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Exploring the dynamics of ‘2P’ wakes with reflective symmetry using point vortices

Published online by Cambridge University Press:  13 October 2017

Saikat Basu Open the ORCID record for Saikat Basu [Opens in a new window]  and
Mark A. Stremler
Saikat Basu*
Affiliation:
Computing and Clinical Research Lab, Department of Otolaryngology, The University of North Carolina at Chapel Hill, NC 27599, USA
Mark A. Stremler
Affiliation:
Department of Biomedical Engineering and Mechanics, Virginia Tech, Blacksburg, VA 24061, USA
*
†Email address for correspondence: saikat25@vt.edu
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Abstract

Wakes formed behind bluff bodies frequently reveal complex patterns of coherent vortical structures, with emergence of streamwise spatial periodicity particularly in the mid-wake region. In some cases, the vortex positions also maintain symmetry about the wake centreline. For the case in which two pairs of vortices are generated per shedding cycle, thereby constituting the so-called ‘2P’ mode wake, assumptions of spatial periodicity and symmetry allow for development of a mathematically tractable model using the point-vortex approximation. Our previous work (Basu & Stremler, Phys. Fluids, vol. 27 (10), 2015, 103603) considered staggered 2P wake configurations with two glide-reflective pairs of vortices shed in each period. Here we investigate the dynamics of a spatially periodic point-vortex street consisting of two pairs of vortices arranged with reflective symmetry about the streamwise centreline. Because of the symmetry, it is possible to model the spatially periodic point-vortex dynamics as an integrable Hamiltonian system. For a particular choice of initial condition, the topological structure of the Hamiltonian level curves is determined by location in a circulation–impulse parameter space. These Hamiltonian level curves delineate multiple regimes of motion, with all vortex motions within one regime being qualitatively identical. This approach thus enables identification and a full classification of all possible vortex motions in this constrained system. There also exist a limited number of equilibrium configurations with no relative vortex motion; some of these relative equilibria are neutrally stable to (appropriate) perturbations. Only one such neutrally stable equilibrium configuration continues to preserve the distinct four-vortex array, and numerical experiments indicate that these configurations are also neutrally stable to small perturbations that break the spatial symmetry. We apply this analysis to identify the parameter values necessary for co-existence of two closely spaced, neutrally stable Kármán vortex streets that preserve the assumed symmetry. Finally, comparison of the model dynamics to a wake pattern reported in the literature suggests that the classification of exotic wakes should be based on more details than just the number of vortices periodically shed by the body.

JFM classification

Vortex Flows: Vortex dynamics Wakes/Jets: Vortex streets Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 831 , 25 November 2017 , pp. 72 - 100
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Copyright
© 2017 Cambridge University Press 

