Bulletin of the American Physical Society
75th Annual Meeting of the Division of Fluid Dynamics
Volume 67, Number 19
Sunday–Tuesday, November 20–22, 2022; Indiana Convention Center, Indianapolis, Indiana.
Session T21: Nonlinear Dynamics: Mathematical |
Hide Abstracts |
Chair: Bartosz Protas, McMaster University Room: 207 |
Monday, November 21, 2022 4:10PM - 4:23PM |
T21.00001: Systematic search for singularities in 3D Euler flows on a periodic domain Xinyu Zhao, Bartosz Protas The local well-posedness of smooth solutions of 3D incompressible Euler equations has been established when the initial data is in the Sobolev space $H^s$ for $s > 5/2$. However, it is still an open question whether these solutions may develop finite-time singularities. In order to systematically search for initial data that may lead to a singularity formation in finite time, we formulate a PDE-constrained optimization problem in which the quantity $|| \boldsymbol{u}(T) ||_{H^3}$, where $\boldsymbol{u}(t)$, $0 \le t \le T$, is the solution of the Euler equation, is maximized subject to the constraint $|| \boldsymbol{u}(0) ||_{H^3} = 1$. This optimization problem is solved numerically using a state-of-the-art Riemannian conjugate gradient method based on Sobolev gradients obtained by solving a suitable adjoint system. In the process, we repeatedly refine the resolution to detect a possible finite-time blow-up indicated by an unbounded growth of the maximized quantity. The behavior of the extreme flows obtained in this way is consistent with the formation of singularities in finite time when $T$ is sufficiently large. |
Monday, November 21, 2022 4:23PM - 4:36PM |
T21.00002: Exact coherent structures in the 2D Euler Equation Dmitriy Zhigunov, Roman O Grigoriev While the formalism of exact coherent structures (ECS) proved highly insightful for transitional flows, extension to fully developed turbulence proved challenging. As $Re$ increases ECSs become more unstable, numerous, and much harder to find. However, investigations of high-$Re$ 2D turbulence in a periodic box produced several surprises. Key among them is that, on large scales, turbulence shadows particular time-periodic solutions of the Euler equation over extremely long temporal intervals, which has serious implications for the lack of universality in the direct cascade. The Euler equation has substantially higher symmetry that Navier-Stokes and, as a result, has far more ECSs. Unexpectedly, these ECSs, at least those describing large scales, are both far easier to find than ECSs of the Navier-Stokes equation and very weakly unstable. Moreover, they come in continuous families spanned by what looks like an infinite number of parameters, which is unlike ECSs of Navier-Stokes where each parameter corresponds to a continuous spatial symmetry (e.g., rotation or translation about an axis). Finally, in Euler, different classes of ECS are all connected, i.e. an equilibrium can be continued to a traveling wave or a time-periodic state. |
Monday, November 21, 2022 4:36PM - 4:49PM |
T21.00003: On Maximum Enstrophy Dissipation in 2D Navier-Stokes Flows in the Limit of Vanishing Viscosity Bartosz Protas, Pritpal Matharu, Tsuyoshi Yoneda We consider enstrophy dissipation in two-dimensional (2D) Navier-Stokes flows and focus on how this quantity behaves in the limit of vanishing viscosity. After recalling a number of a priori estimates providing lower and upper bounds on this quantity, we state an optimization problem aimed at probing the sharpness of these estimates as functions of viscosity. More precisely, solutions of this problem are the initial conditions with fixed palinstrophy and possessing the property that the resulting 2D Navier-Stokes flows locally maximize the enstrophy dissipation over a given time window. This problem is solved numerically with an adjoint-based gradient ascent method and solutions obtained for a broad range of viscosities and lengths of the time window reveal the presence of multiple branches of local maximizers, each associated with a distinct mechanism for the amplification of palinstrophy. The dependence of the maximum enstrophy dissipation on viscosity is shown to be in quantitative agreement with the estimate due to Ciampa, Crippa & Spirito (2021), demonstrating the sharpness of this bound. |
Monday, November 21, 2022 4:49PM - 5:02PM |
T21.00004: Exact invariant solutions capturing large-scale motions in the turbulent asymptotic suction boundary layer Sajjad Azimi, Carlo Cossu, Tobias M Schneider A dynamical systems description of high-Reynolds-number turbulence requires capturing large-scale motions (LSMs), coherent structures that span the entire domain and control global energy and momentum transport. As LSMs emerge as correlated collective structures within a background of small-scale fluctuations, they are unlikely represented by invariant solutions of the full Navier-Stokes equations. Using spatial filtering approaches, we thus associate large-scale motions in the asymptotic suction boundary layer flow with exact invariant solutions of model equations that govern large-scale motions without the background fluctuations in the open boundary layer. We specifically construct several steady and time-periodic invariant self-sustained solutions of the filtered Navier-Stokes equations at the friction Reynolds number of 1168. Interactions of the large-scale streaks and vortices within these solutions are reminiscent of known processes including the streak-vortex quasi-periodic regeneration cycle and large-scale hairpin-like vortices. |
Monday, November 21, 2022 5:02PM - 5:15PM |
T21.00005: Finding heteroclinic connections between simple invariant solutions using automatic differentiation Andrew Cleary, Jacob Page Our mechanistic understanding of fluid turbulence has substantially improved in recent decades due to the discovery of large numbers of unstable simple invariant solutions to the Navier-Stokes equations. Heteroclinic connections between these solutions have been hypothesised to play an important role in high-dissipation, intermittent `bursting' events. However, standard methods of detecting simple invariant solutions are not suited to finding these connecting orbits, and consequently only a few have been discovered to date. Here, we introduce automatic differentiation (AD) as a robust technique for finding connections via gradient-based minimisation of a suitable loss function. We first use a new, fully differentiable point vortex solver [jax-pv] as a playground to test the loss-based approach. We demonstrate that AD can successfully find the connection in the integrable 3 vortex system, and also find connections in non-integrable systems with higher numbers of vortices. We then extend our analysis to a two-dimensional Kolmogorov flow (monochromatically forced on the two-torus) using a fully differentiable Navier-Stokes solver (Kochkov et al, Proc. Nat. Acad. Sci. 118, 2021) to search for "bursting" connections between low dissipation relative equilibria. |
Monday, November 21, 2022 5:15PM - 5:28PM |
T21.00006: Strange nonchaos and crisis-induced intermittency in a forced self-excited jet Zhijian Yang, Yu Guan, Stephane Redonnet, Larry K.B. Li We experimentally study the transition from order to chaos in a prototypical hydrodynamic oscillator, namely a self-excited low-density jet subjected to external harmonic forcing. On increasing the forcing amplitude at an off-resonance frequency, we find that the jet bifurcates through a complex sequence of nonlinear states: period-1 limit cycle $\rightarrow$ 2-frequency quasiperiodic torus $\rightarrow$ strange nonchaotic attractor (SNA) $\rightarrow$ crisis-induced intermittency $\rightarrow$ low-dimensional chaos. We verify the existence of the SNA through the spectral distribution, the correlation dimension, the 0-1 test, and the horizontal visibility graph. We find that the SNA emerges from a loss of smoothness in the quasiperiodic torus. We then verify the existence of crisis-induced intermittency through the scaling laws of the average and instantaneous SNA epoch durations as well as various recurrence measures. We find that the crisis-induced intermittency is caused by a collision between the SNA and a basin of chaotic attraction. Our measurements represent the first experimental evidence of an SNA and crisis-induced intermittency in an open shear flow, contributing to a better understanding of how chaos can arise in forced hydrodynamic oscillators. |
Monday, November 21, 2022 5:28PM - 5:41PM |
T21.00007: Invariant tori in turbulence and chaos Jeremy Parker, Tobias M Schneider One approach to understand the chaotic dynamics of turbulence is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems including turbulence, while higher-dimensional invariant tori representing quasi-periodic dynamics have rarely been considered. Fully developed turbulence is spatiotempoarally chaotic and has a large number of positive Lyapunov exponents, so-called hyperchaos. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; that tori can be numerically identified via bifurcations of unstable periodic orbits and that their parametric continuation and characterization of stability properties is feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos. Our results specifically open avenues toward including tori in a generalized periodic orbit theory aimed at most accurately expressing statistical properties of chaos in terms of expansions over the non-chaotic invariant solutions of the system. |
Monday, November 21, 2022 5:41PM - 5:54PM |
T21.00008: Discrete and Continuous Symmetry Reduction for Minimal Parametrizations of Chaotic Kolmogorov Flows Simon Kneer, Nazmi Burak Budanur Mathematical laws that govern fluid motion preserve their shape under translation, rotation, and reflection of coordinates. Consequently, most hydrodynamic systems of interest exhibit a set of symmetries, the action of which on the fluid states commutes with the dynamics. In complex flows, typical non-laminar fluid states are not invariant under these symmetries. Thus, each solution of the system has many dynamically equivalent symmetry copies. For data-driven model reduction methods, such as undercomplete Autoencoders, this multiplicity is not desired since it results in an artificial inflation of the training data which does not yield any physical insight. We consider this problem in the sinusoidally-driven Navier-Stokes equations in two dimensions, i.e. Kolmogorov flow, which is symmetric under continuous translations as well as discrete rotations and reflections. |
Monday, November 21, 2022 5:54PM - 6:07PM Author not Attending |
T21.00009: Quasi-periodic oscillations, resonance, and chaos in a two-dimensional square-vortex flow Balachandra Suri We present a numerical study of transition to chaos in a laterally bounded two-dimensional flow composed of an array of square vortices. The flow at low Reynolds numbers is invariant under two-fold reflection symmetries. We show that the flow undergoes a sequence of hopf bifurcations leading to the formation of a spatially asymmetric temporally quasi-periodic solution (a 2-torus in phase space) which remains stable inside a narrow O(1) window of Reynolds numbers. Computing the intersections of this 2-torus with a Poincare section, we identify very narrow windows O(0.1) in Reynolds number where the dynamics turn periodic due to resonance. Finally, we show that the flow transitions to chaos via the breaking-up of the 2-torus. |
Monday, November 21, 2022 6:07PM - 6:20PM |
T21.00010: Dynamics of Minimal Networks: Ring of Oscillators Andrea E Biju, Sneha Srikanth, Krishna Manoj, Samadhan A Pawar, R. I. Sujith Several natural and engineering systems exhibiting oscillatory behavior can be modeled as minimal networks of oscillators. In such systems, oscillators that may not be direct neighbors can interact with distance-dependent time delays, resulting in local, nonlocal, or global coupling schemes. The present study examines how different coupling schemes and coupling parameters (number of oscillators, coupling strength, and time delay) affect the dynamics of a minimal ring of Stuart-Landau oscillators. As the coupling scheme changes from local to nonlocal, the variety of splay states and the region of amplitude death increases. Additionally, when all oscillators are coupled, a weak chimera state is observed when the delays are distance-dependent, whereas, generalized splay states are observed when the delays are identical. We anticipate that the insights from this study will help identify the underlying coupling scheme in thermo-fluid and other physical systems. |
Monday, November 21, 2022 6:20PM - 6:33PM |
T21.00011: Breathers and Fermi-Pasta-Ulam-Tsingou recurrence for resonant three-wave interactions Hui Min YIN, Kwok Wing CHOW Resonant three-wave interactions occur frequently for surface and internal waves. Quadratic nonlinearities constitute the dominant features, in sharp contrast with the cubic nonlinear Schrödinger model of a narrow-band wave packet. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers. We study computationally the dynamics beyond the first formation of breathers, and demonstrate repeating patterns of breathers as a manifestation of the Fermi-Pasta-Ulan-Tsingou recurrence (FPUT) phenomenon. A 'cascading mechanism' provides an analytical verification, as the fundamental and sideband modes reach roughly the same order of magnitude at one particular instant, signifying the first occurrence of a breather. A triangular spectrum is also obtained, similar to experimental observations of optical pulses governed by the nonlinear Schrödinger equation. Such energy spectra can also elucidate the spreading of energy among the sidebands and components of the triad resonance. The concept of 'effective energy' can elucidate the 'regular' and 'staggered' FPUT patterns. These results can enhance the understanding the dynamics of the upper ocean. |
Monday, November 21, 2022 6:33PM - 6:46PM |
T21.00012: Non-linear analysis of partially ionized Richtmyer-Meshkov instability Abeer H BAKHSH, Yuan Li, Shahad Alzaidi, Ravi Samtaney Richtmyer-Meshkov instability (RMI) occurs when a perturbed density interface between different fluids is impulsively accelerated by a shock wave. It has been noticed that regardless of the direction of the shock wave propagation, the perturbed interface between two fluids of different densities is unstable in RM. Complex non-equilibrium processes such as dissociation and ionization can influence RMI. In the present work, we investigate the effects of partially or fully ionized plasma on the RMI using two-dimensional nonlinear simulations in planar geometry with a two-component ion-neutral model. In addition, we investigate the effect of Mach numbers and magnetic fields on the ionization process. |
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