We develop negative-norm least-squares methods to solve the three-dimensional Maxwell equations for static and time-harmonic electromagnetic fields in the case of axial symmetry. The methods compute solutions in a two-dimensional cross section of the domain, thereby reducing the dimension of the problem from three to two. To achieve this dimension reduction, we work with weighted spaces in cylindrical coordinates. In this setting, approximation spaces consisting of low order finite element functions and bubble functions are analyzed. In contrast to other methods for axisymmetric Maxwell equations, our leastsquares methods allow for discontinuous coefficients with large jumps and non-convex, irregular polygonal domains discretized by unstructured meshes. The resulting linear systems are of modest size, are symmetric positive definite, and can be solved very efficiently. Computations demonstrate the robustness of the methods with respect to the coefficients and domain shape.

This thesis describes investigations into the use of a technique for improving the efficiency of axial flow turbines. The flow in the turbine component of axial flow machines is complex, with a number of three-dimensional features. In order to extract power from a stream of high pressure and high temperature flow this flow must be turned through a large angle, this high turning introduces a phenomenon know as "secondary flow". This secondary flow introduces additional loss, unsteadiness and regions of high heat transfer into the machine - all of which are undesirable features. Endwall profiling aims to reduce these undesirable features by shaping the end-wall between the turbine blades. The shaping either accelerates the flow which reduces the local static pressure or retards the flow which increases the static pressure. These effects are confined to a region near the endwall so the overall performance of the blade row is not affected. However due to the complexity of the flow it is easy to make things worse rather than better! - careful design is needed. This thesis aims to understand how and why the reductions in may be achieved so that they can be better exploited as well as providing information of the performance of a major engine manufacturers design system. The thesis describes pressure probe measurements inside and outside of the blade passage of a low speed linear cascade with a number of profiled endwall geometries. The aerofoils used in the cascade are already relatively efficient and the overall loss changes are small, accurate measurement is therefore very difficult. The current best profiled endwall reduces secondary loss by 30%±5% compared to the planar case. Hot film measurements have been conducted on the endwalls and suction surface of the blade to determine if these benefits are substantially due to changing the boundary layer state. The results from this thesis indicate that this is not the case. This thesis describes measurements on three generations of profiled endwalls, two of which successfully reduce loss, one does not. The success of the first two endwalls indicates the power of current CFD based design practices, the failure of the third design to reduce loss illustrates some of the shortcomings of current CFD based design practices. The information from this thesis is being used in the design of the next generation of aircraft engines to which non-axisymmetric profiled endwalls are being fitted.

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The finite element process is now used almost routinely as a tool of engineering analysis. From early days, a significant effort has been devoted to developing simple, cost effective elements which adequately fulfill accuracy requirements. In this thesis we describe the development and application of one of the simplest elements available for the statics and dynamics of axisymmetric shells . A semi analytic truncated cone stiffness element has been formulated and implemented in a computer code: it has two nodes with five degrees of freedom at each node, circumferential variations in displacement field are described in terms of trigonometric series, transverse shear is accommodated by means of a penalty function and rotary inertia is allowed for. The element has been tested in a variety of applications in the statics and dynamics of axisymmetric shells subjected to a variety of boundary conditions. Good results have been obtained for thin and thick shell cases .

a b s t r a c t A study concerning the propagation of free non-axisymmetric waves in a homogeneous piezoelectric cylinder of transversely isotropic material with axial polarization is carried out on the basis of the linear theory of elasticity and linear electro-mechanical coupling. The solution of the three dimensional equations of motion and quasi-electrostatic equation is given in terms of seven mechanical and three electric potentials. The characteristic equations are obtained by the application of the mechanical and two types of electric boundary conditions at the surface of the piezoelectric cylinder. A novel method of displaying dispersion curves is described in the paper and the resulting dispersion curves are presented for propagating and evanescent waves for PZT-4 and PZT-7A piezoelectric ceramics for circumferential wave numbers m = 1, 2, and 3. It is observed that the dispersion curves are sensitive to the type of the imposed boundary conditions as well as to the measure of the electro-mechanical coupling of the material.

This document describes the design of a supersonic and a subsonic test section for a high enthalpy wind tunnel. A streamline is tracked through a supersonic test section using the method of characteristics. The specifics of the design program and the design techniques are illustrated for the supersonic section. The section of the paper dealing with the subsonic nozzle has a greatly diverse nature. This section details the inlet and exhaust restrictions and construction elements for the entire low speed system. The system is currently being set up for testing with the subsonic section, and the supersonic will eventually follow.

