Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-24T22:36:23.890Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady separated flow past a circular cylinder at low Reynolds numbers

Published online by Cambridge University Press:  10 February 2009

SUBHANKAR SEN ,
SANJAY MITTAL  and
GAUTAM BISWAS
SUBHANKAR SEN
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India
SANJAY MITTAL*
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India
GAUTAM BISWAS
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The steady two-dimensional laminar flow around a stationary circular cylinder has been investigated via a stabilized finite-element method. The Reynolds number Re is based on the cylinder diameter and free-stream speed. The results have been presented for 6 ≤ Re ≤ 40 and the blockages between 0.000125 and 0.80. The blockage B is the ratio of the cylinder diameter to the domain width. There is large scatter in the value of Res, reported in the literature, marking the onset of the flow separation. From the present study the Res is found to be 6.29, approximately for B = 0.005. The effect of the blockage on the characteristic flow parameters is found to be insignificant for B ≤ 0.01. The bubble length, separation angle and Res exhibit non-monotonic variation with the blockage. It is for the first time that such a behaviour for the separation angle and Res is being reported. Two types of boundary conditions at the lateral walls have been studied: the slip wall and towing tank. In general for high blockage, the results from the slip boundary condition are closer to the ones for the unbounded flow. In that sense, the use of the slip boundary condition as opposed to the towing tank boundary condition on the lateral walls is advocated. The bubble length, separation angle, base suction, total drag, pressure drag, viscous drag and maximum vorticity on the cylinder surface for the steady flow are found to vary as Re, Re−0.5, Re−1, Re−0.5, Re−0.64, Re−0.60 and Re0.5, respectively. The extrapolated results for the steady flow, for higher Re, are found to match quite well with the other results from the literature.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 620 , 10 February 2009 , pp. 89 - 119
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

