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Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate

Published online by Cambridge University Press:  24 June 2014

Mark C. Thompson ,
Alexander Radi ,
Anirudh Rao ,
John Sheridan  and
Kerry Hourigan
Mark C. Thompson*
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia
Alexander Radi
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia
Anirudh Rao
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia
John Sheridan
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia
Kerry Hourigan
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia Division of Biological Engineering, Monash University, Melbourne, Victoria 3800, Australia
*
†Email address for correspondence: mark.thompson@monash.edu
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Abstract

While the wake of a circular cylinder and, to a lesser extent, the normal flat plate have been studied in considerable detail, the wakes of elliptic cylinders have not received similar attention. However, the wakes from the first two bodies have considerably different characteristics, in terms of three-dimensional transition modes, and near- and far-wake structure. This paper focuses on elliptic cylinders, which span these two disparate cases. The Strouhal number and drag coefficient variations with Reynolds number are documented for the two-dimensional shedding regime. There are considerable differences from the standard circular cylinder curve. The different three-dimensional transition modes are also examined using Floquet stability analysis based on computed two-dimensional periodic base flows. As the cylinder aspect ratio (major to minor axis) is decreased, mode A is no longer unstable for aspect ratios below 0.25, as the wake deviates further from the standard Bénard–von Kármán state. For still smaller aspect ratios, another three-dimensional quasi-periodic mode becomes unstable, leading to a different transition scenario. Interestingly, for the 0.25 aspect ratio case, mode A restabilises above a Reynolds number of approximately 125, allowing the wake to return to a two-dimensional state, at least in the near wake. For the flat plate, three-dimensional simulations show that the shift in the Strouhal number from the two-dimensional value is gradual with Reynolds number, unlike the situation for the circular cylinder wake once mode A shedding develops. Dynamic mode decomposition is used to characterise the spatially evolving character of the wake as it undergoes transition from the primary Bénard–von Kármán-like near wake into a two-layered wake, through to a secondary Bénard–von Kármán-like wake further downstream, which in turn develops an even longer wavelength unsteadiness. It is also used to examine the differences in the two- and three-dimensional near-wake state, showing the increasing distortion of the two-dimensional rollers as the Reynolds number is increased.

JFM classification

Instability: Absolute/convective instability Wakes/Jets: Vortex streets Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 751 , 25 July 2014 , pp. 570 - 600
Copyright
© 2014 Cambridge University Press 

