Flow past a circular cylinder at low Reynolds number: Oblique vortex sheddingSuresh Behara and Sanjay Mittal
Citation: Physics of Fluids 22, 054102 (2010); doi: 10.1063/1.3410925 View online: http://dx.doi.org/10.1063/1.3410925 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/22/5?ver=pdfcov Published by the AIP Publishing
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Flow past a circular cylinder at low Reynolds number:Oblique vortex shedding
Suresh Behara and Sanjay MittalaDepartment of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur 208 016, IndiaReceived 15 May 2009; accepted 29 March 2010; published online 4 May 2010
Oblique shedding in the laminar regime for the flow past a nominally two-dimensional circularcylinder has been investigated numerically via a stabilized finite element method. No-slip conditionon one of the sidewalls leads to the formation of a boundary layer which promotes oblique vortexshedding. Computations are carried out for three values of Reynolds number Re: 60, 100, and 150.Cellular shedding is observed in all cases. Three cells are observed along the span for the Re=60flow while only two cells are formed at Re=100 and 150. Spotlike vortex dislocations form at thejunction of the cells. The frequency of the appearance of the dislocations increases with Re. Cellularshedding leads to low frequency modulation in the time histories of aerodynamic coefficients.Lowest value of drag is achieved at a time instant corresponding to the appearance of a newdislocation in the near wake. The vortex shedding frequency as well as the oblique angle of theprimary vortices is found to vary with time for the Re=60 flow. Their variation is also related to theappearance of dislocations in the near wake. It is found that the vortex shedding frequency St isrelated to the frequency observed for parallel shedding St0 and the angle of the oblique vortices by the relation: St=St0 cos . This relationship was proposed earlier for the case when the vortexshedding frequency and the oblique angle do not change with time. The velocity fluctuations arefound to decrease with increase in . For the Re=100 and 150 flow, the oblique angle of the vorticesand the shedding frequency outside the end cell do not change with time. However, and St dependon the aspect ratio of the cylinder. The oblique shedding angle, for various lengths of endplate andRe, is found to vary linearly with the thickness of the boundary layer on the side wall. 2010American Institute of Physics. doi:10.1063/1.3410925
Flow past bluff bodies have received continued researchattention over several years because of its significance inengineering applications as well as rich flow physics. Despiteits simple geometry, the flow past a nominally two-dimensional 2D circular cylinder is associated with most ofthe aspects related to other bluff bodies with more complexgeometries. A comprehensive description of the various re-gimes encountered in the flow past a circular cylinder hasbeen given by Williamson1 in a review article. The steadyflow past a circular cylinder looses stability at Re47 viathe instability of the wake.2 This instability leads to the vonKarman vortex shedding. However, the flow still remainstwo-dimensional. At Re in the range of 180190, the three-dimensional 3D instabilities in the wake appear. Anotherinteresting aspect related to the wake is the angle betweenthe axis of the cylinder and that of the vortices being shed.This leads to oblique versus parallel modes of vortex shed-ding and is largely determined by the experimental condi-tions.
Several experimental studies in the past36 have revealedthat oblique shedding is a result of the end conditions in thewind/water tunnel. The end conditions affect the vortex shed-ding over the entire span of the cylinder even if the length ofcylinder is very large compared with its diameter. They have
a direct effect over a region of the length that is 1020Dfrom the end, where D is diameter of the cylinder. The effecton the remaining span is indirect. The disturbances related tothe bending of vortices are passed on from one shed vortexto another over the span.5 Therefore, the end conditionswhich promote oblique mode of shedding lead to vorticesthat are parallel in the initial transient period. This phenom-enon was corroborated by Albarede and Monkewitz7 viamodeling the flow with the GinzburgLandau equations. It ispossible to obtain parallel shedding by manipulating endconditions via end plates5 or via other techniques, such asusing control cylinders.8
Williamson9 observed that in the laminar regime 49Re178, two curves for Strouhal-Re exist. The lowercurve, with a discontinuity at Re=64, is observed for obliqueshedding. Parallel shedding, induced by manipulating endconditions, leads to the higher Strouhal-Re curve. In anotherexperimental effort Williamson5 established that the angle ofthe vortices with the spanwise axis of the cylinder modifiesthe vortex shedding frequency. He proposed a relation be-tween the frequency of oblique shedding, f, and the obliqueangle, , as f= f0 cos , where f0 is the frequency whenparallel shedding takes place. It was noticed that the obliqueangle increases with the increase in Re and experiences sud-den rise at Re=64, which explains the discontinuity in theSt-Re plot. As Re is increased up to 100, decreases andthereafter, it remains almost constant.aElectronic mail: firstname.lastname@example.org.
PHYSICS OF FLUIDS 22, 054102 2010
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It has been observed in several experimental investiga-tions that the end effects lead to spanwise variation of shed-ding frequency. For example, Gerich and Eckelmann10 no-ticed that near the end of cylinder, ranging at 615 diametersalong the span, the frequency of vortex shedding is 10%15% less than the frequency in the central region. Cellularshedding and the related vortex dislocations have been stud-ied by various researchers in the past.1113 Leweke et al.14proposed a mechanism for the formation of cells. They at-tributed it to the spatial growth of the secondary Eckhausinstability of oblique shedding when the shedding angle ex-ceeds a certain critical value. This causes the vortex to breakalong the span and the resulting cells settle to a lower ob-lique shedding angle. The break region becomes the new endcondition for the next cell. The perturbation at the new endgrows leading to another break if the oblique angle is abovethe stability limit. The cell structure is, therefore, a conse-quence of the successive destabilization of the oblique shed-ding patterns.
Williamson5 observed two kinds of oblique vortex shed-ding modes. For 64Re178, the vortex shedding takesplace in two cells. This is referred to as the periodic obliqueshedding. The smaller cell that is closer to the end plates hasa smaller shedding frequency, while a regular chevronshaped vortex pattern is observed in the middle cell with asingle frequency. However, a different shedding pattern isobserved for Re64. The flow is associated with three fre-quencies along the span. For each of the two halves of thespan the vortex shedding frequency increases as one goesfrom the end-to-middle and then the central cell. This modeof shedding is referred to as the quasiperiodic oblique shed-ding.
Inoue and Sakuragi15 carried out computations for a cyl-inder of finite span with free ends for 40Re300 and0.5L /D100, where L is the spanwise length of the cyl-inder. In a very comprehensive study, they observed a drasticchange in the wake structure depending on the value of Reand L /D. Five basic patterns of vortex shedding were iden-tified. They obtained periodic oblique vortex shedding atRe=150, for L /D=100 and quasiperiodic shedding at Re=60 with L /D=70. The length of the central cell in the pe-riodic oblique shedding pattern was found to decrease withdecrease in L /D. The length of the end cell 10D, how-ever, remains unchanged.
Except for the effort by Mittal16 we are not aware of anyother numerical effort where the oblique vortex shedding in auniform flow has been investigated for a nominally 2D cyl-inder. Mittal16 carried out a numerical study for a cylinderwith aspect ratio 16 at Re=100, 300, and 1000. The endconditions are specified to model the effect of a wall. Theflow for Re=100 was found to be very organized, devoid ofany vortex dislocations and was associated with only one cellalong the cylinder span. The flow at Re=1000 was inter-spersed with vortex dislocations and the vortex sheddingangle varied, both, temporally and along the span. It wa