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The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, we derive convergence rates as known for the conforming finite element method in presence of regular solutions. Numerical examples illustrate the approach and the results.

The requirement for reduced jet noise in order to meet stringent noise legislation (civil aviation), and low infra-red observability and the use of unconventional exhaust nozzle configurations to improve aircraft survivability and performance (military aviation) is driving research to develop a better understanding of jet development and mixing mechanisms. One option open to the engineer is the use of small-scale model testing to investigate jets flows and provide valuable data for the validation of numerical models. Although more economical than large/full scale testing, additional factors that influence jet development may be present which would not be present at full scale and whose influence needs to be fully understood in order to allow small scale–large scale read-across. One such factor is the nozzle exit boundary layer. Although considerable data exist on the influence of nozzle exit boundary layers on low speed jet flows, current information on high speed jet flows is limited. It was, therefore, the aim of this thesis to extend the current understanding of nozzle exit boundary layers and their influence on the jet development for high speed jet flows through a combination of experimental and computational techniques. A combination of pneumatic probe measurements and Laser Doppler Anemometry (LDA) was used to investigate nozzle inlet and exit boundary layers of simple conical nozzles and the influence of adding a parallel extension piece. The measurements showed that the rapid acceleration of the boundary layer within the nozzle significantly reduced its momentum thickness Reynolds number and changed the state of the boundary layer from turbulent to laminar-like. The addition of a parallel extension to the nozzle exit returned the boundary layer to a fully turbulent state. A low Reynolds number RANS CFD approach was used to investigate the flow within the nozzle. Simulations using the Launder-Sharma low Reynolds number k–ε model revealed that the magnitude of the acceleration within the conical nozzles resulted in the boundary layer beginning to relaminarise. Full relaminarisation was not achieved due to the short axial distance over which the acceleration was sustained. The addition of a parallel extension provided a relaxation region in which the boundary layer could recover from the acceleration to become fully turbulent. Measurements of the jet plume originating from nozzles with laminar-like and turbulent boundary layers showed little influence of the boundary layer shape and thickness on shear layer spreading and jet centreline development.

The form and dynamics of ocean turbulence are critical to all marine processes; biological, chemical and physical. The three-dimensional turbulence characteristics of the bottom boundary layer of the coastal ocean are examined using a series of 29,991 instantaneous velocity distributions. These data, recorded by a submersible 3D-PTV system at an elevation of 0.64 m above the seabed, represent conditions typical of moderate tidal flows in the coastal ocean. A complexity associated with submersible 3D-PTV in the coastal ocean is that gaps and noise affect the accuracy of the data collected. To accommodate this, a new Physics-Enabled Flow Restoration Algorithm has been tested for the restoration of gappy and noisy velocity measurements where a standard PTV or PIV laboratory set-up (e.g. concentration / size of the particles tracked) is not possible and the boundary and initial conditions are not known a priori. This is able to restore the physical structure of the flow from gappy and noisy data, in accordance with its hydrodynamical basis. In addition to the restoration of the velocity flow field, PEFRA also estimates the maximum possible deviation of the output from the true flow. 3D-PTV measurements show coherent structures, with the hairpin-like vortices highlighted in laboratory measurements and numerical modelling, were frequently present within the logarithmic layer. These exhibit a modal alignment of 8 degrees from the mean flow and a modal elevation of 27 degrees from the seabed, with a mean period of occurrence of 4.3 sec. These appear to straddle sections of zero-mean along-stream velocity, consistent with an interpretation as packets. From these measurements, it is clear that data collected through both laboratory and numerical experiments are directly applicable to geophysical scales – a finding that will enable the fine-scale details of particle transport and pollutant dispersion to be studied in future. Conditional sampling of the Reynolds shear stress (without using Taylor’s hypothesis) reveals that these coherent structures are responsible for the vertical exchange of momentum and, as such, are the key areas where energy is extracted from the mean flow and into turbulence. The present study offers the first assessment of the magnitude of the errors associated with assuming isotropy on shear-based sensors of the TKE dissipation rate and its consequential effect on the Kolmogorov microscale using 3D-PTV data from the bottom boundary layer of the coastal ocean. The results indicate a high degree of spatial variability associated with the low conditions. The averaged data supports the validity of measurements obtained by horizontal and vertical profilers, however along-stream velocity derivatives underestimate the TKE dissipation rate by more than 40% – a factor of two higher than for the equivalent cross-stream and vertical estimates. This has important implications for the deployment of these sensors and the subsequent interpretation of higher-order statistics. Finally, the data have been processed to test four popular sub-grid scale (SGS) stress models and SGS dissipation rate estimates for Large-Eddy Simulations using these in situ experimental data. When the correlation and SGS model coefficients are assessed, the nonlinear model represents the best stress models to use for the present data, consistent with the substantial anisotropy and inhomogeneity associated with these flows. The detailed measurement and analysis of coherent structures in the coastal ocean undertaken therefore supports the development of numerical models and assists with the understanding of all marine processes.

