Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-09T22:03:59.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Influence of aspect ratio on the dynamics of a freely moving circular disk

Published online by Cambridge University Press:  02 June 2010

A. R. SHENOY  and
C. KLEINSTREUER
A. R. SHENOY*
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
C. KLEINSTREUER
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The influence of aspect ratio (χ = diameter/thickness) on the vortex shedding behaviour of fixed, and freely moving, circular disk has been investigated numerically. The aspect ratio significantly changes the structure of the vortices shed from the disk, thus altering the fluid induced forces. Disks of χ = 2 and 4 were selected, and by choosing Re = 240 periodic behaviour was observed for both the ‘fixed’ and ‘freely’ moving disks. First, the vortex structures shed from a ‘fixed’ circular disk of χ = 2 and 4 were computed for Re = 240. This was followed by a computation of their trajectories falling ‘freely’ under the action of gravity at Re = 240. For the ‘fixed’ disk of χ = 2, periodic shedding of one-sided hairpin-shaped vortex loops was observed. The flow field had a spatial planar symmetry and the vortices were shed from the same location, resulting in an asymmetric lateral force on the disk. The Strouhal number (St), calculated using the fluctuation in the axial velocity in the far-wake, was 0.122. This vortex shedding behaviour is referred to as the ‘single-sided’ vortex shedding mode. For the ‘fixed’ disk of χ = 4, periodic shedding of hairpin-shaped vortex loops was observed from the diametrically opposite location of the disk. The flow field had a spatial planar symmetry, and also a spatio-temporal one, with respect to a plane orthogonal to the spatial symmetry plane. The shed vortices induced a symmetric lateral force on the disk with a zero mean. The computed Strouhal number, was equal 0.122, same as that for χ = 2. This vortex shedding behaviour is referred as the ‘double-sided’ vortex shedding mode. For the ‘freely falling’ disk of χ = 2, an oscillatory motion was observed in a plane with a 83° phase lag between the lateral and angular velocity. The Strouhal number (Stb), calculated using the oscillation frequency of the ‘freely’ falling disk was equal to 0.116, which is comparable to the St of the fixed disk. For a ‘freely falling’ disk of χ = 4, oscillatory motion was observed in a plane with a 21° phase lag between the lateral and angular velocity. The Strouhal number (Stb) was equal to 0.171, which differs from the St observed in the wake of the fixed disk.

JFM classification

Aerodynamics: Flow-structure interactions Vortex Flows: Vortex shedding
Type
Papers
Information
Journal of Fluid Mechanics , Volume 653 , 25 June 2010 , pp. 463 - 487
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Abelev, A. V., Valent, P. J. & Holland, K. T. 2007 Behaviour of a large cylinder in free-fall through water. IEEE J. Ocean. Engng 32 (1), 10–20.CrossRefGoogle Scholar
Achenbach, E. 1974 Vortex shedding from spheres. J. Fluid Mech. 62, 209–221.CrossRefGoogle Scholar
Auguste, F., Fabre, D. & Magnaudet, J. 2010 Bifurcations in the wake of a thick circular disk. Theor. Comput. Fluid Dyn. 24, 305–313.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.CrossRefGoogle Scholar
Barth, T. & Jespersen, D. 1989 The design and application of upwind schemes on unstructured meshes. AIAA Paper 89–0366.CrossRefGoogle Scholar
Belmonte, A., Eisenberg, H. & Moses, E. 1998 From flutter to tumble: inertial drag and Froude similarity in falling paper. Phys. Rev. Lett. 81 (2), 345–348.CrossRefGoogle Scholar
Berger, E., Scholz, D. & Schumm, M. M. 1990 Coherent vortex structures in the wake of a sphere and a circular disk at rest and under forced vibrations. J. Fluids Struct. 4, 231–257.CrossRefGoogle Scholar
Brücker, C. 2001 Spatio-temporal reconstruction of vortex dynamics in axisymmetric wakes. J. Fluids Struct. 15 (3–4), 543–554.CrossRefGoogle Scholar
Crowder, T. M., Rosati, J. A., Schroeter, J. D., Hickey, A. J. & Martonen, T. B. 2002 Fundamental effects of particle morphology on lung delivery: predictions of Stokes law and the particular relevance to dry powder inhaler formulation and development. Pharm. Res. 19 (3), 239–245.CrossRefGoogle ScholarPubMed
Demirdzć, I. & Peric, M. 1988 Space conservation law in finite volume calculations of fluid flow. Intl J. Numer. Methods Fluids 8 (9), 1037–1050.CrossRefGoogle Scholar
Demirdzić, I. & Peric, M. 1990 Finite volume method for prediction of fluid flow in arbitrarily shaped domains with moving boundaries. Intl J. Numer. Methods Fluids 10 (7), 771–790.CrossRefGoogle Scholar
Fernandes, P. C., Ern, P., Risso, F. & Magnaudet, J. 2005 On the zigzag dynamics of freely moving axisymmetric bodies. Phys. Fluids 17 (9), 098107.CrossRefGoogle Scholar
Fernandes, P. C., Ern, P., Risso, F. & Magnaudet, J. 2008 Dynamics of axisymmetric bodies rising along a zigzag path. J. Fluid Mech. 606, 209–223.CrossRefGoogle Scholar
Fernandes, P. C., Risso, F. R., Ern, P. & Magnaudet, J. 2007 Oscillatory motion and wake instability of freely rising axisymmetric bodies. J. Fluid Mech. 573, 479–502.CrossRefGoogle Scholar
Field, S. B., Klaus, M., Moore, M. G. & Nori, F. 1997 Chaotic dynamics of falling disks. Nature 388, 252–254.CrossRefGoogle Scholar
Goldstein, H., Poole, C. & Safko, J. 2002 Classical Mechanics. Addison Wesley.Google Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.CrossRefGoogle Scholar
Jenny, M., Dušek, J. & Bouchet, G. 2004 Instabilities and transition of a sphere falling or ascending freely in a newtonian fluid. J. Fluid Mech. 508, 201–239.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. M. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70.CrossRefGoogle Scholar
Kleinstreuer, C. 2006 Biofluid Dynamics: Principles and Selected Applications. CRC Press, Taylor & Francis Group.Google Scholar
Kleinstreuer, C., Zhang, Z. & Donohue, J. F. 2008 Targeted drug-aerosol delivery in the human respiratory system. Annu. Rev. Biomed. Engng 10 (1), 195–220.CrossRefGoogle ScholarPubMed
Loewenberg, M. 1993 Stokes resistance, added mass, and basset force for arbitrarily oriented, finite-length cylinders. Phys. Fluids 5 (3), 765–767.CrossRefGoogle Scholar
Magarvey, R. H. & Maclatchy, C. S. 1965 Vortices in sphere wakes. Can. J. Phys. 43, 1649–1656.CrossRefGoogle Scholar
Marchildon, E. K., Clamen, A. & Gauvin, W. H. 1964 Drag and oscillatory motion of freely falling cylindrical particles. Can. J. Chem. Engng 42, 178–182.CrossRefGoogle Scholar
Milne-Thomson, L. M. 1996 Theoretical Hydrodynamics. Dover.Google Scholar
Mordant, N. & Pinton, J. F. 2000 Velocity measurement of a settling sphere. Eur. Phys. J. B 18 (2), 343–352.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2002 The generalized Kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow. Intl J. Multiph. Flow 28 (11), 1837–1851.CrossRefGoogle Scholar
Perry, A. E., Lim, T. T. & Chong, M. S. 1980 The instantaneous velocity fields of coherent structures in coflowing jets and wakes. J. Fluid Mech. 101 (02), 243–256.CrossRefGoogle Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical Recipes in FORTRAN: The Art of Scientific Computing. Cambridge University Press.Google Scholar
Rhie, C. M. & Chow, W. L. 1983 Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21 (11), 1525–1532.CrossRefGoogle Scholar
Ross, F. W. & Willmarth, W. W. 1971 Some experimental results on sphere and disk drag. AIAA J. 9, 285–291.CrossRefGoogle Scholar
Shenoy, A. R. & Kleinstreuer, C. 2008 Flow over a thin circular disk at low-to-moderate Reynolds numbers. J. Fluid Mech. 605, 253–262.CrossRefGoogle Scholar
Skotheim, J. M. & Secomb, T. W. 2007 Red blood cells and other nonspherical capsules in shear flow: oscillatory dynamics and the tank-treading-to-tumbling transition. Phys. Rev. Lett. 98 (7), 078301.CrossRefGoogle ScholarPubMed
Stewart, R. E. & List, R. 1983 Gyrational motion of disks during free-fall. Phys. Fluids 26 (4), 920–927.CrossRefGoogle Scholar
Van Doormaal, J. P. & Raithby, G. D. 1984 Enhancements of the SIMPLE method for predicting incompressible flows. Numer. Heat Transfer A 7 (2), 147–163.Google Scholar
Wang, Z., Hopke, P. K., Ahmadi, G., Cheng, Y. & Baron, P. A. 2008 Fibrous particle deposition in human nasal passage: the influence of particle length, flow rate, and geometry of nasal airway. J. Aerosol Sci. 39 (12), 1040–1054.CrossRefGoogle Scholar
Willmarth, W. W., Hawk, N. E. & Harvey, R. L. 1964 Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208.CrossRefGoogle Scholar
Yin, C., Rosendahl, L., Kær, S. K. & Condra, T. J. 2004 Use of numerical modelling in design for co-firing biomass in wall-fired burners. Chem. Engng Sci. 59 (16), 3281–3292.CrossRefGoogle Scholar

Shenoy and Kleinstreuer supplementary movie

Movie 1. Single-sided vortex shedding observed behind a disk of aspect ratio=2 at Re=240 (free-stream oriented along the positive z-direction).

Download Shenoy and Kleinstreuer supplementary movie(Video)
Video 4.6 MB

Shenoy and Kleinstreuer supplementary movie

Movie 2. Double-sided vortex shedding observed behind a disk of aspect ratio=4 at Re=240 (free-stream oriented along the positive z-direction).

Download Shenoy and Kleinstreuer supplementary movie(Video)
Video 3.4 MB

Shenoy and Kleinstreuer supplementary movie

Movie 3. Single-sided vortex shedding observed behind a disk of aspect ratio=2 at Re=240 using azimuthal vorticity contours on the symmetry plane (free-stream oriented along the positive z-direction).

Download Shenoy and Kleinstreuer supplementary movie(Video)
Video 2.7 MB

Shenoy and Kleinstreuer supplementary movie

Movie 4. Double-sided vortex shedding observed behind a disk of aspect ratio=4 at Re=240 using azimuthal vorticity contours on the symmetry plane (free-stream oriented along the positive z-direction).

Download Shenoy and Kleinstreuer supplementary movie(Video)
Video 2.1 MB

Shenoy and Kleinstreuer supplementary movie

Movie 5. Vortex structures behind a freely falling disk, along the negative z-direction, of aspect ratio=2 at Re=240 .

Download Shenoy and Kleinstreuer supplementary movie(Video)
Video 1.1 MB

Shenoy and Kleinstreuer supplementary movie

Movie 6. Vortex structures behind a freely falling disk, along the negative z-direction, of aspect ratio=4 at Re=240.

Download Shenoy and Kleinstreuer supplementary movie(Video)
Video 1.5 MB