The stability of wall driven and thermally driven cavity flow is investigated for a wide range of viscous and viscoelastic fluids. The effect of inertia, elasticity, temperature gradients, viscous heating and Biot boundary conditions are of particular interest. Both destabilisation and bifurcation phenomenon are found. For Newtonian constant viscosity flow the instabilities are characterised by a critical Reynolds number which represents the ratio of inertial forces to viscous forces, and instability occurs when the inertial forces become large. For non-Newtonian viscoelastic fluids the instability is characterised by a critical Weissenberg number, which represents the ratio of elastic forces to viscous forces, and instability also occurs when elastic forces dominate the viscous forces. For thermally driven flow the instability is characterised by a critical Rayleigh number, which represents the ratio of temperature gradient to viscosity, and instability occurs when the Rayleigh number become large. In this case the instability is also characterised by both Eckert and Biot number. The work has relevance to thermal convection and mixing processes which occur in the viscous and viscoelastic fluid within the Earth's mantle. Three-dimensional steady and transient flow in a cylindrical cavity and three dimensional steady flow in a spherical cavity, are also considered for both viscous and viscoelastic fluids. Instabilities in these three-dimensional flow depend on the same parameters as the flow in square cavity.

Obtaining accurate solutions to flows that involve discontinuous features still re- mains one of the most difficult tasks in computational fluid dynamics today. Some discontinuous features, such as shear waves and material interfaces, are quite deli- cate, yet they have a profound effect on the rest of the flow field. The accuracy of the numerical scheme and the quality of the grid discretisation of the flow domain, are both critical when computing multi-dimensional discontinuous solutions. Here, the second order WAF scheme is used in conjuction with an adaptive grid algorithm, which is able to automatically modify the grid in regions of discontinuous features and solid boundaries. The grid algorithm is a combination of two successful ap- proaches, namely Chimera and Cartesian grid Adaptive Mesh Refinement (AMR). The Chimera approach is able to accurately represent non-Cartesian boundaries, whilst the AMR approach yields significant savings in memory storage and cPu time. The combined algorithm has been thoroughly validated for convection test problems in gas dynamics. The computed solutions compare well with other numerical and experimental results. These tests have also been used to assess the efficiency of the grid adaption algorithms. Finally, the approach is applied to axi-symmetric, two- dimensional, two-phase, reactive flows in the context of internal ballistics problems. Again, the computed results are compared with other numerical and experimental results.

Effective and robust high resolution schemes are of vital importance for simulation of viscous and inviscid flows. Since second-order high resolution schemes in practice are inadquate for many applications, large efforts have been put towards developing higher- order accurate schemes in the past. Although some progress has been made, the efforts were frustrated by the lack of effective and robust new schemes. Therefore this thesis is aimed at challenging this difficult but very important issue. Some new theories and methodologies were established during this research, which covers the linear stability analysis for high-order numerical schemes; the fully discrete techniques for model equations; the formulation of conservative high-order schemes and the high-order Total Variation Diminishing (TVD) schemes. According to these theories arbitrary-order high resolution schemes can be developed. To illustrate the methodologies second-, third-, fourth-, and 20th-order schemes are presented. These high resolution schemes were tested and validated by solving some popular test problems for one and two dimensional Euler and incompressible Navier-Stokes equations. The efficiency and robustness are the features of these high-order schemes.

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There have been countless research investigations taking place on Marion Island (MI), both ecological and geological, which have reached conclusions that must necessarily neglect the impacts of wind on the systems under study. Since only the dominant wind direction of the general atmospheric wind is known from weather and satellite data, not much can be said about local wind conditions at ground level. Therefore, a baseline Computational Fluid Dynamics (CFD) model has been developed for simulating wind patterns over Marion and Prince Edward Islands, a South African territory lying in the subantarctic Indian Ocean. A review of the current state of the art of Computational Wind Engineering (CWE) revealed that large-scale Atmospheric Boundary Layer (ABL) simulations have been successfully performed before with varying degrees of success. With ANSYS Fluent chosen as the numerical solver, the Reynolds-Averaged Navier-Stokes (RANS) equations were set up to simulate a total of 16 wind flow headings approaching MI from each of the cardinal compass directions. The standard k-epsilon turbulence closure scheme with modified constants was used to numerically approximate the atmospheric turbulence. A strategy was devised for generating a reusable mesh system to simulate multiple climatic conditions and wind directions around MI. In conjunction with the computational simulations, a wind measurement campaign was executed to install 17 wind data logging stations at key locations around MI. Raw data output from the stations were cleaned and converted into an easily accessible MySQL database format using the Python scripting language. The Marion Island Recorded Experimental Dataset (MIRED) database contains all wind measurements gathered over the span of two years. The decision was taken to focus on validating only three of the 16 cardinal wind directions against the measured wind data; North-Westerly, Westerly and South-Westerly winds. An initial interrogation of the simulation results showed that island-to-island wake interactions could not be ignored as the turbulent stream from MI could definitely be intercepted by its neighbour under the right conditions, and vice versa. An underestimation of the true strength of the Coriolis effect led to larger wind deflection in the simulations than originally expected, thus resulting in the wind flow at surface levels having an entirely different heading to what was intended. The westerly and south-westerly wind validation cases did not seem too badly affected by the lapse in judgement but the north-westerly case suffered strong losses in accuracy. Significant effort was put into quantifying the error present in the simulations. After a full validation exercise, it was finally resolved to apply a conservative uncertainty factor of 35 % when using these simulations to predict actual wind speed conditions. Similarly, the predicted wind direction can only be trusted within the bounds of a 35 degree prediction uncertainty. Under these circumstances, the baseline CFD model was successfully validated against the measured wind data and can thus be used in further research. In terms of post-processing, all the wind direction simulations have been combined into a single wind velocity map, generated by weighting each of the simulations by the frequency of wind prevalence measured in the corresponding wind sector. A second turbulence intensity combined map has been provided using similar techniques. These maps, as well as the individual wind maps showing all 16 cardinal wind directions, are believed to be helpful to many future biological studies on MI as well as any possible forays into wind energy generation on the island. Despite the encountered deficiencies, this project offers significant value to academia by providing a reliable method of predicting fine-scale wind patterns in a location previously devoid of any accurate data. Furthermore, it has highlighted where future CFD attempts can be improved in order to produce a compelling approximation of the realistic atmospheric phenomena occurring in the Marion Island territory. While error cannot be avoided when modelling such complex systems, it has been well quantified and discussed here so that any further research may make informed judgements in future studies. / Dissertation (MEng (Mechanical Engineering))--University of Pretoria, 2021. / South African National Antarctic Programme (SANAP) grant number 110726 / National Research Foundation (NRF) / Mechanical and Aeronautical Engineering / MEng (Mechanical Engineering) / Unrestricted