@Article{cmes.2011.080.179,
AUTHOR = {Sushil Mohan Ratnaker, Sanjay Mittal},
TITLE = {Is the Karman Mode the Least Stable Mode Below the Critical Re?},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {80},
YEAR = {2011},
NUMBER = {3&4},
PAGES = {179--200},
URL = {http://www.techscience.com/CMES/v80n3&4/25752},
ISSN = {1526-1506},
ABSTRACT = {Flow past a circular cylinder looses stability at <i>Re</i> ~ 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 <i>Re</i>, is investigated. In particular, the study explores the possibility of modes other than the Kármán mode having the largest growth rate for <i>Re < Re<sub>cr</sub></i>. To this extent, global linear stability analysis (<i>LSA</i>) of the steady flow past a circular cylinder is carried out for <i>Re</i> = 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, <i>St</i>, 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 <i>St</i> = 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 <i>Re</i> = 45 flow. Of the three modes, the Kármán mode is most stable. Interestingly, the <i>St</i> → 0 mode is found to be least stable. For the <i>Re</i> = 48 flow, the <i>St</i> → 0 mode is most stable followed by the <i>St</i> = 0 mode. The computations are utilized to determine the least stable mode for various <i>Re</i>. The Kármán mode has the largest growth rate for <i>Re</i> ≥ 47.05 while the <i>St</i> → 0 mode is the least stable mode for <i>Re</i> ≤ 46.59. The <i>St</i> = 0 mode dominates for 46.59 < <i>Re</i> < 47.05. The results from the LSA are confirmed via direct time integration (<i>DTI</i>) of the linearized equations.},
DOI = {10.3970/cmes.2011.080.179}
}