Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-29T13:41:05.719Z Has data issue: false hasContentIssue false
  • English
  • Français

An analytical study of transport, mixing and chaos in an unsteady vortical flow

Published online by Cambridge University Press:  26 April 2006

V. Rom-Kedar ,
A. Leonard  and
S. Wiggins
V. Rom-Kedar
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Present address: The James Franck Institute, The University of Chicago, Chicago, IL 60637, USA.
A. Leonard
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
S. Wiggins
Affiliation:
Applied Mechanics, California Institute of Technology, Pasadena, CA 91125, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 214 , May 1990 , pp. 347 - 394
Copyright
© 1990 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.)

References

Arnol'd, V. I. 1965 Sur la topologie des ecoulements stationnaires de fluides parfaits. C.R. Acad. Sci. Paris 261, 17.Google Scholar
Arnol'd, V. I. 1973 Ordinary Differential Equations. M.I.T. Press.
Arnol'd, V. I. 1978 Mathematical Methods of Classical Mechanics. Springer.
Arnol'd, V. I. 1982 Geometrical Methods in the Theory of Ordinary Differential Equations. Springer.
Arnol'd, V. I. & Avez, A. 1968 Ergodic Problems of Classical Mechanics. W. A. Benjamin.
Arnol'd, V. I. & Korkine, E. I. 1983 The growth of a magnetic field in a steady compressible flow. Vestn. Mosk. Univ. Mat. Mekh. 3, 43 (in Russian).Google Scholar
Aref, H. 1983 Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345.Google Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1.Google Scholar
Aref, H. & Balachandar, S. 1986 Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515.Google Scholar
Aref, H. & Kambe, T. 1988 Report on the IUTAM Symposium: fundamental aspects of vortex motion. J. Fluid Mech. 190, 571.Google Scholar
Arter, W. 1983 Ergodic streamlines in steady convection. Phys. Lett. 97A, 171.Google Scholar
Aubry, S. & Le Daeron, P. R. 1983 The discrete Frenkel-Kontorova model and its extensions. 1, Exact results for the ground states. Physica 8D, 381.Google Scholar
Broadwell, J. E. 1989 A model for reactions in turbulent jets: effects of Reynolds, Schmidt, and Damkohler numbers. In Turbulent Reactive Flows. Lecture Notes in Engineering, Vol. 40 (ed. R. Borghi & S. N. B. Murthy), pp. 251, Springer.
Broomhead, D. S. & Ryrie, S. C. 1988 Particle paths in wavy vortices. Nonlinearity 1, 409.Google Scholar
Chaiken, J., Chevray, R., Tabor, M. & Tan, Q. M. 1986 Experimental study of Lagrangian turbulence in Stokes flow.. Proc. R. Soc. London A 408, 105.Google Scholar
Chaiken, J., Chu, C. K., Tabor, M. & Tan, Q. M. 1987 Lagrangian turbulence and spatial complexity in Stokes flow. Phys. Fluids 30, 687.Google Scholar
Chien, W.-L, Rising, H. & Ottino, J. M. 1986 Laminar and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355.Google Scholar
Danckwerts, P. V. 1953 Continuous flow systems of residence times, Chem. Engng Sci. 2, 1.Google Scholar
Dimotakis, P. E. 1989 Turbulent shear layer mixing with fast chemical reactions. In Turbulent Reactive Flows. Lecture Notes in Engineering, Vol. 40 (ed. R. Borghi & S. N. B. Murthy), p. 417, Springer.
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353.Google Scholar
Dyson, F. W. 1893 The potential of an anchor ring–Part II.. Phil. Trans. R. Soc. Lond. A 184, 1041.Google Scholar
Feingold, M., Kadanoff, L. P. & Piro, O. 1988 Passive scalars, 3D volume preserving maps and chaos. J. Statist. Phys. 50, 529.Google Scholar
Fenichel, N. 1971 Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193.Google Scholar
