The issue of local scale and smoothness presents a crucial and daunting challenge for numerical simulation methods in fluid dynamics. Yet in the interests of both accuracy and economy, how can one devise a general technique that efficiently resolves flow features of consequence and discriminates against others which are either ``negligible' or amenable to ``universal' modeling? This is particularly difficult because geometries of engineering interest are complex and multi-dimensional, precluding a priori knowledge of the flowfield. To address this challenge, the current work employs wavelet theory for the local scale decomposition of functions, which provides a natural mechanism for the adaptive compression of data. The resulting technique is known as the Multi-Resolution Discontinuous Galerkin (MRDG) method. This research successfully demonstrates that the multi-resolution framework and the discontinuous Galerkin method are well-suited for a new approach to accuracy and cost as demonstrated by the relative ease of their integration in spatial dimension greater than one. Some specific steps achieved include the implementation of suitable data encoding and compression algorithms, construction of multi-wavelet expansion bases in one and two dimensions, and derivation of the multi-resolution derivative operator that includes an upwind-type correction to the central scheme. Solutions with the MRDG method are observed to adapt to and track both smooth and discontinuous flow features in an entirely solution-driven manner without the need for a priori user knowledge of those flow features. Run-time efficiency and local adaptation characteristics are explored via a series of classic test problems.

Unsteady flow fields over a NACA 0012 at an angle of attack are investigated. The first is the classical pitching motion about the airfoil's quarter chord. The second is the flow over a fixed airfoil immersed in the wake of the pitching airfoil. Large reduced frequencies are considered. Measurements were obtained in a water tunnel by Laser-Doppler velocimetry. Ensemble-averaged velocity measurements were obtained in the vicinity of the trailing edges of both the pitching and the fixed airfoils. The flowfields in the wake and at the trailing edges of both airfoils were studied visually. The validity of the quasi-steady and an extension to an unsteady Kutta condition are examined. A new dynamic similarity parameter is proposed. An analytical method based on the dynamics of discrete vortices is employed. Numerical calculations of the flow over the fixed airfoil are compared with experimental results. / Ph. D.

The influence of tip clearance and angle of attack upon the mid-span unsteady pressure response of a vibrating flat plate was investigated experimentally. Unsteady pressure measurements were taken for a variety of incidence angles, vibration frequencies and tip clearances over a Mach number range of 0.2 to 0.6. It was found that changes in tip clearance had an effect on measured pressure fluctuations at higher angles of attack and larger Mach numbers. It was also observed that the amplitude of the unsteady pressure increased as the incidence angle was increased. The plate was mechanically induced to oscillate in translation, simulating the flISt bending mode. Averaged Fast Fourier Transforms were used to determine pressure oscillation amplitudes and phase lags with respect to the plate motion. / Master of Science

Strong interactions between flow about an aircraft wing and the wing structure can result in aeroelastic phenomena which significantly impact aircraft performance. Time-accurate methods for solving the unsteady Navier-Stokes equations have matured to the point where reliable results can be obtained with reasonable computational costs for complex non-linear flows with shock waves, vortices and separations. The ability to combine such a flow solver with a general finite element structural model is key to an aeroelastic analysis in these flows. Earlier work involved time-accurate integration of modal structural models based on plate elements. A finite element model was developed to handle three-dimensional wing boxes, and incorporated into the flow solver without the need for modal analysis. Static condensation is performed on the structural model to reduce the structural degrees of freedom for the aeroelastic analysis. Direct incorporation of the finite element wing-box structural model with the flow solver requires finding adequate methods for transferring aerodynamic pressures to the structural grid and returning deflections to the aerodynamic grid. Several schemes were explored for handling the grid-to-grid transfer of information. The complex, built-up nature of the wing-box complicated this transfer. Aeroelastic calculations for a sample wing in transonic flow comparing various simple transfer schemes are presented and discussed. / M.S.

