Hostname: page-component-5d84bcc8dc-x5q6h Total loading time: 0 Render date: 2026-09-13T00:38:52.665Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow complexity in open systems: interlacing complexity index based on mutual information

Published online by Cambridge University Press:  21 July 2017

Jose M. Pozo Open the ORCID record for Jose M. Pozo [Opens in a new window] ,
Arjan J. Geers ,
Maria-Cruz Villa-Uriol  and
Alejandro F. Frangi
Jose M. Pozo*
Affiliation:
CISTIB Center for Computational Imaging and Simulation Technologies in Biomedicine, Department of Electronic and Electrical Engineering, The University of Sheffield, Sheffield S1 3JD, UK
Arjan J. Geers
Affiliation:
CISTIB Center for Computational Imaging and Simulation Technologies in Biomedicine, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Tànger, 122-140, 08018 Barcelona, Spain
Maria-Cruz Villa-Uriol
Affiliation:
CISTIB Center for Computational Imaging and Simulation Technologies in Biomedicine, Department of Electronic and Electrical Engineering, The University of Sheffield, Sheffield S1 3JD, UK
Alejandro F. Frangi
Affiliation:
CISTIB Center for Computational Imaging and Simulation Technologies in Biomedicine, Department of Electronic and Electrical Engineering, The University of Sheffield, Sheffield S1 3JD, UK
*
†Email address for correspondence: j.pozo@sheffield.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Flow complexity is related to a number of phenomena in science and engineering and has been approached from the perspective of chaotic dynamical systems, ergodic processes or mixing of fluids, just to name a few. To the best of our knowledge, all existing methods to quantify flow complexity are only valid for infinite time evolution, for closed systems or for mixing of two substances. We introduce an index of flow complexity coined interlacing complexity index (ICI), valid for a single-phase flow in an open system with inlet and outlet regions, involving finite times. ICI is based on Shannon’s mutual information (MI), and inspired by an analogy between inlet–outlet open flow systems and communication systems in communication theory. The roles of transmitter, receiver and communication channel are played, respectively, by the inlet, the outlet and the flow transport between them. A perfectly laminar flow in a straight tube can be compared to an ideal communication channel where the transmitted and received messages are identical and hence the MI between input and output is maximal. For more complex flows, generated by more intricate conditions or geometries, the ability to discriminate the outlet position by knowing the inlet position is decreased, reducing the corresponding MI. The behaviour of the ICI has been tested with numerical experiments on diverse flows cases. The results indicate that the ICI provides a sensitive complexity measure with intuitive interpretation in a diversity of conditions and in agreement with other observations, such as Dean vortices and subjective visual assessments. As a crucial component of the ICI formulation, we also introduce the natural distribution of streamlines and the natural distribution of world-lines, with invariance properties with respect to the cross-section used to parameterize them, valid for any type of mass-preserving flow.

