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.

The electric power system is designed to economically and reliably transmit electricity to homes, industry, and businesses. The economic impact of the electric grid was demonstrated by the 2003 blackout's visible impact in the graph of the yearly gross domestic product of the Unites States. However, because the number of customers is so large and economies of scale are leveraged to keep electricity prices low, utilities are strongly interconnected. Performing comprehensive engineering analyses to ensure reliable operation is still impossible. Instead, heuristics and safety factors are incorporated into planning processes to continually meet demand in a way that complies with Federal regulations. As evidenced by the infrequency of blackouts in the United States, the sophisticated planning processes have up to date been relatively successful. However, the power system is constantly changing. Electrical generators based on renewable energies are a beneficial addition to the grid, but these and other technological changes like high-voltage power electronic converters also come with their own challenges. These systems as currently employed tend to have a different impact on the reliability of operation than traditional fossil fuel based generators. As the system changes, so do the engineering analyses required to ensure reliable operation. In particular, the wind energy conversion systems (WECS) negatively impact the response of the grid to disturbances in certain ways due to inherent challenges harnessing the wind as an energy sources. These negative impacts (and the advent of powerful personal computing) require an increase in the sophistication of power system models. Thus, there are competing challenges: the scale of the power system necessitates computationally efficient modeling, but the complexity of analysis required to maintain reliable operation is also increasing. The primary aim of this study is to develop models and methods for more detailed yet computationally manageable simulation. To this aim, higher order linearizations and the properties of linear systems (graph theory and linear algebra) are exploited. More specifically, this document contains three studies. In the short term planning and situational awareness context, a method is proposed to quickly check credible outages of important grid equipment. This methodology enables the inspection of a wider breadth of system conditions to ameliorate the negative impacts of the unpredictability of the wind. A linear model in the traditional sense is also developed to model any arbitrary number of wind turbines in a wind farm. This enables industry players to study the impacts wind turbine interaction on the dynamic stability of the grid in response to small disturbances. Finally, a wind farm is modeled as a large matrix to model even nonlinear behavior of wind farms. This helps industry players analyze the impact of large disturbances on the grid. / Doctor of Philosophy / The electric power system is designed to economically and reliably transmit electricity to homes, industry, and businesses. The economic impact of the electric grid was demonstrated by the 2003 blackout’s visible impact in the graph of the yearly gross domestic product of the United States. However, because the number of customers is so large and economies of scale are leveraged to keep electricity prices low, utilities are strongly interconnected. Performing comprehensive engineering analyses to ensure reliable operation is still impossible. Instead, heuristics and safety factors are incorporated into planning processes to continually meet demand in a way that complies with Federal regulations. As evidenced by the infrequency of blackouts in the United States, the sophisticated planning processes have up to date been relatively successful. However, the power system is constantly changing. Electrical generators based on renewable energies are a beneficial addition to the grid, but these and other technological changes like high-voltage power electronic converters also come with their own challenges. These systems as currently employed tend to have a different impact on the reliability of operation than traditional fossil fuel based generators. As the system changes, so do the engineering analyses required to ensure reliable operation. In particular, the wind energy conversion systems (WECS) negatively impact the response of the grid to disturbances in certain ways due to inherent challenges harnessing the wind as an energy sources. These negative impacts (and the advent of powerful personal computing) require an increase in the sophistication of power system models. Thus, there are competing challenges: the scale of the power system necessitates computationally efficient modeling, but the complexity of analysis required to maintain reliable operation is also increasing. The primary aim of this study is to develop models and methods for more detailed yet computationally manageable simulation. To this aim, higher order linearizations and the properties of linear systems (graph theory and linear algebra) are exploited. More specifically, this document contains three studies. In the short term planning and situational awareness context, a method is proposed to quickly check credible outages of important grid equipment. This methodology enables the inspection of a wider breadth of system conditions to ameliorate the negative impacts of the unpredictability of the wind. A linear model in the traditional sense is also developed to model any arbitrary number of wind turbines in a wind farm. This enables industry players to study the impacts wind turbine interaction on the dynamic stability of the grid in response to small disturbances. Finally, a wind farm is modeled as a large matrix to model even nonlinear behavior of wind farms. This helps industry players analyze the impact of large disturbances on the grid.

