Koopman Mode Decomposition: Revisiting the Route to Turbulence
Technical Report,09 May 2016,08 Mar 2017
AIMdyn, Inc Santa Barbara United States
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The purpose of the proposed project was to continue the approach we have developed, with ARO support, in collaboration with ARL and Dr.Bryan Glaz, pursue a deeper study of transition to turbulence we have observed and extend it to shed the light on classical theories of turbulence. The proposed work is fundamental for developing better understanding of transition phenomena that has so far eluded theoretical description using low order models, and more generally for understanding dynamical systems phenomena that occur in systems that are being periodically forced through a Hopf and other oscillatory bifurcations. In this project, in collaboration with Dr. Bryan Glaz, we analyzed fluid dynamics induced by periodically forced flow around a cylinder for the case when the forcing frequency is much lower than the von Karman vortex shedding frequency corresponding to the constant flow velocity condition. By using the Koopman Mode Decomposition approach, we found a new normal form equation that extends the classical Hopf bifurcation normal form by a time-dependent term for Reynolds numbers close to the Hopf bifurcation value. The normal form describes the dynamics of an observable and features a forcing control term that multiplies the state, and is thus a parametric - i.e. not an additive - forcing effect. We found that the dynamics of the flow in this regime are characterized by alternating instances of quiescent and strong oscillatory behavior, and that this pattern persists indefinitely. Furthermore, the spectrum of the associated Koopman operator possesses quasi-periodic features. We established the theoretical underpinnings of this phenomenon -- that we name Quasi-Periodic Intermittency -- using the new normal form model and showed that the dynamics are caused by the tendency of the flow to oscillate between the unstable fixed point and the stable limit cycle of the unforced flow.
- Fluid Mechanics