mirror of
https://github.com/ParticulateFlow/CFDEMcoupling-PFM.git
synced 2025-12-08 06:37:44 +00:00
9fd4f62f21
This 2D case of vortex shedding in laminar cross-flow demonstrates the concept of recurrence. This is a bit work-in-progress: please check whether this tutorial runs with the recurrence model and tools of CFDEMcoupling, namely rStatAnalysis. The simulation roughly goes through three stages: * The initial solution computed by potentialFoam * A period of symmetric, steady-state flow * Finally, periodic vortex shedding These three stages are clearly visible in the recurrence plot. * We see how not one of the later velocity fields is similar to the initial one * We see the intermediate stage with a symmetric flow field * We see the periodic vortex shedding
158 lines
6.4 KiB
Markdown
158 lines
6.4 KiB
Markdown
# Laminar flow over cylinder - recurrent flow
|
||
|
||
This OpenFOAM case demonstrates the recurring flow patterns of vortex sheding
|
||
of a cylinder in laminar cross flow. Apart from judging the recurrence using
|
||
the eyeball norm, i.e. by looking at the solution, we are computing the
|
||
recurrence matrix from the computed time steps.
|
||
|
||
The recurrence plot [1], a graphical representation of the recurrence matrix,
|
||
allows us to clearly distinguish between the initial phase of the solution and
|
||
the quasi steady state, which exhibits a nice periodic flow.
|
||
|
||
|
||
## Case description
|
||
|
||
The mesh is created with blockMesh using a simple blocking strategy with 6
|
||
blocks. The domain is meshed using one cell in z direction, thus the flow
|
||
simulation is a 2D simulation. The patch on the left acts as inlet, the patch
|
||
on the right is an outlet. The top and bottom patches are frictionless walls.
|
||
The cylindrical void in the domain employs wall boundary conditions.
|
||
|
||
|
||
## Results
|
||
|
||
The image below, is the recurrence plot of this simulation. The recurrence plot
|
||
is based on the recurrence matrix, which in turn was computed from the velocity
|
||
fields of this simulation.
|
||
|
||
Each entry in the recurrence matrix is computed by comparing two states of the
|
||
simulation, i.e. time steps. In this case the velocity fields of all written
|
||
time steps were compared to each other. For each entry, the magnitude of the
|
||
difference of two velocity fields was computed. This magnitude, which itself is
|
||
still a field quantity was summed up over all cell values, and finally
|
||
normalized. Consequently, the values along the main diagonal, i.e. both indices
|
||
for x and y, are zero, as the corresponding state is compared to itself.
|
||
|
||
|
||
|
||
As the simulation is initialized using *potentialFoam*, we have an initial
|
||
velocity field, which is closer to reality than a trivial initial field, i.e.
|
||
zero velocity everywhere. However, the initial velocity field is quite
|
||
unphysical as it contains artifacts due to the mesh, as shown below.
|
||
Notice, in the recurrence plot, the red bands along the x-axis for low values
|
||
of y, and along the y-axis for low values of x. This is a result of the initial
|
||
velocity field bearing no resemblance to all subsequent velocity fields.
|
||
Thus, a horizonal or vertical high-value band is the result of a state, which
|
||
does not re-occur.
|
||
|
||
Thus, a low value indicates great similarity between two time steps. E.g. the
|
||
value at (500,1000) is very low (shown in blue in the plot), which means that
|
||
the velocity fields at the time 500 s and 1000 s are very similar, as shown
|
||
below.
|
||
The whole blue square in the recurrence plot indicates a certain period in
|
||
which the solution changes very little over time. Thus, one could assume, if
|
||
the simulation were to stop aroung t = 1000 s, that the flow field in the
|
||
simulation had entered its steady state.
|
||
|
||
However, as the simulation progresses, vortex shedding kicks in, and we end up
|
||
having a nice, periodic solution. This is indicated by the narrow bands
|
||
parallel to the main diagonal in the upper right corner of the recurrence plot.
