Skip to main content Accessibility help

Bifurcation and stability analysis of a jet in cross-flow: onset of global instability at a low velocity ratio

  • Miloš Ilak (a1), Philipp Schlatter (a1), Shervin Bagheri (a1) and Dan S. Henningson (a1)


We study direct numerical simulations (DNS) of a jet in cross-flow at low values of the jet-to-cross-flow velocity ratio . We observe that, as the ratio increases, the flow evolves from simple periodic vortex shedding (a limit cycle) to more complicated quasi-periodic behaviour, before finally becoming turbulent, as seen in the simulation of Bagheri et al. (J. Fluid. Mech., vol. 624, 2009b, pp. 33–44). The value of at which the first bifurcation occurs for our numerical set-up is found, and shedding of hairpin vortices characteristic of a shear layer instability is observed. We focus on this first bifurcation, and find that a global linear stability analysis predicts well the frequency and initial growth rate of the nonlinear DNS at the critical value of and that good qualitative predictions about the dynamics can still be made at slightly higher values of where multiple unstable eigenmodes are present. In addition, we compute the adjoint global eigenmodes, and find that the overlap of the direct and the adjoint eigenmode, also known as a ‘wavemaker’, provides evidence that the source of the first instability lies in the shear layer just downstream of the jet.


Corresponding author

Email address for correspondence:


Hide All

Present address: United Technologies Research Center, 411 Silver Lane, MS 129-85, East Hartford, CT 06108, USA.



