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In this thesis, an advanced order model reduction for the incompressible steady-state Navier-Stokes equations is investigated. Initially, some basic principles of the finite element an approach are reviewed. This initial theoretical section offers a chance to connect elegant and intuitive ideas, while also helping to frame the problem into a matrix formulation. Moreover, this primary phase will help acquire a solid understanding on the implementation of the Finite Element Method (FEM) in a vectorized Matlab code, which reduces significantly the computation cost of the MATLAB program. In the fifth chapter, which examines the foundations of reduced order modeling, the thesis makes its first contribution. Moreover, the code will be used to obtain essential training data sets which will serve as the foundational material for constructing velocity and pressure modes for the reduced order model. Finally, these modes will be used as comprehensive shape functions for the original finite element problem, integrating them into the framework to enhance its efficiency and accuracy.

In this study, we investigate a handful of equilibrium states and their hydrodynamic stability for the incompressible flow in a square cavity driven by the tangential simultaneous motion of all of its lids at the same speed. This problem has been investigated in the recent past, albeit only partially and always disregarding the singular boundary conditions at the corners. The implications of the system symmetries have not been rigorously addressed, either. In our study, we introduce a regularized version of the boundary conditions to avoid the corner singularities, in a way that preserves the main features of the original setup. In addition, the Navier–Stokes equations are discretized by means of highly accurate Chebyshev spectral methods that provide exponential convergence of all computed flows and consistently eliminate any potential source of structural instability of the bifurcation scenario. We employ Newton–Krylov solvers, implemented within continuation algorithms, to accurately compute equilibrium solutions. Linear stability analysis of both the primary symmetric base flow and the secondary asymmetric states uncovers new branches of fully asymmetric steady states. The analysis has allowed identification of six previously undetected bifurcations, all of which are associated with the disruption of either the rotational invariance or the reflection symmetry. Some of these bifurcations have been found to be quite clustered in some regions of the parameter space, which points at the underlying action of higher codimension mechanisms. Notably, all the bifurcations reported occur within the range of low to moderate Reynolds numbers, making the regularized four-sided lid-driven cavity flow a reliable benchmark for assessing different numerical schemes in the context of Navier–Stokes equivariant bifurcation theory., Postprint (published version)

Mitigation of waves in shipboard swimming pools excited by ship motions is a topic of current interest that has not yet been extensively investigated. Various damping mechanisms have been applied to mitigate waves excited in tanks, such as porous baffles or different types of internal baffles. However, most of these devices are installed in LNG tanks to prevent serious sloshing effects. Furthermore, these are passive devices and, hence, unable to control wave-induced excitations. Our focus was on the validation and verification of the three-dimensional (3D) fully linear Finite Difference Method (FDM) based on the previously developed method of Qi et al. (2024) for predicting waves in a transversely placed swimming pool excited by ship motions and piston-type actuators. We validated our FDM approach against comparative Computational Fluid Dynamics (CFD) simulations as well as experimental model test measurements. The CFD tools solved the Reynolds-averaged Navier-Stokes (RANS) equations, relied on an appropriate turbulence model and the Volume of Fluid (VOF) method to capture the free surface of the liquid in the model tank, and employed the overset technique to specify the motions of the active actuators. Our FDM considered ship-induced roll excitations as well as simultaneous roll and sway excitations, albeit only at frequencies outside the range of the pool's resonance frequency. For these periods, our approach accurately predicted not only the mitigated waves in the pool, but also the optimum amplitude of the actuators needed to mitigate such waves. Our FDM can be applied to obtain a preview of excited waves under various conditions, leveraging its principal advantage of extreme computational efficiency.

