Hostname: page-component-5c6d5d7d68-thh2z Total loading time: 0 Render date: 2024-08-20T13:41:43.625Z Has data issue: false hasContentIssue false
  • English
  • Français

Influence of optimally amplified streamwise streaks on the Kelvin–Helmholtz instability

Published online by Cambridge University Press:  17 January 2018

Mathieu Marant  and
Carlo Cossu Open the ORCID record for Carlo Cossu [Opens in a new window]
Mathieu Marant
Affiliation:
IMFT, CNRS-INP-UPS, 2 allée du Professeur Camille Soula, 31400 Toulouse, France
Carlo Cossu*
Affiliation:
IMFT, CNRS-INP-UPS, 2 allée du Professeur Camille Soula, 31400 Toulouse, France LHEEA, CNRS – Ecole Centrale de Nantes, 1 rue de la Noé, 44300 Nantes, France
*
†Email address for correspondence: carlo.cossu@imft.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The optimal energy amplifications of streamwise-uniform and spanwise-periodic perturbations of the hyperbolic-tangent mixing layer are computed and found to be very large, with maximum amplifications increasing with the Reynolds number and with the spanwise wavelength of the perturbations. The optimal initial conditions are streamwise vortices and the most amplified structures are streamwise streaks with sinuous symmetry in the cross-stream plane. The leading suboptimal perturbations have opposite (varicose) symmetry. When forced with finite amplitudes these perturbations modify the characteristics of the Kelvin–Helmholtz instability. Maximum temporal growth rates are reduced by optimal sinuous perturbations and are slightly increased by varicose suboptimal ones. In contrast, the onset of absolute instability is delayed by varicose suboptimal perturbations and is slightly promoted by sinuous optimal ones. We show that if, instead of the computed fully nonlinear basic-flow distortions, the stability analysis is based on a shape assumption for the flow distortions, then opposite effects on the flow stability are predicted in most of the considered cases. These strong differences are attributed to the spanwise-uniform component of the nonlinear basic-flow distortion which, we conclude, should be systematically included in sensitivity analyses of the stability of two-dimensional basic flows to three-dimensional basic-flow perturbations. We finally show that the leading-order quadratic sensitivity of the eigenvalues to the amplitude of the streaks is preserved if the effects of the mean flow distortion are included in the sensitivity analysis.

JFM classification

Instability: Absolute/convective instability Boundary Layers: Free shear layers Flow Control: Instability control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 838 , 10 March 2018 , pp. 478 - 500
Check for updates
Copyright
© 2018 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