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References

Aref, H. 2007 Point vortex dynamics: a classical mathematics playground. J. Math. Phys. 48 (6), 065401.Google Scholar
Aref, H. & Pomphrey, N. 1982 Integrable and chaotic motions of four vortices I. The case of identical vortices. Proc. R. Soc. Lond. A 380 (1779), 359–387.Google Scholar
Aref, H. & Stremler, M. A. 1999 Four-vortex motion with zero total circulation and impulse. Phys. Fluids 11 (12), 3704–3715.CrossRefGoogle Scholar
Aref, H. & Stremler, M. A. 1996 On the motion of three point vortices in a periodic strip. J. Fluid Mech. 314, 1–25.CrossRefGoogle Scholar
Aref, H., Stremler, M. A. & Ponta, F. L. 2006 Exotic vortex wakes – point vortex solutions. J. Fluids Struct. 22 (6–7), 929–940.Google Scholar
Balakumar, B. J., Orlicz, G. C., Tomkins, C. D. & Prestridge, K. P. 2008 Dependence of growth patterns and mixing width on initial conditions in Richtmyer–Meshkov unstable fluid layers. Phys. Scr. T132, 014013.Google Scholar
Basu, S.2014 Dynamics of vortices in complex wakes: modeling, analysis, and experiments. PhD thesis, Virginia Polytechnic Institute and State University.Google Scholar
Basu, S. & Stremler, M. A. 2013 Point vortex modeling of symmetric four vortex wakes. Bull. Am. Phys. Soc. 58 (18), M12.3.Google Scholar
Basu, S. & Stremler, M. A. 2015 On the motion of two point vortex pairs with glide-reflective symmetry in a periodic strip. Phys. Fluids 27 (10), 103603.Google Scholar
Birkhoff, G. & Fisher, J. 1959 Do vortex sheets roll up? Rendiconti del Circolo matematico di Palermo 8 (1), 77–90.Google Scholar
Charney, J. G. 1963 Numerical experiments in atmospheric hydrodynamics. In Experimental Arithmetic, High Speed Computing and Mathematics, Proceedings of Symposia in Applied Mathematics, vol. 15, pp. 289–310. American Mathematical Society.CrossRefGoogle Scholar
Fayed, M., Portaro, R., Gunter, A.-L., Abderrahmane, H. A. & Ng, H. D. 2011 Visualization of flow patterns past various objects in two-dimensional flow using soap film. Phys. Fluids 23 (9), 091104.CrossRefGoogle Scholar
Friedmann, A. & Poloubarinova, P.1928 Über fortschreitende Singularitäten der ebenen Bewegung einer inkompressiblen Flüssigkeit. Recueil de Géophysique Tome V, Fascicule II, Leningrad, 9–23, Russian with German summary.Google Scholar
Helmholtz, H. 1858 Über die integrale der hydrodynamischen gleichungen, welche wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55; transl. by P. G. Tait 1867 On integrals of the hydrodynamical equations which express vortex-motion. Phil. Mag. 33 485–512.Google Scholar
Hoeijmakers, H. W. M. & Vaatstra, W. 1983 A higher order panel method applied to vortex sheet roll-up. AIAA J. 21 (4), 516–523.Google Scholar
von Kármán, T. 1911 Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfärt. 1. Teil. Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 509–517; reprinted in: Collected works of Theodore von Kármán (Butterworth, London, 1956) 1, 324–330.Google Scholar
von Kármán, T. 1912 Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfärt. 2. Teil. Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 547–556; reprinted in: Collected works of Theodore von Kármán (Butterworth, London, 1956) 1, 331–338.Google Scholar
von Kármán, T. & Rubach, H. 1912 Über den Mechanismus des Flüssigkeits- und Luftwiderstandes. Phys. Z. 13, 49–59; reprinted in: Collected works of Theodore von Kármán (Butterworth, London, 1956) 1 339–358.Google Scholar
Kumar, S., Cantu, C. & Gonzalez, B. 2011a Flow past a rotating cylinder at low and high rotation rates. Trans. ASME J. Fluids Engng 133 (4), 041201.Google Scholar
Kumar, S. & Gonzalez, B.2010 Symmetry breaking in the wake behind two rotating cylinders. American Physical Society – Division of Fluid Dynamics Virtual Pressroom, https://www.aps.org/units/dfd/pressroom/gallery/2010/kumar10.cfm.Google Scholar
Kumar, S., Gonzalez, B. & Probst, O. 2011b Flow past two rotating cylinders. Phys. Fluids 23 (1), 014102.Google Scholar
Meleshko, V. V. & Aref, H. 2007 A bibliography of vortex dynamics 1858–1956. Adv. Appl. Mech. 41, 197–292.Google Scholar
Mikaelian, K. O. 1996 Numerical simulations of Richtmyer–Meshkov instabilities in finite-thickness fluid layers. Phys. Fluids 8 (5), 1269–1292.Google Scholar
Newton, P. K. 2001 The N-Vortex Problem: Analytical Techniques. Springer.Google Scholar
Oler, J. W. & Goldschmidt, V. W. 1982 A vortex-street model of the flow in the similarity region of a two-dimensional free turbulent jet. J. Fluid Mech. 123, 523–535.Google Scholar
Ploumhans, P., Winckelmans, G. S., Salmon, J. K., Leonard, A. & Warren, M. S. 2002 Vortex methods for direct numerical simulation of three-dimensional bluff body flows: application to the sphere at Re = 300, 500, and 1000. J. Comput. Phys. 178 (2), 427–463.CrossRefGoogle Scholar
Sarpkaya, T. & Schoaff, R. L. 1979 Inviscid model of two-dimensional vortex shedding by a circular cylinder. AIAA J. 17 (11), 1193–1200.CrossRefGoogle Scholar
Silva, A. L. F., Lima, E., Silveira-Neto, A. & Damasceno, J. J. R. 2003 Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method. J. Comput. Phys. 189 (2), 351–370.CrossRefGoogle Scholar
Stremler, M. A. 2010 On relative equilibria and integrable dynamics of point vortices in periodic domains. Theor. Comput. Fluid Dyn. 24 (1–4), 25–37.Google Scholar
Stremler, M. A. & Basu, S. 2014 On point vortex models of exotic bluff body wakes. Fluid Dyn. Res. 46 (6), 061410.Google Scholar
Stremler, M. A., Salmanzadeh, A., Basu, S. & Williamson, C. H. K. 2011 A mathematical model of 2P and 2C vortex wakes. J. Fluids Struct. 27 (5–6), 774–783.Google Scholar
Williamson, C. H. K. & Govardhan, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413–455.Google Scholar
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2, 355–381.Google Scholar