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Inviscid coupling between point symmetric bodies and singular distributions of vorticity

Published online by Cambridge University Press:  08 October 2007

I. EAMES ,
M. LANDERYOU  and
J. B. FLÓR
I. EAMES
Affiliation:
Departments of Mechanical Engineering and Mathematics, University College London, Torrington Place, London, WC1E 7JE, UK
M. LANDERYOU
Affiliation:
Departments of Mechanical Engineering and Mathematics, University College London, Torrington Place, London, WC1E 7JE, UK
J. B. FLÓR
Affiliation:
Laboratoires des Ecoulement Geophysiques et Industriels, BP 53, 38041, Grenoble Cedex 09, France
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Abstract

We study the inviscid coupled motion of a rigid body (of density ρb, in a fluid of density ρ) and singular distributions of vorticity in the absence of gravity, using for illustration a cylinder moving near a point vortex or dipolar vortex, and the axisymmetric interaction between a vortex ring and sphere.

The coupled motion of a cylinder (radius a) and a point vortex, initially separated by a distance R and with zero total momentum, is governed by the parameter R4/(ρb/ρ+1)a4. When R4/(ρb/ρ+1)a4,≤,1, a (positive) point vortex moves anticlockwise around the cylinder which executes an oscillatory clockwise motion, with a mixture of two frequencies, centred around its initial position. When R4/(ρb/ρ+1)a4≫1, the initial velocity of the cylinder is sufficiently large that the dynamics become uncoupled, with the cylinder moving off to infinity. The final velocity of the cylinder is related to the permanent displacement of the point vortex.

The interaction between a cylinder (initially at rest) and a dipolar vortex starting at infinity depends on the distance of the vortex from the centreline (h), the initial separation of the vortical elements (2d), and ρb/ρ. For a symmetric encounter (h=0) with a dense cylinder, the vortical elements pass around the cylinder and move off to infinity, with the cylinder being displaced a finite distance forward. However, when ρb/ρ<1, the cylinder is accelerated forward to such an extent that the vortex cannot overtake. Instead, the cylinder ‘extracts’ a proportion of the impulse from the dipolar vortex. An asymmetric interaction (h>0) leads to the cylinder moving off in the opposite direction to the dipolar vortex.

To illustrate the difference between two- and three-dimensional flows, we consider the axisymmetric interaction between a vortex ring and a rigid sphere. The velocity perturbation decays so rapidly with distance that the interaction between the sphere and vortex ring is localized, but the underlying processes are similar to two-dimensional flows.

