Skip to main content
×
×
Home

Three-dimensional instability of a flow past a sphere: Mach evolution of the regular and Hopf bifurcations

  • A. Sansica (a1) (a2), J.-Ch. Robinet (a2), F. Alizard (a3) and E. Goncalves (a4)
Abstract

A fully three-dimensional linear stability analysis is carried out to investigate the unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper technique is used to compute both equilibrium states and leading eigenmodes. In agreement with previous studies, the numerical results reveal a regular bifurcation under the action of a steady mode and a supercritical Hopf bifurcation that causes the onset of unsteadiness but also illustrate the limitations of previous linear approaches, based on parallel and axisymmetric base flow assumptions, or weakly nonlinear theories. The evolution of the unstable bifurcations is investigated up to low-supersonic speeds. For increasing Mach numbers, the thresholds move towards higher Reynolds numbers. The unsteady fluctuations are weakened and an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds number, the regular bifurcation disappears and the flow directly passes from an unsteady planar-symmetric solution to a stationary axisymmetric stable one when the Mach number is increased. A stability map is drawn by tracking the bifurcation boundaries for different Reynolds and Mach numbers. When supersonic conditions are reached, the flow becomes globally stable and switches to a noise-amplifier system. A continuous Gaussian white noise forcing is applied in front of the shock to examine the convective nature of the flow. A Fourier analysis and a dynamic mode decomposition show a modal response that recalls that of the incompressible unsteady cases. Although transition in the wake does not occur for the chosen Reynolds number and forcing amplitude, this suggests a link between subsonic and supersonic dynamics.

