Four Decades of Studying Global Linear Instability: Progress and Challenges

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two(and periodic in one)or all three spatial directions.After a brief exposition of the theory,some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton–Krylov time-stepping method into a standard finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse matrix treatment, all these problems may now be solved on standard desktop computers

Order 10 4 speedup in global linear instability analysis using matrix formation

A unified solution framework is presented for one-, two- or three-dimensional complex non-symmetric eigenvalue problems, respectively governing linear modal instability of incompressible fluid flows in rectangular domains having two, one or no homogeneous spatial directions. The solution algorithm is based on subspace iteration in which the spatial discretization matrix is formed, stored and inverted serially. Results delivered by spectral collocation based on the Chebyshev-Gauss-Lobatto (CGL) points and a suite of high-order finite-difference methods comprising the previously employed for this type of work Dispersion-Relation-Preserving (DRP) and Padé finite-difference schemes, as well as the Summationby- parts (SBP) and the new high-order finite-difference scheme of order q (FD-q) have been compared from the point of view of accuracy and efficiency in standard validation cases of temporal local and BiGlobal linear instability. The FD-q method has been found to significantly outperform all other finite difference schemes in solving classic linear local, BiGlobal, and TriGlobal eigenvalue problems, as regards both memory and CPU time requirements. Results shown in the present study disprove the paradigm that spectral methods are superior to finite difference methods in terms of computational cost, at equal accuracy, FD-q spatial discretization delivering a speedup of ð (10 4). Consequently, accurate solutions of the three-dimensional (TriGlobal) eigenvalue problems may be solved on typical desktop computers with modest computational effort.

Matrix-free time-stepping methods for the solution of TriGlobal instability problems

La inmensa mayoría de los flujos de relevancia ingenieril permanecen sin estudiar en el marco de la teoría de estabilidad global. Esto es debido a dos razones fundamentalmente, las dificultades asociadas con el análisis de los flujos turbulentos y los inmensos recursos computacionales requeridos para obtener la solución del problema de autovalores asociado al análisis de inestabilidad de flujos tridimensionales, también conocido como problema TriGlobal. En esta tesis se aborda el problema asociado con la tridimensionalidad. Se ha desarrollado una metodología general para obtener soluciones de problemas de análisis modal de las inestabilidades lineales globales mediante el acoplamiento de métodos de evolución temporal, desarrollados en este trabajo, con códigos de mecánica de fluidos computacional de segundo orden, utilizados de forma general en la industria. Esta metodología consiste en la resolución del problema de autovalores asociado al análisis de inestabilidad mediante métodos de proyección en subespacios de Krylov, con la particularidad de que dichos subespacios son generados por medio de la integración temporal de un vector inicial usando cualquier código de mecánica de fluidos computacional. Se han elegido tres problemas desafiantes en función de la exigencia de recursos computacionales necesarios y de la complejidad física para la demostración de la presente metodología: (i) el flujo en el interior de una cavidad tridimensional impulsada por una de sus tapas, (ii) el flujo alrededor de un cilindro equipado con aletas helicoidales a lo largo su envergadura y (iii) el flujo a través de una cavidad abierta tridimensinal en ausencia de homogeneidades espaciales. Para la validación de la tecnología se ha obtenido la solución del problema TriGlobal asociado al flujo en la cavidad tridimensional, utilizando el método de evolución temporal desarrollado acoplado con los operadores numéricos de flujo incompresible del código CFD OpenFOAM (código libre). Los resultados obtenidos coinciden plentamente con la literatura. La aplicación de esta metodología al estudio de inestabilidades globales de flujos abiertos tridimensionales ha proporcionado por primera vez, información sobre la transición tridimensional de estos flujos. Además, la metodología ha sido adaptada para resolver problemas adjuntos TriGlobales, permitiendo el control de flujo basado en modificaciones de las inestabilidades globales. Finalmente, se ha demostrado que la cantidad moderada de los recursos computacionales requeridos para la solución del problema de valor propio TriGlobal usando este método numérico, junto a su versatilidad al poder acoplarse a cualquier código aerodinámico, permite la realización de análisis de inestabilidad global y control de flujos complejos de relevancia industrial. Abstract Most flows of engineering relevance still remain unexplored in a global instability theory context for two reasons. First, because of the difficulties associated with the analysis of turbulent flows and, second, for the formidable computational resources required for the solution of the eigenvalue problem associated with the instability analysis of three-dimensional base flows, also known as TriGlobal problem. In this thesis, the problem associated with the three-dimensionality is addressed by means of the development of a general approach to the solution of large-scale global linear instability analysis by coupling a time-stepping approach with second order aerodynamic codes employed in industry. Three challenging flows in the terms of required computational resources and physical complexity have been chosen for demonstration of the present methodology; (i) the flow inside a wall-bounded three-dimensional lid-driven cavity, (ii) the flow past a cylinder fitted with helical strakes and (iii) the flow over a inhomogeneous three-dimensional open cavity. Results in excellent agreement with the literature have been obtained for the three-dimensional lid-driven cavity by using this methodology coupled with the incompressible solver of the open-source toolbox OpenFOAM®, which has served as validation. Moreover, significant physical insight of the instability of three-dimensional open flows has been gained through the application of the present time-stepping methodology to the other two cases. In addition, modifications to the present approach have been proposed in order to perform adjoint instability analysis of three-dimensional base flows and flow control; validation and TriGlobal examples are presented. Finally, it has been demonstrated that the moderate amount of computational resources required for the solution of the TriGlobal eigenvalue problem using this method enables the performance of instability analysis and control of flows of industrial relevance.

