Partial differential equations for self-organization in cellular and developmental biology

Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field

Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design

This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a liquid-solid material and its moving interface. This physical process is mathematically formulated as a diffusion partial differential equation (PDE) evolving on a time-varying spatial domain described by an ordinary differential equation (ODE). The state-dependency of the moving interface makes the coupled PDE-ODE system a nonlinear and challenging problem. We propose a full-state feedback control law, an observer design, and the associated output-feedback control law via the backstepping method. The designed observer allows estimation of the temperature profile based on the available measurement of solid phase length. The associated output-feedback controller ensures the global exponential stability of the estimation errors, the H1- norm of the distributed temperature, and the moving interface to the desired setpoint under some explicitly given restrictions on the setpoint and observer gain. The exponential stability results are established considering Neumann and Dirichlet boundary actuations.Comment: 16 pages, 11 figures, submitted to IEEE Transactions on Automatic Contro

Sketch-To-Solution: An Exploration of Viscous CFD with Automatic Grids

Numerical simulation of the Reynolds-averaged NavierStokes (RANS) equations has become a critical tool for the design of aerospace vehicles. However, the issues that affect the grid convergence of three dimensional RANS solutions are not completely understood, as documented in the AIAA Drag Prediction Workshop series. Grid adaption methods have the potential for increasing the automation and discretization error control of RANS solutions to impact the aerospace design and certification process. The realization of the CFD Vision 2030 Study includes automated management of errors and uncertainties of physics-based, predictive modeling that can set the stage for ensuring a vehicle is in compliance with a regulation or specification by using analysis without demonstration in flight test (i.e., certification or qualification by analysis). For example, the Cart3D inviscid analysis package has automated Cartesian cut-cell gridding with output-based error control. Fueled by recent advances in the fields of anisotropic grid adaptation, error estimation, and geometry modeling, a similar work flow is explored for viscous CFD simulations; where a CFD application engineer provides geometry, boundary conditions, and flow parameters, and the sketch-to-solution process yields a CFD simulation through automatic, error-based, grid adaptation

Large eddy simulations para modelado de combustión de hidrógeno. Aplicaciones a unidades balísticas de reducción de arrastre de base y análisis de secuencias de accidentes nucleares

