Open Access

ARTICLE

Is the Karman Mode the Least Stable Mode Below the Critical Re?

Sushil Mohan Ratnaker¹, Sanjay Mittal^1,2

1 Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, U.P., India -208 016
2 smittal@iitk.ac.in, Phone: +91 512 2597906, Fax: +91 512 2597561

Computer Modeling in Engineering & Sciences 2011, 80(3&4), 179-200. https://doi.org/10.3970/cmes.2011.080.179

Abstract

Flow past a circular cylinder looses stability at Re ~ 47 via Hopf bifurcation. The eigenmode responsible for the instability leads to the von Kármán vortex shedding. In this work the linear stability of the flow to other modes, near the critical Re, is investigated. In particular, the study explores the possibility of modes other than the Kármán mode having the largest growth rate for Re < Re_cr. To this extent, global linear stability analysis (LSA) of the steady flow past a circular cylinder is carried out for Re = 45 and 48. In addition to the Kármán modes, two other modes are tracked. The eigenvalue of one of them is associated with a very small imaginary part; the mode is referred to as the St → 0 mode. The Strouhal number, St, is the non-dimensional vortex shedding frequency and is related to the imaginary part of the eigenvalue. The other mode is real and is referred to as the St = 0 mode. The modes also differ in regard to their symmetry about the wake centerline. Unlike the Kármán mode, the two modes are very sensitive to the spatial extent of the computational domain. Computations are carried out with domains of varying spatial extent and their results are utilized to estimate the growth rate and St for the unbounded flow. All the modes are stable for the Re = 45 flow. Of the three modes, the Kármán mode is most stable. Interestingly, the St → 0 mode is found to be least stable. For the Re = 48 flow, the St → 0 mode is most stable followed by the St = 0 mode. The computations are utilized to determine the least stable mode for various Re. The Kármán mode has the largest growth rate for Re ≥ 47.05 while the St → 0 mode is the least stable mode for Re ≤ 46.59. The St = 0 mode dominates for 46.59 < Re < 47.05. The results from the LSA are confirmed via direct time integration (DTI) of the linearized equations.

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