Discrete-orthogonal wavelets
For the purpose of decomposing flow fields with respect to space and scale we make use of the wavelet formalism. We concentrate upon
discrete-orthogonal wavelets for various reasons:
- A Parseval relation lets us argument in terms of energy
contributions from scales/positions.
- Filtering/manipulation in wavelet space is possible.
- The size of the data is not increased as it would be if the
continuous transform were applied in breadth.
Here we document the technical points of the different types of
transform which we have used in the course of our project. Note that
there has been a strong involvement of J.
Fröhlich from the U. of Karlsruhe.
- A simple example of wavelet compression, i.e. non-linear
filtering in wavelet space according to coefficient magnitude, applied
to data from homogeneous-isotropic turbulence at modest grid size. Go
to this
page.
- A parallel algorithm for the discrete orthogonal wavelet
transform, using the slice data model and the MPI
library. Adapted for long filters, i.e. work in Fourier space, like
those associated with spline wavelets. This is PIK Report No. 68 as of
September 2000. It is available as
pdf
document (1.3MB). The images can be viewed
online.
- Wavelets based upon Legendre polynomials, useful for analyzing
data on the interval (as opposed to the classic wavelets for the real
line or the torus). PIK
Report
No. 72 as of July 2001 by J. Fröhlich and M. Uhlmann (29MB).
- Analysis of data on non-uniform grid in bounded domain. A
realistic example of fluid dynamics data: vortex-dipole rebound from a
no-slip wall at circulation-based Reynolds number of ReGamma=771.
- A short
report describing the simulation which uses a Fourier/B-splines method.
- An animated visualization of the vorticity in
MPEG
format (800KB).
- Its 2D wavelet coefficient scalogram, using spline wavelets
in the horizontal direction and Legendre wavelets
vertically. Animations in MPEG format (approx. 1MB) are available
for two different multi-resolution analyses (MRA):
"classical",
i.e. square 2D MRA;
rectangular
2D MRA.
- Wavelets & Turbulent Flow Analysis: an overview in form of a
slide
show
(pdf, 5MB).
- "Local Spectra in Plane Channel Flow Using Wavelets Designed for
the Interval" by M. Uhlmann and J. Fröhlich (Proc. 9th ETC, Southampton, 2002, pp. 111-114). This conference
contribution
shows some applications of the Legendre wavelet basis to turbulent
plane channel flow (local spectra and intermittency index) as well
as outlining some perspectives for achieving a better space
localization. The slides
and the text of
the conference presentation are accessible.
- "Analysis of channel flow using improved polynomial wavelets for
the interval" by M. Uhlmann and J. Fröhlich (Proc. 3rd TSFP,
Sendai, 2003, pp. 841-846). This conference
contribution
shows applications of the new localized Legendre wavelet basis to
turbulent plane channel flow.
markus.uhlmann AT kit.edu
