There are two critical research needs for the study of hydrocarbon combustion in high speed flows: 1) combustion diagnostics with adequate temporal and spatial resolution, and 2) mathematical techniques that can extract key information from large datasets. The goal of this work is to address these needs, respectively, by the use of high speed and multi-perspective chemiluminescence and advanced mathematical algorithms. To obtain the measurements, this work explored the application of high speed chemiluminescence diagnostics and the use of fiber-based endoscopes (FBEs) for non-intrusive and multi-perspective chemiluminescence imaging up to 20 kHz. Non-intrusive and full-field imaging measurements provide a wealth of information for model validation and design optimization of propulsion systems. However, it is challenging to obtain such measurements due to various implementation difficulties such as optical access, thermal management, and equipment cost. This work therefore explores the application of FBEs for non-intrusive imaging to supersonic propulsion systems. The FBEs used in this work are demonstrated to overcome many of the aforementioned difficulties and provided datasets from multiple angular positions up to 20 kHz in a supersonic combustor. The combustor operated on ethylene fuel at Mach 2 with an inlet stagnation temperature and pressure of approximately 640 degrees Fahrenheit and 70 psia, respectively. The imaging measurements were obtained from eight perspectives simultaneously, providing full-field datasets under such flow conditions for the first time, allowing the possibility of inferring multi-dimensional measurements. Due to the high speed and multi-perspective nature, such new diagnostic capability generates a large volume of data and calls for analysis algorithms that can process the data and extract key physics effectively. To extract the key combustion dynamics from the measurements, three mathematical methods were investigated in this work: Fourier analysis, proper orthogonal decomposition (POD), and wavelet analysis (WA). These algorithms were first demonstrated and tested on imaging measurements obtained from one perspective in a sub-sonic combustor (up to Mach 0.2). The results show that these algorithms are effective in extracting the key physics from large datasets, including the characteristic frequencies of flow—flame interactions especially during transient processes such as lean blow off and ignition. After these relatively simple tests and demonstrations, these algorithms were applied to process the measurements obtained from multi-perspective in the supersonic combustor. compared to past analyses (which have been limited to data obtained from one perspective only), the availability of data at multiple perspective provide further insights into the flame and flow structures in high speed flows. In summary, this work shows that high speed chemiluminescence is a simple yet powerful combustion diagnostic. Especially when combined with FBEs and the analyses algorithms described in this work, such diagnostics provide full-field imaging at high repetition rate in challenging flows. Based on such measurements, a wealth of information can be obtained from proper analysis algorithms, including characteristic frequency, dominating flame modes, and even multi-dimensional flame and flow structures. / Ph. D.

In this thesis we conduct a numerical study of the 1D viscous Burgers' equation and several Reduced Order Models (ROMs) over a range of parameter values. This study is motivated by the need for robust reduced order models that can be used both for design and control. Thus the model should first, allow for selection of optimal parameter values in a trade space and second, identify impacts from changes of parameter values that occur during development, production and sustainment of the designs. To facilitate this study we apply a Finite Element Method (FEM) and where applicable, the Group Finite Element Method (GFE) due its demonstrated stability and reduced complexity over the standard FEM. We also utilize Proper Orthogonal Decomposition (POD) as a model reduction technique and modifications of POD that include Global POD, and the sensitivity based modifications Extrapolated POD and Expanded POD. We then use a single baseline parameter in the parameter range to develop a ROM basis for each method above and investigate the error of each ROM method against a full order "truth" solution for the full parameter range. / Master of Science