The thesis studies homogenization of Dirichlet boundary value problems for divergence type elliptic operators, and the associated boundary layer issues. This type of problems for operators with periodically oscillating coeffcients, and fixed boundary data are by now a classical topic largely due to the celebrated work by Avellaneda and Lin from late 80's. The case when the operator and the Dirichlet boundary data exhibit periodic oscillations simultaneously was a longstanding open problem, and a progress in this direction has been achieved only very recently, in 2012, by Gerard-Varet and Masmoudi who proved a homogenization result for the simultaneously oscillating case with an algebraic rate of convergence in L2. Aimed at understanding the homogenization process of oscillating boundary data, in the first part of the thesis we introduce and develop Fourier-analytic ideas into the study of homogenization of Dirichlet boundary value problems for elliptic operators in divergence form. In smooth and bounded domains, for fixed operator and periodically oscillating boundary data we prove pointwise, as well as Lp convergence results the homogenization problem. We then investigate the optimality (sharpness) of our Lp upper bounds. Next, for the above mentioned simultaneously oscillating problem studied by Gerard-Varet and Masmoudi, we establish optimal Lp bounds for homogenization in some class of operators. For domains with non smooth boundary, we study similar boundary value homogenization problems for scalar equations set in convex polygonal domains. In the vein of smooth boundaries, here as well for problems with fixed operator and oscillating Dirichlet data we prove pointwise, and Lp convergence results, and study the optimality of our Lp bounds. Although the statements are somewhat similar with the smooth setting, challenges for this case are completely different due to a radical change in the geometry of the domain. The second part of the work is concerned with the analysis of boundary layers arising in periodic homogenization. A key difficulty toward the homogenization of Dirichlet problem for elliptic systems in divergence form with periodically oscillating coefficients and boundary condition lies in identification of the limiting Dirichlet data corresponding to the effective problem. This question has been addressed in the aforementioned work by Gerard-Varet and Masmoudi on the way of proving their main homogenization result. Despite the progress in this direction, some very basic questions remain unanswered, for instance the regularity of this effective data on the boundary. This issue is directly linked with the up to the boundary regularity of homogenized solutions, but perhaps more importantly has a potential to cast light on the homogenization process. We initiate the study of this regularity problem, and prove certain Lipschitz continuity result. The work also comprises a study on asymptotic behaviour of solutions to boundary layer systems set in halfspaces. By a new construction we show that depending on the normal direction of the hyperplane, convergence of the solutions toward their tails far away from the boundaries can be arbitrarily slow. This last result, combined with the previous studies gives an almost complete picture of the situation.

