1,486 research outputs found

    Linear Amplification in Nonequilibrium Turbulent Boundary Layers

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    Resolvent analysis is applied to nonequilibrium incompressible adverse pressure gradient (APG) turbulent boundary layers (TBL) and hypersonic boundary layers with high temperature real gas effects, including chemical nonequilibrium. Resolvent analysis is an equation-based, scale-dependent decomposition of the Navier Stokes equations, linearized about a known mean flow field. The decomposition identifies the optimal response and forcing modes, ranked by their linear amplification. To treat the nonequilibrium APG TBL, a biglobal resolvent analysis approach is used to account for the streamwise and wall-normal inhomogeneities in the streamwise developing flow. For the hypersonic boundary layer in chemical nonequilibrium, the resolvent analysis is constructed using a parallel flow assumption, incorporating N₂, O₂, NO, N, and O as a mixture of chemically reacting gases. Biglobal resolvent analysis is first applied to the zero pressure gradient (ZPG) TBL. Scaling relationships are determined for the spanwise wavenumber and temporal frequency that admit self-similar resolvent modes in the inner layer, mesolayer, and outer layer regions of the ZPG TBL. The APG effects on the inner scaling of the biglobal modes are shown to diminish as their self-similarity improves with increased Reynolds number. An increase in APG strength is shown to increase the linear amplification of the large-scale biglobal modes in the outer region, similar to the energization of large scale modes observed in simulation. The linear amplification of these modes grows linearly with the APG history, measured as the streamwise averaged APG strength, and relates to a novel pressure-based velocity scale. Resolvent analysis is then used to identify the length scales most affected by the high-temperature gas effects in hypersonic TBLs. It is shown that the high-temperature gas effects primarily affect modes localized near the peak mean temperature. Due to the chemical nonequilibrium effects, the modes can be linearly amplified through changes in chemical concentration, which have non-negligible effects on the higher order modes. Correlations in the components of the small-scale resolvent modes agree qualitatively with similar correlations in simulation data. Finally, efficient strategies for resolvent analysis are presented. These include an algorithm to autonomously sample the large amplification regions using a Bayesian Optimization-like approach and a projection-based method to approximate resolvent analysis through a reduced eigenvalue problem, derived from calculus of variations.</p

    Advancements in Fluid Simulation Through Enhanced Conservation Schemes

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    To better understand and solve problems involving the natural phenomenon of fluid and air flows, one must understand the Navier-Stokes equations. Branching several different fields including engineering, chemistry, physics, etc., these are among the most important equations in mathematics. However, these equations do not have analytic solutions save for trivial solutions. Hence researchers have striven to make advancements in varieties of numerical models and simulations. With many variations of numerical models of the Navier-Stokes equations, many lose important physical meaningfulness. In particular, many finite element schemes do not conserve energy, momentum, or angular momentum. In this thesis, we will study new methods in solving the Navier-Stokes equations using models which have enhanced conservation qualities, in particular, the energy, momentum, and angular momentum conserving (EMAC) scheme. The EMAC scheme has gained popularity in the mathematics community over the past few years as a desirable method to model fluid flow. It has been proven to conserve energy, momentum, angular momentum, helicity, and others. EMAC has also been shown to perform better and maintain accuracy over long periods of time compared to other schemes. We investigate a fully discrete error analysis of EMAC and SKEW. We show that a problematic dependency on the Reynolds number is present in the analysis for SKEW, but not in EMAC under certain conditions. To further explore this concept, we include some numerical experiments designed to highlight these differences in the error analysis. Additionally, we include other projection methods to measure performance. Following this, we introduce a new EMAC variant which applies a differential spatial filter to the EMAC scheme, named EMAC-Reg. Standard models, including EMAC, require especially fine meshes with high Reynold\u27s numbers. This is problematic because the linear systems for 3D flows will be far too large and take an extraordinary amount of time to compute. EMAC-Reg not only performs better on a coarser mesh, but maintains conservation properties as well. Another topic in fluid flow computing that has been gaining recognition is reduced order models. This method uses experimental data to create new models of reduced computational complexity. We introduce the concept of consistency between a full order and a reduced order model, i.e., using the same numerical scheme for the full order and reduced order model. For inconsistency, one could use SKEW in the full order model and then EMAC for the reduced order model. We explore the repercussions of having inconsistency between these two models analytically and experimentally. To obtain a proper linear system from the Navier-Stokes equations, we must solve the nonlinear problem first. We will explore a method used to reduce iteration counts of nonlinear problems, known as Anderson acceleration. We will discuss how we implemented this using the finite element library deal.II \cite{dealII94}, measure the iteration counts and time, and compare against Newton and Picard iterations

