Skip to main content Accessibility help
×
Home

Successive bifurcations in a fully three-dimensional open cavity flow

  • F. Picella (a1), J.-Ch. Loiseau (a1), F. Lusseyran (a2), J.-Ch. Robinet (a1), S. Cherubini (a3) and L. Pastur (a4)...

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.

Copyright

Corresponding author

Email address for correspondence: jean-christophe.robinet@ensam.eu

References

Hide All
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.
Albensoeder, S. & Kuhlmann, H. C. 2006 Nonlinear three-dimensional flow in the lid-driven square cavity. J. Fluid Mech. 569, 465480.
Albensoeder, S., Kuhlmann, H. C. & Rath, H. J. 2001 Three-dimensional centrifugal-flow instabilities in the lid-driven cavity problem. Phys. Fluids 13, 121135.
Antkowiak, A. & Brancher, P. 2007 On vortex rings around vortices: an optimal mechanism. J. Fluid Mech. 578, 295304.
Arnoldi, W. E. 1951 The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Maths 9 (1), 1729.
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), 10571068.
Bagheri, S., Schlatter, P., Schmid, P. J. & Henningson, D. S. 2009b Global stability of a jet in crossflow. J. Fluid Mech. 624, 3344.
Barbagallo, A., Sipp, D. & Schmid, P. J. 2009 Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641, 150.
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57 (9), 14351458.
Barkley, D., Gomes, G. M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167190.
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.
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, 546578.
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, 147155.
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, 271304.
Bres, G. & Colonius, T. 2008 Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599, 309339.
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.
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), 586591.
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, 113140.
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.
Conway, S. L., Shinbrot, T. & Glasser, B. J. 2004 A Taylor vortex analogy in granular flows. Nature 431 (7007), 433437.
Denham, MK & Patrick, MA 1974 Laminar flow over a downstream-facing step in a two-dimensional flow channel. Trans. Inst. Chem. Engrs 52 (4), 361367.
Ding, Y. & Kawahara, M. 1998 Linear stability of incompressible fluid flow in a cavity using finite element method. Intl J. Numer. Meth. Fluids 27, 139157.
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.
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.
Douay, C. L., Pastur, L. R. & Lusseyran, F. 2016b Centrifugal instabilities in an experimental open cavity flow. J. Fluid Mech. 788, 670694.
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, 82102.
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, 395410.
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, 169184.
Feldman, Y. & Gelfgat, A. Yu. 2010 Oscillatory instability of a three-dimensional lid-driven flow in a cube. Phys. Fluids 22 (9), 093602.
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.
Gharib, M. & Roshko, A. 1987 The effect of flow oscillations on cavity drag. J. Fluid Mech. 177, 501530.
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), 223234.
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, 169199.
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, 442445.
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, 94121.
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.
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243251.
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.
Loiseau, J.-Ch.2014 Dynamics and global stability of three-dimensional flows. PhD thesis, Arts & Métiers ParisTech.
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, 175211.
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.
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46 (1), 493517.
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221252.
Meseguer-Garrido, F., de Vicente, J., Valero, E. & Theofilis, V. 2014 On linear instability mechanisms in incompressible open cavity flow. J. Fluid Mech. 752, 219236.
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2001 Instability and equilibration of centrifugally stable stratified Taylor–Couette flow. Phys. Rev. Lett. 86, 5270.
Non, E., Pierre, R. & Gervais, J.-J. 2006 Linear stability of the three-dimensional lid-driven cavity. Phys. Fluids 18 (8), 084103.
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, 438447.
Ramanan, N. & Homsy, G. M. 1994 Linear stability of lid-driven cavity flow. Phys. Fluids 6, 26902701.
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.
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, 315346.
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.
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, 333358.
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.
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.
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.
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.
Theofilis, V., Duck, P. W. & Owen, J. 2004 Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505, 249286.
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, 32293246.
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, 219236.
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, 134165.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Successive bifurcations in a fully three-dimensional open cavity flow

  • F. Picella (a1), J.-Ch. Loiseau (a1), F. Lusseyran (a2), J.-Ch. Robinet (a1), S. Cherubini (a3) and L. Pastur (a4)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed