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The Use of Fractional Brownian Motion in the Generation of Bed Topography for Bodies of Water Coupled with the Lattice Boltzmann Method

Authors: Elysia G. Barker

Abstract:

Collecting data to determine the topography of a riverbed has many flaws, a few being observer bias, erosion over time, and inaccuracies due to debris. To overcome these issues, Fractional Brownian Motion (FBM), a generalization of Brownian Motion, is used to generate a fractal-like structure to model the bed topography in a randomized, nonlinear way. This allows us to overcome the issues arising when needing to take measurements of the bed topography by estimating the bed in a realistic manner. The main benefits being this would present a more efficient method as physical measurements would no longer be needed, the method changes each time it is run, due to its fractal-like nature, which represents the changing bed topography due to erosion and other external factors surrounding the changing of riverbeds, and simulations may still be run for areas we currently have no topographical measurements of. To test this method, simulations of shallow water are run over the FBM generated bed topography and the traditional method of modelling bed topography using data points. The Lattice Boltzmann Method is used to solve the shallow water equations in this case. The results of these simulations are compared side by side to establish accuracy and similarity between the two methods, whether the FBM generated bed topography is any more reliable than the traditional method, and to test the efficiency of the methods. Different flow cases will be tested such as stationary flow, non-stationary steady flow, and more complex flows to thoroughly test the method. Pure water is used in the simulations but eventually, the collection and transportation of dirt and debris will be considered as this may affect the overall movement of the water. MATLAB is used to write a program which generates a bed topography using FBM and compares this against the original bed data collected in the traditional method. The generated bed must be less than five percent different than the traditional bed for it to be acceptable to use. FBM considers the action and movement of the previous data point which allows for constraints to be placed on the generation of the bed topography, such as the average height above sea level, the starting and ending bed elevation and the amount of data points required. In turn, this allows for the generation of accurate, fractal-like bed topography which produces results without requiring physical measurements of the bed topography prior to simulation.

Keywords: bed topography, fractional brownian motion, lattice boltzmann method, shallow water, simulations

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