    Boussinesq modeling of the influence of wave energy converters on nearshore circulation

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    In Chapter 4 it was found that particle transport is much more sensitive to waves originating in the infra-gravity (IG) band. These are waves which typically have much smaller amplitudes than the typical waves seen on the ocean, and their long period and wavelength makes them hard to spot while looking at out at sea. Nevertheless it was found that the movement of particles have a significant component in IG frequency offshore. As the waves break in the surf zone the IG band grows in importance before the movement associated with them are dominant in the surf zone. This explains multiple effects that has been found to be dominated by IG frequencies such as the location of the river plume, net movement of inertial particles, and sand flux. As it was found that the phase-resolving nearshore wave model Boussinesq Ocean and surf zone model (BOSZ) can generate these kind of waves, it also makes sense that the model can be used to somewhat accurately predict the movement of actual particle paths. In Chapter 5 this was put to the test by comparing the simulated paths in BOSZ with actual orange paths outside Sylt. Oranges was tracked through pictures taken with a stereo camera and combined with GPS data to recreate their position in a coordinate system. Combining this data about their position with data regarding the bathymetry, tide levels, and wave compositions it was possible to numerically recreate the wave conditions of the day. In the resulting simulated movements we find that the numerical paths matches the actual paths to a satisfactory degree. Further it was found that the drifters moved slowly in one direction while having much faster oscillating movements while travelling there. This of course follows from the discussion that the slowly varying IG waves dominate the movements. It was also found that these IG waves is critical to accurately simulate particles in the surf zone, as the phases of the waves determined the direction of movement of the particles. Only by averaging paths by different phases could we get the right paths, a results which is necessary to include in all further work in tracking near shore particles. Along the coast multiple large scale fluid patterns can be found. There exists regions of flow which primarly move towards the beach or out of the beach, the latter of which is known as rip currents, as well as alongshore flow and lastly large scale circular patterns called vortexes. In Chapter 6 and 7 it was found that vortexes in the surf zone are dependent upon the tide level, the mean direction the waves were coming from, and if they all came in at the same angle or not. The radii, strength and the number of these circular movements for different values of these parameters were compared, and it was found through statistical testing that some of the parameters only influenced the radii of the vortexes, while others influenced the strength or number of vortexes. Oscillating wave surge converter (OWSC) devices are a relatively new type of wave energy converter of recent interest as these can capture a large percentage of the energy of the waves (capture factor). In Chapter 7 a method of analytically calculating the capture factor for a wave was retold. A way of numerically calculate this factor was developed, and how to include this in the BOSZ model was explained. The inclusion of a farm of OWSCs in the bathymetry had the effect of reducing all the radii and circulatory strength in the surf zone, while keeping most of the relationships from the case with an open beach. A better understanding of the vortex structures can shed light into the intensity of mixing. This goes beyond this project but we might be able to recalculate the vertical exchange of water by knowing the size and intensity of eddies. The fact that we found in Chapter 6 that vortexes at the beach match the fundamental Rankine vortex yields a better feeling of what to expect from computations. It might be possible to use the computed results in combination with analytical solutions to create better estimates for mixing ratios in the future. This can among other help to understand the influence OWSC farms have on the local ecology and morphology.Masteroppgave i anvendt og beregningsorientert matematikkMAB399MAMN-MA

    On the Numerical Integration of Singular Initial and Boundary Value Problems for Generalised Lane-Emden and Thomas-Fermi Equations

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    We propose a geometric approach for the numerical integration of singular initial value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane-Emden equations and the Thomas-Fermi equation.Comment: 29 pages, 9 figure

    Drift-diffusion models for innovative semiconductor devices and their numerical solution

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    We present charge transport models for novel semiconductor devices which may include ionic species as well as their thermodynamically consistent finite volume discretization

