Koopman theory is widely used for data-driven modeling of nonlinear dynamical systems. One of the well-known algorithms that stem from this approach is the Extended Dynamic Mode Decomposition (EDMD), a data-driven algorithm for uncontrolled systems. In this thesis, we will start by discussing the EDMD algorithm. We will discuss how this algorithm encompasses Dynamic Mode Decomposition (DMD), a widely used data-driven algorithm. Then we will extend our discussion to input-output systems and identify ways to extend the Koopman Operator Theory to input-output systems. We will also discuss how various algorithms can be identified as instances of this framework. Special care is given to Wavelet-based Dynamic Mode Decomposition (WDMD). WDMD is a variant of DMD that uses only the input and output data. WDMD does that by generating auxiliary states acquired from the Wavelet transform. We will show how the action of the Koopman operator can be simplified by using the Wavelet transform and how the WDMD algorithm can be motivated by this representation. We will also introduce a slight modification to WDMD that makes it more robust to noise. / Master of Science / To analyze a real-world phenomenon we first build a mathematical model to capture its behavior. Traditionally, to build a mathematical model, we isolate its principles and encode it into a function. However, when the phenomenon is not well-known, isolating these principles is not possible. Hence, rather than understanding its principles, we sample data from that phenomenon and build our mathematical model directly from this data by using approximation techniques. In this thesis, we will start by focusing on cases where we can fully observe the phenomena, when no external stimuli are present. We will discuss how some algorithms originating from these approximation techniques can be identified as instances of the Extended Dynamic Mode Decomposition (EDMD) algorithm. For that, we will review an alternative approach to mathematical modeling, called the Koopman approach, and explain how the Extended DMD algorithm stems from this approach. Then we will focus on the case where there is external stimuli and we can only partially observe the phenomena. We will discuss generalizations of the Koopman approach for this case, and how various algorithms that model such systems can be identified as instances of the EDMD algorithm adapted for this case. Special attention is given to the Wavelet-based Dynamic Mode Decomposition (WDMD) algorithm. WDMD builds a mathematical model from the data by borrowing ideas from Wavelet theory, which is used in signal processing. In this way, WDMD does not require the sampling of the fully observed system. This gives WDMD the flexibility to be used for cases where we can only partially observe the phenomena. While showing that WDMD is an instance of EDMD, we will also show how Wavelet theory can simplify the Koopman approach and thus how it can pave the way for an easier analysis.

Detonation is a supersonic mode of combustion that is modeled by a system of conservation laws of compressible fluid mechanics coupled with the equations describing thermodynamic and chemical properties of the fluid. Mathematically, these governing equations admit steady-state travelling-wave solutions consisting of a leading shock wave followed by a reaction zone. However, such solutions are often unstable to perturbations and rarely observed in laboratory experiments. The goal of this work is to study the stability of travelling-wave solutions of detonation models by the following novel approach. We linearize the governing equations about a base travelling-wave solution and solve the resultant linearized problem using high-order numerical methods. The results of these computations are postprocessed using dynamic mode decomposition to extract growth rates and frequencies of the perturbations and predict stability of travelling-wave solutions to infinitesimal perturbations. We apply this approach to two models based on the reactive Euler equations for perfect gases. For the first model with a one-step reaction mechanism, we find agreement of our results with the results of normal-mode analysis. For the second model with a two-step mechanism, we find that both types of admissible travelling-wave solutions exhibit the same stability spectra. Then we investigate the Fickett’s detonation analogue coupled with a particular reaction-rate expression. In addition to the linear stability analysis of this model, we demonstrate that it exhibits rich nonlinear dynamics with multiple bifurcations and chaotic behavior.

A research-scale solid-fuel ramjet test article has been developed to study the secondary combustion zone of solid fuel ramjets. Tests were performed at a constant core air mass flowrate of 0.77 kg/s with 0%, 15%, and 30% bypass ratios. The propulsive performance analysis results indicate that the 0% bypass case had the highest regression rate and fuel mass flowrate. The regression rate and fuel mass flowrate of fuel without carbon black was the lowest. The specific impulse with air mass flowrate included was highest for the 0% bypass case reaching 130 s and lowest for the 30% bypass case reaching 110 s. For specific impulse with air mass flowrate excluded, the 30% bypass case achieved 2,800 s while the 0% bypass case achieved 1,800 s. The characteristic velocity was greatest for 0% bypass reaching 1,025 m/s and lowest for 30% bypass reaching 900 m/s. The combustion efficiency was highest for the 15% bypass case with carbon black addition approaching 0.82. 50 kHz and 75 kHz CH* chemiluminescence imaging was performed. Analyzing thin slivers of the images over 40,001 frames with frequency-domain techniques showed that most of the high amplitude content occurred below 1-5kHz with small peaks near 20 kHz and 30 kHz. Dynamic mode decomposition (DMD) was performed on sets of 10,001 spatially-calibrated images and their corresponding uncalibrated, uncropped images. Most of the tests exhibited low-frequency axial pumping, transverse modes, and other mode shapes indicative of the secondary injection. The prominence of transverse and other jet-related modes over axial modes appeared to be related to increasing bypass ratio. High-frequency axial modes also appeared in a case thought to have high core-flow momentum that did not appear at these high frequencies for other cases. The DMD modes for 0% bypass were indiscernible due to high soot content. Most of the modes corresponding to the calibrated images also appeared in the uncalibrated images, however, with different mode amplitude rankings. PIV was performed at 5 kHz for one test at 15% bypass. The instantaneous vector fields for these tests displayed local velocities up to 600 m/s. The mean images showed velocities up to 250 m/s. The two-dimensional turbulent kinetic energies reached 200 m2/s2 in several regions throughout the flowfield. The turbulence intensity exceeded 0.20 near the bottom of the flowfield.