Galloway, D. & Frisch, U. 1986 Dynamo action in a family of flows with chaotic streamlines. Geophys. Astrophys. Fluid Dyn. 36, 53.Google Scholar
Goldhirsch, I., Sulem, P. L. & Orszag, S. A. 1987 Stability and Lyapunov stability of dynamical systems: a different approach and a numerical method. Physica 27D, 311.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Non-Linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Hama, F. R. 1962 Streaklines in a perturbed shear flow. Phys. Fluids 5, 644.Google Scholar
Hénon, M. 1966 Sur la topologie des lignes de courant dans un cas particular.. C. R. Acad. Sci. Paris A 262, 312.Google Scholar
Hirsch, M. W., Pugh, C. C. & Shub, M. 1977 Invariant Manifolds. Lecture Notes in Mathematics, vol. 583, Springer.
Irwin, M. C. 1980 Smooth Dynamical Systems. Academic.
Kerr, R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31.Google Scholar
Kerstein, A. R. & Ashurst, W. T. 1984 Lognormality of gradients of diffusive scalars in homogeneous, two dimensional mixing systems. Phys. Fluids 27, 2819.Google Scholar
Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 419.Google Scholar
MacKay, R. S., Meiss, J. D. & Percival, I. C. 1984 Transport in Hamiltonian systems,. Physica D 13, 55.Google Scholar
Marble, F. E. 1985 Growth of a diffusion flame in the field of a vortex. In Recent Advances in the Aerospace Sciences (ed. C. Casci), pp. 395, Plenum.
Mather, J. N. 1984 Non-existence of invariant circles. Ergod. Theory Dyn. Syst. 4, 301.Google Scholar
Melnikov, V. K. 1963 On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, 1.Google Scholar
Moffatt, H. K. & Proctor, M. R. E. 1985 Topological constraints associated with fast dynamo action. J. Fluid Mech. 154, 493.Google Scholar
Moser, J. 1973 Stable and Random Motions in Dynamical Systems. Princeton University Press.
Ottino, J. M. 1988 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.
Ottino, J. M., Leong, C. W., Rising, H. & Swanson, P. D. 1988 Morphological structures produced by mixing in chaotic flows. Nature 333, 419.Google Scholar
Percival, I. C. 1980 Variational principles for invariant tori and cantori. AIP Conf. Proc., Nonlinear Dynamics and the Beam-Beam Interaction, Vol. 57 (ed. M. Mouth & J. C. Herrera). New York: Am. Inst. Phys.
Pope, S. B. 1987 Turbulent premixed flames. Ann. Rev. Fluid Mech. 19, 237.Google Scholar
Ralph, M. E. 1986 Oscillatory flows in wavy-walled tubes, J. Fluid Mech. 168, 515.Google Scholar
Rom-Kedar, V. 1988 Part I: An analytical study of transport, mixing and chaos in an unsteady vortical flow. Part II: Transport in two dimensional maps. Ph.D. thesis, California Institute of Technology, Pasadena, CA.
Rom-Kedar, V. & Wiggins, S. 1989 Transport in two-dimensional maps. Arch. Rat. Mech. Anal. (to appear).Google Scholar
Shariff, K. 1989 Dynamics of a class of vortex rings. Ph.D thesis. Dept. of Mech. Engng, Stanford University.
Shariff, K., Leonard, A., Zabusky, N. J. & Ferziger, J. H. 1988 Acoustics and dynamics of coaxial interacting vortex rings. Fluid Dyn. Res. 3, 337.Google Scholar
Sobey, I. J. 1985 Dispersion caused by separation during oscillatory flow through a furrowed channel. Chem. Engng Sci. 40, 2129.Google Scholar
Suresh, A. 1985 Point vortex interactions, Ph.D thesis, Princeton University.
Wiggins, S. 1987 Chaos in the quasiperiodically forced duffing oscillator. Phys. Lett. 124A, 138.Google Scholar
Wiggins, S. 1988 Global Bifurcations and Chaos–Analytical Methods. Springer.
Yamada, H. & Matsui, T. 1978 In An Album of Fluid Motion (ed. M. Van Dyke), p. 46. Parabolic.