Chan, Tak Shing = 粗糙表面的熱湍流對流的Nusselt數和雷諾數的測量 / 陳德城. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 63-67). / Abstracts in English and Chinese. / Chan, Tak Shing = Cu cao biao mian de re tuan liu dui liu de Nusselt shu he Leinuo shu de ce liang / Chen Decheng. / Table of Contents --- p.v / List of Figures --- p.xi / List of Tables --- p.xii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- What is turbulence ? --- p.1 / Chapter 1.2 --- Rayleigh Benard convection system --- p.3 / Chapter 1.2.1 --- Oberbeck-Boussinesq approximation and equations of Rayleigh- Benard system --- p.5 / Chapter 1.2.2 --- Some coherent structures of Rayleigh-Benard convection system --- p.7 / Chapter 1.3 --- Motivation --- p.8 / Chapter 2 --- Experimental methods and setups --- p.12 / Chapter 2.1 --- Convection cell --- p.12 / Chapter 2.2 --- Temperature measurement --- p.15 / Chapter 2.3 --- Experimental techniques --- p.16 / Chapter 2.3.1 --- Heat leakage prevention --- p.16 / Chapter 2.3.2 --- Water absorption of Dipropylene Glycol --- p.21 / Chapter 2.3.3 --- Particle Image Velocimetry --- p.22 / Chapter 3 --- Heat flux measurement --- p.25 / Chapter 3.1 --- Water Results --- p.26 / Chapter 3.1.1 --- Experimental procedures --- p.26 / Chapter 3.1.2 --- Heat leakage/ heat absorption estimation --- p.27 / Chapter 3.1.3 --- Results and discussions --- p.29 / Chapter 3.2 --- Dipropylene Glycol Results --- p.32 / Chapter 3.2.1 --- Experimental procedures --- p.32 / Chapter 3.2.2 --- Heat leakage/ heat absorption estimation --- p.33 / Chapter 3.2.3 --- Result and discussions --- p.34 / Chapter 3.3 --- More discussion --- p.41 / Chapter 4 --- Large scale circulation and Reynolds number measurement --- p.44 / Chapter 4.1 --- Flow pattern of turbulent Rayleigh-Benard convection over rough plates --- p.46 / Chapter 4.2 --- Reynolds number measurement --- p.48 / Chapter 4.2.1 --- Reynolds number determined from oscillation of temper- ature signals --- p.48 / Chapter 4.2.2 --- Reynolds number determined from velocity measurement near sidewall --- p.55 / Chapter 5 --- Conclusion --- p.61 / Chapter 5.1 --- Conclusion --- p.61 / Bibliography --- p.63

Ko, Tze Cheung = 均勻湍流對流殼模型內的熱傳送及雷諾數標度律 / 高子翔. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 76-78). / Abstracts in English and Chinese. / Ko, Tze Cheung = Jun yun tuan liu dui liu ke mo xing nei de re chuan song ji Leinuo shu biao du lü / Gao Zixiang. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Description of Rayleigh-Benard convection --- p.2 / Chapter 1.2 --- Interesting issues in turbulent Rayleigh-Benard convection --- p.3 / Chapter 2 --- Earlier studies of heat transport in Rayleigh-Benard convection --- p.6 / Chapter 2.1 --- Marginal stability arguments --- p.7 / Chapter 2.2 --- The Chicago mixing zone model --- p.8 / Chapter 2.3 --- Shraiman and Siggia theory --- p.10 / Chapter 2.4 --- Grossmann and Lohse theory --- p.12 / Chapter 2.4.1 --- Estimating the kinetic dissipation rates due to boundary layer and bulk --- p.13 / Chapter 2.4.2 --- Estimating the thermal dissipation rates due to boundary layer and bulk --- p.13 / Chapter 2.4.3 --- The four regimes --- p.15 / Chapter 2.5 --- The asymptotic limit of very high Ra --- p.17 / Chapter 3 --- The shell model used --- p.20 / Chapter 3.1 --- Background of shell models of turbulence --- p.20 / Chapter 3.2 --- The model used --- p.22 / Chapter 3.2.1 --- The Brandenburg model --- p.22 / Chapter 3.2.2 --- The requirement of a large scale drag term --- p.23 / Chapter 3.3 --- Previous work on the Brandenburg model --- p.24 / Chapter 4 --- "Definitions of Ra, Nu, and Re and two exact results" --- p.26 / Chapter 4.1 --- Heat transport study using shell model --- p.26 / Chapter 4.2 --- Two exact results --- p.28 / Chapter 5 --- Results and discussions --- p.29 / Chapter 5.1 --- Parameters used --- p.29 / Chapter 5.2 --- "Nu(Ra,Pr) and Re(Ra,Pr) scaling results" --- p.29 / Chapter 5.3 --- "Scaling results of ε, εdrag and x" --- p.32 / Chapter 5.4 --- Physical meaning of the drag term --- p.35 / Chapter 5.5 --- Understanding the dependence of ε on Re --- p.36 / Chapter 5.6 --- Understanding