Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-08T03:15:00.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation analysis and frequency prediction in shear-driven cavity flow

Published online by Cambridge University Press:  23 July 2019

Y. Bengana Open the ORCID record for Y. Bengana [Opens in a new window] ,
J.-Ch. Loiseau Open the ORCID record for J.-Ch. Loiseau [Opens in a new window] ,
J.-Ch. Robinet Open the ORCID record for J.-Ch. Robinet [Opens in a new window]  and
L. S. Tuckerman Open the ORCID record for L. S. Tuckerman [Opens in a new window]
Y. Bengana
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS; ESPCI Paris, PSL Research University; Sorbonne Université; Univ. Paris Diderot, 75005 Paris, France
J.-Ch. Loiseau
Affiliation:
Laboratoire DynFluid, Arts et Métiers ParisTech, 75013 Paris, France
J.-Ch. Robinet
Affiliation:
Laboratoire DynFluid, Arts et Métiers ParisTech, 75013 Paris, France
L. S. Tuckerman*
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS; ESPCI Paris, PSL Research University; Sorbonne Université; Univ. Paris Diderot, 75005 Paris, France
*
†Email address for correspondence: laurette@pmmh.espci.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

A comprehensive study of the two-dimensional incompressible shear-driven flow in an open square cavity is carried out. Two successive bifurcations lead to two limit cycles with different frequencies and different numbers of structures which propagate along the top of the cavity and circulate in its interior. A branch of quasi-periodic states produced by secondary Hopf bifurcations transfers the stability from one limit cycle to the other. A full analysis of this scenario is obtained by means of nonlinear simulations, linear stability analysis and Floquet analysis. We characterize the temporal behaviour of the limit cycles and quasi-periodic state via Fourier transforms and their spatial behaviour via the Hilbert transform. We address the relevance of linearization about the mean flow. Although here the nonlinear frequencies are not very far from those obtained by linearization about the base flow, the difference is substantially reduced when eigenvalues are obtained instead from linearization about the mean and in addition, the corresponding growth rate is small, a combination of properties called RZIF (real zero imaginary frequency). Moreover growth rates obtained by linearization about the mean of one limit cycle are correlated with relative stability to the other limit cycle. Finally, we show that the frequencies of the successive modes are separated by a constant increment.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Boundary Layers: Free shear layers Vortex Flows: Vortex shedding
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 875 , 25 September 2019 , pp. 725 - 757
Copyright
© 2019 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

