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An Embedding Method for Bluff Body Flows: Interactions of Two Side-by-Side Cylinder Wakes

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Abstract

We introduce a simple method for the numerical simulation of bluff body flows where the solid object is represented by a distributed body force in the Navier–Stokes equations. The body force density is found at every time step to reduce the velocity within the computational cells occupied by the rigid body to a prescribed value. The method combines certain ideas from the immersed boundary method which was developed to treat biofluid mechanical flows and the volume-of-fluid method for simulating flows with fluid–fluid interfaces. The main advantage of this embedding method is that the computations can be effected on a regular Cartesian grid, without the need to fit the grid to the bluff body surfaces. Thus, flow past several complex bodies can be treated as easily as flow past a single body. The method is validated by reproducing well-established results for vortex shedding from a stationary cylinder. The flow past two side-by-side cylinders is then investigated. When the distance between the cylinders is small, they are seen to shed vortices in-phase, whereas for larger distances, the shedding occurs in anti-phase. For intermediate distances, various shedding patterns are observed, including quasi-periodic, asymmetric and chaotic regimes. Mean values and phase portraits associated with the cylinder lift and drag coefficients, as well as spectral analysis of the same data, are used to describe the flow. A transition diagram that can be compared with experiments or models outlines the various dynamical regimes as a function of the distance between the cylinders and the Reynolds number.

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References

  1. Barkley, D., and Henderson, R.D. (1996). Secondary instability in the wake of a circular cylinder. J. Fluid Mech., 322, 215–241.

    Google Scholar 

  2. Bearman, P.W., and Wadcock, A.J. (1973). The interaction between a pair of circular cylinders normal to a stream. J. Fluid Mech., 61, 499–511.

    Google Scholar 

  3. Biermann, D., and Herrnstein, W.H. (1933). The interference between struts in various combinations. NACA Technical Report (1933); Spivak, H.M. (1946). Vortex frequency and flow pattern in the wake of two parallel cylinders varied spacings normal to an airstream. J. Aero. Sci., 13, 289–297. Hori, H. (1959). Experiments on flow around a pair of parallel circular cylinders. Proc. 9th JNCAM, Tokyo, 231–234. Ishigai, S., Nishikawa, E., Nishimura, K., and Cho, K. (1972). Experimental study of structure of gas flow in tube banks with tube axes normal to flow (Part I, Kármán vortex flow from two tubes at various spacings). Bull. JSME, 15, 949–956. Quadflieg, H. (1977). Vortex induced load on the cylinder pair at high Re. Forsch. Ing. Wes., 43, 9–18.

    Google Scholar 

  4. Bruneau, C.H., Greffier, O., and Kellay, H. (1999). Numerical study of grid turbulence in two dimensions and comparisons with experiments on turbulent soap films. Physica Rev. E, Part A60(2), R1162–1165.

    Google Scholar 

  5. Chang, K.J., and Song, C.J. (1990). Interactive vortex shedding from a pair of circular cylinders in a transverse arrangement. Int. J. Numer. Methods Fluid, 11, 317–329.

    Google Scholar 

  6. Clift, R., Grace, J.R., and Weber, M.E. (1978). Bubbles, Drops and Particles. Academic Press, New York. ISBN 0-12-176950-X.

  7. Dowell, E.H., and Hall, K.C. (2001). Modeling of fluid–structure interaction. Ann. Rev. Fluid Mech., 33, 445–490.

    Google Scholar 

  8. Evangelinos, C., Lucor, D., and Karniadakis, G.E. (2000). DNS-derived distribution on flexible cylinders subject to vortex-induced vibration. J. Fluids Struct., 14(3), 429–440.

    Google Scholar 

  9. Fadlun, E.A., Verzicco, R., Orlandi, P., and Mohd-Yusof, J. (2000). Combined immersed boundary finite-difference methods for three-dimensional complex simulations. J. Comput. Phys., 161(2), 35–60.

    Google Scholar 

  10. Gad-el-Hak, M. (1999). The fluid mechanics of microdevices – the freeman scholar lecture. J. Fluids Eng., 121, 5–33.

    Google Scholar 

  11. Gillies, E.A. (1998). Low dimensional control of circular cylinder wake. J. Fluid Mech., 371, 157–178.

    Google Scholar 

  12. Goldburg, W.I., Rutgers, M.A., and Wu, X.-L. (1997). Experiments on turbulence in soap films. Physica A, 239, 340–349.

    Google Scholar 

  13. Gueyffier, D., Lie, J., Nadim, A., Scardovelli, R., and Zaleski, S. (1999). Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comp. Phys., 152, 423–456.

    Google Scholar 

  14. Guillaume, D.W., and LaRue, J.C. (1999). Investigation of the flopping regime with two-, three- and four-cylinder arrays. Exp. Fluids, 27, 145–156.