Thesis (PhD)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Numerical simulations are widely used for predicting and optimising the exploitation of aquifers. They are also used to determine certain physical parameters, for example soil conductivity, by inverse calculations, where the model parameters are changed until the model results correspond optimally to measurements taken on site. The Richards’ equation describes the movement of an unsaturated fluid through porous media, and is characterised as a non-linear partial differential equation. The equation is subject to a number of parameters and is typically computationally expensive to solve. To determine the parameters in the Richards’ equation, inverse modelling studies often need to be undertaken. In these studies, the parameters of a numerical model are varied until the numerical response matches a measured response. Inverse modelling studies typically require 100’s of simulations, which implies that parameter optimisation in unsaturated case studies is common only in small or 1D problems in the literature. As a solution to overcome the computational expense incurred in inverse modelling, the use of Proper Orthogonal Decomposition (POD) as a Reduced Order Modelling (ROM) method is proposed in this thesis to speed-up individual simulations. An explanation of the Finite Element Method (FEM) is given using the Galerkin method, followed by a detailed explanation of the Galerkin POD approach. In the development of the Galerkin POD approach, the method of reducing matrices and vectors is shown, and the treatment of Neumann and Dirichlet boundary values is explained. The Galerkin POD method is applied to two case studies. The first case study is the Kogelberg site in the Table Mountain Group near Cape Town in South Africa. The response of the site is modelled at one well over the period of 2 years, and is assumed to be governed by saturated flow, making it a linear problem. The site is modelled as a 3D transient, homogeneous site, using 15 layers and ≈ 20000 nodes, using the FEM implemented on the open-source software FreeFem++. The model takes the evapotranspiration of the fynbos vegetation at the site into consideration, allowing the calculation of annual recharge into the aquifer. The ROM is created from high-fidelity responses taken over time at different parameter points, and speed-up times of ≈ 500 are achieved, corresponding to speed-up times found in the literature for linear problems. The purpose of the saturated groundwater model is to demonstrate that a POD-based ROM can approximate the full model response over the entire parameter domain, highlighting the excellent interpolation qualities and speed-up times of the Galerkin POD approach, when applied to linear problems. A second case study is undertaken on a synthetic unsaturated case study, using the Richards’ equation to describe the water movement. The model is a 2D transient model consisting of ≈ 5000 nodes, and is also created using FreeFem++. The Galerkin POD method is applied to the case study in order to replicate the high-fidelity response. This did not yield in any speed-up times, since the full matrices of non-linear problems need to be recreated at each time step in the transient simulation. Subsequently, a method is proposed in this thesis that adapts the Galerkin POD method by linearising the non-linear terms in the Richards’ equation, in a method named the Linearised Galerkin POD (LGP) method. This method is applied to the same 2D synthetic problem, and results in speed-up times in the range of 10 to 100. The adaptation, notably, does not use any interpolation techniques, favouring a code intrusive, but physics-based, approach. While the use of an intrusively linearised POD approach adds to the complexity of the ROM, it avoids the problem of finding kernel parameters typically present in interpolative POD approaches. Furthermore, the interpolation and possible extrapolation properties inherent to intrusive POD-based ROM’s are explored. The good extrapolation properties, within predetermined bounds, of intrusive POD’s allows for the development of an optimisation approach requiring a very small Design of Experiments (DOE) sets (e.g. with improved Latin Hypercube sampling). The optimisation method creates locally accurate models within the parameter space using Support Vector Classification (SVC). The region inside of the parameter space in which the optimiser is allowed to move is called the confidence region. This confidence region is chosen as the parameter region in which the ROM meets certain accuracy conditions. With the proposed optimisation technique, advantage is taken of the good extrapolation characteristics of the intrusive POD-based ROM’s. A further advantage of this optimisation approach is that the ROM is built on a set of high-fidelity responses obtained prior to the inverse modelling study, avoiding the need for full simulations during the inverse modelling study. In the methodologies and case studies presented in this thesis, initially infeasible