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Journal of Mechanical Science and Technology

, Volume 33, Issue 4, pp 1731–1741 | Cite as

Numerical study of rectangular spectral collocation method on flow over a circular cylinder

  • B. Smith
  • R. Laoulache
  • A. R. H. Heryudono
  • J. LeeEmail author
Article
  • 2 Downloads

Abstract

Laminar flow over a circular cylinder has been widely used as a classical benchmark test for various numerical methods for partial differential equations. One of the popular ways is by using the vorticity-streamfunction formulation of the Navier-Stokes equations on a bounded numerical domain. The partial differential equations are solved numerically by the method of lines, where space is discretized using the standard collocation method, subject to multiple boundary conditions in the form of Dirichlet at the inlet and Neumann at the outlet. The resulting system of equations are then advanced in time using multistep methods. Fourier-Chebyshev pseudospectral method is used to approximate the solutions in space and Adams-Bashforth third-order backward differentiation method is employed as a time-stepping method. The rectangular spectral collocation method, developed by Driscoll and Hale, is applied to solve the ambiguity in imposing multiple boundary conditions on the same boundary points. The numerical simulations show very good agreement with similar studies for the Strouhal number, drag and lift coefficients over Reynolds numbers ranging from 50 to 150.

Keywords

Flow over cylinder Multiple boundary conditions Navier-Stokes problem Rectangular spectral collocation Time stepping 