JFM classification

Biological Fluid Dynamics: Blood flow Mixing: Chaotic advection Mixing: Mixing

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 704 - 742
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alemaskin, K., Camesasca, M., Manas-Zloczower, I. & Kaufman, M. 2004 Entropic measures of mixing tailored for various applications. In Materials Processing and Design: Modeling, Simulation and Applications — NUMIFORM 2004, AIP Conference Proceedings, vol. 712, pp. 169–173. AIP Publishing.Google Scholar
Aneurist Consortium 2010 The @neurIST Integrated Project. http://www.aneurist.org.Google Scholar
Aref, H., Blake, J. R., Budišić, M., Cartwright, J. H. E., Clercx, H. J. H., Feudel, U., Golestanian, R., Gouillart, E., Guer, Y. L., van Heijst, G. F. et al. 2014 Frontiers of chaotic advection. arXiv:1403.2953.Google Scholar
Arzani, A., Gambaruto, A. M., Chen, G. & Shadden, S. C. 2016 Lagrangian wall shear stress structures and near-wall transport in high-Schmidt-number aneurysmal flows. J. Fluid Mech. 790, 158–172.CrossRefGoogle Scholar
Bogunović, H., Pozo, J. M., Villa-Uriol, M. C., Majoie, C. B. L. M., van den Berg, R., Gratama van Andel, H. A. F., Macho, J. M., Blasco, J., San Román, L. & Frangi, A. F. 2011 Automated segmentation of cerebral vasculature with aneurysms in 3DRA and TOF-MRA using geodesic active regions: an evaluation study. Med. Phys. 38, 210.Google Scholar
Brandani, G. B., Schor, M., Macphee, C. E., Grubmüller, H., Zachariae, U. & Marenduzzo, D. 2013 Quantifying disorder through conditional entropy: an application to fluid mixing. PLoS ONE 8 (6), e65617.Google Scholar
Cahill, N. D. 2010 Normalized measures of mutual information with general definitions of entropy for multimodal image registration. In Biomedical Image Registration, Lecture Notes in Computer Science, vol. 6204, pp. 258–268. Springer.CrossRefGoogle Scholar
Camesasca, M., Kaufman, M. & Manas-Zloczower, I. 2006 Quantifying fluid mixing with the Shannon entropy. Macromol. Theory Simul. 15 (8), 595–607.CrossRefGoogle Scholar
Camesasca, M. & Manas-Zloczower, I. 2009 Danckwerts revisited – the use of entropy to define scale and intensity of segregation. Macromol. Theory Simul. 18 (2), 87–96.Google Scholar
Cebral, J. R., Mut, F., Weir, J. & Putman, C. M. 2011 Association of hemodynamic characteristics and cerebral aneurysm rupture. Am. J. Neuroradiol. 32 (2), 264–270.Google Scholar
Chen, S., Ma, B. & Zhang, K. 2009 On the similarity metric and the distance metric. Theor. Comput. Sci. 410 (24), 2365–2376.Google Scholar
Childs, P. R. N. 2010 Rotating Flow. Elsevier.Google Scholar
Danckwerts, P. V. 1952 The definition and measurement of some characteristics of mixtures. Appl. Sci. Res. A 3 (4), 279–296.Google Scholar
Domingues, N., Gaspar-Cunha, A. & Covas, J. A. 2008 Global mixing indices for single screw extrusion. Intl J. Mater. Forming 1 (1), 723–726.Google Scholar
Ford, M. D., Hoi, Y., Piccinelli, M., Antiga, L. & Steinman, D. A. 2009 An objective approach to digital removal of saccular aneurysms: technique and applications. Brit. J. Radiol. 82, S55–S61; special issue 1.Google Scholar
Funakoshi, M. 2008 Chaotic mixing and mixing efficiency in a short time. Fluid Dyn. Res. 40 (1), 1–33.CrossRefGoogle Scholar
Geers, A. J., Larrabide, I., Radaelli, A. G., Bogunovic, H., Kim, M., van Andel, H. A. F. G., Majoie, C. B., VanBavel, E. & Frangi, A. F. 2011 Patient-specific computational hemodynamics of intracranial aneurysms from 3D rotational angiography and CT angiography: an in vivo reproducibility study. Am. J. Neuroradiol. 32 (3), 581–586.Google Scholar
