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We propose a novel cluster-based reduced-order modelling (CROM) strategy for unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's group (Burkardt, Gunzburger & Lee, Comput. Meth. Appl. Mech. Engng, vol. 196, 2006a, pp. 337-355) and transition matrix models introduced in fluid dynamics in Eckhardt's group (Schneider, Eckhardt & Vollmer, Phys. Rev. E, vol. 75, 2007, art. 066313). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Second, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e. g. using finite-time Lyapunov exponent (FTLE) and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

Many previous studies have shown that the turbulent mixing layer under periodic forcing tends to adopt a lock-on state, where the major portion of the fluctuations in the flow are synchronized at the forcing frequency. The goal of this experimental study is to apply closed-loop control in order to provoke the lock-on state, using information from the flow itself. We aim to determine the range of frequencies for which the closed-loop control can establish the lock-on, and what mechanisms are contributing to the selection of a feedback frequency. In order to expand the solution space for optimal closed-loop control laws, we use the genetic programming control (CPC) framework. The best closed-loop control laws obtained by CPC are analysed along with the associated physical mechanisms in the mixing layer flow. The resulting closed-loop control significantly outperforms open-loop forcing in terms of robustness to changes in the free-stream velocities. In addition, the selection of feedback frequencies is not locked to the most amplified local mode, but rather a range of frequencies around it.

Mixing layer manipulation experiment from open-loop forcing to closed-loop machine learning control
(2015)

Ordinary differential equations (ODEs) have been studied for centuries as a means to model complex dynamical processes from the real world. Nevertheless, their application to sound synthesis has not yet been fully exploited. In this article we present a systematic approach to sound synthesis based on first-order complex and real ODEs. Using simple time-dependent and nonlinear terms, we illustrate the mapping between ODE coefficients and physically meaningful control parameters such as pitch, pitch bend, decay rate, and attack time. We reveal the connection between nonlinear coupling terms and frequency modulation, and we discuss the implications of this scheme in connection with nonlinear synthesis. The ability to excite a first-order complex ODE with an external input signal is also examined; stochastic or impulsive signals that are physically or synthetically produced can be presented as input to the system, offering additional synthesis possibilities, such as those found in excitation/filter synthesis and filter-based modal synthesis.

Small- and large-scale characterization and mixing properties in a thermally driven thin liquid film
(2015)

We study aqueous, freestanding, thin films stabilized by a surfactant with respect to mixing and dynamical systems properties. With this special setup, a two-dimensional fluid can be realized experimentally. The physics of the system involves a complex interplay of thermal convection and interface and gravitational forces. Methodologically, we characterize the system using two classical dynamical systems properties: Lyapunov exponents and entropies. Our experimental setup produces convection with two stable eddies by applying a temperature gradient in one spot that yields weakly turbulent mixing. From dynamical systems theory, one expects a relation of entropies, Lyapunov exponents, a prediction with little experimental support. We can confirm the corresponding statements experimentally, on different scales using different methods. On the small scale the motion and deformation of fluid filaments of equal size (color imaging velocimetry) are used to compute Lyapunov exponents. On the large scale, entropy is computed by tracking the left-right motion of the center fluid jet at the separatrix between the two convection rolls. We thus combine here dynamical systems methods with a concrete application of mixing in a nanoscale freestanding thin film.

Wave energy harvesting could be a substantial renewable energy source without impact on the global climate and ecology, yet practical attempts have struggled with the problems of wear and catastrophic failure. An innovative technology for ocean wave energy harvesting was recently proposed, based on the use of soft capacitors. This study presents a realistic theoretical and numerical model for the quantitative characterization of this harvesting method. Parameter regions with optimal behavior are found, and novel material descriptors are determined, which dramatically simplify analysis. The characteristics of currently available materials are evaluated, and found to merit a very conservative estimate of 10 years for raw material cost recovery.

Characterization and calibration of piezoelectric polymers in situ measurements of body vibrations
(2011)

Piezoelectric polymers are known for their flexibility in applications, mainly due to their bending ability, robustness, and variable sensor geometry. It is an optimal material for minimal-invasive investigations in vibrational systems, e.g., for wood, where acoustical impedance matches particularly well. Many applications may be imagined, e. g., monitoring of buildings, vehicles, machinery, alarm systems, such that our investigations may have a large impact on technology. Longitudinal piezoelectricity converts mechanical vibrations normal to the polymer-film plane into an electrical signal, and the respective piezoelectric coefficient needs to be carefully determined in dependence on the relevant material parameters. In order to evaluate efficiency and durability for piezopolymers, we use polyvinylidene fluoride and measure the piezoelectric coefficient with respect to static pressure, amplitude of the dynamically applied force, and long-term stability. A known problem is the slow relaxation of the material towards equilibrium, if the external pressure changes; here, we demonstrate how to counter this problem with careful calibration. Since our focus is on acoustical measurements, we determine accurately the frequency response curve - for acoustics probably the most important characteristic. Eventually, we show that our piezopolymer transducers can be used as a calibrated acoustical sensors for body vibration measurements on a wooden musical instrument, where it is important to perform minimal-invasive measurements. A comparison with the simultaneously recorded airborne sound yields important insight of the mechanism of sound radiation in comparison with the sound propagating in the material. This is especially important for transient signals, where not only the long-living eigenmodes contribute to the sound radiation. Our analyses support that piezopolymer sensors can be employed as a general tool for the determination of the internal dynamics of vibrating systems.

A key technology for large eddy simulation (LES) of complex flows is an appropriate wall modeling strategy. In this paper we apply for the first time a fully nonparametric procedure for the estimation of generalized additive models (GAM) by conditional statistics. As a database, we use DNS and wall-resolved LES data of plane channel flow for Reynolds numbers, Re = 2800, 4000 (DNS) and 10,935, 22,776 (LES). The statistical method applied is a quantitative tool for the identification of important model terms, allowing for an identification of some of the near-wall physics. The results are given as nonparametric functions which cannot be attained by other methods. We investigated a generalized model which includes Schumann's and Piomelli et al.'s model. A strong influence of the pressure gradient in the viscous sublayer is found; for larger wall distances the spanwise pressure gradient even dominates the tau(w,zy). component. The first a posteriori LES results are given.

We investigate localized periodic solutions (breathers) in a lattice of parametrically driven, nonlinear dissipative oscillators. These breathers are demonstrated to be exponentially localized, with two characteristic localization lengths. The crossover between the two lengths is shown to be related to the transition in the phase of the lattice oscillations.