We study the equilibrium configurations for several fluid-structure interaction problems. The fluid is confined in a 2D unbounded channel that contains a body, free to move inside the channel with rigid motions (transversal translations and rotations). The motion of the fluid is generated by a Poiseuille inflow/outflow at infinity and governed by the stationary Navier-Stokes equations. For a model where the fluid is the air and the body represents the cross-section of a suspension bridge, therefore also subject to restoring elastic forces, we prove that for small inflows there exists a unique equilibrium position, while for large inflows we numerically show the appearance of additional equilibria. A similar uniqueness result is also obtained for a discretized 3D bridge, consisting in a finite number of cross-sections interacting with the adjacent ones. The very same model, but without restoring forces, is used to describe the mechanism of the Leonardo da Vinci ferry, which is able to cross a river without engines. We numerically determine the optimal orientation of the ferry that allows it to cross the river in minimal time., SCOPUS: ar.j, info:eu-repo/semantics/published

The integration in time of the semi-discrete Navier-Stokes equations has been historically bounded by the classic CFL condition, and later on, an innovative method has been defined by ensuring the eigenvalues of the method lay in the boundary of the stability region, yet the effects of this in the resolution have not been tested. In this abstract, an efficiency region in a set of schemes is defined so that the method can alternate among methods in order to keep the maximum efficiency subject to a set of conditions, such as a minimum order of accuracy or a minimum numerical dissipation., The investigations presented in this paper are supported by the Ministerio de Economía y Competitividad, Spain, RETO twin project (PDC2021-120970-I00). J.P-R. is also supported by the Catalan Agency for Management of University and Research Grants (AGAUR).The authors thankfully acknowledge these institutions., Postprint (published version)

This project is dedicated to the study of turbulence models applied to the Navier-Stokes equations to predict turbulent flows. For such purpose, the turbulence is firstly approached from a mathematical point of view, being the reasoning behind each modeling technique explored and the formulation of each approach mathematically explained. Two different types of modeling approaches are considered: the Reynolds Averaged Navier-Stokes equations and the Large Eddy Simulation. The most widely used models in industry of each approach are tested in a canonical case: the turbulent channel flow. In that case, the performance that each model has is analysed, detecting the advantages and shortcomings of each solution and the specific characteristics that the engineer has to consider when using these models. With that, the best-performing models for the case are selected and tested in a more challenging case: the backward-facing step. In this case, with a shear layer present as well as a wall-bounded flow, the performance of the models is again assessed. Furthermore, a compromise solution that has been used in industry between RANS and LES named Delayed Detached Eddy Simulation (DDES) is tested for this more complex case. To improve the performance of the model, a modification on a function (called shielding function) characteristic of this type of model is implemented in OpenFOAM and tested to check whether better performance is reached.

In this paper a novel approach to the computation of the eigenbounds of the system of ordinary differential equations is presented. Afterwards, this approach is applied for different Runge-Kutta methods with various properties among which the energy conservation in time is considered. By doing so, the energy budgets of these schemes applied to an inviscid flow as well as a three-dimensional Taylor-Green vortex are presented., The investigations presented in this paper are supported by the Ministerio de Economía y Competitividad, Spain, RETOtwin project (PDC2021-120970- I00). J.P-R. is also supported by the Catalan Agency for Management of University and Research Grants (AGAUR). The authors thankfully acknowledge these institutions., Postprint (published version)

Direct numerical simulations of the incompressible Navier-Stokes equations at high Reynolds numbers are not yet feasible, so dynamically less complex mathematical formulations such as Large Eddy Simulation (LES) have been developed. For the well-known eddy viscosity models for LES, the computational method is based on the combination of invariants of a symmetric tensor that depends on the gradient of the resolved velocity field, G = ¿u. Several models (namely S3PQR) have been developed using the first three principal invariants of the symmetric tensor GGT with excellent results. Therefore, in this work, we will focus on the application of the S3PQR and other LES models on the free boundary layer case. Then, we will test their performances over a fully developed boundary layer wind farm, using a simplified wind turbine model., Postprint (published version)

This master's thesis explores the implementation of convective schemes in incompressible flow modeling, emphasizing their impact on energy conservation, momentum, and angular momentum. Specifically the skew-symmetric term and the EMAC term recently proposed. The study begins with a literature review on incompressible fluid simulation and introduces the variational multiscale method, applied specifically to stabilize formulations in incompressible fluid flow. Then the study of conservation properties for the convective terms and finally some numerical tests.