the dependence of x and εdrag on Re and Pr --- p.40 / Chapter 5.7 --- The form of the added drag term --- p.41 / Chapter 6 --- Possible changes of Nu and Re due to non-Boussinesq effects --- p.43 / Chapter 6.1 --- Background --- p.43 / Chapter 6.2 --- Method of study --- p.44 / Chapter 6.3 --- Effects due to the temperature dependence of kinematic viscosity --- p.45 / Chapter 6.4 --- Effects due to the temperature dependence of thermal diffusivity --- p.50 / Chapter 6.5 --- Effects due to the temperature dependence of volume expansion coefficient --- p.55 / Chapter 6.6 --- Understanding the scaling behavior under non-Boussinesq effects --- p.61 / Chapter 6.6.1 --- Scaling behavior of x on Re --- p.61 / Chapter 6.6.2 --- Scaling behavior of εtotai on Re --- p.65 / Chapter 6.6.3 --- Scaling behavior of Nu and Re on Ra --- p.66 / Chapter 6.7 --- Summary and future work --- p.68 / Chapter 7 --- Conclusion --- p.69 / Chapter A --- Height independence of Nu for homogeneous turbulent convection with periodic boundary conditions --- p.73 / Chapter B --- "Height independence of (uz)A,t for homogeneous turbulent convection with periodic boundary conditions" --- p.75 / Bibliography --- p.76

Tese de mestrado integrado. Engenharia Química. Faculdade de Engenharia. Universidade do Porto. 2010

La méthode Lattice Boltzmann (LBM) est une alternative viable à la simulation directe (DNS) des équations de Navier et Stokes, particulièrement en Mécanique des Fluides. La clé de son succès se situe dans l’exactitude, la simplicité et la propriété conforme de parallélisation de l’algorithme stream-collision. L’inconvénient majeur de cette méthode provient de la limitation aux mailles cubiques spatialement uniformes. Pour y remédier, plusieurs extensions de la LBM aux mailles non-homogènes ont été proposées. Ces techniques ont été revisitées dans la thèse. La revue de maillage montre que la meilleure technique de raffinement remplit certains critères: elle doit satisfaire aux lois de conservation et doit être stable. Elle suggère l’adoption des approches de type Volumes Finis (FV LBM). Une revue de ces techniques a permis de conclure que bien qu’intéressantes, elles présentent de nombreux inconvénients. Cette étude présente une méthode de discrétisation de type FV pour Lattice Boltzmann de haute précision et avec un faible coût de calcul. Afin d’évaluer la performance de la méthode FV nous effectuons une comparaison systématique axée sur la précision et les performances de calcul avec la méthode de Lattice Boltzmann standard (ST). En particulier, nous cherchons à clarifier si et dans quelles conditions l’algorithme proposé et plus généralement tout algorithme FV peut être considéré comme la méthode de choix pour les simulations en Mécanique des Fluides. Nous présentons la première simulation des écoulements convectifs à haut nombre de Rayleigh réalisée avec une méthode Lattice Boltzmann de type FV avec des mailles réduites près de la paroi. / Lattice Boltzmann Method (LBM) has become a viable alternative to Navier-Stokes Direct Numerical Simulations (DNS) in fluid dynamics research. The key of this success is the accuracy/simplicity and parallelization compliant property of the stream-collision algorithm. One shortcoming however, comes from the limitation to spatially uniform cubic grids. To overcome this, several LBM extension to non-homogeneous grids have been proposed. These techniques have been reviewed in this thesis. Such review suggests that a better refinement technique should fulfill some properties: obey conservation laws and be stable. This suggests a pathway to adopt Finite Volume approaches (FV LBM). A review on such volumetric approach to LBM concludes that although interesting, at present such methods suffer from several drawbacks. In this study, a new FV discretization method for the Lattice Boltzmann equation that combines high accuracy with limited computational cost is presented. In order to assess the performance of the FV method we carry out a systematic comparison, focused on accuracy and computational performances, with the standard streaming (ST) Lattice Boltzmann equation algorithm. In particular we aim at clarifying whether and in which conditions the proposed algorithm, and more generally any FV algorithm, can be taken as the method of choice in fluid-dynamics LB simulations. We report the first successful simulation of high-Rayleigh number convective flow performed by a Lattice Boltzmann FV based algorithm with wall grid refinement.