<div>Initial experiments on separated shock/boundary-layer interactions were carried out within the Boeing/AFOSR Mach-6 Quiet Tunnel. Measurements were made of hypersonic laminar-turbulent transition within the separation above a compression corner. This wind tunnel features freestream fluctuations that are similar to those in</div><div>flight. The present work focuses on the role of traveling instabilities within the shear layer above the separation bubble.</div><div>A 7 degree half-angle cone with a slice and a finite-span compression ramp was designed and tested. Due to a lack of space for post-reattachment sensors, early designs of this</div><div>generic geometry did not allow for measurement of a post-reattachment boundary layer. Oil flow and heat transfer measurements showed that by lengthening the ramp, the post-reattachment boundary layer could be measured. A parametric study was completed to determine that a 20 degree ramp angle caused reattachment at 45% of the</div><div>total ramp length and provided the best flow field for boundary-layer transition measurements.</div><div>Surface pressure fluctuation measurements showed post-reattachment wave packets and turbulent spots. The presence of wave packets suggests that a shear-layer</div><div>instability might be present. Pressure fluctuation magnitudes showed a consistent transition Reynolds numbers of 900000, based on freestream conditions and distance</div><div>from the nosetip. Pressure fluctuations grew exponentially from less than 1% to roughly 10% of tangent-wedge surface pressure during transition.</div><div>A high-voltage pulsed plasma perturber was used to introduce controlled disturbances into the boundary layer. The concept was demonstrated on a straight 7 degree half-angle circular cone. The perturbations successfully excited the second-mode instability at naturally unstable frequencies. The maximum second-mode amplitudes prior to transition were measured to be about 10% of the mean surface static pressure. </div><div>The plasma perturber was then used to disturb the boundary layer just upstream of the separation bubble on the cone with the slice and ramp. A traveling instability was measured post-reattachment but the transition location did not change for any tested condition. It appears that the excited shear-layer instability was not the dominant mechanism of transition.</div>

<div>Pressure-gain combustion in wave rotors offer the opportunity for substantial improvement in gas turbine efficiency and power, while controlling emissions with fuel flexibility, if provided rapid and reliable ignition of lean mixtures. In addition, tightening emission regulations and increasing availability of gas fuels for internal-combustion engines require more reliable ignition for ultra-lean operation to avoid high peak combustion temperature. Turbulent jet ignition (TJI) is able to address the ignition challenges of lean premixed combustion. Especially, the turbulent hot jet results in faster ignition penetration for wave rotor pressure-gain combustors that have high-frequency operation and fast-burn requirements. Controllability of TJI needs better understanding of the chemistry and fluid mechanics in the jet mixing region, particularly the estimation of ignition delay time and identifying the location of the ignition onset. </div><div>In the present work, numerical and analytical methods are employed to develop models capable of estimating the ignition characteristics that the turbulent hot jet exhibits as it is issued to a cold stoichiometric CH4-H2-Air mixture with varied fuel reactivity blends. Numerical models of the starting turbulent jet are developed by Reynolds-averaged and large-eddy simulation of Navier-Stokes and scalar transport equations in a high-resolution computational domain, with major focus on ignition of high-reactivity fuel blends in the jet near-field due to computational resource limitations. The chemical reactions are modeled using detailed chemistry by well-stirred and partially stirred reactor approaches. Numerical models describe the temporal evolution of jet mixture fraction, scalar dissipation rate, flow strain rate, and thermochemical quantities of the flow.</div><div>For faster estimation of ignition characteristics, analytical methods are developed to explicitly solve governing equations for the transient evolution of the near field and the leading vortex of the starting hot jet. First, the transient radial evolution of the turbulent shear-layer of a round transient jet is analytically investigated in the near-field of the nozzle, where the momentum potential core exists. The methods approximate the mixing and chemical processes in the jet shear and mixing layer. The momentum equation is integrated analytically, with a mixing-length turbulence model to represent the variation of effective viscosity due to the velocity gradients. The analytic predictions of the velocity field and mass entrainment rate of the jet are compared with numerical predictions and experimental findings. In addition, the transport equation of conserved scalars in the jet near-field is solved analytically for the history of the jet mixture fraction. This analytic solution for temperature and species is used, together with available models for instantaneous chemical induction time, to create an analytic ignition model that provides the time and radial location of the ignition onset.</div><div>Lastly, the ignition mechanism within the vortex ring, which leads the starting turbulent jet, is modeled using prior understanding about the mixing characteristics of the vortex. This mechanism is more relevant to low-reactivity fuel blends. Due to the presence of strong mixing at the large-scale, the vortex ring is treated as a homogeneous batch-reactor, which contains certain levels of the jet mixture fraction. This assumption provides the initial composition and temperature of the reactor in which ignition ensues. </div><div>This article-dissertation is developed as a collection of 4 articles published in peer-reviewed journals, one submitted article, and additional unpublished work. The study is laid out in 6 chapters with the following contributions:</div><div>Chapter 1: This chapter numerically investigates the three-dimensional behavior of a transient hot jet as modeled using the Reynolds-averaged turbulence flow. The study aims at providing an insight towards the role of mixing in the ignition progress and how the operating conditions such as fuel mixture and pre-chamber pressure ratio can influence the ignition success. An ignition prediction criterion is developed in this chapter, which helps to predict the ignition success under a broad range of operating conditions.</div><div>Chapter 2: In this chapter, the large-eddy simulation (LES) of hot jet ignition is reported in conjunction with detailed kinetics mechanism and adaptive-mesh refinement. The correlation between local values of mixture fraction gradient and ignition is discussed. Furthermore, the role of methane-hydrogen ratio on the heat release pattern is studied for two specific mixtures.</div><div>Chapter 3: The LES of CH4-H2-Air ignition is extended in this chapter to account for multivariable evaluation of ignition. Joint probability assessment of ignition explains the role of important scalars on the formation and growth of ignition. Also, the effect of CH4-H2 ratio on the spatial distribution of ignition is assessed and discussed.</div><div>Chapter 4: In this chapter, the rate of mass entrainment into the jet in the near-field region is studied. Characterization of the mass entrainment illuminates the understanding of mixing behavior of the starting turbulent jets. Through an exact solution of the momentum equation, this chapter includes a model of the diffusive transport in a round transient jet at high Reynolds numbers.</div><div>Chapter 5: This chapter proposes a method to evaluate the mass/heat exchange between a transient-turbulent jet and a quiescent environment. To analyze the transport phenomena in the jet near-field, the transient diffusion equation in cylindrical coordinates is explicitly solved and its solution is compared with the empirical findings. The transport solution then enables an ignition model to describe the spatiotemporal characteristics of ignition in the near-field.</div><div>Chapter 6: The development of ignition within the vortex ring of the transient jet is investigated in this chapter. The initiation, growth, and departure of the vortex ring are studied using the available empirical correlations and the LES. Using a perfectly-stirred, zero-dimensional representation of the vortex, chemical kinetic calculations provide estimates of ignition delay for various fuel mixtures.</div><div><br></div>

