points in the shear layer emerging from the
backward facing step over several acoustic cycles. A passive control strategy
is then developed by injecting a steady stream of secondary air targeting the
optimal locations thus identified. After implementing the control action, the
flow-field is also analysed using LCS to understand the key differences in flow
dynamics. We find that the shear layer emerging from the dump plane are
deflected in a direction almost parallel to the axis of the combustor after the
control action. This deflection in turn prevents the shear layer from
enveloping the vortex and impinging on the combustor walls, resulting in a
drastic reduction of the amplitude of the acoustic field.
