Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T09:24:47.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Space–time dynamics of optimal wavepackets for streaks in a channel entrance flow

Published online by Cambridge University Press:  06 April 2018

F. Alizard Open the ORCID record for F. Alizard [Opens in a new window] ,
A. Cadiou ,
L. Le Penven ,
B. Di Pierro  and
M. Buffat
F. Alizard*
Affiliation:
LMFA, CNRS, Ecole Centrale de Lyon, Université Lyon 1, INSA Lyon, 43 Boulevard du 11 Novembre 1918, 69100 Villeurbanne, France DynFluid-CNAM, 151 Boulevard de l’Hôpital, 75013 Paris, France
A. Cadiou
Affiliation:
LMFA, CNRS, Ecole Centrale de Lyon, Université Lyon 1, INSA Lyon, 43 Boulevard du 11 Novembre 1918, 69100 Villeurbanne, France
L. Le Penven
Affiliation:
LMFA, CNRS, Ecole Centrale de Lyon, Université Lyon 1, INSA Lyon, 43 Boulevard du 11 Novembre 1918, 69100 Villeurbanne, France
B. Di Pierro
Affiliation:
LMFA, CNRS, Ecole Centrale de Lyon, Université Lyon 1, INSA Lyon, 43 Boulevard du 11 Novembre 1918, 69100 Villeurbanne, France
M. Buffat
Affiliation:
LMFA, CNRS, Ecole Centrale de Lyon, Université Lyon 1, INSA Lyon, 43 Boulevard du 11 Novembre 1918, 69100 Villeurbanne, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The laminar–turbulent transition of a plane channel entrance flow is revisited using global linear optimization analyses and direct numerical simulations. The investigated case corresponds to uniform upstream velocity conditions and a moderate value of Reynolds number so that the two-dimensional developing flow is linearly stable under the parallel flow assumption. However, the boundary layers in the entry zone are capable of supporting the development of streaks, which may experience secondary instability and evolve to turbulence. In this study, global optimal linear perturbations are computed and studied in the nonlinear regime for different values of streak amplitude and optimization time. These optimal perturbations take the form of wavepackets having either varicose or sinuous symmetry. It is shown that, for short optimization times, varicose wavepackets grow through a combination of Orr and lift-up effects, whereas for longer target times, both sinuous and varicose wavepackets exhibit an instability mechanism driven by the presence of inflection points in the streaky flow. In addition, while the optimal varicose modes obtained for short optimization times are localized near the inlet, where the base flow is strongly three-dimensional, when the target time is increased, the sinuous and varicose optimal modes are displaced farther downstream, in the nearly parallel streaky flow. Finally, the optimal wavepackets are found to lead to turbulence for sufficiently high initial amplitudes. It is noticed that the resulting turbulent flows have the same wall-shear stress, whether the wavepackets have been obtained for short or for long time optimization.

JFM classification

Boundary Layers: Boundary Layers Instability: Instability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 844 , 10 June 2018 , pp. 669 - 706
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