Copyright
Corresponding author
Email address for correspondence: Andrea.Sansica@ensam.eu
References
Hide All
Åkervik, E., Brandt, L., Henningson, D. S., Hoefpffner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18, 068102.
Arnoldi, W. E. 1951 The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Maths 9, 1729.
Bagheri, S., Åkervik, E., Brandt, L. & Henningson, D. S. 2009 Matrix-free methods for the stability and control of boundary layers. AIAA J. 45, 10571068.
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75, 750756.
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57, 14351458.
Beneddine, S., Mettot, C. & Sipp, D. 2015 Global stability analysis of underexpanded screeching jets. Eur. J. Mech. (B/Fluids) 49, 392399.
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 485504.
Bouchet, G., Mebarek, M. & Duŝek, J. 2006 Hydrodynamic forces acting on a rigid fixed sphere in early transitional regimes. Eur. J. Mech. (B/Fluids) 25, 321336.
Brès, G. A. & Colonius, T. 2008 Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599, 309339.
Citro, V., Giannetti, F., Luchini, P. & Auteri, F. 2015 Global stability and sensitivity analysis of boundary-layer flows past a hemispherical roughness element. Phys. Fluids 27 (8), 084110.
Citro, V., Siconolfi, L., Fabre, D., Giannetti, F. & Luchini, P. 2017 Stability and sensitivity analysis of the secondary instability in the sphere wake. AIAA J. 55 (11), 36613668.
Crouch, J. D., Garbaruk, A. & Magidov, D. 2007 Predicting the onset of flow unsteadiness based on global instability. J. Comput. Phys. 224 (2), 924940.
Edwards, W. S., Tuckerman, L. S., Friesner, R. A. & Sorensen, D. 1994 Krylov methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 110, 82101.
Fabre, D., Auguste, F. & Magnaudet, J. 2008 Bifurcations and symmetry breaking in the wake of axisymmetric bodies. Phys. Fluids 20, 14.
Ghidersa, B. & Dusek, J. 2000 Breaking of axisymmetry and onset of unsteadiness in the wake of a sphere. J. Fluid Mech. 423, 3369.
Goncalves, E. & Houdeville 2009 Numerical simulations of a transport-aircraft configuration. J. Comput. Fluid Dyn. 23 (6), 449459.
Guiho, F., Alizard, F. & Robinet, J.-C. 2016 Instabilities in oblique shock wave/laminar boundary-layer interactions. J. Fluid Mech. 789, 135.
Gumowski, K., Miedzik, J., Goujon-Durand, S., Jenffer, P. & Wesfreid, J. E. 2008 Transition to a time-dependent state of fluid flow in the wake of a sphere. Phys. Rev. Lett. 77, 055308(R).
Jameson, A.1991 Time-dependent calculations using multigrid with applications to unsteady flows past airfoils and wings. AIAA Paper, 10th Computational Fluid Dynamics Conference, Honolulu, HI, USA. AIAA.
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 1970.
Jovanovic, M. R., Schmid, P. J. & Nichols, J. W. 2014 Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26, 024103.
Lehoucq, R. B., Sorensen, D. C. & Yang, C.1997 Arpack user’s guide: Solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. Tech. Note.
Loiseau, J.-C., Robinet, J.-C., Cherubini, S. & Leriche, E. 2014 Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech. 760, 175211.
Lomax, H. & Steger, J. L. 1975 Relaxation methods in fluid mechanics. Annu. Rev. Fluid Mech. 7, 6388.
Mack, C. J., Schmid, P. J. & Sesterhenn, J. L. 2008 Global stability of swept flow around a parabolic body: connecting attachment-line and crossflow modes. J. Fluid Mech. 611, 205214.
Magarvey, R. H. & Bishop, R. L. 1961 Transition ranges for three-dimensional wakes. Can. J. Phys. 39, 14181422.
Magnaudet, J., Rivero, M. & Fabre, J. 1995 Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow. J. Fluid Mech. 284, 97135.
Meliga, P., Sipp, D. & Chomaz, J.-M. 2007 Unsteadiness in the wake of the sphere: receptivity and weakly nonlinear global stability analysis. In Proc. 5th Conference on Bluff Body Wakes and Vortex-Induced Vibrations (Bahia, Brazil). BBVIV.
Meliga, P., Sipp, D. & Chomaz, J.-M. 2009 Unsteadiness in the wake of disks and spheres: instability, receptivity and control using direct and adjoint global stability analyses. J. Fluids Struct. 25, 601616.
Meliga, P., Sipp, D. & Chomaz, J.-M. 2010 Effect of compressibility on the global stability of axisymmetric wake flows. J. Fluid Mech. 660, 499526.
Morzyński, M.2009 Global flow stability results for the flow around a sphere. http://stanton.ice.put.poznan.pl/morzynski/2009/08/24/global-flow-stability-results.
Nagata, T., Nonomura, T., Takahashi, S. & Fukuda, K. 2016 Investigation on subsonic to supersonic flow around a sphere at low Reynolds number of between 50 and 300 by direct numerical simulation. Phys. Fluids 28, 056101.
Nagata, T., Nonomura, T., Takahashi, S., Mizuno, Y. & Fukuda, K. 2018 Direct numerical simulation of flow past a sphere at a Reynolds number between 500 and 1000 in compressible flows. In AIAA 2018-0381 (ed. AIAA SciTech Forum AIAA Aerospace Sciences Meeting) Kissimmee, FL.
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323344.
Nichols, J. W., Lele, S. K. & Moin, P. 2009 Global mode decomposition of supersonic jet noise. In CTR Annu. Res. Briefs. Center for Turbulence Research.
Ormières, D. & Provansal, M. 1999 Transition to turbulence in the wake of a sphere. Phys. Rev. Lett. 81 (1), 8083.
Pier, B. 2008 Local and global instabilities in the wake of a sphere. J. Fluid Mech. 603, 3961.
Ray, P. K. & Lele, S. K. 2007 Sound generated by instability wave/shock-cell interaction in supersonic jets. J. Fluid Mech. 587, 173215.
Robinet, J.-C. 2007 Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach. J. Fluid Mech. 579, 85112.
Roe, P. 1981 Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (2), 357372.
Sartor, F., Mettoc, C., Bur, R. & Sipp, D. 2015 Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis. J. Fluid Mech. 781, 550577.
Schmid, P. J. 2010 Dynamic mode decomposition of numerical experimental data. J. Fluid Mech. 656, 528.
Schouveiler, L. & Provansal, M. 2002 Self-sustained oscillations in the wake of a sphere. Phys. Fluids 14, 38463854.
Szaltys, P., Chrust, M., Goujon-Durand, S., Tuckerman, L. S. & Wesfreid, J. E. 2012 Nonlinear evolution of instabilities behind spheres and disks. J. Fluids Struct. 28, 483487.
Taneda, S. 1956 Experimental investigation of the wake behind a sphere at low Reynolds numbers. J. Phys. Soc. Japan 11 (10), 11041108.
Taneda, S. 1978 Visual observations of the flow past a sphere at Reynolds numbers between 104 and 106 . J. Fluid Mech. 85, 187192.
Tezuka, A. & Suzuki, K. 2006 Three-dimensional global linear stability analysis of flow around a spheroid. AIAA J. 44 (8), 16971708.
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319352.
Tomboulides, A. & Orszag, S. A. 2000 Numerical investigation of transitional and weak turbulent flow past a sphere. J. Fluid Mech. 416, 4573.
Tomboulides, A. G., Orszag, S. A. & Karniadakis, G. E. 1993 Direct and large-eddy simulation of axisymmetric wakes. In AIAA 31st Aerospace Sciences Meeting & Exhibit, Reno, NV, USA. AIAA.
Wu, J.-S. & Faeth, G. M. 1993 Sphere wakes in still surroundings at intermediate Reynolds numbers. AIAA J. 31, 14481455.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

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