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Network-theoretic approach to sparsified discrete vortex dynamics

Published online by Cambridge University Press:  10 March 2015

Aditya G. Nair  and
Kunihiko Taira
Aditya G. Nair
Affiliation:
Department of Mechanical Engineering and Florida Center for Advanced Aero-Propulsion, Florida State University, Tallahassee, FL 32310, USA
Kunihiko Taira*
Affiliation:
Department of Mechanical Engineering and Florida Center for Advanced Aero-Propulsion, Florida State University, Tallahassee, FL 32310, USA
*
†Email address for correspondence: [email protected]
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Abstract

We examine discrete vortex dynamics in two-dimensional flow through a network-theoretic approach. The interaction of the vortices is represented with a graph, which allows the use of network-theoretic approaches to identify key vortex-to-vortex interactions. We employ sparsification techniques on these graph representations based on spectral theory to construct sparsified models and evaluate the dynamics of vortices in the sparsified set-up. Identification of vortex structures based on graph sparsification and sparse vortex dynamics is illustrated through an example of point-vortex clusters interacting amongst themselves. We also evaluate the performance of sparsification with increasing number of point vortices. The sparsified-dynamics model developed with spectral graph theory requires a reduced number of vortex-to-vortex interactions but agrees well with the full nonlinear dynamics. Furthermore, the sparsified model derived from the sparse graphs conserves the invariants of discrete vortex dynamics. We highlight the similarities and differences between the present sparsified-dynamics model and reduced-order models.

JFM classification

Mathematical Foundations: Mathematical Foundations Vortex Flows: Vortex dynamics Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 768 , 10 April 2015 , pp. 549 - 571
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Copyright
© 2015 Cambridge University Press 