inverse modelling problems are made possible by the use of the POD-based ROM’s. The speed up times and extrapolation properties of POD-based ROM’s are also shown to be favourable. In this research, the use of POD as a groundwater management tool for saturated and unsaturated sites is evident, and allows for the quick evaluation of different scenarios that would otherwise not be possible. It is proposed that a form of POD be implemented in conventional groundwater software to significantly reduce the time required for inverse modelling studies, thereby allowing for more effective groundwater management. / AFRIKAANSE OPSOMMING: Die Richards vergelyking beskryf die beweging van ’n vloeistof deur ’n onversadigde poreuse media, en word gekenmerk as ’n nie-lineêre parsiële differensiaalvergelyking. Die vergelyking is onderhewig aan ’n aantal parameters en is tipies berekeningsintensief om op te los. Om die parameters in die Richards vergelyking te bepaal, moet parameter optimering studies dikwels onderneem word. In hierdie studies, word die parameters van ’n numeriese model verander totdat die numeriese resultate die gemete resultate pas. Parameter optimering studies vereis in die orde van honderde simulasies, wat beteken dat studies wat gebruik maak van die Richards vergelyking net algemeen is in 1D probleme in die literatuur. As ’n oplossing vir die berekingskoste wat vereis word in parameter optimering studies, is die gebruik van Eie Ortogonale Ontbinding (POD) as ’n Verminderde Orde Model (ROM) in hierdie tesis voorgestel om individuele simulasies te versnel in die optimering konteks. Die Galerkin POD benadering is aanvanklik ondersoek en toegepas op die Richards vergelyking, en daarna is die tegniek getoets op verskeie gevallestudies. Die Galerkin POD metode word gedemonstreer op ’n hipotetiese gevallestudie waarin water beweging deur die Richards-vergelyking beskryf word. As gevolg van die nie-lineêre aard van die Richards vergelyking, het die Galerkin POD metode nie gelei tot beduidende vermindering in die berekeningskoste per simulasie nie. ’n Verdere gevallestudie word gedoen op ’n ware grootskaalse terrein in die Tafelberg Groep naby Kaapstad, Suid-Afrika, waar die grondwater beweging as versadig beskou word. Weens die lineêre aard van die vergelyking wat die beweging van versadigde water beskryf, is merkwaardige versnellings van > 500 in die ROM waargeneem in hierdie gevallestudie. Daarna was die die Galerkin POD metode aangepas deur die nie-lineêre terme in die Richards vergelyking te lineariseer. Die tegniek word die geLineariserde Galerkin POD (LGP) tegniek genoem. Die aanpassing het goeie resultate getoon, met versnellings groter as 50 keer wanneer die ROM met die oorspronklike simulasie vergelyk word. Al maak die tegniek gebruik van verder lineariseering, is die metode nogsteeds ’n fisika-gebaseerde benadering, en maak nie gebruik van interpolasie tegnieke nie. Die gebruik van ’n fisika-gebaseerde POD benaderings dra by tot die kompleksiteit van ’n volledige numeriese model, maar die kompleksiteit is geregverdig deur die merkwaardige versnellings in parameter optimerings studies. Verder word die interpolasie eienskappe, en moontlike ekstrapolasie eienskappe, inherent aan fisika-gebaseerde POD ROM tegnieke ondersoek in die navorsing. In die navorsing word ’n tegniek voorgestel waarin hierdie inherente eienskappe gebruik word om plaaslik akkurate modelle binne die parameter ruimte te skep. Die voorgestelde tegniek maak gebruik van ondersteunende vektor klassifikasie. Die grense van die plaaslik akkurate model word ’n vertrouens gebeid genoem. Hierdie vertrouens gebied is gekies as die parameter ruimte waarin die ROM voldoen aan vooraf uitgekiesde akkuraatheidsvereistes. Die optimeeringsbenadering vermy ook die uitvoer van volledige simulasies tydens die parameter optimering, deur gebruik te maak van ’n ROM wat gebaseer is op die resultate van ’n stel volledige simulasies, voordat die parameter optimering studie gedoen word. Die volledige simulasies word tipies uitgevoer op parameter punte wat gekies word deur ’n proses wat genoem word die ontwerp van eksperimente. Verdere hipotetiese grondwater gevallestudies is onderneem om die LGP en die plaaslik akkurate tegnieke te toets. In hierdie gevallestudies is die grondwater beweging weereens beskryf deur die Richards vergelyking. In die gevalle studie word komplekse en tyd-rowende modellerings probleme vervang deur ’n POD gebaseerde ROM, waarin individuele simulasies merkwaardig vinniger is. Die spoed en interpolasie/ekstrapolasie eienskappe blyk baie gunstig te wees. In hierdie navorsing is die gebruik van verminderde orde modelle as ’n grondwaterbestuursinstrument duidelik getoon, waarin voorsiening geskep word vir die vinnige evaluering van verskillende modellering situasies, wat andersins nie moontlik is nie. Daar word voorgestel dat ’n vorm van POD in konvensionele grondwater sagteware geïmplementeer word om aansienlike versnellings in parameter studies moontlik te maak, wat na meer effektiewe bestuur van grondwater sal lei.

