MEASURED AND PREDICTED LOSS GENERATION IN TRANSONIC TURBINE BLADING.

For a typical transonic turbine rotor blade, designed for use with coolant ejection, the trailing edge, or base loss is three to four times the profile boundary layer loss. The base region of such a profile is dominated by viscous effects and it seems essential to attack the problem of loss prediction by solving the compressible Navier-Stokes equations. However, such an approach is inevitably compromised by both numerical accuracy and turbulence modelling constraints. This paper describes a Navier-Stokes solver written for 2D blade-blade flows and employing a simple two-layer mixing length eddy viscosity model. Then, measured and predicted losses and base pressures are presented for two transonic rotor blades and attempts are made to assess the capabilities of the Navier-Stokes solver and to outline areas for future work.

Flow and wakes in large wind farms in complex terrain and offshore

Power losses due to wind turbine wakes are of the order of 10 and 20% of total power output in large wind farms. The focus of this research carried out within the EC funded UPWIND project is wind speed and turbulence modelling for large wind farms/wind turbines in complex terrain and offshore in order to optimise wind farm layouts to reduce wake losses and loads.

Análisis de los factores de adopción de comercio electrónico en segmentos de consumidores finales. Aplicación al caso español

El comercio electrónico ha experimentado un fuerte crecimiento en los últimos años, favorecido especialmente por el aumento de las tasas de penetración de Internet en todo el mundo. Sin embargo, no todos los países están evolucionando de la misma manera, con un espectro que va desde las naciones pioneras en desarrollo de tecnologías de la información y comunicaciones, que cuentan con una elevado porcentaje de internautas y de compradores online, hasta las rezagadas de rápida adopción en las que, pese a contar con una menor penetración de acceso, presentan una alta tasa de internautas compradores. Entre ambos extremos se encuentran países como España que, aunque alcanzó hace años una tasa considerable de penetración de usuarios de Internet, no ha conseguido una buena tasa de transformación de internautas en compradores. Pese a que el comercio electrónico ha experimentado importantes aumentos en los últimos años, sus tasas de crecimiento siguen estando por debajo de países con características socio-económicas similares. Para intentar conocer las razones que afectan a la adopción del comercio por parte de los compradores, la investigación científica del fenómeno ha empleado diferentes enfoques teóricos. De entre todos ellos ha destacado el uso de los modelos de adopción, proveniente de la literatura de adopción de sistemas de información en entornos organizativos. Estos modelos se basan en las percepciones de los compradores para determinar qué factores pueden predecir mejor la intención de compra y, en consecuencia, la conducta real de compra de los usuarios. Pese a que en los últimos años han proliferado los trabajos de investigación que aplican los modelos de adopción al comercio electrónico, casi todos tratan de validar sus hipótesis mediante el análisis de muestras de consumidores tratadas como un único conjunto, y del que se obtienen conclusiones generales. Sin embargo, desde el origen del marketing, y en especial a partir de la segunda mitad del siglo XIX, se considera que existen diferencias en el comportamiento de los consumidores, que pueden ser debidas a características demográficas, sociológicas o psicológicas. Estas diferencias se traducen en necesidades distintas, que sólo podrán ser satisfechas con una oferta adaptada por parte de los vendedores. Además, por contar el comercio electrónico con unas características particulares que lo diferencian del comercio tradicional –especialmente por la falta de contacto físico entre el comprador y el producto– a las diferencias en la adopción para cada consumidor se le añaden las diferencias derivadas del tipo de producto adquirido, que si bien habían sido consideradas en el canal físico, en el comercio electrónico cobran especial relevancia. A la vista de todo ello, el presente trabajo pretende abordar el estudio de los factores determinantes de la intención de compra y la conducta real de compra en comercio electrónico por parte del consumidor final español, teniendo en cuenta el tipo de segmento al que pertenezca dicho comprador y el tipo de producto considerado. Para ello, el trabajo contiene ocho apartados entre los que se encuentran cuatro bloques teóricos y tres bloques empíricos, además de las conclusiones. Estos bloques dan lugar a los siguientes ocho capítulos por orden de aparición en el trabajo: introducción, situación del comercio electrónico, modelos de adopción de tecnología, segmentación en comercio electrónico, diseño previo del trabajo empírico, diseño de la investigación, análisis de los resultados y conclusiones. El capítulo introductorio justifica la relevancia de la investigación, además de fijar los objetivos, la metodología y las fases seguidas para el desarrollo del trabajo. La justificación se complementa con el segundo capítulo, que cuenta con dos elementos