    Google Scholar 

  15. Hakim, V., and Rappel, W.J. (1992). Dynamics of the globally coupled complex Ginzburg–Landau equation. Phys. Rev. A, 46, 7347–7350.

    Google Scholar 

  16. Harlow, H.H., and Welch, J.E. (1965). Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids, 8(12), 2182–2189.

    Google Scholar 

  17. Hiejima, S., and Nomura, T. (1999). Numerical study of the effect of periodic velocity excitation on aerodynamic characteristics of an oscillating circular cylinder. Int. J. Comput. Fluid Dyn., 12(3–4), 269–278.

    Google Scholar 

  18. Hirt, C.W., and Nichols, B.W. (1981). Volume of fluid (VOF) method for dynamics of free boundaries. J. Comput. Phys., 39, 201–225.

    Google Scholar 

  19. Jendrzejczyk, J.A., and Chen, S.S. (1986). Fluid forces on two circular cylinders in cross-flow. In Flow-Induced Vibration – 1986 (S.S. Chen, J.C. Simonis, Y.S. Shin eds.) (Pressure Vessels and Piping Conference and Exhibition, Chicago, Illinois, 20–24 July 1986), pp. 1–13. PVP Vol. 104. ASME, New York.

  20. Kamemoto, K. (1976). Formation and interaction of two parallel vortex streets. Bulletin of the JSME, 19, 283–290.

    Google Scholar 

  21. Karniadakis, G.E., and Triantafyllou, G.S. (1989). Frequency selection and asymptotic states in laminar wakes. J. Fluid Mech., 199, 441–469.

    Google Scholar 

  22. Karniadakis, G.E., and Triantafyllou, G.S. (1992). Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. J. Fluid Mech., 238, 1–30.

    Google Scholar 

  23. Khalak, A., and Williamson, C.H.K. (1997). Flow forces and dynamics of a hydroelastic structure with very low mass and damping. J. Fluids Struct., 11(8), 973–982.

    Google Scholar 

  24. Kim, H.J., and Durbin, P.A. (1988). Investigation of the flow between a pair of circular cylinders in the flopping regime. J. Fluid Mech., 196, 431–448.

    Google Scholar 

  25. Lai, M.C., and Peskin, C.S. (2000). An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys., 160(2), 705–719.

    Google Scholar 

  26. Le, Gal P., Chauve, M.P., Lima, R., and Rezende, J. (1990). Coupled wakes behind two circular cylinders. Phys. Rev. A, 41(8), R4566–4569.

    Google Scholar 

  27. Le Gal, P., Nadim, A., and Thompson, M. (2001). Hysteresis in the forced response of the subcritical Stuart–Landau equation: application to vortex shedding from a cylinder. J. Fluids Struct., 15, 445–457.

    Google Scholar 

  28. Mathis, C., Provansal, M., and Boyer, L. (1984). The Bénard–von Karman instability: an experimental study near the threshold. J. Phys. Lett., 45, L483–491.

    Google Scholar 

  29. Meneghini, J.R., Saltar, F., Siqueira, C.L.R., and Ferrari, J.A. (2001). Numerical simulation of flow interference between two circular cylinders in tandem and side-by-side arrangements. J. Fluids Struct., 15, 327–350.

    Google Scholar 

  30. Nayfeh, A.H., and Balachandran, B. (1995). Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley-Interscience, New York.

  31. Ng, C.W., and Ko, N.W.M. (1995). Flow interaction behind two circular cylinders of equal diameter – a numerical study. J. Wind Eng. Ind. Aerodyn., 54–55, 277–287.

    Google Scholar 

  32. Norberg, C. (2001). Flow around a circular cylinder: aspects of fluctuating lift. J. Fluids Struct., 15, 459–469.

    Google Scholar 

  33. Ohya, Y., Okajima, A., and Hayashi, M. (1988). Wake interference and vortex shedding. In Encyclopedia of Fluid Mechanics (N.P. Cheremisinoff, ed.), Vol. 8, Chapt. 10. Gulf, Houston.

  34. Olinger, D.J., and Sreenivasan, K.R. (1988). Nonlinear dynamics of the wake of an oscillating cylinder. Phys. Rev. Lett., 60(9), 797–800.

    Google Scholar 

  35. Parkinson, G. (1989). Phenomena and modelling of flow-induced vibrations of bluff bodies. Prog. Aerospace Sci., 26, 169–224.

    Google Scholar 

  36. Persillon, H., and Braza, M. (1998). Physical analysis of the transition to turbulence in the wake of a circular cylinder by three-dimensional Navier–Stokes simulation. J. Fluid Mech., 365, 23–88.

