Adjoint-based phase reduction analysis of incompressible periodic flows

Abstract

We establish the theoretical framework for adjoint-based phase reduction analysis for incompressible periodic flows. Through this adjoint-based method, we obtain spatiotemporal phase sensitivity fields through a single pair of forward and backward direct numerical simulations, as opposed to the impulse-based method that requires a very large number of simulations. Phase-based analysis involves perturbation analysis about a periodically varying base state and hence is tailored for the analysis of periodic flows. We formulate the phase description of periodic flows with respect to the potential and vortical perturbations in the flow field. The current phase-reduction analysis can also be implemented consistently in the immersed boundary projection method, which facilitates the analysis over arbitrarily-shaped bodies. We demonstrate the strength of the phase-based analysis for periodic flows over circular cylinder and symmetric airfoils at high incidence angles. The critical regions for phase modification in the cylinder flow are investigated and the locations of flow separation are shown to be the most sensitive regions. Further, the results reveal the influence of the angle of attack and airfoil thickness on the phase-sensitivity distribution of flows over various airfoils. The phase for such flows is defined based on the lift coefficient, and hence is influenced by the vortical structures responsible for lift production. The present framework sheds light on the connection between phase-sensitivity and vortex formation dynamics.

18 pages, 10 figures

References in corpus (7)

• Synchronization in complex networks
• An Introduction to Adjoint Problems
• Coupling functions: Universal insights into dynamical interaction mechanisms
• Two-dimensional compressible viscous flow around a circular cylinder
• Data-driven resolvent analysis
• Jacobian-free algorithm to calculate the phase sensitivity function in the phase reduction theory and its applications to Kármán's vortex street
• Phase-locking of laminar wake to periodic vibrations of a circular cylinder

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• Phase autoencoder for rapid data-driven synchronization of rhythmic spatiotemporal patterns
• Phase reduction of reaction-diffusion systems with delay
• Phase reduction analysis of traveling breathers in reaction--diffusion systems