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References

Albarede, P. & Monkewitz, P. A. 1992 A model for the formation of oblique shedding and Chevron patterns in cylinder wakes. Phys. Fluids A 4 (4), 744–756.Google Scholar
Aleksyuk, A. I., Shkadova, V. P. & Shkadov, V. Y. 2012 Formation, evolution, and decay of a vortex street in the wake of a streamlined body. Moscow Univ. Mech. Bull. 67 (3), 53–61.CrossRefGoogle Scholar
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.Google Scholar
Behara, S. & Mittal, S. 2010 Wake transition in flow past a circular cylinder. Phys. Fluids 22 (11), 114104.CrossRefGoogle Scholar
Bénard, H. 1908 Formation de centres de giration à l’àrriere d’un obstacle en mouvement. C. R. Acad. Sci. Paris 147, 839–842.Google Scholar
Chaurasia, H. K. & Thompson, M. C. 2012 Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate. J. Fluid Mech. 681, 411–433.Google Scholar
Chorin, A. J. 1968 Numerical solution of the Navier–Stokes equations. Maths Comput. 22, 745–762.CrossRefGoogle Scholar
Cimbala, J. M.1984 Large structure in the far wakes of two-dimensional bluff bodies. PhD thesis, Graduate Research Laboratories.Google Scholar
Cimbala, J. M., Nagib, H. M. & Roshko, A. 1988 Large structure in the far wakes of two-dimensional bluff bodies. J. Fluid Mech. 190, 265–298.Google Scholar
Durgin, W. W. & Karlsson, S. K. F. 1971 On the phenomenon of vortex street breakdown. J. Fluid Mech. 48 (03), 507–527.CrossRefGoogle Scholar
Dusek, J., Le Gal, P. & Fraunie, D. P. 1994 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 59–80.CrossRefGoogle Scholar
Eisenlohr, H. & Eckelmann, H. 1989 Vortex splitting and its consequences in the vortex street wake of cylinders at low Reynolds number. Phys. Fluids A 1 (2), 189–192.Google Scholar
Fage, A. & Johansen, F. C. 1927 On the flow of air behind an inclined flat plate of infinite span. Proc. R. Soc. Lond. A 116 (773), 170–197.Google Scholar
Griffith, M. D., Leweke, T., Thompson, M. C. & Hourigan, K. 2008 Steady inlet flow in stenotic geometries: convective and absolute instabilities. J. Fluid Mech. 616, 111–133.Google Scholar
Griffith, M. D., Leweke, T., Thompson, M. C. & Hourigan, K. 2010 Convective instability in steady stenotic flow: optimal transient growth and experimental observation. J. Fluid Mech. 655, 504–514.CrossRefGoogle Scholar
Henderson, R. D. 1995 Details of the drag curve near the onset of vortex shedding. Phys. Fluids 7 (9), 2102–2104.CrossRefGoogle Scholar
Henderson, R. D. 1997 Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech. 352 (1), 65–112.Google Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Johnson, S. A., Thompson, M. C. & Hourigan, K. 2004 Predicted low frequency structures in the wake of elliptical cylinders. Eur. J. Mech. (B/Fluids) 23 (1), 229–239.Google Scholar
Karasudani, T. & Funakoshi, M. 1994 Evolution of a vortex street in the far wake of a cylinder. Fluid Dyn. Res. 14 (6), 331–352.CrossRefGoogle Scholar
Kármán, Th. V. 1911 Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüsseigkeit erfährt. Gött. Nachr. 509–511.Google Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414–443.CrossRefGoogle Scholar
Karniadakis, G. E. & Sherwin, S. J. 2005 Spectral/HP Methods for Computational Fluid Dynamics. Oxford University Press.Google Scholar
Karniadakis, G. E. & Triantafyllou, G. S. 1992 Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. J. Fluid Mech. 238, 1–30.Google Scholar
Kumar, B. & Mittal, S. 2006 Effect of blockage on critical parameters for flow past a circular cylinder. Intl J. Numer. Meth. Fluids 50 (8), 987–1001.CrossRefGoogle Scholar
Kumar, B. & Mittal, S. 2012 On the origin of the secondary vortex street. J. Fluid Mech. 711, 641–666.CrossRefGoogle Scholar
Le Gal, P., Nadim, A. & Thompson, M. C. 2001 Hysteresis in the forced Stuart–Landau equation: application to vortex shedding from an oscillating cylinder. J. Fluids Struct. 15, 445–457.CrossRefGoogle Scholar
Leontini, J. S., Lo Jacono, D. & Thompson, M. C. 2013 Wake states and frequency selection of a streamwise oscillating cylinder. J. Fluid Mech. 730, 162–192.Google Scholar
Leontini, J. S., Thompson, M. C. & Hourigan, K. 2007 Three-dimensional transition in the wake of a transversely oscillating cylinder. J. Fluid Mech. 577, 79–104.Google Scholar
Mamun, C. K. & Tuckerman, L. S. 1995 Asymmetry and Hopf-bifurcation in spherical Couette flow. Phys. Fluids 7, 80–91.Google Scholar
Miller, G. D. & Williamson, C. H. K. 1994 Control of three-dimensional phase dynamics in a cylinder wake. Exp. Fluids 18 (1), 26–35.CrossRefGoogle Scholar
Modi, V. J. & Dikshit, A. K. 1975 Near-wakes of elliptic cylinders in subcritical flow. AIAA J. 13 (4), 490–497.CrossRefGoogle Scholar
Modi, V. J. & Wiland, E. 1970 Unsteady aerodynamics of stationary elliptic cylinders in subcritical flow. AIAA J. 8 (10), 1814–1821.CrossRefGoogle Scholar
