Numerical Error Prediction and its applications in CFD using tau-estimation

Nowadays, Computational Fluid Dynamics (CFD) solvers are widely used within the industry to model fluid flow phenomenons. Several fluid flow model equations have been employed in the last decades to simulate and predict forces acting, for example, on different aircraft configurations. Computational time and accuracy are strongly dependent on the fluid flow model equation and the spatial dimension of the problem considered. While simple models based on perfect flows, like panel methods or potential flow models can be very fast to solve, they usually suffer from a poor accuracy in order to simulate real flows (transonic, viscous). On the other hand, more complex models such as the full Navier- Stokes equations provide high fidelity predictions but at a much higher computational cost. Thus, a good compromise between accuracy and computational time has to be fixed for engineering applications. A discretisation technique widely used within the industry is the so-called Finite Volume approach on unstructured meshes. This technique spatially discretises the flow motion equations onto a set of elements which form a mesh, a discrete representation of the continuous domain. Using this approach, for a given flow model equation, the accuracy and computational time mainly depend on the distribution of nodes forming the mesh. Therefore, a good compromise between accuracy and computational time might be obtained by carefully defining the mesh. However, defining an optimal mesh for complex flows and geometries requires a very high level expertize in fluid mechanics and numerical analysis, and in most cases a simple guess of regions of the computational domain which might affect the most the accuracy is impossible. Thus, it is desirable to have an automatized remeshing tool, which is more flexible with unstructured meshes than its structured counterpart. However, adaptive methods currently in use still have an opened question: how to efficiently drive the adaptation ? Pioneering sensors based on flow features generally suffer from a lack of reliability, so in the last decade more effort has been made in developing numerical error-based sensors, like for instance the adjoint-based adaptation sensors. While very efficient at adapting meshes for a given functional output, the latter method is very expensive as it requires to solve a dual set of equations and computes the sensor on an embedded mesh. Therefore, it would be desirable to develop a more affordable numerical error estimation method. The current work aims at estimating the truncation error, which arises when discretising a partial differential equation. These are the higher order terms neglected in the construction of the numerical scheme. The truncation error provides very useful information as it is strongly related to the flow model equation and its discretisation. On one hand, it is a very reliable measure of the quality of the mesh, therefore very useful in order to drive a mesh adaptation procedure. On the other hand, it is strongly linked to the flow model equation, so that a careful estimation actually gives information on how well a given equation is solved, which may be useful in the context of _ -extrapolation or zonal modelling. The following work is organized as follows: Chap. 1 contains a short review of mesh adaptation techniques as well as numerical error prediction. In the first section, Sec. 1.1, the basic refinement strategies are reviewed and the main contribution to structured and unstructured mesh adaptation are presented. Sec. 1.2 introduces the definitions of errors encountered when solving Computational Fluid Dynamics problems and reviews the most common approaches to predict them. Chap. 2 is devoted to the mathematical formulation of truncation error estimation in the context of finite volume methodology, as well as a complete verification procedure. Several features are studied, such as the influence of grid non-uniformities, non-linearity, boundary conditions and non-converged numerical solutions. This verification part has been submitted and accepted for publication in the Journal of Computational Physics. Chap. 3 presents a mesh adaptation algorithm based on truncation error estimates and compares the results to a feature-based and an adjoint-based sensor (in collaboration with Jorge Ponsín, INTA). Two- and three-dimensional cases relevant for validation in the aeronautical industry are considered. This part has been submitted and accepted in the AIAA Journal. An extension to Reynolds Averaged Navier- Stokes equations is also included, where _ -estimation-based mesh adaptation and _ -extrapolation are applied to viscous wing profiles. The latter has been submitted in the Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. Keywords: mesh adaptation, numerical error prediction, finite volume Hoy en día, la Dinámica de Fluidos Computacional (CFD) es ampliamente utilizada dentro de la industria para obtener información sobre fenómenos fluidos. La Dinámica de Fluidos Computacional considera distintas modelizaciones de las ecuaciones fluidas (Potencial, Euler, Navier-Stokes, etc) para simular y predecir las fuerzas que actúan, por ejemplo, sobre una configuración de aeronave. El tiempo de cálculo y la precisión en la solución depende en gran