G.A. Akponasa

Doctor of Philosophy, St. Edmund Hall, University of Oxford, Trinity Term 1992

Solution of the Contravariant Shallow Water Equations (SWEs) Using Boundary-Fitted Coordinate Systems


Rivers, lakes and estuaries are usually shallow with respect to their depth. This thesis presents a derivation of the 2-dimensional contravariant () form of the shallow water equations (where and are contravariant depth-averaged velocity components and is the free surface elevation above still water lever). The equations are formulated for solution on non-orthogonal curvilinear coordinate systems and allow for a free surface. This formulation has the advantage of being able to represent curved or irregular boundaries within a finite difference model because all calculations are performed in the transformed domain which is made up of square grid cells.

Non-orthogonal curvilinear grids are generated using Stone's (1968) strongly implicit procedure (SIP). This procedure was found to converge much faster than successive over- relaxation (SOR) and alternating direction implicity (ADI) methods. The shallow water equations (SWEs) are discretised by finite differences on a staggered () mesh and solved by a semi-implicit method. The scheme is stabilized by applying a temporal filter to the three variables, and , at every half-time step; and by using second order up wind differences or weighted central differences for the cross-advection terms depending on the direction of release.

The contravariant SWE solver has been successfully validated against: Uniform flow in a rectangular channel, wind-induced free surface gradient in a rectangular basin, laminar flow in a rectangular channel, laminar flow past a backward-facing step, jet-formed flow in circular and square reservoirs with symmetrical inflow/outflow stems, and jet-forced flow in a circular reservoir with asymmetrical inflow/outflow stem. The results show that the model is stable, and does not give rise to undue amounts of numerical diffusion within the range of tests considered.

The model should be useful for simulating flows in shallow natural watercourses of arbitrary shape like wide rivers, lakes, estuaries and harbours.

(no thesis available)