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Flow instabilities in the wake of a rounded square cylinder

Published online by Cambridge University Press:  23 March 2016

Doohyun Park  and
Kyung-Soo Yang
Doohyun Park
Affiliation:
Department of Mechanical Engineering, Inha University, Incheon 22212, Republic of Korea
Kyung-Soo Yang*
Affiliation:
Department of Mechanical Engineering, Inha University, Incheon 22212, Republic of Korea
*
†Email address for correspondence: ksyang@inha.ac.kr
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Abstract

Instabilities in the flow past a rounded square cylinder have been numerically studied in order to clarify the effects of rounding the sharp edges of a square cylinder on the primary and secondary instabilities associated with the flow. Rounding the edges was done by inscribing a quarter circle of radius $r$ in each edge of a square cylinder of height $d$. Nine cases of rounding were considered, ranging from a square cylinder ($r/d=0$) to a circular cylinder ($r/d=0.5$) with an increment of 0.0625. Each cross-section was numerically implemented in a Cartesian grid system by using an immersed boundary method. The key parameters are the Reynolds number (Re) and the edge-radius ratio ($r/d$). For low Re, the flow is steady and symmetric with respect to the centreline. Over the first critical Reynolds number ($Re_{c1}$), the flow undergoes a Hopf bifurcation to a time-periodic flow, termed the primary instability. As Re further increases, the onset of the secondary instability of three-dimensional (3D) nature is detected beyond the second critical Reynolds number ($Re_{c2}$). Rounding the sharp edges of a square cylinder significantly affects the flow topology, leading to noticeable changes in both instabilities. By employing the Stuart–Landau equation, we investigated the criticality of the primary instability depending upon $r/d$. The onset of the 3D secondary instability was detected by using Floquet stability analysis. The temporal and spatial characteristics of the dominant modes (A, B, QP) were described. The neutral stability curves of each mode were computed depending upon $r/d$.

JFM classification

Instability: Instability Wakes/Jets: Separated flows Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 793 , 25 April 2016 , pp. 915 - 932
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Copyright
© 2016 Cambridge University Press 

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References

Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Blackburn, H. M. & Lopez, J. M. 2003 On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids 15, L57–L60.CrossRefGoogle Scholar
Blackburn, H. M., Marques, F. & Lopez, J. M. 2005 Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395–411.CrossRefGoogle Scholar
Blackburn, H. M. & Sheard, G. J. 2010 On quasiperiodic and subharmonic Floquet wake instabilities. Phys. Fluids 22, 031701.CrossRefGoogle Scholar
Braza, M., Chassaing, P. & Minh, H. H. 1986 Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech. 165, 79–130.CrossRefGoogle Scholar
Briggs, W. L., Henson, V. E. & McCormick, S. F. 2000 A Multigrid Tutorial, 2nd edn. SIAM.CrossRefGoogle Scholar
Choi, C.-B., Jang, Y.-J. & Yang, K.-S. 2012 Secondary instability in the near-wake past two tandem square cylinders. Phys. Fluids 24, 024102.CrossRefGoogle Scholar
Choi, C.-B. & Yang, K.-S. 2013 Three-dimensional instability in the flow past two side-by-side square cylinders. Phys. Fluids 25, 074107.CrossRefGoogle Scholar
Choi, C.-B. & Yang, K.-S. 2014 Three-dimensional instability in flow past a rectangular cylinder ranging from a normal flat plate to a square cylinder. Phys. Fluids 26, 061702.CrossRefGoogle Scholar
De, A. K. & Dalal, A. 2006 Numerical simulation of unconfined flow past a triangular cylinder. Intl J. Numer. Meth. Fluids 52, 801–821.Google Scholar
Ding, Y. & Kawahara, M. 1999 Three-dimensional linear stability analysis of incompressible viscous flows using the finite element method. Intl J. Numer. Meth. Fluids 31, 451–479.3.0.CO;2-O>CrossRefGoogle Scholar
Giannetti, F., Camarri, S. & Luchini, P. 2010 Structural sensitivity of the secondary instability in the wake of a circular cylinder. J. Fluid Mech. 651, 319–337.CrossRefGoogle Scholar
Kim, D.-H., Yang, K.-S. & Senda, M. 2004 Large eddy simulation of turbulent flow past a square cylinder confined in a channel. Comput. Fluids 33, 81–96.CrossRefGoogle Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.CrossRefGoogle Scholar
Kumar, B. & Mittal, S. 2006 Prediction of the critical Reynolds number for flow past a circular cylinder. Comput. Meth. Appl. Mech. Engng 195, 6046–6058.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1976 Mechanics, 3rd edn. Pergamon.Google Scholar
Leontini, J. S., Jacono, D. L. & Thompson, M. C. 2014 Stability analysis of the elliptic cylinder wake. J. Fluid Mech. 763, 302–321.CrossRefGoogle Scholar
Luo, S. C., Tong, X. H. & Khoo, B. C. 2007 Transition phenomena in the wake of a square cylinder. J. Fluids Struct. 23, 227–248.CrossRefGoogle Scholar
Marques, F., Lopez, J. M. & Blackburn, H. M. 2004 Bifurcations in systems with $Z_{2}$ spatio-temporal and $O(2)$ spatial symmetry. Physica D 189, 247–276.CrossRefGoogle Scholar
Norberg, C. 2001 Flow around a circular cylinder: aspects of fluctuating lift. J. Fluids Struct. 15, 459–469.CrossRefGoogle Scholar
Park, D. S. 1994 Theoretical analysis of feedback control of Karman vortex shedding at slightly supercritical Reynolds numbers. Eur. J. Mech. (B/Fluids) 13, 387–399.Google Scholar
Park, J., Kwon, K. & Choi, H. 1998 Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160. KSME Intl J. 12, 1200–1205.CrossRefGoogle Scholar
Persillon, H. & 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.CrossRefGoogle Scholar
Robichaux, J., Balachandar, S. & Vanka, S. P. 1999 Three-dimensional Floquet instability of the wake of square cylinder. Phys. Fluids 11, 560–578.CrossRefGoogle Scholar
Saiki, E. M. & Biringen, S. 1996 Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method. J. Comput. Phys. 123, 450–465.CrossRefGoogle Scholar
Schumm, M., Berger, E. & Monkewitz, P. A. 1994 Self-excited oscillations in the wake of two-dimensional bluff bodies and their control. J. Fluid Mech. 271, 17–53.CrossRefGoogle Scholar
Sharma, A. & Eswaran, V. 2004 Heat and fluid flow across a square cylinder in the two-dimensional laminar flow regime. Numer. Heat Tranf. A 45, 247–269.CrossRefGoogle Scholar
Sheard, G. J. 2011 Wake stability features behind a square cylinder: focus on small incidence angles. J. Fluids Struct. 27, 734–742.CrossRefGoogle Scholar
Sheard, G. J., Fitzgerald, M. J. & Ryan, K. 2009 Cylinders with square cross-section: wake instabilities with incidence angle variation. J. Fluid Mech. 630, 43–69.CrossRefGoogle Scholar
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2003 From spheres to circular cylinders: the stability and flow structures of bluff ring wakes. J. Fluid Mech. 492, 147–180.CrossRefGoogle Scholar
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2004 From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings. J. Fluid Mech. 506, 45–78.CrossRefGoogle Scholar
Sohankar, A., Norberg, C. & Davidson, L. 1998 Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition. Intl J. Numer. Meth. Fluids 26, 39–56.3.0.CO;2-P>CrossRefGoogle Scholar
Stuart, J. T. 1971 Nonlinear stability theory. Annu. Rev. Fluid Mech. 3, 347–370.CrossRefGoogle Scholar
Williamson, C. H. K. 1988 The existence of two stages in the transition to three-dimensionality of a circular cylinder wake. Phys. Fluids 31, 3165–3168.CrossRefGoogle Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.CrossRefGoogle Scholar
Yang, D., Pettersen, B., Andersson, H. I. & Narasimhamurthy, V. D. 2013 Floquet stability analysis of the wake of an inclined flat plate. Phys. Fluids 25, 094103.CrossRefGoogle Scholar
Yang, J. & Balaras, E. 2006 An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries. J. Comput. Phys. 215, 12–40.CrossRefGoogle Scholar
Yang, K.-S. 2000 Numerical investigation of instability and transition in an obstructed channel flow. AIAA J. 38, 1173–1178.CrossRefGoogle Scholar
Yoon, D.-H., Yang, K.-S. & Choi, C.-B. 2010 Flow past a square cylinder with an angle of incidence. Phys. Fluids 22, 043603.CrossRefGoogle Scholar
Yoon, D.-H., Yang, K.-S. & Choi, C.-B. 2012 Three-dimensional wake structures and aerodynamic coefficients for flow past an inclined square cylinder. J. Wind Engng Ind. Aerodyn. 101, 34–42.CrossRefGoogle Scholar
Zhang, H.-Q., Fey, U., Noack, B. R., Konig, M. & Eckelmann, H. 1995 On the transition of the cylinder wake. Phys. Fluids 7, 779–794.CrossRefGoogle Scholar