Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-15T10:02:49.101Z Has data issue: false hasContentIssue false
  • English
  • Français

Development of the wake behind a circular cylinder impulsively started into rotatory and rectilinear motion

Published online by Cambridge University Press:  26 April 2006

Yen-Ming Chen ,
Yuh-Roung Ou  and
Arne J. Pearlstein
Yen-Ming Chen
Affiliation:
NASA-Langley Research Center, Hampton, VA 23665, USA Present Address: General Electric Aircraft Engines, 1 Neumann Way, Cincinnati, OH 45215, USA.
Yuh-Roung Ou
Affiliation:
Institute for Computer Applications in Science and Engineering, NASA-Langley Research Center, Hampton, VA 23665, USA Present Address: Interdisciplinary Center for Applied Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA.
Arne J. Pearlstein
Affiliation:
Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The temporal development of two-dimensional viscous incompressible flow generated by a circular cylinder impulsively started into steady rotatory and rectilinear motion at Re = 200 (based on the cylinder diameter 2a and the magnitude U of the rectilinear velocity) is studied computationally. We use an explicit finite-difference/pseudospectral technique and a new implementation of the Biot–Savart law to integrate a velocity/vorticity formulation of the Navier–Stokes equations. Results are presented for the four angular: rectilinear speed ratios α = Ωa/U (where Ω is the angular speed) considered experimentally by Coutanceau & Ménard (1985). For α ≤ 1, extension of the computations to dimensionless times larger than achieved either in the experimental work or in the computations of Badr & Dennis (1985) allows for a more complete discussion of the temporal development of the wake. Using the frame-invariant vorticity distribution, we discuss several aspects of the vortex kinematics and dynamics not revealed by the earlier work, in which vortex cores were identified from frame-dependent streamline and streamfunction information. Consideration of the flow in the absence of sidewalls confirms the artifactual nature of the trajectory of the first vortex reported by Coutanceau & Ménard for α = 3.25. For α greater than unity (the largest value considered by Badr & Dennis), our results indicate that at Re = 200 shedding of more than one vortex does indeed occur for α = 3.25 (and possibly for larger α), in contrast to the conclusion of Coutanceau & Ménard. Moreover, the shedding process is very different from that associated with the usual Kármán vortex street for α = 0. Specifically, consecutive vortices can be shed from one side of the cylinder and be of the same sense, in contrast to the non-rotating case, in which mirror-image vortices of opposite sense are shed alternately from opposite sides of the cylinder. The results are discussed in relation to the possibility of suppressing vortex shedding by open- or closed-loop control of the rotation rate.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 253 , August 1993 , pp. 449 - 484
Copyright
© 1993 Cambridge University Press

