We study computational theory and methods for finding multiple unstable solutions (corresponding to saddle points) to three types of nonlinear variational elliptic systems: cooperative, noncooperative, and Hamiltonian. We first propose a new Lorthogonal selection in a product Hilbert space so that a solution manifold can be defined. Then, we establish, respectively, a local characterization for saddle points of finite Morse index and of infinite Morse index. Based on these characterizations, two methods, called the local min-orthogonal method and the local min-max-orthogonal method, are developed and applied to solve those three types of elliptic systems for multiple solutions. Under suitable assumptions, a subsequence convergence result is established for each method. Numerical experiments for different types of model problems are carried out, showing that both methods are very reliable and efficient in computing coexisting saddle points or saddle points of infinite Morse index. We also analyze the instability of saddle points in both single and product Hilbert spaces. In particular, we establish several estimates of the Morse index of both coexisting and non-coexisting saddle points via the local min-orthogonal method developed and propose a local instability index to measure the local instability of both degenerate and nondegenerate saddle points. Finally, we suggest two extensions of an L-orthogonal selection for future research so that multiple solutions to more general elliptic systems such as nonvariational elliptic systems may also be found in a stable way.

Large-eddy simulations of turbulent flows past a circular cylinder have been performed at sub-, critical and super-critical Re using an orthogonal curvilinear grid solver, CFDship-Iowa version 6.2. An extensive verification and validation study has been carried out. Various aspects of the flow field have been investigated. The aspect ratio of the computational domain has major effects on the results. In general, large aspect ratio produced best results for the sub-critical Re. Small dependency on both aspect ratio and grid resolution was observed for the critical Re. Small aspect ratio and conservative scheme produced best results for the super-critical Re. Overall flow features and the drag crisis phenomenon have been correctly predicted. A lot of experimental and numerical studies of flow past a circular cylinder were collected and used for the validation of the present LES study. Integral and local variables were in fairly good agreement for the sub-critical Re. Sharp behavior including drag crisis was predicted for the critical Re. Although some discrepancy including early formation of turbulent separation was observed, local flow structures including separation bubble were observed for the super-critical Re. The formation of secondary vortex near the cylinder wall and its evolution into separation bubble were observed. The spectral analysis showed that the separation bubble had the instabilities close to the shear layer frequency. The proximity of shear layer to the cylinder enhanced the mixing process of boundary layer and shear layer and led to the formation of separation bubble. A snapshot POD method was used to extract flow structures in the boundary layer, shear layer and wake. In the boundary layer, the secondary vortices and separation bubble were successfully extracted. Due to the weak TKE distribution, specific flow structures were hard to find in the shear layer. Large two-dimensional flow structures representing the Karman shedding vortices were extracted for the sub- and super-critical Re.

The objectives of this project were to synthesise semi-conducting ceramic nanoparticles including zinc oxide (ZnO) and aluminium doped zinc oxide (AZO) through a wet chemistry route to obtain nanoparticles with a controlled size and morphology. Wet chemistry methods (co-precipitation method and hydrothermal method) were used to synthesise ZnO and AZO particles. In the synthesis, various compounds and morphologies were synthesised. ZnO, Zn(OH)2 and unknown phases were co-precipitated, with only ZnO obtained following hydrothermal treatment. Morphologies ranging from platelets, flower-like, nanorods and microflowers were obtained. Particle sizes as small as 11 nm were characterised. Nanorod and nanosphere AZO particles were also synthesised with the results indicated the average grain size decreasing with increasing Al atomic content. Three orthogonal arrays were carried out to investigate the effects of the reaction parameters on the size and morphology of ZnO particles. The applicability of the orthogonal array was successful, with the optimum parameters of both hydrothermal experiments showing an increase in aspect ratio. The L/D ratio of ZnO nanorods obtained in the confirmation experiment increased to 9.4 which was larger than the ZnO synthesised using other reaction conditions (1.0 8.0). Scanning electron microscopy, transmission electron microscopy and X-ray diffraction were used to characterise the properties of the obtained particles. Morphology, crystallinity and particle size were all characterised.