    Data-Driven Exploration of Coarse-Grained Equations: Harnessing Machine Learning

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    In scientific research, understanding and modeling physical systems often involves working with complex equations called Partial Differential Equations (PDEs). These equations are essential for describing the relationships between variables and their derivatives, allowing us to analyze a wide range of phenomena, from fluid dynamics to quantum mechanics. Traditionally, the discovery of PDEs relied on mathematical derivations and expert knowledge. However, the advent of data-driven approaches and machine learning (ML) techniques has transformed this process. By harnessing ML techniques and data analysis methods, data-driven approaches have revolutionized the task of uncovering complex equations that describe physical systems. The primary goal in this thesis is to develop methodologies that can automatically extract simplified equations by training models using available data. ML algorithms have the ability to learn underlying patterns and relationships within the data, making it possible to extract simplified equations that capture the essential behavior of the system. This study considers three distinct learning categories: black-box, gray-box, and white-box learning. The initial phase of the research focuses on black-box learning, where no prior information about the equations is available. Three different neural network architectures are explored: multi-layer perceptron (MLP), convolutional neural network (CNN), and a hybrid architecture combining CNN and long short-term memory (CNN-LSTM). These neural networks are applied to uncover the non-linear equations of motion associated with phase-field models, which include both non-conserved and conserved order parameters. The second architecture explored in this study addresses explicit equation discovery in gray-box learning scenarios, where a portion of the equation is unknown. The framework employs eXtended Physics-Informed Neural Networks (X-PINNs) and incorporates domain decomposition in space to uncover a segment of the widely-known Allen-Cahn equation. Specifically, the Laplacian part of the equation is assumed to be known, while the objective is to discover the non-linear component of the equation. Moreover, symbolic regression techniques are applied to deduce the precise mathematical expression for the unknown segment of the equation. Furthermore, the final part of the thesis focuses on white-box learning, aiming to uncover equations that offer a detailed understanding of the studied system. Specifically, a coarse parametric ordinary differential equation (ODE) is introduced to accurately capture the spreading radius behavior of Calcium-magnesium-aluminosilicate (CMAS) droplets. Through the utilization of the Physics-Informed Neural Network (PINN) framework, the parameters of this ODE are determined, facilitating precise estimation. The architecture is employed to discover the unknown parameters of the equation, assuming that all terms of the ODE are known. This approach significantly improves our comprehension of the spreading dynamics associated with CMAS droplets

    Exponential integrators: tensor structured problems and applications

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    The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

    Near-continuous time Reinforcement Learning for continuous state-action spaces

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    We consider the Reinforcement Learning problem of controlling an unknown dynamical system to maximise the long-term average reward along a single trajectory. Most of the literature considers system interactions that occur in discrete time and discrete state-action spaces. Although this standpoint is suitable for games, it is often inadequate for mechanical or digital systems in which interactions occur at a high frequency, if not in continuous time, and whose state spaces are large if not inherently continuous. Perhaps the only exception is the Linear Quadratic framework for which results exist both in discrete and continuous time. However, its ability to handle continuous states comes with the drawback of a rigid dynamic and reward structure. This work aims to overcome these shortcomings by modelling interaction times with a Poisson clock of frequency ε1\varepsilon^{-1}, which captures arbitrary time scales: from discrete (ε=1\varepsilon=1) to continuous time (ε0\varepsilon\downarrow0). In addition, we consider a generic reward function and model the state dynamics according to a jump process with an arbitrary transition kernel on Rd\mathbb{R}^d. We show that the celebrated optimism protocol applies when the sub-tasks (learning and planning) can be performed effectively. We tackle learning within the eluder dimension framework and propose an approximate planning method based on a diffusive limit approximation of the jump process. Overall, our algorithm enjoys a regret of order O~(ε1/2T+T)\tilde{\mathcal{O}}(\varepsilon^{1/2} T+\sqrt{T}). As the frequency of interactions blows up, the approximation error ε1/2T\varepsilon^{1/2} T vanishes, showing that O~(T)\tilde{\mathcal{O}}(\sqrt{T}) is attainable in near-continuous time

