  In this paper, we present a framework to study the synchronization of flow
velocity with acoustic pressure and heat-release rate in technically-premixed
swirl flames. The framework uses the extended proper orthogonal decomposition
to identify regions of the velocity field where velocity and heat release
fluctuations are highly correlated. We apply this framework to study coupled
interactions associated with period-1 and period-2 type thermoacoustic
instability in a technically-premixed, swirl stabilized gas turbine model
combustor operated with hydrogen-enriched natural gas. We find the structures
in flame surface and heat release rate that are correlated to the dominant
coherent structures of the flow field using extended POD. We observe that the
correlated structures in the flow velocity, flame surface and heat release rate
fields share the same spatial regions during thermoacoustic instability with
period-1 oscillations. In the case of period-2 oscillations, the structures
from flame surface and heat release rate field are strongly correlated.
However, these structures contribute less to the coherent structures of the
flow field. Using the temporal coefficients of the dominant POD modes of the
flow velocity field, we also observed 1:1 and 2:1 frequency locking behaviour
among the time series of acoustic pressure, heat release rate and the temporal
coefficients of the first two dominating POD modes of velocity field during the
state of period-1 and period-2 oscillations, respectively. These
frequency-locked states, which indicate the underlying phase-synchronization
states, are then correlated with coherent structures in the flow velocity
field.