Thesis (PhD) -- Stellenbosch University, 2014. / ENGLISH ABSTRACT: With increasing environmental concerns and the drive for a greener economy comes an increased desire to improve turbine engine fuel efficiency and reduce emissions. Unfortunately weight reduction techniques used increase the blade loading, which in turn increases the losses. Non-axisymmetric endwall contouring is one of several techniques being investigated to reduce loss in a turbine. An investigation at Durham University produced a non-axisymmetric endwall design for a linear cascade. An adaption of the most promising endwall was investigated in an annular rotating test rig at the CSIR using steady state instrumentation. The current investigation extends those investigations into the unsteady time domain. Previous investigations found that a generic rotor endwall contour improved efficiency by controlling the endwall secondary flow vortex system in both a linear cascade and an annular 1½ stage rotating test turbine. The current research was aimed at determining if there were any unsteady effects introduced by the contoured endwall. The approach was unique in that it investigated the unsteady effects of an endwall contour originally designed for a linear cascade both experimentally and numerically at three incidence angles (positive, zero and negative to represent increased load, design load and decreased load respectively), the results of which are openly available. Unsteady experimental hotfilm results showed that the endwall contour made the velocity profile more radially uniform by reducing the strength of the endwall secondary flow vortex system. The fluctuations in the velocity were also reduced producing a more temporally uniform velocity profile. The FFT magnitude of the velocity at the blade passing frequency below midspan was also reduced. It was found that the reduction in the endwall secondary flow vortex system due to the contour increased with increasing loading. Numerical results showed that the oscillations in the flow were small and did not penetrate the boundary layer. The contoured rotor was forward and aft loaded when compared to the annular rotor, resulting in a weaker cross passage pressure gradient which allowed the endwall secondary flow vortex system to be less tightly wrapped. Numerical results did not show a significant difference in the oscillations observed in the annular and contoured rotor. A new objective function for use in the endwall optimisation process was proposed that acts as a proxy for efficiency, but is less prone to uncertainty in the results. When used on the current results it shows the same trend as efficiency. It remains to be used to design an endwall for full validation. / AFRIKAANSE OPSOMMING: Met ’n toenemende omgewingsbesorgdheid en die strewe na ’n groener ekonomie kom ’n toenemende behoefte om turbine enjin brandstofdoeltreffendheid te verbeter en vrystellings te verlaag. Ongelukkig het gewigsbesparingstegnieke wat gebruik is die lemlading verhoog, wat op sy beurt die verliese verhoog. Nie-assimmetriese endwandprofilering is een van verskeie tegnieke wat ondersoek word om verliese in ’n turbine te verminder. ’n Ondersoek by die Universiteit van Durham het ’n nie-assimmetriese endwandontwerp vir ’n lineêre kaskade gelewer. ’n Aanpassing van die mees belowende endwand is in ’n annulêre roterende toetsopstelling by die WNNR getoets, deur gebruik te maak van bestendige toestand instrumentasie. Die huidige ondersoek brei daardie ondersoeke uit na die nie-bestendige verwysingsraamwerk . Vorige ondersoeke het bevind dat die generiese rotor endwandprofiel doeltreffendheid verbeter as gevolg van die beheer van die endwand sekondêre vloei draaikolkstelsel in beide ’n lineêre kaskade sowel as ’n annulêre 1½ stadium roterende toetsturbine. Die huidige navorsing was daarop gemik om vas te stel of die endwandprofiel enige onbestendige effekte tot gevolg gehad het. Die benadering was uniek in die sin dat dit die onbestendige effekte ondersoek het van ’n endwandprofiel wat oorspronklik ontwerp is vir ’n lineêre kaskade beide eksperimenteel en numeries op drie invalsshoeke (positief, nul en negatief om onderskeidelik verhoogde lading, ontwerplading en verlaagde lading te verteenwoordig), waarvan die resultate algemeen beskikbaar is. Onbestendige eksperimentele warmfilm resultate het getoon dat die endwandprofiel die snelheidsprofiel meer radiaal uniform gemaak het deur die vermindering van die sterkte van die endwand sekondêre vloei werwelstelsel. Die skommelinge in die snelheid is ook verminder wat ’n meer tyduniforme snelheidsprofiel gelewer het. Die FFT (Fast Fourier Transform) grootte van die snelheid van die lem verbygaan frekwensie onder lem midbestek het ook verminder. Daar was bevind dat die vermindering in die endwand sekondêre vloei draaikolkstelsel as gevolg van die endwandprofiel toeneem met toenemende lading. Numeriese resultate het getoon dat die ossilasie in die vloei klein was en nie die grenslaag binnegedring het nie. Die rotor met gevormde wand het ’n voor- en agterlading gehad in vergelyking met die rotor met annulêre wand, wat tot ’n laer drukgradient dwarsop die vloeirigting gelei het, die endwand sekondêre vloei draaikolkstelsel minder beperk het. Numeriese resultate het nie ’n beduidende verskil in die ossilasies tussen die annulêre en gevormde rotorwand getoon nie. ’n Nuwe doelwitfunksie vir gebruik in die endwand optimersproses is voorgestel wat dien as ’n plaasvervanger vir doeltreffendheid, maar minder geneig is tot onsekerheid in die resultate. Wanneer dit gebruik word op die huidige resultate toon dit dieselfde tendens as doeltreffendheid. Dit moet nog gebruik word in die ontwerp van ’n endwand vir volledige bevestiging.

