A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for the degree of Master of Science. 21 March 2016. / The problem of turbulent hydraulic fracturing is considered. Despite it being a known phenomenon, limited mathematical literature exists in this field. Prandtl’s mixing length model is utilised to describe the eddy viscosity and a mathematical model is developed for two distinct cases: turbulence where the kinematic viscosity is sufficiently small to be neglected and the case where it is not. These models allow for the examination of the fluid’s behaviour and its effect on the fracture’s evolution through time. The Lie point symmetries of both cases are obtained, and a wide range of analytical and numerical solutions are explored. Solutions of physical significance are calculated and discussed, and approximate solutions are constructed for ease of fracture estimation. The non-classical symmetries of these equations are also investigated. It was found that the incorporation of the kinematic viscosity within the modelling process was important and necessary. / MT2016

This thesis aims to answer the question, “What physical process dominates the formation of plasma irregularities, capable of directly or indirectly causing GPS L1 band scintillation, in polar cap plasma patches during magnetic storm conditions?.” A novel modelling technique utilising an ionospheric imaging algorithm is developed and used to elucidate the relative importance of the two most commonly discussed processes. These are the Gradient Drift Instability (GDI) and turbulence induced by electric field mapping to the ionosphere from the magnetosphere. The results show that in magnetic storm conditions, at times the GDI process is dominant, but that at other times turbulence may be as significant as the GDI in determining how the plasma within a polar cap patch behaves, possibly more so. This in turn suggests that further study of the turbulence process is necessary in order to fully understand how big a role it plays in causing GPS L1 band scintillation in the polar cap. The success of the modelling technique developed here shows the utility of ionospheric imaging as a tool for understanding physical problems of the ionosphere; efforts to improve it and to apply it in other contexts would be worthwhile.

Orientador: Gerson Francisco / Banca: Jayme Vicente de Luca Filho / Banca: Reynaldo Roberto Rosa / Resumo: Nesse trabalho procuramos desenvolver e avaliar o experimento de Reynolds. Fazemos, para tal, uma análise teórica do experimento, sob o cenário de Ruelle-Takens, destacando suas diferenças com o de Landau. Por fim, fazemos uma simulação numérica usando o software FIDAP, afim de comparar com os resultados preditos pela teoria e concluímos que, para condições não-assintóticas, o modelo de Ruelle-Takens funciona bem / Abstract: In this work, we develop the Reynolds' Experiment, under the Ruelle-Takens-Newhouse scenario, we show the difference between this scenario and the Landau scenario. We make a numerical simulation using the FIDAP software, comparing the results with the theoretical prediction. Our conclusion is that, for non assymptotics conditions, the Ruelle-Takens-Newhouse is in agreement with simulation / Mestre

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Song Hao = 湍流熱對流中的熱傳導測量 / 宋浩. / Thesis submitted in: December 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 65-70). / Text in English; abstracts in English and Chinese. / Song Hao = Tuan liu re dui liu zhong de re zhuan dao ce liang / Song Hao. / Abstract (in English) --- p.i / Abstract (in Chinese) --- p.ii / Acknowledgements --- p.iii / Table of Contents --- p.iv / List of Figures --- p.vi / List of Tables --- p.ix / Chapters / Chapter 1. --- Introduction --- p.1 / Chapter 2. --- Experimental Setup and Methods --- p.7 / Chapter 2.1 --- The rough boundary cell / Chapter 2.1.1 --- Convection Cell --- p.7 / Chapter 2.1.2 --- Temperature Probes --- p.9 / Chapter 2.1.3 --- Working Fluids --- p.10 / Chapter 2.1.4 --- Temperature-stabilized Box --- p.12 / Chapter 2.2 --- Rectangular and Square Cell --- p.13 / Chapter 2.3 --- The Big Cell --- p.13 / Chapter 2.4 --- The Thermal Measurements --- p.16 / Chapter 3. --- Nusselt Number Measurement in the Rough Boundary Cell --- p.19 / Chapter 3.1 --- Nu correction --- p.19 / Chapter 3.2 --- Non-Boussinesq Effect --- p.25 / Chapter 3.3 --- Experimental Results --- p.28 / Chapter 3.3.1 --- Water --- p.28 / Chapter 3.3.2 --- 1-Pentonal --- p.29 / Chapter 3.3.3 --- Dipropylene Glycol --- p.30 / Chapter 3.3.4 --- Triethylene Glycol --- p.32 / Chapter 3.4 --- Discussion on the Results --- p.34 / Chapter 3.4.1 --- Nusselt number --- p.34 / Chapter 3.4.2 --- Comparison with the smooth cell --- p.35 / Chapter 3.4.3 --- Normalized Nusselt number enhancement --- p.37 / Chapter 3.4.4 --- Nu~Pr-Relation --- p.40 / Chapter 3.4.5 --- Effects of Roughness Size --- p.43 / Chapter 3.4.6 --- Temperature Fluctuation Measurement --- p.44 / Chapter 4. --- Geometry Dependence of Nusselt Number and Temperature Fluctuation --- p.47 / Chapter 4.1 --- Nusselt Number Measurement --- p.48 / Chapter 4.2 --- Temperature Fluctuation's Dependence on Geometry --- p.50 / Chapter 5. --- Nusselt Number in the Big Cell --- p.53 / Chapter 5.1 --- The Big Cell --- p.53 / Chapter 5.2 --- Correction for Big Cell --- p.57 / Chapter 5.3 --- Results and Discussion --- p.61 / Chapter 6. --- Conclusions --- p.63 / References --- p.65

