Lattice Boltzmann modelling of supercritical shallow water flows

Abstract

The Lattice Boltzmann Method (LBM) is a promising tool to model fluid flows. This thesispresents a summary of the investigations carried out to apply the LBM to study run-up ofwaves induced by bores on beaches. The thesis starts with a critical review of the commonnumerical models used in fluid mechanics with a specific focus on the origin and historicaladvances in the LBM. This indicated that at the outset of this study it was accepted thatLBM application was limited to flows with subcritical regime. Hence, modelling supercriticalrun-up flow did not appear possible with LBM. The major achievement of current work is aone-dimensional Lattice Boltzmann Model which is developed to solve the shallow waterequations for steady and unsteady flows within both the subcritical and supercriticalregimes. The asymmetric LBM proposed by Chopard et al. (2013) is extended through ageneralised Galilean transformation applied to the standard LBM scheme. Thetransformation yields a general asymmetric Lattice Boltzmann Model scheme which cansuccessfully model a wide range of subcritical and supercritical flows, and enablesimplementation of the asymmetric model for practical purposes.In the current work a new set of the Equilibrium Distribution Functions, boundaryconditions and the external force weights are derived for the generalised transformedscheme. A new stability region is also defined, allowing selection of a lattice speed thatmaintains numerical stability for a wider range of sub- and supercritical flows andcombinations of those flow conditions, compared to the previous scheme with fixedasymmetry. The model is validated against a range of benchmark cases in open-channelhydraulics that demonstrate the applicability of the new model. The applicability of themodel to solve nearshore problems, such as wave run-up, is studied further by a criticalreview of existing shoreline treatment techniques and developing a new wetting-dryingboundary condition.A wetting-drying boundary condition is developed using LBM fundamentals, which is amodified version of the technique proposed by Liu and Zhou (2014), to accommodate thetransformation. The modified algorithm is successfully implemented in the transformedscheme. However, due to very shallow depths that inevitably occur in nearshore zone, theflow conditions in that area fall outside the numerical stability zone defined for thetransformed scheme, resulting in instability. It is concluded that while the transformedscheme can successfully be applied to both subcritical and supercritical regimes, in itscurrent form it has limited applicability to problems involving very shallow flows where theFroude and lattice Froude numbers will not be encompassed by the stable zone.Declaration by authorThis thesis is composed of my original work, and contains no material previously