principales: en primer lugar se define el concepto de comercio electrónico y se hace una breve retrospectiva desde sus orígenes hasta la situación actual en un contexto global; en segundo lugar, el análisis estudia la evolución del comercio electrónico en España, mostrando su desarrollo y situación presente a partir de sus principales indicadores. Este apartado no sólo permite conocer el contexto de la investigación, sino que además permite contrastar la relevancia de la muestra utilizada en el presente estudio con el perfil español respecto al comercio electrónico. Los capítulos tercero –modelos de adopción de tecnologías– y cuarto –segmentación en comercio electrónico– sientan las bases teóricas necesarias para abordar el estudio. En el capítulo tres se hace una revisión general de la literatura de modelos de adopción de tecnología y, en particular, de los modelos de adopción empleados en el ámbito del comercio electrónico. El resultado de dicha revisión deriva en la construcción de un modelo adaptado basado en los modelos UTAUT (Unified Theory of Acceptance and Use of Technology, Teoría unificada de la aceptación y el uso de la tecnología) y UTAUT2, combinado con dos factores específicos de adopción del comercio electrónico: el riesgo percibido y la confianza percibida. Por su parte, en el capítulo cuatro se revisan las metodologías de segmentación de clientes y productos empleadas en la literatura. De dicha revisión se obtienen un amplio conjunto de variables de las que finalmente se escogen nueve variables de clasificación que se consideran adecuadas tanto por su adaptación al contexto del comercio electrónico como por su adecuación a las características de la muestra empleada para validar el modelo. Las nueve variables se agrupan en tres conjuntos: variables de tipo socio-demográfico –género, edad, nivel de estudios, nivel de ingresos, tamaño de la unidad familiar y estado civil–, de comportamiento de compra – experiencia de compra por Internet y frecuencia de compra por Internet– y de tipo psicográfico –motivaciones de compra por Internet. La segunda parte del capítulo cuatro se dedica a la revisión de los criterios empleados en la literatura para la clasificación de los productos en el contexto del comercio electrónico. De dicha revisión se obtienen quince grupos de variables que pueden tomar un total de treinta y cuatro valores, lo que deriva en un elevado número de combinaciones posibles. Sin embargo, pese a haber sido utilizados en el contexto del comercio electrónico, no en todos los casos se ha comprobado la influencia de dichas variables respecto a la intención de compra o la conducta real de compra por Internet; por este motivo, y con el objetivo de definir una clasificación robusta y abordable de tipos de productos, en el capitulo cinco se lleva a cabo una validación de las variables de clasificación de productos mediante un experimento previo con 207 muestras. Seleccionando sólo aquellas variables objetivas que no dependan de la interpretación personal del consumidores y que determinen grupos significativamente distintos respecto a la intención y conducta de compra de los consumidores, se obtiene un modelo de dos variables que combinadas dan lugar a cuatro tipos de productos: bien digital, bien no digital, servicio digital y servicio no digital. Definidos el modelo de adopción y los criterios de segmentación de consumidores y productos, en el sexto capítulo se desarrolla el modelo completo de investigación formado por un conjunto de hipótesis obtenidas de la revisión de la literatura de los capítulos anteriores, en las que se definen las hipótesis de investigación con respecto a las influencias esperadas de las variables de segmentación sobre las relaciones del modelo de adopción. Este modelo confiere a la investigación un carácter social y de tipo fundamentalmente exploratorio, en el que en muchos casos ni siquiera se han encontrado evidencias empíricas previas que permitan el enunciado de hipótesis sobre la influencia de determinadas variables de segmentación. El capítulo seis contiene además la descripción del instrumento de medida empleado en la investigación, conformado por un total de 125 preguntas y sus correspondientes escalas de medida, así como la descripción de la muestra representativa empleada en la validación del modelo, compuesta por un grupo de 817 personas españolas o residentes en España. El capítulo siete constituye el núcleo del análisis empírico del trabajo de investigación, que se compone de dos elementos fundamentales. Primeramente se describen las técnicas estadísticas aplicadas para el estudio de los datos que, dada la complejidad del análisis, se dividen en tres grupos fundamentales: Método de mínimos cuadrados parciales (PLS, Partial Least Squares): herramienta estadística de análisis multivariante con capacidad de análisis predictivo que se emplea en la determinación de las relaciones estructurales de los modelos propuestos. Análisis multigrupo: conjunto de técnicas que permiten comparar los resultados obtenidos con el método PLS entre dos o más grupos derivados del uso de una o más variables de segmentación. En este caso se emplean cinco métodos de comparación, lo que permite asimismo comparar los rendimientos de cada uno de los