Advances in global instability computations: from incompressible to hypersonic flow

Esta tesis constituye un gran avance en el conocimiento del estudio y análisis de inestabilidades hidrodinámicas desde un punto de vista físico y teórico, como consecuencia de haber desarrollado innovadoras técnicas para la resolución computacional eficiente y precisa de la parte principal del espectro correspondiente a los problemas de autovalores (EVP) multidimensionales que gobiernan la inestabilidad de flujos con dos o tres direcciones espaciales inhomogéneas, denominados problemas de estabilidad global lineal. En el contexto del trabajo de desarrollo de herramientas computacionales presentado en la tesis, la discretización mediante métodos de diferencias finitas estables de alto orden de los EVP bidimensionales y tridimensionales que se derivan de las ecuaciones de Navier-Stokes linealizadas sobre flujos con dos o tres direcciones espaciales inhomogéneas, ha permitido una aceleración de cuatro órdenes de magnitud en su resolución. Esta mejora de eficiencia numérica se ha conseguido gracias al hecho de que usando estos esquemas de diferencias finitas, técnicas eficientes de resolución de problemas lineales son utilizables, explotando el alto nivel de dispersión o alto número de elementos nulos en las matrices involucradas en los problemas tratados. Como más notable consecuencia cabe destacar que la resolución de EVPs multidimensionales de inestabilidad global, que hasta la fecha necesitaban de superordenadores, se ha podido realizar en ordenadores de sobremesa. Además de la solución de problemas de estabilidad global lineal, el mencionado desarrollo numérico facilitó la extensión de las ecuaciones de estabilidad parabolizadas (PSE) lineales y no lineales para analizar la inestabilidad de flujos que dependen fuertemente en dos direcciones espaciales y suavemente en la tercera con las ecuaciones de estabilidad parabolizadas tridimensionales (PSE-3D). Precisamente la capacidad de extensión del novedoso algoritmo PSE-3D para el estudio de interacciones no lineales de los modos de estabilidad, desarrollado íntegramente en esta tesis, permite la predicción de transición en flujos complejos de gran interés industrial y por lo tanto extiende el concepto clásico de PSE, el cuál ha sido empleado exitosamente durante las pasadas tres décadas en el mismo contexto para problemas de capa límite bidimensional. Típicos ejemplos de flujos incompresibles se han analizado en este trabajo sin la necesidad de recurrir a restrictivas presuposiciones usadas en el pasado. Se han estudiado problemas vorticales como es el caso de un vórtice aislado o sistemas de vórtices simulando la estela de alas, en los que la homogeneidad axial no se impone y así se puede considerar la difusión viscosa del flujo. Además, se ha estudiado el chorro giratorio turbulento, cuya inestabilidad se utiliza para mejorar las características de funcionamiento de combustores. En la tesis se abarcan adicionalmente problemas de flujos compresibles. Se presenta el estudio de inestabilidad de flujos de borde de ataque a diferentes velocidades de vuelo. También se analiza la estela formada por un elemento rugoso aislado en capa límite supersónica e hipersónica, mostrando excelentes comparaciones con resultados obtenidos mediante simulación numérica directa. Finalmente, nuevas inestabilidades se han identificado en el flujo hipersónico a Mach 7 alrededor de un cono elíptico que modela el vehículo de pruebas en vuelo HIFiRE-5. Los resultados comparan favorablemente con experimentos en vuelo, lo que subraya aún más el potencial de las metodologías de análisis de estabilidad desarrolladas en esta tesis. ABSTRACT The present thesis constitutes a step forward in advancing the frontiers of knowledge of fluid flow instability from a physical point of view, as a consequence of having been successful in developing groundbreaking methodologies for the efficient and accurate computation of the leading part of the spectrum pertinent to multi-dimensional eigenvalue problems (EVP) governing instability of flows with two or three inhomogeneous spatial directions. In the context of the numerical work presented in this thesis, the discretization of the spatial operator resulting from linearization of the Navier-Stokes equations around flows with two or three inhomogeneous spatial directions by variable-high-order stable finite-difference methods has permitted a speedup of four orders of magnitude in the solution of the corresponding two- and three-dimensional EVPs. This improvement of numerical performance has been achieved thanks to the high-sparsity level offered by the high-order finite-difference schemes employed for the discretization of the operators. This permitted use of efficient sparse linear algebra techniques without sacrificing accuracy and, consequently, solutions being obtained on typical workstations, as opposed to the previously employed supercomputers. Besides solution of the two- and three-dimensional EVPs of global linear instability, this development paved the way for the extension of the (linear and nonlinear) Parabolized Stability Equations (PSE) to analyze instability of flows which depend in a strongly-coupled inhomogeneous manner on two spatial directions and weakly on the third. Precisely the extensibility of the novel PSE-3D algorithm developed in the framework of the present thesis to study nonlinear flow instability permits transition prediction in flows of industrial interest, thus extending the classic PSE concept which has been successfully employed in the same context to boundary-layer type of flows over the last three decades. Typical examples of incompressible flows, the instability of which was analyzed in the present thesis without the need to resort to the restrictive assumptions used in the past, range from isolated vortices, and systems thereof, in which axial homogeneity is relaxed to consider viscous diffusion, as well as turbulent swirling jets, the instability of which is exploited in order to improve flame-holding properties of combustors. The instability of compressible subsonic and supersonic leading edge flows has been solved, and the wake of an isolated roughness element in a supersonic and hypersonic boundary-layer has also been analyzed with respect to its instability: excellent agreement with direct numerical simulation results has been obtained in all cases. Finally, instability analysis of Mach number 7 ow around an elliptic cone modeling the HIFiRE-5 flight test vehicle has unraveled flow instabilities near the minor-axis centerline, results comparing favorably with flight test predictions.

