Simulation Studies of Charge Transport on Resistive Structures in Gaseous Ionization Detectors

We developed a tool for the simulation of charge transport on a conducting plate of finite dimensions. This tool is named Chani. Main motivation of developing Chani was to provide a tool for the optimization of the dimensions and resistivity of the anode electrodes in spark-protected Micropattern Gaseous Detectors (MPGD). In this thesis, we start with the general description of the LHC and the ATLAS Experiment. Then, we review the gaseous ionization detector technologies and in particular, the micromegas technology. We then present the working principles of Chani along with the example calculations. These examples include comparisons with the analytically solvable problems which shows that the simulation results are reasonable.Comment: MSc Thesis submitted to Istanbul Technical University, Institute of Science and Technology in June 2012. 105 pages, 28 figure

Heteroclinic path to spatially localized chaos in pipe flow

In shear flows at transitional Reynolds numbers, localized patches of turbulence, known as puffs, coexist with the laminar flow. Recently, Avila et al., Phys. Rev. Let. 110, 224502 (2013) discovered two spatially localized relative periodic solutions for pipe flow, which appeared in a saddle-node bifurcation at low speeds. Combining slicing methods for continuous symmetry reduction with Poincar\'e sections for the first time in a shear flow setting, we compute and visualize the unstable manifold of the lower-branch solution and show that it contains a heteroclinic connection to the upper branch solution. Surprisingly this connection even persists far above the bifurcation point and appears to mediate puff generation, providing a dynamical understanding of this phenomenon.Comment: 10 pages, 5 figure

Prediction and control of spatiotemporal chaos by learning conjugate tubular neighborhoods

I present a data-driven predictive modeling tool that is applicable to high-dimensional chaotic systems with unstable periodic orbits. The basic idea is using deep neural networks to learn coordinate transformations between the trajectories in the periodic orbits' neighborhoods and those of low-dimensional linear systems in a latent space. I argue that the resulting models are partially interpretable since their latent-space dynamics is fully understood. To illustrate the method, I apply it to the numerical solutions of the Kuramoto--Sivashinsky partial differential equation in one dimension. Besides the forward-time predictions, I also show that these models can be leveraged for control.Comment: 22 pages, 11 figures. Revised Manuscript under consideration for publication in APL Machine Learnin

Scale-dependent Error Growth in Navier--Stokes Simulations

We estimate the maximal Lyapunov exponent at different resolutions and Reynolds numbers in large eddy (LES) and direct numerical simulations (DNS) of sinusoidally-driven Navier--Stokes equations in three dimensions. Independent of the Reynolds number when nondimensionalized by Kolmogorov units, the LES Lyapunov exponent diverges as an inverse power of the effective grid spacing showing that the fine scale structures exhibit much faster error growth rates than the larger ones. Effectively, i.e., ignoring the cut-off of this phenomenon at the Kolmogorov scale, this behavior introduces an upper bound to the prediction horizon that can be achieved by improving the precision of initial conditions through refining of the measurement grid.Comment: 13 pages, 4 figure

Complexity of the laminar-turbulent boundary in pipe flow

Over the past decade, the edge of chaos has proven to be a fruitful starting point for investigations of shear flows when the laminar base flow is linearly stable. Numerous computational studies of shear flows demonstrated the existence of states that separate laminar and turbulent regions of the state space. In addition, some studies determined invariant solutions that reside on this edge. In this paper, we study the unstable manifold of one such solution with the aid of continuous symmetry-reduction, which we formulate here for the first time for the simultaneous quotiening of axial and azimuthal symmetries. Upon our investigation of the unstable manifold, we discover a previously unknown traveling wave solution on the laminar-turbulent boundary with a relatively complex structure. By means of low-dimensional projections, we visualize different dynamical paths that connect these solutions to the turbulence. Our numerical experiments demonstrate that the laminar-turbulent boundary exhibits qualitatively different regions whose properties are influenced by the nearby invariant solutions.Comment: 15 pages, 11 figure

State space geometry of the chaotic pilot-wave hydrodynamics

We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the formation of a global attractor. In this final regime, the droplet's angular momentum spontaneously changes its sign as observed in the experiments of Perrard et al. (Phys. Rev. Lett., 113(10):104101, 2014).Comment: Accepted for publication in Chaos: An Interdisciplinary Journal of Nonlinear Scienc

Geometry of transient chaos in streamwise-localized pipe flow turbulence

In pipes and channels, the onset of turbulence is initially dominated by localized transients, which lead to sustained turbulence through their collective dynamics. In the present work, we study the localized turbulence in pipe flow numerically and elucidate a state space structure that gives rise to transient chaos. Starting from the basin boundary separating laminar and turbulent flow, we identify transverse homoclinic orbits, the presence of which necessitates a homoclinic tangle and chaos. A direct consequence of the homoclinic tangle is the fractal nature of the laminar-turbulent boundary, which was conjectured in various earlier studies. By mapping the transverse intersections between the stable and unstable manifold of a periodic orbit, we identify the 'gateways' that promote an escape from turbulence.Comment: Accepted for publication in Physical Review Fluids as a Rapid Communicatio

Coarse graining the state space of a turbulent flow using periodic orbits

We show that turbulent dynamics that arise in simulations of the three-dimensional Navier--Stokes equations in a triply-periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the original system with more than $10^5$ degrees of freedom to a 17-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system's observables as weighted sums over the periodic orbits.Comment: Supplementary video and data will be made available following publication in a journa

Symmetry-reduced Dynamic Mode Decomposition of Near-wall Turbulence

Data-driven dimensionality reduction methods such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) have proven to be useful for exploring complex phenomena within fluid dynamics and beyond. A well-known challenge for these techniques is posed by the continuous symmetries, e.g. translations and rotations, of the system under consideration as drifts in the data dominate the modal expansions without providing an insight into the dynamics of the problem. In the present study, we address this issue for the pressure-driven flow in a rectangular channel by formulating a continuous symmetry reduction method that eliminates the translations simultaneously in the streamwise and spanwise directions. As an application, we consider turbulence in a minimal flow unit at a Reynolds number (based on the centerline velocity and half-channel height) Re = 2000 and compute the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding data windows of varying durations. SRDMD of channel flow reveals episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.Comment: 10 pages, 6 figure