métodos. Determinación de segmentos no identificados a priori: en el caso de algunas de las variables de segmentación no existe un criterio de clasificación definido a priori, sino que se obtiene a partir de la aplicación de técnicas estadísticas de clasificación. En este caso se emplean dos técnicas fundamentales: análisis de componentes principales –dado el elevado número de variables empleadas para la clasificación– y análisis clúster –del que se combina una técnica jerárquica que calcula el número óptimo de segmentos, con una técnica por etapas que es más eficiente en la clasificación, pero exige conocer el número de clústeres a priori. La aplicación de dichas técnicas estadísticas sobre los modelos resultantes de considerar los distintos criterios de segmentación, tanto de clientes como de productos, da lugar al análisis de un total de 128 modelos de adopción de comercio electrónico y 65 comparaciones multigrupo, cuyos resultados y principales consideraciones son elaboradas a lo largo del capítulo. Para concluir, el capítulo ocho recoge las conclusiones del trabajo divididas en cuatro partes diferenciadas. En primer lugar se examina el grado de alcance de los objetivos planteados al inicio de la investigación; después se desarrollan las principales contribuciones que este trabajo aporta tanto desde el punto de vista metodológico, como desde los punto de vista teórico y práctico; en tercer lugar, se profundiza en las conclusiones derivadas del estudio empírico, que se clasifican según los criterios de segmentación empleados, y que combinan resultados confirmatorios y exploratorios; por último, el trabajo recopila las principales limitaciones de la investigación, tanto de carácter teórico como empírico, así como aquellos aspectos que no habiendo podido plantearse dentro del contexto de este estudio, o como consecuencia de los resultados alcanzados, se presentan como líneas futuras de investigación. ABSTRACT Favoured by an increase of Internet penetration rates across the globe, electronic commerce has experienced a rapid growth over the last few years. Nevertheless, adoption of electronic commerce has differed from one country to another. On one hand, it has been observed that countries leading e-commerce adoption have a large percentage of Internet users as well as of online purchasers; on the other hand, other markets, despite having a low percentage of Internet users, show a high percentage of online buyers. Halfway between those two ends of the spectrum, we find countries such as Spain which, despite having moderately high Internet penetration rates and similar socio-economic characteristics as some of the leading countries, have failed to turn Internet users into active online buyers. Several theoretical approaches have been taken in an attempt to define the factors that influence the use of electronic commerce systems by customers. One of the betterknown frameworks to characterize adoption factors is the acceptance modelling theory, which is derived from the information systems adoption in organizational environments. These models are based on individual perceptions on which factors determine purchase intention, as a mean to explain users’ actual purchasing behaviour. Even though research on electronic commerce adoption models has increased in terms of volume and scope over the last years, the majority of studies validate their hypothesis by using a single sample of consumers from which they obtain general conclusions. Nevertheless, since the birth of marketing, and more specifically from the second half of the 19th century, differences in consumer behaviour owing to demographic, sociologic and psychological characteristics have also been taken into account. And such differences are generally translated into different needs that can only be satisfied when sellers adapt their offer to their target market. Electronic commerce has a number of features that makes it different when compared to traditional commerce; the best example of this is the lack of physical contact between customers and products, and between customers and vendors. Other than that, some differences that depend on the type of product may also play an important role in electronic commerce. From all the above, the present research aims to address the study of the main factors influencing purchase intention and actual purchase behaviour in electronic commerce by Spanish end-consumers, taking into consideration both the customer group to which they belong and the type of product being purchased. In order to achieve this goal, this Thesis is structured in eight chapters: four theoretical sections, three empirical blocks and a final section summarizing the conclusions derived from the research. The chapters are arranged in sequence as follows: introduction, current state of electronic commerce, technology adoption models, electronic commerce segmentation, preliminary design of the empirical work, research design, data analysis and results, and conclusions. The introductory chapter offers a detailed justification of the relevance of this study in the context of e-commerce adoption research; it also sets out the objectives, methodology and research stages. The second chapter further expands and complements the