Abstract
The present work is concerned with the numerical modelling of largeamplitude interfacial waves produced by metal pad roll instability in the aluminium reduction cells. A semiconservative twolayer shallowwater model containing a novel, fully nonlinear equation for electric potential is developed and solved using an original finite difference scheme. The latter is based on the twodimensional Lax WendroffRichtmyer scheme, which is adopted and extended to the twolayer system containing interfacial pressure. Twodimensional Poissontype equations for pressure and electric potential are solved using an original highlyefficient 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 onedimensional twolayer system and various conservative forms of shallowwater 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 socalled lockexchange 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 shallowwater approximation. The onedimensional solution is further used to validate twodimensional numerical code by considering onedimensional 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 smallamplitude 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 largeamplitude quasiequilibrium 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 largeamplitude rotating interfacial wave induces a global countercirculation in the top and bottom layers. Numerical results indicate that the increase of the shear velocity above the critical value results in the KelvinHelmholtz type of instability which eventually causes the interface to break down.
Date of Award  Sep 2020 

Original language  English 
Awarding Institution 

Supervisor  Janis Priede (Supervisor) & Alex Pedcenko (Supervisor) 