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References

  1. [1]
    F. S. Hover, H. Tvedt and M. S. Triantafyllou, Vortex-induced vibrations of a cylinder with tripping wires, Journal of Fluid Mechanics, 448 (2001) 175–195.CrossRefzbMATHGoogle Scholar
  2. [2]
    N. Ren and J. Ou, Aerodynamic interference effect between large wind turbine blade and tower, Proc. of the International Symposium on Computational Structural Engineering, Shanghai, China (2009) 489–495.CrossRefGoogle Scholar
  3. [3]
    D. Sumner, M. D. Richards and O. O. Akosile, Strouhal number data for two staggered circular cylinders, Journal of Wind Engineering and Industrial Aerodynamics, 96(6) (2008) 859–871.CrossRefGoogle Scholar
  4. [4]
    H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, USA (1979).zbMATHGoogle Scholar
  5. [5]
    S. J. D. D’Alessio and S. C. R. Dennis, A vorticity model for viscous flow past a cylinder, Computers & Fluids, 23(2) (1994) 279–293.CrossRefzbMATHGoogle Scholar
  6. [6]
    D. Newman and G. E. Karniadakis, Simulations of flow over a flexible cable: A comparison of forced and flow-induced vibration, Journal of Fluids and Structures, 10(5) (1996) 439–453.CrossRefGoogle Scholar
  7. [7]
    H. M. Blackburn and R. D. Henderson, A study of two-dimensional flow past an oscillating cylinder, Journal of Fluid Mechanics, 385 (1999) 255–286.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S. Kumar, C. Cantu and B. Gonzalez, Flow past a rotating cylinder at low and high rotation rates, Journal of Fluids Engineering, 133(4) (2011) 041201–041209.CrossRefGoogle Scholar
  9. [9]
    D. J. Tritton, Experiments on the flow past a circular cylinder at low Reynolds numbers, Journal of Fluid Mechanics, 6(4) (1959) 547–567.CrossRefzbMATHGoogle Scholar
  10. [10]
    F. M. White and I. Corfield, Viscous Fluid Flow, McGraw-Hill, New York, USA, 3 (2006).Google Scholar
  11. [11]
    C. Norberg, Fluctuating lift on a circular cylinder: Review and new measurements, Journal of Fluids and Structures, 17(1) (2003) 57–96.CrossRefGoogle Scholar
  12. [12]
    C. Norberg, Flow around a circular cylinder: aspects of fluctuating lift, Journal of Fluids and Structures, 15(3) (2001) 459–469.CrossRefGoogle Scholar
  13. [13]
    M. H. Wu, C. Y. Wen, R. H. Yen, M. C. Weng and A. B. Wang, Experimental and numerical study of the separation angle for flow around a circular cylinder at low Reynolds number, Journal of Fluid Mechanics, 515 (2004) 233–260.CrossRefzbMATHGoogle Scholar
  14. [14]
    T. E. Tezduyar and J. Liou, On the downstream boundary conditions for the vorticity-stream function formulation of two-dimensional incompressible flows, Computer Methods in Applied Mechanics and Engineering, 85(2) (1991) 207–217.CrossRefzbMATHGoogle Scholar
  15. [15]
    O. Daube, Vorticity boundary conditions for the Navier-Stokes equation in velocity-vorticity formulation, NATO Science Series C: Mathematical and Physical Science, 395 (1993) 117–127.MathSciNetzbMATHGoogle Scholar
  16. [16]
    N. Hasan, S. F. Anwer and S. Sanghi, On the outflow boundary condition for external incompressible flows: A new approach, Journal of Computational Physics, 206(2) (2005) 661–683.CrossRefzbMATHGoogle Scholar
  17. [17]
    B. N. Rajani, A. Kandasamy and S. Majumdar, Numerical simulation of laminar flow past a circular cylinder, Applied Mathematical Modelling, 33(3) (2009) 1228–1247.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    L. Qu, C. Norberg, L. Davidson, S. H. Peng and F. Wang, Quantitative numerical analysis of flow past a circular cylinder at Reynolds number between 50 and 200, Journal of Fluids and Structures, 39 (2013) 347–370.CrossRefGoogle Scholar
  19. [19]
    H. H. Yang, B. R. Seymour and B. D. Shizgal, A Chebyshev pseudospectral multi-domain method for steady flow past a cylinder up to Re=150, Computers & Fluids, 23(6) (1994) 829–851.CrossRefzbMATHGoogle Scholar
  20. [20]
    R. Gautier, D. Biau and E. Lamballais, A reference solution of the flow over a circular cylinder at Re = 40, Computers & Fluids, 75 (2013) 103–111.CrossRefzbMATHGoogle Scholar
  21. [21]
    B. Fornberg, A numerical study of steady viscous flow past a circular cylinder, Journal of Fluid Mechanics, 98(4) (1980) 819–855.CrossRefzbMATHGoogle Scholar
  22. [22]
    M. Kotovshchikova, Numerical simulation of 2D flow past a dimpled cylinder using a pseudospectral method, Master’s Thesis, University of Manitoba, Winnipeg, Canada (2006).Google Scholar
  23. [23]
    T. A. Driscoll and N. Hale, Rectangular spectral collocation, IMA Journal of Numerical Analysis, 36(1) (2016) 108–132.MathSciNetzbMATHGoogle Scholar
  24. [24]
    B. Smith, R. Laoulache and A. Heryudono, Implementation of Neumann boundary condition with influence matrix method for viscous annular flow using pseudospectral collocation, Journal of Computational and Applied Mathematics, 285 (2015) 100–115.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Revised Ed., Springer-Verlag, Berlin, Germany (1999).CrossRefzbMATHGoogle Scholar