This thesis explores the application of Radial Basis Functions (RBFs) to fluid dynamical problems. In particular, stationary Stokes and Navier-Stokes equations are solved using RBF collocation method. An existing approach from the literature, is enchanced by an additional polynomial basis and a new preconditioner. A faster method based on the partition of unity is introduced for stationary Stokes equations. Finally, a global method based on Picard linearization is introduced for stationary Navier-Stokes equations., Denna avhandling utforskar tillämpningen av Radial Basis Functions (RBF) på dynamiska problem med vätskor. I synnerhet löses stationära Stokes och Navier-Stokes ekvationer lösas med hjälp av RBF-samlokaliseringsmetoden. En befintlig metod från litteraturen, förbättras genom en ytterligare polynombas och en ny förkonditionering. En snabbare metod baserad på enhetens partition introduceras för stationära Stokes-ekvationer. Slutligen introduceras en global metod baserad på Picard linjärisering för stationära Navier-Stokes ekvationer.

The simulation of fluid flow is a challenging and important problem in science and engineering. This thesis primarily focuses on developing finite element methods for simulating subsonic two-phase flows with varying densities, described by the variable density incompressible Navier-Stokes equations. These equations are commonly used to model a wide range of phenomena, including aerodynamic forces around vehicles, climate and weather prediction, combustion and the spread of pollution. Incompressible flow is characterized by the velocity field satisfying the divergence-free condition. However, numerically satisfying this condition is one of the main challenges in simulating such flows. In practice, this condition is rarely satisfied exactly, which can result in stability and conservation issues in computations. Moreover, enforcing the divergence-free condition is a primary computational bottleneck for incompressible flow solvers. To improve computational efficiency, we explore and develop artificial compressibility techniques, which regularize this constraint. Additionally, we develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to enhance conservation properties when the divergence-free condition is not strongly enforced, leading to significantly improved accuracy and robustness. Another primary difficulty in simulating fluid flows arises from the challenge of accurately representing underresolved flows, where the mesh resolution cannot capture the gradient of the true solution. This leads to stability issues unless appropriate stabilization techniques are used. In this thesis, we develop new high-order accurate artificial viscosity techniques to deal with this issue. Furthermore, we thoroughly investigate the properties of viscous regularizations, ensuring that kinetic energy stability is guaranteed when using artificial viscosity.

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Holder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Holder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. The most precise estimates are given in weighted spaces, where the weight depends on the distance to the boundary. This allows for the deficit to be captured precisely in the vicinity of irregularities of the boundary. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier-Stokes equations in time dependent domains.

Consider a velocity field $u$ solving the incompressible Navier-Stokes equations on $[0,T]\times\mathbb R^d$ ($d\geq3$) and satisfying $\|u(t)\|_X\leq A$ for all times, where the norm $X$ is critical with respect to the Navier-Stokes scaling. We prove several theorems to the effect that the regularity of the solution can be controlled explicitly in terms of $A$, building upon Tao's pioneering work on the case $d=3$, $X=L^3(\mathbb R^3)$. First we prove a generalization to the critical Lebesgue space in any number of spatial dimensions ($d\geq4$, $X=L^d(\mathbb R^d)$). Then we show a variety of circumstances under which Tao's bounds can be strengthened, including the case in which the solution is nearly axisymmetric. For exactly axisymmetric solutions, we prove regularity in terms of the weak norm $X=L^{3,\infty}(\mathbb R^3)$ which implies effective bounds on approximately self-similar behavior.

This work consists of understanding the Navier Stokes equations by programming codes, made in MatLab, that simulate fluid problems where the continuity and momentum equations take place. Firstly, other types of simpler cases will be studied to understand concepts such as heat conduction or the terms of convection and diffusion. This last concept will be studied by developing a code of two different cases: diagonal flow and Smith-Hutton flow. Once these simpler cases have been analyzed and studied, a special case will be reached for the study of the aforementioned equations at different Reynolds numbers, that is, for laminar and turbulent fluids. Finally, the final budget of the study is analyzed, and the impact it has had on the environment, for example, the carbon monoxide emissions emitted when carrying out the project.

Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We consider proper orthogonal decomposition (POD) methods to approximate the incompressible Navier–Stokes equations. We study the case in which one discretization for the nonlinear term is used in the snapshots (that are computed with a full order method (FOM)) and a different discretization of the nonlinear term is applied in the POD method. We prove that an additional error term appears in this case, compared with the case in which the same discretization of the nonlinear term is applied for both the FOM and the POD methods. However, the added term has the same size as the error coming from the FOM so that the rate of convergence of the POD method is barely affected. We analyze the case in which we add grad–div stabilization to both the FOM and the POD methods because it allows to get error bounds with constants independent of inverse powers of the viscosity. We also study the case in which no stabilization is added. Some numerical experiments support the theoretical analysis.

Scientific applications face the challenge of efficiently exploiting increasingly complex parallel and distributed systems. Developing hand-tuned codes is a time-consuming, tedious and hardly reusable task. Reaching high performance appears detrimental to productivity and portability and unreasonable to expect from scientists. Domain-Specific Languages (DSLs) are collaborative environments aiming to overcome such difficulties by decoupling the problem description from the algorithmic implementation. However, developing a competitive tool in High-Performance Computing (HPC) is challenging: DSLs for HPC environments have two additional critical requirements, performance and scalability. Moreover, documented and successful cases are few, making it difficult to popularise DSLs as problem-solving environments for scientific HPC code development. In this context, Saiph is a task-based DSL easing the simulation of physical phenomena from Computational Fluid Dynamics (CFD), developed to meet HPC productivity and performance requirements. This work reports the tuning and evaluation of Saiph using the Taylor–Green Vortex (TGV) problem as a case study. We assess Saiph’s productivity, numerical methods, and high-performance strategies to illustrate its use and demonstrate its competitiveness, viability and benefits for CFD software developments in HPC environments. Hence, we contribute to the popularisation of HPC DSLs as suitable problem-solving environments able to unify modern computational and scientific knowledge., This work has been supported by the Generalitat de Catalunya (contract 2021-SGR-01007), the Spanish Ministry of Science and Technology (contract PID2019-107255 GB), and Severo Ochoa (CEX2021- 001148-S/MCIN/AEI/10.13039/ 501100011033)., Peer Reviewed, Postprint (author's final draft)

El present treball exposa un estudi avançat de la resolució numèrica de les equacions multidimensionals de Navier-Stokes mitjançant mètodes computacionals. El projecte ha constat d’un detallat anàlisi teòric, a partir del qual s’han desenvolupat diferents simulacions de casos de referència amb el llenguatge de programació C++. Finalment, s’ha portat a terme una aplicació en el camp d’enginyeria de la climatització; en concret, en fluxos turbulents donats en la dinàmica de fluids de cortines d’aire. L’estudi de les equacions de Navier-Stokes ha sorgit de voler contribuir a la seva recerca científica degut a la seva complexitat. A partir de simulacions CFD, s’han obtingut solucions numèriques des de casos elementals a més avançats, podent-se endinsar en règims turbulents. Dels casos fonamentals, s’ha pogut verificar i validar la resolució numèrica considerant una varietat de geometries, condicions de contorn, dimensions de malla i esquemes numèrics. I dels casos més avançats, s’ha ampliat l’anàlisi a tres dimensions i a règims dinàmics turbulents, els quals no haguessin sigut possible sense la implementació de models de turbulència, ressaltant el model de Smagorinsky. A més, cal destacar d’ells el paper de la viscositat cinemàtica turbulenta en el tipus de simulació LES, que filtra els petits remolins de turbulència en l’espai i només resol directament les escales grans. L’aplicació desenvolupada simula un cas simplificat del flux turbulent que produeix la cortina d’aire, un equip que impulsa aire condicionat que té com a funció separar tèrmicament dos espais. En l’anàlisi realitzat, s’han avaluat la influència de la temperatura i la velocitat en els resultats, dels quals es conclou que una cortina d’aire isoterma és més energèticament eficient comparada amb una calefactada perquè presenta més pèrdues de calor, tot i que ajudi a mantenir millor la calor

This project starts by studying the finite element method for the steady state Navier-Stokes equations, afterwards it is implemented in Matlab and optimized via Vectorization achieving up to 5000x speed-up in some calculations. Then a reduced order model is studied to decrease the computational time of performing different simulations with slight modifications to the input parameters. Finally, the results obtained are compared against an already tested FEM code, Kratos Multiphysics, and against literature, and the performance of the developed solver for the equations is analyzed. It has been observed that the results obtained with the present work’s solver are almost equal to those made by the reference alternatives.

