Lattice-Boltzmann for Aeronautical Flows: An introduction to and evaluation of the Lattice-Boltzmann Method
Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Program
Applied mechanics (MPAME), MSc
Publicerad
2024
Författare
Ellénius, Emil
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
Lattice-Boltzmann , CFD , aeronautical , Euler equations , non-reflecting