introductory chapter, focusing on two elements: the concept of electronic commerce and its evolution from a general point of view, and the evolution of electronic commerce in Spain and main indicators of adoption. This section is intended to allow the reader to understand the research context, and also to serve as a basis to justify the relevance and representativeness of the sample used in this study. Chapters three (technology acceptance models) and four (segmentation in electronic commerce) set the theoretical foundations for the study. Chapter 3 presents a thorough literature review of technology adoption modelling, focusing on previous studies on electronic commerce acceptance. As a result of the literature review, the research framework is built upon a model based on UTAUT (Unified Theory of Acceptance and Use of Technology) and its evolution, UTAUT2, including two specific electronic commerce adoption factors: perceived risk and perceived trust. Chapter 4 deals with client and product segmentation methodologies used by experts. From the literature review, a wide range of classification variables is studied, and a shortlist of nine classification variables has been selected for inclusion in the research. The criteria for variable selection were their adequacy to electronic commerce characteristics, as well as adequacy to the sample characteristics. The nine variables have been classified in three groups: socio-demographic (gender, age, education level, income, family size and relationship status), behavioural (experience in electronic commerce and frequency of purchase) and psychographic (online purchase motivations) variables. The second half of chapter 4 is devoted to a review of the product classification criteria in electronic commerce. The review has led to the identification of a final set of fifteen groups of variables, whose combination offered a total of thirty-four possible outputs. However, due to the lack of empirical evidence in the context of electronic commerce, further investigation on the validity of this set of product classifications was deemed necessary. For this reason, chapter 5 proposes an empirical study to test the different product classification variables with 207 samples. A selection of product classifications including only those variables that are objective, able to identify distinct groups and not dependent on consumers’ point of view, led to a final classification of products which consisted on two groups of variables for the final empirical study. The combination of these two groups gave rise to four types of products: digital and non-digital goods, and digital and non-digital services. Chapter six characterizes the research –social, exploratory research– and presents the final research model and research hypotheses. The exploratory nature of the research becomes patent in instances where no prior empirical evidence on the influence of certain segmentation variables was found. Chapter six also includes the description of the measurement instrument used in the research, consisting of a total of 125 questions –and the measurement scales associated to each of them– as well as the description of the sample used for model validation (consisting of 817 Spanish residents). Chapter 7 is the core of the empirical analysis performed to validate the research model, and it is divided into two separate parts: description of the statistical techniques used for data analysis, and actual data analysis and results. The first part is structured in three different blocks: Partial Least Squares Method (PLS): the multi-variable analysis is a statistical method used to determine structural relationships of models and their predictive validity; Multi-group analysis: a set of techniques that allow comparing the outcomes of PLS analysis between two or more groups, by using one or more segmentation variables. More specifically, five comparison methods were used, which additionally gives the opportunity to assess the efficiency of each method. Determination of a priori undefined segments: in some cases, classification criteria did not necessarily exist for some segmentation variables, such as customer motivations. In these cases, the application of statistical classification techniques is required. For this study, two main classification techniques were used sequentially: principal component factor analysis –in order to reduce the number of variables– and cluster analysis. The application of the statistical methods to the models derived from the inclusion of the various segmentation criteria –for both clients and products–, led to the analysis of 128 different electronic commerce adoption models and 65 multi group comparisons. Finally, chapter 8 summarizes the conclusions from the research, divided into four parts: first, an assessment of the degree of achievement of the different research objectives is offered; then, methodological, theoretical and practical implications of the research are drawn; this is followed by a discussion on the results from the empirical study –based on the segmentation criteria for the research–; fourth, and last, the main limitations of the research –both empirical and theoretical– as well as future avenues of research are detailed.