This dissertation aims to use theory and numerical simulations to quantify the propagation of fronts, which consist of autocatalytic reaction fronts, fronts with feedback and pattern forming fronts in Rayleigh-Bénard convection. The velocity and geometry of fronts are quantified for fronts traveling through straight parallel convection rolls, spatiotemporally chaotic rolls, and weakly turbulent rolls. The front velocity is found to be dependent on the competing influence of the orientation of the convection rolls and the geometry of the wrinkled front interface which is quantified as a fractal having a non-integer box-counting dimension. Front induced solutal and thermal feedback to the convective flow field is then studied by solving an exothermic autocatalytic reaction where the products and the reactants can vary in density. A single self-organized fluid roll propagating with the front is created by the solutal feedback while a pair of propagating counterrotating convection rolls are formed due to heat release from the reaction. Depending on the relative change in density induced by the solutal and thermal feedback, cooperative and antagonistic feedback scenarios are quantified. It is found that front induced feedback enhances the front velocity and reactive mixing length and induces spatiotemporal oscillations in the front and fluid dynamics. Using perturbation expansions, a transition in symmetry and scaling behavior of the front and fluid dynamics for larger values of feedback is studied. The front velocity, flow structure, front geometry and reactive mixing length scales for a range of solutal and thermal feedback are quantified. Lastly, pattern forming fronts of convection rolls are studied and the wavelength and velocity selected by the front near the onset of convective instability are investigated. This research was partially supported by DARPA Grant No. HR0011-16-2-0033. The numerical computations were done using the resources of the Advanced Research Computing center at Virginia Tech. / Doctor of Philosophy / Quantification of transport of reacting species in the presence of a flow field is important in many problems of engineering and science. A front is described as a moving interface between two different states of a system such as between the products and reactants in a chemical reaction. An example is a line of wildfire which separates burnt and fresh vegetation and propagates until all the fresh vegetation is consumed. In this dissertation the propagation of reacting fronts in the presence of convective flow fields of varying complexity is studied. It is found that the spatial variations in a convective flow field affects the burning and propagation of fronts by reorienting the geometry of the front interface. The velocity of the propagating fronts and its dependence on the spatial variation of the flow field is quantified. In certain scenarios the propagating front feeds back to the flow by inducing a local flow that interacts with the background convection. The rich and emergent dynamics resulting from this front induced feedback is quantified and it is found that feedback enhances the burning and propagation of fronts. Finally, the properties of pattern forming fronts are studied for fronts which leave a trail of spatial structures behind as they propagate for example in dendritic solidification and crystal growth. Pattern forming fronts of convection rolls are studied and the velocity of the front and spatial distribution of the patterns left behind by the front is quantified. This research was partially supported by DARPA Grant No. HR0011-16-2-0033. The numerical computations were done using the resources of the Advanced Research Computing center at Virginia Tech.