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

Badr, H. M. & Dennis, S. C. R. 1985 Time-dependent viscous flow past an impulsively started rotating and translating circular cylinder. J. Fluid Mech. 158, 447–488.Google Scholar
Badr, H. M., Dennis, S. C. R. & Young, P. J. S. 1989 Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers. Computers & Fluids 17, 579–609.Google Scholar
Badr, H., Coutanceau, M., Dennis, S. & Ménard, C. 1986 On the phenomenon of vortex transposition of coalescence in separated flows. C. R. Acad. Sci. Paris 302(II), 1127–1130.Google Scholar
Badr, H. M., Coutanceau, M., Dennis, S. C. R. & Ménard, C. 1990 Unsteady flow past a rotating circular cylinder at Reynolds numbers 103 and 104. J. Fluid Mech. 220, 459–484.Google Scholar
Chang, C.-C. & Chern, R.-L. 1992 Vortex shedding from an impulsively started rotating and translating cylinder. J. Fluid Mech. 235, 265–298.Google Scholar
Chen, Y.-M. 1989 Numerical simulation of the unsteady two-dimensional flow in a time-dependent doubly-connected domain. PhD dissertation, University of Arizona.
Chen, Y.-M., Koniges, A. E. & Anderson, D. V. 1989 ILUBCG2-11: Solution of 11-banded nonsymmetric linear equation systems by a preconditioned biconjugate gradient routine. Comput. Phys. Commun. 55, 359–365.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1973 Flow past an impulsively started circular cylinder. J. Fluid Mech. 60, 105–127.Google Scholar
Coutanceau, M. & Ménard, C. 1985 Influence of rotation on the near-wake development behind an impulsively started circular cylinder. J. Fluid Mech. 158, 399–446.Google Scholar
Díaz, F., Gavaldá, J., Kawall, J. G., Keffer, J. F. & Giralt, F. 1983 Vortex shedding from a spinning cylinder. Phys. Fluids 26, 3454–3460.Google Scholar
Eaton, B. E. 1987 Analysis of laminar vortex shedding behind a circular cylinder by computer-aided flow visualization. J. Fluid Mech. 180, 117–145.Google Scholar
Fasel, H. 1976 Investigation of the stability of boundary layers by a finite-difference model of the Navier–Stokes equations. J. Fluid Mech. 78, 355–383.Google Scholar
Fornberg, B. 1980 A numerical study of steady viscous flow past a circular cylinder. J. Fluid Mech. 98, 819–855.Google Scholar
Glauert, M. B. 1957a A boundary layer theorem, with applications to rotating cylinders. J. Fluid Mech. 2, 89–99.Google Scholar
Glauert, M. B. 1957b The flow past a rapidly rotating circular cylinder. Proc. R. Soc. Lond. A 242, 108–115.Google Scholar
Ingham, D. B. 1983 Steady flow past a rotating cylinder. Computers & Fluids 11, 351–366.Google Scholar
Ingham, D. B. & Tang, T. 1990 A numerical investigation into the steady flow past a rotating circular cylinder at low and intermediate Reynolds numbers. J. Comput. Phys. 87, 91–107.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
Kimura, T., Tsutahara, M. & Wang, Z. 1992 Wake of a rotating circular cylinder. AIAA J. 30, 555–556.Google Scholar
Klewicki, J. C. & Falco, R. E. 1991 On accurately measuring statistics associated with small-scale turbulent boundary layers using hot-wire probes. J. Fluid Mech. 219, 119–142.Google Scholar
Koromilas, C. A. & Telionis, D. P. 1980 Unsteady laminar separation: an experimental study. J. Fluid Mech. 97, 347–384.Google Scholar
Krahn, E. 1955 The laminar boundary layer on a rotating cylinder in crossflow. NAVORD Rep. 4022.
Kuo, B. 1975 Automatic Control Systems, 3rd Edn. Prentice-Hall.
Lugt, H. J. 1979 The dilemma of defining a vortex. In Recent Developments in Theoretical and Experimental Fluid Mechanics (ed. U. Müller, K. G. Roesner & B. Schmidt), pp. 309–321. Springer.
Lugt, H. J. & Haussling, H. J. 1974 Laminar flow past an abruptly accelerated elliptic cylinder at 45° incidence. J. Fluid Mech. 65, 711–734.Google Scholar
Lyulka, V. A. 1977 Numerical solution of the problem of the rotation of a cylinder in a flow of a viscous incompressible fluid. Vyschisl. Mat. mat. Fiz. 17, 470–480. (Translated in USSR Comput. Maths and Math. Phys. 17(2), 178–188, 1978.)Google Scholar
Mo, J. 1989 An investigation of wake flow around a cylinder with rotational oscillations. PhD dissertation, University of Tennessee Space Institute.
Modi, V. J., Mokhtarian, F. & Yokomizo, T. 1990 Effect of moving surfaces on the airfoil boundary-layer control. J. Aircraft 27, 42–50.Google Scholar
Moore, D. W. 1957 The flow past a rapidly rotating cylinder in a uniform stream. J. Fluid Mech. 2, 541–550.Google Scholar
Okajima, A., Takata, H. & Asanuma, T. 1975 Viscous flow around a rotationally oscillating circular cylinder. Rep. 532. Institute of Space and Aeronautical Science, University of Tokyo.
Orszag, S. A. 1980 Spectral methods for problems in complex geometries. J. Comput. Phys. 37, 70–92.Google Scholar
Payne, R. B. 1958 Calculations of unsteady flow past a circular cylinder. J. Fluid Mech. 4, 81–86.Google Scholar