Arratia, C., Caulfield, C. P. & Chomaz, J.-M. 2013 Transient perturbation growth in time-dependent mixing layers. J. Fluid Mech. 717, 90–133.Google Scholar
Arratia, C. & Chomaz, J.-M. 2013 On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation. J. Fluid Mech. 737, 387–411.Google Scholar
Bell, J. H. & Mehta, R. D. 1993 Effects of imposed spanwise perturbations on plane mixing-layer structure. J. Fluid Mech. 257, 33–63.Google Scholar
Bers, A. 1983 Space-time evolution of plasma instabilities – absolute and convective. In Handbook of Plasma Physics (ed. Rosenbluth, M. N. & Sagdeev, R. Z.), vol. 1, pp. 451–517. North-Holland.Google Scholar
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.Google Scholar
Boujo, E., Fani, A. & Gallaire, F. 2015 Second-order sensitivity of parallel shear flows and optimal spanwise-periodic flow modifications. J. Fluid Mech. 782, 491–514.Google Scholar
Brandt, L., Cossu, C., Chomaz, J.-M., Huerre, P. & Henningson, D. S. 2003 On the convectively unstable nature of optimal streaks in boundary layers. J. Fluid Mech. 485, 221–242.Google Scholar
Bridges, J. & Brown, C. A.2004 Parametric testing of chevrons on single flow hot jets. In Proceedings of the 10th AIAA/CEAS Aeroacoustics Conference, Manchester (UK): AIAA Paper 2004–2824.Google Scholar
Choi, H., Jeon, W. P. & Kim, J. 2008 Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40, 113–139.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: nonnormality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1988 Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60, 25–28.Google Scholar
Cossu, C.2014 On the stabilizing mechanism of 2D absolute and global instabilities by 3D streaks. arXiv:1404.3191.Google Scholar
Cossu, C. & Brandt, L. 2002 Stabilization of Tollmien–Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer. Phys. Fluids 14, L57–L60.Google Scholar
Cossu, C. & Brandt, L. 2004 On Tollmien–Schlichting waves in streaky boundary layers. Eur. J. Mech. (B/Fluids) 23, 815–833.Google Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.CrossRefGoogle Scholar
Delbende, I. & Chomaz, J.-M. 1998 Nonlinear convective/absolute instabilities of parallel two-dimensional wakes. Phys. Fluids 10, 2724–2736.Google Scholar
Delbende, I., Chomaz, J.-M. & Huerre, P. 1998 Absolute and convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.Google Scholar
Del Guercio, G., Cossu, C. & Pujals, G. 2014a Optimal perturbations of non-parallel wakes and their stabilizing effect on the global instability. Phys. Fluids 26, 024110.CrossRefGoogle Scholar
Del Guercio, G., Cossu, C. & Pujals, G. 2014b Optimal streaks in the circular cylinder wake and suppression of the global instability. J. Fluid Mech. 752, 572–588.Google Scholar
Del Guercio, G., Cossu, C. & Pujals, G. 2014c Stabilizing effect of optimally amplified streaks in parallel wakes. J. Fluid Mech. 739, 37–56.CrossRefGoogle Scholar
Drazin, P. G. & Howard, L. N. 1962 The instability to long waves of unbounded parallel inviscid flow. J. Fluid Mech. 14 (02), 257–283.Google Scholar
Fransson, J., Brandt, L., Talamelli, A. & Cossu, C. 2005 Experimental study of the stabilisation of Tollmien–Schlichting waves by finite amplitude streaks. Phys. Fluids 17, 054110.Google Scholar
Fransson, J., Talamelli, A., Brandt, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96, 064501.CrossRefGoogle ScholarPubMed
Garcia, R. V. 1956 Barotropic waves in straight parallel flow with curved velocity profile. Tellus 8 (1), 82–93.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Gudmundsson, K. & Colonius, T. 2006 Linear stability analysis of chevron jet profiles. In Proceedings of FEDSM 2006. ASME Joint U.S.-European Fluids Engineering Summer Meeting July 17–20, 2006, Miami, FL.Google Scholar
Gudmundsson, K. & Colonius, T.2007 Spatial stability analysis of chevron jet profiles. In 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference), AIAA Paper 2007-3599.Google Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.CrossRefGoogle Scholar
Herbert, Th. 1988 Secondary instability of boundary-layers. Annu. Rev. Fluid Mech. 20, 487–526.Google Scholar
Hill, D. C.1992 A theoretical approach for analyzing the restabilization of wakes. AIAA Paper 92-0067.Google Scholar
Huerre, P & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Hwang, Y. & Choi, H. 2006 Control of absolute instability by basic-flow modification in a parallel wake at low Reynolds number. J. Fluid Mech. 560, 465–475.CrossRefGoogle Scholar
Hwang, Y., Kim, J. & Choi, H. 2013 Stabilization of absolute instability in spanwise wavy two-dimensional wakes. J. Fluid Mech. 727, 346–378.Google Scholar
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173–195.Google Scholar
Kachanov, Y. S. & Tararykin, O. I. 1987 Experimental investigation of a relaxating boundary layer. Izv. SO AN SSSR, Ser. Tech. Nauk 18, 9–19.Google Scholar
Marant, M., Cossu, C. & Pujals, G. 2017 Optimal streaks in the wake of a blunt-based axisymmetric bluff body and their influence on vortex shedding. C. R. Méc. 345, 378–385.Google Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech. 19, 543–556.CrossRefGoogle Scholar
Monkewitz, P. A. 1988 The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers. Phys. Fluids 31, 999–1006.Google Scholar
OpenCFD 2007 OpenFOAM – The Open Source CFD Toolbox – User’s Guide, 1st edn. OpenCFD Ltd., United Kingdom.Google Scholar
Park, J., Hwang, Y. & Cossu, C. 2011 On the stability of large-scale streaks in turbulent Couette and Poiseulle flows. C. R. Méc. 339, 1–5.Google Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21, 015109.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J., Baggett, J. S. & Henningson, D. S. 1998 On the stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Tammisola, O. 2017 Optimal wavy surface to suppress vortex shedding using second-order sensitivity to shape changes. Eur. J. Mech. B 62, 139–148.Google Scholar
Tammisola, O., Giannetti, F., Citro, V. & Juniper, M. P. 2014 Second-order perturbation of global modes and implications for spanwise wavy actuation. J. Fluid Mech. 755, 314–335.Google Scholar
Tanner, M. 1972 A method of reducing the base drag of wings with blunt trailing edges. Aeronaut. Q. 23, 15–23.Google Scholar
Tombazis, N. & Bearman, P. W. 1997 A study of three-dimensional aspects of vortex shedding from a bluff body with a mild geometric disturbance. J. Fluid Mech. 330, 85–112.Google Scholar
Willis, A. P., Hwang, Y. & Cossu, C. 2010 Optimally amplified large-scale streaks and drag reduction in the turbulent pipe flow. Phys. Rev. E 82, 036321.Google Scholar
Zaman, K., Bridges, J. & Huff, D. 2011 Evolution from ‘tabs’ to ‘chevron technology’ – a review. Intl J. Aeroacoust. 10 (5–6), 685–710.Google Scholar
Zaman, K. B. M. Q., Reeder, M. F. & Samimy, M. 1994 Control of an axisymmetric jet using vortex generators. Phys. Fluids 6, 778–793.Google Scholar
Zdravkovich, M. M. 1981 Review and classification of various aerodynamic and hydrodynamic means for suppressing vortex shedding. J. Wind Engng Ind. Aerodyn. 7, 145–189.Google Scholar