The thesis deals with the development of a new residual error estimator and adaptive algorithms based on the error estimator for steady and unsteady fluid flows in a finite volume framework. The aposteriori residual error estimator referred to as R--parameter, is a measure of the local truncation error and is derived from the imbalance arising from the use of an exact operator on the numerical solution for conservation laws. A detailed and systematic study of the R--parameter on linear and non--linear hyperbolic problems, involving continuous flows and discontinuities is performed. Simple theoretical analysis and extensive numerical experiments are performed to establish the fact that the R--parameter is a valid estimator at limiter--free continuous flow regions, but is rendered inconsistent at discontinuities and with limiting. The R--parameter is demonstrated to work equally well on different mesh topologies and detects the sources of error, making it an ideal choice to drive adaptive strategies. The theory of the error estimation is also extended for unsteady flows, both on static and moving meshes. The R--parameter can be computed with a low computational overhead and is easily incorporated into existing finite volume codes with minimal effort. Adaptive refinement algorithms for steady flows are devised employing the residual error estimator. For continuous flows devoid of limiters, a purely R--parameter based adaptive algorithm is designed. A threshold length scale derived from the estimator determines the refinement/derefinement criterion, leading to a self--evolving adaptive algorithm devoid of heuristic parameters. On the other hand, for compressible flows involving discontinuities and limiting, a hybrid adaptive algorithm is proposed. In this hybrid algorithm, error indicators are used to flag regions for refinement, while regions of derefinement are detected using the R--parameter. Two variants of these algorithms, which differ in the computation of the threshold length scale are proposed. The disparate behaviour of the R--parameter for continuous and discontinuous flows is exploited to design a simple and effective discontinuity detector for compressible flows. For time--dependent flow problems, a two--step methodology is proposed for adaptive grid refinement. In the first step, the ``best" mesh at any given time instant is determined. The second step involves predicting the evolution of flow phenomena over a period of time and refines regions into which the flow features would progress into. The latter step is implemented using a geometric--based ``Refinement Level Projection" strategy which guarantees that the flow features remain in adapted zones between successive adaptive cycles and hence uniform solution accuracy. Several numerical experiments involving inviscid and viscous flows on different grid topologies are performed to illustrate the success of the proposed adaptive algorithms. Appendix 1 Candidate's response to the comments/queries of the examiners The author would like to thank the reviewers for their appreciation of the work embodied in the thesis and for their comments. The clarifications to the comments and queries posed in the reviews are summarized below. Referee 1 Q: The example of mesh refinement for RANS solution with shock was performed with isotropic mesh, while the author claims that it is appropriate with anisotropic mesh. If this is the case, why did he not demonstrate that ? As the author knows well, in the case of full 3--D configuration, isotropic adaptation will lead to substantial grid points. The large mesh will hamper timely turnaround time of simulation. Therefore it would be a significant contribution to the aero community if this point is investigated at a later date. Response: The author is of the view that for most practical situations, a pragmatic approach to mesh adaptation for RANS computations would merely involve generating a viscous padding of adequate fineness around the body and allowing for grid adaptation only in the outer potential region. Of course, this method would allow for grid adaptation in the outer layers of viscous padding only to the extent the smoothness criterion is satisfied while adapting the grids in the potential region. This completely obviates point addition to the wall (CAD surface) and there by avoids all complexities (like loss in automation) resulting from the interaction with the surface modeler while adding point on the wall. This method is expected to do well for attached flows and mildly separated flows. This method is expected to do well even for problems involving shock - boundary layer interaction, owing to the fact that the shock is normal to the wall surface (recall, a flow aligned grid is ideal to capture such shocks), as long as the interaction does not result in a massive separation. This approach has already been demonstrated in section 4.5.3 where in adaptive high-lift computations have been performed. Isotropic adaptation retains the goodness of the zero level grid and therefore the robustness of the solver does not suffer through successive levels of grid adaptation. This procedure may result in large number of volumes. On the other hand, the anisotropic refinement may result in significantly less number of volumes, but the mesh quality may have badly degenerated during successive levels of adaptation leading to difficulties in convergence. Therefore, the choice of either of these strategies is effectively dictated by requirements on grid quality and grid size. Also, it is generally understood that building tools for anisotropic adaptation are more complicated as compared to those required for isotropic adaptation, while anisotropic refinement may not require point addition on the wall. Considering these facts, in the view of the author, this issue is an open issue and his personal preference would be to use isotropic refinement or a hybrid strategy employing a combination of these methodologies, particularly considering aspects of solution quality. Finally, in both the examples cited by the reviewer (sections 6.4.5 & 6.4.6) the objective was to demonstrate the efficacy of the new adaptive algorithm (using error indicators and the residual estimator), rather than evaluating the pros & cons of isotropic and anisotropic refinement strategies. In the sections cited above, the author has merely highlighted the advantages of the refinement strategies in specific context of the problem considered and these statements need not be considered as general. Referee 2 Q: For convection problems, a good error estimator must be able to distinguish between locally generated error and convected error. The thesis says the residual error estimator is able to do this and some numerical evidence is presented, but can the candidate comment how the estimator is able to achieve this ? Response: The ultimate aim of any AMR strategy is to reduce the global error. The residual error estimator proposed in this work measures the local truncation error. It has been shown in the context of a linear convective equation that the global error in a cell consists of two parts--the locally generated error in the cell (which is the R--parameter) and the local error transported from other cells in the domain. Either of these errors are dependent on the local error itself and any algorithm that reduces the local truncation error (sources of error) will reduce the global error in the domain. This conclusion is supported by the test case of isentropic flow past an airfoil (Chapter 3, C, Pg 79), where refinement based on the R--parameter leads to lower global error levels than a global error based refinement itself. Q: While analysing the R--parameter in Section 3.3, the operator δ2 is missing. Response: The analysis in Section 3.3 is based on Eq.(3.3) (Pg 58) which provides the local truncation error. As can be seen from Eq.(3.14), the LHS represents the discrete operator acting on the numerical solution (which is zero) and the first term on the RHS is the exact operator acting on the numerical solution (which is I[u]). Consequently the truncation terms T1 and T2 contribute to the truncation error R1 . However, from the viewpoint of computing the error estimate on a discretised domain, we need to replace the exact operator I by a higher order discrete operator δ2 . This gives the R-parameter, which has contributions from R1 as well as discretisation errors due to the higher order operator, R2 . When the latter is negligible compared to the former, the R--parameter is an estimate of the local truncation error. The truncation error depends on the accuracy of the reconstruction procedure used in obtaining the numerical solution and hence on the discrete operator δ1. On very similar lines, it can be shown that operator δ2 leads to a formal second order accuracy and this operator is only required in computing the residual error estimate. Q: What does the phrase "exact derivatives of the numerical solution" mean ? Response: This statement exemplifies the fact that the numerical solution is the exact solution to the modified partial differential equation and that the truncation terms T1 and T2 that constitute the R--parameter are functions of the derivatives of this numerical solution. Q: For the operator δ2 quadratic reconstruction is employed. Is the exact or numerical flux function used ? Response: The operator δ2 is a higher order discrete approximation to the exact operator I. Therefore, a quadratic polynomial with a three--point Gauss quadrature has been used in the error estimation procedure. Error estimation does not involve issues with convergence associated with the flow solver and therefore an exact flux function has been employed with the δ2 operator. Nevertheless, it is also possible to use the same