We examine the related nature of two identification algorithms, subspace identification (SID) and Dynamic Mode Decomposition (DMD), and their correctness properties over a broad range of problems. This investigation begins by noting the strong relationship between the two algorithms, both drawing significantly on the pseudoinverse calculation using singular value decomposition, and ultimately revealing that DMD can be viewed as a substep of SID. We then perform extensive computational studies, characterizing the performance of SID on problems of various model orders and noise levels. Specifically, we generate 10,000 random systems for each model order and noise level, calculating the average identification error for each case, and then repeat the entire experiment to ensure the results are, in fact, consistent. The results both quantify the intrinsic algorithmic error at zero-noise, monotonically increasing with model complexity, as well as demonstrate an asymptotically linear degradation to noise intensity, at least for the range under study. Finally, we close by demonstrating DMD's ability to recover system matrices, because its access to full state measurements makes them identifiable. SID, on the other hand, can't possibly hope to recover the original system matrices, due to their fundamental unidentifiability from input-output data. This is true even when SID delivers excellent performance identifying a correct set of equivalent system matrices.

No description available.

Data-driven intelligent transportation systems (ITS) are increasingly playing a critical role in improving the efficiency of the existing transportation network and addressing traffic challenges in large cities, such as safety and road congestion. This dissertation employs data dimensionality reduction, reinforcement learning, and discrete outcome models to improve traffic flow and transportation safety. First, we propose a novel data-driven technique based on Koopman operator theory and dynamic mode decomposition (DMD) to address the complex nonlinear dynamics of signalized intersections. This approach not only provides a better understanding of intersection behavior but also offers faster computation times, making it a valuable tool for system identification and controller design. It represents a significant step towards more efficient and effective traffic management solutions. Second, we propose an innovative phase-switching approach for traffic light control using deep reinforcement learning, enhancing the efficiency of signalized intersections. The novel reward function, based on speed, waiting time, deceleration, and time to collision (TTC) for each vehicle, maximizes traffic flow and safety through real-time optimization. Finally, we introduce a mixed spline indicator pooled model, an approach for multivariate crash severity prediction, addressing the limitations of previous models by capturing temporal instability. It carefully incorporates additional independent variables to measure parameter slope changes over time, enhancing data fit and predictive accuracy. The developed models are estimated and validated using data from the Central Florida region.

Unsteady flowfields are integral to high-speed applications, demanding precise modelling to characterize their unsteady features accurately. The simulation of unsteady supersonic and hypersonic flows is inherently computationally expensive, requiring a highly refined mesh to capture these unsteady effects. While anisotropic metric-based adaptive mesh refinement has proven effective in achieving accuracy with much less complexity, current algorithms are primarily tailored for steady flow fields. This thesis presents a novel approach to address the challenges of anisotropic grid adaptation of unsteady flows by leveraging a data-driven technique called Dynamic Mode Decomposition (DMD). DMD has proven to be a powerful tool to model complex nonlinear flows, given its links to the Koopman operator, and also its easy mathematical implementation. This research proposes the integration of DMD into the process of anisotropic grid adaptation to dynamically adjust the mesh in response to evolving flow features. The effectiveness of the proposed approach is demonstrated through numerical experiments on representative unsteady flow configurations, such as a cylinder in a subsonic flow and a cylinder in a supersonic channel flow. Results indicate that the incorporation of DMD enables an accurate representation of unsteady flow dynamics. Overall, this thesis contributes to making advances in the adaptation of unsteady flows. The novel framework proposed makes it computationally tractable to track the evolution of the main coherent features of the flowfield without losing out on accuracy by using a data-driven method. / Master of Science / Simulating unsteady, high-speed fluid flows around objects like aircraft and rockets poses a significant computational challenge. These flows exhibit rapidly evolving, intricate pattern structures that demand highly refined computational meshes to capture accurately. However, using a statically refined mesh for the entire simulation is computationally prohibitive. This research proposes a novel data-driven approach to enable efficient anisotropic mesh adaptation for such unsteady flow simulations. It leverages a technique called Dynamic Mode Decomposition (DMD) to model the dominant coherent structures and their evolution from snapshot flow field data. DMD has shown powerful capabilities in identifying the most energetic flow features and their time dynamics from numerical or experimental data. By integrating DMD into the anisotropic mesh adaptation process, the computational mesh can be dynamically refined anisotropically just in regions containing critical time-varying flow structures. The efficacy of this DMD-driven anisotropic adaptation framework is demonstrated in representative test cases - an unsteady subsonic flow over a circular cylinder and a supersonic channel flow over a cylinder. Results indicate that it enables accurate tracking and resolution of the key unsteady flow phenomena like vortex shedding using far fewer computational cells compared to static mesh simulations. In summary, this work makes anisotropic mesh adaptation computationally tractable for unsteady flow simulations by leveraging data-driven DMD modelling of the evolving coherent structures. The developed techniques pave the way for more accurate yet efficient unsteady CFD simulations.