medida de los modelos utilizados, así como de la dimensión espacial del problema considerado. Mientras que modelos simples basados en flujos perfectos, como modelos de flujos potenciales, se pueden resolver rápidamente, por lo general aducen de una baja precisión a la hora de simular flujos reales (viscosos, transónicos, etc). Por otro lado, modelos más complejos tales como el conjunto de ecuaciones de Navier-Stokes proporcionan predicciones de alta fidelidad, a expensas de un coste computacional mucho más elevado. Por lo tanto, en términos de aplicaciones de ingeniería se debe fijar un buen compromiso entre precisión y tiempo de cálculo. Una técnica de discretización ampliamente utilizada en la industria es el método de los Volúmenes Finitos en mallas no estructuradas. Esta técnica discretiza espacialmente las ecuaciones del movimiento del flujo sobre un conjunto de elementos que forman una malla, una representación discreta del dominio continuo. Utilizando este enfoque, para una ecuación de flujo dado, la precisión y el tiempo computacional dependen principalmente de la distribución de los nodos que forman la malla. Por consiguiente, un buen compromiso entre precisión y tiempo de cálculo se podría obtener definiendo cuidadosamente la malla, concentrando sus elementos en aquellas zonas donde sea estrictamente necesario. Sin embargo, la definición de una malla óptima para corrientes y geometrías complejas requiere un nivel muy alto de experiencia en la mecánica de fluidos y el análisis numérico, así como un conocimiento previo de la solución. Aspecto que en la mayoría de los casos no está disponible. Por tanto, es deseable tener una herramienta que permita adaptar los elementos de malla de forma automática, acorde a la solución fluida (remallado). Esta herramienta es generalmente más flexible en mallas no estructuradas que con su homóloga estructurada. No obstante, los métodos de adaptación actualmente en uso todavía dejan una pregunta abierta: cómo conducir de manera eficiente la adaptación. Sensores pioneros basados en las características del flujo en general, adolecen de una falta de fiabilidad, por lo que en la última década se han realizado grandes esfuerzos en el desarrollo numérico de sensores basados en el error, como por ejemplo los sensores basados en el adjunto. A pesar de ser muy eficientes en la adaptación de mallas para un determinado funcional, este último método resulta muy costoso, pues requiere resolver un doble conjunto de ecuaciones: la solución y su adjunta. Por tanto, es deseable desarrollar un método numérico de estimación de error más asequible. El presente trabajo tiene como objetivo estimar el error local de truncación, que aparece cuando se discretiza una ecuación en derivadas parciales. Estos son los términos de orden superior olvidados en la construcción del esquema numérico. El error de truncación proporciona una información muy útil sobre la solución: es una medida muy fiable de la calidad de la malla, obteniendo información que permite llevar a cabo un procedimiento de adaptación de malla. Está fuertemente relacionado al modelo matemático fluido, de modo que una estimación precisa garantiza la idoneidad de dicho modelo en un campo fluido, lo que puede ser útil en el contexto de modelado zonal. Por último, permite mejorar la precisión de la solución resolviendo un nuevo sistema donde el error local actúa como término fuente (_ -extrapolación). El presenta trabajo se organiza de la siguiente manera: Cap. 1 contiene una breve reseña de las técnicas de adaptación de malla, así como de los métodos de predicción de los errores numéricos. En la primera sección, Sec. 1.1, se examinan las estrategias básicas de refinamiento y se presenta la principal contribución a la adaptación de malla estructurada y no estructurada. Sec 1.2 introduce las definiciones de los errores encontrados en la resolución de problemas de Dinámica Computacional de Fluidos y se examinan los enfoques más comunes para predecirlos. Cap. 2 está dedicado a la formulación matemática de la estimación del error de truncación en el contexto de la metodología de Volúmenes Finitos, así como a un procedimiento de verificación completo. Se estudian varias características que influyen en su estimación: la influencia de la falta de uniformidad de la malla, el efecto de las no linealidades del modelo matemático, diferentes condiciones de contorno y soluciones numéricas no convergidas. Esta parte de verificación ha sido presentada y aceptada para su publicación en el Journal of Computational Physics. Cap. 3 presenta un algoritmo de adaptación de malla basado en la estimación del error de truncación y compara los resultados con sensores de featured-based y adjointbased (en colaboración con Jorge Ponsín del INTA). Se consideran casos en dos y tres dimensiones, relevantes para la validación en la industria aeronáutica. Este trabajo ha sido presentado y aceptado en el AIAA Journal. También se incluye una extensión de estos métodos a las ecuaciones RANS (Reynolds Average Navier- Stokes), en donde adaptación de malla basada en _ y _ -extrapolación son aplicados a perfiles con viscosidad de alas. Este último trabajo se ha presentado en los Actas de la Institución de Ingenieros Mecánicos, Parte G: Journal of Aerospace Engineering. Palabras clave: adaptación de malla, predicción del error numérico, volúmenes finitos