There are still many open questions regarding spatiotemporal chaos although many well developed theories exist for chaos in time. Rayleigh-B'enard convection is a paradigmatic example of spatiotemporal chaos that is also experimentally accessible. Discoveries uncovered using numerics can often be compared with experiments which can provide new physical insights. Lyapunov diagnostics can provide important information about the dynamics of small perturbations for chaotic systems. Covariant Lyapunov vectors reveal the true direction of perturbation growth and decay. The degree of hyperbolicity can also be quantified by the covariant Lyapunov vectors. To know whether a dynamical system is hyperbolic is important for the development of a theoretical understanding. In this thesis, the degree of hyperbolicity is calculated for chaotic Rayleigh-B'enard convection. For the values of the Rayleigh number explored, it is shown that the dynamics are non-hyperbolic. The spatial distribution of the covariant Lyapunov vectors is different for the different Lyapunov vectors. Localization is used to quantify this variation. The spatial localization of the covariant Lyapunov vectors has a decreasing trend as the order of the Lyapunov vector increases. The spatial localization of the covariant Lyapunov vectors are found to be related to the instantaneous Lyapunov exponents. The correlation is stronger as the order of the Lyapunov vector decreases. The covariant Lyapunov vectors are also computed using a spectral element approach. This allows an exploration of the covariant Lyapunov vectors in larger domains and for experimental conditions. The finite conductivity and finite thickness of the lateral boundaries of an experimental convection domain is also studied. Results are presented for the variation of the Nusselt number and fractal dimension for different boundary conditions. The fractal dimension changes dramatically with the variation of the finite conductivity. / Ph. D. / There are still many open questions regarding chaos. Rayleigh-Bènard convection is a type of natural convection which occurs when a fluid is placed between a hot bottom plate and a cold top plate. Rayleigh-Bènard convection is a classical model to explore chaos in space and time. The major application of Rayleigh-Bènard convection is weather prediction which is an extremely difficult problem of intense interest. The governing equations can only be solved using supercomputing resources. The main reason for this difficulty is the presence of a very large number of degrees of freedom that may influence the weather. To reduce the number of degrees of freedom by only including ones that contribute significantly is a difficult problem. In this thesis, vectors describing the growth of disturbances have been calculated for Rayleigh-Bènard convection. These vectors give us information about which regions in space are more important than others. For weather example, the knowledge of these vectors would tell us which regions are important. With this information, scientists and engineers can focus on the important regions and possibly improve their long term predictions. These vectors also yield the number of degrees of freedom to characterize a chaotic system, on average. In this thesis, this number is also explored for Rayleigh-Bènard convection. This thesis extends the calculation of these vectors to a realistic fluid model which gives us new insights into fundamental questions about chaos in space and time.

In many systems, instabilities can lead to time-dependent behavior, and instabilities can act as mechanisms for sustained chaos; an understanding of the dynamical modes governing instability is thus essential for prediction and/or control in such systems. In this thesis work, we have developed an approach toward characterizing instabilities quantitatively, from experiments on the prototypical Rayleigh-Bénard convection system. We developed an experimental technique for preparing a given convection pattern using rapid optical actuation of pressurized SF6, a greenhouse gas. Real-time analysis of convection patterns was developed as part of the implementation of closed-loop control of straight roll patterns. Feedback control of the patterns via actuation was used to guide patterns to various system instabilities. Controlled, spatially localized perturbations were applied to the prepared states, which were observed to excite the dominant system modes. We extracted the spatial structure and growth rates of these modes from analysis of the pattern evolutions. The lifetimes of excitations were also measured, near a particular instability; a critical wavenumber was found from the observed dynamical slowing near the bifurcation. We will also describe preliminary results of using a state estimation algorithm (LETKF) on experimentally prepared non-periodic patterns in a cylindrical convection cell.