Gopalakrishnan, S. S., Pier, B. & Biesheuvel, A. 2014 Dynamics of pulsatile flow through model abdominal aortic aneurysms. J. Fluid Mech. 758, 150–179.Google Scholar
Guida, A., Nienow, A. W. & Barigou, M. 2010 Shannon entropy for local and global description of mixing by Lagrangian particle tracking. Chem. Engng Sci. 65 (10), 2865–2883.Google Scholar
Hollander, M., Wolfe, D. A. & Chicken, E. 2013 Nonparametric Statistical Methods. John Wiley & Sons.Google Scholar
Jang, B. & Funakoshi, M. 2010 Chaotic mixing in a helix-like pipe with periodic variations in curvature and torsion. Fluid Dyn. Res. 42 (3), 035506.Google Scholar
Jaynes, E. T. 1968 Prior probabilities. IEEE Trans. Syst. Sci. Cybern. 4 (3), 227–241.CrossRefGoogle Scholar
Jaynes, E. T. 2003 Probability Theory: The Logic of Science. Cambridge University Press.Google Scholar
Koenker, R. 1981 A note on studentizing a test for heteroscedasticity. J. Econ. 17 (1), 107–112.Google Scholar
Kolmogorov, A. 1956 On the Shannon theory of information transmission in the case of continuous signals. IRE Trans. Inform. Theory 2 (4), 102–108.Google Scholar
Kruijt, P. G. M., Galaktionov, O. S., Anderson, P. D., Peters, G. W. M. & Meijer, H. E. H. 2001 Analyzing mixing in periodic flows by distribution matrices: mapping method. AIChE J. 47 (5), 1005–1015.Google Scholar
Lee, K., Parker, K., Caro, C. & Sherwin, S. 2008 The spectral/hp element modelling of steady flow in non-planar double bends. Intl J. Numer. Methods Fluids 57 (5), 519–529.Google Scholar
Lin, Z., Thiffeault, J.-L. & Doering, C. R. 2011 Optimal stirring strategies for passive scalar mixing. J. Fluid Mech. 675, 465–476.CrossRefGoogle Scholar
Ling, F. H. 1994 Interpolated Poincaré map and application to mixing problems. Chaos, Solitons Fractals 4 (5), 681–692.Google Scholar
Loskutov, A. 2010 Fascination of chaos. Phys. Usp. 53 (12), 1257.CrossRefGoogle Scholar
Lucas, D. & Kerswell, R. 2014 Spatiotemporal dynamics in two-dimensional kolmogorov flow over large domains. J. Fluid Mech. 750, 518–554.CrossRefGoogle Scholar
Lueptow, R. M., Docter, A. & Min, K. 1992 Stability of axial flow in an annulus with a rotating inner cylinder. Phys. Fluids A 4 (11), 2446–2455.CrossRefGoogle Scholar
Mann, H. B. & Whitney, D. R. 1947 On a test of whether one of two random variables is stochastically larger than the other. Ann. Math. Stat. 18 (1), 50–60.Google Scholar
Mathew, G. & Mezić, I. 2011 Metrics for ergodicity and design of ergodic dynamics for multi-agent systems. Physica D 240 (4), 432–442.Google Scholar
Mathew, G., Mezić, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211 (1–2), 23–46.Google Scholar
McDaid, A. F., Greene, D. & Hurley, N.2011 Normalized mutual information to evaluate overlapping community finding algorithms. arXiv:1110.2515.Google Scholar
Millan, R. D., Dempere-Marco, L., Pozo, J. M., Cebral, J. R. & Frangi, A. F. 2007 Morphological characterization of intracranial aneurysms using 3-D moment invariants. IEEE Trans. Med. Imaging 26 (9), 1270–1282.Google Scholar
Nuttall, A. H. 1975 Some integrals involving the Q M function (corresp.). IEEE Trans. Inform. Theory 21 (1), 95–96.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge Texts in Applied Mathematics, vol. 3. Cambridge University Press.Google Scholar
Ottino, J. M. 1990 Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech. 22 (1), 207–254.Google Scholar
Paninski, L. 2003 Estimation of entropy and mutual information. Neural Comput. 15 (6), 1191–1253.Google Scholar
Piccinelli, M., Veneziani, A., Steinman, D., Remuzzi, A. & Antiga, L. 2009 A framework for geometric analysis of vascular structures: application to cerebral aneurysms. IEEE Trans. Med. Imaging 28 (8), 1141–1155.Google Scholar