On the Use of Matrix-Free Shift-Invert Strategies For Global Flow Instability Analysis

A novel time-stepping shift-invert algorithm for linear stability analysis of laminar flows in complex geometries is presented. This method, based on a Krylov subspace iteration, enables the solution of complex non-symmetric eigenvalue problems in a matrix-free framework. Validations and comparisons to the classical exponential method have been performed in three different cases: (i) stenotic flow, (ii) backward-facing step and (iii) lid-driven swirling flow. Results show that this new approach speeds up the required Krylov subspace iterations and has the capability of converging to specific parts of the global spectrum. It is shown that, although the exponential method remains the method of choice if leading eigenvalues are sought, the performance of the present method could be dramatically improved with the use of a preconditioner. In addition, as opposed to other methods, this strategy can be directly applied to any time-stepper, regardless of the temporal or spatial discretization of the latter.

A gradient estimate to a degenerate parabolic equation with a singular absorption term: The global quenching phenomena

We prove global existence of nonnegative solutions to the one dimensional degenerate parabolic problems containing a singular term. We also show the global quenching phenomena for L1 initial datums. Moreover, the free boundary problem is considered in this paper.

El plan global frente al riesgo de inundación en la ribera del Júcar

El Plan Global del Júcar frente al riesgo de inundación pretende minimizar los daños causados por las avenidas que periódicamente afectan a la Ribera del Júcar. En el presente artículo se describen y analizan las actuaciones de este Plan, la génesis del mismo, el proceso de elaboración y sus resultados. La problemática es compleja, ya que el riesgo de inundación abarca un territorio muy poblado, con un importante desarrollo industrial y agrícola. Además, la existencia de espacios naturales de elevado valor ambiental, hace que las intervenciones derivadas del Plan consideren al mismo nivel la defensa frente a avenidas y las demandas sociales y ambientales, tras el consenso alcanzado en un proceso de participación pública. El Plan incorpora aspectos innovadores en ingeniería hidráulica, cuyo resultado es una actuación integrada y sostenible, que compatibiliza la protección frente a avenidas con la recuperación y mejora del ámbito fluvial, y su función como corredor vertebrador del territorio.

Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

In recent times the Douglas–Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a well-quasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems.

The Enlarged European Union and its Eastern Neighbours: Problems and Solutions. OSW Point of View, October 2003

The EU enlargement is scheduled to take place in 2004. After this date, it should be a priority for the EU to develop a coherent and comprehensive policy towards its nearest neighbours, i.e. countries bordering the Member States, which cannot join the EU in the nearest future due to their location or weaknesses of their political and economic systems. There are at least three reasons for this. Firstly, good relations with neighbours will underlie the broadly understood security of the Community. Relations with the nearest neighbours will determine both military security of the EU (including the combating of terrorism) and its ability to prevent other threats such as illegal migration, smuggling, etc. Secondly, good economic relations with neighbours may contribute to the Member States' economic growth in the longer term. And finally, the EU's ability to develop an effective and adequate policy towards its nearest neighbours will demonstrate its competence as a subject of international politics. In other words, the EU will not be recognised as a reliable political player in the global scene until it develops an effective strategy for its neighbourhood.

On the periphery of global jihad. The North Caucasus: the illusion of stabilisation. OSW Point of View Number 45, November 2014

The situation in the North Caucasus has stabilised, in comparison with previous years, mainly as regards the activity of the Islamic military underground. This is an effect of ideological changes among the militants which have led to a dilution of the Caucasian armed struggle and its marginalisation in global jihad, since top priority has been granted to the Middle Eastern front. The factors which have contributed to this stabilisation are the organisational crisis in the Caucasus Emirate and the outflux of militants to the Middle East, as well the successful ‘carrot and stick’ policy adopted by Moscow. However, the partial stabilisation in the Caucasus is inherently precarious, being a contingent outcome of the situation rather than the result of systemic change. The region’s pressing problems remain unresolved; and these problems are generating chronic instability and cauing the Caucasus to drift away from Russia in civilisational terms. An economic or political crisis in the Russian Federation may result in the conflicts in the Caucasus unfreezing, including a reactivation of the idea of Chechen independence as well as the idea of the Caucasus Emirate, which is a part of global jihad.

America’s global defence predicament – Why the Asia ‘rebalancing’ has little significance for European security. Egmont Paper No. 72, December 2014

Summary. This paper argues that the Obama Administration’s Asia ‘pivot’ or ‘rebalancing’ is mainly a rhetorical construct that has very little substantive effect on US security policy. In contrast to the widely held belief that a renewed emphasis on Asia would place meaningful restrictions on America’s willingness and ability to deal with European security problems there is little evidence to support this argument nor is it well-grounded conceptually. Furthermore, undue emphasis on this aspect of American policy has detracted attention from more important developments in Washington’s capacity to effectively project military power.

America's global defence predicament-Why the Asia 'rebalancing' has little significance for European security. Egmont Paper No. 72, December 2014

This paper argues that the Obama Administration’s Asia ‘pivot’ or ‘rebalancing’ is mainly a rhetorical construct that has very little substantive effect on US security policy. In contrast to the widely held belief that a renewed emphasis on Asia would place meaningful restrictions on America’s willingness and ability to deal with European security problems there is little evidence to support this argument nor is it well-grounded conceptually. Furthermore, undue emphasis on this aspect of American policy has detracted attention from more important developments in Washington’s capacity to effectively project military power.

Russia marks the 70th anniversary of the victory over Nazism: What significance does it have in an epoch of global confrontation? OSW Commentary No. 171, 20 May 2015

The Victory Day celebrations held in Russia on 9 May 2015 were special for marking the seventieth anniversary of the end of World War II but the particular international and domestic context they were set in was of yet greater importance. The element which set the celebrations in 2015 apart from those in the preceding years was how the military and moral aspects of Soviet victory over Nazi Germany was made part of the current geopolitical confrontation with the West concerning the Ukrainian crisis. The escalation of the aggressive rhetoric on Europe and the USA and accusations that the West is destabilising the international situation and striving for conflict was accompanied by a display of the increasing military power of the Russian Federation; the display itself was stronger than has been seen in preceding years. This was a clear sign that Moscow is ready to protect its national interests in the area of foreign policy by any means. At the same time, the creation of an atmosphere of threat and stoking patriotic feelings was intended to mobilise the Russian public around the political leadership while the country’s economic problems are deteriorating further.