  26. [26]
    L. Halpern and M. Schatzman, Artificial boundary conditions for incompressible viscous flows, SIAM Journal of Mathematical Analysis, 20(2) (1989) 308–353.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    T. Yoshida, T. Watanabe and I. Nakamura, Numerical analysis of open boundary conditions for incompressible viscous flow past a square cylinder, Transactions of the Japan Society of Mechanical Engineers Series B, 59 (1993) 2799–2806.CrossRefGoogle Scholar
  28. [28]
    A. Sohankar, C. Norberg and L. Davidson, Low-Reynolds-number flow around a square cylinder at incidence: Study of blockage, onset of vortex shedding and outlet boundary condition, International Journal for Numerical Methods in Fluids, 26(1) (1998) 39–56.CrossRefzbMATHGoogle Scholar
  29. [29]
    L. N. Trefethen, Spectral Methods in Matlab, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (2000).CrossRefzbMATHGoogle Scholar
  30. [30]
    L. N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (2013).zbMATHGoogle Scholar
  31. [31]
    J. P. Berrut and L. N. Trefethen, Barycentric Lagrange interpolation, SIAM Review, 46(3) (2004) 501–517.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    T. A. Driscoll, N. Hale and L. N. Trefethen, Chebfun Guide, Pafnuty Publications, Oxford, UK (2014).Google Scholar
  33. [33]
    R. Peyret, Spectral Methods for Incompressible Viscous Flow, Springer-Verlag, New York, USA (2002).CrossRefzbMATHGoogle Scholar
  34. [34]
    H. E. Salzer, Lagrangian interpolation at the Chebyshev points Xn, ν = cos(νπ/n), ν = 0(1)n; some unnoted advantages, The Computer Journal, 15 (1972) 156–159.MathSciNetCrossRefGoogle Scholar
  35. [35]
    J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second Edition, Dover Publications, Mineola, USA (2001).zbMATHGoogle Scholar
  36. [36]
    M. Uhlmann, The Need for De-aliasing in a Chebyshev Pseudo-spectral Method, PIK Report No. 60, Potsdam Institute for Climate Impact Research (PIK), Potsdam, Germany (2000).Google Scholar
  37. [37]
    C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, Germany (2006).zbMATHGoogle Scholar
  38. [38]
    S. A. Orszag, On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components, Journal of Atmospheric Science, 28(6) (1971) 1074–1074.CrossRefGoogle Scholar
  39. [39]
    M. Braza, P. Chassaing and H. H. Minh, Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, Journal of Fluid Mechanics, 165 (1986) 79–130.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    H. Ding, C. Shu, K. S. Yeo and D. Xu, Numerical simulation of flows around two circular cylinders by mesh-free least square-based finite difference methods, International Journal for Numerical Methods in Fluids, 53 (2007) 305–332.CrossRefzbMATHGoogle Scholar
  41. [41]
    C. Liu, X. Zheng and C. H. Sung, Preconditioned multi-grid methods for unsteady incompressible flows, Journal of Computational Physics, 139 (1998) 35–57.CrossRefzbMATHGoogle Scholar
  42. [42]
    O. Posdziech and R. Grundmann, A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder, Journal of Fluids and Structures, 23 (2007) 479–499.CrossRefGoogle Scholar
  43. [43]
    M. Ramšak, L. Škerget, M. Hriberšek and Z. Žunič, A multidomain boundary element method for unsteady laminar flow using stream function-vorticity equations, Engineering Analysis with Boundary Elements, 29 (2005) 1–14.CrossRefzbMATHGoogle Scholar
  44. [44]
    E. Stålberg, A. Brüger, P. Lötstedt, A. V. Johansson and D. S. Henningson, High order accurate solution of flow past a circular cylinder, Journal of Scientific Computing, 27(1–3) (2006) 431–441.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    M. Van Dyke, An Album of Fluid Motion, Stanford, Calif.: Parabolic Press (1982).CrossRefGoogle Scholar
  46. [46]
    G. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press (1970).Google Scholar
  47. [47]
    J. P. Boyd, Two comments on filtering (artificial viscosity) for Chebyshev and Legendre spectral and spectral element methods: Preserving boundary conditions and interpretation of the filter as a diffusion, Journal of Computational Physics, 143(1) (1998) 283–288.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© KSME & Springer 2019

Authors and Affiliations

  • B. Smith
    • 1
  • R. Laoulache
    • 1
  • A. R. H. Heryudono
    • 2
  • J. Lee
    • 2
    • 3
    Email author
  1. 1.Department of Mechanical EngineeringUniversity of MassachusettsDartmouthUSA
  2. 2.Department of MathematicsUniversity of MassachusettsDartmouthUSA
  3. 3.Department of Mechanical and Design EngineeringHongik UniversitySejongKorea

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