numerical flux function as employed in the flow solver for error estimation also. Q: The same stencil of grid points is used for the solution update and the error estimation. Does this not lead to an increased stencil size ? Response: In comparison to reconstruction using higher degree polynomials such as cubic and quartic reconstruction, quadratic reconstruction involves only a smaller stencil of points consisting of the node--sharing neighbours of a cell. The use of such a support stencil is sufficient for linear reconstruction also and adds to the robustness of the flow solver, although a linear reconstruction can, in principle, work with a smaller support stencil. A possible alternative to using quadratic reconstruction (and hence a slightly larger stencil) is to adopt a Defect Correction strategy to obtain derivatives to higher order accuracy and needs to be explored in detail. Q: How is the R--parameter computed for viscous flows ? Response: The computation of the R--parameter for viscous flows is on the same lines as for inviscid flows. The gradients needed for viscous flux computation at the face centers are obtained using quadratic reconstruction. The procedure for calculation of the R--parameter for steady flows (both inviscid and viscous) is the step--by--step algorithm in Section 3.5. Q: In some cases, regions ahead of the shock show no coarsening. Response: The adaptive algorithm proposed in this work does not allow for coarsening of the initial mesh, and regions ahead of the shock remain unaffected (because of uniform flow) at all levels of refinement. Q: Do adaptation strategies terminate automatically atleast for steady flows ? Response: The adaptation strategies (RAS and HAS) must, in principle by virtue of construction of the algorithm, automatically terminate for steady flows. In the HAS algorithms though, there are certain heuristic criteria for termination of refinement especially at shocks/turbulent boundary layers. In this work, a maximum of four cycles of refinement/derefinement have only been carried out and therefore an automatic termination of the adaptive strategies were no studied. Q: How do residual--based adaptive strategies compare and contrast with adjoint--based approaches which are now becoming popular for goal--oriented adaptation ? Adjoint--based methods involve solution to the adjoint problem in addition to solving the primal problem, which represents a substantial computational cost. A timing study for a typical 3D problem[2] indicates that the solution of the adjoint problem (which needs the computation of the Jacobian and sensitivities of the functional) could require as much as one--half of the total time needed to compute the flow solution. On the contrary, R--parameter based refinement involves no additional information than that required by the flow solver and is roughly equivalent to one explicit iteration of the flow solver (Section 3.5.1). For practical 3--D applications, adjoint--based approaches will lead to a prohibitively high cost, and more so for dynamic adaptation. This is also exemplified by the fact that there has been only few recent works on 3D adaptive computations based on adjoint error estimation (which consider only inviscid flows)[1,2]. Goal--oriented adaptation involves reducing the error in some functional of interest. This can be achieved within the framework of R--parameter based adaptation, by introducing additional termination criteria based on integrated quantities. Within an automated adaptation loop, such an algorithm would terminate when the integrated quantities do not change appreciably with refinement levels. This is in contrast to the adjoint--based approach which strives to reduce the error in the functional below a certain threshold. Considering the fact that reducing the residual leads to reducing the global error itself, the R--parameter based adaptive algorithm would also lead to accurate estimates of the integrated quantities (which depend on the numerical solution). This is also reflected in the fact that the R--parameter based adaptation for the three--element NHLP configuration predicts the lift and drag coefficients to reasonable accuracy, as shown in Section 4.5.3. The author is of the belief that the R--parameter based adaptive algorithm holds huge promise for adaptive simulations of flow past complex geometries, both in terms of computational cost and solution accuracy. This is exemplified by successful adaptive simulations of inviscid flow past ONERA M6 wing as well as a conventional missile configuration[3]. A more concrete comparison of the R--parameter based and adjoint--based approaches would involve systematically solving a set of problems by both approaches and has not been considered in this thesis. [1] Nemec and Aftosmis,``Adjoint error estimation and adaptive refinement for embedded--boundary cartesian meshes", AIAA Paper 2007--4187, 2007. [2] Wintzer, Nemec and Aftosmis,``Adjoint--based adaptive mesh refinement for sonic boom prediction", AIAA Paper 2008--6593, 2008. [3] Nikhil Shende, ``A general purpose flow solver for Euler equations", Ph.D. Thesis, Dept. of Aerospace Engg., Indian Institute of Science, 2005.