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References

Ahuja, S. & Rowley, C. W. 2010 Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. J. Fluid Mech. 645, 447–478.Google Scholar
Bagheri, S., Hœpffner, J., Schmid, P. & Henningson, D. 2009 Input–output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62 (2), 020803.Google Scholar
Batchelor, G. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Benczúr, A. & Karger, D. R. 1996 Approximating $s{-}t$ minimum cuts in $O(N^{2})$ time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 47–55. ACM.CrossRefGoogle Scholar
Berkooz, G., Holmes, P. & Lumley, J. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1), 539–575.Google Scholar
Bollobás, B. 1998 Modern Graph Theory. Springer.Google Scholar
Cauchemez, S., Bhattarai, A., Marchbanks, T. L., Fagan, R. P., Ostroff, S., Ferguson, N. M., Swerdlow, D. & working group, Pennsylvania H1N1 2011 Role of social networks in shaping disease transmission during a community outbreak of 2009 H1N1 pandemic influenza. Proc. Natl Acad. Sci. USA 108 (7), 2825–2830.CrossRefGoogle ScholarPubMed
Chen, W. K. 2004 The Electrical Engineering Handbook. Elsevier Science.Google Scholar
Chung, F. 1997 Spectral Graph Theory. American Mathematical Society.Google Scholar
Cottet, G.-H. & Koumoutsakos, P. D. 2000 Vortex Methods: Theory and Practice. Cambridge University Press.CrossRefGoogle Scholar
Duarte-Cavajalino, J. M., Jahanshad, N., Lenglet, C., McMahon, K. L., de Zubicaray, G. I., Martin, N. G., Wright, M. J., Thompson, P. M. & Sapiro, G. 2012 Hierarchical topological network analysis of anatomical human brain connectivity and differences related to sex and kinship. NeuroImage 59, 3784–3804.CrossRefGoogle Scholar
Ellens, W., Spieksma, F. M., Van Mieghem, P., Jamakovic, A. & Kooij, R. E. 2011 Effective graph resistance. Linear Algebr. Applics. 435 (10), 2491–2506.Google Scholar
Glass, R. J., Glass, L. M., Beyeler, W. E. & Min, H. J. 2006 Targeted social distancing design for pandemic influenza. Emerg. Infect. Diseases 12 (11), 1671–1681.Google Scholar
Greengard, L. & Rokhlin, V. 1987 A fast algorithm for particle summations. J. Comput. Phys. 73, 325–348.Google Scholar
Hemati, M., Eldredge, J. D. & Speyer, J. L. 2014 Improving vortex models via optimal control theory. J. Fluids Struct. 49, 91–111.CrossRefGoogle Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Kaiser, E., Noack, B. R., Cordier, L., Spohn, A., Segond, M., Abel, M., Daviller, G. & Niven, R. K. 2014 Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754, 365–414.CrossRefGoogle Scholar
Kelner, J. & Levin, A. 2011 Spectral sparsification in the semi-streaming setting. In Leibniz International Proceedings in Informatics (LIPIcs) Series, vol. 9, pp. 440–451.Google Scholar
Klein, D. J. & Randić, M. 1993 Resistance distance. J. Math. Chem. 12 (1), 81–95.CrossRefGoogle Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37 (3), 289–335.CrossRefGoogle Scholar
Lloyd-Smith, J., Schreiber, S., Kopp, P. & Getz, W. 2005 Superspreading and the effect of individual variation on disease emergence. Nature 438, 355–359.Google Scholar
Ma, Z., Ahuja, S. & Rowley, C. W. 2009 Reduced order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. 25 (1), 233–247.Google Scholar
Mieghem, P. V. 2011 Graph Spectra for Complex Networks. Cambridge University Press.Google Scholar
Mohar, B. 1991 The Laplacian spectrum of graphs. In Graph Theory, Combinatorics, and Applications (ed. Alavi, Y., Chartrand, G., Ollermann, O. & Schwenk, A.), pp. 871–898. Wiley.Google Scholar
Morris, M. 1993 Epidemiology and social networks – modeling structured diffusion. Sociol. Method Res. 22, 99–126.CrossRefGoogle Scholar
Newman, M. E. J. 2004 Fast algorithm for detecting community structure in networks. Phys. Rev. E 69, 066133.Google Scholar
Newman, M. E. J. 2010 Networks: An Introduction. Oxford University Press.CrossRefGoogle Scholar
Newton, P. K. 2001 The N-Vortex Problem: Analytical Techniques, Applied Mathematical Sciences, vol. 145. Springer.CrossRefGoogle Scholar
Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365.Google Scholar
Owen, J. P., Li, Y.-O., Ziv, E., Strominger, Z., Gold, J., Bukhpun, P., Wakahiro, M., Friedman, E. J., Sherr, E. H. & Mukherjee, P. 2013 The structural connectome of the human brain in agenesis of the corpus callosum. NeuroImage 70, 340–355.Google Scholar
Peleg, D. & Ullman, J. 1989 An optimal synchronizer for the hypercube. SIAM J. Sci. Comput. 18 (4), 740–747.Google Scholar
Porter, M. A., Mucha, P. J., Newman, M. E. J. & Warmbrand, C. M. 2005 A network analysis of committees in the US House of Representatives. Proc. Natl Acad. Sci. USA 102 (20), 7057–7062.Google Scholar
Robinson, K., Cohen, T. & Colijn, C. 2012 The dynamics of sexual contact networks: effects on disease spread and control. Theor. Popul. Biol. 81, 89–96.Google Scholar
Rowley, C., Colonius, T. & Murray, R. 2004 Model reduction for compressible flows using POD and Galerkin projection. Physica D 189 (1), 115–129.Google Scholar
Rowley, C. W. 2005 Model reduction for fluids, using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 997–1013.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Salathé, M. & Jones, J. H. 2010 Dynamics and control of diseases in networks with community structure. PLoS Comput. Biol. 6 (4), e1000736.Google Scholar
Spielman, D. A. & Srivastava, N. 2011 Graph sparsification by effective resistances. SIAM J. Sci. Comput. 40 (6), 1913–1926.Google Scholar
Spielman, D. A. & Teng, S.-H. 2011 Spectral sparsification of graphs. SIAM J. Sci. Comput. 40 (4), 981–1025.CrossRefGoogle Scholar
Srivastava, N.2010 Spectral sparsification and restricted invertibility. PhD thesis, Yale University.Google Scholar
Wang, C. & Eldredge, J. 2013 Low-order phenomenological modeling of leading-edge vortex formation. J. Theor. Comput. Fluid Dyn. 27 (5), 577–598.Google Scholar