Lattice-Boltzmann for Aeronautical Flows: An introduction to and evaluation of the Lattice-Boltzmann Method

Abstract

In aircraft design, there is a need for accurate, efficient and robust computational fluid dynamics (CFD) simulations. Industry dominated methods are based on the non-linear Navier-Stokes equations which are rather computationally expensive to solve. The Lattice-Boltzmann method (LBM) is an alternative CFD method that has risen in popularity lately due to the promised performance gain resulting from its linear equations. The method describes the evolution of a particle distribution function (PDF) at the meso-scale through the Boltzmann equation. The PDF is a statistical function describing the probability of finding a particle with a certain velocity at a certain location in time and space and is connected to the macro-scale through integrals over velocity space. In the standard LBM, the discretisation of the Boltzmann equation involves expressing the PDF at equilibrium through a truncated polynomial expansion. This allows for exact computation of the macroscopic density and fluid velocity through finite sums and a limited set of particle velocities. However, the truncation introduces an error scaling with the Mach number, limiting the method to Ma ≲ 0.3. There is also a correlation between the viscosity, grid spacing and time step. To simulate high Reynolds number (Re) flows the grid must therefore be very fine, which adds computational cost. In this master’s thesis, the standard LBM has been evaluated for aeronautical applications. It was implemented in Python, where part of the work focused on increasing the performance resulting in 30 times faster code. The Euler equations were used as a baseline, but since the standard LBM is always viscous there were difficulties reaching good correspondence. Partly, this was due to using simple boundary conditions (BCs), but a great improvement could be shown through a proposed modification. The limitation in Re was still an issue, however, and the conclusion is that more advanced BCs should be used for arbitrary geometries. Through a minor modification to the equilibrium PDF, an Euler equation test case for isentropic vortex convection was successfully simulated, although with some viscous dissipation present. The stability of the method was also explored, finding that the Ma limit was stricter at low viscosities since the method operates closer to its stability limit there. Lastly, the initialisation proved another challenge due to the interplay between the macro- and meso-scales, often leading to polluting the solution with numerical acoustic noise. It is possible to create non-reflecting BCs, but stability problems where the solution diverges were encountered when using established methods, leading to the development of new boundary treatments.