Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-26T21:55:16.486Z Has data issue: false hasContentIssue false
  • English
  • Français

On the frequency selection mechanism of the low-Re flow around rectangular cylinders

Published online by Cambridge University Press:  06 January 2022

A. Chiarini Open the ORCID record for A. Chiarini [Opens in a new window] ,
M. Quadrio Open the ORCID record for M. Quadrio [Opens in a new window]  and
F. Auteri Open the ORCID record for F. Auteri [Opens in a new window]
A. Chiarini
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156Milano, Italy
M. Quadrio
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156Milano, Italy
F. Auteri*
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156Milano, Italy
*
†Email address for correspondence: franco.auteri@polimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

In the flow past elongated rectangular cylinders at moderate Reynolds numbers, vortices shedding from the leading- and trailing-edge corners are frequency locked by the impinging leading-edge vortex instability. The present work investigates how the chord-based Strouhal number varies with the aspect ratio of the cylinder at a Reynolds number (based on the cylinder thickness and the free-stream velocity) of $Re=400$, i.e. when locking is strong. Several two-dimensional, nonlinear simulations are run for rectangular and D-shaped cylinders, with the aspect ratio ranging from $1$ to $11$, and a global linear stability analysis of the flow is performed. The shedding frequency observed in the nonlinear simulations is predicted fairly well by the eigenfrequency of the leading eigenmode. The inspection of the structural sensitivity confirms the central role of the trailing-edge vortex shedding in the frequency locking, as already assumed by other authors. Surprisingly, however, the stepwise increase of the Strouhal number with the aspect ratio reported by several previous works is not fully reproduced. Indeed, with increasing aspect ratio, two distinct flow behaviours are observed, associated with two flow configurations where the interaction between the leading- and trailing-edge vortices is different. These two configurations are fully characterised, and the mechanism of selection of the flow configuration is discussed. Lastly, for aspect ratios close to the jump between two consecutive shedding modes, the Strouhal number is found to present hysteresis, implying the existence of multiple stable configurations. Continuing the lower shedding-mode branch by increasing the aspect ratio, we found that the periodic configuration loses stability via a Neimark–Sacker bifurcation leading to different Arnold tongues. This hysteresis can explain, at least partially, the significant scatter of existing experimental and numerical data.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex shedding Wakes/Jets: Separated flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A44
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.CrossRefGoogle Scholar
Bayly, B.J., Orszag, S.A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Annu. Rev. Fluid Mech. 20 (1), 359–391.CrossRefGoogle Scholar
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 485–504.CrossRefGoogle Scholar
Blackburn, H.M. & Lopez, J.M. 2003 On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids 15 (8), L57–L60.CrossRefGoogle Scholar
Boghosian, M.E. & Cassel, K.W. 2016 On the origins of vortex shedding in two-dimensional incompressible flows. Theor. Comput. Fluid Dyn. 30 (6), 511–527.CrossRefGoogle ScholarPubMed
Brezzi, F. 1974 On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. ESAIM Math Model. Numer. Anal. 8 (R2), 129–151.Google Scholar
Brezzi, F. & Fortin, M. 1991 Mixed and Hybrid Finite Element Methods. Springer.CrossRefGoogle Scholar
Chaurasia, H.K. & Thompson, M.C. 2011 Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate. J. Fluid Mech. 681, 411–433.CrossRefGoogle Scholar
Chiarini, A., Quadrio, M. & Auteri, F. 2021 Linear stability of the steady flow past rectangular cylinders. J. Fluid Mech. 929, A36.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 (6), 061702.CrossRefGoogle Scholar
Citro, V., Luchini, P., Giannetti, F. & Auteri, F. 2017 Efficient stabilization and acceleration of numerical simulation of fluid flows by residual recombination. J. Comput. Phys. 344, 234–246.CrossRefGoogle Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97–128.CrossRefGoogle Scholar
