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Successive bifurcations in a fully three-dimensional open cavity flow

Published online by Cambridge University Press:  12 April 2018

F. Picella Open the ORCID record for F. Picella [Opens in a new window] ,
J.-Ch. Loiseau Open the ORCID record for J.-Ch. Loiseau [Opens in a new window] ,
F. Lusseyran ,
J.-Ch. Robinet ,
S. Cherubini Open the ORCID record for S. Cherubini [Opens in a new window]  and
L. Pastur
F. Picella
Affiliation:
DynFluid – Arts et Métiers ParisTech, 151Bd. de l’Hôpital, 75013 Paris, France
J.-Ch. Loiseau
Affiliation:
DynFluid – Arts et Métiers ParisTech, 151Bd. de l’Hôpital, 75013 Paris, France
F. Lusseyran
Affiliation:
LIMSI, CNRS, Université Paris-Saclay, Bat. 508, Rue John Von Neumann, Campus Universitaire, 91403 Orsay, France
J.-Ch. Robinet*
Affiliation:
DynFluid – Arts et Métiers ParisTech, 151Bd. de l’Hôpital, 75013 Paris, France
S. Cherubini
Affiliation:
DMMM, Politecnico di Bari, Via Re David 200, 70126 Bari, Italy
L. Pastur
Affiliation:
LIMSI, CNRS, Université Paris-Sud, Université Paris-Saclay, Bat. 508, Rue John Von Neumann, 91403 Orsay, France
*
†Email address for correspondence: jean-christophe.robinet@ensam.eu
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Abstract

The transition to unsteadiness of a three-dimensional open cavity flow is investigated using the joint application of direct numerical simulations and fully three-dimensional linear stability analyses, providing a clear understanding of the first two bifurcations occurring in the flow. The first bifurcation is characterized by the emergence of Taylor–Görtler-like vortices resulting from a centrifugal instability of the primary vortex core. Further increasing the Reynolds number eventually triggers self-sustained periodic oscillations of the flow in the vicinity of the spanwise end walls of the cavity. This secondary instability causes the emergence of a new set of Taylor–Görtler vortices experiencing a spanwise drift directed toward the spanwise end walls of the cavity. While a two-dimensional stability analysis would fail to capture this secondary instability due to the neglect of the lateral walls, it is the first time to our knowledge that this drifting of the vortices can be entirely characterized by a three-dimensional linear stability analysis of the flow. Good agreements with experimental observations and measurements strongly support our claim that the initial stages of the transition to turbulence of three-dimensional open cavity flows are solely governed by modal instabilities.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Instability: Instability Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 844 , 10 June 2018 , pp. 855 - 877
Copyright
© 2018 Cambridge University Press 