Aidun, C. K., Triantafillopoulos, N. G. & Benson, J. D. 1991 Global stability of a lid-driven cavity with throughflow: flow visualization studies. Phys. Fluids A 3 (9), 2081–2091.Google Scholar
Åkervik, E., Brandt, L., Henningson, D. S., Hœpffner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18 (6), 068102.Google 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.Google Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.Google Scholar
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.Google Scholar
Basley, J., Pastur, L. R., Lusseyran, F., Faure, T. M. & Delprat, N. 2011 Experimental investigation of global structures in an incompressible cavity flow using time-resolved PIV. Exp. Fluids 50 (4), 905–918.Google 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.Google 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.Google Scholar
Bengana, Y.2019. Numerical simulations for frequency prediction via mean flows. PhD thesis, PSL Research University. Available at: https://hal.archives-ouvertes.fr/tel-02170483.Google Scholar
Bengana, Y. & Tuckerman, L. S. 2019 Spirals and ribbons in counter-rotating Taylor–Couette flow: frequencies from mean flows and heteroclinic orbits. Phys. Rev. Fluids 4, 044402.Google Scholar
Chien, W-L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355–377.Google Scholar
Cunha, G., Passaggia, P-Y. & Lazareff, M. 2015 Optimization of the selective frequency damping parameters using model reduction. Phys. Fluids 27 (9), 094103.Google Scholar
Douay, C. L., Pastur, L. R. & Lusseyran, F. 2016 Centrifugal instabilities in an experimental open cavity flow. J. Fluid Mech. 788, 670–694.Google Scholar
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008 Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255–274.Google 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 (1), 82–102.Google Scholar
Fani, A., Citro, V., Giannetti, F. & Auteri, F. 2018 Computation of the bluff-body sound generation by a self-consistent mean flow formulation. Phys. Fluids 30 (3), 036102.Google Scholar
Faure, T. M., Adrianos, P., Lusseyran, F. & Pastur, L. 2007 Visualizations of the flow inside an open cavity at medium range Reynolds numbers. Exp. Fluids 42 (2), 169–184.Google Scholar
Faure, T. M., Pastur, L., Lusseyran, F., Fraigneau, Y. & Bisch, D. 2009 Three-dimensional centrifugal instabilities development inside a parallelepipedic open cavity of various shape. Exp. Fluids 47 (3), 395–410.Google Scholar
Fischer, P. F., Lottes, J. W. & Kerkemeir, S. G.2008 Nek5000 Web pages.http://nek5000.mcs.anl.gov.Google Scholar
Fortin, A., Jardak, M., Gervais, J. J. & Pierre, R. 1997 Localization of Hopf bifurcations in fluid flow problems. Intl J. Numer. Meth. Fluids 24 (11), 1185–1210.Google Scholar
Gioria, R. d. S., Jabardo, P. J. S., Carmo, B. S. & Meneghini, J. R. 2009 Floquet stability analysis of the flow around an oscillating cylinder. J. Fluids Struct. 25 (4), 676–686.Google Scholar
Gloerfelt, X. 2009 Cavity noise. VKI Lecture Series vol. 3, pp. 1–169.Google Scholar
Hall, K. C., Thomas, J. P. & Clark, W. S. 2002 Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA J. 40 (5), 879–886.Google Scholar
Hammond, D. A. & Redekopp, L. G. 1997 Global dynamics of symmetric and asymmetric wakes. J. Fluid Mech. 331, 231–260.Google Scholar
Hwang, Y. & Cossu, C. 2010 Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.Google Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.Google Scholar
Jordi, B. E., Cotter, C. J. & Sherwin, S. J. 2014 Encapsulated formulation of the selective frequency damping method. Phys. Fluids 26 (3), 034101.Google Scholar
Jordi, B. E., Cotter, C. J. & Sherwin, S. J. 2015 An adaptive selective frequency damping method. Phys. Fluids 27 (9), 094104.Google Scholar
Kulikovskii, A. G. 1966 On the stability of homogeneous states. (Z. Angew. Math. Mech.) J. Appl. Math. Mech. 30 (1), 180–187.Google Scholar
Kuznetsov, Y. A. 1998 Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, vol. 112. Springer.Google Scholar
Loiseau, J.-Ch. & Brunton, S. L. 2018 Constrained sparse Galerkin regression. J. Fluid Mech. 838, 42–67.Google Scholar
Lopez, J. M., Welfert, B. D., Wu, K. & Yalim, J. 2017 Transition to complex dynamics in the cubic lid-driven cavity. Phys. Rev. Fluids 2 (7), 074401.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1 (5), 521–539.Google Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113 (8), 084501.Google Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2015 A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake. Phys. Fluids 27 (7), 074103.Google Scholar
