Bulletin of the American Physical Society
70th Annual Meeting of the APS Division of Fluid Dynamics
Volume 62, Number 14
Sunday–Tuesday, November 19–21, 2017; Denver, Colorado
Session F1: Nonlinear Dynamics: Turbulence and TransitionNonlinear Turbulence
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Chair: Sebastian Altmeyer, Institute of Science and Technology Austria Room: 401 |
Monday, November 20, 2017 8:00AM - 8:13AM |
F1.00001: Invariant solutions organizing pipe flow turbulence Sebastian Altmeyer, Björn Hof Invariant unstable solutions such as (relative) periodic orbits (RPOs) and travelling waves (TWs) have been suggested to act as building blocks of turbulence in basic shear flows. A large number of such invariant solutions have been determined in recent years, yet most observed states typically possess spatial symmetry (e.g. to rotation, reflection or translation) due to artificial symmetry restrictions. In contrast turbulence does not have any of such symmetry in general. Commonly used recurrence methods are unlikely to capture orbits in full space due to their complexity and the short visiting times of turbulent trajectories. Nevertheless, looking for periodic modulations instead of full recurrences we have been able to extract asymmetric invariant solutions, RPOs and TWs, dynamically embedded in turbulence. Compared to other invariant solutions in subspaces, neither TWs or orbits the isosurfaces looks less smooth indicating various different length scales within these structures. This is a typical observation in turbulent flows which also show strong fluctuations, e.g. in the internal arrangement of high and low velocity streaks. The complexity of the underlying manifold results in closed curves either in Re and k defining parameters with up to four solutions of a single RPO. [Preview Abstract] |
Monday, November 20, 2017 8:13AM - 8:26AM |
F1.00002: Calculation of minimal seeds in stabilised pipe flows Elena Marensi, Ashley P. Willis The minimal seed is the initial perturbation of lowest energy that triggers transition to turbulence. Variational methods are used to construct fully nonlinear optimisation problems that seek the minimal seed in stabilised pipe flows. By introducing a body force that flattens the mean streamwise velocity profile, the minimal seed is shown to move towards higher values of the critical initial energy $E_c$. In the unforced case, we apply a spectral filter to the minimal seed at $Re$=2000. The structure of the minimal seed is found to be robust to quite severe spectral filtering as well as to changes in the base flow. To clarify the relevance and realisability of the minimal seed in experiments, a statistical study is performed, where energy is scattered randomly over the allowed wavenumbers and the probability of transition calculated as a function of the initial energy $E_0$$>$$E_c$. The initial conditions generated with this analysis are fed into direct numerical simulations with a localised forcing introduced to mimic the presence of a baffle in the core of the flow. The resulting curve $E_c$=$E_c(Re)$ is compared to the one obtained in the uncontrolled case. [Preview Abstract] |
Monday, November 20, 2017 8:26AM - 8:39AM |
F1.00003: Statespace geometry of puff formation in pipe flow Nazmi Burak Budanur, Bjoern Hof Localized patches of chaotically moving fluid known as puffs play a central role in the transition to turbulence in pipe flow. Puffs coexist with the laminar flow and their large-scale dynamics sets the critical Reynolds number: When the rate of puff splitting exceeds that of decaying, turbulence in a long pipe becomes sustained in a statistical sense (Avila \textit{et al.}, Science \textbf{333}, 192--196 (2011)). Since puffs appear despite the linear stability of the Hagen-Poiseuille flow, one expects them to emerge from the bifurcations of finite-amplitude solutions of Navier-Stokes equations. In numerical simulations of pipe flow, Avila \textit{et al.}, Phys. Rev. Let. \textbf{110}, 224502 (2013) discovered a pair of streamwise localized relative periodic orbits, which are time-periodic solutions with spatial drifts. We combine symmetry reduction and Poincar\'e section methods to compute the unstable manifolds of these orbits, revealing statespace structures associated with different stages of puff formation. [Preview Abstract] |
Monday, November 20, 2017 8:39AM - 8:52AM |
