One of the most challenging problems in the development of air breathing propulsion systems is the characterization of the complex, two-phase reacting flow in the combustor. Combustor flows in high-speed engines can be highly turbulent in the fuel mixing regions, where the flow is further complicated by the turbulent interaction of the liquid fuel droplets and the reacting gas. Increasingly powerful computers and novel computational methods and analysis enable detailed analysis and consequent improved understanding of flow structures, droplet dynamics and chemical reaction. Within this complex class of flow, we aim to elucidate the transport of fuel and air mass in separated flows in complex combustors geometries at subsonic speeds using our recent and break-through Lagrangian Coherent Structure (LCS) theory and in-house high-fidelity methods .

Discontinuous Galerkin (DG) spectral element methods have shown to be effective for direct computation of turbulent flows in complex geometries. Spectral element methods, in general, are capable of handling unstructured grids, giving them a distinct advantage over methods that rely on structured grids, such as standard spectral methods. DG methods also have the added advantage of localized solution dependencies. Flux derivatives only depend on fluxes in adjacent cells, thereby making parallelization of these algorithms easy and efficient. Efficiency is further improved through exponential convergence with higher-order polynomial interpolants.

We have been part of a team that has made ground-breaking developments in the theoretical analysis of flow separation and flow structures in the Lagrangian frame. Through Lagrangian analysis one can extract attracting and repelling transport barriers from velocity field data which form LCS. These LCS are determined via the finite time Lyapunov exponent (FTLE), where the FTLE quantifies the local stretching of particle paths in a flow field. When a FTLE field is calculated over a flow field, regions of FTLE maxima appear around boundaries to turbulent structures such as eddies. These maxima represent transport barrier in which fluid particles do not cross and constitute the boundaries of the LCS in the flow. The distinctness of the transport enables quantification of mass in control volumes whose bounds are formed by the LCS.

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