Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-17T10:42:38.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Inviscid instability of a unidirectional flow sheared in two transverse directions

Published online by Cambridge University Press:  15 July 2019

Kengo Deguchi Open the ORCID record for Kengo Deguchi [Opens in a new window]
Get access Rights & Permissions [Opens in a new window]

Abstract

Linear inviscid stability of general unidirectional flows sheared in one transverse direction has long been investigated by numerous researchers using the Rayleigh equation. However, unlike the simple shear flow considered in this equation, most physically relevant unidirectional flows vary in two transverse directions. Here the inviscid instability of such flows is studied by the large-Reynolds-number limit asymptotic analysis. We derive an a priori necessary condition for the existence of a limiting neutral mode, and develop a new numerical method to accurately capture singular neutral modes.

JFM classification

Aerodynamics: High-speed flow Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 979 - 994
Check for updates
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

Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. S. 2001 On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 29–60.10.1017/S0022112000002421Google Scholar
Balmforth, N. J. & Morrison, P. J. 1995 Normal modes and continuous spectra. Ann. N.Y. Acad. Sci. 773, 80–94.10.1111/j.1749-6632.1995.tb12163.xGoogle Scholar
Benney, D. 1984 The evolution of disturbances in shear flows at high Reynolds numbers. Stud. Appl. Maths 70, 1–19.10.1002/sapm19847011Google Scholar
Blackaby, N. D. & Hall, P. 1995 The nonlinear evolution of the inviscid secondary instability of streamwise vortex structures. Phil. Trans. R Soc. Lond. A 352, 483–502.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Brown, P. G., Brown, S. N., Smith, F. T. & Timoshin, S. N. 1993 On the starting process of strongly nonlinear vortex/Rayleigh interactions. Mathematika 40, 7–29.10.1112/S0025579300013693Google Scholar
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech. 234, 511–527.10.1017/S0022112092000892Google Scholar
Deguchi, K. 2017 Scaling of small vortices in stably stratified shear flows. J. Fluid Mech. 821, 582–594.10.1017/jfm.2017.213Google Scholar
Deguchi, K. 2019 High-speed shear driven dynamos. Part 1. Asymptotic analysis. J. Fluid Mech. 868, 176–211.10.1017/jfm.2019.178Google Scholar
Deguchi, K. & Hall, P. 2016 On the instability of vortex–wave interaction states. J. Fluid Mech. 802, 634–666.10.1017/jfm.2016.480Google Scholar
Deguchi, K. & Walton, A. G. 2018 Bifurcation of nonlinear Tollmien–Schlichting waves in a high-speed channel flow. J. Fluid Mech. 843, 53–97.10.1017/jfm.2018.137Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Goldstein, M. E. 1976 Aeroacoustics. McGraw-Hill.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51, 139–161.10.1002/sapm1972512139Google Scholar
Hall, P. 2018 Vortex–wave interaction arrays: a sustaining mechanism for the log layer? J. Fluid Mech. 850, 46–82.10.1017/jfm.2018.425Google Scholar
Hall, P. & Horseman, N. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.10.1017/S0022112091003725Google Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.10.1017/S0022112010002892Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.10.1017/S0022112091000289Google Scholar
Henningson, D. S.1987 Stability of parallel inviscid shear flow with mean spanwise variation. Tech. Rep. FFA-TN 1987–57. Aeronautical Research Institute of Sweden.Google Scholar
Hirota, M., Morrison, P. J. & Hattori, Y. 2014 Variational necessary and sufficient stability conditions for inviscid shear flow. Proc. R. Soc. Lond. A 470, 20140322.Google Scholar
Hocking, L. M. 1968 Long wavelength disturbances to non-planar parallel flow. J. Fluid Mech. 31 (4), 625–634.10.1017/S0022112068000364Google Scholar
Karniadakis, G. E. & Sherwin, S. J. 2005 Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press.10.1093/acprof:oso/9780198528692.001.0001Google Scholar
Li, Y. 2011 Stability criteria of 3D inviscid shears. Q. Appl. Maths 69, 379–387.10.1090/S0033-569X-2011-01213-7Google Scholar
Li, F. & Malik, M. R. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100.10.1017/S0022112095003016Google Scholar
Lin, C. C. 1945 On the stability of two-dimensional parallel flows. Part III – stability in a viscous fluid. Q. Appl. Maths 3, 277–301.10.1090/qam/14894Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.Google Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.10.1017/S0022112090000829Google Scholar
Olvera, D. & Kerswell, R. R. 2017 Exact coherent structures in stably stratified plane Couette flow. J. Fluid Mech. 826, 583–614.10.1017/jfm.2017.447Google Scholar
Slattery, J. C. 1999 Advanced Transport Phenomena. Cambridge University Press.10.1017/CBO9780511800238Google Scholar
Tollmien, W. 1935 Ein allgemeines Kriterium der Instabilität laminarer Geschwindigkeitsverteilungen. Nachr. Ges. Wiss. Göttingen 1, 79–114.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.10.1063/1.869185Google Scholar
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states: transition and control. Phys. Rev. Lett. 98, 204501.10.1103/PhysRevLett.98.204501Google Scholar
Wu, X. & Cowley, S. J. 1995 On the nonlinear evolution of instability modes in unsteady shear layers: the Stokes layer as a paradigm. Q. J. Mech. Appl. Maths 48 (2), 159–188.10.1093/qjmam/48.2.159Google Scholar
Wundrow, D. W. & Goldstein, M. E.1994 Nonlinear instability of a uni-directional transversely sheared mean flow. NASA TM 106779.Google Scholar
Xu, D., Zhang, Y. & Wu, X. 2017 Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances. J. Fluid Mech. 829, 681–730.10.1017/jfm.2017.572Google Scholar