Optimization-Based Discovery of A Non-Attracting Flow State in An Oscillating-Cylinder Wake

Abstract

In the flow past a stationary circular cylinder, the classical Karman vortex street arises from a Hopf bifurcation of the steady flow at the critical Reynolds number. Although this solution becomes dynamically unstable beyond this point, it remains an exact solution of the governing equations. Motivated by this observation, we investigates whether similar non-attracting flow solutions exist in the flow past a forced oscillating cylinder at supercritical Re. In the present study, while employing PINNs to investigate the flow past a forced oscillating cylinder, we identify a class of flow solutions that are inaccessible through direct time-stepping simulations. The obtained solution remains phase-locked with the cylinder oscillation frequency, despite the corresponding parameters lying outside the lock-in regime. To verify this solution, the obtained PINNs solution is used as the initial guess for an optimization based on the optimizing a discrete loss (ODIL) framework. The results show that the solution can be consistently maintained during the optimization process. This indicates that the solution is self-consistent in the optimization sense, although it does not an attracting state of the original dynamical system. To understand the reason, we compare the numerical evolution mechanisms of each solvers. The results indicate that, flow states that satisfy the governing equations but are dynamically non-attracting can be identified and maintained as minima of the optimization problem. For the flow past a forced oscillating cylinder, non-attracting periodic solutions that satisfy the governing equations exist in addition to the attracting states obtained by conventional time-stepping simulations. Optimization-based solvers can therefore reveal such flow states that are difficult to obtain through direct time integration, providing a new perspective for understanding complex wake dynamics.