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Validation of a Numerical Solution of the Fully Non-linear Weakly Dispersive Serre Equations for Steep Gradient Flows

Christopher Zoppou

Mathematical Sciences Institute,ÌýCollege of Physical and Mathematical Sciences,ÌýAustralian National University

Date and time:Ìý

Tuesday, August 25, 2015 - 4:00pm

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ECOT 226

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Rapidly-varying free-surface flows are characterized by large surface gradients. These flows can be found for example, in hydraulic jumps, tsunamis, tidal bores and releases from power stations. Large surface gradients produce vertical accelerations of fluid particles and a nonhydrostatic pressure distribution.

System of equations that describe the behaviour of these flows are obtained from the threedimensional Euler equations describing incompressible free-surface flows with constant density. By integrating the Euler equations over the water depth results in system of equations that are more amenable for solution by numerical techniques and for solving practical problems. If nonhydrostatic pressure distribution is assumed, one arrives at a system of equations that contain dispersive terms.

The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. This includes mixed spatial and temporal derivative flux terms which are difficult to handle numerically. By introducing a new conserved variable the Serre equations can be written in conservation law form and the numerical techniques for solving hyperbolic equations can be used to solve the Serre equations. The water depth and new conserved quantities are evolved using a second-order finite volume scheme. The remaining primitive variable, the fluid velocity, is obtained by solving a second-order elliptic equation using simple finite differences. The advantage of this approach is that problems with steep gradients can now be solved efficiently. For example, our modelling approach was used to produce the following results. The simulated results are shown in red and the blue line are the laboratory data from Hammack and Segur, Journal of Fluid Mechanics, 1978.

We demonstrate how these results and others were achieved and that it is straightforward to express other dispersion-type equations in conservation law form so that rapidly varying flows can be modelled using conventional techniques accurately and efficiently.