Numerical Simulation of Flows in Domains with Moving Boundaries
Dipl.-Math. Martin Engel, Prof. Dr. Michael Griebel
In many applications of Computational Fluid Dynamics (CFD) the domain
under consideration is not constant in time, but the boundaries are
moving. There exists a variety of different numerical methods for the
solution of flow problems in time-dependent domains. One widespread
approach is to discretize the Eulerian formulation of the
Navier-Stokes equations on fixed structured grids. The actual fluid
domain is embedded into the grid and described by additional, in
general passively transported, quantities, e.g. massless marker
particles (Marker and Cell), scalar color functions (Volume of Fluid)
or level-sets of higher-dimensional functions. Most of these
techniques allow for an easy tracking of the fluid domain even when it
undergoes changes in topology like merging or breaking up of
interfaces. They are well suited for the simulation of free surface or
multiphase flows. However, since the interface is only implicitly given,
prescription of boundary conditions is difficult and depends on the accurate
reconstruction of the interface.
In this project we use another approach, which corresponds to the
discretization of the Arbitrary Lagrange Euler formulation of the
Navier-Stokes equations. A fixed logical domain built of unit cubes
is mapped onto the fluid domain in physical space by a
time-dependent coordinate transformation. The time-derivative of this
mapping enters the transformed equations as the grid velocity. To
achieve a conservative discretization, additional so-called Geometric
Conservation Laws (GCL) have to be fulfilled.
We implemented a flow solver for the parallel solution of the
incompressible Navier-Stokes equations in three space dimensions. The solver
is based on a Chorin projection method for decoupling velocity and
pressure variables. The discretization is carried out on a staggered
mesh using a finite volume method and weighted grid-oriented velocity
components as the primary variables. The GCL is reformulated and its
discretization is used to compute the mesh velocity. The whole
solution process is parallelized by decomposing the fluid domain in
several structured zones.
The above image demonstrates the block-structured approach which is used to
handle complicated geometries and provides a means for parallelization as well.
Shown is a transported scalar quantity in a flow field around a cylinder.
Click the image to view the corresponding mpeg-animation (mpg, 4.4 MB).
The following series of images shows color-codings of the horizontal
velocity component for a flow through a channel with a moving
indentation. Only the part of the channel downstream from the
indentation is shown. Click on an image to view the mpeg animation (mpg, 350 KB).
M. Engel, M. Griebel. Flow simulation on moving boundary-fitted grids and application to fluid-structure interaction problems. International Journal for Numerical Methods in Fluids, 2005. To Appear.
M. Engel. Numerische Simulation von Strömungen in zeitabhängigen Gebieten und Anwendung auf Fluid-Struktur-Wechselwirkungsprobleme. Diplomarbeit, Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany, 2002.
- Numerical Simulation of Fluid-Structure Interaction