Alamo, J. C. Del & Jimenez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 5, 41–44.10.1063/1.1570830Google Scholar
Alizard, F. 2015 Linear stability of optimal streaks in the log-layer of turbulent channel flows. Phys. Fluids 27, 105103.10.1063/1.4932178Google Scholar
Alizard, F., Robinet, J.-C. & Filliard, G. 2015 Sensitivity analysis of optimal transient growth for turbulent boundary layers. Eur. J. Mech. (B/Fluids) 49, 373–386.10.1016/j.euromechflu.2014.04.013Google Scholar
Alizard, F., Robinet, J.-C. & Gloerfelt, X. 2012 A domain decomposition matrix-free method for global linear stability. Comput. Fluids 66, 63–84.10.1016/j.compfluid.2012.05.017Google Scholar
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
Asai, M. & Floryan, J. M. 2004 Certain aspects of channel entrance flow. Phys. Fluids 16, 1160–1163.10.1063/1.1649340Google Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57, 1435–1458.10.1002/fld.1824Google Scholar
Bertolotti, F. P., Herbert, T. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474.10.1017/S0022112092002453Google Scholar
Biau, D. 2008 Linear stability of channel entrance flow. Eur. J. Mech. (B/Fluids) 27, 579–590.10.1016/j.euromechflu.2007.11.004Google Scholar
Brandt, L. 2007 Numerical studies of the instability and breakdown of a boundary-layer low-speed streak. Eur. J. Mech. (B/Fluids) 26, 64–82.10.1016/j.euromechflu.2006.04.008Google 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, 221–242.10.1017/S0022112003004427Google Scholar
Brandt, L. & Henningson, D. S. 2002 Transition of streamwise streaks in zero-pressure-gradient boundary layers. J. Fluid Mech. 472, 229–261.10.1017/S0022112002002331Google Scholar
Brandt, L. & de Lange, H. C. 2008 Streak interactions and breakdown in boundary layer flows. Phys. Fluids 20, 024107.10.1063/1.2838594Google Scholar
Brandt, L. & Schlatter, P. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198.10.1017/S0022112004000941Google Scholar
Buffat, M., Le Penven, L. & Cadiou, A. 2011 An efficient spectral method based on an orthogonal decomposition of the velocity for transition analysis in wall bounded flow. Comput. Fluids 42, 62–72.10.1016/j.compfluid.2010.11.003Google Scholar
Buffat, M., Le Penven, L., Cadiou, A. & Montagnier, J. 2014 DNS of bypass transition in entrance channel flow induced by boundary layer interaction. Eur. J. Mech. (B/Fluids) 43, 1–13.10.1016/j.euromechflu.2013.06.009Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A4, 1637–1650.10.1063/1.858386Google Scholar
Chen, T. S. & Sparrow, E. M. 1967 Stability of the developing laminar flow in a parallel-plate channel. J. Fluid Mech. 30, 209–224.10.1017/S0022112067001399Google Scholar
Cherubini, S., De Tullio, M. D., De Palma, P. & Pascazio, G. 2013 Transient growth in the flow past a three-dimensional smooth roughness element. J. Fluid Mech. 724, 642–670.10.1017/jfm.2013.177Google Scholar
Cherubini, S., Palma, P. D., Robinet, J. C. & Bottaro, A. 2010 Rapid path to transition via nonlinear optimal perturbations in a boundary-layer flow. Phys. Rev. E 82, 066302.Google Scholar
Cossu, C. & Brandt, L. 2004 On Tollmien–Schlichting-like waves in streaky boundary layers. Eur. J. Mech. (B/Fluids) 23, 815–833.10.1016/j.euromechflu.2004.05.001Google Scholar
Cossu, C., Brandt, L., Bagheri, S. & Henningson, D. S. 2011 Secondary threshold amplitudes for sinuous streak breakdown. Phys. Fluids 23, 074103.10.1063/1.3614480Google Scholar
Cossu, C., Chevalier, M. & Henningson, D. S. 2007 Optimal secondary energy growth in a plane channel flow. Phys. Fluids 19, 058107.10.1063/1.2736678Google Scholar
Darbandi, M. & Schneider, G. E. 1999 Numerical study of the flow behavior in the uniform velocity entry flow problem. Numer. Heat Transfer A 34, 479–494.10.1080/10407789808913999Google Scholar
Dixit, S. A. & Ramesh, O. N. 2010 Large-scale structures in turbulent and reverse-transitional sink flow boundary layers. J. Fluid Mech. 649, 233–273.10.1017/S0022112009993430Google Scholar
Draad, A. A., Kuiken, G. D. C. & Nieuwstadt, F. T. M. 1998 Laminar–turbulent transition in pipe flow for Newtonian and non-Newtonian fluids. J. Fluid Mech. 377, 267–312.10.1017/S0022112098003139Google Scholar
Duguet, Y., Schlatter, P., Henningson, D. S. & Eckhardt, B. 2012 Self-sustained localized structures in a boundary-layer flow. Phys. Rev. Lett. 108, 044501.10.1103/PhysRevLett.108.044501Google Scholar
Durbin, P. & Wu, X. 2007 Transition beneath vortical disturbances. Annu. Rev. Fluid Mech. 39, 107–128.10.1146/annurev.fluid.39.050905.110135Google Scholar
Durst, F., Ray, S., Ünsal, B. & Bayoumi, O. A. 2005 The development lengths of laminar pipe and channel flows. Trans. ASME J. Fluids Engng 127, 1154–1160.10.1115/1.2063088Google Scholar
Eitel-Amor, G., Flores, O. & Schlatter, P. 2014 Hairpin vortices in turbulent boundary layers. J. Phys. 506, 012008.Google Scholar