    Google Scholar 

  37. Peschard, I. (1995). De l’oscillateur aux sillages couplés. Ph.D. Thesis, Université Aix-Marseille II.

  38. Peschard, I., and Le Gal, P. (1996). Coupled wakes of cylinders. Phys. Rev. Lett., 77(15), 3122–3125.

  39. Peskin, C.S. (1977). Numerical analysis of blood flow in the heart. J. Comput. Phys., 25, 220–252. Mc Queen, D.M., and Peskin, C.S. (1989). A 3-dimensional computational method for blood-flow in the heart. 1. Immersed elastic fibers in a viscous incompressible fluid J. Comput. Phys., 81(2), 372–405.

  40. Pozrikidis, C. (1997). Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press, Oxford. ISBN 0-19-509320-8.

  41. Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (1992). Numerical Recipes in C. Cambridge University Press, Cambridge.

  42. Ravoux, J.F., and Le Gal, P. (1998). Biorthogonal decomposition analysis and reconstruction of spatiotemporal chaos generated by coupled wakes. Phys. Rev. E, Part A, 58(5), R5233–5236.

  43. Roussopoulos, K., and Monkewitz, P.A. (1996). Nonlinear modelling of vortex shedding control in cylinder wakes. Phys. D, 97, 264–273.

    Google Scholar 

  44. Scardovelli, R., and Zaleski, S. (1999). Direct numerical simulation of free-surface and interfacial flows. Ann. Rev. Fluid Mech., 31, 567–603.

    Google Scholar 

  45. Sen, M., Wajerski, D., and Gad-el-Hak, M. (1996). A novel pump for MEMS applications. ASME J. Fluids Eng., 118, 624–627.

    Google Scholar 

  46. Slaouti, A., and Stansby, P.K. (1992). Flow around two circular cylinders by the random-vortex method. J. Fluids Struct., 6, 641–670.

    Google Scholar 

  47. Sumner, D., Wong, S.S.T., Price, S.J., and Païdoussis, M.P. (1999). Flow behaviour of side-by-side circular cylinders in steady cross-flow. J. Fluids Struct., 13, 309–338.

    Google Scholar 

  48. Tang, S., and Aubry, N. (1997). On the symmetry breaking instability leading to vortex shedding. Phys. Fluids, 9, 2550–2561.

    Google Scholar 

  49. Tezduyar, T.E., Liou, J., Ganjoo, D.K., and Behr, M. (1990). Solution techniques for the vorticity-streamfunction formulation of two-dimensional unsteady incompressible flows. Int. J. Numer. Meth. Fluid, 11, 515–539.

    Google Scholar 

  50. Vorobieff, P., and Ecke, R.E. (1999). Cylinder wakes in flowing soap films. Phys. Rev. E., 60(3), 2953–2956.

    Google Scholar 

  51. Williamson, C.H.K. (1985). Evolution of a single wake behind a pair of bluff bodies. J. Fluid Mech., 159, 1–18.

    Google Scholar 

  52. Williamson, C.H.K. (1988). Defining a universal and continuous Strouhal–Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys. Fluids, 31(10), 2742–2744.

    Google Scholar 

  53. Williamson, C.H.K. (1996). Vortex dynamics in the cylinder wake. Ann. Rev. Fluid Mech., 28, 477–539.

    Google Scholar 

  54. Xiao, F. (1999). A computational model for suspended large rigid bodies in 3D unsteady viscous flows. J. Comput. Phys., 155(2), 348–379.

    Google Scholar 

  55. Zdravkovich, M.M. (1977). Review of flow interference between two circular cylinders in various arrangements. J. Fluids Eng., 99, 618–633.

    Google Scholar 

  56. Zhang, H.Q., Fey, U., Noack, B.R., König, M., and Eckelmann, H. (1995). On the transition of the cylinder wake. Phys. Fluids, 7(4), 779–794.

    Google Scholar 

  57. Zhou, C.Y., So, R.M.C., and Lam, K. (1999). Vortex-induced vibrations of an elastic circular cylinder. J. Fluids Struct., 13(4), 165–189.

    Google Scholar 

  58. Zhou, Y., Zhang, H.J., and Yiu, M.W. (2002). The turbulent wake of two side-by-side circular cylinders. J. Fluid Mech., 458, 303–332.

    Google Scholar 

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Authors and Affiliations

  1. Center for BioDynamics, Boston University, 02215, Boston, MA, U.S.A.

    J.F. Ravoux

  2. Keck Graduate Institute and Claremont Graduate University, 91711, Claremont, CA, U.S.A.

    A. Nadim

  3. Mechanical and Aerospace Engineering, University of Virginia, 22903, Charlottesville, VA, U.S.A.

    H. Haj-Hariri

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  1. J.F. Ravoux
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  2. A. Nadim
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M.Y. Hussaini

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Ravoux , J., Nadim , A. & Haj-Hariri , H. An Embedding Method for Bluff Body Flows: Interactions of Two Side-by-Side Cylinder Wakes. Theoret Comput Fluid Dynamics 16, 433–466 (2003). https://doi.org/10.1007/s00162-003-0090-4

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