<p>Every engineered system behaves slightly differently because of manufacturing and operational uncertainties. The ability to build system-specific predictive models that adapt to manufactured systems, also known as digital twins, opens up many possibilities for reducing operating and maintenance costs. Nonlinear dynamical systems with unknown governing equations and states characterize many engineered systems. As a result, learning their dynamics from data has become both the current research area and one of the biggest challenges. In this thesis, we do an investigation of different data driven approaches for modeling various engineered systems. Firstly, we develop a model to predict the transient and steady-state behavior of a turbocharger turbine using the Koopman operator which can be helpful for modelling, analysis and control design. Our approach is as follows. We use experimental data from a Cummins heavy-duty diesel engine to develop a turbine model using Extended Dynamic Mode Decomposition (EDMD), which approximates the action of the Koopman operator on a finite-dimensional subspace of the space of observables. The results demonstrate comparable performance with a tuned nonlinear autoregressive network with an exogenous input (NARX) model widely used in the literature. The performance of these two models is analyzed based on their ability to predict turbine transient and steady-state behavior. Furthermore, we assess the ability of liquid time-constant (LTC) networks to learn the dynamics of various oscillatory systems using noisy data. In this study, we analyze and compare the performance of the LTC network with various commonly used recurrent neural network (RNN) architectures like long short-term memory (LSTM) network, and gated recurrent units (GRU). Our approach is as follows. We first systematically generate synthetic data by exciting the system of interest with a band-limited white noise and simulating it using a forward Euler discretization scheme. After the output has been simulated, we then corrupt it with different levels of noise to replicate a practically measured signal and train the RNN architectures with that corrupted output. The model is then tested on various types of forcing excitations to analyze the robustness of these networks in capturing different behaviors exhibited by the system. We also analyze the ability of these networks to capture the resonance effect for various parameter settings. Case studies discussing standard benchmark oscillatory systems (i.e., spring-mass-damper (S-M-D) system, single degree of freedom (DOF) Bouc-Wen oscillator, and forced Van der pol oscillator) are used to test the performance of these methodologies. The results reveal that the LTC network showed better performance in modeling the S-M-D system and 1-DOF Bouc-Wen oscillator as compared to an LSTM network but was outperformed by the GRU network. None of the networks were able to model the forced Van der pol oscillator with a reasonable accuracy. Since the GRU network outperformed other networks in terms of the computational time and the model accuracy for most of the scenarios, we applied it to a real world experimental dataset (i.e. turbocharger turbine dynamics) to compare it against the EDMD and NARX model. The results showed better performance of the GRU network in modeling the transient behaviours of the turbine. However, it failed to predict the turbine outlet temperature with a reasonable accuracy in most of the regions for the steady state dataset. As future work, we plan to consider training the GRU network with a data sampling frequency of 100 Hz for a fair comparison with the NARX and the Koopman approach.</p>

<p>Train transportation is a vital part of the transportation system of today anddue to its safe and environmental friendly concept it will be even more impor-tant in the future. The speeds of trains have increased continuously and withhigher speeds the aerodynamic effects become even more important. One aero-dynamic effect that is of vital importance for passengers’ and track workers’safety is slipstream, i.e. the flow that is dragged by the train. Earlier ex-perimental studies have found that for high-speed passenger trains the largestslipstream velocities occur in the wake. Therefore the work in this thesis isdevoted to wake flows. First a test case, a surface-mounted cube, is simulatedto test the analysis methodology that is later applied to a train geometry, theAerodynamic Train Model (ATM). Results on both geometries are comparedwith other studies, which are either numerical or experimental. The comparisonfor the cube between simulated results and other studies is satisfactory, whiledue to a trip wire in the experiment the results for the ATM do not match.The computed flow fields are used to compute the POD and Koopman modes.For the cube this is done in two regions of the flow, one to compare with a priorpublished study Manhart & Wengle (1993) and another covering more of theflow and especially the wake of the cube. For the ATM, a region containing theimportant flow structures is identified in the wake, by looking at instantaneousand fluctuating velocities. To ensure converged POD modes two methods toinvestigate the convergence are proposed, tested and applied. Analysis of themodes enables the identification of the important flow structures. The flowtopologies of the two geometries are very different and the flow structures arealso different, but the same methodology can be applied in both cases. For thesurface-mounted cube, three groups of flow structures are found. First groupis the mean flow and then two kinds of perturbations around the mean flow.The first perturbation is at the edge of the wake, relating to the shear layerbetween the free stream and the disturbed flow. The second perturbation isinside the wake and is the convection of vortices. These groups would then betypical of the separation bubble that exists in the wake of the cube. For theATM the main flow topology consists of two counter rotating vortices. Thiscan be seen in the decomposed modes, which, except for the mean flow, almostonly contain flow structures relating to these vortices.</p> / QC 20100518 / Gröna Tåget