[SPA] Esta tesis doctoral se presenta bajo la modalidad de compendio de publicaciones. En este trabajo se han desarrollado modelos de simulación mediante herramientas de mecánica de fluidos computacional (CFD) utilizando modelado de turbulencia Large Eddy Simulation (LES) para abordar el análisis de problemas en los que, tradicionalmente, se han utilizado de forma extendida simulaciones con modelado de turbulencia Reynolds Averaged Navier Stokes Equations (RANS), en las que los resultados alcanzados presentan ,en muchas ocasiones, diferencias significativas comparados con datos experimentales. En la actualidad, simulaciones CFD con modelado de turbulencia LES se están convirtiendo en una atractiva alternativa a simulaciones RANS, siendo abordable en términos de coste computacional y tiempo de simulación para muchas aplicaciones industriales, debido principalmente a la evolución y avances en materia de recursos y potencia computacional. En ese contexto, el objetivo principal de este trabajo consiste en desarrollar y validar modelos y estrategias de simulación CFD para ser aplicados y extraer conclusiones relevantes en problemas donde tradicionalmente simulaciones con modelos RANS han sido ampliamente aplicadas, pero con limitaciones en su validación experimental. Estos problemas son el análisis de balística exterior incluyendo unidades de reducción de resistencia de base mediante tecnología Base Bleed, así como el estudio de problemas de combustión en secuencias de accidente nuclear. Ambas aplicaciones tienen en común que involucran procesos de combustión hidrógeno-aire en condiciones de flujo turbulento. Para cada una de estas aplicaciones, diferentes metodologías y estrategias numéricas han sido desarrolladas y validadas. Adicionalmente, junto al desarrollo de estos modelos, se proponen metodologías para optimizar el coste computacional con limitado impacto en la precisión de los resultados alcanzados. La tecnología conocida como Base Bleed ha sido, y es, ampliamente utilizada con el objetivo de reducir la resistencia aerodinámica de cuerpos esbeltos mediante la destilación de gases (procedentes de una combustión) en su zona posterior. Los modelos desarrollados en este trabajo permiten estimar el coeficiente de resistencia aerodinámica (CD) cuando el cuerpo, con unidad de Base Bleed (activa o no), posee rotación axial (spin) y se considera vuelo cuasi - estacionario en régimen transónico y supersónico (Mach 0.99-1.5). Se han comparado los resultados de varios modelos bidimensionales y tridimensionales con datos experimentales obtenidos mediante técnicas de trayectografía. Los resultados alcanzados evidencian que los modelos de turbulencia RANS y Detached-Eddy Simulation (DES) obtienen buenas predicciones de CD en ausencia de unidades Base Bleed. Sin embargo, el efecto de reducción de resistencia provocado por estas no aparece reflejado en las predicciones de CD calculados con estos modelos de turbulencia. En cambio, con modelos de turbulencia LES, se obtienen predicciones más realistas. En relación al estudio de procesos de combustión en secuencias de accidente nuclear, estos precisan de simulaciones de combustión premezclada turbulenta en espacios confinados, simulaciones que presentan comúnmente la limitación del elevado coste computacional requerido, así como el reducido número de datos experimentales disponibles para la validación. De forma general, ciertos modelos de combustión turbulenta basados en RANS han obtenido resultados satisfactorios para predecir parámetros globales de la combustión, pero presentan limitaciones para modelar correctamente algunos fenómenos transitorios, especialmente interacciones dinámicas de los frentes de llama en un medio turbulento y su influencia en la combustión. En este contexto, los modelos de combustión basados en LES se presentan como una alternativa eficiente en términos de coste computacional para analizar secuencias de accidente involucrando la combustión del hidrógeno. En este trabajo, dos modelos diferentes han sido desarrollados y propuestos para analizar la evolución de la velocidad de combustión de deflagraciones y la interacción de estas en medios turbulentos. Estas estrategias han sido, un modelo de variable de progreso (Flamelet Progress Variable, LES-FPV) y otro con modelado de tasa de reacción química de gases multicomponente (Finite-Rate chemistry model) denominado Thickened Flame Model (LES-TFM) en el que se pretende modelar la interacción entre el mecanismo de cinética química con la turbulencia. Se ha llevado a cabo la validación de estos modelos para predecir fenómenos tales como la velocidad de combustión, aceleración turbulenta y evolución de la presión. Adicionalmente, se han propuesto técnicas para reducir el coste computacional y para hacer abordable su aplicación en problemas industriales, de mayor escala que los ensayos de laboratorio para validación. Estas técnicas incluyen: Dynamic Adaptive Chemistry (DAC), in-situ Adaptive Tabulation (ISAT) y mallados dinámicos adaptativos. Esta última técnica tiene el objetivo de aumentar la resolución espacial localmente en el frente de llama, manteniendo un coste computacional y tiempos de simulación abordables. Finalmente, se ha aplicado los modelos previamente validados para analizar dos secuencias de pérdidas de vacío en ITER (Loss Of Vacuum Accident, LOVA). Con ellos se han obtenido conclusiones relevantes sobre dichos accidentes. Adicionalmente, otra aproximación basada en la hipótesis de “Reactor Perfectamente Agitado” (Perfectly Stirred Reactor, PSR) ha sido propuesta y validada para predicción de variables globales en secuencias de combustión de hidrógeno-aire premezclado. Esta aproximación tiene la ventaja de una menor complejidad desde el punto de vista de modelado, a expensas de requerir un mayor coste computacional, además de presentar una aplicabilidad limitada en determinados regímenes de combustión. Se ha llevado a cabo una validación y evaluación de estos modelos comparando con datos experimentales y con otros estudios numéricos de aceleración de llama en un canal con obstáculos. Los resultados permiten identificar las principales deficiencias a tener en cuenta al utilizar esta aproximación y evaluar las incertidumbres relacionadas con el uso de diferentes modelos de turbulencia sub-grid scale. Por último, se ha desarrollado un modelo, para simular problemas de combustión bifásicos de flujos reactivos en presencia de partículas de grafito a partir de los modelos LES-TFM. La modelización numérica de la combustión turbulenta de mezclas de H2-aire con partículas sólidas de grafito es un reto clave en muchos problemas industriales, incluyendo el ámbito de la seguridad nuclear. El modelo se basa en una aproximación Euler-Euler acoplada con diferentes cinéticas químicas detalladas para simular la combustión de mezclas de gases y partículas. El modelo se ha empleado para predecir la evolución transitoria de las secuencias de combustión turbulenta de mezclas de H2, aire y partículas de grafito en condiciones de baja concentración de este último, obteniendo resultados que se ajustan a los experimentales obtenidos en una bomba esférica. El modelo permite predecir ciertas tendencias experimentales, como la composición de productos de la combustión, mostrando que una baja concentración inicial de partículas de grafito (~96 g/m3) influye en la dinámica de la combustión del H2 para mezclas de 20% en volumen de H2 en aire. En estas condiciones, se aumentaron los niveles de presión alcanzados en las paredes de la esfera y se redujo el tiempo de combustión respecto al caso sin presencia de partículas. Los resultados muestran la viabilidad de utilizar este tipo de modelado para caracterizar parámetros globales como la evolución temporal de la presión en las paredes. [ENG] This doctoral dissertation has been presented in the form of thesis by publication. In this work, Computational Fluid Dynamics (CFD) simulations using Large Eddy