  The direct monitoring of a rotating detonation engine (RDE) combustion
chamber has enabled the observation of combustion front dynamics that are
composed of a number of co- and/or counter-rotating coherent traveling shock
waves whose nonlinear mode-locking behavior exhibit bifurcations and
instabilities which are not well understood. Computational fluid dynamics
simulations are ubiquitous in characterizing the dynamics of RDE's reactive,
compressible flow. Such simulations are prohibitively expensive when
considering multiple engine geometries, different operating conditions, and the
long-time dynamics of the mode-locking interactions. Reduced-order models
(ROMs) provide a critically enabling simulation framework because they exploit
low-rank structure in the data to minimize computational cost and allow for
rapid parameterized studies and long-time simulations. However, ROMs are
inherently limited by translational invariances manifest by the combustion
waves present in RDEs. In this work, we leverage machine learning algorithms to
discover moving coordinate frames into which the data is shifted, thus
overcoming limitations imposed by the underlying translational invariance of
the RDE and allowing for the application of traditional dimensionality
reduction techniques. We explore a diverse suite of data-driven ROM strategies
for characterizing the complex shock wave dynamics and interactions in the RDE.
Specifically, we employ the dynamic mode decomposition and a deep Koopman
embedding to give new modeling insights and understanding of combustion wave
interactions in RDEs.