|
||
A low-value band parallel to the main diagonal means that the same sequence of
|
||
states can be observed with a certain time shift. A purely periodic system will
|
||
show the exactly same state after completing a full cycle.
|
||
|
||
The narrow bands in this recurrence plot strongly suggests that the laminar
|
||
vortex shedding exhibits periodic behaviour.
|
||
|
||
|
||
### Initial velocity field
|
||
|
||
|
||
t = 0 s
|
||
|
||
### Velocity fields at t = 500 s and t = 1000 s
|
||
|
||
The following two velocity fields are very similar to each other. In the
|
||
recurrence matrix the value for indicating the similarity of these two velocity
|
||
fields is 0.001.
|
||
|
||
|
||
t = 500 s
|
||
|
||
|
||
t = 1000 s
|
||
|
||
### Velocity fields at t = 1500 s, t = 1850 s and t = 1950 s
|
||
|
||
The following two velocity fields are very similar to each other. In the
|
||
recurrence matrix the value for indicating the similarity of these two velocity
|
||
fields is 0.047.
|
||
|
||
|
||
t = 1500 s
|
||
|
||
|
||
t = 1850 s
|
||
|
||
The following velocity field is very dissimilar to the preceeding two, although
|
||
it looks very alike. However, whereas the fields at t = 1500 s and t = 1850 s
|
||
are nearly at the same part of the cycle, these two fields and the field at
|
||
t = 1950 s are on opposite sides of the cycle. The value of the recurrence
|
||
matrix, which indicates similarity is 0.496, respectively 0.488.
|
||
|
||
|
||
t = 1950 s
|
||
|
||
|
||
### Additional notes
|
||
|
||
Apart from running the meshing tools *blockMesh* and *renumberMesh*, the solvers
|
||
*pimpleFoam* and *potentialFoam*, this case runs the tool *rStatAnalysis*, which
|
||
is part of the recurrence model.
|
||
|
||
|
||
## Post-processing
|
||
|
||
This tutorial uses a number of third party software tools for post-processing.
|
||
|
||
[Octave](https://www.gnu.org/software/octave/) is open source under the GPL and
|
||
is available for all platforms. Octave is widely compatible to MATLAB, so you
|
||
can run the post-processing script also with MATLAB.
|
||
|
||
[gnuplot](http://www.gnuplot.info/) is a portable command-line driven graphing
|
||
utility for Linux, MS Windows, OSX, and many other platforms. It is open source
|
||
under its own licence.
|
||
|
||
[TeX Live](https://www.tug.org/texlive/) is an easy way to get up and running
|
||
with the TeX document production system. It provides a comprehensive TeX system
|
||
with binaries for most flavors of Unix, including GNU/Linux, and also Windows.
|
||
It includes all the major TeX-related programs, macro packages, and fonts that
|
||
are free software, including support for many languages around the world.
|
||
|
||
[pdfCrop](https://ctan.org/pkg/pdfcrop) is part of a larger collection
|
||
[texlive-extra-utils](https://launchpad.net/ubuntu/+source/texlive-extra) of
|
||
auxiliary tools for the TeX system. This tool, however, is non essential to
|
||
the functioning of this tutorial. If it is not present or available on your
|
||
system, simply comment out the call to it.
|
||
|
||
The post-processing script has been tested with:
|
||
|
||
* GNU Octave, version 4.0.0
|
||
* gnuplot 5.0
|
||
* pdfTeX 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian)
|
||
* texlive-extra-utils 2015.20160320-1
|
||
|
||
|
||
## Tested
|
||
|
||
This collection of cases has been tested with:
|
||
|
||
* OpenFOAM-5.0
|
||
|
||
|
||
## References
|
||
|
||
[1] T. Lichtenegger and S. Pirker. Recurrence CFD – A novel approach to simulate
|
||
multiphase flows with strongly separated time scales. Chemical Engineering Science,
|
||
153:394-410, 2016.
|