Identifying non-subjective thresholds for scalar indicators of coherent structures with percolation analysis

This page has been setup to support the preprint titled Geometry and organization of coherent structures in stably stratified atmospheric boundary layers which can be found in arXiv: https://arxiv.org/abs/2110.02253

This idea originated in the work of Moisy & Jiménez 2004, JFM [1] while inspecting vorticity structures in isotropic turbulence and later adopted by Del Alamo & Jiménez 2006, JFM [2] for wall-bounded flows. It makes use of the observation that the ratio of the volume of the largest structure in the domain ($V_{max}$) over the volume of all structures ($V$) undergoes a percolation transition for increasing threshold values ($\tau$). At the lowest $\tau$, a seemingly infinite cluster appears as $V_{max} = V$ (a large structure spans the entire domain) and as $\tau$ is increased, the cluster breaks down into simpler structures. The percolation threshold is defined as the threshold at which the slope of $V_{max}/V$ is maximum.

For wall-bounded flows, a necessary condition is that the scalar indicator has to be normalized with its root-mean-square over wall-parallel planes to ensure that a single, global threshold can be chosen to highlight the structures uniformly (see eq 3.1 of [2]). This was further explored for other scalar indicators of coherent structures such as low-and high-speed streaks, sweeps, ejections, shear layers, backs, and bulges in the work of Harikrishnan et al., 2021 [3].

If you use this code for your work, please cite our arXiv preprint [4]:

@article{harikrishnan2021geometry,
  title={Geometry and organization of coherent structures in stably stratified atmospheric boundary layers},
  author={Harikrishnan, Abhishek and Ansorge, Cedrick and Klein, Rupert and Vercauteren, Nikki},
  journal={arXiv preprint arXiv:2110.02253},
  year={2021}
}

Installation

The code is available as a package from PyPI: https://pypi.org/project/percolationAnalysis/ and acts as a wrapper to the extraction script: https://github.com/Phoenixfire1081/CoherentStructureExtraction. First, install the latter script,

pip install extractstructuresMC

followed by,

pip install percolationAnalysis

Example - Calculating the percolation threshold from the given test data

Download testData.bin from the GitHub website first.

import numpy as np
import array
from percolationAnalysis import percolationAnalysis

# Select file to run tests on
_filenameRead = 'testData.bin'

# What indexing does it use?
# This refers to array order
# See here for a full description: https://docs.oracle.com/cd/E19957-01/805-4940/z400091044d0/index.html
# Python uses C-order indexing (or row-major order). For this set _zFastest = True
_zFastest = False

# Set data related parameters
xlen = 100 
ylen = 50
zlen = 200

# Set precision of binary data. 'f' is 32-bit floating point data.
# For others, see here: https://docs.python.org/3/library/array.html
precision = 'f'

# Use array to read data. This is fast for binary files.
data = array.array(precision)
fr = open(_filenameRead, 'rb')
data.fromfile(fr, (xlen*ylen*zlen))
fr.close()

# Convert to numpy array
data = np.array(data, dtype = np.float32)

# Threshold values for percolation analysis
# Split evenly and take the first 50 values or so
# The transition can be usually seen within this range
# To see a sharper transition, reduce the number of evenly spaced numbers
_threshVals = np.linspace(0, data.max(), 500)[:49]

# Print out details
_verbose = True

# Use Marching Cubes neighbor correction (more computation power required)
_marchingCubesExt = True

percolationObj = percolationAnalysis(_threshVals, data, xlen, ylen, zlen, \
_zFastest, _verbose, _marchingCubesExt)
percolationObj.processThresholds()

# Plot the data and identify the percolation threshold
filename = 'Percolation_threshold.txt'
_min, _max, _mean = percolationObj.plotPercolation(filename)

Running the above script on testData.bin produces the following loglog plot:

The region of percolation transition is clearly visible. The plot highlights the thresholds corresponding to the maximum slope of $V_{max}/V$. Normally, the mean value is used. However, the minimum and maximum are also acceptable choices of threshold.

References

[1] Moisy, Frédéric, and Javier Jiménez. "Geometry and clustering of intense structures in isotropic turbulence." Journal of fluid mechanics 513 (2004): 111-133.

[2] Del Alamo, Juan C., et al. "Self-similar vortex clusters in the turbulent logarithmic region." Journal of Fluid Mechanics 561 (2006): 329-358.

[3] Harikrishnan, Abhishek, et al. "Geometry and organization of coherent structures in stably stratified atmospheric boundary layers." arXiv preprint arXiv:2110.02253 (2021).