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

  • V. Rom-Kedar (a1) (a2), A. Leonard (a3) and S. Wiggins (a4)

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.

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Arnol'd, V. I. 1965 Sur la topologie des ecoulements stationnaires de fluides parfaits. C.R. Acad. Sci. Paris 261, 17.
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).
Aref, H. 1983 Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345.
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1.
Aref, H. & Balachandar, S. 1986 Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515.
Aref, H. & Kambe, T. 1988 Report on the IUTAM Symposium: fundamental aspects of vortex motion. J. Fluid Mech. 190, 571.
Arter, W. 1983 Ergodic streamlines in steady convection. Phys. Lett. 97A, 171.
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.
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.
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.
Chaiken, J., Chu, C. K., Tabor, M. & Tan, Q. M. 1987 Lagrangian turbulence and spatial complexity in Stokes flow. Phys. Fluids 30, 687.
Chien, W.-L, Rising, H. & Ottino, J. M. 1986 Laminar and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355.
Danckwerts, P. V. 1953 Continuous flow systems of residence times, Chem. Engng Sci. 2, 1.
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.
Dyson, F. W. 1893 The potential of an anchor ring–Part II.. Phil. Trans. R. Soc. Lond. A 184, 1041.
Feingold, M., Kadanoff, L. P. & Piro, O. 1988 Passive scalars, 3D volume preserving maps and chaos. J. Statist. Phys. 50, 529.
Fenichel, N. 1971 Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193.
Galloway, D. & Frisch, U. 1986 Dynamo action in a family of flows with chaotic streamlines. Geophys. Astrophys. Fluid Dyn. 36, 53.
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.
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.
Hénon, M. 1966 Sur la topologie des lignes de courant dans un cas particular.. C. R. Acad. Sci. Paris A 262, 312.
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.
Kerstein, A. R. & Ashurst, W. T. 1984 Lognormality of gradients of diffusive scalars in homogeneous, two dimensional mixing systems. Phys. Fluids 27, 2819.
Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 419.
MacKay, R. S., Meiss, J. D. & Percival, I. C. 1984 Transport in Hamiltonian systems,. Physica D 13, 55.
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.
Melnikov, V. K. 1963 On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, 1.
Moffatt, H. K. & Proctor, M. R. E. 1985 Topological constraints associated with fast dynamo action. J. Fluid Mech. 154, 493.
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.
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.
Ralph, M. E. 1986 Oscillatory flows in wavy-walled tubes, J. Fluid Mech. 168, 515.
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).
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.
Sobey, I. J. 1985 Dispersion caused by separation during oscillatory flow through a furrowed channel. Chem. Engng Sci. 40, 2129.
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.
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.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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