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References

Åkervik, E., Brandt, L., Henningson, D. S., Hoepffner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the navier-stokes equations by selective frequency damping. Phys. Fluids 18 (6), 068102.10.1063/1.2211705Google Scholar
Albensoeder, S. & Kuhlmann, H. C. 2006 Nonlinear three-dimensional flow in the lid-driven square cavity. J. Fluid Mech. 569, 465–480.10.1017/S0022112006002758Google Scholar
Albensoeder, S., Kuhlmann, H. C. & Rath, H. J. 2001 Three-dimensional centrifugal-flow instabilities in the lid-driven cavity problem. Phys. Fluids 13, 121–135.10.1063/1.1329908Google Scholar
Antkowiak, A. & Brancher, P. 2007 On vortex rings around vortices: an optimal mechanism. J. Fluid Mech. 578, 295–304.10.1017/S0022112007005198Google Scholar
Arnoldi, W. E. 1951 The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Maths 9 (1), 17–29.10.1090/qam/42792Google Scholar
Bagheri, S., Åkervik, E., Brandt, L. & Henningson, D. S. 2009a Matrix-free methods for the stability and control of boundary layers. AIAA J. 47 (5), 1057–1068.10.2514/1.41365Google Scholar
Bagheri, S., Schlatter, P., Schmid, P. J. & Henningson, D. S. 2009b Global stability of a jet in crossflow. J. Fluid Mech. 624, 33–44.10.1017/S0022112009006053Google Scholar
Barbagallo, A., Sipp, D. & Schmid, P. J. 2009 Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641, 1–50.10.1017/S0022112009991418Google Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57 (9), 1435–1458.10.1002/fld.1824Google Scholar
Barkley, D., Gomes, G. M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190.10.1017/S002211200200232XGoogle Scholar
Basley, J., Pastur, L. R., Delprat, N. & Lusseyran, F. 2013 Space-time aspects of a three-dimensional multi-modulated open cavity flow. Phys. Fluids 25, 064105.10.1063/1.4811692Google Scholar
Basley, J., Pastur, L. R., Lusseyran, F., Soria, J. & Delprat, N. 2014 On the modulating effect of three-dimensional instabilities in open cavity flows. J. Fluid Mech. 759, 546–578.10.1017/jfm.2014.576Google Scholar
Beaudoin, J.-F., Cadot, O., Aider, J.-L. & Wesfreid, J. E. 2004 Three-dimensional stationary flow over a backward-facing step. Eur. J. Mech. (B/Fluids) 23, 147–155.Google Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.10.1017/S0022112008001109Google Scholar
Bres, G. & Colonius, T. 2008 Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599, 309–339.10.1017/S0022112007009925Google Scholar
Cherubini, S., Robinet, J.-Ch., De Palma, P. & Alizard, F. 2010 The onset of three-dimensional centrifugal global modes and their nonlinear development in a recirculating flow over a flat surface. Phys. Fluids 22 (11), 114102.10.1063/1.3500677Google Scholar
Chicheportiche, J., Merle, X., Gloerfelt, X. & Robinet, J.-Ch. 2008 Direct numerical simulation and global stability analysis of three-dimensional instabilities in a lid-driven cavity. C. R. Mech. 336 (7), 586–591.10.1016/j.crme.2008.04.007Google Scholar
Citro, V., Giannetti, F., Brandt, L. & P., Luchini 2015a Linear three-dimensional global and asymptotic stability analysis of incompressible open cavity flow. J. Fluid Mech. 768, 113–140.10.1017/jfm.2015.72Google Scholar
Citro, V, Giannetti, F, Luchini, P & Auteri, F 2015b Global stability and sensitivity analysis of boundary-layer flows past a hemispherical roughness element. Phys. Fluids 27 (8), 084110.10.1063/1.4928533Google Scholar
Conway, S. L., Shinbrot, T. & Glasser, B. J. 2004 A Taylor vortex analogy in granular flows. Nature 431 (7007), 433–437.10.1038/nature02901Google Scholar
Denham, MK & Patrick, MA 1974 Laminar flow over a downstream-facing step in a two-dimensional flow channel. Trans. Inst. Chem. Engrs 52 (4), 361–367.Google Scholar
Ding, Y. & Kawahara, M. 1998 Linear stability of incompressible fluid flow in a cavity using finite element method. Intl J. Numer. Meth. Fluids 27, 139–157.10.1002/(SICI)1097-0363(199801)27:1/4<139::AID-FLD655>3.0.CO;2-D3.0.CO;2-D>Google Scholar
Douay, C.2014 Etude expérimentale paramétrique des propriétés et transitions de l’écoulement intra-cavitaire en cavité ouverte et contrôle de l’écoulement. PhD thesis, Université Pierre et Marie Curie.Google Scholar
Douay, C. L., Lusseyran, F. & Pastur, L. R. 2016a The onset of centrifugal instability in an open cavity flow. Fluid Dyn. Res. 48 (6), 061410.10.1088/0169-5983/48/6/061410Google Scholar
Douay, C. L., Pastur, L. R. & Lusseyran, F. 2016b Centrifugal instabilities in an experimental open cavity flow. J. Fluid Mech. 788, 670–694.10.1017/jfm.2015.726Google Scholar
Edwards, W. S., Tuckerman, L. S., Friesner, R. A. & Sorensen, D. C. 1994 Krylov methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 110, 82–102.10.1006/jcph.1994.1007Google Scholar
Faure, M. T., Pastur, L., Lusseyran, F., Fraigneau, Y. & Bish, D. 2009 Three-dimensional centrifugal instabilities development inside a parallelepipedic open cavity of various shape. Exp. Fluids 47, 395–410.10.1007/s00348-009-0671-0Google Scholar
Faure, T., Adrianos, P., Lusseyran, F. & Pastur, L. 2007 Visualizations of the flow inside an open cavity at medium range Reynolds numbers. Exp. Fluids 42, 169–184.10.1007/s00348-006-0188-8Google Scholar
Feldman, Y. & Gelfgat, A. Yu. 2010 Oscillatory instability of a three-dimensional lid-driven flow in a cube. Phys. Fluids 22 (9), 093602.10.1063/1.3487476Google Scholar
