Bulletin of the American Physical Society
72nd Annual Meeting of the APS Division of Fluid Dynamics
Volume 64, Number 13
Saturday–Tuesday, November 23–26, 2019; Seattle, Washington
Session Q10: Nonlinear Dynamics: General 
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Chair: Burak Budanur, IST  Austria Room: 3A 
Tuesday, November 26, 2019 7:45AM  7:58AM 
Q10.00001: Synchronization in periodically forced oscillator flows Benjamin Herrmann, Steven Brunton, Richard Semaan We investigate the synchronization properties of the turbulent wake past a Dshaped bluff body with periodic Coanda blowing. Time series from experimental measurements of the base pressure are used to study the response of the global vortex shedding mode for different actuation frequencies and amplitudes. Multiple regions of synchronization are found, resulting in the socalled Arnold tongues, where the oscillation frequency of the global mode lockson to a rational multiple of the forcing frequency. We construct a model using a sparse regression and its structure explains these nonlinear couplings as an anharmonic parametric excitation of the global mode. The model is further analyzed using phase reduction theory, predicting the presence of $2:n$ synchronization and revealing the boundaries of the respective Arnold tongues. We postulate that this phenomena is universal to periodically forced oscillator flows, and arises from resonant wavetriads that transfer of energy from the $n^{\text{th}}$ harmonic of the forcing, i.e., the parent wave, to the conjugate pair of vortex shedding modes, i.e., the daughter waves with halffrequency and higher wavenumber. [Preview Abstract] 
Tuesday, November 26, 2019 7:58AM  8:11AM 
Q10.00002: Dynamics and stability of confined interfacial droplets Stuart Thomson, Matthew Durey, Miles Couchman, Ruben Rosales, John Bush Millimetric droplets may bounce on the surface of a vibrating liquid bath or ``walk" by means of selfpropulsion through a resonant interaction with their own wave field. The propulsive wave force exerted by the bath is balanced by the droplet's inertia and dissipation in the form of drag. In this talk we present the results of a combined experimental and theoretical study in which we consider the dynamics and stability of droplets confined to an annular ring. Single droplets are observed to destabilize to a pulsating walking state and then to a randomwalk like instability as the vibrational acceleration of the bath is increased. When multiple droplets are brought into close proximity, they may bind together to form onedimensional lattices which exhibit several features characteristic of driven dissipative oscillator systems including periodic oscillations and a striking solitary wavelike instability. Our experimental system provides a highly tunable apparatus to study dynamical systems driven far from equilbrium at the macroscale. [Preview Abstract] 
Tuesday, November 26, 2019 8:11AM  8:24AM 
Q10.00003: Extention of Arnold's stability theory for planar viscous shear flows Harry Lee, Shixiao Wang A viscous extension of Arnold's inviscid theory for planar parallel noninflectional shear flows is developed and a viscous Arnold's identity is obtained. Special forms of our viscous Arnold's identity are revealed that are closely related to the perturbation's enstrophy identity derived by Synge (1938) (see also Fraternale {\it et al.} 2018). Firstly, an alternative derivation of the perturbation's enstrophy identity for strictly parallel shear flows is acquired based on our viscous Arnold's identity. The alternative derivation induces a weight function, inspired by which, a novel weighted perturbation's enstrophy identity is established that extends the previously known enstrophy identity to include general nonparallel streamwise translationinvariant shear flows imposed with relaxed wall boundary conditions. As an application of the enstrophy identity, we quantitatively investigate the mechanism of linear instability/stability within the normal modal framework. The investigation finds that the critical layer is always a primary source of damping in disturbance's enstrophy and thus it enhances stability. Moreover, a control scheme is proposed that transitions the wall settings from the noslip condition to the freeslip condition, through which a flow is quickly stabilized. [Preview Abstract] 
Tuesday, November 26, 2019 8:24AM  8:37AM 
Q10.00004: Nonlinear stability of wallbounded viscous flows Shixiao Wang, Harry Lee A viscous extension of Arnold's inviscid theory for planar shear flows is developed and a viscous Arnold's identity is obtained. The viscous Arnold's identity is revealed to be closely related to the perturbation's enstrophy identity (Synge 1938). The mechanism of linear instability/stability of wallbounded shear flows has been reexamined by the viscous Arnold's identity and the perturbation's enstrophy identity. It was found that the role of nonslip wall boundary condition in a planar shear flow is strikingly different from that in the circular TaylorCouette flow confined between two concentric rotating cylinders. For the former, the perturbation's enstrophy is generated at the walls by the nonslip induced flow rubbing effect. For the latter, however, the perturbation's flow circulation is rigidly fixed by the nonslip wall condition and thus it effectively stabilizes the flow globally under axisymmetric disturbances within a subdomain of the inviscid linear stability regime. A remarkable feature of the global stability for the circular TaylorCouette flow is its independence of the $Re$ number. [Preview Abstract] 