Xu, Xiaoqi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 91-92). / Abstracts in English and Chinese. / Xu, Xiaoqi. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Experimental measurements --- p.10 / Chapter 3 --- Review of earlier work --- p.13 / Chapter 3.1 --- Moments of {xP fT) --- p.13 / Chapter 3.2 --- Earlier results of Xr --- p.18 / Chapter 4 --- Probability Density Functions --- p.24 / Chapter 5 --- Scaling behavior of the moments --- p.43 / Chapter 5.1 --- Longest time scale in the problem --- p.43 / Chapter 5.2 --- The maximum order of the moment that can be calculated from a given set of data --- p.47 / Chapter 5.3 --- Moments of the surrogate XT --- p.50 / Chapter 5.4 --- The surrogate XT using water measurements and helium measurements --- p.59 / Chapter 5.4.1 --- PDFs comparison --- p.59 / Chapter 5.4.2 --- Scaling behavior of moments --- p.61 / Chapter 5.5 --- Investigation --- p.64 / Chapter 5.5.1 --- Helium measurements and water measurements --- p.64 / Chapter 5.5.2 --- Xr and Xfr using water measurements --- p.69 / Chapter 5.6 --- Conclusion --- p.73 / Chapter 6 --- Conditional statistics of temperature fluctuations --- p.74 / Chapter 6.1 --- Estimating the maximum order of moment --- p.74 / Chapter 6.2 --- Conditional temperature structure functions using Xfr and Xr --- p.78 / Chapter 6.3 --- Conditional temperature structure functions using x/r and XT at various r in the temperature derivatives --- p.83 / Chapter 7 --- Conclusion --- p.89 / Bibliography --- p.91

Cheng, Wai Chi = 利用殼模型對熱對流湍流的研究 / 鄭偉智. / "September 2007." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 72-74). / Text in English; abstracts in English and Chinese. / Cheng, Wai Chi = Li yong qiao mo xing dui re dui liu tuan liu de yan jiu / Zheng Weizhi. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- What is turbulence? --- p.1 / Chapter 1.1.1 --- The governing equation --- p.1 / Chapter 1.1.2 --- Richardson cascade and the K41 theory --- p.2 / Chapter 1.2 --- Thermal turbulence --- p.4 / Chapter 1.2.1 --- Entropy cascade and the Bolgiano-0bukhov scaling --- p.5 / Chapter 1.2.2 --- Interesting issues in turbulent convection --- p.7 / Chapter 1.2.3 --- Shell model of turbulence --- p.8 / Chapter 1.3 --- Motivations and structure of thesis --- p.11 / Chapter 2 --- Different scaling behavior in different shell models of turbulent convection --- p.13 / Chapter 2.1 --- Introduction --- p.13 / Chapter 2.1.1 --- Model dependence of scaling behavior --- p.15 / Chapter 2.1.2 --- Bolgiano scale and the dynamical significance of buoyancy --- p.26 / Chapter 2.2 --- Summary --- p.34 / Chapter 3 --- Scaling behavior in Brandenburg's model --- p.35 / Chapter 3.1 --- Introduction --- p.35 / Chapter 3.2 --- Scaling behavior in Brandenburg's model with different forcing mechanisms and parameters --- p.36 / Chapter 3.3 --- Summary --- p.43 / Chapter 4 --- Understanding the scaling behavior in Brandenburg's model --- p.45 / Chapter 4.1 --- Introduction --- p.45 / Chapter 4.2 --- Theory --- p.46 / Chapter 4.3 --- Summary --- p.48 / Chapter 5 --- Testing our theory against numerical results --- p.49 / Chapter 5.1 --- Introduction --- p.49 / Chapter 5.2 --- Testing of the hierarchical structure --- p.49 / Chapter 5.3 --- "Testing ζp, and тp with our prediction" --- p.52 / Chapter 5.4 --- Scaling behavior with fixed entropy transfer rate --- p.55 / Chapter 5.5 --- Summary --- p.57 / Chapter 6 --- Distinguishing feature for active and passive scalars --- p.59 / Chapter 6.1 --- Introduction --- p.59 / Chapter 6.2 --- Distinguishing feature of active and passive scalar --- p.60 / Chapter 6.3 --- Scaling behavior of the auxiliary scalar --- p.66 / Chapter 6.4 --- Summary --- p.69 / Chapter 7 --- Conclusion --- p.70 / Bibliography --- p.72 / A Constraint equations on the parameters in the extended GOY model --- p.75

