Throughout the past decade, interest has grown in the use of boundary-fitted coordinate systems in many areas of computational fluid dynamics. The boundary-fitted technique provides an exact method of implementing finite-difference numerical schemes in curved flow geometries and offers an alternative solution procedure to the finite-element method. The unavoidable large bandwidth of the global stiffness matrix, employed in finite-element algorithms, means that they are computationally less efficient than corresponding finite-difference schemes. As a consequence, the boundary-fitted method offers a more efficient process for solving partial differential flow equations in awkwardly shaped regions. This thesis describes a versatile finite-difference numerical scheme for the solution of the shallow water equations on arbitrary boundary-fitted non-orthogonal curvilinear grids. The model is capable of simulating flows in irregular geometries typically encountered in river basin management. Validation tests have been conducted against the severe condition of jet-forced flow in a circular reservoir with vertical side walls, where initial reflections of free surface waves pose major problems in achieving a stable solution. Furthermore, the validation exercises have been designed to test the computer model for artificial diffusion which may be a consequence of the numerical scheme adopted to stabilise the shallow water equations. The thesis also describes two subsidiary numerical studies of jet-forced recirculating flow in circular cylinders. The first of these implements a Biot-Savart discrete vortex method for simulating the vorticity in the shear layers of the inflow jet, whereas the second employs a stream function/vorticity-transport finite-difference procedure for solving the two-dimensional Navier-Stokes equations on a distorted orthogonal polar mesh. Although the predictions from the stream function/vorticity-transport model are confined to low Reynolds number flows, they provide a valuable set of benchmark velocity fields which are used to confirm the validity of the boundary-fitted shallow water equation solver.

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The frequent presence of ripples on the free surface of water. on both thin film flows and ponds or lakes motivates this theoretical investigation into the propagation of ripples on gravity waves. These ripples are treated as "slowly-varying" waves in a reference frame where the gravity wave flow is steady. The methods used are those of the averaged Lagrangian (Whitham 1965,1967,1974) and the averaged equations of motion (Phillips 1966) which are shown to be equivalent. The capillary wave modulation is taken to be steady in the reference frame which brings the gravity wave, or gravity driven flow, to rest. Firstly the motion over ponds or lakes is considered. Linear capillary-gravity waves are examined in order to set the scene. Crapper's (1957) exact finite-amplitude waves are examined next to show the actual behaviour of the flow field. The underlying gravity driven flow is that of pure gravity waves over an' "infinite" depth liquid. These gravity waves are modelled with "numerically exact" solutions for periodic plane-waves. The initial studies are inviscid and show that steep gravity waves either "absorb" or "sweep-up" a range of capillary waves or, alternatively, cause them to break in the vicinity of gravity wave crests. Improvements on the theory are made by including viscous dissipation of wave energy. This leads to a number of solutions approaching "stopping velocities" or the "stopped waves solution". In addition to these effects "higher-order dispersion" is introduced for weakly nonlinear waves near linear caustics. This clarifies aspects of the dissipation results and shows that wave reflection sometimes occurs. Secondly, waves on thin film flows are considered. Linear capillary-gravity waves are again examined in order to set the scene. Kinnersley's (1957) exact finite-amplitude waves are examined next to show the actual behaviour of the flow field. The underlying gravity driven flow is given by shallow water gravity waves. No modelling of these is necessary simply because they are included within Whitham's or Phillips' equations ab initio. This study is inviscid and shows the unexpected presence of critical velocities at which pairs of solution branches originate. iii

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The applicability of computational fluid dynamics (CFD) modelling schemes to turbulent wall-bounded flows is a matter of concern. In the near-wall region of bounded flows, the standard high Reynolds number k-e model is not valid and requires the use of empirical wall models to mimic the behaviour of this region. A theoretical study of the physics of prevalent wall modelling techniques showed that the velocity distribution took no account of the pressure gradient. To determine the effect of this shortcoming, a typical transient three-dimensional flow was analysed using current CFD methods and the results compared with experimental flow measurements. Consideration of these results showed that the 'traditional' wall model was unable to replicate observed flow features in the near-wall region: further analysis of the computational results confirmed that these poor flow predictions arose from the inability of the model to consider local pressure gradient effects. Consequently, a strong case was made for a more robust wall model for use in conjunction with the standard high Reynolds number k-e model. A number of boundary layer analyses were reviewed and Coles' law of the wake (1956) presented as a viable candidate for the development of a new wall modelling scheme. In theory, Coles' law (1956) provides a description of bounded flows under arbitrary pressure gradients up to the point of near-separation and may be extended to the study of reversed flows. A generic algorithm for Coles' law was prepared and used to study the fundamental test cases of U-bend and backward facing step flows. In a comparison between documented experimentation, 'conventional' CFD modelling and Coles' law models of these flows, the Coles' law model was shown to provide a viable alternative to 'traditional' schemes. Consequently, the Coles' law model of the near-wall region, being valid for pressure-driven flows, offers an extension to the range of flows for which the standard high Reynolds number k-e model may be used.