Hide All
1. Acarlar, M. & Smith, C. 1987a A study of hairpin vortices in a laminar boundary layer. Part 1. Hairpin vortices generated by a hemisphere protuberance. J. Fluid Mech. 175, 141.
2. Acarlar, M. & Smith, C. 1987b A study of hairpin vortices in a laminar boundary layer. Part 2. Hairpin vortices generated by fluid injection. J. Fluid Mech. 175, 4383.
3. 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, 068102.
4. Alves, L., Kelly, R. E. & Karagozian, A. R. 2007 Local stability analysis of an inviscid transverse jet. J. Fluid Mech. 581, 401418.
5. Alves, L., Kelly, R. E. & Karagozian, A. R. 2008 Transverse-jet shear-layer instabilities. Part 2. Linear analysis for large jet-to-crossflow velocity ratio. J. Fluid Mech. 602, 383401.
6. Bagheri, S. 2010 Analysis and control of transitional shear flows using global modes. PhD thesis, Royal Institute of Technology.
7. Bagheri, S., Brandt, L. & Henningson, D. S. 2009a Input–output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620, 263298.
8. Bagheri, S., Schlatter, P., Schmid, P. J. & Henningson, D. S. 2009b Global stability of a jet in cross-flow. J. Fluid Mech. 624, 3344.
9. Bertolotti, F., Herbert, T. & Spalart, P. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441474.
10. Blanchard, J., Brunet, Y. & Merlen, A. 1999 Influence of a counter rotating vortex pair on the stability of a jet in a cross flow: an experimental study by flow visualizations. Exp. Fluids 26, 6374.
11. Chevalier, M., Schlatter, P., Lundbladh, A. & Henningson, D. S. 2007 SIMSON: a pseudo-spectral solver for incompressible boundary layer flows. Tech Rep. TRITA-MEK 2007:07. KTH Mechanics.
12. Chomaz, J. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.
13. Coelho, S. L. V. & Hunt, J. C. R. 1989 The dynamics of the near field of strong jets in crossflows. J. Fluid Mech. 200, 95120.
14. Cortelezzi, L. & Karagozian, A. R. 2001 On the formation of the counter-rotating vortex pair in transverse jets. J. Fluid Mech. 446, 347373.
15. Davitian, J., Getsinger, D., Hendrickson, C. & Karagozian, A. R. 2010 Transition to global instability in transverse-jet shear layers. J. Fluid Mech. 661, 294315.
16. Fric, T. F. & Roshko, A. 1994 Vortical structure in the wake of a transverse jet. J. Fluid Mech. 279, 147.
17. Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167197.
18. Grout, R., Gruber, A., Yoo, C. & Chen, J. 2010 Direct numerical simulation of flame stabilization downstream of a transverse fuel jet in cross-flow. Proceedings of the Combustion Institute 33, 16291637.
19. Hammond, D. & Redekopp, L. G. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech. B/Fluids 17 (2), 145164.
20. Hill, D. C. 1992 A theoretical approach for analysing the restabilization of wakes. AIAA Paper 92-0067.
21. Hill, D. C. 1995 Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183204.
22. Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
23. Ilak, M., Schlatter, P., Bagheri, S., Chevalier, M. & Henningson, D. S. 2011 Stability of a jet in crossflow. Phys. Fluids 23, 091113.
24. Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.
25. Jovanović, M. B. 2006 Film cooling through imperfect holes. PhD thesis, Eindhoven University of Technology.
26. Karagozian, A. 2010 Transverse jets and their control. Prog. Energy Combust. Sci. 30, 123.
27. Kelso, R. M., Lim, T. T. & Perry, A. E. 1996 An experimental study of round jets in cross-flow. J. Fluid Mech. 306, 111144.
28. Kelso, R. M. & Smits, A. J. 1995 Horseshoe vortex systems resulting from the interaction between a laminar boundary layer and a transverse jet. Phys. Fluids 7 (1).
29. Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users’ Guide. Society for Industrial and Applied Mathematics.
30. Li, Q., Schlatter, P. & Henningson, D. S. 2008 Spectral simulations of wall-bounded flows on massively parallel computers. Tech Rep. KTH Mechanics.
31. Lim, T. T., New, T. H. & Luo, S. C. 2001 On the development of large-scale structures of a jet normal to a cross flow. Phys. Fluids 13 (3), 770775.
32. Luchini, P., Giannetti, F. & Pralits, J. O. 2008 Structural sensitivity of linear and nonlinear global modes. AIAA Paper 2008-4227.
33. Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221252.
34. M’Closkey, R. T., King, J. M., Cortelezzi, L. & Karagozian, A. 2002 The actively controlled jet in crossflow. J. Fluid Mech. 452.
35. Megerian, S., Davitian, J., Alves, L. S. d. B. & Karagozian, A. R. 2007 Transverse-jet shear-layer instabilities. Part 1. Experimental studies. J. Fluid Mech. 593, 93129.
36. Muldoon, F. & Acharya, S. 2010 Direct numerical simulation of pulsed jets-in-crossflow. Comput. Fluids 39, 17451773.
37. Muppidi, S. & Mahesh, K. 2005 Study of trajectories of jets in crossflow using direct numerical simulations. J. Fluid Mech. 530, 81100.
38. Muppidi, S. & Mahesh, K. 2006 Two-dimensional model problem to explain counter-rotating vortex pair formation in a transverse jet. Phys. Fluids 18, 085103.
39. Muppidi, S. & Mahesh, K. 2007 Direct numerical simulation of round turbulent jets in crossflow. J. Fluid Mech. 574, 5984.
40. Noack, B., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335363.
41. Perry, A. E. & Lim, T. T. 1978 Coherent structures in coflowing jets and wakes. J. Fluid Mech. 88, 451463.
42. Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407417.
43. Pier, B. 2008 Local and global instabilities in the wake of a sphere. J. Fluid Mech. 603, 3961.
44. Pralits, J. O., Brandt, L. & Giannetti, F. 2010 Instability and sensitivity of the flow around a rotating circular cylinder. J. Fluid Mech. 650, 513536.
45. Rodriguez, D. & Theofilis, V. 2010 Structural changes of laminar separation bubbles induced by global linear instability. J. Fluid Mech. 655, 280305.
46. Salewski, M., Stankovic, D. & Fuchs, L. 2008 Mixing in circular and non-circular jets in crossflow. Flow Turbul. Combust. 80, 255283.
47. Schlatter, P., Bagheri, S. & Henningson, D. S. 2011 Self-sustained global oscillations of a jet in crossflow. Theor. Comput. Fluid Dyn. 25, 129146.
48. Smith, S. H. & Mungal, M. G. 1998 Mixing, structure and scaling of the jet in crossflow. J. Fluid Mech. 357, 83122.
49. Tammisola, O., Lundell, F., Schlatter, P., Wehrfritz, A. & Söderberg, L. D. 2011 Global linear and nonlinear stability of viscous confined plane wakes with co-flow. J. Fluid Mech. 675, 397434.
50. Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aeronaut. Sci. 39, 249315.
51. Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319352.
52. Theofilis, V., Hein, S. & Dallmann, U. 2000 On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. R. Soc. Lond. A 358, 32293246.
53. Yuan, L. L., Street, R. L. & Ferziger, J. H. 1999 Large-eddy simulations of a round jet in crossflow. J. Fluid Mech. 379, 71104.
54. Ziefle, J. 2007 Large-eddy simulation of complex massively-separated turbulent flows. PhD thesis, ETH Zurich, Diss. no. 17846.
55. Ziefle, J. & Kleiser, L. 2009 Large-eddy simulation of a round jet in crossflow. AIAA J. 47 (5), 11581172.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Bifurcation and stability analysis of a jet in cross-flow: onset of global instability at a low velocity ratio

  • Miloš Ilak (a1), Philipp Schlatter (a1), Shervin Bagheri (a1) and Dan S. Henningson (a1)


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