The proper orthogonal decomposition (POD) has been widely used in fluid dynamic applications for extracting dominant flow features. The “gappy” POD is an extension to this method that allows the consideration of incomplete data sets. In this paper, the gappy POD is extended to handle unsteady flow reconstruction problems, such as those encountered when limited flow measurement data is available. In addition, a systematic approach for effective sensor placement is formulated within the gappy framework. Two applications are considered. The first aims to reconstruct the unsteady flow field using a small number of surface pressure measurements for a subsonic airfoil undergoing plunging motion. The second considers estimation of POD modal content of a cylinder wake flow for active control purposes. In both cases, using the dominant POD basis vectors and a small number of sensor signals, the gappy approach is found to yield accurate flow reconstruction results. / Singapore-MIT Alliance (SMA)

In this paper, a new methodology for predicting fluid free surface shape using Model Order Reduction (MOR) is presented. Proper Orthogonal Decomposition combined with a linear interpolation procedure for its coefficient is applied to a problem involving bubble dynamics near to a free surface. A model is developed to accurately and efficiently capture the variation of the free surface shape with different bubble parameters. In addition, a systematic approach is developed within the MOR framework to find the best initial locations and pressures for a set of bubbles beneath the quiescent free surface such that the resultant free surface attained is close to a desired shape. Predictions of the free surface in two-dimensions and three-dimensions are presented. / Singapore-MIT Alliance (SMA)

This thesis presents a study of reduced-order models based on proper orthogonal decomposition applied to non-isothermal transport phenomena in °uidized beds. A numerical °ow solver called Multiphase Flow with Interphase eXchanges (MFIX) was used to generate a database of solution snapshots for proper orthogonal decomposi- tion (POD). Using POD, time independent basis functions were extracted from the data and the governing equations of the numerical solver were projected onto the basis functions to generate reduced-order models. A reduced-order model was constructed that simulates multi-phase isothermal and non-isothermal °ow. In the reduced-order models (ROMs) the large number of partial di®erential equations were replaced by a much smaller number of ordinary di®erential equations. These reduced-order models were applied to two reference cases, a time extrapolation case and a time-dependent period boundary condition case. Three additional acceleration techniques were devel- oped to further improve computational e±ciency of the POD based ROM: 1) Database splitting, 2) Freezing the matrix of the linear system and 3) Time step adjustment. Detailed numerical analysis of both the full-order model, MFIX and the POD-based ROM, including estimating the number of operations and the CPU time per iteration, was performed as part of this study. The results of this investigation show that the reduced-order models are capable of producing qualitatively accurate results with less than 5% error with a two-order of magnitude reduction of computational costs.

This thesis presents advances in reduced-order modeling based on proper orthogonal decomposition (POD) for single and two-phase flows. Reduced-order models (ROMs) are generated for two-phase gas-solid flows. A multiphase numerical flow solver, MFIX, is used to generate a database of solution snapshots for proper orthogonal decomposition. Time-independent basis functions are extracted using POD from the data, and the governing equations of the MFIX are projected onto the basis functions to generate the multiphase POD-based ROMs. Reduced-order models are constructed to simulate multiphase two-dimensional non-isothermal flow and isothermal flow particle kinetics and three-dimensional isothermal flow. These reduced-order models are applied to three reference cases. The results of this investigation show that the two-dimensional reduced-order models are capable of producing qualitatively accurate results with less than 5 percent error with at least an order of magnitude reduction of computational costs. The three-dimensional ROM shows improvements in computational costs. This thesis also presents an algorithm based on mathematical morphology used to extract discontinuities present in quasi-steady and unsteady flows for POD basis augmentation. Both MFIX and a Reynolds Average Navier-Stokes (RANS) flow solver, UNS3D, are used to generate solution databases for feature extraction. The algorithm is applied to bubbling uidized beds, transonic airfoils, and turbomachinery seals. The results of this investigation show that all of the important features are extracted without loss in accuracy.

