# Numerical Solution of the One-dimensional and Cylindrical Serre Equations for Rapidly Varying Free Surface Flows

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2014

## Authors

Zoppou, Christopher

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## Abstract

Rapidly-varying free surface flows that arise, for example, from; rapid reservoir releases, dam-breaks, mud slides, tidal bores, storm surges, tsunamis and flows over variable topography, are characterized by abrupt changes in the water surface. These changes produce vertical acceleration of the fluid particles. These vertical accelerations manifest as a series of oscillating waves, called dispersive waves, which follow abrupt changes in the water surface. These dispersive waves can have a significant influence on the water depth which impacts on the area inundated by these flows. A system of equations that are capable of describing dispersive waves are the Serre equations. They are applicable to a wide range of problems involving small to large amplitude waves in both shallow and relatively deep water. For practical problems, these equations must be solved using computer programs. Unlike the shallow water wave equations, the Serre equations contain terms which make the solution of the Serre equations computationally expensive. A large number of efficient and accurate computer programs have been developed for solving the shallow water wave equations. Unfortunately, the shallow water wave equations are not capable of modelling dispersive waves. Because the shallow water wave equations are a subset of the Serre equations, it should be possible to adapt these efficient computer programs to solve the Serre equations. This has been achieved by rewriting the Serre equations in a form that resembles the shallow water wave equations. Efficient computer programs used to solve the shallow water wave equations have been adapted to solve the reformulated Serre equations. Results from these computer programs are validated using an analytical solution, laboratory flume data and the simulation of the dam-break problem. Two hypothetical examples, one that includes bathymetry and the second involving the circular dam-break problem in two-dimensions, are also used to validate the computer programs. Most of these problems involve rapidly-varying flows that produce dispersive waves. Solving the Serre equations is only slightly more expensive than solving the shallow water wave equations. Comparing the results from the solution of the Serre equations with the results from the solution of the shallow water wave equations, demonstrates the importance of including dispersive terms when simulating rapidly-varying flows. These examples demonstrate the accuracy, robustness and versatility of the Serre equations in modelling dispersive waves. The computer programs developed are simple to implement, efficient and stable for a range of problems, including rapidly-varying free surface flows.

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Unsteady free surface flows, conservation laws, Serre equations, finite volume methods, dispersive waves

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Thesis (PhD)