Bulletin of the American Physical Society
67th Annual Meeting of the APS Division of Fluid Dynamics
Volume 59, Number 20
Sunday–Tuesday, November 23–25, 2014; San Francisco, California
Session A17: Nonlinear Dynamics I: Coherent Structures I |
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Chair: Steven Brunton, University of Washington Room: 2002 |
Sunday, November 23, 2014 8:00AM - 8:13AM |
A17.00001: A comprehensive investigation of exact coherent states in Newtonian channel flow Jae Sung Park, Rashad Moarref, Beverley J. McKeon, Michael D. Graham Exact coherent states have been intensively investigated for better understanding of the transition to turbulence and the complex dynamics in shear flows. Here, we present five families of nonlinear traveling wave solutions in Newtonian channel flow. A Prandtl-von K\'arm\'an plot is used to characterize the solutions, in comparison to previously discovered solutions in the same geometry. One solution family shows very intriguing behavior in terms of mean profiles: its upper and lower branches appear to approach the classical Newtonian and viscoelastic turbulent profiles, respectively. On the lower branch of this solution, a spatially subharmonic bifurcation arises, giving rise to period doubling. The solutions are then considered in state space to identify connections to turbulent flow trajectories and paths of an intermittent bursting phenomenon. Lastly, a low-order representation of our exact coherent states is obtained using the resolvent mode decomposition of McKeon \& Sharma (JFM, 2010). While lower branch solutions can be approximated by a few resolvent modes, typically one dominant mode, upper branch solutions need a larger number of modes. The dominant features of leading resolvent modes and the dependence of Reynolds number on those modes are further discussed. [Preview Abstract] |
Sunday, November 23, 2014 8:13AM - 8:26AM |
A17.00002: An infinite hierarchy of guage particle solutions to a regularized Euler equation: numerical methods and beyond Henry Jacobs, Colin Cotter, Darryl Holm, David Meier In this talk we present an infinite hierarchy of exact solutions to a regularized form of Euler's fluid equations. Each of these solutions is isomorphic to the motion of finitely many guage-theoretic particles, wherein each particle stores internal Lie group structures which correspond to higher-order deformation gradients of the Lagrangian flow map. Collision experiments suggest that two particles at one level in the hierarchy can asymptotically merge into a single particle at a higher-level in the hierarchy. We will display some of these collisions and provide a formal argument to explain this phenomena. These collision events are interpreted as a cascade to smaller scales. [Preview Abstract] |
Sunday, November 23, 2014 8:26AM - 8:39AM |
A17.00003: Exact coherent states in a reduced model of parallel shear flows Cedric Beaume, Edgar Knobloch, Greg Chini, Keith Julien In plane Couette flow, the lower branch Nagata solution follows simple streamwise dynamics at large Reynolds numbers. A decomposition of this solution into Fourier modes in this direction yields modes whose amplitudes scale with inverse powers of the Reynolds number, with exponents that increase with increasing mode number (Wang et al., Phys. Rev. Lett. 98, 204501 (2007)). In this work, we use this scaling to derive a reduced model for exact coherent structures in general parallel shear flows. The reduced model describes the dynamics of the streamwise-averaged flow and of the fundamental fluctuations and is regularized by retaining higher order viscous terms for the fluctuations. Numerical methods are designed to find good approximates of nontrivial solutions which are then converged using a preconditioned Newton method. This procedure captures both lower branch and upper branch solutions and demonstrates that these branches are connected via a saddle-node bifurcation. [Preview Abstract] |
Sunday, November 23, 2014 8:39AM - 8:52AM |
A17.00004: Homotopy between plane Couette flow and Pipe flow Masato Nagata, Kengo Deguchi In order to investigate symmetry connections between two canonical shear flows, i.e. plane Couette (PCF) and pipe flow (PF), which are linearly stable for all Reynolds numbers and therefore undergo subcritical transition, we take annular Poiseuille-Couette flow (APCF) as an intermediary Although PCF and PF are very different geometrically, APCF recovers PCF by taking the narrow gap limit, and also PF by taking the limit of vanishing inner cylinder where a homotopy of the basis functions from no-slip to regular conditions at the centre is considered. We show that the double-layered mirror-symmetric solutions in sliding Couette flow (APCF without axial pressure gradient) found by Deguchi \& Nagata (2011) can be traced back to the mirror-symmetric solutions in PCF. Also we show that only the double-layered solution successfully reaches the PF limit, reproducing the mirror-symmetric solution in PF classified as M1 by Pringle \& Kerswell (2007). [Preview Abstract] |
Sunday, November 23, 2014 8:52AM - 9:05AM |
A17.00005: Evolution of Power and Structure in an Electroconvective Transition Marcus Daum, Zrinka Greguri\'c Feren\v{c}ek, John Cressman Electroconvecting liquid crystals support a wide range of states which are characterized by the system's ability to create, support, and annihilate structure. Through the creation of electroconvective rolls and defects, the liquid crystal sample is able to absorb and dissipate more energy. By simultaneously acquiring optical and electrical data we are able to accurately compare structure of the sample and injected electrical power. Here we report on spatiotemporal interactions as we abruptly force the sample from an initial state to defect turbulence. By observing the power and the structure of the sample, we have identified qualitatively different transient behaviors based on initial structure within the sample. Using these characterizations, we are able to draw parallels between structural dynamics and large fluctuations in power. [Preview Abstract] |
