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Burgers’ equation: numerical models and filtering

Bosman, R. (2008) Burgers’ equation: numerical models and filtering. Master's Thesis / Essay, Mathematics.

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Direct Numerical Simulation is of great interest to the world of Computational Science due to its proven precision. There are people who state that everything has a drawback and Direct Numerical Simulation (DNS) is no exception to this statement. The drawback of this method in combination with computers nowadays lies in speed. This method is rather slow compared to other methods. If we want to keep the precision of the method we have to come up with a way to decrease the computation time of DNS. In this paper we will apply four different DNS methods to Burgers’ equation, two Lagrangian methods and two Spectro-Consistent methods. We want to determine which of these methods is most suitable to use for Burgers’ equation. First we will look at a simple one-dimensional flow problem, without time, in order to show which type of discretization method ought to be used to discretize both convective as well as diffusive terms of the Navier-Stokes equations. Part of this proces is to find out whether the fourth-order versions of these methods outclass the second-order versions. Based on these one-dimensional results we will investigate how our programs handles turbulence by looking at Burgers’ equation, still in one dimension, hence adding time. From this investigation of different numerical methods we will see that the Spectro-Consistent methods are always stable as opposed to the Lagrangian methods, which are not always stable. The second part of this paper will focus on applying filtering to the convective term of Burgers’ equation. Reason for this is to be able to compute less small scales of motion and therefore use coarser grids. We will first take a look at filtering in physical space and finally take a look at filtering in spectral space. In spectral space we will take a look the energy for different wavenumbers. The structure of this document can be summarized by the following scheme: • Decide what discretization methods to use for convective as well as diffusive terms based on a one-dimensional flow problem; • Decide to choose either second- or fourth order discretization methods; • Apply a filter in physical space to the chosen discretization method; • Convert Burgers’ equation to spectral space and apply filtering; • Extend Burgers’ equation to two dimensions and apply a DNS method; • And finally report our findings and conclude whether or not filtering should be applied to these kind of problems and how filtering can efficiently used to speed up DNS methods. To obtain a complete image of

Item Type: Thesis (Master's Thesis / Essay)
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Feb 2018 07:47
Last Modified: 15 Feb 2018 07:47

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