How to get out of the crisis without reforms? The Belarusian authorities confront growing economic problems. OSW Commentary NUMBER 202/21.03.2016

2015 saw a drop in Belarus’s GDP for the first time in almost 20 years, which is primarily the result of a significant reduction in levels of production and export. As a consequence, there was also a serious depletion of the country’s foreign exchange reserves, as well as a progressive weakening of the Belarusian rouble. The macroeconomic figures from January and February 2016 show that these trends are not only continuing, but they are also becoming even more severe, which confirms that Belarus now finds itself in a prolonged economic crisis. On one hand, the reason for this state of affairs is the protracted economic recession in Russia, which is Belarus’s main economic partner, together with the drastic global decline in prices for fuel, which is a key Belarusian export. On the other hand, meanwhile, an equally important reason for the current crisis is the failure of the Belarusian economic model. President Aleksandr Lukashenko, out of fear that his authoritarian system of government will be dismantled and that public discontent will rise, has categorically rejected the proposals for even partial reforms put forward by some of his entourage, who are aware of the need for the immediate transformation of the country’s anachronistic and very costly economic model, based as it still is on quasi-Soviet management policies.

Essays on coordination problems in economics

No estudo da economia, há diversas situações em que a propensão de um indivíduo a tomar determinada ação é crescente na quantidade de outras pessoas que este indivíduo acredita que tomarão a mesma ação. Esse tipo de complementaridade estratégica geralmente leva à existência de múltiplos equilíbrios. Além disso, o resultado atingido pelas decisões decentralizadas dos agentes pode ser ineficiente, deixando espaço para intervenções de política econômica. Esta tese estuda diferentes ambientes em que a coordenação entre indivíduos é importante. O primeiro capítulo analisa como a manipulação de informação e a divulgação de informação afetam a coordenação entre investidores e o bem-estar em um modelo de corridas bancárias. No modelo, há uma autoridade reguladora que não pode se comprometer a revelar a verdadeira situação do setor bancário. O regulador observa informações idiossincráticas dos bancos (através de um stress test, por exemplo) e escolhe se revela essa informação para o público ou se divulga somente um relatório agregado sobre a saúde do sistema financeiro como um todo. O relatório agregado pode ser distorcido a um custo – um custo mais elevado significa maior credibilidade do regulador. Os investidores estão cientes dos incentivos do regulador a esconder más notícias do mercado, mas a manipulação de informação pode, ainda assim, ser efetiva. Se a credibilidade do regulador não for muito baixa, a política de divulgação de informação é estado-contingente, e existe sempre um conjunto de estados em que há manipulação de informação em equilíbrio. Se a credibilidade for suficientemente baixa, porém, o regulador opta por transparência total dos resultados banco-específicos, caso em que somente os bancos mais sólidos sobrevivem. Uma política de opacidade levaria a uma crise bancária sistêmica, independentemente do estado. O nível de credibilidade que maximiza o bem-estar agregado do ponto de vista ex ante é interior. O segundo e o terceiro capítulos estudam problemas de coordenação dinâmicos. O segundo capítulo analisa o bem-estar em um ambiente em que agentes recebem oportunidades aleatórias para migrar entre duas redes. Os resultados mostram que sempre que a rede de pior qualidade (intrínseca) prevalece, isto é eficiente. Na verdade, um planejador central estaria ainda mais inclinado a escolher a rede de pior qualidade. Em equilíbrio, pode haver mudanças ineficientes que ampliem a rede de qualidade superior. Quando indivíduos escolhem entre dois padrões ou redes com níveis de qualidade diferentes, se todos os indivíduos fizessem escolhas simultâneas, a solução eficiente seria que todos adotassem a rede de melhor qualidade. No entanto, quando há fricções e os agentes tomam decisões escalonadas, a solução eficiente difere ix do senso comum. O terceiro capítulo analisa um problema de coordenação dinâmico com decisões escalonadas em que os agentes são heterogêneos ex ante. No modelo, existe um único equilíbrio, caracterizado por thresholds que determinam as escolhas para cada tipo de agente. Apesar da heterogeneidade nos payoffs, há bastante conformidade nas ações individuais em equilíbrio. Os thresholds de diferentes tipos de agentes coincidem parcialmente contanto que exista um conjunto de crenças arbitrário que justifique esta conformidade. No entanto, as estratégias de equilíbrio de diferentes tipos nunca coincidem totalmente. Além disso, a conformidade não é ineficiente. A solução eficiente apresentaria estratégias ainda mais similares para tipos distintos em comparação com o equilíbrio decentralizado.