Le développement de l'accès à l'espace s'inscrit dans le contexte scientifique et économique actuel comme un enjeu majeur de l'industrie et de la recherche. Un des objectifs principaux est d'augmenter la capacité et le confort de la charge utile pour réduire les coûts de transport vers l'espace. L'exploitation des données en vol du lanceur Ariane 5 a mis en évidence la présence de fluctuations de pression pouvant induire des efforts instationnaires repris par les vérins du moteur Vulcain. Ces efforts s'exercent dans la zone décollée du culot d'un lanceur normalement à l'axe de la poussée, et sont qualifiés de charges latérales. Cette observation conduit à l'étude de deux aspects. Dans un premier temps, la dynamique des écoulements massivement décollés d'arrière-corps, à haut nombre de Reynolds et en régime compressible est considérée au moyen de simulations numériques ZDES. Une analyse approfondie des champs instantanés, moyens et fluctuants est réalisée au moyen de post-traitements avancés. Notamment, le champ tri-dimensionnel de pression fluctuante dans la région du culot fait l'objet d'analyses spectrales (Fourier) et modales (DMD) massives. Ensuite, le contrôle des phénomènes potentiellement nuisibles au confort de la charge utile à savoir le buffeting culot et l'ovalisation de tuyère est abordé. L'analyse physique de l'écoulement sur une géométrie tri-corps permet la conception d'un dispositif de contrôle adapté. Enfin, l'effet de deux dispositifs retenus (quatre jets équirépartis en azimut sur le corps central et augmentation de la section de passage entre les trois corps) est évalué au moyen des outils d'analyse de la dynamique de la configuration non contrôlée.

In this thesis I investigate a number of problems in the nonlinear stability of density stratified plane Couette flow. I begin by describing the history of transient growth phenomena, and in particular the recent application of adjoint based optimisation to find nonlinear optimal perturbations and associated minimal seeds for turbulence, the smallest amplitude perturbations that are able to trigger transition to turbulence. I extend the work of Rabin et al. (2012) in unstratified plane Couette flow to find minimal seeds in both vertically and horizontally sheared stratified plane Couette flow. I find that the coherent states visited by such minimal seed trajectories are significantly altered by the stratification, and so proceed to investigate these states both with generalised Koopman mode analysis and by stratifying the self-sustaining process described by Waleffe (1997). I conclude with an introductory problem I considered that investigates the linear Taylor instability of layered stratified plane Couette flow, and show that the nonlinear evolution of the primary Taylor instability is not coupled to the form of the linearly unstable mode, in contrast to the Kelvin-Helmholtz instability, for example. I also include an appendix in which I describe joint work conducted with Professor Neil Balmforth of UBC during the 2015 WHOI Geophysical Fluid Dynamics summer programme, investigating stochastic homoclinic bifurcations.

In the thesis we acquaint ourselves with the terms from ergodic theory and re- presentation theory of topological groups. We pay attention particularly to terms unitary representation, realizability by an action, dual group, unitary equivalence and Kazhdan's property (T). We achieve a result regarding unitary representati- ons realizable by an action on finite abelian groups according to article [5] and show that it is possible to generalize it to all finite groups at the end of the thesis according to article [6]. A large part of the text subsequently deals with proper- ties of unitary representations and their relations. We connect the terms compact topological group and Kazhdan's property (T).