Simulation (LES) turbulence modeling are proposed for analyzing problems where traditionally Reynolds Averaged Navier Stokes Equations (RANS) have been extensively used, but with results that did not find good agreement when compared with experimental data. Nowadays, as a consequence of the increase in computational efficiency and power during last years, LES models has become an affordable alternative for being applied on a lot of fluid-dynamics problems even from an industrial perspective. This work is focused on two problems: external ballistics for slender bodies with drag reduction (Base Bleed) units, and nuclear accident sequences. Both problems have in common that involve hydrogen-air combustion processes under turbulent flow conditions. For each application, different approaches have been developed and tested, and methodologies for improving computational cost with low (or not) penalty on the results accuracy have been analyzed and proposed. Base Bleed technology is a common strategy used for body drag reduction. This work studied analyzes CFD models to estimate the drag coefficient of slender bodies with spin and Base Bleed technology under transonic and supersonic (Mach number 0.99–1.5) quasi-steady conditions. 2-dimensional and 3-dimensional numerical models based on RANS, Detached Eddy Simulation (DES) and LES models were presented and benchmarked against ad-hoc experimental flight measurements performed with both active and inactive Base Bleed units. Results show that RANS and DES models predict well the drag coefficient in the absence of Base Bleed units. However, they have a very limited accuracy in drag prediction when facing a problem involving a high temperature jet mixing layer with a transonic wake as in the case of active Base Bleed. Notwithstanding, a reasonable agreement is found between numerical predictions of drag reduction and experimental data for the case of LES. On the other hand, the modelling of premixed combustion in three-dimensional confined scenarios is also studied in this work. Accurate modelling of combustion sequences is difficult due to computational costs and the limited ad-hoc experiments available to validate the models. RANS based combustion models have shown to be successful in predicting gross features of combustion, nevertheless, they have serious deficiencies to predict transient phenomena, such as combustion instabilities, cycle-to-cycle variations, self-ignition, and pollutant emission. LES seems to be a cost-effective method to reach this goal when analyzing H2 combustion dynamics in accident sequences. In this work, two different LES models have been proposed and assessed for predicting flame combustion acceleration and interaction in the presence of turbulence: a Flamelet Progress Variable (LES-FPV) and a Thickened Flame Model (LES-TFM). With the aim of reducing computational costs, Dynamic Adaptive Chemistry (DAC) and in-situ adaptive tabulation (ISAT) methods have been exploited when facing detailed kinetic mechanism for hydrogen combustion. Moreover, an adaptive meshing technique was used with the aim of tracking the flame front to ensure an adequate local spatial resolution, where the model requires such level discretization. Experimental validation was performed to assess the ability of the different studied approaches to predict the flame burning speed, flame acceleration, and pressure evolution for lean H2-Air volume percent mixtures from 16 to 28% propagating within a turbulent field. Results revealed that both approaches led to accurate predictions in terms of flame burning speed. When considering DAC and ISAT methods with detailed chemistry, LES-TFM model was found to be a cost-efficient solution, which relies less on experimental inputs than the LES-FPV alternative. Once this model has been validated, it is used to analyze two loss of vacuum accident (LOVA) sequences within the International Thermonuclear Experimental Reactor (ITER) Vacuum Vessel. Results permitted to get key insights into these accidents. Additionally, LES turbulence with perfectly stirred reactor (PSR) assumption and detailed chemistry have been also assessed to predict global variables of unsteady, premixed, hydrogen combustion sequences. This approach requires less modeling efforts but increases the need of computational resources and it shows application limitations. The assessment is faced by benchmarking the model with hydrogen-air experimental tests and with numerical data of flame acceleration in an obstructed channel obtained with other models. Results permit to identify major shortcomings that should be addressed with this approach and to assess the uncertainties linked to the use of different sub-models. Finally, LES-TFM approach have been proposed for modeling two-phase combustion problems to describe reacting flows in presence of graphite particles. The model proposed is benchmarked against experimental combustion data obtained in a spherical bomb. The numerical modelling of turbulent combustion of H2-air mixtures with solid graphite particles is a challenging and key issue in many industrial problems including nuclear safety. The model relies in an Eulerian–Eulerian approach coupled with different detailed chemical kinetics to simulate the combustion of mixtures of gases and particles. The model is applied to predict the transient evolution of turbulent combustion sequences of mixtures of hydrogen, air, and a low concentration of graphite particles. Results show a good agreement between experimental and numerical data. Moreover, the model is able to predict some key experimental tendencies and reveals that the presence of a low concentration of graphite particles (~96 g/m3) influences the hydrogen combustion dynamics for mixtures of 20% (in volume) of hydrogen in air. Under these conditions, pressure levels reached at the walls of the sphere are increased and the combustion time is shortened. The results also show the viability of using this kind of models for obtaining global combustion parameters such as the temporal evolution of the wall pressure.Esta tesis doctoral se presenta bajo la modalidad de compendio de publicaciones. Está formada por un total de cuatro artículos: Article 1.- F. Nicolás-Pérez, F.J.S. Velasco, J.R. García-Cascales, R.A. Otón-Martínez, A. López-Belchí, D. Moratilla, F. Rey, A. Laso, On the accuracy of RANS, DES and LES turbulence models for predicting drag reduction with Base Bleed technology, Aerospace Science and Technology, Volume 67, 2017, Pages 126-140, ISSN 1270-9638, https://doi.org/10.1016/j.ast.2017.03.031. Article 2.- F. Nicolás-Pérez, F.J.S. Velasco, José R. García-Cascales, Ramón A. Otón-Martínez, Ahmed Bentaib, Nabiha Chaumeix, Evaluation of different models for turbulent combustion of hydrogen-air mixtures. Large Eddy Simulation of a LOVA sequence with hydrogen deflagration in ITER Vacuum Vessel, Fusion Engineering and Design, Volume 161, 2020, 111901, ISSN 0920-3796, https://doi.org/10.1016/j.fusengdes.2020.111901. Article 3.- F. Nicolás-Pérez, F.J.S. Velasco, Ramón A. Otón-Martínez, José R. García-Cascales, Ahmed Bentaib and Nabiha Chaumeix, Capabilities and limitations of Large Eddy Simulation with perfectly stirred reactor assumption for engineering applications of unsteady, hydrogen combustion sequences, Engineering Applications of Computational Fluid Mechanics (TCFM) 2021 https://doi.org/10.1080/19942060.2021.1974092. Article 4.- F. Nicolás-Pérez, F.J.S. Velasco, Ramón A. Otón-Martínez, José R. García-Cascales, Ahmed Bentaib and Nabiha Chaumeix, Mathematical Modelling of Turbulent Combustion of Two-Phase Mixtures of Gas and Solid Particles with a Eulerian–Eulerian Approach: The Case of Hydrogen Combustion in the Presence of Graphite Particles, Mathematics 2021, 9(17), 2017; https://doi.org/10.3390/math9172017.Escuela Internacional de Doctorado de la Universidad Politécnica de CartagenaPrograma de Doctorado en Energías Renovables y Eficiencia Energétic