Wind energy is one of the significant sources of renewable energy, yet a number of challenges preclude optimal operation of wind plants. Research is warranted in order to minimize the power losses and improve the productivity of wind plants. Here, a framework combining turbulence theory and data mining techniques is built to elucidate physics and mechanisms driving the energy extraction of the wind plants under a number of atmospheric/operating conditions. The performance of wind turbines is subjected to adverse effects caused by wake interactions. Therefore, it is crucial to understand wake-to-wake interactions as well as wake-to-atmospheric boundary layer interactions. Experimental and numerical data sets are examined in order to provide descriptions of the wakes and extract relevant features. As wakes merge, it is of interest to observe characteristics within the turbulent velocity signal obtained via wind tunnel experiments. Higher order moments, structure functions, intermittency and multifractality analysis are investigated to distinguish the flow dynamics. In this manner, considered approaches highlight the flow deceleration induced by the wind turbines, which subsequently changes the energy transfer rate imposed by the coherent eddies, and adapt the equilibrium range in the energy cascade. Also, wind turbines induce scale interactions and cause the intermittency that lingers at large and small scales. When wind plants interact dynamically with small scales, the flow becomes highly intermittent and multifractality is increased, especially near the rotor. Multifractality parameters, including the Hurst exponent and the combination factor, show the ability to describe the flow state in terms of its development. Based on Markov theory, the time evolution of the probability density function of the velocity is described via the Fokker-Planck equation and its Kramers-Moyal coefficients. Stochastic analysis proves the non-universality of the turbulent cascade immediate to the rotor, and the impact of the generation mechanism on flow cascade. Classifying the wake flow based the velocity and intermittency signs emphasizes that a negative correlation is dominant downstream from the rotor. These results reflect large-scale organization of the velocity-intermittency events corresponding to a recirculation region near the hub height and bottom tip. A linear regression approach based on the Gram-Charlier series expansion of the joint probability density function successfully models the contribution of the second and fourth quadrants. Thus, the model is able to predict the imbalance in the velocity and intermittency contribution to momentum transfer. Via large eddy simulations, the structure of the turbulent flow within the array under stratified conditions is quantified through the use of the Reynolds stress anisotropy tensor, proper orthogonal decomposition and cluster-based modeling. Perturbations induced by the turbine wakes are absorbed by the background turbulence in the unstable and neutrally stratified cases. Contrary, the flow in the stable stratified case is fully dominated by the presence of turbines and extremely influenced by the Coriolis force. Also, during the unstable period the turbulent kinetic energy is maximum. Thus, leading to fast convergence of the cumulative energy with only few modes. Reynolds stress anisotropy tensor reveals that under unstable thermal stratification the turbulence state tends to be more isotropic. The turbulent mixing due to buoyancy determines the degree of anisotropy and the energy distribution between the flow layers. The wakes of the turbines display large degree of anisotropy due to the correlation with the turbulent kinetic energy production. A combinatorial technique merging image segmentation via K-Means clustering and colormap of the barycentric map is posed. Clustering aids in extracting identical features from the spatial distribution of anisotropy colormap images by minimizing the sum of squared error over all clusters. Clustering also enables to highlight the wake expansion and interaction as produced by the wind turbines as a function of thermal stratification. A cluster-based reduced-order dynamical model is proposed for flow field and passive scalars; the model relies on full-state measurements. The dynamical behavior is predicted through the cluster transition matrix and modeled as a Markov process. The geometric nature of the attractor shows the ability to assess the quality of the clustering and identify transition regions. Periodical trends in the cluster transition matrix characterize the intrinsic periodical behavior of the wake. The modeling strategy points out a feasible path for future design and control that can be used to maximize power output. In addition, characterization of intermittency with power integration model can allow for power fluctuation arrangement/prediction in wind plants.

No description available.

Master of Engineering / Turbulent propagating premixed flames are encountered in spark ignition engines, gas turbines, industrial burners, as well as in vented gas explosions. In all these applications, the flame fronts interact with complex solid boundaries which not only distort the flame structure but directly affect the propagation rate in ways that are not yet fully understood. This thesis aims to provide both a quantitative and qualitative understanding of the link between overpressure, flame front wrinkling and turbulence levels generated in the propagating medium. This is an issue of importance for the provision of improved sub-models for the burning rates of premixed flames. An experimental chamber was constructed where controlled premixed flames were ignited from rest to propagate past solid obstacles and/or baffle plates strategically positioned in the chamber. Laser Doppler Anemometry was used to measure the velocity field and turbulence fields while pressure transducers were used to obtain pressure-time traces. In addition to this Laser-Induced Fluorescence of the Hydroxyl radical is was to image the flame front as it consumes the unburnt fuel captured in the re-circulation zone behind the main obstruction. The thesis reports on the effects of various parameters such as the inclusion of grids and obstructions, blockage ratio, and repeated obstacles to explore possible correlations between the pressure and the flow-fields. Pressure, velocity and LIF images were correlated and analysed to prove the significance of grid location and number on overall turbulence intensity. Corresponding flow field parameters such as flame front wrinkling, peak overpressure and RMS all combine to conclusively demonstrate their interaction and influence to turbulence intensity. By progressively positioning more grids further downstream, consequent rises in the flow field parameters and the establishment of positive trends indicates the overall significance of kernel development and flow disturbances in relation to turbulence generation.