To view the electricity supply in our society as just sockets mountedin our walls with a constant voltage output is far from the truth. Inreality, the power system supplying the electricity or the grid, is themost complex man-made dynamical system there is. It demands severecontrol and safety measures to ensure a reliable supply of electric power.Throughout the world, incidents of widespread power grid failures havebeen continuously reported. The state where electricity delivery to customersis terminated by a disturbance is called a blackout. From a stateof seemingly stable operating conditions, the grid can fast derail intoan uncontrollable state due to cascading failures. Transmission linesbecome automatically disconnected due to power flow redirections andparts of the grid become isolated and islands are formed. An islandedsub-grid incapable of maintaining safe operation conditions experiencesa blackout. A widespread blackout is a rare, but an extremely costlyand hazardous event for society.During recent years, many methods to prevent these kinds of eventshave been suggested. Controlled islanding has been a commonly suggestedstrategy to save the entire grid or parts of the grid from a blackout.Controlled islanding is a strategy of emergency control of a powergrid, in which the grid is intentionally split into a set of islanded subgridsfor avoiding an entire collapse. The key point in the strategy is todetermine appropriate separation boundaries, i.e. the set of transmissionlines separating the grid into two or more isolated parts.The power grid exhibits highly nonlinear response in the case oflarge failures. Therefore, this thesis proposes a new controlled islandingmethod for power grids based on the nonlinear Koopman Mode Analysis(KMA). The KMA is a new analyzing technique of nonlinear dynamicsbased on the so-called Koopman operator. Based on sampled data followinga disturbance, KMA is used to identify suitable partitions of thegrid.The KMA-based islanding method is numerically investigated withtwo well-known test systems proposed by the Institute of Electrical andElectronics Engineers (IEEE). By simulations of controlled islanding inthe test system, it is demonstrated that the grid’s response following afault can be improved with the proposed method.The proposed method is compared to a method of partitioning powergrids based on spectral graph theory which captures the structural propertiesof a network. It is shown that the intrinsic structural propertiesof a grid characterized by spectral graph theory are also captured by theKMA. This is shown both by numerical simulations and a theoreticalanalysis.

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Train transportation is a vital part of the transportation system of today anddue to its safe and environmental friendly concept it will be even more impor-tant in the future. The speeds of trains have increased continuously and withhigher speeds the aerodynamic effects become even more important. One aero-dynamic effect that is of vital importance for passengers’ and track workers’safety is slipstream, i.e. the flow that is dragged by the train. Earlier ex-perimental studies have found that for high-speed passenger trains the largestslipstream velocities occur in the wake. Therefore the work in this thesis isdevoted to wake flows. First a test case, a surface-mounted cube, is simulatedto test the analysis methodology that is later applied to a train geometry, theAerodynamic Train Model (ATM). Results on both geometries are comparedwith other studies, which are either numerical or experimental. The comparisonfor the cube between simulated results and other studies is satisfactory, whiledue to a trip wire in the experiment the results for the ATM do not match.The computed flow fields are used to compute the POD and Koopman modes.For the cube this is done in two regions of the flow, one to compare with a priorpublished study Manhart &amp; Wengle (1993) and another covering more of theflow and especially the wake of the cube. For the ATM, a region containing theimportant flow structures is identified in the wake, by looking at instantaneousand fluctuating velocities. To ensure converged POD modes two methods toinvestigate the convergence are proposed, tested and applied. Analysis of themodes enables the identification of the important flow structures. The flowtopologies of the two geometries are very different and the flow structures arealso different, but the same methodology can be applied in both cases. For thesurface-mounted cube, three groups of flow structures are found. First groupis the mean flow and then two kinds of perturbations around the mean flow.The first perturbation is at the edge of the wake, relating to the shear layerbetween the free stream and the disturbed flow. The second perturbation isinside the wake and is the convection of vortices. These groups would then betypical of the separation bubble that exists in the wake of the cube. For theATM the main flow topology consists of two counter rotating vortices. Thiscan be seen in the decomposed modes, which, except for the mean flow, almostonly contain flow structures relating to these vortices. / QC 20100518 / Gröna Tåget