Piessens, R. 1983 Quadpack: A Subroutine Package for Automatic Integration, Springer Series in Computational Mathematics, vol. 1. Springer.CrossRefGoogle Scholar
R Core Team 2013 R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing.Google Scholar
Reymond, P., Merenda, F., Perren, F., Rüfenacht, D. & Stergiopulos, N. 2009 Validation of a one-dimensional model of the systemic arterial tree. Am. J. Physiol. Heart Circ. Physiol. 297 (1), H208–H222.Google Scholar
Robert, C. & Casella, G. 2013 Monte Carlo Statistical Methods. Springer Science & Business Media.Google Scholar
Sandri, M. 1996 Numerical calculation of Lyapunov exponents. Mathematica J. 6 (3), 78–84.Google Scholar
Schnell, S., Ansari, S. A., Vakil, P., Wasielewski, M., Carr, M. L., Hurley, M. C., Bendok, B. R., Batjer, H., Carroll, T. J., Carr, J. et al. 2014 Three-dimensional hemodynamics in intracranial aneurysms: influence of size and morphology. J. Magn. Reson. Imaging 39 (1), 120–131.CrossRefGoogle ScholarPubMed
Schroeder, W., Martin, K. & Lorensen, B. 2003 The Visualization Toolkit: An Object Oriented Approach to 3D Graphics, 3rd edn. Kitware Inc.Google Scholar
Scott, S. E., Redd, T. C., Kuznetsov, L., Mezić, I. & Jones, C. K. R. T. 2009 Capturing deviation from ergodicity at different scales. Physica D 238 (16), 1668–1679.Google Scholar
Shannon, C. E. 1948 A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423.Google Scholar
Shannon, C. E. & Weaver, W. 1949 The Mathematical Theory of Information. University of Illinois Press.Google Scholar
Tang, X. Z. & Boozer, A. H. 1996 Finite time Lyapunov exponent and advection-diffusion equation. Physica D 95 (3), 283–305.CrossRefGoogle Scholar
Tarantola, A. 2005 Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics.Google Scholar
Tarantola, A. & Mosegaard, K.2007 Mapping of probabilities theory for the interpretation of uncertain physical measurements. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.177.9042.Google Scholar
Thiffeault, J.-L. 2012 Using multiscale norms to quantify mixing and transport. Nonlinearity 25 (2), R1.CrossRefGoogle Scholar
Tucker, C. L. & Peters, G. W. M. 2003 Global measures of distributive mixing and their behavior in chaotic flows. Korea-Australia Rheol. J. 15 (4), 197–208.Google Scholar
Villa-Uriol, M. C., Berti, G., Hose, D. R., Marzo, A., Chiarini, A., Penrose, J., Pozo, J. M., Schmidt, J. G., Singh, P., Lycett, R. et al. 2011 @neurist complex information processing toolchain for the integrated management of cerebral aneurysms. Interface Focus 1 (3), 308–319.CrossRefGoogle ScholarPubMed
Vinh, N. X., Epps, J. & Bailey, J. 2010 Information theoretic measures for clusterings comparison: variants, properties, normalization and correction for chance. J. Machine Learning Res. 11, 2837–2854.Google Scholar
VTK community 2014 VTK visualization toolkit. http://www.vtk.org.Google Scholar
Wereley, S. T. & Lueptow, R. M. 1999 Velocity field for Taylor–Couette flow with an axial flow. Phys. Fluids 11 (12), 3637–3649.CrossRefGoogle Scholar
Wiggins, S. & Ottino, J. M. 2004 Foundations of chaotic mixing. Phil. Trans. R. Soc. Lond. A 362 (1818), 937–970.Google Scholar
Xia, H. M., Shu, C., Chew, Y. T. & Wang, Z. P. 2010 Approximate mapping method for prediction of chaotic mixing in spatial-periodic microchannel. Chem. Engng Res. Des. 88 (10), 1419–1426.Google Scholar
Zu, P., Chen, L. & Xin, J. 2015 A computational study of residual kpp front speeds in time-periodic cellular flows in the small diffusion limit. Physica D 311, 37–44.Google Scholar