- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Speed of a von Kármán point vortex street in a weakly compressible fluid

Darren G. Crowdy* and Vikas S. Krishnamurthy

  • Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

  • *d.crowdy@imperial.ac.uk
PDF

Phys. Rev. Fluids 2, 114701 – Published 3 November, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.114701

Abstract

Analytical expressions are obtained for the change in speed of translation of the von Kármán point vortex streets of given aspect ratios due to the effects of weak compressibility in subsonic flow of an isentropic fluid. We also clarify the nature of the force-free condition on a weakly compressible point vortex in equilibrium. For staggered streets, it is found that the speed of a compressible point vortex street can both increase and decrease relative to its incompressible counterpart of the same aspect ratio. Compressibility increases the speeds of streets with aspect ratios less than the critical value of 0.38187, at which no change in speed occurs to first order in the (squared) Mach number. In particular, the compressible counterpart to the neutrally stable incompressible point vortex street of aspect ratio 0.28056 is found to propagate with increased speed. Streets with aspect ratios larger than 0.38187 slow down under the effects of compressibility, with the slowdown becoming maximal at a street aspect ratio of 0.52630. On the other hand, the speed of unstaggered streets always increases, with the first-order correction in speed relative to its incompressible value increasing maximally at aspect ratios around κ=0.36216.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (41)

  1. C. H. K. Williamson, Vortex dynamics in the cylinder wake, Ann. Rev. Fluid Mech. 28, 477 (1996).
  2. S. K. Lele, Compressibility effects on turbulence, Ann. Rev. Fluid Mech. 26, 211 (1994).
  3. P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, UK, 1992).
  4. P. K. Newton, The N-Vortex Problem: Analytical Techniques (Springer-Verlag, New York, 2001).
  5. J.-Z. Wu, H.-Y. Ma, and M. D. Zhou, Vorticity and Vortex Dynamics (Springer-Verlag, New York, 2006).
  6. D. Bershader, Compressible vortices, in Fluid Dynamics and Its Applications, edited by S. Green (1995), Vol. 30, p. 291.
  7. J.-P. Hickey, F. Hussain, and X. Wu, Compressibility effects on the structural evolution of transitional high-speed planar wakes, J. Fluid Mech. 796, 5 (2016).
  8. S. Mowlavi, C. Arratia, and F. Gallaire, Spatio-temporal stability of the Kármán vortex street and the effect of confinement, J. Fluid Mech. 795, 187 (2016).
  9. M. S. Howe, Theory of Vortex Sound, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, UK, 2002).
  10. T. Colonius, S. Lele, and P. Moin, The scattering of sound waves by a vortex: Numerical simulations and analytical solutions, J. Fluid Mech. 260, 271 (1994).
  11. R. Ford and S. G. Llewellyn Smith, Scattering of acoustic waves by a vortex, J. Fluid Mech. 386, 305 (1999).
  12. T. Colonius, S. Lele, and P. Moin, The free compressible viscous vortex, J. Fluid Mech. 230, 45 (1991).
  13. K. D. von EllenRieder and B. J. Cantwell, Self-similar, slightly compressible, free vortices, J. Fluid Mech. 423, 293 (2000).
  14. M. J. Mandella, Experimental and analytical studies of compressible vortices, Ph.D. thesis, Dept. of Applied Physics, Stanford University, Stanford, CA, 1987 (unpublished).
  15. D. W. Moore, The effect of compressibility on the speed of a vortex ring, Proc. Roy. Soc. London, Ser. A 397, 87 (1985).
  16. D. W. Moore and D. I. Pullin, The compressible vortex pair, J. Fluid Mech. 185, 171 (1987).
  17. A. Barsony-Nagy, J. Er-El, and S. Yungster, Compressible flow past a contour and stationary vortices, J. Fluid Mech. 178, 367 (1987).
  18. F. G. Leppington, The field due to a pair of line vortices in a compressible fluid, J. Fluid Mech. 559, 45 (2006).
  19. K. Ardalan, D. Meiron, and D. Pullin, Steady compressible vortex flows: The hollow-core vortex array, J. Fluid Mech. 301, 1 (1995).
  20. G. K. O'Reilly and D. I. Pullin, Smooth transonic flow in an array of counter-rotating vortices, J. Fluid Mech. 524, 197 (2005).
  21. T. von Kármán and H. L. Rubach, On the mechanisms of fluid resistance, Physik Z. 13, 49 (1912).
  22. T. von Kármán, Collected works of Theodore von Kármán (Butterworths, London, 1956), Vol. 1.
  23. P. G. Saffman and J. C. Schatzman, Stability of a vortex street of finite vortices, J. Fluid Mech. 117, 171 (1982).
  24. D. I. Meiron, P. G. Saffman, and J. C. Schatzman, The linear two-dimensional stability of inviscid vortex streets of finite-cored vortices, J. Fluid Mech. 147, 187 (1984).
  25. S. Kida, Stabilizing effects of finite core on Kármán vortex street, J. Fluid Mech. 122, 487 (1982).
  26. J. Jiménez, On the linear stability of the inviscid Kármán vortex street, J. Fluid Mech. 178, 177 (1987).
  27. R. von Mises, Mathematical Theory of Compressible Fluid Flow (Dover, New York, 2004).
  28. S.-I. Pai, Introduction to the Theory of Compressible Flow (D. Van Nostrand, Princeton, NJ, 1959).
  29. C. Jacob, Introduction Mathématique a la Mécanique des Fluides (Gauthier-Villars, Paris, 1959).
  30. H. C. Pocklington, The configuration of a pair of equal and opposite hollow straight vortices, of finite cross-section, moving steadily through fluid, Proc. Camb. Phil. Soc. 8, 178 (1895).
  31. G. I. Taylor, Recent work on the flow of compressible fluids, J. Lond. Math. Soc. s1-5, 224 (1930).
  32. S. G. Llewellyn Smith, How do singularities move in potential flow? Phys. D (Amsterdam, Neth.) 240, 1644 (2011).
  33. A. Barsony-Nagy, Extension of the Blasius force theorem to subsonic speeds, AIAA J. 23, 1811 (1985).
  34. L. M. Milne-Thomson, Theoretical Hydrodynamics (Dover, New York, 2013).
  35. I. Imai, On the flow of a compressible fluid past a circular cylinder, Proc. Phys. Math. Soc. Jpn. 32, 636 (1938).
  36. I. Imai, On the flow of a compressible fluid past a circular cylinder, II, Proc. Phys. Math. Soc. Jpn. 23, 180 (1941).
  37. I. Imai, A new method of successive approximations for dealing with the two-dimensional subsonic flow of a compressible fluid, Proc. Phys. Math. Soc. Jpn. 24, 120 (1942).
  38. P. G. Saffman and J. C. Schatzman, Properties of a vortex street of finite vortices, SIAM J. Sci. Stat. Comput. 2, 285 (1981).
  39. D. G. Crowdy and C. C. Green, Analytical solutions for von Kármán streets of hollow vortices, Phys. Fluids 23, 126602 (2011).
  40. D. G. Crowdy, S. G. Llewellyn Smith, and D. Freilich, Translating hollow vortex pairs, Eur. J. Mech B/Fluids 37, 180 (2012).
  41. D. G. Crowdy and V. S. Krishnamurthy, The effect of core size on the speed of compressible hollow vortex streets, J. Fluid Mech. (submitted).

Outline

Information

Phys. Rev. Fluids 2, 114701– Published 3 November, 2017
  • Received 9 June 2017
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

©2017 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation