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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
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 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 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 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 is part of a larger collection texlive-extra-utils 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.