Sunday, November 23, 2014 9:05AM - 9:18AM |
A17.00006: First experimental measurement of the Melnikov function Patrice Meunier, Peter Huck, Emmanuel Villermaux The problem of scalar mixing in a 2D flow has been extensively studied numerically by following Lagrangian tracers or theoretically using the tools of dynamical systems (KAM tori, quasi-periodic orbits, chaotic attractors...). However, in all these modelisations, the diffusion of the scalar is usually neglected for the purposes of the numerical/theoretical tools. We present here an experiment with an exactly 2D flow, which allows to study properly the diffusive and mixing problem at very large Peclet number. To avoid any 3D flow, the fluid is stratified with a linear density gradient using salted water. Moreover, the viscosity of the water is decreased of an order of magnitude by adding 10\% ucon oil in the water. The flow under study is created by the co-rotation of two vertical cylinders, leading to a homoclinc point at the center. This base flow is perturbed periodically by a third oscillating cylinder. The dye injected at the center settles on the stable manifold of the homoclinic point. The distance between the stable and the unstable manifold is measured as half the distance between the maximum and the minimum of the dye's undulation. The results are in good quantitative agreement with the theoretical prediction of the Melnikov function for this flow. [Preview Abstract] |
Sunday, November 23, 2014 9:18AM - 9:31AM |
A17.00007: Linear analysis of the mean flow of thermosolutal travelling waves Sam Turton, Laurette Tuckerman We carry out a stability analysis on the mean flow extracted from 2D travelling waves in thermosolutal convection over a range of values for separation parameter S, Lewis number Le and Prandtl number Pr. Consistent with similar analyses performed on the mean flow of the cylinder wake, we find that the eigenfrequency provides an accurate measure of the frequency of the travelling waves, in contrast to the frequency obtained by linearizing about the unstable conductive state. The linear growth rates are close to zero just beyond the Hopf bifurcation, and in the case of large Pr, remain so for larger values of the thermal Rayleigh number, implying that the travelling wave mean flow is marginally stable in these regimes. [Preview Abstract] |
Sunday, November 23, 2014 9:31AM - 9:44AM |
A17.00008: Reynolds Number Effects on Mixing Due to Topological Chaos Spencer Smith, Sangeeta Warrier Topological chaos has emerged as a powerful modeling tool to investigate fluid mixing. While this theory can guarantee a lower bound on the stretching rate of certain material lines, it does not indicate what fraction of the fluid actually participates in this minimally mandated mixing. Indeed, the area in which effective mixing takes place depends on physical parameters such as the Reynolds number. To help clarify this dependency, we numerically simulate the effects of a batch stirring device on a 2D incompressible Newtonian fluid in the laminar regime. In particular, we calculate the finite time Lyapunov exponent (FTLE) field for two different stirring protocols, one topologically complex (pseudo Anosov) and one simple (finite order), over a range of viscosities. After extracting appropriate measures indicative of mixing from the FTLE field, we see a clearly defined range of Reynolds numbers for which the relative efficacy of the pseudo Anosov protocol over the finite order protocol justifies the application of topological chaos. The Reynolds number dependance of these mixing measures also reveals several other intriguing phenomena. [Preview Abstract] |
Sunday, November 23, 2014 9:44AM - 9:57AM |
A17.00009: A Jacobian-free Newton-Krylov solver for determination of scaling laws in coherent Rayleigh-B\'{e}nard convection David Sondak, Leslie Smith, Fabian Waleffe, Anakewit Boonkasame Computational studies of \emph{coherent} Rayleigh-B\'{e}nard convection in a two-dimensional channel with no-slip top and bottom walls are performed in order to determine scaling laws for a range of Rayleigh ($Ra$) and Prandtl ($Pr$) numbers. Since these coherent states are unstable, a Jacobian-free Newton-GMRES algorithm is developed. This approach allows us to determine scaling of the Nusselt number ($Nu$) with $Ra$ by tracking unstable solutions to the Boussinesq equations. Scaling laws are presented for the primary solution that bifurcates from the conducting state at $Ra \sim 1708$, becomes unstable in a Hopf bifurcation at $Ra \sim 5.4\times 10^{4}$ but have been computed up to $Ra \sim 5\times 10^{6}$. We also determine scaling laws for the optimal heat transport up to $Ra\sim 10^{8}$. Mechanisms for the observed behavior are discussed including the relationship between the optimal solution and the primary solution as well as the effect of $Pr$. We explore properties of the algorithm and review its potential as a tool in determining scaling laws for thermal convection as well as some areas for improvement. Extensions of this work to three-dimensional Rayleigh-B\'{e}nard convection will be discussed. [Preview Abstract] |
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