Skip to main content

Three-dimensional instabilities in oscillatory flow past elliptic cylinders

  • José P. Gallardo (a1), Helge I. Andersson (a2) and Bjørnar Pettersen (a1)

We investigate the early development of instabilities in the oscillatory viscous flow past cylinders with elliptic cross-sections using three-dimensional direct numerical simulations. This is a classical hydrodynamic problem for circular cylinders, but other configurations have received only marginal attention. Computed results for some different aspect ratios ${\it\Lambda}$ from 1 : 1 to 1 : 3, all with the major axis of the ellipse aligned in the main flow direction, show good qualitative agreement with Hall’s stability theory (J. Fluid Mech., vol. 146, 1984, pp. 347–367), which predicts a cusp-shaped curve for the onset of the primary instability. The three-dimensional flow structures for aspect ratios larger than 2 : 3 resemble those of a circular cylinder, whereas the elliptical cross-section with the lowest aspect ratio of 1 : 3 exhibits oblate rather than tubular three-dimensional flow structures as well as a pair of counter-rotating spanwise vortices which emerges near the tips of the ellipse. Contrary to a circular cylinder, instabilities for an elliptic cylinder with sufficiently high eccentricity emerge from four rather than two different locations in accordance with the Hall theory.

Corresponding author
Email address for correspondence:
Hide All
An H., Cheng L. & Zhao M. 2011 Direct numerical simulation of oscillatory flow around a circular cylinder at low Keulegan–Carpenter number. J. Fluid Mech. 666, 77103.
Badr H. M. & Kocabiyik S. 1997 Symmetrically oscillating viscous flow over an elliptic cylinder. J. Fluids Struct. 11, 745766.
Bearman P. W., Downie M. J., Graham J. M. R. & Obasaju E. D. 1985 Forces on cylinders in viscous oscillatory flow at low Keulegan–Carpenter numbers. J. Fluid Mech. 154, 337356.
Elston J. R., Blackburn H. M. & Sheridan J. 2006 The primary and secondary instabilities of flow generated by an oscillating circular cylinder. J. Fluid Mech. 550, 359389.
Hall P. 1984 On the stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech. 146, 347367.
Honji H. 1981 Streaked flow around an oscillating circular cylinder. J. Fluid Mech. 107, 509520.
Jeong J. & Hussain F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.
Keulegan G. H. & Carpenter L. H. 1958 Forces on cylinders and plates in an oscillating fluid. J. Res. Natl Bur. Stand. 60, 423440.
Manhart M. 2004 A zonal grid algorithm for DNS of turbulent boundary layers. Comput. Fluids 33, 435461.
Manhart M., Tremblay F. & Friedrich R. 2001 MGLET: a parallel code for efficient DNS and LES of complex geometries. In Parallel Computational Fluid Dynamics-Trends and Applications, pp. 449456. Elsevier.
Morison J. R., O’Brien M. P., Johnson J. W. & Schaaf S. A. 1950 The force exerted by surface waves on piles. Petrol. Trans. AIME 189, 149157.
Peller N., Le Duc A., Tremblay F. & Manhart M. 2006 High-order stable interpolations for immersed boundary methods. Intl J. Numer. Meth. Fluids 52, 11751193.
Sarpkaya T. 1986 Force on a circular cylinder in viscous oscillatory flow at low Keulegan–Carpenter numbers. J. Fluid Mech. 165, 6171.
Sarpkaya T. 2002 Experiments on the stability of sinusoidal flow over a circular cylinder. J. Fluid Mech. 457, 157180.
Sarpkaya T. 2005 On the parameter 𝛽 = Re/KC = D 2/𝜈T . J. Fluids Struct. 21, 435440.
Sarpkaya T. 2006 Structures of separation on a circular cylinder in periodic flow. J. Fluid Mech. 567, 281297.
Stone H. L. 1968 Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM J. Numer. Anal. 5, 530558.
Suthon P. & Dalton C. 2011 Streakline visualization of the structures in the near wake of a circular cylinder in sinusoidally oscillating flow. J. Fluids Struct. 27, 885902.
Suthon P. & Dalton C. 2012 Observations on the Honji instability. J. Fluids Struct. 32, 2736.
Tatsuno M. & Bearman P. W. 1990 A visual study of the flow around an oscillating circular cylinder at low Keulegan–Carpenter numbers and low Stokes numbers. J. Fluid Mech. 211, 157182.
White F. M. 2006 Viscous Fluid Flow, 3rd edn. McGraw-Hill.
Williamson C. H. K. 1985 Sinusoidal flow relative to circular cylinders. J. Fluid Mech. 155, 141174.
Williamson J. H. 1980 Low-storage Runge–Kutta schemes. J. Comput. Phys. 56, 4856.
Yang K.2014 Oscillatory flow past cylinders at low inline-graphic $KC$ numbers. PhD thesis, The University of Western Australia.
Yang K., Cheng L., An H., Bassom A. P. & Zhao M. 2014 Effects of an axial flow component on the Honji instability. J. Fluids Struct. 49, 614639.
Zhang J. & Dalton C. 1999 The onset of three-dimensionality in an oscillating flow past a fixed circular cylinder. Intl J. Numer. Meth. Fluids 42, 1942.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 2
Total number of PDF views: 151 *
Loading metrics...

Abstract views

Total abstract views: 246 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 15th December 2017. This data will be updated every 24 hours.