Diagnosing Causes of Water Scarcity in Complex Water Resources Systems and Identifying Risk Management Actions

From the water management perspective, water scarcity is an unacceptable risk of facing water shortages to serve water demands in the near future. Water scarcity may be temporary and related to drought conditions or other accidental situation, or may be permanent and due to deeper causes such as excessive demand growth, lack of infrastructure for water storage or transport, or constraints in water management. Diagnosing the causes of water scarcity in complex water resources systems is a precondition to adopt effective drought risk management actions. In this paper we present four indices which have been developed to evaluate water scarcity. We propose a methodology for interpretation of index values that can lead to conclusions about the reliability and vulnerability of systems to water scarcity, as well as to diagnose their possible causes and to propose solutions. The described methodology was applied to the Ebro river basin, identifying existing and expected problems and possible solutions. System diagnostics, based exclusively on the analysis of index values, were compared with the known reality as perceived by system managers, validating the conclusions in all cases

GreenDisc: a HW/SW energy optimization framework in globally distributed computation

In recent future, wireless sensor networks (WSNs) will experience a broad high-scale deployment (millions of nodes in the national area) with multiple information sources per node, and with very specific requirements for signal processing. In parallel, the broad range deployment of WSNs facilitates the definition and execution of ambitious studies, with a large input data set and high computational complexity. These computation resources, very often heterogeneous and driven on-demand, can only be satisfied by high-performance Data Centers (DCs). The high economical and environmental impact of the energy consumption in DCs requires aggressive energy optimization policies. These policies have been already detected but not successfully proposed. In this context, this paper shows the following on-going research lines and obtained results. In the field of WSNs: energy optimization in the processing nodes from different abstraction levels, including reconfigurable application specific architectures, efficient customization of the memory hierarchy, energy-aware management of the wireless interface, and design automation for signal processing applications. In the field of DCs: energy-optimal workload assignment policies in heterogeneous DCs, resource management policies with energy consciousness, and efficient cooling mechanisms that will cooperate in the minimization of the electricity bill of the DCs that process the data provided by the WSNs.

GreenDisc: a HW/SW energy optimization framework in globally distributed computation

In recent future, wireless sensor networks ({WSNs}) will experience a broad high-scale deployment (millions of nodes in the national area) with multiple information sources per node, and with very specific requirements for signal processing. In parallel, the broad range deployment of {WSNs} facilitates the definition and execution of ambitious studies, with a large input data set and high computational complexity. These computation resources, very often heterogeneous and driven on-demand, can only be satisfied by high-performance Data Centers ({DCs}). The high economical and environmental impact of the energy consumption in {DCs} requires aggressive energy optimization policies. These policies have been already detected but not successfully proposed. In this context, this paper shows the following on-going research lines and obtained results. In the field of {WSNs}: energy optimization in the processing nodes from different abstraction levels, including reconfigurable application specific architectures, efficient customization of the memory hierarchy, energy-aware management of the wireless interface, and design automation for signal processing applications. In the field of {DCs}: energy-optimal workload assignment policies in heterogeneous {DCs}, resource management policies with energy consciousness, and efficient cooling mechanisms that will cooperate in the minimization of the electricity bill of the DCs that process the data provided by the WSNs.

Land cover dynamics and climate change implications on water resources in South Pacific Costa Rica

Water is fundamental to human life and the availability of freshwater is often a constraint on human welfare and economic development. Consequently, the potential effects of global changes on hydrology and water resources are considered among the most severe and vital ones. Water scarcity is one of the main problems in the rural communities of Central America, as a result of an important degradation of catchment areas and the over-exploitation of aquifers. The present Thesis is focused on two critical aspects of global changes over water resources: (1) the potential effects of climate change on water quantity and (2) the impacts of land cover and land use changes on the hydrological processes and water cycle. Costa Rica is among the few developing countries that have recently achieved a land use transition with a net increase in forest cover. Osa Region in South Pacific Costa Rica is an appealing study site to assess water supply management plans and to measure the effects of deforestation, forest transitions and climate change projections reported in the region. Rural Community Water Supply systems (ASADAS) in Osa are dealing with an increasing demand of freshwater due to the growing population and the change in the way of life in the rural livelihoods. Land cover mosaics which have resulted from the above mentioned processes are characterized by the abandonment of marginal farmland with the spread over these former grasslands of high return crops and the expansion of secondary forests due to reforestation initiatives. These land use changes have a significant impact on runoff generation in priority water-supply catchments in the humid tropics, as evidenced by the analysis of the Tinoco Experimental Catchment in the Southern Pacific area of Costa Rica. The monitoring system assesses the effects of the different land uses on the runoff responses and on the general water cycle of the basin. Runoff responses at plot scale are analyzed for secondary forests, oil palm plantations, forest plantations and grasslands. The Oil palm plantation plot presented the highest runoff coefficient (mean RC=32.6%), twice that measured under grasslands (mean RC=15.3%) and 20-fold greater than in secondary forest (mean RC=1.7%). A Thornthwaite-type water balance is proposed to assess the impact of land cover and climate change scenarios over water availability for rural communities in Osa Region. Climate change projections were obtained by the downscaling of BCM2, CNCM3 and ECHAM5 models. Precipitation and temperature were averaged and conveyed by the A1B, A2 and B1 IPCC climate scenario for 2030, 2060 and 2080. Precipitation simulations exhibit a positive increase during the dry season for the three scenarios and a decrease during the rainy season, with the highest magnitude (up to 25%) by the end of the 21st century under scenario B1. Monthly mean temperature simulations increase for the three scenarios throughout the year with a maximum increase during the dry season of 5% under A1B and A2 scenarios and 4% under B1 scenario. The Thornthwaite-type Water Balance model indicates important decreases of water surplus for the three climate scenarios during the rainy season, with a maximum decrease on May, which under A1B scenario drop up to 20%, under A2 up to 40% and under B1 scenario drop up to almost 60%. Land cover scenarios were created taking into account current land cover dynamics of the region. Land cover scenario 1 projects a deforestation situation, with forests decreasing up to 15% due to urbanization of the upper catchment areas; land cover scenario 2 projects a forest recovery situation where forested areas increase due to grassland abandonment on areas with more than 30% of slope. Deforestation scenario projects an annual water surplus decrease of 15% while the reforestation scenario projects a water surplus increase of almost 25%. This water balance analysis indicates that climate scenarios are equal contributors as land cover scenarios to future water resource estimations.

Irrigated agriculture: Water resources management for a sustainable environment

In the last decade, research on irrigation has mainly been aimed at reducing crop water consumption. In arid and semi-arid environments, in relation to the limited water resources, the use of low quality water in agriculture has also been investigated in order to detect their effects on soil physical properties and on crop production. More recently, even the reduction of energy consumption in agriculture, as well as the effects of external factors, climate change and agricultural policies, have been major research interests. All these objectives have been considered in the papers included in this special issue. However, in the last years, approaches aimed at reducing crop water requirements have significantly changed. Remote sensing with satellites or unmanned vehicles, and vegetation spectral measurements, among others, represent in fact the newest frontier of existing technologies. Knowledge of soil hydraulic properties, often forgotten because of the difficulty of their estimation, can also be considered as a new way to reduce water consumption.

Pressure management in water distribution systems: current status, proposals, and future trends

Pressure management (PM) is commonly used in water distribution systems (WDSs). In the last decade, a strategic objective in the field has been the development of new scientific and technical methods for its implementation. However, due to a lack of systematic analysis of the results obtained in practical cases, progress has not always been reflected in practical actions. To address this problem, this paper provides a comprehensive analysis of the most innovative issues related to PM. The methodology proposed is based on a case-study comparison of qualitative concepts that involves published work from 140 sources. The results include a qualitative analysis covering four aspects: (1) the objectives yielded by PM; (2) types of regulation, including advanced control systems through electronic controllers; (3) new methods for designing districts; and (4) development of optimization models associated with PM. The evolution of the aforementioned four aspects is examined and discussed. Conclusions regarding the current status of each factor are drawn and proposals for future research outlined