DNS of a hot-wire anemometer in a turbulent channel flow

A body with a shape similar to a hot wire with its sheath, but no prongs, has been placed close to the wall of a turbulent channel at Re_tau = 600. The results of the channel flow, without the wire, agree with previous published ones, despite the modest resolution and domain size. A simplified, two-dimensional version of the wire at the same Reynolds number has been studied to compare the dynamic response of cold and hot wires, where a slightly bigger perturbation is seen in the hot case, but an almost identical dynamic response. The cold wire seems to be able to measure instantaneous velocity with total drag after proper calibration. Being a DNS, the complete description of the flow field around the wire is obtained.

Local numerical modelling of magnetoconvection and turbulence - implications for mean-field theories

During the last decades mean-field models, in which large-scale magnetic fields and differential rotation arise due to the interaction of rotation and small-scale turbulence, have been enormously successful in reproducing many of the observed features of the Sun. In the meantime, new observational techniques, most prominently helioseismology, have yielded invaluable information about the interior of the Sun. This new information, however, imposes strict conditions on mean-field models. Moreover, most of the present mean-field models depend on knowledge of the small-scale turbulent effects that give rise to the large-scale phenomena. In many mean-field models these effects are prescribed in ad hoc fashion due to the lack of this knowledge. With large enough computers it would be possible to solve the MHD equations numerically under stellar conditions. However, the problem is too large by several orders of magnitude for the present day and any foreseeable computers. In our view, a combination of mean-field modelling and local 3D calculations is a more fruitful approach. The large-scale structures are well described by global mean-field models, provided that the small-scale turbulent effects are adequately parameterized. The latter can be achieved by performing local calculations which allow a much higher spatial resolution than what can be achieved in direct global calculations. In the present dissertation three aspects of mean-field theories and models of stars are studied. Firstly, the basic assumptions of different mean-field theories are tested with calculations of isotropic turbulence and hydrodynamic, as well as magnetohydrodynamic, convection. Secondly, even if the mean-field theory is unable to give the required transport coefficients from first principles, it is in some cases possible to compute these coefficients from 3D numerical models in a parameter range that can be considered to describe the main physical effects in an adequately realistic manner. In the present study, the Reynolds stresses and turbulent heat transport, responsible for the generation of differential rotation, were determined along the mixing length relations describing convection in stellar structure models. Furthermore, the alpha-effect and magnetic pumping due to turbulent convection in the rapid rotation regime were studied. The third area of the present study is to apply the local results in mean-field models, which task we start to undertake by applying the results concerning the alpha-effect and turbulent pumping in mean-field models describing the solar dynamo.

