Intrinsic phase based proper orthogonal decomposition (IPhaB POD): a method for physically interpretable modes in near-periodic systems

Abstract: Fluid dynamics systems driven by dominant, nearly periodic large-scale dynamics are common across wakes, jets, rotating machinery, and high-speed flows. Traditional decomposition techniques such as proper orthogonal decomposition and dynamic mode decomposition have been used to gain insight into these flows, but can require many modes to represent physical processes. With the aim of generating modes that intuitively convey the underlying physical mechanisms, we propose an intrinsic phase-based proper orthogonal decomposition (IPhaB POD) method. IPhaB POD creates energetically ranked modes that evolve along a characteristic cycle of a driving near-periodic large scale. Our proposed formulation is set in the time domain, which is particularly useful in cases where the large-scale dynamics are imperfectly periodic. We formally derive IPhaB POD within a proper orthogonal decomposition framework, and it inherits the optimal representation inherent to POD. As part of this derivation, a dynamical systems representation of the large scale is utilized, facilitating a definition of phase relative to the large scale within the time domain. An expectation operator and inner product are also constructed relative to this definition of phase in a manner that allows for the various cycles within the data to demonstrate imperfect periodicity. The formulation is tested on two sample problems: a simple, low Reynolds number airfoil wake, and a complex, high-speed pulsating shock wave problem. The resulting modes are shown to better isolate the large-scale dynamics in the first mode than space-only proper orthogonal decomposition, and to highlight meaningful small-scale dynamics in higher modes for the shock flow problem.