Two coordinate systems description of viscous flow past a circular cylinder

F.P.H. van Beckum

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    Abstract

    Two dimensional steady viscous exterior flow is often treated as an elliptic problem within a finite region, bounded by a big circle. Solutions by means of Fourier expansions or discretisation on a polar grid will only give reliable results for low Reynolds numbers. In the present paper an example of a transformation is given, that maps the entire vorticity field onto a rectangle so that: (a) the vorticity equation can develop its parabolic character, (b) the wake in the far field comes out properly, and (c) the restriction to low Reynolds numbers is removed. For the elliptic stream-function equation the polar system is retained.

    This two coordinate systems description has been studied for a linearized version of the Navier-Stokes equations obtained by assuming a fixed velocity field v(o) in the convection term. The first results ζ and ψ are qualitatively so similar to the final solutions known from literature, that an iteration process, suggested-by deriving a new velocity field v(1) from ϕ, can be expected to show rapid convergence.
    Original languageEnglish
    Title of host publicationProceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics June 28 – July 2, 1976 Twente University, Enschede
    EditorsAdriaan I. van de Vooren, Pieter J. Zandbergen
    Place of PublicationBerlin, Heidelberg
    PublisherSpringer
    Pages101-106
    ISBN (Electronic)978-3-540-37548-7
    ISBN (Print)978-3-540-08004-6
    DOIs
    Publication statusPublished - 1976
    Event5th International Conference on Numerical Methods in Fluid Dynamics, NMFD 1976 - Twente University, Enschede, Netherlands
    Duration: 28 Jun 19762 Jul 1976
    Conference number: 5

    Publication series

    NameLecture Notes in Physics
    PublisherSpringer
    Volume59
    ISSN (Print)0075-8450
    ISSN (Electronic)1616-6361

    Conference

    Conference5th International Conference on Numerical Methods in Fluid Dynamics, NMFD 1976
    Abbreviated titleNMFD
    CountryNetherlands
    CityEnschede
    Period28/06/762/07/76

    Fingerprint

    Low Reynolds number
    Circular Cylinder
    Viscous Flow
    Vorticity
    Velocity Field
    Fourier Expansion
    Elliptic function
    Stream Function
    Wake
    Far Field
    Elliptic Problems
    Rectangle
    Convection
    Navier-Stokes Equations
    Circle
    Discretization
    Entire
    Restriction
    Grid
    Iteration

    Keywords

    • Convection term
    • Polar system
    • Vorticity equation
    • Vorticity distribution
    • Polar grid

    Cite this

    van Beckum, F. P. H. (1976). Two coordinate systems description of viscous flow past a circular cylinder. In A. I. van de Vooren, & P. J. Zandbergen (Eds.), Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics June 28 – July 2, 1976 Twente University, Enschede (pp. 101-106). (Lecture Notes in Physics; Vol. 59). Berlin, Heidelberg: Springer. https://doi.org/10.1007/3-540-08004-X_303
    van Beckum, F.P.H. / Two coordinate systems description of viscous flow past a circular cylinder. Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics June 28 – July 2, 1976 Twente University, Enschede. editor / Adriaan I. van de Vooren ; Pieter J. Zandbergen. Berlin, Heidelberg : Springer, 1976. pp. 101-106 (Lecture Notes in Physics).
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    abstract = "Two dimensional steady viscous exterior flow is often treated as an elliptic problem within a finite region, bounded by a big circle. Solutions by means of Fourier expansions or discretisation on a polar grid will only give reliable results for low Reynolds numbers. In the present paper an example of a transformation is given, that maps the entire vorticity field onto a rectangle so that: (a) the vorticity equation can develop its parabolic character, (b) the wake in the far field comes out properly, and (c) the restriction to low Reynolds numbers is removed. For the elliptic stream-function equation the polar system is retained.This two coordinate systems description has been studied for a linearized version of the Navier-Stokes equations obtained by assuming a fixed velocity field v(o) in the convection term. The first results ζ and ψ are qualitatively so similar to the final solutions known from literature, that an iteration process, suggested-by deriving a new velocity field v(1) from ϕ, can be expected to show rapid convergence.",
    keywords = "Convection term, Polar system, Vorticity equation, Vorticity distribution, Polar grid",
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    van Beckum, FPH 1976, Two coordinate systems description of viscous flow past a circular cylinder. in AI van de Vooren & PJ Zandbergen (eds), Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics June 28 – July 2, 1976 Twente University, Enschede. Lecture Notes in Physics, vol. 59, Springer, Berlin, Heidelberg, pp. 101-106, 5th International Conference on Numerical Methods in Fluid Dynamics, NMFD 1976, Enschede, Netherlands, 28/06/76. https://doi.org/10.1007/3-540-08004-X_303

    Two coordinate systems description of viscous flow past a circular cylinder. / van Beckum, F.P.H.

    Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics June 28 – July 2, 1976 Twente University, Enschede. ed. / Adriaan I. van de Vooren; Pieter J. Zandbergen. Berlin, Heidelberg : Springer, 1976. p. 101-106 (Lecture Notes in Physics; Vol. 59).

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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    N2 - Two dimensional steady viscous exterior flow is often treated as an elliptic problem within a finite region, bounded by a big circle. Solutions by means of Fourier expansions or discretisation on a polar grid will only give reliable results for low Reynolds numbers. In the present paper an example of a transformation is given, that maps the entire vorticity field onto a rectangle so that: (a) the vorticity equation can develop its parabolic character, (b) the wake in the far field comes out properly, and (c) the restriction to low Reynolds numbers is removed. For the elliptic stream-function equation the polar system is retained.This two coordinate systems description has been studied for a linearized version of the Navier-Stokes equations obtained by assuming a fixed velocity field v(o) in the convection term. The first results ζ and ψ are qualitatively so similar to the final solutions known from literature, that an iteration process, suggested-by deriving a new velocity field v(1) from ϕ, can be expected to show rapid convergence.

    AB - Two dimensional steady viscous exterior flow is often treated as an elliptic problem within a finite region, bounded by a big circle. Solutions by means of Fourier expansions or discretisation on a polar grid will only give reliable results for low Reynolds numbers. In the present paper an example of a transformation is given, that maps the entire vorticity field onto a rectangle so that: (a) the vorticity equation can develop its parabolic character, (b) the wake in the far field comes out properly, and (c) the restriction to low Reynolds numbers is removed. For the elliptic stream-function equation the polar system is retained.This two coordinate systems description has been studied for a linearized version of the Navier-Stokes equations obtained by assuming a fixed velocity field v(o) in the convection term. The first results ζ and ψ are qualitatively so similar to the final solutions known from literature, that an iteration process, suggested-by deriving a new velocity field v(1) from ϕ, can be expected to show rapid convergence.

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    van Beckum FPH. Two coordinate systems description of viscous flow past a circular cylinder. In van de Vooren AI, Zandbergen PJ, editors, Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics June 28 – July 2, 1976 Twente University, Enschede. Berlin, Heidelberg: Springer. 1976. p. 101-106. (Lecture Notes in Physics). https://doi.org/10.1007/3-540-08004-X_303