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Transport and stirring induced by vortex formation

Published online by Cambridge University Press:  23 November 2007

S. C. SHADDEN
Affiliation:
Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA
K. KATIJA
Affiliation:
Graduate Aeronautical Laboratories & Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA
M. ROSENFELD
Affiliation:
Department of Fluid Mechanics and Heat Transfer, Tel Aviv University, Ramat Aviv, 69978, Israel
J. E. MARSDEN
Affiliation:
Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA
J. O. DABIRI
Affiliation:
Graduate Aeronautical Laboratories & Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA

Abstract

The purpose of this study is to analyse the transport and stirring of fluid that occurs owing to the formation and growth of a laminar vortex ring. Experimental data was collected upstream and downstream of the exit plane of a piston-cylinder apparatus by particle-image velocimetry. This data was used to compute Lagrangian coherent structures to demonstrate how fluid is advected during the transient process of vortex ring formation. Similar computations were performed from computational fluid dynamics (CFD) data, which showed qualitative agreement with the experimental results, although the CFD data provides better resolution in the boundary layer of the cylinder. A parametric study is performed to demonstrate how varying the piston-stroke length-to-diameter ratio affects fluid entrainment during formation. Additionally, we study how regions of fluid are stirred together during vortex formation to help establish a quantitative understanding of the role of vortical flows in mixing. We show that identification of the flow geometry during vortex formation can aid in the determination of efficient stirring. We compare this framework with a traditional stirring metric and show that the framework presented in this paper is better suited for understanding stirring/mixing in transient flow problems. A movie is available with the online version of the paper.

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Papers
Copyright
Copyright © Cambridge University Press 2007

