Session D8: Nonlinear Dynamics: Topology & Theoretical

Characterizing Mixing in a Quasi-Two-Dimensional Flow using Persistent Homology

Fluid mixing is a tremendously important phenomenon present in numerous physical systems, both natural and human-made. Describing, understanding, and predicting the mixing behavior of fluid flows poses an immense challenge. In this work, we explore the utility of topological data analysis in quantifying fluid mixing. We analyze Eulerian and Lagrangian quantities obtained from a quasi-two-dimensional flow realized by driving a thin layer of fluid with electromagnetic forces. Our analysis employs persistent homology, which offers a unique framework for quantifying topological features associated with connectivity in the fluid flow. Preliminary results suggest that this topological approach offers new physical insight, complementing existing methods for quantifying fluid mixing.   

Topological Chaos in a Three-Dimensional Spherical Vortex

Topological techniques have proven to be powerful tools for characterizing the complexity of advection in many 2D fluid flows. However, the path to extending many techniques to three dimensions is filled with roadblocks, which prevent their application to a wider variety of interesting flows. We successfully extend the homotopic lobe dynamics (HLD) technique, previously developed for 2D area-preserving flows, to 3D volume-preserving flows. Specifically, we use intersecting two-dimensional stable and unstable invariant manifolds to construct a symbolic representation of the topological dynamics. This symbolic representation can be used to classify the trajectories of passively advected particles and to compute mixing measures, such as the topological entropy. In this talk, we apply the 3D HLD technique to an explicit numerical example: a time-periodic perturbation of Hill's spherical vortex, modified to break both rotational symmetry and integrability. For this system, the 3D HLD technique is able to detect a distinction between the topologically forced 2D stretching rate of material surfaces and the 1D stretching rate of material curves, illustrating the truly 3D nature of our approach.   

Transition to turbulence: highway through the edge of chaos is charted by Koopman modes

We present evidence of low-dimensional dynamical state-space structures enabling transition to turbulence using an extension of the recently advanced operator-theoretic approach to turbulence of Mezić (2005). To do this, we use the dynamic-mode-decomposition (DMD) algorithm of Schmid & Sesterhenn (2008) on the minimal seed trajectories in plane Couette flow of Rabin et al. (2012) and Eaves & Caulfield (2015), which transition to turbulence via the most energy-efficient finite amplitude perturbation from the laminar state. The methodology enables identification of low dimensional structures associated with stable and unstable manifolds of exact solutions to the Navier-Stokes equations, even though the state space is very high-dimensional. In consequence, the results provide a low-dimensional representation of the transition to turbulence and also identify the first known dynamical signature of the importance of edge states in this transition.   

Using Persistent Homology to Identify Localised Defects in Rayleigh B\'enard Convection

Complex spatiotemporal convective roll patterns are observed when a sufficiently large temperature gradient is created across a thin layer of fluid. These roll patterns are often characterized by the presence of localised defects such as centers, spirals, disclinations, grain boundaries, which play a crucial dynamical role. Our research focuses on using persistent homology (a branch of algebraic topology) to identify these defects in an experimental realization of the Rayleigh B\'enard convection in a cylindrical container. Persistent homology provides a powerful mathematical formalism in which the topological characteristics of a pattern (shadowgraph image in our case) are encoded in a so-called persistence diagram. By identifying several instants in the experiment that correspond to the appearance of a certain type of defect and computing the persistence diagrams for the corresponding shadowgraph images, we extract signatures in the persistence diagram which characterize the defect. Then, for a spatiotemporally resolved series of shadowgraph images we show that using signatures from the persistence diagrams one can automate identifying the instants when localized defects appear.   

An effective diffusivity model based on Koopman mode decomposition

In the previous work, we had shown that the Koopman mode decomposition (KMD) can be used to analyze mixing of passive tracers in time-dependent flows. In this talk, we discuss the extension of this type of analysis to the case of advection-diffusion transport for passive scalar fields. Application of KMD to flows with complex time-dependence yields a decomposition of the flow into mean, periodic and chaotic components. We briefly discuss the computation of these components using a combination of harmonic averaging and Discrete Fourier Transform. We propose a new effective diffusivity model in which the advection is dominated by mean and periodic components whereas the effect of chaotic motion is absorbed into an effective diffusivity tensor. The performance of this model is investigated in the case of lid-driven cavity flow.   

Could time itself be logarithmic?