Numerical CFD modelling of non-neutral atmospheric boundary layers for offshore wind resource assessment based on Monin-Obukhov theory

The presented works aim at proposing a methodology for the simulation of offshore wind conditions using CFD. The main objective is the development of a numerical model for the characterization of atmospheric boundary layers of different stability levels, as the most important issue in offshore wind resource assessment. Based on Monin-Obukhov theory, the steady k-ε Standard turbulence model is modified to take into account thermal stratification in the surface layer. The validity of Monin-Obukhov theory in offshore conditions is discussed with an analysis of a three day episode at FINO-1 platform.

Study of industrially relevant boundary layer and axisymmetric flows, including swirl and turbulence

Micropolar and RNG-based modelling of industrially relevant boundary layer and recirculating swirling flows is described. Both models contain a number of adjustable parameters and auxiliary conditions that must be either modelled or experimentally determined, and the effects of varying these on the resulting flow solutions is quantified. To these ends, the behaviour of the micropolar model for self-similar flow over a surface that is both stretching and transpiring is explored in depth. The simplified governing equations permit both analytic and numerical approaches to be adopted, and a number of closed form solutions (both exact and approximate) are obtained using perturbation and order of magnitude analyses. Results are compared with the corresponding Newtonian flow solution in order to highlight the differences between the micropolar and classical models, and significant new insights into the behaviour of the micropolar model are revealed for this flow. The behaviour of the RNG-bas based models for swirling flow with vortex breakdown zones is explored in depth via computational modelling of two experimental data sets and an idealised breakdown flow configuration. Meticulous modeling of upstream auxillary conditions is required to correctly assess the behavior of the models studied in this work. The novel concept of using the results to infer the role of turbulence in the onset and topology of the breakdown zone is employed.

A theory of process modelling impact

Process modelling – the design and use of graphical documentations of an organisation’s business processes – is a key method to document and use information about business processes. Still, despite current interest in process modelling, this research area faces essential challenges. Key unanswered questions concern the impact of process modelling in organisational practice, and the mechanisms through which impacts are developed. To answer these questions and to provide a better understanding of process modelling impact, I turn to the concept of affordances. Affordances describe the possibilities for goal-oriented action that technical objects offer to specified users. This notion has received growing attention from IS researchers. I report on my efforts to further develop the IS discipline’s understanding of affordances and impacts from informational objects, such as process models used by analysts for purposes of information systems analysis and design. Specifically, I seek to extend existing theory on the emergence and actualisation of affordances. I develop a research model that describes the process by which affordances are perceived and actualised and explain their dependence on available information and actualisation effort. I present my plans for operationalising and testing this research model empirically, and provide details about my design of a full-cycle, mixed methods study currently in progress.

General theory of stigmergy : modelling stigma semantics

Stigmergy is a biological term used when discussing a sub-set of insect swarm-behaviour describing the apparent organisation seen during their activities. Stigmergy describes a communication mechanism based on environment-mediated signals which trigger responses among the insects. This phenomenon is demonstrated in the behavior of ants and their food gathering process when following pheromone trails, where the pheromones are a form of environment-mediated communication. What is interesting with this phenomenon is that highly organized societies are achieved without an apparent management structure. Stigmergy is also observed in human environments, both natural and engineered. It is implicit in the Web where sites provide a virtual environment supporting coordinative contributions. Researchers in varying disciplines appreciate the power of this phenomenon and have studied how to exploit it. As stigmergy becomes more widely researched we see its definition mutate as papers citing original work become referenced themselves. Each paper interprets these works in ways very specific to the research being conducted. Our own research aims to better understand what improves the collaborative function of a Web site when exploiting the phenomenon. However when researching stigmergy to develop our understanding we discover a lack of a standardized and abstract model for the phenomenon. Papers frequently cited the same generic descriptions before becoming intimately focused on formal specifications of an algorithm, or esoteric discussions regarding sub-facets of the topic. None provide a holistic and macro-level view to model and standardize the nomenclature. This paper provides a content analysis of influential literature documenting the numerous theoretical and experimental papers that have focused on stigmergy. We establish that stigmergy is a phenomenon that transcends the insect world and is more than just a metaphor when applied to the human world. We present from our own research our general theory and abstract model of semantics of stigma in stigmergy. We hope our model will clarify the nuances of the phenomenon into a useful road-map, and standardise vocabulary that we witness becoming confused and divergent. Furthermore, this paper documents the analysis on which we base our next paper: Special Theory of Stigmergy: A Design Pattern for Web 2.0 Collaboration.