This thesis aims to make advances in the accurate simulation of the ows in and around clustered module plug nozzles. The resulting simulations presented in this thesis are, as far as can be ascertained from available data, the most detailed to date in Europe. A comparison is made with results from other sources for clarication of this point. In the process of producing these solutions, two ow solvers have been developed. NSAXIMB is a general 2D multi-block ow solver,developed by the author, for the axisymmetric, Reynolds averaged Navier-Stokes equations. It was developed to allow simulation of axisymmetic plug nozzle congurations and the investigation of the effects of turbulence modelling on such ows. MERLIN is a general 3D, implicit, multi-block ow solver again for the RANS equations. MERLIN was developed by the Centre for Computational uid Dynamic at Craneld. Signicant input from this work has included a large portion of the structure of the mean ow solver and the extension of the advanced two equation turbulence modelling, incorporated in NSAX- IMB, to three dimensions. Of the turbulence models investigated the zonal models of Menter prove to be most effective in reproducing experimental results. These models out perform a more advanced non-linear eddy viscosity formulation, based on the work of Abid. In an effort to improve solution accuracy, grid adaptation software, based on node redistribution techniques has been developed for use in conjunction with the 3D ow solver. This work is demonstrated in conjunction with a basic test case before application to the clustered module plug nozzle conguration. Results for the complex block topology adopted in the 3D test case are shown to cause the adaptation process to fail. Further, it is shown that such a process may not be generalised for arbitrary topologies.

With the continued economic and socio-political pressure on aircraft manufacturers to produce more profitable and environmentally-friendly aircraft, the drive towards increasingly more efficient aircraft engines remains of prime importance to aircraft engine manufacturers. While the majority of axial flow turbomachines use cylindrically shaped endwalls between the blades on the hub or shroud, non-axisymmetric endwall contouring is a reasonably recent technique which relaxes this constraint, and allows the geometry of the endwalls to depart from that of a plain cylinder. Although a number of studies have shown non-axisymmetric endwall contouring to be an effective mechanism for the reduction of secondary flows (and the losses associated with them), within the open literature there still remains a general lack of detailed information relating to the optimal design of these devices. Among some of the most important issues which remain unresolved, are uncertainties such as: “What is the best way to identify and thereafter quantify the strength of turbine secondary flows?”, and thereafter, as a natural progression from this, “Of the metrics which are currently found within the literature, which are best for use in the design of secondary loss mitigating endwall contours for a real turbine?”. Some of the reasons for the lack of information as described above, result from the undertaking of many of the investigations into the design of endwall contours by or on behalf of the major engine manufacturers, and therefore, a general inability or perhaps even unwillingness to divulge many of the specific details related to the methodologies and quantities used as a result of the commercial sensitivity of these investigations. In addition to this, as a result of the relatively large number and diverse nature of groups involved in non-axisymmetric endwall contouring research, within the literature which has been made available, there exists a wide variety of different test geometries as well as conditions which have been used, making a neutral determination of the most successful approach to endwall contouring considerably more difficult. This thesis documents the design and testing of a number of different non-axisymmetric endwall configurations intended to produce flow conditions optimized using a selection of the metrics commonly found in the literature, for the rotor of a low speed, research turbine, whose baseline as well as performance using contoured endwalls has been reported on previously, in order to establish which of these metrics is the most effective. As part of this process, a fully validated computational fluid dynamics model of the turbine downstream of the first nozzle was developed and incorporated into an automated non-axisymmetric end- wall design routine, capable of producing endwall contours optimized for various objective functions. Numerical testing showed that, in order to distinguish accurately between the various endwall configurations, relatively fine computational meshes were required and therefore, as a result of corresponding computational expense associated with these meshes, the implementation of a surrogate modelling procedure in which part of this computational cost is offset by mathematical modelling, was necessary. Altogether, a total of 8 endwall designs were produced - 6 using a single metric each as the basis of their objective functions (the ‘simple’ designs) and a further 2 so-called ‘compound’ designs. Of the simple designs, the best performing endwalls in terms of improvements to the rotor exit efficiency were the ηtt-, Cske- & βdev-based designs, which were based in turn on the rotor total-total efficiency (ηtt), coefficient of secondary kinetic energy (Cske) and flow deviation from design angle (βdev) respectively. All three of these designs were predicted to result in very similar changes to the secondary flow characteristics although the increasing bias towards flow correction was found to have an inverse correlation with the overall efficiencies predicted for each rotor. Of these designs, the numerical predictions for both the ηtt- & Cske-based designs (which were included in the experimental subset), were found to be validated, at both the rotor exit as well as downstream measurement planes. Further to this (with the exception of the Cp0,rel-based case), although the remainder of the simple designs (i.e. the SKEH & ηde-based designs) were also predicted to improve the overall rotor efficiency, either the form or the performance of these endwalls resulted in the final corresponding designs for these metrics being considered unsatisfactory. Finally, the two ‘compound’ metrics were both formulated to to include a term designed to target the secondary flow within the target blade row, as well as an additional term which was designed to promote improvement in the flow into the downstream blade row. While both designs produced using the compound design objective functions were predicted to improve both the conditions for the target blade row, as well as the flow quality at the exit of the blade row, flow separations at the exit of the contoured regions for both designs resulted in only partial validation of each design when tested experimentally. Finally, although both designs were once again predicted to perform very well at the ‘mixed-out’ measurement plane, these predictions were found to be only partially validated by the experiment.

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