Acrivos, A., Leal, L. G., Snowden, D. D. & Pan, F. 1968 Further experiments on steady separated flows past bluff objects. J. Fluid Mech. 34, 25–48.CrossRefGoogle Scholar
Acrivos, A., Snowden, D. D., Grove, A. S. & Peterson, E. E. 1965 The steady separated flow past a circular cylinder at large Reynolds numbers. J. Fluid Mech. 21, 737–760.Google Scholar
Allen, D. N. & Southwell, R. V. 1955 Relaxation methods applied to determine the motion in two dimensions of a viscous fluid past a fixed cylinder. Quart. J. Mech. Appl. Math. 8, 129–145.CrossRefGoogle Scholar
Apelt, C. J. 1961 The steady flow of a viscous fluid past a circular cylinder at Reynolds numbers 40 and 44. Aeronaut. Res. Counc. Lond. R & M 3175, 1–28.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.CrossRefGoogle Scholar
Chen, J. H. 2000 Laminar separation of flow past a circular cylinder between two parallel plates. Proc. Natl Sci. Counc. ROC A 24, 341–351.Google Scholar
Coutanceau, M. & Bouard, R. 1977 Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 1. Steady flow. J. Fluid Mech. 79, 231–256.CrossRefGoogle Scholar
Dennis, S. C. R. & Chang, G. Z. 1970 Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. J. Fluid Mech. 42, 471–489.CrossRefGoogle Scholar
Dennis, S. C. R. & Shimshoni, M. 1965 The steady flow of a viscous fluid past a circular cylinder. Aero. Res. Counc. Current Papers, no. 797.Google Scholar
Fornberg, B. 1980 A numerical study of steady viscous flow past a circular cylinder. J. Fluid Mech. 98, 819–855.Google Scholar
Fornberg, B. 1985 Steady viscous flow past a circular cylinder up to Reynolds number 600. J. Comput. Phys. 61, 297–320.Google Scholar
Fornberg, B. 1991 Steady incompressible flow past a row of circular cylinders. J. Fluid Mech. 225, 655–671.CrossRefGoogle 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
Ghia, K. N., Ghia, U., Osswald, G. A. & Liu, C. A. 1986 Simulation of separated flow past a bluff body using Navier-Stokes equations. In Proc. of the IUTAM Symp. on Boundary Layer Separation. London.Google Scholar
Grove, A. S., Shair, F. H., Peterson, E. E. & Acrivos, A. 1964 An experimental investigation of the steady separated flow past a circular cylinder. J. Fluid Mech. 19, 60–80.CrossRefGoogle Scholar
Hamielec, A. E. & Raal, J. D. 1969 Numerical studies of viscous flow around circular cylinders. Phys. Fluids 12, 11–17.CrossRefGoogle Scholar
Henderson, R. D. 1995 Details of the drag curve near the onset of vortex shedding. Phys. Fluids 7, 2102–2104.Google Scholar
Homann, F. 1936 Einfluss grösser Zähigkeit bei Strömung um Zylinder. Forsch. Ing. Wes. 7, 1–10.CrossRefGoogle 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.), 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. Part 5. 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
Kawaguti, M. 1953 Numerical solution of the Navier–Stokes equations for the flow around a circular cylinder at Reynolds number 40. J. Phys. Soc. Jpn 8, 747–757.Google Scholar
Kawaguti, M. & Jain, P. 1966 Numerical study of a viscous fluid flow past a circular cylinder. J. Phys. Soc. Jpn 21, 2055–2062.Google Scholar
Keller, H. B. & Takami, H. 1966 Numerical studies of viscous flow about cylinders. In Numerical Solutions of Nonlinear Differential Equations (ed. Greenspan, D.), pp. 115–140. Wiley.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.CrossRefGoogle 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
Mittal, S. 2000 On the performance of high aspect ratio elements for incompressible flows. Comput. Methods Appl. Mech. Engng 188, 269–287.CrossRefGoogle Scholar
Nieuwstadt, F. & Keller, H. B. 1973 Viscous flow past circular cylinders. Comput. Fluids 1, 59–71.Google Scholar
Nishioka, M. & Sato, H. 1974 Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 65, 97–112.CrossRefGoogle Scholar
Nisi, H. & Porter, A. W. 1923 On eddies in air. Phil. Mag. 46, 754–768.CrossRefGoogle Scholar
Posdziech, O. & Grundmann, R. 2007 A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder. J. Fluids Struct. 23, 479–499.CrossRefGoogle Scholar
Prasanth, T. K., Behara, S., Singh, S. P., Kumar, R. & Mittal, S. 2006 Effect of blockage on vortex-induced vibrations at low Reynolds numbers. J. Fluids Struct. 22, 865–876.Google Scholar
Pruppacher, H. R., Clair, B. P. Le & Hamielec, A. E. 1970 Some relations between drag and flow past a sphere and a cylinder at low and intermediate Reynolds numbers. J. Fluid Mech. 44, 781–790.CrossRefGoogle 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
Sahin, M. & Owens, R. G. 2004 A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder. Phys. Fluids 16, 1305–1320.CrossRefGoogle Scholar
Singh, S. P. & Mittal, S. 2005 Flow past a cylinder: shear layer instability and drag crisis. Intl J. Numer. Methods Fluids 47, 75–98.CrossRefGoogle Scholar
Smith, F. T. 1979 Laminar flow of an incompressible fluid past a bluff body: the separation, reattachment, eddy properties and drag. J. Fluid Mech. 92, 171–205.CrossRefGoogle Scholar
Smith, F. T. 1981 Comparisons and comments concerning recent calculations for flow past a circular cylinder. J. Fluid Mech. 113, 407–410.Google Scholar
Sobey, I. J. 2000 Introduction to Interactive Boundary Layer Theory. Oxford University Press.CrossRefGoogle Scholar
Son, J. S. & Hanratty, T. J. 1969 Numerical solution for the flow around a cylinder at Reynolds numbers of 40, 200 and 500. J. Fluid Mech. 35, 369–386.Google Scholar
Srinivasan, K. 2006 On a separation criterion for symmetric elliptic bluff body flows. http://arxiv.org/pdf/physics/0511250.Google Scholar
Takami, H. & Keller, H. B. 1969 Steady two-dimensional viscous flow of an incompressible fluid past a circular cylinder. Phys. Fluids Suppl. 12, II 51–II 56.Google Scholar
Taneda, S. 1956 Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. J. Phys. Soc. Jpn 11, 302–307.CrossRefGoogle 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
Thom, A. 1933 The flow past circular cylinders at low speeds. Proc. R. Soc. Lond. A 141, 651–669.Google Scholar
Thoman, D. C. & Szewczyk, A. A. 1969 Time-dependent viscous flow over a circular cylinder. Phys. Fluids Suppl. 12, II 76–II 86.Google Scholar
Tomotika, S. & Aoi, T. 1950 The steady flow of viscous fluid past a sphere and circular cylinder at small Reynolds numbers. Quart. J. Mech. Appl. Math. 3, 140–161.Google Scholar
Tritton, D. J. 1959 Experiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 6, 547–567.CrossRefGoogle Scholar
Tuann, S. Y. & Olson, M. D. 1978 Numerical studies of the flow around a circular cylinder by a finite element method. Comput. Fluids 6, 219–240.Google Scholar
Underwood, R. L. 1969 Calculation of incompressible flow past a circular cylinder at moderate Reynolds numbers. J. Fluid Mech. 37, 95–114.CrossRefGoogle Scholar
Wu, M. H., Wen, C. Y., Yen, R. H., Weng, M. C. & Wang, A. B. 2004 Experimental and numerical study of the separation angle for flow around a circular cylinder at low Reynolds number. J. Fluid Mech. 515, 233–260.Google Scholar
Yamada, H. 1954 On the slow motion of viscous liquid past a circular cylinder. Rep. Res. Inst. Appl. Mech. Kyushu Univ. 3, 11–23.Google Scholar
Zdravkovich, M. M. 1997 Flow Around Circular Cylinders, vol. 1. Oxford University Press.Google Scholar