Marques, F., Lopez, J. M. & Shen, J. 2002 Mode interactions in an enclosed swirling flow: a double Hopf bifurcation between azimuthal wavenumbers 0 and 2. J. Fluid Mech. 455, 263–281.Google Scholar
Maurel, A., Pagneux, V. & Wesfreid, J. E. 1995 Mean-flow correction as non-linear saturation mechanism. Europhys. Lett. 32 (3), 217–222.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.Google Scholar
McMullen, M., Jameson, A. & Alonso, J. 2006 Demonstration of nonlinear frequency domain methods. AIAA J. 44 (7), 1428–1435.Google Scholar
McMullen, M. S. & Jameson, A. 2006 The computational efficiency of non-linear frequency domain methods. J. Comput. Phys. 212 (2), 637–661.Google Scholar
Meliga, P. 2017 Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description. J. Fluid Mech. 826, 503–521.Google Scholar
Meliga, P., Gallaire, F. & Chomaz, J.-M. 2012 A weakly nonlinear mechanism for mode selection in swirling jets. J. Fluid Mech. 699, 216–262.Google Scholar
Mittal, S. 2008 Global linear stability analysis of time-averaged flows. Intl J. Numer. Meth. Fluids 58 (1), 111–118.Google Scholar
Morris, P. J. 1976 The spatial viscous instability of axisymmetric jets. J. Fluid Mech. 77 (3), 511–529.Google Scholar
Picella, F., Loiseau, J.-Ch., Lusseyran, F., Robinet, J.-Ch., Cherubini, S. & Pastur, L. 2018 Successive bifurcations in a fully three-dimensional open cavity flow. J. Fluid Mech. 844, 855–877.Google Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.Google Scholar
Poliashenko, M. & Aidun, C. K. 1995 A direct method for computation of simple bifurcations. J. Comput. Phys. 121 (2), 246–260.Google Scholar
Rockwell, D. 1977 Prediction of oscillation frequencies for unstable flow past cavities. Trans. ASME J. Fluids Engng 99 (2), 294–299.Google Scholar
Rockwell, D. & Knisely, C. 1980 Vortex-edge interaction: mechanisms for generating low frequency components. Phys. Fluids 23 (2), 239–240.Google Scholar
Rockwell, D. & Naudascher, E. 1978 Self-sustaining oscillations of flow past cavities. Trans. ASME J. Fluids Engng 100 (2), 152–165.Google Scholar
Rossiter, J. E.1964 Wind tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Tech. Rep. Ministry of Aviation; Royal Aircraft Establishment; RAE Farnborough.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.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
Shankar, P. N. & Deshpande, M. D. 2000 Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech. 32 (1), 93–136.Google 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.Google 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.Google Scholar
Smith, J. O. 2007 Mathematics of the Discrete Fourier Transform (DFT). W3K Publishing.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4 (1), 1–21.Google Scholar
Symon, S., Rosenberg, K., Dawson, S. T. M. & McKeon, B. J. 2018 Non-normality and classification of amplification mechanisms in stability and resolvent analysis. Phys. Rev. Fluids 3 (5), 053902.Google 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.Google Scholar
Tiesinga, G., Wubs, F. W. & Veldman, A. E. P. 2002 Bifurcation analysis of incompressible flow in a driven cavity by the Newton–Picard method. J. Comput. Appl. Maths 140 (1–2), 751–772.Google Scholar
Tuerke, F., Pastur, L., Fraigneau, Y., Sciamarella, D., Lusseyran, F. & Artana, G. 2017 Nonlinear dynamics and hydrodynamic feedback in two-dimensional double cavity flow. J. Fluid Mech. 813, 1–22.Google Scholar
Tuerke, F., Sciamarella, D., Pastur, L. R., Lusseyran, F. & Artana, G. 2015 Frequency-selection mechanism in incompressible open-cavity flows via reflected instability waves. Phys. Rev. E 91 (1), 013005.Google 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.Google 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.Google Scholar
Yu, Y. H. 1977 Measurements of sound radiation from cavities at subsonic speeds. J. Aircraft. 14 (9), 838–843.Google Scholar
Zielinska, B. J. A., Goujon-Durand, S., Dusek, J. & Wesfreid, J. E. 1997 Strongly nonlinear effect in unstable wakes. Phys. Rev. Lett. 79 (20), 3893.Google Scholar

Bengana et al. supplementary movie 1

Vertical velocity fluctuations for limit cycle $LC_2$

Download Bengana et al. supplementary movie 1(Video)
Video 216.9 KB

Bengana et al. supplementary movie 2

Vertical velocity fluctuations for limit cycle $LC_3$

Download Bengana et al. supplementary movie 2(Video)
Video 243.4 KB