Hecht, F. 2012 New development in FreeFem++. J. Numer. Maths 20 (3–4), 251–266.Google Scholar
Hourigan, K., Mills, R., Thompson, M.C., Sheridan, J., Dilin, P. & Welsh, M.C. 1993 Base pressure coefficients for flows around rectangular plates. J. Wind Engng Ind. Aerodyn. 49 (1), 311–318.CrossRefGoogle Scholar
Hourigan, K., Thompson, M.C. & Tan, B.T. 2001 Self-sustained oscillations in flows around long blunt plates. J. Fluids Struct. 15 (3), 387–398.CrossRefGoogle Scholar
Kuznetsov, Y. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.CrossRefGoogle Scholar
Lehoucq, R.B., Sorensen, D.C. & Yang, C. 1998 ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.CrossRefGoogle Scholar
Mills, R., Sheridan, J. & Hourigan, K. 2002 Response of base suction and vortex shedding from rectangular prisms to transverse forcing. J. Fluid Mech. 461, 25–49.CrossRefGoogle Scholar
Mills, R., Sheridan, J. & Hourigan, K. 2003 Particle image velocimetry and visualization of natural and forced flow around rectangular cylinders. J. Fluid Mech. 478, 299–323.CrossRefGoogle Scholar
Mills, R., Sheridan, J., Hourigan, K. & Welsh, M.C. 1995 The mechanism controlling vortex shedding from rectangular bluff bodies. In Proceeding Twelfth Australas. Fluid Mech. Conf., pp. 227–230. Sidney University Press.Google Scholar
Nakamura, Y. & Nakashima, M. 1986 Vortex excitation of prisms with elongated rectangular, H and $\vdash$ cross-sections. J. Fluid Mech. 163, 149–169.CrossRefGoogle Scholar
Nakamura, Y., Ohya, Y. & Tsuruta, H. 1991 Experiments on vortex shedding from flat plates with square leading and trailing edges. J. Fluid Mech. 222, 437–447.CrossRefGoogle Scholar
Naudascher, E. & Rockwell, D. 1994 Flow-Induced Vibrations : An Engineering Guide. Hydraulic Structures Design Manual 7. A.A. Balkema, 1994.Google Scholar
Oberleithner, K., Rukes, L. & Soria, J. 2014 Mean flow stability analysis of oscillating jet experiments. J. Fluid Mech. 757, 1–32.CrossRefGoogle Scholar
Okajima, A. 1982 Strouhal numbers of rectangular cylinders. J. Fluid Mech. 123, 379–398.CrossRefGoogle Scholar
Ozono, S., Ohya, Y., Nakamura, Y. & Nakayama, R. 1992 Stepwise increase in the Strouhal number for flows around flat plates. Intl J. Numer. Meth. Fluids 15 (9), 1025–1036.CrossRefGoogle Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.CrossRefGoogle Scholar
Prasanth, T.K. & Mittal, S. 2008 Vortex-induced vibrations of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 594, 463–491.CrossRefGoogle Scholar
Rai, M.M. & Moin, P. 1991 Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Phys. 96, 15.Google Scholar
Robichaux, J., Balachandar, S. & Vanka, S.P. 1999 Three-dimensional Floquet instability of the wake of square cylinder. Phys. Fluids 11 (3), 560–578.CrossRefGoogle Scholar
Roshko, A. 1954 On the Drag and Shedding Frequency of Two-Dimensional Bluff Bodies. National Advisory Committee for Aeronautics.Google Scholar
Ryan, K., Thompson, M.C. & Hourigan, K. 2005 Three-dimensional transition in the wake of bluff elongated cylinders. J. Fluid Mech. 538, 1–29.CrossRefGoogle Scholar
Saad, Y. 2011 Numerical Methods for Large Eigenvalue Problems. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.CrossRefGoogle Scholar
Stokes, A.N. & Welsh, M.C. 1986 Flow-resonant sound interaction in a duct containing a plate, II: square leading edge. J. Sound Vib. 104 (1), 55–73.CrossRefGoogle Scholar
Tamura, T., Miyagi, T. & Kitagishi, T. 1998 Numerical prediction of unsteady pressures on a square cylinder with various corner shapes. J. Wind Engng Ind. Aerodyn. 74, 531–542.CrossRefGoogle Scholar
Tan, B.T., Thompson, M. & Hourigan, K. 1998 Simulated flow around long rectangular plates under cross flow perturbations. Intl J. Fluid Dyn. 2, 1.Google Scholar
Tan, B.T., Thompson, M.C. & Hourigan, K. 2004 Flow past rectangular cylinders: receptivity to transverse forcing. J. Fluid Mech. 515, 33–62.CrossRefGoogle Scholar
Thompson, M.C. 2012 Effective transition of steady flow over a square leading-edge plate. J. Fluid Mech. 698, 335–357.CrossRefGoogle Scholar
Turton, S.E., Tuckerman, L.S. & Barkley, D. 2015 Prediction of frequencies in thermosolutal convection from mean flows. Phys. Rev. E 91 (4), 043009.CrossRefGoogle ScholarPubMed