Fischer, P., Kruse, J., Mullen, J., Tufo, H., Lottes, J. & Kerkemeier, S.2008 Open source spectral element CFD solver. https://nek5000.mcs.anl.gov/index.php/MainPage.Google Scholar
Gharib, M. & Roshko, A. 1987 The effect of flow oscillations on cavity drag. J. Fluid Mech. 177, 501–530.10.1017/S002211208700106XGoogle Scholar
Gómez, F., Gómez, R. & Theofilis, V. 2014 On three-dimensional global linear instability analysis of flows with standard aerodynamics codes. Aerosp. Sci. Technol. 32 (1), 223–234.10.1016/j.ast.2013.10.006Google Scholar
Guermond, J.-L., Migeon, C., Pineau, G. & Quartapelle, L. 2002 Start-up flows in a three-dimensional rectangular driven cavity of aspect ratio 1 :  1 :  2 at Re = 1000. J. Fluid Mech. 450, 169–199.10.1017/S0022112001006383Google Scholar
Gurnett, D. A., Persoon, A. M., Kurth, W. S., Groene, J. B., Averkamp, T. F., Dougherty, M. K. & Southwood, D. J. 2007 The variable rotation period of the inner region of Saturn’s plasma disk. Science 316, 442–445.10.1126/science.1138562Google Scholar
Ilak, M., Schlatter, P., Bagheri, S. & Henningson, D. S. 2012 Bifurcation and stability analysis of a jet in cross-flow: onset of global instability at a low velocity ratio. J. Fluid Mech. 686, 94–121.10.1017/jfm.2012.10Google Scholar
Kuhlmann, H. C. & Albensoeder, S. 2014 Stability of the steady three-dimensional lid-driven flow in a cube and the supercritical flow dynamics. Phys. Fluids 26 (2), 024104.10.1063/1.4864264Google Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.10.1017/S0022112080000122Google Scholar
Liu, Q., Gómez, F. & Theofilis, V. 2016 Linear instability analysis of low-incompressible flow over a long rectangular finite-span open cavity. J. Fluid Mech. 799, R2.10.1017/jfm.2016.391Google Scholar
Loiseau, J.-Ch.2014 Dynamics and global stability of three-dimensional flows. PhD thesis, Arts & Métiers ParisTech.Google Scholar
Loiseau, J.-Ch., Robinet, J.-Ch., Cherubini, S. & Leriche, E. 2014 Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech. 760, 175–211.10.1017/jfm.2014.589Google Scholar
Loiseau, J.-Ch., Robinet, J.-Ch. & Leriche, E. 2016 Intermittency and transition to chaos in the cubical lid-driven cavity flow. Fluid Dyn. Res. 48 (6), 061421.10.1088/0169-5983/48/6/061421Google Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46 (1), 493–517.10.1146/annurev-fluid-010313-141253Google Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.10.1017/S0022112008003662Google Scholar
Meseguer-Garrido, F., de Vicente, J., Valero, E. & Theofilis, V. 2014 On linear instability mechanisms in incompressible open cavity flow. J. Fluid Mech. 752, 219–236.10.1017/jfm.2014.253Google Scholar
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2001 Instability and equilibration of centrifugally stable stratified Taylor–Couette flow. Phys. Rev. Lett. 86, 5270.10.1103/PhysRevLett.86.5270Google Scholar
Non, E., Pierre, R. & Gervais, J.-J. 2006 Linear stability of the three-dimensional lid-driven cavity. Phys. Fluids 18 (8), 084103.10.1063/1.2335153Google Scholar
Peplinski, A., Schlatter, P. & Henningson, D. S. 2015 Global stability and optimal perturbation for a jet in cross-flow. Eur. J. Mech. (B/Fluids) 49, 438–447.10.1016/j.euromechflu.2014.06.001Google Scholar
Ramanan, N. & Homsy, G. M. 1994 Linear stability of lid-driven cavity flow. Phys. Fluids 6, 2690–2701.10.1063/1.868158Google Scholar
Rossiter, J. E.1964 Wind tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Tech. Rep., Royal Aircraft Establishment, Farnborough. Ministry of Aviation.Google Scholar
Rowley, C. W., Colonius, T. & Basu, A. J. 2002 On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities. J. Fluid Mech. 455, 315–346.10.1017/S0022112001007534Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.10.1017/S0022112010001217Google 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, 333–358.10.1017/S0022112007008907Google Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.10.1115/1.4001478Google Scholar
Sisan, D. R., Mujica, N., Tillotson, W. A., Huang, Y.-M., Dorland, W., Hassam, A. B., Antonsen, T. M. & Lathrop, D. P. 2004 Experimental observation and characterization of the magnetorotational instability. Phys. Rev. Lett. 93, 114502.10.1103/PhysRevLett.93.114502Google Scholar
Sun, Y., Nair, A. G., Taira, K., Cattafesta, L. N., Bres, G. A. & Ukeiley, L. S.2014 Numerical simulations of subsonic and transonic open-cavity flows. AIAA Paper 3092.10.2514/6.2014-3092Google Scholar
Theofilis, V. & Colonius, T.2003 An algorithm for the recovery of 2-D and 3-D BiGlobal instabilities of compressible flow over 2-D open cavities. AIAA Paper 2003-4143.10.2514/6.2003-4143Google Scholar
Theofilis, V., Duck, P. W. & Owen, J. 2004 Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505, 249–286.10.1017/S002211200400850XGoogle Scholar
Theofilis, V., Hein, S. & Dallmann, U. 2000 On the origin of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. R. Lond. A 358, 3229–3246.10.1098/rsta.2000.0706Google Scholar
de Vicente, J., Basley, J., Meseguer-Garrido, F., Soria, J. & Theofilis, V. 2014 Three-dimensional instabilities over a rectangular open cavity: from linear stability analysis to experimentation. J. Fluid Mech. 748, 219–236.10.1017/jfm.2014.126Google Scholar
Yamouni, S., Sipp, D. & Jacquin, L. 2013 Interaction between feedback aeroacoustic and acoustic resonance mechanisms in a cavity flow: a global stability analysis. J. Fluid Mech. 717, 134–165.10.1017/jfm.2012.563Google Scholar