Une étude théorique et expérimentale de la convection de Rayleigh-Bénard pour un fluide non-Newtonien rhéofluidifiant a été effectuée. L’approche théorique consiste en une analyse linéaire et faiblement non linéaire de l’instabilité thermo-convective d’une couche horizontale d’un fluide non-Newtonien, d’étendue supposée infinie dans le plan horizontal, chauffée par le bas et refroidie par le haut. Le comportement rhéofluidifiant est décrit par le modèle de Carreau. Pour ce modèle, les conditions critiques d’instabilité du régime conductif sont les mêmes que pour un fluide Newtonien. L’objectif de l’analyse faiblement non linéaire consiste à déterminer d’une part la valeur critique du degré de rhéofluidification à partir duquel la bifurcation primaire devient sous critique et d’autre part l’influence de rhéofluidification sur la sélection du motif de convection au voisinage des conditions critiques, en tenant compte d’un éventuel glissement à la paroi, d’une conductivité thermique finie de celle-ci et de la thermodépendance de la viscosité. Les conséquences sur le champ de viscosité et l’évolution du nombre de Nusselt sont caractérisées. L’approche expérimentale consiste à visualiser par ombroscopie les motifs de convection qui se développent dans une cellule cylindrique. Deux rapports d’aspect ont été considérés : AR = 3 et AR = 4. Les fluides utilisés sont des solutions aqueuses de Xanthan à différentes concentrations. L’influence du degré de rhéofluidification combiné avec la thermodépendance de la viscosité sur le domaine de stabilité des rouleaux et des hexagones ainsi que sur la zone de transitions rouleaux hexagones est mise en évidence / Theoretical and experimental study of Rayleigh-Bénard convection in a non-Newtonian shear-thinning fluid was performed. The theoretical approach consists in a linear and a weakly nonlinear of thermo-convective instability in a horizontal layer of a non-Newtonian fluid, assumed infinite in extent, heated from below and cooled from above. The rheological behavior of the fluid is described by the Carreau model. For this rheological model, the critical threshold is the same as for a Newtonian fluid. The objective of the weakly non linear analysis is to determine on one hand the critical value of the shear-thinning degree above which the bifurcation becomes subcritical and on the other hand, the influence of shear-thinning effects on the pattern selection near the onset, taking into account the possibility of wall slip, a finite thermal conductivity of the walls as well as the thermo-dependency of the viscosity. The impact on the viscosity field and on the evolution of the Nusselt number are characterized. The experimental approach consists in visualizing the convection patterns using the shadowgraph method in a cylindrical cell. Two aspect ratios were considered : AR = 3 and AR = 4. The fluids used are aqueous solutions of xanthan-gum at different concentrations. The influence of shear-thinning effects combined with the thermo-dependency of the viscosity on the stability domain of rolls and hexagons as well as on the transition between rolls and hexagons is highlighted

Dans ces travaux , nous avons deux principaux objectifs physique et numérique. Le problème physique consiste à trouver la solution de Rayleigh-Bénard pour des fluides newtoniens et non-newtoniens. Dans la présente étude, une présentation générale des résultats de la convection de Rayleigh-Bénard (RBC) est donnée dans le cas des fluides newtoniens et non-newtoniens tels que des fluides rhéofluidifiants modélisés par la loi puissance et des fluides viscoplastiques (fluides de Bingham, Herschel-Bulkley et Casson), en régime permanent et transitoire. Dans le cas des fluides viscoplastiques, les modèles macroscopiques ne prenant pas bien en compte la réalité physique de la contrainte seuil ont fait l'objet d'une modélisation. Un modèle mesoscopique proposé par Hébraud et Lequeux a été utilisé. Le problème numérique consiste à développer la méthode de résolution PGD (Proper Generalized Decomposition) pour résoudre les modèles non linéaires couplés transitoires, dans le cas du problème de Rayleigh-Bénard. Cette méthode est également utilisée pour résoudre le problème RBC paramétrique en y