Merger of coherent structures in time-periodic viscous flows

Abstract

Inertia-induced changes in transport properties of an incompressible viscous time-periodic flow are studied in terms of the topological properties of volume-preserving maps. In the noninertial limit, the flow admits one constant of motion and thus relates to a so-called one-action map. However, the invariant surfaces corresponding to the constant of motion are topologically equivalent to spheres rather than the common case of tori. This has fundamental ramifications for the effect of inertia and leads to a new kind of response scenario: resonance-induced merger of coherent structures.

Comparative numerical-experimental analysis of the universal impact of arbitrary perturbations on transport in three-dimensional unsteady flows

“…Perturbation of the system by weak fluid inertia, inducing nonzero secondary flow u , (6), destroys the invariant surfaces and leads to significantly different tracer dynamics [13,16,17]. Tracers exhibit distinct behavior depending on their proximity to elliptic segments of periodic lines.…”

“…The topology of the tracer trajectories in the 3D cylinder flow has been extensively examined before [13][14][15][16][17]. Two features important in the current context are confinement of tracers released in the base flow u s to closed streamlines symmetric about the planes x = 0 and y = 0 [ Fig.…”

“…The secondary flow u can be introduced numerically in two ways. First, by explicitly considering a weak fluid inertia (nonzero Re 1) and a numerical resolution of the full base flow u s using the aforementioned spectral flow solver [13,17], the secondary flow can be implicitly defined as…”

“…Second, we further unravel the mechanisms underlying tube formation. The studies in [13,[16][17][18] put forth emergence of certain types of isolated periodic points as a trigger for this phenomenon. However, the fundamental question of whether the tubes are merely individual spiraling orbits or indeed, as conjectured in [16], constitute distinct coherent structures parametrized by a (local) adiabatic invariant remains open.…”

“…To this end, we adopt the 3D time-periodic flow inside a lid-driven cylinder according to [13][14][15][16][17] as a representative experimentally realizable system. Numerical studies of this configuration exposed remarkable responses of the Lagrangian flow structure of the stroboscopic map in the Stokes limit (Re = 0) to minute inertial perturbations [Re ∼ O(10 −3 -10 −2 )]: transition of a global family of nested spheroidal invariant surfaces to intricate coherent structures, each formed by merger of (remnants of) two spheroids and one tube, embedded in chaos [13,16,17]. Observations of a similar behavior in a completely different system, an external 3D time-periodic flow driven by a rotating sphere, strongly suggest this is a universal phenomenon [18].…”