In this thesis, a reduced order model (ROM) based on the proper orthogonal decomposition (POD) for the transport phenomena in fluidized beds has been developed. The reduced order model is tested first on a gas-only flow. Two different strategies and implementations are described for this case. Next, a ROM for a two-dimensional gas-solids fluidized bed is presented. A ROM is developed for a range of diameters of the solids particles. The reconstructed solution is calculated and compared against the full order solution. The differences between the ROM and the full order solution are smaller than 3.2% if the diameters of the solids particles are in the range of diameters used for POD database generation. Otherwise, the errors increase up to 10% for the cases presented herein. The computational time of the ROM varied between 25% and 33% of the computational time of the full order solution. The computational speed-up depended on the complexity of the transport phenomena, ROM methodology and reconstruction error. In this thesis, we also investigated the accuracy of the reduced order model based on the POD. When analyzing the accuracy, we used two simple sets of governing partial differential equations: a non-homogeneous Burgers' equation and a system of two coupled Burgers' equations.

abstract: Structural features of canonical wall-bounded turbulent flows are described using several techniques, including proper orthogonal decomposition (POD). The canonical wall-bounded turbulent flows of channels, pipes, and flat-plate boundary layers include physics important to a wide variety of practical fluid flows with a minimum of geometric complications. Yet, significant questions remain for their turbulent motions' form, organization to compose very long motions, and relationship to vortical structures. POD extracts highly energetic structures from flow fields and is one tool to further understand the turbulence physics. A variety of direct numerical simulations provide velocity fields suitable for detailed analysis. Since POD modes require significant interpretation, this study begins with wall-normal, one-dimensional POD for a set of turbulent channel flows. Important features of the modes and their scaling are interpreted in light of flow physics, also leading to a method of synthesizing one-dimensional POD modes. Properties of a pipe flow simulation are then studied via several methods. The presence of very long streamwise motions is assessed using a number of statistical quantities, including energy spectra, which are compared to experiments. Further properties of energy spectra, including their relation to fictitious forces associated with mean Reynolds stress, are considered in depth. After reviewing salient features of turbulent structures previously observed in relevant experiments, structures in the pipe flow are examined in greater detail. A variety of methods reveal organization patterns of structures in instantaneous fields and their associated vortical structures. Properties of POD modes for a boundary layer flow are considered. Finally, very wide modes that occur when computing POD modes in all three canonical flows are compared. The results demonstrate that POD extracts structures relevant to characterizing wall-bounded turbulent flows. However, significant care is necessary in interpreting POD results, for which modes can be categorized according to their self-similarity. Additional analysis techniques reveal the organization of smaller motions in characteristic patterns to compose very long motions in pipe flows. The very large scale motions are observed to contribute large fractions of turbulent kinetic energy and Reynolds stress. The associated vortical structures possess characteristics of hairpins, but are commonly distorted from pristine hairpin geometries. / Dissertation/Thesis / Ph.D. Mechanical Engineering 2012

The main objective of this research is to develop an innovative multi-fidelity multi-disciplinary design, analysis and optimization suite that integrates certain solution generation codes and newly developed innovative tools to improve the overall optimization process. The research performed herein is divided into two parts: (1) the development of an MDAO framework by integration of variable fidelity physics-based computational codes, and (2) enhancements to such a framework by incorporating innovative features extending its robustness. The first part of this dissertation describes the development of a conceptual Multi-Fidelity Multi-Strategy and Multi-Disciplinary Design Optimization Environment (M3 DOE), in context of aircraft wing optimization. M3 DOE provides the user a capability to optimize configurations with a choice of (i) the level of fidelity desired, (ii) the use of a single-step or multi-step optimization strategy, and (iii) combination of a series of structural and aerodynamic analyses. The modularity of M3 DOE allows it to be a part of other inclusive optimization frameworks. The M3 DOE is demonstrated within the context of shape and sizing optimization of the wing of a Generic Business Jet aircraft. Two different optimization objectives, viz. dry weight minimization, and cruise range maximization are studied by conducting one low-fidelity and two high-fidelity optimization runs to demonstrate the application scope of M3 DOE. The second part of this dissertation describes the development of an innovative hybrid optimization framework that extends the robustness of M3 DOE by employing a proper orthogonal decomposition-based design-space order reduction scheme combined with the evolutionary algorithm technique. The POD method of extracting dominant modes from an ensemble of candidate configurations is used for the design-space order reduction. The snapshot of candidate population is updated iteratively using evolutionary algorithm technique of fitness-driven retention. This strategy capitalizes on the advantages of evolutionary algorithm as well as POD-based reduced order modeling, while overcoming the shortcomings inherent with these techniques. When linked with M3 DOE, this strategy offers a computationally efficient methodology for problems with high level of complexity and a challenging design-space. This newly developed framework is demonstrated for its robustness on a non-conventional supersonic tailless air vehicle wing shape optimization problem. / Ph. D.