Monkewitz, P. A. 1988 The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers. Phys. Fluids 31 (5), 999–1006.Google Scholar
Najjar, F. M. & Balachandar, S. 1998 Low-frequency unsteadiness in the wake of a normal flat plate. J. Fluid Mech. 370, 101–147.Google Scholar
Norberg, C.1987 Effects of Reynolds number and a low-intensity free stream turbulence on the flow around a circular cylinder. PhD thesis, Chalmers University of Technology.Google Scholar
Ota, T., Nishiyama, H. & Taoka, Y. 1987 Flow around an elliptic cylinder in the critical Reynolds number regime. Trans. ASME J. Fluids Engng 109 (2), 149–155.Google Scholar
Radi, A., Thompson, M. C., Sheridan, J. & Hourigan, K. 2013 From the circular cylinder to the flat plate wake: the variation of Strouhal number with Reynolds number for elliptical cylinders. Phys. Fluids 25, 101706.Google Scholar
Rao, A., Leontini, J. S., Thompson, M. C. & Hourigan, K. 2013a Three-dimensionality in the wake of a rotating cylinder in a uniform flow. J. Fluid Mech. 717, 1–29.Google Scholar
Rao, A., Thompson, M. C., Leweke, T. & Hourigan, K. 2013b Dynamics and stability of the wake behind tandem cylinders sliding along a wall. J. Fluid Mech. 722, 291–316.Google Scholar
Roshko, A. 1954 On the Drag and Shedding Frequency of Two-Dimensional Bluff Bodies. National Aeronautics and Space Administration.Google Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127.CrossRefGoogle Scholar
Ryan, K., Thompson, M. C. & Hourigan, K. 2005 Three-dimensional transition in the wake of elongated bluff bodies. J. Fluid Mech. 538, 1–29.Google Scholar
Saha, A. K. 2007 Far-wake characteristics of two-dimensional flow past a normal flat plate. Phys. Fluids 19, 128110.Google Scholar
Saha, A. K. 2013 Direct numerical simulation of two-dimensional flow past a normal flat plate. J. Engng Mech. ASCE 12, 1894–1901.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Schmid, P. J. 2011 Application of the dynamic mode decomposition to experimental data. Exp. Fluids 50, 1123–1130.CrossRefGoogle Scholar
Shintani, K., Umemura, A. & Takano, A. 1983 Low-Reynolds-number flow past an elliptic cylinder. J. Fluid Mech. 136 (1), 277–289.CrossRefGoogle Scholar
Stewart, B. E., Thompson, M. C., Leweke, T. & Hourigan, K. 2010 The wake behind a cylinder rolling on a wall at varying rotation rates. J. Fluid Mech. 648, 225–256.Google Scholar
Strouhal, V. 1878 Üeber eine besondere Art der Tonerregung. Ann. Phys. 241 (10), 216–251.Google Scholar
Strykowski, P. J. & Sreenivasan, K. R. 1990 On the formation and suppression of vortex ‘shedding’ at low Reynolds numbers. J. Fluid Mech. 218, 71–107.Google Scholar
Taneda, S. 1959 Downstream development of the wakes behind cylinders. J. Phys. Soc. Japan 14 (6), 843–848.Google Scholar
Thompson, M. C., Hourigan, K., Cheung, A. & Leweke, T. 2006a Hydrodynamics of a particle impact on a wall. Appl. Math. Model. 30, 1356–1369.Google Scholar
Thompson, M. C., Hourigan, K., Ryan, K. & Sheard, G. J. 2006b Wake transition of two-dimensional cylinders and axisymmetric bluff bodies. J. Fluids Struct. 22 (6–7), 793–806.Google Scholar
Thompson, M. C., Hourigan, K. & Sheridan, J. 1996 Three-dimensional instabilities in the wake of a circular cylinder. Exp. Therm. Fluid Sci. 12 (2), 190–196.Google Scholar
Thompson, M. C., Leweke, T. & Hourigan, K. 2007 Sphere-wall collisions: vortex dynamics and stability. J. Fluid Mech. 575, 121–148.Google Scholar
Thompson, M. C., Leweke, T. & Williamson, C. H. K. 2001 The physical mechanism of transition in bluff body wakes. J. Fluids Struct. 15 (3–4), 607–616.CrossRefGoogle Scholar
Tsuboi, K. & Oshima, Y. 1985 Merging of two-dimensional vortices by the discrete vortex method. J. Phys. Soc. Japan 54 (6), 2137–2145.Google Scholar
Vorobieff, P., Georgiev, D. & Ingber, M. S. 2002 Onset of the second wake: dependence on the Reynolds number. Phys. Fluids 14 (7), L53–L56.Google Scholar
Williamson, C. H. K. 1988 The existence of two stages in the transition to three-dimensionality of a cylinder wake. Phys. Fluids 31 (11), 3165–3168.Google Scholar
Williamson, C. H. K. 1989 Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 206, 579–627.Google Scholar
Williamson, C. H. K. 1996a Three-dimensional wake transition. J. Fluid Mech. 328, 345–407.Google Scholar
Williamson, C. H. K. 1996b Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28 (1), 477–539.Google Scholar
Williamson, C. H. K. & Prasad, A. 1993 A new mechanism for oblique wave resonance in the far wake. J. Fluid Mech. 256, 269–313.Google Scholar
Yang, D., Narasimhamurthy, V. D., Pettersen, B. & Andersson, H. I. 2012 Three-dimensional wake transition behind an inclined flat plate. Phys. Fluids 24 (9), 094107.Google Scholar
Zhang, H.-Q., Fey, U., Noack, B. R., Konig, M. & Eckelmann, H. 1995 On the transition of the cylinder wake. Phys. Fluids 7 (4), 779–794.Google Scholar