This presentation hypothesizes that increments of time may be logarithmic and measured from an initial instant -- the log of absolute time if you will. In this alternative view all equations involving time must be written with $\ln t/t_o$ where $t$ is measured in linear increments from the beginning of the universe and $t_o$ is the universal time scale. All equations involving time derivatives must be written not as $d/dt$ but $d/d \ln t/to = t d/dt$. An immediate consequence, for example, is that our definition of mass in Newton's Law must change as well: from $m dv/dt = F$ to $m_* d v / d \ln t/to = m_* t dv/dt =F$ where $F$ is force applied and $v$ is velocity (however defined). $m_* =m/t$ is the 'true' or absolute mass. Since we have been measuring for only about 500 years and the universe is estimated to be about 18 billion years (millions of billions of seconds) old, the differences are impossible to measure; i.e., $ln (t + \delta t) - \ln t \approx \delta t / t$. It is only when we look backwards towards the beginning of the universe that we notice the difference -- mass, $m = m_* t$, appears to be missing. So we need ``dark matter'' to make our equations balance -- when in fact it might be our ``linear-time'' equations and definitions that are wrong.   

Metriplectic simulated annealing for quasigeostrophic flow

Metriplectic dynamics [1,2] is a general form for dynamical systems that embodies the first and second laws of thermodynamics, energy conservation and entropy production. The formalism provides an $H$-theorem for relaxation to nontrivial equilibrium states. Upon choosing enstrophy as entropy and potential vorticity of the form $q= \nabla^2\psi +T(x)$, recent results of computations, akin to those of [3], will be described for various topography functions $T(x)$, including ridge ($T=\exp(-x^2/2)$) and random functions. Interpretation of the results, in particular their sensitivity to the chosen entropy function will be discussed. \\ \noindent [1] P.J.~Morrison, Physica D {\bf18}, 410 (1986).\\ \noindent [2] A.M.~Bloch, P.J.~Morrison, and T.S. Ratiu, in Recent Trends\\ in Dynamical Systems {\bf35}, 371 (2013).\\ \noindent [3] G.R. Flierl and P.J, Morrison, Physica D {\bf240}, 212 (2011).   

Data-driven discovery of partial differential equations

Fluid dynamics is inherently governed by spatial-temporal interactions which can be characterized by partial differential equations (PDEs). Emerging sensor and measurement technologies allowing for rich, time-series data collection motivate new data-driven methods for discovering governing equations. We present a novel computational technique for discovering governing PDEs from time series measurements. A library of candidate terms for the PDE including nonlinearities and partial derivatives is computed and sparse regression is then used to identify a subset which accurately reflects the measured dynamics. Measurements may be taken either in a Eulerian framework to discover field equations or in a Lagrangian framework to study a single stochastic trajectory. The method is shown to be robust, efficient, and to work on a variety of canonical equations. Data collected from a simulation of a flow field around a cylinder is used to accurately identify the Navier-Stokes vorticity equation and the Reynolds number to within 1\%. A single trace of Brownian motion is also used to identify the diffusion equation. Our method provides a novel approach towards data enabled science where spatial-temporal information bolsters classical machine learning techniques to identify physical laws.   

Approximate Solutions to the Linearized Navier-Stokes Equations

The linearized Navier-Stokes equations for incompressible channel flow are considered in which the flow is homogeneous in two directions. We study the initial-value problem for $ v$ and $\omega_y$, where $y$ is the coordinate normal to the wall. After a Laplace transform in time and a double Fourier transform in space we use the WKB approximation on the resulting system of ODE’s in $y$. For example, for the inviscid equations we can construct analytically the Green’s function for such solutions in terms of the Bessel functions $J_{+1/3}, J_{-1/3}, J_1,$ and $Y_1$ and their modified counterparts. In this approach the critical layer or the $y$ location where $ U(y) = \omega/k_x$ requires special attention, as might be expected, as well as the location of the turning point where $d^2U/dy^2 = (k_x^2+ k_z^2)(\omega/k_x - U(y))$, if it exists.   

Test-filter scale effects on spectral energy transfer in direct numerical simulations of stratified turbulence

The spectral kinetic and potential energy transfers around a test-filter scale will be presented in this talk. We use direct numerical simulations of stratified turbulence and study the up- and downscale energy transfers when different test-filter scales are applied. Our results suggest that the spectral energy transfer depends on the buoyancy Reynolds number $Re_b$ and test-filter scale $\Delta_{test}$. In particular, an up-scale energy transfer (i.e. backscatter) from sub-filter scales to intermediate scales are seen when $\Delta_{test}$ is around the dissipation scale $L_d$. However, we find that this spectral backscatter is due to viscous effects and not a turbulent mechanisms of stratified turbulence. In addition, our results demonstrate that effective turbulent Prandtl number spectra show constant values around $Pr_t \approx 1$ for the local energy transfer or when the buoyancy Reynolds number is large.