In this article, we study the behaviour of the Abels-Garcke-Grün Navier-Stokes-Cahn- Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. For the diffuse-interface model to approach a classical sharp-interface model in the limit, the so-called mobility parameter in the diffuse-interface model must scale appropriately with the interface-thickness parameter. In the literature various scaling relations in the range to have been proposed, but the optimal order to pass to the limit has not been explored previously. Our primary objective is to elucidate this optimal order of the - scaling relation in terms of the rate of convergence of the diffuse-interface solution to the sharp-interface solution. Additionally, we examine how the convergence rate is affected by a sub-optimal parameter scaling. We centre our investigation around the case of an oscillating droplet. To provide reference limit solutions, we derive new analytical expressions for small-amplitude oscillations of a viscous droplet in a viscous ambient fluid in two dimensions. For two distinct modes of oscillation, we probe the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations by means of an adaptive finite-element method. The adaptive-refinement procedure enables us to consider diffuse-interface thicknesses that are significantly smaller than other relevant length scales in the droplet-oscillation problem, allowing an exploration of the asymptotic regime.

El treball consisteix en el desenvolupament de codis en C++ per a la resolució de les equacions multidimensionals de Navier-Stokes, el que es coneix com a CFD (“Computational Fluid Dynamics”). A través de l’ús de mètodes numèrics, concretament de tècniques de volums finits i “solvers” d’equacions lineals, s’han obtingut els resultats per a una sèrie de casos d’estudi: Smith-Hutton, Lid Driven Cavity i Differentially Heated Cavity. Finalment, s’ha desenvolupat un cas propi que tracta sobre la propagació del fum originat en un incendi a l’interior d’un túnel. Al llarg del treball es desenvolupen les diferents equacions que s’utilitzaran en cada cas i es profunditza en cada un dels termes que les componen. Per altra banda, es mostra el procés de discretització temporal i espacial per a cada una de les equacions. En tots els casos estudiats, que disposen de solucions analítiques o de referència, es comparen els resultats obtinguts amb els de referència i es comprova que el codi desenvolupat té com a resultat uns valors molt pròxims als teòrics. El primer cas (Smith-Hutton) que es resol tracta sobre el transport d’una propietat en un fluid i, per resoldre’l, es fa ús de l’equació de la convecció-difusió. En aquest cas es parla tant de la resolució de l’equació com dels diferents esquemes convectius que es poden utilitzar. En el segon i el tercer cas es resolen les equacions de Navier-Stokes utilitzant el “Fractional Step Method”. En el segon cas (Lid Driven Cavity) s’analitza l’efecte que té una velocitat externa sobre un fluid que es troba contingut en un espai delimitat. En el tercer cas (Differentially Heated Cavity) s’analitza la convecció lliure a través de l’aproximació de Boussinesq i com aquesta modelitza els corrents que s’indueixen a causa dels canvis de temperatura en el fluid. En l‘últim cas s’introdueix el món de la turbulència i s’analitza com es propaga el fum en l’interior d’un recinte tancat. En aquest cas, s’analitza el model de Smagorinsky, un model de

This work deals with the investigation of bifurcating fluid phenomena using a reduced order modelling setting aided by artificial neural networks. We discuss the POD-NN approach dealing with non-smooth solutions set of nonlinear parametrized PDEs. Thus, we study the Navier–Stokes equations describing: (i) the Coanda effect in a channel, and (ii) the lid driven triangular cavity flow, in a physical/geometrical multi-parametrized setting, considering the effects of the domain’s configuration on the position of the bifurcation points. Finally, we propose a reduced manifold-based bifurcation diagram for a non-intrusive recovery of the critical points evolution. Exploiting such detection tool, we are able to efficiently obtain information about the pattern flow behaviour, from symmetry breaking profiles to attaching/spreading vortices, even in the advection-dominated regime.