Garg, V. K. & Gupta, S. C. 1981 Nonparallel effects on the stability of developing flow in a channel. Phys. Fluids 24, 1752.10.1063/1.863596Google Scholar
Hack, M. J. P. & Zaki, T. A. 2014 Streak instabilities in boundary layers beneath free-stream turbulence. J. Fluid Mech. 741, 280–315.10.1017/jfm.2013.677Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.10.1017/S0022112095000978Google Scholar
Hifdi, A., Touhami, M. O. & Naciri, J. K. 2004 Channel entrance flow and its linear stability. J. Stat. Mech. 2004 (06), P06003.Google Scholar
Hoepffner, J., Brandt, L. & Henningson, D. S. 2005 Transient growth on boundary layer streaks. J. Fluid Mech. 537, 91–100.10.1017/S0022112005005203Google Scholar
John, M. O., Obrist, D. & Kleiser, L. 2016 Secondary instability and subcritical transition of the leading-edge boundary layer. J. Fluid Mech. 792, 682–711.10.1017/jfm.2016.117Google Scholar
Kapila, A. K., Ludford, G. S. S. & Olunloyo, V. O. S. 1972 Entry flow in a channel. Part 3. Inlet in a uniform stream. J. Fluid Mech. 57, 769–784.10.1017/S0022112073002004Google Scholar
Landhal, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flow. J. Fluid Mech. 98, 243–251.10.1017/S0022112080000122Google Scholar
Mandal, A. C., Venkatakrishnan, L. & Dey, J. 2010 A study of boundary layer transition induced by freestream turbulence. J. Fluid Mech. 660, 114–146.10.1017/S0022112010002600Google Scholar
Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.10.1017/S0022112000002810Google Scholar
Monokrousos, A., Akervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181–214.10.1017/S0022112009993703Google Scholar
Moser, R. D., Kim, J. & Mansour, N. N. 1998 Direct numerical simulation of turbulent channel flow up to Re 𝜏 = 590. Phys. Fluids 11, 943–945.10.1063/1.869966Google Scholar
Moser, R. D., Moin, P. & Leonard, A. 1983 A spectral numerical method for the Navier–Stokes equations with applications to Taylor–Couette flow. J. Comput. Phys. 52, 524–544.10.1016/0021-9991(83)90006-2Google Scholar
Nordstrom, J., Nordin, N. & Henningson, D. 1999 The fringe region technique and the Fourier method used in the direct numerical simulation of spatially evolving viscous flows. SIAM J. Sci. Comput. 20, 1365–1393.10.1137/S1064827596310251Google Scholar
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: a perfect liquid. Part II: a viscous liquid. Proc. R. Irish Acad. A 27, 9–37.Google Scholar
Passaggia, P. Y., Leweke, T. & Ehrenstein, U. 2012 Transverse instability and low-frequency flapping in incompressible separated boundary layer flows: an experimental study. J. Fluid Mech. 703, 363–373.10.1017/jfm.2012.225Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105, 154502.10.1103/PhysRevLett.105.154502Google Scholar
Sadri, R. M. & Floryan, J. M. 2002 Entry in a channel. Comput. Fluids 31, 133–157.10.1016/S0045-7930(01)00023-8Google Scholar
Sayadi, T., Hamman, C. W. & Moin, P. 2013 Direct numerical simulation of complete h-type and k-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480–509.10.1017/jfm.2013.142Google Scholar
Schlatter, P. & Orlu, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126.10.1017/S0022112010003113Google Scholar
Schmid, P. J. & Henningson, D. S. 1992 A new mechanism for rapid transition involving a pair of oblique waves. Phys. Fluids 4, 1986–1989.10.1063/1.858367Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, Applied Mathematical Sciences, vol. 142. Springer.10.1007/978-1-4613-0185-1Google Scholar
Schmidt, O. T., Hosseini, S. M., Rist, U., Hanifi, A. & Henningson, D. S. 2015 Optimal wavepackets in streamwise corner flow. J. Fluid Mech. 766, 405–435.10.1017/jfm.2015.18Google Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.10.1017/S002211200100667XGoogle Scholar
Spalart, P. R., Moser, R. D. & Rogers, M. M. 1990 Spectral methods for the Navier–Stokes equations with one infinite and two periodic conditions. J. Comput. Phys. 96, 297–324.10.1016/0021-9991(91)90238-GGoogle Scholar
Sparrow, E. M. & Anderson, C. E. 1977 Effect of upstream flow processes on hydrodynamic development in a duct. Trans. ASME J. Fluids Engng 99, 556–560.10.1115/1.3448846Google Scholar
Van Dyke, M. 1970 Entry flow in a channel. J. Fluid Mech. 44, 813–823.10.1017/S0022112070002173Google Scholar
Vaughan, N. J. & Zaki, T. A. 2011 Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks. J. Fluid Mech. 681, 116–153.10.1017/jfm.2011.177Google Scholar
Wilson, S. D. R. 1971 Entry flow in a channel. Part 2. J. Fluid Mech. 46, 787–799.10.1017/S0022112071000855Google Scholar
Zaki, T. A. & Durbin, P. A. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85–111.10.1017/S0022112005003800Google Scholar
Zhou, J., Adrian, R. J., Balachandar, S. & Kendall, T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396.10.1017/S002211209900467XGoogle Scholar