We briefly discuss the general implications of these results for turbulent multiphase flows.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 589 , 25 October 2007 , pp. 33 - 56
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Auton, T. R. 1987 The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199–218.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.Google Scholar
Borisov, A. V. & Mamaev, I. S. 2003 An integrability of the problem on the motion of cylinder and vortex in the ideal fluid. Reg. Chaot. Dyn. 8, 163–166.CrossRefGoogle Scholar
Dhanak, M. R. 1981 Interaction between a vortex filament and an approaching rigid sphere. J. Fluid Mech. 110, 129–147.CrossRefGoogle Scholar
Eames, I. & Dalziel, S. B. 2000 Dust resuspension by the flow around an impacting sphere. J. Fluid Mech. 403, 305–328.CrossRefGoogle Scholar
Eames, I. & Hunt, J. C. R. 2004 Forces on bodies moving unsteadily in rapidly compressed flows. J. Fluid Mech. 505, 349–364.CrossRefGoogle Scholar
Flór, J. B. & Eames, I. 2002 Dynamics of monopolar vortices on a topographic beta-plane. J. Fluid Mech. 456, 353–376.CrossRefGoogle Scholar
Govardhan, R. & Williamson, C. H. K. 2002 Resonance forever: existence of a critical mass and an infinite regime of resonance in vortex-induced vibration. J. Fluid Mech. 473, 147–166.CrossRefGoogle Scholar
Hill, R. J., Koch, D. L., Ladd, A. J. C. 2001 Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243–278.CrossRefGoogle Scholar
Howe, M. S. 1995 On the force and moment on a body in an incompressible fluid, with application to rigid bodies and bubbles at high and low Reynolds numbers. Q. J. Mech. Appl. Maths 48, 401–426.CrossRefGoogle Scholar
Howe, M. S. 2003 Theory of Vortex Sound. Cambridge University Press.Google Scholar
Howe, M. S., Lauchle, G. C., Wang, J. 2001 Aerodynamic lift and drag fluctuations of a sphere. J. Fluid Mech. 436, 41–57.CrossRefGoogle Scholar
Hunt, J. C. R. 1987 Vorticity and vortex dynamics in complex turbulent flows. Trans. Can. Soc. Mech. Engrs 11, 21–35.CrossRefGoogle Scholar
Hunt, J. C. R. & Durbin, P. A. 1999 Perturbed vortical layers and shear sheltering. Fluid Dyn. Res. 24, 375–404.CrossRefGoogle Scholar
Juckes, M. N. & McIntyre, M. E. 1987 A high-resolution one-layer model of breaking planetary waves in the stratosphere. Nature 305, 593–600.Google Scholar
Kim, I., Elghobashi, S. & Sirignano, W. A. 1997 Unsteady flow interaction between a pair of advected vortex tubes and a rigid sphere. Intl J. Multiphase Flow 23, 1–23.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Dover.Google Scholar
Lighthill, M. J. 1956 a The image system of a vortex element in a rigid sphere. Proc. Cam. Phil. Soc. 52, 317–321.CrossRefGoogle Scholar
Lighthill, M. J. 1956 b Drift. J. Fluid Mech. 1, 31–53. (Corrigenda 1957 J. Fluid Mech. 2, 311–312.)CrossRefGoogle Scholar
Lighthill, M. J. 1958 Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press.CrossRefGoogle Scholar
Linden, P. F. 1973 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60, 467–480.CrossRefGoogle Scholar
McLain, T. W., Rock, S. M. 1996 Experiments in the hydrodynamic modeling of an underwater manipulator. In Symp. Autonomous Underwater Vehicle Technology, IEEE, pp. 463–469.Google Scholar
Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.CrossRefGoogle Scholar
Masoudi, M. & Sirignano, W. 2002 Collision of a vortex with a vaporising droplet. Intl J. Multiphase Flow 26, 1925–1949.CrossRefGoogle Scholar
Milne-Thompson, L. M. 1968 Theoretical Hydrodynamics. 4th edn. Macmillan.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2002 Path instability of a rising bubble. Phys. Rev. Lett. 88, 1, 014502.CrossRefGoogle ScholarPubMed
Obasaju, E. D., Bearman, P. W., Graham, J. M. R. 1988 A study of forces, circulation and vortex patterns around a circular cylinder in oscillating flow. J. Fluid Mech. 196, 467–494.CrossRefGoogle Scholar
Orlandi, P. 1993 Vortex dipoles impinging circular cylinders. Phys. Fluids. 5, 2196–2206.CrossRefGoogle Scholar
Pedrizzetti, G. 1992 Close interaction between a vortex filament and a rigid sphere. J. Fluid Mech. 245, 701–722.CrossRefGoogle Scholar
Ramodanov, S. M. 2001 Motion of a circular cylinder and a vortex in an ideal fluid. Reg. & Chaot. Dyn. 6, 33–38.CrossRefGoogle Scholar
Ramodanov, S. M. 2002 Motion of a circular cylinder and N point vortices in a perfect fluid. Reg. Chaot. Dyn. 7, 291–298.CrossRefGoogle Scholar
Rossby, C. G. 1948 On displacements and intensity changes of atmospheric vortices. J. Mar. Res. 7, 175–187.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sarpkaya, T. 1963 Lift, drag and added-mass coefficients for a circular cylinder immersed in a time-dependent flow. J. Appl. Mech. 30, 13–18.CrossRefGoogle Scholar
Sarpkaya, T. 1968 An analytical study of separated flow about circular cylinders. Trans. ASME D: J. Basic Engng 90, 511–520.CrossRefGoogle Scholar
Sarpkaya, T. & Garrison, C. J. 1963 Vortex formation and resistance in unsteady flow. J. Appl. Mech. 30, 16–24.CrossRefGoogle Scholar
Shashikanth, B. N. 2005 Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and N point vortices: the case of arbitrary smooth cylinder shapes. Reg. Chaot. Dyn. 10, 1–14.CrossRefGoogle Scholar
Shashikanth, B. N. 2006 Symmetric pairs of point vortices interacting with a neutrally buoyant two-dimensional circular cylinder. Phys. Fluids 18, 127103.CrossRefGoogle Scholar
Shashikanth, B. N., Marsden, J., Burdick, J., Kelly, S. 2002 The Hamilitonian structure of a 2-D rigid cylinder interacting dynamically with N point vortices. Phys. Fluids 14, 1214–1228.CrossRefGoogle Scholar
Smedman, A.-S., Hogstrom, U., Hunt, J. C. R. 2003 Effects of shear sheltering in a stable atmospheric boundary layer with strong shear. Q. J. R. Met. Soc. 130 (A-596), 31–50.CrossRefGoogle Scholar
Squires, K. S. & Simonin, O. 2002 Application of DNS and LES to dispersed turbulent two-phase turbulent flow. Proc. Tenth Workshop on Two-Phase Flow Predictions, Halle, Germany.Google Scholar
Wells, J. C. 1996 A geometrical interpretation of force on a translating body in rotational flow. Phys. Fluids 8, 442–450.CrossRefGoogle Scholar