ajoutant quelques variables physiques comme coordonnées supplémentaires. Par ailleurs, dans le cas des fluides non-newtoniens, nous avons utilisé la PGD pour résoudre les équations mesoscopiques et macroscopiques couplées. / In this research work we are looking for two main physical and numerical purposes. The physical problem is to find the solution of Rayleigh Bénard convection for several conditions dependent on fluid thermo-physical properties such as temperature, viscosity and initial and boundary conditions. Continuing previous research works in this study we have provided the results of Rayleigh Bénard convection for Newtonian, Power-law and viscoplastic fluids (Bingham, Herschel-Bulkley and Casson) and for steady state and transient conditions. We also solve this problem for Nano and soft glassy materials. In some cases the results are interesting not only as a part of the Rayleigh Bénard convection analysis but also on a larger scale as a part of the heat transfer and mechanical fluid analysis such as viscoplastic and soft glassy material studies. Numerically, it was interesting to develop Proper Generalized Decomposition (PGD) method for solving transient coupled non-linear models, in particular the one related to the Rayleigh–Bénard flow. This model also was used to solve RBC problem parametrically by adding some physical properties as extra coordinates. For soft glassy material we used PGD to connect micro and macro equations together.

La convection thermique dans les objets naturels de grande taille est associée à de fortes variations de la pression, hydrostatique au premier ordre. C’est le cas pour l’atmosphère de la Terre (et d’autres planètes), les planètes gazeuses géantes, les étoiles, mais aussi l’intérieur des planètes telluriques. De part l’importance des effets de compressibilité, l’approximation de Boussinesq n’y est pas vérifiée et d’autres modèles, comportant également des approximations, sont utilisés : les modèles anélastiques. Toutefois, peu d’expériences ont été réalisées pour les vérifier. Cette thèse présente une expérience dont les paramètres ont été optimisés afin d’obtenir des effets de compressibilité importants en laboratoire. Pour ce faire, une gravité apparente forte est obtenue à l’aide d’une centrifugeuse et du xénon gazeux est utilisé, nous permettant d’atteindre un nombre de dissipation significatif. Ces expériences ont permis l’observation en laboratoire d’un gradient adiabatique de 3 K/cm et d’un exposant de 0,3 pour la loi de puissance caractérisant le transfert thermique turbulent entre le nombre de Nusselt et le nombre de Rayleigh superadiabatique.L’étude des fluctuations de pression et de température montrant que l’écoulement est quasi-geostrophique dû à la forte rotation imposée par la centrifugeuse, un modèle anélastique quasi-géostrophique est développé afin de réaliser des simulations numériques bidimensionnelles relatives à l’expérience. / In large natural objects, thermal convection is associated with large pressure differences, mainly due to hydrostatic balance. This is true in the atmosphere of the Earth (and other planets), in gas giant planets, in stars, but also in the interior of telluric planets. Boussinesq approximation is not valid owing to large compressibility effects, and other approximate models can be used to model these objects, like the anelastic approximation. However, very few experiments have been performed to assess these models. In the present PhD thesis, an experiment is shown, with parameters designed to maximize compressibility effects in a laboratory. In this perspective, an enhanced apparent gravity is obtained using a centrifuge, and Xenon gas is used, allowing us to reach a significant dissipation parameter. In our experiments, we have observed an adiabatic gradient of 3~K/cm and the power law between the superadiabatic Rayleigh number and the Nusselt number measuring the turbulent heat transfer is characterized by an exponent 0.3.Measurements of temperature and pressure fluctuations show that the flow is quasi-geostrophic as a result of the strong rotation rate of the centrifuge. An anelastic, quasi-geostrophic model has then been developed and solved numerically in the same configuration as the experiments.