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References

Adrian, R. J. 1991 Particle-imaging techniques for experimental fluid-mechanics. Annu. Rev. Fluid Mech. 23, 261304.CrossRefGoogle Scholar
Aref, H. 2002 The development of chaotic advection. Phys. Fluids 14 (4), 1350–1325.CrossRefGoogle Scholar
Arratia, P. E., Voth, G. A. & Gollub, J. P. 2005 Stretching and mixing of non-Newtonian fluids in time-periodic flows. Phys. Fluids 17 (5), 053102.CrossRefGoogle Scholar
Chella, R. & Ottino, J. M. 1985 Stretching in some classes of fluid motions and asymptotic mixing efficiencies as a measure of flow classification. Arch. Rat. Mech. Anal. 90, 1542.CrossRefGoogle Scholar
Dabiri, J. O. 2005 On the estimation of swimming and flying forces from wake measurements. J. Exp Biol. 208, 35193532.CrossRefGoogle ScholarPubMed
Dabiri, J. O. & Gharib, M. 2004 Fluid entrainment by isolated vortex rings. J. Fluid Mech. 511, 311331.CrossRefGoogle Scholar
Didden, N. 1979 On the formation of vortex rings: rolling-up and production of circulation. Z. Angew. Math. Phys. 30, 101116.CrossRefGoogle Scholar
Dudley, R. 2002 The Biomechanics of Insect Flight: Form, Function, Evolution. Princeton University Press.Google Scholar
Eckart, C. 1948 An analysis of the stirring and mixing processes in incompressible fluids. J. Mar. Res. 7, 265275.Google Scholar
Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121140.CrossRefGoogle Scholar
Green, M. A., Rowley, C. W. & Haller, G. 2007 Detection of Lagrangian coherent structures in three-dimensional turbulence. J. Fluid Mech. 572, 111120.CrossRefGoogle Scholar
Haller, G. 2001 Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149, 248277.Google Scholar
Haller, G. 2002 Lagrangian coherent structures from approximate velocity data. Phys. Fluids 14 (6), 18511861.CrossRefGoogle Scholar
Haller, G. & Poje, A. C. 1998 Finite time transport in aperiodic flows. Physica D 119, 352380.Google Scholar
Haller, G. & Yuan, G. 2000 Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147, (3–4), 352370.Google Scholar
Jones, C. K. R. T. & Winkler, S. 2002 Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere. In Handbook of Dynamical Systems (ed. Fiedler, B.), vol. 2, pp. 5592. Elsevier.Google Scholar
Joseph, B. & Legras, B. 2002 Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex. J. Atmos. Sci. 59, 11981212.2.0.CO;2>CrossRefGoogle Scholar
Khakhar, D. V. & Ottino, J. M. 1986 Fluid mixing (stretching) by time periodic sequences of weak flows. Phys. Fluids 29 (11), 35033505.CrossRefGoogle Scholar
Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 419451.CrossRefGoogle Scholar
Koh, T. Y. & Legras, B. 2002 Hyperbolic lines and the stratospheric polar vortex. Chaos 12, 382394.CrossRefGoogle ScholarPubMed
Krasny, R. & Nitsche, M. 2002 The onset of chaos in vortex sheet flow. J. Fluid Mech. 454, 4769.CrossRefGoogle Scholar
Lekien, F., Coulliette, C., Mariano, A. J., Ryan, E. H., Shay, L. K., Haller, G. & Marsden, J. 2005 Pollution release tied to invariant manifolds: a case study for the coast of Florida. Physica D 210, 120.Google Scholar
Leonard, A., Rom-Kedar, V. & Wiggins, S. 1987 Fluid mixing and dynamical systems. Proc. Intl Conf. on the Physics of Chaos and Systems Far from Equilibium, Nuclear Physics B 2, 179190.Google Scholar
Lugt, H. J. 1983 Vortex Flow in Nature and Technology. John Wiley.Google Scholar
Malhotra, N., Mezić, I. & Wiggins, S. 1998 Patchiness: a new diagnostic for Lagrangian trajectory analysis in time-dependent fluid flows. Intl J. Bifurcation Chaos 8, 10531093.CrossRefGoogle Scholar
Mancho, A. M., Small, D., Wiggins, S. & Ide, K. 2003 Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields. Physica D 182, 188222.Google Scholar
Mathiew, G., Mezić, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211, 2346.Google Scholar
Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465495.CrossRefGoogle Scholar
Mezić, I. & Wiggins, S. 1999 A method for visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos 9, 213218.CrossRefGoogle ScholarPubMed
Moore, D. W. & Saffman, P. G. 1973 Axial flow in laminar trailing vortices. Proc. R. Soc. Lond. A 333, 491508.CrossRefGoogle Scholar
Müller, E. A. & Didden, N. 1980 Zur erzeugung der zirkulation bei der bildung eines ringwirbels an einer dusenmundung. Stroj. Casop. 31, 363372.Google Scholar
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139161.CrossRefGoogle Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Ottino, J. M. 1990 Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech. 22, 207253.CrossRefGoogle Scholar
Padberg, K. 2005 Numerical analysis of chaotic transport in dynamical systems. PhD thesis, University of Paderborn.Google Scholar
Pierrehumbert, R. T. & Yang, H. 1993 Global chaotic mixing on isentropic surfaces. J. Atmos. Sci. 50, 24622480.2.0.CO;2>CrossRefGoogle Scholar
Poje, A. C. & Haller, G. 1999 Geometry of cross-stream mixing in a double-gyre ocean model. J. Phys. Oceanogr. 29, 16491665.2.0.CO;2>CrossRefGoogle Scholar
Pullin, D. I. 1979 Vortex ring formation at tube and orifice openings. Phys. Fluids 22, 401403.CrossRefGoogle Scholar
Pullin, D. I. & Saffman, P. G. 1998 Vortex dynamics in turbulence. Annu. Rev. Fluid Mech. 30, 3151.CrossRefGoogle Scholar
Romkedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347394.CrossRefGoogle Scholar
Rosenfeld, M., Rambod, E. & Gharib, M. 1998 Circulation and formation number of laminar vortex rings. J. Fluid Mech. 376, 297318.CrossRefGoogle Scholar
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 721733.CrossRefGoogle Scholar
Shadden, S. C. 2006 A dynamical systems approach to unsteady systems. PhD thesis, California Institute of Technology.Google Scholar
Shadden, S. C., Lekien, F. & Marsden, J. E. 2005 Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212, 271304.Google Scholar
Shadden, S. C., Dabiri, J. O. & Marsden, J. E. 2006 Lagrangian analysis of fluid transport in empirical vortex rings. Phys. Fluids 18 (4), 0471051–11.CrossRefGoogle Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24, U235U279.CrossRefGoogle Scholar
Spedding, G. R., Rosen, M. & Hedenstrom, A. 2003 A family of vortex wakes generated by a thrush nightingale in free flight in a wind tunnel over its entire natural range of flight speeds. J. Exp. Biol. 206, 23132344.CrossRefGoogle Scholar
Tambasco, M. & Steinman, D. A. 2002 On assessing the quality of particle tracking through computational fluid dynamic models. J. Biomech. Engng 124, 166175.CrossRefGoogle ScholarPubMed
Voth, G. A., Haller, G. & Gollub, J. P. 2002 Experimental measurements of stretching fields in fluid mixing. Phys. Rev. Lett. 88, 254501.CrossRefGoogle ScholarPubMed
Weihs, D. 1973 Hydromechanics of fish schooling. Nature 241, 290291.CrossRefGoogle Scholar
Wiggins, S. 2005 The dynamical systems approach to Lagrangian transport in oceanic flows. Annu. Rev. Fluid Mech. 37, 295328.CrossRefGoogle Scholar
Willert, C. E. & Gharib, M. 1991 Digital particle image velocimetry. Exps Fluids 10 (4), 181193.CrossRefGoogle Scholar
Zabusky, N. J. 1999 Vortex paradigm for accelerated inhomogeneous flows: visiometrics for the Rayleigh–Taylor and Richtmyer–Meshkov environments. Annu. Rev. Fluid Mech. 31, 495536.CrossRefGoogle Scholar

Shadden et al. supplementary movie

Movie 1. Forward-time finite-time Lyapunov exponent (FTLE) field computed from axisymmetric CFD vortex formation data for L/D = 2. The revealed Lagrangian coherent structures show how fluid is entrained into the vortex. Also apparent is the rear boundary to the stopping vortex. The integration time used to compute the FTLE was T = 3, and the x- and y-axes are plotted in metres. The location of the cylinder is represented by the solid rectangle. The movie spans from τ = 0 to τ = 1.725.

Download Shadden et al. supplementary movie(Video)
Video 674 KB

Shadden et al. supplementary movie

Movie 1. Forward-time finite-time Lyapunov exponent (FTLE) field computed from axisymmetric CFD vortex formation data for L/D = 2. The revealed Lagrangian coherent structures show how fluid is entrained into the vortex. Also apparent is the rear boundary to the stopping vortex. The integration time used to compute the FTLE was T = 3, and the x- and y-axes are plotted in metres. The location of the cylinder is represented by the solid rectangle. The movie spans from τ = 0 to τ = 1.725.

Download Shadden et al. supplementary movie(Video)
Video 751 KB
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