Görtler vortices evolve in boundary layers over concave surfaces as a result of the imbalance between centrifugal forces and radial pressure gradients. Depending on various geometrical and free-stream flow conditions, these instabilities may lead to secondary instabilities and early transition to turbulence. In this thesis, a control algorithm based on the boundary region equations is applied to reduce the strength of the Görtler instabilities by controlling the energy of the fully developed vortices, using either local wall deformations or blowing/suction at the wall. A proportional-integral control scheme is utilized to deform the wall or to provide transpiration velocity, where the inputs are either the wall-normal or streamwise velocity components in a plane that is parallel to the wall. The results show that the control based on wall deformation using wall-normal velocity components is more effective in tempering the vortex during its streamwise growth by almost one or two orders of magnitude.

This thesis presents a modern approach to two Integral Boundary Layer methods implemented in the Python programming language. This work is based on two 2D boundary layer methods: Thwaites' method for laminar boundary layer flows and Head's method for turbulent boundary layer flows. Several novel enhancements improve the quality and usability of the results. These improvements include: a common ordinary differential equation (ODE) integration framework that generalizes computational implementations of Integral Boundary Layer methods; the use of a dense output Runge-Kutta ODE solver that allows for querying of simulation results at any point with accuracy to the same order as that of the solver; and an edge velocity treatment method using cubic spline interpolation that improves the simulation performance using only points from an inviscid edge velocity distribution. Both the laminar and turbulent methods are shown to benefit from smoothing of the edge velocity distribution. The choice of ODE solver alleviates the need to artificially limit step sizes. Comparisons against analytic solutions, experimental data and XFOIL results provide a wide varity of verification and validation cases with which to compare. The implementation of Thwaites' method in this thesis avoids simplifications made in other implementations of this method, which results in more robust results. The implementation of Head's method produces high-quality results typically found in other implementations while utilizing the common ODE integration framework. Utilizing the common ODE framework results in significantly less code needed to implement Thwaites' and Head's methods. In addition, the boundary layer solvers produce results in seconds for all results presented here. Boundary layer transition and separation criteria are implemented as a proof of concept, but require future work.