Hostname: page-component-7479d7b7d-jwnkl Total loading time: 0 Render date: 2024-07-10T21:38:43.612Z Has data issue: false hasContentIssue false
  • English
  • Français

Global stability of flow past a cylinder with centreline symmetry

Published online by Cambridge University Press:  27 July 2009

BHASKAR KUMAR ,
JACOB JOHN KOTTARAM ,
AMIT KUMAR SINGH  and
SANJAY MITTAL
BHASKAR KUMAR
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, UP 208 016, India
JACOB JOHN KOTTARAM
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, UP 208 016, India
AMIT KUMAR SINGH
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, UP 208 016, India
SANJAY MITTAL*
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, UP 208 016, India
*
†Email address for correspondence: smittal@iitk.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

Global absolute and convective stability analysis of flow past a circular cylinder with symmetry conditions imposed along the centreline of the flow field is carried out. A stabilized finite element formulation is used to solve the eigenvalue problem resulting from the linearized perturbation equation. All the computations carried out are in two dimensions. It is found that, compared to the unrestricted flow, the symmetry conditions lead to a significant delay in the onset of absolute as well as convective instability. In addition, the onset of absolute instability is greatly affected by the location of the lateral boundaries and shows a non-monotonic variation. Unlike the unrestricted flow, which is associated with von Kármán vortex shedding, the flow with centreline symmetry becomes unstable via modes that are associated with low-frequency large-scale structures. These lead to expansion and contraction of the wake bubble and are similar in characteristics to the low-frequency oscillations reported earlier in the literature. A global linear convective stability analysis is utilized to find the most unstable modes for different speeds of the disturbance. Three kinds of convectively unstable modes are identified. The ones travelling at very low streamwise speed are associated with large-scale structures and relatively low frequency. Shear layer instability, with relatively smaller scale flow structures and higher frequency, is encountered for disturbances travelling at relatively larger speed. For low blockage a new type of instability is found. It travels at relatively high speed and resembles a swirling flow structure. As opposed to the absolute instability, the convective instability appears at much lower Re and its onset is affected very little by the location of the lateral boundaries. Analysis is also carried out for determining the convective stability of disturbances that travel in directions other than along the free stream. It is found that the most unstable disturbances are not necessarily the purely streamwise travelling ones. Disturbances that move purely in the cross-stream direction can also be convectively unstable. The results from the linear stability analysis are confirmed by carrying out direct time integration of the linearized disturbance equations. The disturbance field shows transient growth by several orders of magnitude confirming that such flows act as amplifiers. Direct time integration of the Navier–Stokes equation is carried out to track the time evolution of both the large-scale low-frequency oscillations and small-scale shear layer instabilities. The critical Re for the onset of convective instability is compared with earlier results from local analysis. Good agreement is found.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009 , pp. 273 - 300
Copyright
Copyright © Cambridge University Press 2009