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

Application of method of lines in chemical engineering problems

In this work, two problems in chemical engineering are studied and solved. Estimation of an important parameter of dust explosions, the deflagration index kST , and a study of unsteady state with axial diffusion Plug Flow Reactors are presented. Both problems are approached by characterizing the physical phenomena involved with suitable transport equations. Such equations have been developed with the synergy of both consolidated theoretical models and ad hoc assumptions and semi-empiric approaches, according to the specific problem analyzed. The final equation systems result in a system of non-linear Partial Differential Equations. The numerical solution of such equations has been performed by implementing the Method of Lines, a numerical method based on the discretization of spatial derivative operators, transforming a system of PDEs into a system of ODEs or DAEs. The resulting ODEs/DAEs systems have been implemented and solved inside MAT LABTMenvironment. The Method of Lines is presented for uniform and non-uniform grids, generalized with the use of spatial derivatives discretization stencils of several orders of accuracy. For the estimation of kST , we validated the model with 8 organic dust: Aspirin, Cork, Corn starch, Niacin, Polyethylene, Polystyrene, Sugar and Wheat flour. Results showed an interesting match between experimental and simulated data: predictions for the deflagration index were good, while the evolution of process variables (such as the temperature of the gas phase), still leaves room for improvements. For the PFR study, we propose 1-D models, taking in account the reactor start-up, thermal and material axial diffusion, and the presence of a heating/cooling system. In order to judge the quality of the results, we took as case study a reaction well studied in the literature over the years: the oxidation of Naphthalene. We developed the so-called Runaway Boundaries for the reaction considered. Our results found good matches with the available literature data and analysis. We also noticed a shifting of the Runaway Boundaries when considering a more realistic heating/cooling system