Perry, A. E., Chong, M. S. & Lim, T. T. 1982 The vortex-shedding process behind two-dimensional bluff bodies. J. Fluid Mech. 116, 77–90.Google Scholar
Prandtl, L. 1925 The Magnus effect and windpowered ships. Naturwissenschaften 13, 93–108.Google Scholar
Prandtl, L. & Tietjens, O. G. 1934 Applied Hydro- and Aeromechanics (transl. J. P. Den Hartog 1957). Dover.
Reddy, R. N. & Thompson, J. F. 1977 Numerical solution of incompressible Navier–Stokes equations in the integro-differential formulation using boundary-fitted coordinate systems. AIAA Paper 77-650.
Shkadova, V. P. 1982 Rotating cylinder in a flowing viscous incompressible fluid. Akad. Nauk SSSR, Izv. Mekh. Zhidk. Gaza (1), 16–21. (Translated in Fluid Dyn. 17 (1), 12–16, 1982.)Google Scholar
Simuni, L. M. 1967 Solution of certain problems in flow of a viscous fluid, associated with a cylinder and a sphere. Izv. Sib. Otdel. Akad. Nauk SSSR, Ser. Tekh. Nauk, (2), 23–28.Google Scholar
Swanson, W. M. 1961 The Magnus effect: a summary of investigations to date. Trans. ASME D: J. Basic Engng 83, 461–470.Google Scholar
Ta Phuoc, Loc 1975 Étude numérique de l'écoulement d'un fluide visqueux incompressible autour d'un cylinder fixe ou en rotation. Effet Magnus. J. Méc. 14, 109–134.Google Scholar
Taneda, S. 1977 Visual study of unsteady separated flows around bodies. Prog. Aero. Sci. 17, 287–348.Google Scholar
Taneda, S. 1978 Visual observations of the flow past a circular cylinder performing a rotatory oscillation. J. Phys. Soc. Japan 45, 1038–1043.Google Scholar
Taneda, S. 1980 Visualization of unsteady flow separation. In Flow Visualization II (ed. W. Merzkirch), pp. 253–257. Hemisphere.
Tang, T. & Ingham, D. B. 1991 On steady flow past a rotating circular cylinder at Reynolds numbers 60 and 100. Computers & Fluids 19, 217–230.Google Scholar
Taslim, M. E., Kinney, R. B. & Paolino, M. A. 1984 Analysis of two-dimensional viscous flow over cylinders in unsteady motion. AIAA J. 22, 586–594.Google Scholar
Telionis, D. P. 1981 Unsteady Viscous Flows. Springer.
Tennant, J. S., Johnson, W. S. & Krothapalli, A. 1976 Rotating cylinder for circulation control on an airfoil. J. Hydronaut. 10, 102–105.Google Scholar
Ting, L. 1983 On the application of the integral invariants and decay laws of vorticity distributions. J. Fluid Mech. 127, 497–506.Google Scholar
Tokumaru, P. T. & Dimotakis, P. E. 1991 Rotary oscillation control of a cylinder wake. J. Fluid Mech. 224, 77–90.Google Scholar
Townsend, P. 1980 A numerical simulation of Newtonian and visco-elastic flow past stationary and rotating cylinders. J. Non-Newtonian Fluid Mech. 6, 219–243.Google Scholar
Vinokur, M. 1983 On one-dimensional stretching functions for finite-difference calculations. J. Comput. Phys. 50, 215–234.Google Scholar
Walowit, J., Tsao, S. & DiPrima, R. C. 1964 Stability of flow between arbitrarily spaced concentric cylindrical surfaces including the effect of a radial temperature gradient. Trans. ASME E: J. Appl. Mech. 31, 585–593.Google Scholar
Wambecq, A. 1978 Rational Runge–Kutta methods for solving systems of ordinary differential equations. Computing 20, 333–342.Google Scholar
Wang, C. M. & Wu, J. C. 1986 Numerical solution of steady Navier–Stokes problems using integral representations. AIAA J. 24, 1305–1312.Google Scholar
Werlé, H. 1984 Hydrodynamic visualization of the flow around a streamlined cylinder with suction. Cousteau-Malavard turbine sail model. La Recherche Aérospatiale (English Edition) 4, 29–38.Google Scholar
Weston, R. P. & Liu, C. H. 1982 Approximate boundary condition procedure for the two-dimensional numerical solution of vortex wakes. AIAA Paper 82-0951.
Williamson, C. H. K. 1985 Sinusoidal flow relative to circular cylinders. J. Fluid Mech. 155, 141–174.Google Scholar
Williamson, C. H. K. 1988 The existence of two stages in the transition to three-dimensionality on a cylinder wake. Phys. Fluids 31, 3165–3168.Google Scholar
Wood, W. W. 1957 Boundary layers whose streamlines are closed. J. Fluid Mech. 2, 77–87.Google Scholar
Wu, J., Mo, J. & Vakili, A. 1989 On the wake of a cylinder with rotational oscillations. AIAA Paper 89-1024.
Wu, J. C. 1975 Velocity and extraneous boundary conditions of viscous flow problems. AIAA Paper 75-47.
Wu, J. C. 1976 Numerical boundary conditions for viscous flow problems. AIAA J. 14, 1042–1049.Google Scholar
Wu, J. C. & Sampath, S. 1976 A numerical study of viscous flow around an airfoil. AIAA Paper 76-337.
Wu, J. C. & Thompson, J. F. 1973 Numerical solutions of time-dependent incompressible Navier-Stokes equations using an integro-differential formulation. Computers & Fluids 1, 197–215.Google Scholar
Zang, T. A., Wong, Y. S. & Hussaini, M. Y. 1982 Spectral multigrid methods for elliptic equations. J. Comput. Phys. 48, 485–501.Google Scholar
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21, 155–165.Google Scholar