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

Alam, M. & Sandham, N. S. 2000 Direct numerical simulation of a ‘short’ laminar separation bubbles with turbulent reattachment. J. Fluid Mech. 410, 1–28.Google Scholar
Blackburn, H. M., Marques, F. & Lopez, J. M. 2005 Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395–411.Google Scholar
Bloor, M. S. 1964 The transition to turbulence in the wake of a circular cylinder. J. Fluid Mech. 19, 290–304.Google Scholar
Brooks, A. N. & Hughes, T. J. R. 1982 Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Engng 32, 199–259.Google Scholar
Castro, I. P. 2002 Weakly stratified laminar flow past normal flat plates. J. Fluid Mech. 454, 21–46.Google Scholar
Castro, I. P. 2005 The stability of laminar symmetric separated wakes. J. Fluid Mech. 532, 389–411.Google Scholar
Chen, J. H., Pritchard, W. G. & Tavener, S. J. 1995 Bifurcation for flow past a cylinder between parallel planes. J. Fluid Mech. 284, 23–41.Google Scholar
Chernyshenko, S. 1988 The asymptotic form of the stationary separated circumfluence of a body at high Reynolds number. Appl. Math. Mech. 52, 746–753.CrossRefGoogle Scholar
Ding, Y. & Kawahara, M. 1999 three-dimensional linear stability analysis of incompressible viscous flows using the finite element method. Intl J. Numer. Methods Fluids 31, 451–479.Google Scholar
Fasel, H. F. & Postl, D. 2006 Interaction of separation and transition in boundary layers: direct numerical simulations. In Sixth IUTAM Symposium on Laminar-Turbulent Transition, pp. 71–88, Springer.Google Scholar
Fornberg, B. 1985 Steady viscous flow past a circular cylinder up to Reynolds number 600. J. Comput. Phys. 98, 297–320.Google Scholar
Fornberg, B. 1991 Steady incompressible flow past a row of circular cylinders. J. Fluid Mech. 225, 655–671.Google Scholar
Gajjar, J. S. B. & Azzam, N. A. 2004 Numerical solution of the Navier–Stokes equations for the flow in a cylinder cascade. J. Fluid Mech. 520, 51–82.CrossRefGoogle Scholar
Gerrard, J. H. 1978 The wakes of cylindrical bluff bodies at low Reynolds number. Phil. Trans. R. Soc. Lond. A 288, 351–382.Google Scholar
Griffith, M. D., Thompson, M. C., Leweke, T., Hourigan, K. & Anderson, W. P. 2007 Wake behaviour and instability of flow through a partially blocked channel. J. Fluid Mech. 582, 319–340.Google Scholar
Healey, J. J. 2006 A new convective instability of the rotating-disk boundary layer with growth normal to the disk. J. Fluid Mech. 560, 279–310.Google Scholar
Hudy, L. M., Naguib, A. M. & Humphries, W. M. 2003 Wall-pressure-array measurements beneath a separating/reattaching flow region, Phys. Fluids 15, 706–717.Google Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Hughes, T. J. R. & Brooks, A. N. 1979 A multi-dimensional upwind scheme with no crosswind diffusion. In Finite Element Methods for Convection Dominated Flows (ed. Hughes, T. J. R), AMD-vol. 34, pp. 19–35. ASME.Google Scholar
Hughes, T. J. R., Franca, L. P. & Balestra, M. 1986 A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Engng 59, 85–99.Google Scholar
Hultgren, L. S. & Aggarwal, A. K. 1978 Absolute instability of the Gaussian wake profile. Phys. Fluids 30, 3383–3387.Google Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.Google Scholar
Juniper, M. P. 2007 The full impulse response of two-dimensional jet/wake flows and implications for confinement. J. Fluid Mech. 590, 163–185.Google Scholar
Koch, K. 1985 Local instability characteristics and frequency determination of self-excited wake flows. J. Sound Vib. 99, 53–83.Google Scholar
Kumar, B. & Mittal, S. 2006 a Effect of blockage on critical parameters for flow past a circular cylinder. Intl J. Numer. Methods Fluids 50, 987–1001.Google Scholar
Kumar, B. & Mittal, S. 2006 b Prediction of the critical Reynolds number for flow past a circular cylinder. Comput. Methods Appl. Mech. Engng 195, 6046–6058.Google Scholar
Manhart, M. & Fredrich, R. 2001 DNS of a turbulent boundary layer with separation. In Proceedings of the Second Conf. on Turbulence and Shear Flow Phenomena, KTH, Stockholm.Google Scholar
Marsden, J. E. & McCracken, M. 1976 The Hopf Bifurcation and Its Applications. Springer.Google Scholar
Mittal, S. 2008 Instability of the separated shear layer in flow past a cylinder: forced excitation. Intl J. Numer. Methods Fluids 56, 687–702.CrossRefGoogle Scholar
Mittal, S., Kottaram, J. J. & Kumar, B. 2008 Onset of shear layer instability in flow past a cylinder. Phys. Fluids 20, 054102–1–10.CrossRefGoogle Scholar
Mittal, S. & Kumar, B. 2003 Flow past a rotating cylinder. J. Fluid Mech. 476, 303–334.Google Scholar
Mittal, S. & Kumar, B. 2007 A stabilized finite element method for global analysis of convective instabilities in nonparallel flows. Phys. Fluids 19, 088105–1–4.Google Scholar
Morzynski, M., Afanasiev, K. & Thiele, F. 1999 Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Methods Appl. Mech. Engng 169, 161–176.Google Scholar
Norberg, C. 1994 An experimental investigation of the flow around a circular cylinder: influence of aspect ratio. J. Fluid Mech. 258, 287–316.Google Scholar
Norberg, C. 2001 Flow around a circular cylinder: aspects of fluctuating lift. J. Fluids Struct. 15, 459–469.Google Scholar
Prasad, A. & Williamson, C. H. K. 1997 The instability of the shear layer separating from a bluff body. J. Fluid Mech. 333, 375–402.Google Scholar
Rajagopalan, S. & Antonia, R. A. 2005 Flow around a circular cylinder-structure if the near wake shear layer. Exp. Fluids 38, 393–402.Google Scholar
Saad, Y. & Schultz, M. 1986 GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869.Google Scholar
Sadovskii, V. 1971 On local properties of vortex flows. Uch. Zap. TsAGI 4, 117–120.Google Scholar
Smith, F. 1979 Laminar flow of an incompressible fluid past a bluff body: the separation, reattachment, eddy properties and drag. J. Fluid Mech. 92, 171–205.Google Scholar
Stewart, G. W. 1975 Methods of simultaneous iteration for calculating eigenvectors of matrices. In Topics in Numerical Analysis II (ed. J. H. H. Miller), pp. 169–185. Academic.CrossRefGoogle Scholar
Tang, S. & Aubry, A. 1997 On the symmetry breaking instability leading to vortex shedding. Phys. Fluids 9, 2550–2561.Google Scholar
Tezduyar, T. E., Mittal, S., Ray, S. E. & Shih, R. 1992 Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput. Methods Appl Mech. Engng, 95, 221–242.Google Scholar
Unal, M. F. & Rockwell, D. 1988 On the vortex formation from a cylinder. Part 1. The initial instability. J. Fluid Mech. 190, 491–512.Google Scholar
Williamson, C. H. K. 1989 Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 206, 579–627.Google Scholar
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Math. 21, 155–165.CrossRefGoogle Scholar