Cavitation can be defined as the breakdown of a liquid (either static or in motion) medium under very low pressure. The hydrodynamic happened in high-speed flow, where local pressure in liquid falls under the saturating pressure thus the liquid vaporizes to form the cavity. During the evolution and collapsing of cavitation bubbles, extreme physical conditions like high-temperature, high-pressure, shock-wave, and high-speed micro-jets can be generated. Such a phenomenon shall be prevented in hydraulic or astronautical machinery due to the induced erosion and noise, while it can be utilized to intensify some treatment processes of chemical, food, and pharmaceutical industries, to shorten sterilization times and lower energy consumption. Advances in the understanding of the physical processes of cavitating flows are challenging, mainly due to the lack of quantitative experimental data on the two-phase structures and dynamics inside the opaque cavitation areas. This dissertation is aimed at finding out the physical mechanisms governing the cavitation instabilities and making contributions in controlling hydraulic cavitation for engineering applications. In this thesis, cavitation developed in various convergent-divergent (Venturi) channels was studied experimentally using the ultra-fast synchrotron X-ray imaging, LIF Particle Image Velocimetry, and high-speed photography techniques, to (1) investigate the internal structures and evolution of bubble dynamics in cavitating flows, with velocity information obtained for two phases; (2) measure the slip velocity between the liquid and the vapor to provide the validation data for the numerical cavitation models; (3) consider the thermodynamic effects of cavitation to establish the relation between the cavitation extent and the fluid temperature, then and optimize the cavitation working condition in water; (4) seek the coherent structures of the complicated high-turbulent cavitating flow to reduce its randomness using data-driven methods. / Doctor of Philosophy / When the pressure of a liquid is below its saturation pressure, the liquid will be vaporized into vapor bubbles which can be called cavitation. In many hydraulic machines like pumps, propulsion systems, internal combustion engines, and rocket engines, this phenomenon is quite common and could induce damages to the mechanical systems. To understand the mechanisms and further control cavitation, investigation of the bubble inception, deformation, collapse, and flow regime change is mandatory. Here, we performed the fluid mechanics experiment to study the unsteady cavitating flow underlying physics as it occurs past the throat of a Venturi nozzle. Due to the opaqueness of this two-phase flow, an X-ray imaging technique is applied to visualize the internal flow structures in micrometer scales with minor beam scattering. Finally, we provided the latest physical model to explain the different regimes that appear in cavitation. The relationship between the cavitation length and its shedding regimes, and the dominant mechanism governing the transition of regimes are described. A combined suppression parameter is developed and can be used to enhance or suppress the cavitation intensity considering the influence of temperature.

No description available.

In axial compressors with a relatively large blade tip clearance, an unsteady phenomenon denoted as rotating instability (RI) can be detected when the compressor is throttled to the operating points near the stability limit. In the frequency domain, RIs are shown as a hump lower than the blade passing frequency. This indicates an increase in noise level and might cause blade vibration and other undesirable structural issues. In this thesis, a comprehensive study on RIs is performed based on an axial compressor rotor row of the Low Speed Research Compressor at Technische Universität Dresden. Three blade tip clearances are investigated, and a groove casing treatment is mounted over the shroud for flow control. Methods of numerical modeling are evaluated, and zonal large eddy simulation is selected as the numerical model. By analyzing the flow properties and applying the dynamic mode decomposition, the coherent flow structure corresponding to the dominant frequency of RIs is extracted and visualized as the waves located in the blade tip region. The criteria for the appearance of RIs in the investigated research object are concluded.