A GRID-FREE LAGRANGIAN DILATATION ELEMENT METHOD WITH APPLICATION TO COMPRESSIBLE FLOW
A GRID-FREE LAGRANGIAN DILATATION ELEMENT METHOD WITH APPLICATION TO COMPRESSIBLE FLOW
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Date
2004-11-19
Authors
Shen, Jun
Advisor
Bernard, Peter S.
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Abstract
In the computational fluid dynamics research, grid-free methods are
getting more attention as an alternative to traditional grid-based
methods due to two important reasons. First, grid-free methods can
be very easily adapted into applications involving complicated
geometries. Secondly, they are less vulnerable to numerical
diffusion introduced by spatial discretization than in grid-based
schemes.
A new grid-free Lagrangian dilatation element method for
compressible flow has been developed in this research as an
extension of incompressible vortex methods. It differs from
grid-based numerical methods in a number of ways. The discretization
is represented by a group of Lagrangian particles that are convected
with the fluid flow velocities instead of a fixed spatial grid
system. The velocity of the flow field, necessary in each time step
to move the computational elements, is recovered from the dilatation
distribution similar to the 'Biot-Savart' law used in incompressible
vortex methods. The Fast Multi-pole Method (FMM) is used to speed up
the process and reduce the cost from $O(N^2)$ down to $O(N\log N)$.
Each computational particle carries physical properties such as
dilatation, temperature, density and geometric volume. These
properties are governed by the Lagrangian governing equations
derived from the Navier-Stokes equations. While the computational
elements are convected in the flow, their properties are updated by
integrating their corresponding governing equations. The spatial
derivatives appearing in the Lagrangian governing equations are
evaluated by using moving least-square fitting. The implementation
of several different boundary conditions has been developed in this
research. The non-penetration wall boundary condition is implemented
by adding a potential velocity field to that recovered from the
dilatation elements so as to cancel the normal component at the
wall. The zero-gradient of properties at the wall such as
temperature and density is enforced by a technique called particle
reflection. The inflow and outflow conditions are implemented with
the help of the characteristic waves moving up and down-stream. The
addition and removal of Lagrangian computational elements at the
inlet and outlet are implemented to ensure that the computational
domain is fully covered by an approximately uniform distribution of
particles with roughly comparable volumes.
The new grid-free dilatation method is applied to the compressible
oscillating waves in an enclosed tube and a subsonic nozzle flow.
Both one-dimensional and two-dimensional results are shown and
compared with either the exact solutions or the solutions given by
other proven numerical schemes. Good agreement of these results
helps to establish the correctness of the present method. Future
work will accommodate viscous terms and shock waves, which is given
a brief discussion at the end of this thesis.