F1.00004: Hairpin exact coherent states in channel flow Michael Graham, Ashwin Shekar Questions remain over the role of hairpin vortices in fully developed turbulent flows. Studies have shown that hairpins play a role in the dynamics away from the wall but the question still persists if they play any part in (near wall) fully developed turbulent dynamics. In addition, the robustness of the hairpin vortex regeneration mechanism is still under investigation. Recent studies have shown the existence of nonlinear traveling wave solutions to the Navier-Stokes equations, also known as exact coherent states (ECS), that capture many aspects of near-wall turbulent structures. Previously discovered ECS in channel flow have a quasi-streamwise vortex structure, with no indication of hairpin formation. Here we present a family of traveling wave solutions for channel flow that displays hairpin vortices. They have a streamwise vortex --streak structure near the wall with a spatially localized hairpin head near the channel centerline, attached to and sustained by the near wall structures. This family of solutions emerges through a transcritical bifurcation from a branch of traveling wave solutions with y and z reflectional symmetry. We also look into the instabilities that lead to the development of hairpins also explore its connection to turbulent dynamics. [Preview Abstract] |
Monday, November 20, 2017 8:52AM - 9:05AM |
F1.00005: The origin of turbulent stripes in plane Poiseuille flow Bjorn Hof, Chaitanya Paranjape, Yohann Duguet Spatio-temporal complexity is a defining feature of turbulence and sets it apart from ordinary chaotic systems. Even for geometrically simple cases such as planar channel flow the multitude of spatial degrees of freedom dominate the dynamics at onset: here turbulence appears in irregular, continuously changing stripe patterns. In direct numerical simulations we identify travelling wave and periodic orbit solutions consisting of streak-vortex pairs arranged in an ordered stripe pattern surrounded by laminar flow. As Re is increased these elementary building blocks undergo a sequence of bifurcations giving rise to chaotic motion, much like in low dimensional dynamical systems. However, due to the spatial degrees of freedom infinitely many routes to chaos co-exist. While temporal chaos develops following standard routes, the spatial complexity originates from the multiplicity of elementary flow states. [Preview Abstract] |
Monday, November 20, 2017 9:05AM - 9:18AM |
F1.00006: One-dimensional hydrodynamic equation generating turbulent scaling laws and self-similar singular solutions Takashi Sakajo, Takeshi Matsumoto One of the remarkable features of the stochastic laws of fluid turbulence is the emergence of the inertial range in the energy density spectrum on which the energy cascades self-similarly. In the Kolmogorov's theory of fluid turbulence, it is suggested by Onsager that singular solutions of the Navier-Stokes equations in the inviscid limit or the Euler equations that does not conserve the energy play an important role. The existence of energy dissipating weak solution of the Euler equations with $1/3$-H\"{o}lder continuity has recently been established by Buckmaster et al. Nevertheless, it remains a theoretical challenge to deduce the stochastic laws from such singular solutions. To gain an insight into this problem, we propose a one-dimensional hydrodynamic nonlinear equation based on the Constantin-Lax-Majda-DeGregorio model. The equation admits an invariant quantity and a finite-time blowup solution in the inviscid case, while with the viscous term and a steady forcing, we obtain a singular steady solution in its inviscid limit. In addition, there emerges the inertial range corresponding to the cascade of the inviscid invariant under a random forcing. In the presentation, we provide recent results on the relation between the turbulent stochastic laws and the singular solutions. [Preview Abstract] |
Monday, November 20, 2017 9:18AM - 9:31AM |
F1.00007: A Hybrid Monte Carlo importance sampling of rare events in Turbulence and in Turbulent Models Georgios Margazoglou, Luca Biferale, Rainer Grauer, Karl Jansen, David Mesterhazy, Tillmann Rosenow, Raffaele Tripiccione Extreme and rare events is a challenging topic in the field of turbulence. Trying to investigate those instances through the use of traditional numerical tools turns to be a notorious task, as they fail to systematically sample the fluctuations around them. On the other hand, we propose that an importance sampling Monte Carlo method can selectively highlight extreme events in remote areas of the phase space and induce their occurrence. We present a brand new computational approach, based on the path integral formulation of stochastic dynamics, and employ an accelerated Hybrid Monte Carlo (HMC) algorithm for this purpose. Through the paradigm of stochastic one-dimensional Burgers' equation, subjected to a random noise that is white-in-time and power-law correlated in Fourier space, we will prove our concept and benchmark our results with standard CFD methods. Furthermore, we will present our first results of constrained sampling around saddle-point instanton configurations (optimal fluctuations). [Preview Abstract] |