Thesis (MEng) -- Stellenbosch University, 2014. / ENGLISH ABSTRACT: The use of air-cooled steam condensers (ACSCs) is the preferred cooling method in the chemical and power industry due to stringent environmental and water use regulations. The performance of ACSCs is however highly dependent on the influence of windy conditions. Research has shown that the presence of wind reduces the performance of ACSCs. It has been found that cross-winds (wind perpendicular to the longest side of the ACSC) cause distorted inlet flow conditions, particularly at the upstream peripheral fans near the symmetry plane of the ACSC. These fans are subjected to what is referred to as '2-D' wind conditions, which are characterised by flow separation on the upstream edge of the fan inlets. Experimental investigations into inlet flow distortion have simulated these conditions by varying the fan platform height. Low platform heights resulted in higher levels of inlet flow distortion, as also found to exist with high cross-wind speeds. This investigation determines the performance of various fan configurations (representative of configurations used in the South- African power industry) subjected to distorted inlet flow conditions through experimental and numerical investigations. The similarity between platform height and cross-wind effects is also investigated and a correlation between system volumetric effectiveness, platform height and cross-wind velocity is found. / AFRIKAANSE OPSOMMING: Die gebruik van lugverkoelde stoom kondensors (LVSK's) word verkies as 'n verkoelingsmetode in die chemiese- en kragvoorsieningsindustrie as gevolg van streng omgewings- en waterverbruiksregulasies. Die werkverrigting van LVSK's word egter grootliks beïnvloed deur die teenwoordigheid van wind. Navorsing het gewys dat die teenwoordigheid van wind die werkverrigting van LVSK's verminder. Daar was gevind dat kruiswinde (wind loodreg tot die langste sy van die LVSK) versteurde inlaat vloeitoestande veroorsaak, veral by waaiers wat aan die stroomop kant van die LVSK naby die simmetrievlak geleë is. Hierdie waaiers word blootgestel aan na wat verwys word as '2-D' windtoestande wat gekenmerk word deur vloeiwegbreking wat plaasvind by die stroomop rand van die waaierinlate. Eksperimentele ondersoeke van inlaat vloeiversteurings het hierdie toestande gesimuleer deur die waaier platformhoogte te verstel. Lae platform hoogtes het gelei tot hoër vlakke van inlaat vloeiversteuring, soortgelyk aan wat gevind word met hoë kruiswindsnelhede. Hierdie ondersoek gebruik numeriese en eksperimentele metodes om die werkverrigting van verskeie waaierkon gurasies (verteenwoordigend van kon- gurasies wat gebruik word in die Suid-Afrikaanse kragvoorsieningsindustrie) wat blootgestel word aan versteurde inlaat vloeitoestande te bepaal. Die ooreenkoms tussen platformhoogte en kruiswind e ekte word ook ondersoek en 'n korrelasie tussen die sisteem volumetriese e ektiwiteit, platformhoogte en kruiswindsnelheid word bepaal.