    A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System

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    We introduce an efficient stochastic interacting particle-field (SIPF) algorithm with no history dependence for computing aggregation patterns and near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis system in three space dimensions (3D). The KS solutions are approximated as empirical measures of particles coupled with a smoother field (concentration of chemo-attractant) variable computed by the spectral method. Instead of using heat kernels causing history dependence and high memory cost, we leverage the implicit Euler discretization to derive a one-step recursion in time for stochastic particle positions and the field variable based on the explicit Green's function of an elliptic operator of the form Laplacian minus a positive constant. In numerical experiments, we observe that the resulting SIPF algorithm is convergent and self-adaptive to the high gradient part of solutions. Despite the lack of analytical knowledge (e.g. a self-similar ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study the emergence of finite time blowup in 3D by only dozens of Fourier modes and through varying the amount of initial mass and tracking the evolution of the field variable. Notably, the algorithm can handle at ease multi-modal initial data and the subsequent complex evolution involving the merging of particle clusters and formation of a finite time singularity

    Extension de la méthode des Différences Spectrales à la combustion

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    L'amélioration des outils d'ingénierie utilisés dans le design des dispositifs industriels de combustion est indispensable afin de respecter les demandes de plus en plus restrictives pour réduire les émissions de gaz à effets de serre. Parmi eux, la mécanique des fluides numériques (CFD) est devenue essentielle pour étudier et optimiser les chambres de combustion au cours des dernières décennies. Elle se complète parfaitement aux expériences réelles qui peuvent être très couteuses et avec lesquelles il est impossible d'obtenir des informations sur n'importe quelle quantité d'intérêt en tout point de la chambre de combustion. En utilisant les simulations aux grandes échelles (LES), la CFD décrit directement l'interaction entre les flammes et les structures turbulentes avec une faible modélisation. La qualité des résultats LES est ainsi très dépendante de la discrétisation utilisée incluant à la fois le maillage et également les propriétés de dissipation et de dispersion des méthodes numériques utilisées. Cependant, la plupart des codes LES employés de nos jours dans l'industrie utilisent des schémas de discrétisation spatiale de basordre (LO) à cause de leur faible coût de calcul et leur facilité d'implémentation sur des maillages complexes. Pourtant, les méthodes numériques d'ordres élevés (HO) pour la LES sont développées depuis deux décennies et ont été appliquées sur des écoulements non-réactifs amenant à des résultats plus précis que les méthodes LO avec un plus faible coût de calcul. Bien que les méthodes HO semblent très prometteuses en combustion, en particulier pour mieux décrire le front de flamme, leur utilisation pour des écoulements réactifs restent encore à être démontrée. Au cours de ces travaux, les avantages et les bénéfices des méthodes HO en combustion sont évalués en utilisant la méthode des Différences Spectrales (SD) avec du raffinement hphp. Premièrement, il est démontré que la formulation originelle des SD est instable pour des écoulements multi-espèces avec des propriétés thermodynamiques variant avec la température et la composition. Il a été constaté que calculer les variables primitives aux points solutions puis de les extrapoler aux points flux, au lieu de faire l'inverse en extrapolant d'abord les variables conservatives, rend stable la méthode SD dans ce cas-ci. De plus, une nouvelle méthodologie, également plus stable pour calculer les flux diffusifs aux interfaces des cellules est détaillée. Enfin, les conditions aux limites caractéristiques et de murs ont été étendues aux écoulements multiespèces dans le formalisme SD. Avec ces développements, des flammes laminaires pré-mélangées 1D et 2D ont été simulées avec des mécanismes réduits à 2 réactions ou des mécanismes réduits analytiquement. Les résultats sont très proches de ceux obtenus avec des solveurs de référence bien établis en combustion. Il est montré que pour un même niveau d'erreur, il est plus efficace d'utiliser des maillages grossiers avec des grandes valeurs de pp et non l'inverse. Par conséquent, le raffinement local en pp, qui applique des grandes valeurs de pp dans les régions d'intérêts seulement, permet de garder une bonne précision à un coût de calcul plus faible. Ceci est particulièrement intéressant pour des simulations de combustion où le front de flamme est très localisé et requiert une plus grande précision que le reste de l'écoulement. Il est également observé sur ces cas simples 1D et 2D que la méthode SD est moins sensible à la discrétisation du front de flamme que les solveurs volumes finis comme AVBP. Pour terminer, deux différentes configurations de flammes 3D turbulentes ont été simulées avec l'algorithme des SD étendu aux écoulements réactifs
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