Tuesday, November 26, 2019 8:37AM  8:50AM 
Q10.00005: Stability and bifurcation of a freelyrotating discontinuity in a symmetrically driven square cavity Gongqiang He, Chenguang Zhang, Krishnaswamy Nandakumar Using a directforcing immersed boundary method at fully resolved grid resolutions, we study the interaction between the fluid in a symmetrically driven square cavity (i.e., top and bottom lids sliding at identical velocity), and the response of a rectangular block inside. The block is made thin enough to approximate an ideal discontinuity; it is fixed at the cavity center but can freely rotate. We scanned up to moderate Reynolds numbers using different block length L, and a phase diagram (Re, L) is created. For a fixed Reynolds number, a short block stabilizes at the vertical orientation (theta $=$ 90 degree). As the block becomes longer, the vertical orientation losses stability and bifurcates into a pair of new stable orientations that are symmetric regarding the vertical direction. The critical lengths for different Reynolds numbers are found, and the reason for the loss of stability is explained by energy argument and analysis of the flow patterns. [Preview Abstract] 
Tuesday, November 26, 2019 8:50AM  9:03AM 
Q10.00006: Koopman Control of Point Vortex Dynamics using Invariants Kartik Krishna, Aditya Nair, Eurika Kaiser, Steve Brunton We seek to manipulate the behaviour of a planar system of point vortices governed by the BiotSavart law, which is often used to simulate fluid flows in an inviscid and incompressible setting. Inspired by recent advances in Koopman operator theory, we recast the original BiotSavart law in terms of well known invariants of vortex dynamics (e.g. the Hamiltonian). This change of variables helps us obtain a linear representation of nonlinear dynamics and reduces the dimensionality for fluid flows, where the number of vortices is very large. We then leverage tools from control theory to manipulate vortex dynamics using “virtual cylinders” (vorticity generating actuators). In particular, we show that increasing (decreasing) the Hamiltonian enables the clustering (declustering) of multiple vortices. We are currently extending this work to dissipative flows for discrete vortex control. [Preview Abstract] 

Q10.00007: ABSTRACT WITHDRAWN 
Tuesday, November 26, 2019 9:16AM  9:29AM 
Q10.00008: Transit Time in the AreaPreserving Hénon Maps Ibere Caldas, Vitor Oliveira, David Ciro Chaotic solutions of unbounded areapreserving maps usually consist of an incoming regular path, a transitory irregular motion and a regular exit path. In simple situations the irregular motion occurs within a localized region of phase space and individual orbits do not access the whole chaotic domain. During its irregular portion an orbit spends a transient time wandering in a subregion of the chaotic domain which is determined by its income path. We show that, for the areapreserving Hénon map, the transit time pattern is influenced by the distribution of intersections of invariant manifolds in the chaotic domain. To corroborate this assertion, we use an adaptive refinement procedure to obtain approximated sets of homoclinic and heteroclinic intersections. As the control parameter increases stickiness gets reduced and both homoclinic and heteroclinic sets have a similar distribution, indicating a transition to a more uniform type of transitory chaotic motion. [Preview Abstract] 
Tuesday, November 26, 2019 9:29AM  9:42AM 
Q10.00009: A 3D Poor Man's Boltzmann Equation J. M. McDonough, H. W. Yu A 1D ``synthetic'' distribution function for the poor man’s Boltzmann equation (PMBE) has been studied previously; but real applications must be in three space dimensions. In this presentation we outline derivation of the 3D PMBE and study bifurcations of the corresponding discrete dynamical system (DDS) for this case. In particular, we first provide a brief singlemode Galerkin derivation of the PMBE, and then present time series, power spectra and regime maps to demonstrate its consistency with expected fluid flow behaviors—in particular, existence of Ruelle \& Takens, Feigenbaum, and Pomeau \& Manneville bifurcation sequences, as well as combinations of these. We also suggest how such a DDS can be used to produce very efficient subgrid scale synthetic distribution function models for turbulence simulations within a latticeBoltzmann/largeeddy simulation framework. [Preview Abstract] 
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