Analysis of instability patterns in non-Boussinesq mixed convection using a direct numerical evaluation of disturbance integrals

The Fourier integrals representing linearised disturbances arising from an initially localised source are evaluated numerically for natural and mixed convection flows between two differentially heated plates. The corresponding spatio-temporal instability patterns are obtained for strongly non-Boussinesq high-temperature convection of air and are contrasted to their Boussinesq counterparts. A drastic change in disturbance evolution scenarios is found when a large cross-channel temperature gradient leads to an essentially nonlinear variation of the fluid's transport properties and density. In particular, it is shown that non-Boussinesq natural convection flows are convectively unstable while forced convection flows can be absolutely unstable. These scenarios are opposite to the ones detected in classical Boussinesq convection. It is found that the competition between two physically distinct instability mechanisms which are due to the action of the shear and the buoyancy are responsible for such a drastic change in spatio-temporal characteristics of instabilities. The obtained numerical results confirm and complement semi-analytical conclusions of Suslov 2007 on the absolute/convective instability transition in non-Boussinesq mixed convection. Generic features of the chosen numerical approach are discussed and its advantages and shortcomings are reported

Gradient-based quantitative image reconstruction in ultrasound-modulated optical tomography: first harmonic measurement type in a linearised diffusion formulation

Ultrasound-modulated optical tomography is an emerging biomedical imaging modality which uses the spatially localised acoustically-driven modulation of coherent light as a probe of the structure and optical properties of biological tissues. In this work we begin by providing an overview of forward modelling methods, before deriving a linearised diffusion-style model which calculates the first-harmonic modulated flux measured on the boundary of a given domain. We derive and examine the correlation measurement density functions of the model which describe the sensitivity of the modality to perturbations in the optical parameters of interest. Finally, we employ said functions in the development of an adjoint-assisted gradient based image reconstruction method, which ameliorates the computational burden and memory requirements of a traditional Newton-based optimisation approach. We validate our work by performing reconstructions of optical absorption and scattering in two- and three-dimensions using simulated measurements with 1% proportional Gaussian noise, and demonstrate the successful recovery of the parameters to within +/-5% of their true values when the resolution of the ultrasound raster probing the domain is sufficient to delineate perturbing inclusions.Comment: 12 pages, 6 figure

Control of fluid flows and other systems governed by partial differential-algebraic equations

The motion of fluids, such as air or water, is central to many engineering systems of significant economic and environmental importance. Examples range from air/fuel mixing in combustion engines to turbulence induced noise and fatigue on aircraft. Recent advances in novel sensor/actuator technologies have raised the intriguing prospect of actively sensing and manipulating the motion of the fluid within these systems, making them ripe for feedback control, provided a suitable control model exists. Unfortunately, the models for many of these systems are described by nonlinear, partial differential-algebraic equations for which few, if any, controller synthesis techniques exist. In stark contrast, the majority of established control theory assumes plant models of finite (and typically small) state dimension, expressed as a linear system of ordinary differential equations. Therefore, this thesis explores the problem of how to apply the mainstream tools of control theory to the class of systems described by partial differential-algebraic equations, that are either linear, or for which a linear approximation is valid. The problems of control system design for infinite-dimensional and algebraically constrained systems are treated separately in this thesis. With respect to the former, a new method is presented that enables the computation of a bound on the n-gap between a discretisation of a spatially distributed plant, and the plant itself, by exploiting the convergence rate of the v-gap metric between low-order models of successively finer spatial resolution. This bound informs the design, on loworder models, of H[infinity] loop-shaping controllers that are guaranteed to robustly stabilise the actual plant. An example is presented on a one-dimensional heat equation. Controller/estimator synthesis is then discussed for finite-dimensional systems containing algebraic, as well as differential equations. In the case of fluid flows, algebraic constraints typically arise from incompressibility and the application of boundary conditions. A numerical algorithm is presented, suitable for the semi-discrete linearised Navier-Stokes equations, that decouples the differential and algebraic parts of the system, enabling application of standard control theory without the need for velocity-vorticity type methods. This algorithm is demonstrated firstly on a simple electrical circuit, and secondly on the highly non-trivial problem of flow-field estimation in the transient growth region of a flat-plate boundary layer, using only wall shear measurements. These separate strands are woven together in the penultimate chapter, where a transient energy controller is designed for a channel-flow system, using wall mounted sensors and actuators

Global turbulence simulations of the tokamak edge region with GRILLIX

Turbulent dynamics in the scrape-off layer (SOL) of magnetic fusion devices is intermittent with large fluctuations in density and pressure. Therefore, a model is required that allows perturbations of similar or even larger magnitude to the time-averaged background value. The fluid-turbulence code GRILLIX is extended to such a global model, which consistently accounts for large variation in plasma parameters. Derived from the drift reduced Braginskii equations, the new GRILLIX model includes electromagnetic and electron-thermal dynamics, retains global parametric dependencies and the Boussinesq approximation is not applied. The penalisation technique is combined with the flux-coordinate independent (FCI) approach [F. Hariri and M. Ottaviani, Comput.Phys.Commun. 184:2419, (2013); A. Stegmeir et al., Comput.Phys.Commun. 198:139, (2016)], which allows to study realistic diverted geometries with X-point(s) and general boundary contours. We characterise results from turbulence simulations and investigate the effect of geometry by comparing simulations in circular geometry with toroidal limiter against realistic diverted geometry at otherwise comparable parameters. Turbulence is found to be intermittent with relative fluctuation levels of up to 40% showing that a global description is indeed important. At the same time via direct comparison, we find that the Boussinesq approximation has only a small quantitative impact in a turbulent environment. In comparison to circular geometry the fluctuations are reduced in diverted geometry, which is related to a different zonal flow structure. Moreover, the fluctuation level has a more complex spatial distribution in diverted geometry. Due to local magnetic shear, which differs fundamentally in circular and diverted geometry, turbulent structures become strongly distorted in the perpendicular direction and are eventually damped away towards the X-point

Development of an implicit framework for the two-fluid model on unstructured grids

The two-fluid model is an efficient method for simulating multiphase flows, based on an averaged description of the phases as interpenetrating and interacting continua. It is particularly attractive for the simulation of dispersed gas-solid flows in which the large number of particles in practical devices can impose an insurmountable computational burden for particle tracking methods, given currently available computing resources. Whilst the two-fluid model is more efficient than particle tracking methods, it results in large, strongly coupled and highly non-linear systems of equations, placing a premium on efficient solution algorithms. Additionally, the constitutive models used to describe the solid phase introduce stiff source terms, requiring a robust solution algorithm to handle them. In this thesis a fully-coupled algorithm is developed for the two-fluid model, based on a Newton linearisation of the underlying equation system, resulting in an algorithm treating all inter-equation couplings implicitly. For comparison, a semi-coupled algorithm, based on a Picard linearisation of the two-fluid model is also implemented, yielding a smaller implicitly coupled pressure-velocity system and a segregated system for the transport of phase concentrations. Motivating this work is the highly non-linear nature of the two-fluid model and the stiff source terms arising in the models of the dispersed phase, these are treated explicitly in the semi-coupled algorithm and may impose stability limits on the algorithm. By treating these terms implicitly, it is expected that the fully-coupled solution algorithm will be more robust. The algorithms are compared by application to test cases ranging from academic problems to problems representative of industrial applications of the two-fluid model. These comparisons show that with increasing problem complexity, the robustness of the fully-coupled algorithm leads to an overall more efficient solution than the semi-coupled algorithm.Open Acces

Finite volume solutions for electromagnetic induction processing

A new method is presented for numerically solving the equations of electromagnetic induction in conducting materials using native, primary variables and not a magnetic vector potential. Solving for the components of the electric field allows the meshed domain to cover only the processed material rather than extend further out in space. Together with the finite volume discretisation this makes possible the seamless coupling of the electromagnetic solver within a multi-physics simulation framework. After validation for cases with known results, a 3-dimensional industrial application example of induction heating shows the suitability of the method for practical engineering calculation

Multi-Adaptive Time-Integration

Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are localised to different components, corresponding to localisation in space for a PDE, efficient time integration thus requires that we use different time steps for different components. We present an overview of the multi-adaptive Galerkin methods mcG(q) and mdG(q) recently introduced in a series of papers by the author. In these methods, the time step sequence is selected individually and adaptively for each component, based on an a posteriori error estimate of the global error. The multi-adaptive methods require the solution of large systems of nonlinear algebraic equations which are solved using explicit-type iterative solvers (fixed point iteration). If the system is stiff, these iterations may fail to converge, corresponding to the well-known fact that standard explicit methods are inefficient for stiff systems. To resolve this problem, we present an adaptive strategy for explicit time integration of stiff ODEs, in which the explicit method is adaptively stabilised by a small number of small, stabilising time steps