CFD codes that are used for industrial applications commonly use a collocated grid arrangement to calculate the physical flow variables. When using a central differencing gradient (CDG) to discretise the continuous operators of the Navier-Stokes equations, a wide stencil is obtained for the Laplacian operator. This wide stencil, in turn, leads to a decoupling between odd and even grid points of the pressure field that results from the pressure Poisson equation. This decoupling can lead to non-physical, spurious modes in the solution, a problem commonly known as the checkerboard problem (Ferziger et al., 2002)., This work is supported by the FusionCAT project (001-P001722) co-financed by the European Union Regional Development Fund within the framework of the ERDFOperational Program of Catalonia 2014-2020 with agrant of 50% of total cost eligible.The work is part of the RETO twin project (PDC2021-120970-I00) of Ministerio de Economía y Competitividad,Spain. J.A.H. is supported by the predoctoral grant FI2022 (2022 FI_B1 00204) of the Catalan Agency for Management of University and Research Grants (AGAUR)., Postprint (author's final draft)

Mathematical Physics, Delft Institute of Applied Mathematics

We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the sense that the finite dimensional function spaces form a discrete de Rham complex. Conservation of mass, kinetic energy and helicity in the absence of dissipative terms is proven at the discrete level. Proper dissipation rates of kinetic energy and helicity in the viscous case are also proven. Numerical tests supporting the method are provided., Structural Optimization and Mechanics, Aerodynamics

Close to solid walls, at high Reynolds numbers, fluids may develop steep gradients which require a fine mesh for an accurate simulation of the turbulent boundary layer. An often used cure is to use a wall model instead of a fine mesh, with the drawback that modeling is introduced, leading to possibly unstable numerical schemes. In this paper, we leave the modeling aside, take it for granted, and propose a new set of provably energy stable boundary procedures for the incompressible Navier-Stokes equations. We show that these new boundary procedures lead to numerical results with high accuracy even for coarse meshes where data is partially obtained from a wall model., Funding: Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC)

In machine learning, if the training data is independently and identically distributed as the test data then a trained model can make an accurate predictions for new samples of data. Conventional machine learning has a strong dependence on massive amounts of training data which are domain specific to understand their latent patterns. In contrast, Domain adaptation and Transfer learning methods are sub-fields within machine learning that are concerned with solving the inescapable problem of insufficient training data by relaxing the domain dependence hypothesis. In this contribution, this issue has been addressed and by making a novel combination of both the methods we develop a computationally efficient and practical algorithm to solve boundary value problems based on nonlinear partial differential equations. We adopt a meshfree analysis framework to integrate the prevailing geometric modelling techniques based on NURBS and present an enhanced deep collocation approach that also plays an important role in the accuracy of solutions. We start with a brief introduction on how these methods expand upon this framework. We observe an excellent agreement between these methods and have shown that how fine-tuning a pre-trained network to a specialized domain may lead to an outstanding performance compare to the existing ones. As proof of concept, we illustrate the performance of our proposed model on several benchmark problems. © 2022, The Author(s).

The present thesis consists on the computational study and development of a 3D reactive flows solver. The main objective is the creation of C++ programming codes able to simulate the combustion chemical reactions. To do so, the project follows a series of steps towards the final goal. First, the theoretical and numerical basis and implementation of the incompressible Navier Stokes equations is addressed. Advanced temporal integration methods are used. Additionally, parallelisation techniques are implemented and tested. Two CFD canonical study cases are presented as a verification for the Navier Stokes solver. Once the first step is completed, mass transport phenomena is tackled. The theoretical framework and numerical implementation of the corresponding equations is addressed. A study case on water vapour transportation in air is presented as a verification process. Finally, the reactive flows solver is completed by adding the chemical reaction phenomena using a one step global mechanism. Premixed CH4/Air combustion in incompressible flow is analysed.

