AbstractThe present work is concerned with the numerical modelling of large-amplitude interfacial waves produced by metal pad roll instability in the aluminium reduction cells. A semi-conservative two-layer shallow-water model containing a novel, fully non-linear equation for electric potential is developed and solved using an original finite difference scheme. The latter is based on the two-dimensional Lax- Wendroff-Richtmyer scheme, which is adopted and extended to the two-layer system containing interfacial pressure. Two-dimensional Poisson-type equations for pressure and electric potential are solved using an original highly-efficient algorithm based on the combination of the tridiagonal matrix factorisation (Thomas algorithm) and the fast discrete cosine transform.
The development of the model and numerical schemes is started by considering purely hydrodynamic one-dimensional two-layer system and various conservative forms of shallow-water equations describing conservation of circulation or momentum in addition to that of mass. Using the method of characteristics, a novel analytical solution is found to the so-called lock-exchange problem. This exact solution is used to validate the ability of various numerical schemes to handle hydraulic shocks which are expected to develop in the shallow-water approximation. The one-dimensional solution is further used to validate two-dimensional numerical code by considering one-dimensional initial interface perturbations along two perpendicular sides of the rectangular container.
In addition, linear stability analysis of various basic models of aluminium reduction cells is revisited and extended to rectangular geometries. Linear stability analysis shows that in the case of negligible viscous friction, the cells with aspect ratios squared equal to the ratio of two odd numbers are inherently unstable and can be destabilised by arbitrary weak electromagnetic effect. The growth rates of small-amplitude electromagnetically destabilised interfacial waves produced by the numerical simulation agree very well with the linear stability results. Numerical results show that the growth rate decreases as the amplitude of unstable rolling interfacial disturbance grows with the time. A large-amplitude quasi-equilibrium state is reached without the interface touching the upper electrode. In this strongly nonlinear stage, the wave amplitude still keeps growing, however the growth rate is much slower than during the linear instability stage. At the same time, the nonlinear streaming effect produced by the large-amplitude rotating interfacial wave induces a global counter-circulation in the top and bottom layers. Numerical results indicate that the increase of the shear velocity above the critical value results in the Kelvin-Helmholtz type of instability which eventually causes the interface to break down.
|Date of Award||Sept 2020|
|Supervisor||Janis Priede (Supervisor) & Alex Pedcenko (Supervisor)|