On the Sub-Critical Bifurcation of Anti-Phase and In-Phase Synchronized Vortex Shedding Forms

Yih Ferng Peng

Department of Civil Engineering, National Chi Nan University, Puli 545, Taiwan, China..

DOI: 10.4236/jmp.2013.45B015                

Abstract

Transition of flows past a pair of side-by-side circular cylinders are investigated by numerical simulations and the bifurcation analysis of the numerical results. Various flow patterns behind the cylinder-pair have been identified by the gap ratio (G) and Reynolds number (Re). This study focus on transition of in-phase and anti-phase vortex shedding synchronized forms. A nested Cartesian-grid formulation, in combination with an effective immersed boundary method and a two-step fractional-step procedure, has been adopted to simulate the flows. Numerical results reveal that the in-phase and anti-phase vortex shedding flows at Re = 100 can co-exist at 2.08 ≤G≤ 2.58. Hysteresis loop with increasing/decreasing G at constant Reynolds number Re = 100 is reported.

Keywords

In-phase Vortex Shedding; Anti-phase Vortex Shedding; Hysteresis

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	H. M. Spivac, “Vortex Frequency and Flow Pattern in the Wake of Two Parallel Cylinders at Varied Spacing Normal to An Airstream,” Journal of the Aeronautical Sciences, Vol. 13, 1946, pp. 289-301.
[2]	P. W. Bearman and A. J. Wadcock, “The Interference between A Pair of Circular Cylinders Normal to A Stream,” Journal of Fluid Mechanics, Vol. 61. No. 3, 1973, pp. 499-511. doi:10.1017/S0022112073000832
[3]	M. M. Zdravkovich, “Review of Flow Interference Between Two Circular Cylinders in Various Arrangements,” ASME, Vol. 99, No. 4, 1977, pp. 618-633. doi:10.1115/1.3448871
[4]	C. H. K. Williamson, “Evolution of a Single Wake behind A Pair of Bluff Bodies,” Journal of Fluid Mechanics, Vol. 159, 1985, pp. 1-18. doi:10.1017/S002211208500307X
[5]	H. J. Kim and P. A. Durbin, “Investigation of the Flow between a Pair of Circular Cylinders in the Flopping Regime,” Journal of Fluid Mechanics, Vol. 196, 1988, pp. 431-448. doi:10.1017/S0022112088002769
[6]	D. Sumner, S. S. T. Wong, S. J. Price and M. P. Päidoussis, “Fluid Behavior of Side-by-side Circular Cylinders in Steady Cross-flow,” Journal of Fluids and Structures, Vol. 13. No. 3, 1999, pp. 309-339. doi:10.1006/jfls.1999.0205
[7]	Y. Zhou, H. J. Zhang and M. W. Yiu, “The Turbulent Wake of Two Side-by-side Circular Cylinders,” Journal of Fluid Mechanics，Vol. 458, 2002, pp. 303-332. doi:10.1017/S0022112002007887
[8]	S. J. Xu, Y. Zhou and R. M. C. So, “Reynolds Number Effects on the Flow Structure behind Two Side-by-side Cylinders,” Physics of Fluids, Vol. 15. No. 5, 2003, pp. 1214-1219. doi:10.1063/1.1561614
[9]	S. Kang, “Characteristics of Flow over Two Circular Cylinders in A Side-by-side Arrangement at Low Reynolds Numbers,” Physics of Fluids, Vol. 15, 2003.
[10]	S. Kumar, B. Gonzalez and O. Probst, “Flow Past Two Rotating Cylinders,” Physics of Fluids, Vol. 23, No. 1, 2011, 01402. doi:10.1063/1.3528260
[11]	Md Mahbub Alam, Y. Zhou, and X. W. Wang, “The Wake of Two Side-by-side Square Cylinders,” Journal of Fluid Mechanics, Vol. 669, 2011, pp. 432-471. doi:10.1017/S0022112010005288
[12]	D. Sumner, “Two Circular Cylinders in Cross-flow: A Review,” Journal of Fluids and Structures, Vol. 26. No. 6, 2010, pp. 849-899. doi:10.1016/j.jfluidstructs.2010.07.001
[13]	J. Mizushima and Y. Ino, “Stability of Flows Past A Pair of Circular Cylinders in A Side-by-side Arrangement,” Journal of Fluid Mechanics, Vol. 595, 2008, pp. 491-507. doi:10.1017/S0022112007009433
[14]	Y. F. Peng, A. Sau, R. R. Hwang, W. C. Yang and C. M. Hsieh, “Criticality of Flow Transition behind Two Side-by-side Elliptic Cylinders,” Physics of Fluids, Vol. 24. No. 3, 2012, 034102. doi:10.1063/1.3687450
[15]	Y. F. Peng, R. Mittal, A. Sau and R. Hwang, “Ested Cartesian Grid Method in Incompressible Viscous Fluid Flow,”Journal of Computational Physics, Vol. 229, No.19, 2010, pp. 7072-7101. doi:10.1016/j.jcp.2010.05.041
[16]	M. Provansal, C. Mathis and L. Boyer, “Be’nard-von Ka’rma’n Instability: Transient and Forced Regimes,” Journal of Fluid Mechanics, Vol. 182, 1987, pp. 1-22. doi:10.1017/S0022112087002222
[17]	C. H. K. Wil-liamson, “Oblique and Parallel Modes of Vortex Shedding in the Wake of a Circular Cylinder at Low Reynolds Numbers,” Journal of Fluid Mechanics, Vol. 206, 1989, pp. 579-627. doi:10.1017/S0022112089002429
[18]	C. Norberg, “An Experimental Investigation of the Flow around a Circular Cylinder: Influence of Aspect Ratio,” Journal of Fluid Mechanics, Vol. 258, 1994, pp. 287-316. doi:10.1017/S0022112094003332
[19]	C. P. Jackson, “A Fi-nite-element Study of the Onset of Vortex Shedding in Flow past Variously Shaped Bodies,” J ournal of Fluid Mechanics, Vol. 182, 1987, pp. 23-45. doi:10.1017/S0022112087002234
[20]	B, Kumar and S. Mittal, “Effect of Blockage on Critical Parameters for Flow past A Circular Cylinder,” International Journal for Numerical Methods in Fluids, Vol. 50. No. 8, 2006, pp. 987-1001. doi:10.1002/fld.1098
[21]	J. Dusek, P. Le Gal and P. Fraunie, “A Numerical and Theoretical Study of the First Hopf Bifurcation in A Cylinder Wake,” Journal of Fluid Mechanics, Vol. 264, 1994, pp. 59-80. doi:10.1017/S0022112094000583