The main objective of that project is the development and posterior verification of C++ codes able to solve the incompressible Navier-Stokes equations. Nonetheless, before attaching the mainconcerned objective other Computational Fluid Dynamics cases must be solved as a learning tool. That is why the report is structured in chapters, which show both the learning evolution and the physical phenomenons studied. Firstly, an introduction to the project is shown where the main objectives, scope and requirements are established, as well as the justification for its development and the state of the art. Secondly, a numerical methods background, in which all the thesis theoretical basis regarding discretization field are indicated. Thirdly, the resolution of problems in which the energy transport mechanism is the diffusion, focusing on solid materials heat transfer. Fourthly, the numerical resolution of the Convection-Diffusion equation, verifying the simulated results with some benchmark cases proposed by the CTTC. Fifthly, the numerical resolution of the NavierStokes equations using the Fractional Step Method where the classical problems Lid-Driven Cavity, Differentially-Heated Cavity, flow between parallel flat plates and the external flow around a square cylinder will be solved as a verification case. Finally, a study of the turbulence will be undertaken using the Burgers equation in Fourier space, which is a simplified mathematical model that shares many aspects with the Navier-Stokes equations.

This article presents error bounds for a velocity–pressure segregated POD reduced order model discretization of the Navier–Stokes equations. The stability is proven in L∞(L2) and energy norms for velocity, with bounds that do not depend on the viscosity, while for pressure it is proven in a semi-norm of the same asymptotic order as the L2 norm with respect to the mesh size. The proposed estimates are calculated for the two flow problems, the flow past a cylinder and the lid-driven cavity flow. Their quality is then assessed in terms of the predicted logarithmic slope with respect to the velocity POD contribution ratio. We show that the proposed error estimates allow a good approximation of the real errors slopes and thus a good prediction of their rate of convergence.

Microplastic (MP) particles are commonly found in freshwater environments such as rivers and lakes, negatively affecting aquatic organisms and potentially causing water quality issues. Understanding the transport and fate of MP particles in these environments is a key prerequisite to mitigate the problem. For standing water bodies (lakes, ponds) the terminal settling velocity (TSV) is a key parameter, which determines particle residence times and exposure times of organisms to MP in lakes. Here we systematically investigate the effects of the physical parameters density, volume, shape and roundness, surface roughness and hydrophobicity and lake water temperature on the TSV of a large number of particles with regular and irregular shapes (equivalent diameters: 0.5 to 2.5mm) and different polymer densities using computational fluid dynamics (CFD) simulations. Simulation results are compared to laboratory settling experiments and used to evaluate existing, semi-empirical relationships to estimate TSV. The semi-empirical relationships were generally found to be in reasonable agreement with the CFD simulations (R2 > 0.92). Deviations were attributed to simplifications in their descriptions of particle shapes. Overall the CFD simulations also matched the TSVs from the experiments quite well, (R2 > 0.82), but experimental TSVs were generally slower than model TSVs with the largest differences for the irregular particles made from biodegradable polymers. The deviations of up to 58% were found to be related to the attachment of air bubbles on irregularities in the particle surfaces caused by the hydrophobicity of the MP particles. Overall, density was the most decisive parameter for TSV with increases in TSV of up to 400% followed by volume (200%), water temperature (47%) and particle roundness (45%). Our simulation results provide a frame of reference for an improved evaluation of the relative effects of different particle characteristics on their TSV in lakes

In this paper, we explore the stability and dynamical relevance of a wide variety of steady, time-periodic, quasiperiodic, and chaotic flows arising between orthogonally stretching parallel plates. We first explore the stability of all the steady flow solution families formerly identified by Ayats et al. [“Flows between orthogonally stretching parallel plates,” Phys. Fluids 33, 024103 (2021)], concluding that only the one that originates from the Stokesian approximation is actually stable. When both plates are shrinking at identical or nearly the same deceleration rates, this Stokesian flow exhibits a Hopf bifurcation that leads to stable time-periodic regimes. The resulting time-periodic orbits or flows are tracked for different Reynolds numbers and stretching rates while monitoring their Floquet exponents to identify secondary instabilities. It is found that these time-periodic flows also exhibit Neimark–Sacker bifurcations, generating stable quasiperiodic flows (tori) that may sometimes give rise to chaotic dynamics through a Ruelle–Takens–Newhouse scenario. However, chaotic dynamics is unusually observed, as the quasiperiodic flows generally become phase-locked through a resonance mechanism before a strange attractor may arise, thus restoring the time-periodicity of the flow. In this work, we have identified and tracked four different resonance regions, also known as Arnold tongues or horns. In particular, the 1¿:¿4 strong resonance region is explored in great detail, where the identified scenarios are in very good agreement with normal form theory., Postprint (author's final draft)