Modelling the intention to adopt cloud computing services : a transaction cost theory perspective

This paper uses transaction cost theory to study cloud computing adoption. A model is developed and tested with data from an Australian survey. According to the results, perceived vendor opportunism and perceived legislative uncertainty around cloud computing were significantly associated with perceived cloud computing security risk. There was also a significant negative relationship between perceived cloud computing security risk and the intention to adopt cloud services. This study also reports on adoption rates of cloud computing in terms of applications, as well as the types of services used.

A theory of process modelling affordances

This thesis studies how conceptual process models - that is, graphical documentations of an organisation's business processes - can enable and constrain the actions of their users. The results from case study and experiment indicate that model design decisions and people's characteristics influence how these opportunities for action are perceived and acted upon in practice.

Some recent advances in the theory of homogeneous isotropic turbulence

We review some advances in the theory of homogeneous, isotropic turbulence. Our emphasis is on the new insights that have been gained from recent numerical studies of the three-dimensional Navier Stokes equation and simpler shell models for turbulence. In particular, we examine the status of multiscaling corrections to Kolmogorov scaling, extended self similarity, generalized extended self similarity, and non-Gaussian probability distributions for velocity differences and related quantities. We recount our recent proposal of a wave-vector-space version of generalized extended self similarity and show how it allows us to explore an intriguing and apparently universal crossover from inertial- to dissipation-range asymptotics.

A closure theory of intermittency of turbulence

The variational approach to the closure problem of turbulence theory, proposed in an earlier article [Phys. Fluids 26, 2098 (1983); 27, 2229 (1984)], is extended to evaluate the flatness factor, which indicates the degree of intermittency of turbulence. Since the flatness factor is related to the fourth moment of a turbulent velocity field, the corresponding higher-order terms in the perturbation solution of the Liouville equation have to be considered. Most closure methods discard these higher-order terms and fail to explain the intermittency phenomenon. The computed flatness factor of the idealized model of infinite isotropic turbulence ranges from 9 to 15 and has the same order of magnitude as the experimental data of real turbulent flows. The intermittency phenomenon does not necessarily negate the Kolmogorov k−5/3 inertial range spectrum. The Kolmogorov k−5/3 law and the high degree of intermittency can coexist as two consistent consequences of the closure theory of turbulence. The Kolmogorov 1941 theory [J. Fluid Mech. 62, 305 (1974)] cannot be disqualified merely because the energy dissipation rate fluctuates.

Variational approach to the closure problem of turbulence theory

A new method is proposed to solve the closure problem of turbulence theory and to drive the Kolmogorov law in an Eulerian framework. Instead of using complex Fourier components of velocity field as modal parameters, a complete set of independent real parameters and dynamic equations are worked out to describe the dynamic states of a turbulence. Classical statistical mechanics is used to study the statistical behavior of the turbulence. An approximate stationary solution of the Liouville equation is obtained by a perturbation method based on a Langevin-Fokker-Planck (LFP) model. The dynamic damping coefficient eta of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution; this leads to a convergent integral equation for eta to replace the divergent response equation of Kraichnan's direct-interaction (DI) approximation, thereby solving the closure problem without appealing to a Lagrangian formulation. The Kolmogorov constant Ko is evaluated numerically, obtaining Ko = 1.2, which is compatible with the experimental data given by Gibson and Schwartz, (1963).