Gert Bartholomeeusen

Ecole Nationale des Ponts et Chaussées,

gert.bartholomeeusen@cermes.enpc.fr

gert.bartholomeeusen@cermes.enpc.fr

Shock Waves and Sedimentation

Kynch (1952) presented the theory of sedimentation, and since it has been applied in a wide variety of engineering fields, eg chemical and geotechnical engineering. Sedimentation is a flux driven process in which the settling process is controlled by the local concentration only. The governing equation for sedimentation is the continuity equation

(1)

where u is the concentration and f(u) the flux function. The hyperbolic nature of equation 1 gives rises to shock waves, and therefore, numerical solutions of equation 1 in a fixed spatial grid are far from trivial, see for instance Leveque (2002). During the presentation video animations of the numerical solution of equation 1 are presented, addressing the following issues.

In the first part is concentrated on the numerical issues that arise in hyperbolic systems,
such as mass conservation, Gibbs phenomenon, first and second order accurate Finite Volume
Methods (FVM). As example, the Inviscid Burgers equation, equation 1 with f(u) = u^{2}/2,
is solved numerically and compared to the analytical solution.

In the second part a convex flux function f(u) = u(1 - u)/2 and a non-convex flux function f(u) = u^{2} (1 - u)/2 are used to illustrate the influence of the non-convexity on the sedimentation problem. The sediment-water interface is marked by a regular shock wave, while the upwards travelling shock wave can be a regular shock wave, a rarefaction wave or a compound shock wave.

In the third and final part of the presentation a numerical prediction of the sedimentation behaviour of a Red mud suspension is presented and compared to experimental results. The flux function used is derived from experimental data, and appeared to be non-convex and highly-peaked (Bartholomeeusen et al., In Print).

References

- Bartholomeeusen, G., De Sterck, H., & Sills, G.C. In Print. Non-convex flux func-tions and compound shock waves in sediment beds. In: Hou, T., & Tadmor, E. (eds), Proceedings of the Ninth International Conference on Hyperbolic Problems. International series of numerical mathematics.
- Kynch, G.J. 1952. A theory of sedimentation. Transactions faraday society, 48, 166-176.
- Leveque, R.J. 2002. Finite volume methods for hyperbolic problems. Cambridge University Press.