Monday, November 20, 2017 9:31AM - 9:44AM |
F1.00008: Critical Transitions in Thin Layer Turbulence Santiago Benavides, Alexandros Alexakis We investigate a model of thin layer turbulence that follows the evolution of the two-dimensional motions ${\bf u}_{_{2D}} (x,y)$ along the horizontal directions $(x,y)$ coupled to a single Fourier mode along the vertical direction ($z$) of the form ${\bf u}_q(x, y, z)=[v_x(x,y) \sin(qz), v_y(x,y)\sin(qz), v_z(x,y)\cos(qz)\, ]$, reducing thus the system to two coupled, two-dimensional equations. Its reduced dimensionality allows a thorough investigation of the transition from a forward to an inverse cascade of energy as the thickness of the layer $H=\pi/q$ is varied.Starting from a thick layer and reducing its thickness it is shown that two critical heights are met (i) one for which the forward unidirectional cascade (similar to three-dimensional turbulence) transitions to a bidirectional cascade transferring energy to both small and large scales and (ii) one for which the bidirectional cascade transitions to a unidirectional inverse cascade when the layer becomes very thin (similar to two-dimensional turbulence). The two critical heights are shown to have different properties close to criticality that we are able to analyze with numerical simulations for a wide range of Reynolds numbers and aspect ratios. [Preview Abstract] |
Monday, November 20, 2017 9:44AM - 9:57AM |
F1.00009: Bifurcation induced by the aspect ratio in a turbulent von Karman swirling flow Javier Burguete, Olivier Liot Two counter-rotating propellers are used to develop turbulence in a cylindrical cavity filled with water. The counter-rotating swirling flow can be the place of multistability, memory effects, and long time dynamics. De la Torre and Burguete [Phys. Rev. Lett. 99, 054101 (2007)] observed a symmetry breaking of the mean flow where the shear layer between the two counter-rotating cells of the flow does not remain in the middle of the cavity. Moreover, this shear layer can spontaneously jump from one side of the cavity to the other with a long residence time (typically 1000 s) compared to the turbulent time scales. But what is/are the problem parameter(s) which fix(es) the position of the shear layer and the spontaneous reversals? Here we analyze this bifurcation: It appears modifying the aspect ratio $\Gamma = H/D$. Whereas for low $\Gamma$ the shear layer position has a smooth evolution when turning the asymmetry between the rotation frequency of the propellers, for high $\Gamma$ the transition becomes abrupt and a symmetry breaking appears. Secondly we observe that the spontaneous reversals with large residence times exist only in a narrow window of aspect ratio. We present a phenomenological model that describres these features. [Preview Abstract] |
Monday, November 20, 2017 9:57AM - 10:10AM |
F1.00010: Koopman decomposition of Burgers’ equation: What can we learn? Jacob Page, Rich Kerswell Burgers’ equation is a well known 1D model of the Navier-Stokes equations and admits a selection of equilibria and travelling wave solutions. A series of Burgers’ trajectories are examined with Dynamic Mode Decomposition (DMD) to probe the capability of the method to extract coherent structures from ``run-down'' simulations. The performance of the method depends critically on the choice of observable. We use the Cole-Hopf transformation to derive an observable which has linear, autonomous dynamics and for which the DMD modes overlap exactly with Koopman modes. This observable can accurately predict the flow evolution beyond the time window of the data used in the DMD, and in that sense outperforms other observables motivated by the nonlinearity in the governing equation. The linearizing observable also allows us to make informed decisions about often ambiguous choices in nonlinear problems, such as rank truncation and snapshot spacing. A number of rules of thumb for connecting DMD with the Koopman operator for nonlinear PDEs are distilled from the results. Related problems in low Reynolds number fluid turbulence are also discussed. [Preview Abstract] |
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