Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T17:37:19.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Edge state modulation by mean viscosity gradients

Published online by Cambridge University Press:  16 January 2018

Enrico Rinaldi Open the ORCID record for Enrico Rinaldi [Opens in a new window] ,
Philipp Schlatter Open the ORCID record for Philipp Schlatter [Opens in a new window]  and
Shervin Bagheri
Enrico Rinaldi*
Affiliation:
Linné FLOW Centre, KTH Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Philipp Schlatter
Affiliation:
Linné FLOW Centre, KTH Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Swedish e-Science Research Centre (SeRC), SE-100 44 Stockholm, Sweden
Shervin Bagheri
Affiliation:
Linné FLOW Centre, KTH Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
*
†Email address for correspondence: erinaldi@mech.kth.se
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by the relevance of edge state solutions as mediators of transition, we use direct numerical simulations to study the effect of spatially non-uniform viscosity on their energy and stability in minimal channel flows. What we seek is a theoretical support rooted in a fully nonlinear framework that explains the modified threshold for transition to turbulence in flows with temperature-dependent viscosity. Consistently over a range of subcritical Reynolds numbers, we find that decreasing viscosity away from the walls weakens the streamwise streaks and the vortical structures responsible for their regeneration. The entire self-sustained cycle of the edge state is maintained on a lower kinetic energy level with a smaller driving force, compared to a flow with constant viscosity. Increasing viscosity away from the walls has the opposite effect. In both cases, the effect is proportional to the strength of the viscosity gradient. The results presented highlight a local shift in the state space of the position of the edge state relative to the laminar attractor with the consequent modulation of its basin of attraction in the proximity of the edge state and of the surrounding manifold. The implication is that the threshold for transition is reduced for perturbations evolving in the neighbourhood of the edge state in the case that viscosity decreases away from the walls, and vice versa.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Nonlinear instability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 838 , 10 March 2018 , pp. 379 - 403
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

Avila, K., Moxey, D., De Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192–196.Google Scholar
Avila, M., Mellibovsky, F., Roland, N. & Hof, B. 2013 Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110 (22), 224502.Google Scholar
Barker, S. J. & Gile, D. 1981 Experiments on heat-stabilized laminar boundary layers in water. J. Fluid Mech. 104, 139–158.Google Scholar
Chantry, M. & Schneider, T. M. 2014 Studying edge geometry in transiently turbulent shear flows. J. Fluid Mech. 747, 506–517.Google Scholar
Cherubini, S., Palma, P. D., Robinet, J.-C. & Bottaro, A. 2011 Edge states in a boundary layer. Phys. Fluids 23 (5), 051705.Google Scholar
Chevalier, M., Schlatter, P., Lundbladh, A. & Henningson, D. S.2007 A pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep. TRITA-MEK 2007:07. KTH Mechanics, Stockholm, Sweden.Google Scholar
Chikkadi, V., Sameen, A. & Govindarajan, R. 2005 Preventing transition to turbulence: a viscosity stratification does not always help. Phys. Rev. Lett. 95 (26), 264504.Google Scholar
De Lozar, A., Mellibovsky, F., Avila, M. d & Hof, B. 2012 Edge state in pipe flow experiments. Phys. Rev. Lett. 108 (21), 214502.Google Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D. S. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25 (8), 084103.Google Scholar
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008a Relative periodic orbits in transitional pipe flow. Phys. Fluids 20 (11), 114102.Google Scholar
Duguet, Y., Schlatter, P. & Henningson, D. S. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21 (11), 111701.Google Scholar
Duguet, Y., Schlatter, P., Henningson, D. S. & Eckhardt, B. 2012 Self-sustained localized structures in a boundary-layer flow. Phys. Rev. Lett. 108 (4), 044501.CrossRefGoogle Scholar
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008b Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255–274.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.Google Scholar
Govindarajan, R., L’vov, V. S. & Procaccia, I. 2001 Retardation of the onset of turbulence by minor viscosity contrasts. Phys. Rev. Lett. 87 (17), 174501.CrossRefGoogle ScholarPubMed
Govindarajan, R. & Sahu, K. C. 2014 Instabilities in viscosity-stratified flow. Annu. Rev. Fluid Mech. 46, 331–353.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Hof, B., De Lozar, A., Kuik, D. J. & Westerweel, J. 2008 Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow. Phys. Rev. Lett. 101 (21), 214501.CrossRefGoogle ScholarPubMed
Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 59–62.Google Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.Google Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.Google Scholar
Khapko, T., Duguet, Y., Kreilos, T., Schlatter, P., Eckhardt, B. & Henningson, D. S. 2014 Complexity of localised coherent structures in a boundary-layer flow. Eur. Phys. J. E 37 (32), 1–12.Google Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S. 2013 Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech. 717, R6.CrossRefGoogle Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S. 2016 Edge states as mediators of bypass transition in boundary-layer flows. J. Fluid Mech. 801, R2.Google Scholar
Kreilos, T., Khapko, T., Schlatter, P., Duguet, Y., Henningson, D. S. & Eckhardt, B. 2016 Bypass transition and spot nucleation in boundary layers. Phys. Rev. Fluids 1, 043602.Google Scholar
Kreilos, T., Veble, G., Schneider, T. M. e & Eckhardt, B. 2013 Edge states for the turbulence transition in the asymptotic suction boundary layer. J. Fluid Mech. 726, 100–122.CrossRefGoogle Scholar
Lam, K. & Banerjee, S. 1992 On the condition of streak formation in a bounded turbulent flow. Phys. Fluids 4 (2), 306–320.Google Scholar
Lauchle, G. C. & Gurney, G. B. 1984 Laminar boundary layer transition on a heated underwater body. J. Fluid Mech. 144, 79–101.Google Scholar
Liepmann, H. W. & Fila, G. H.1947 Investigations of effects of surface temperature and single roughness elements on boundary-layer transition. NACA Tech. Rep. 890.Google Scholar
Nouar, C., Bottaro, A. & Brancher, J. P. 2007 Delaying transition to turbulence in channel flow: revisiting the stability of shear-thinning fluids. J. Fluid Mech. 592, 177–194.Google Scholar
Park, J. S. & Graham, M. D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430–454.Google Scholar
Patel, A., Boersma, B. J. & Pecnik, R. 2016 The influence of near-wall density and viscosity gradients on turbulence in channel flows. J. Fluid Mech. 809, 793–820.Google Scholar
Patel, A., Peeters, J. W. R., Boersma, B. J. & Pecnik, R. 2015 Semi-local scaling and turbulence modulation in variable property turbulent channel flows. Phys. Fluids 27 (9), 095101.Google 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 (15), 154502.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Roland, N., Plaut, E. & Nouar, C. 2010 Petrov–Galerkin computation of nonlinear waves in pipe flow of shear-thinning fluids: first theoretical evidences for a delayed transition. Comput. Fluids 39 (9), 1733–1743.Google Scholar
Sameen, A. & Govindarajan, R. 2007 The effect of wall heating on instability of channel flow. J. Fluid Mech. 577, 417–442.Google Scholar
Schäfer, P. & Herwig, H. 1993 Stability of plane Poiseuille flow with temperature dependent viscosity. Intl J. Heat Mass Transfer 36 (9), 2441–2448.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition Shear Flows. Springer.Google Scholar
Schneider, T. M., De Lillo, F., Buehrle, J., Eckhardt, B., Dörnemann, T., Dörnemann, K. & Freisleben, B. 2010a Transient turbulence in plane Couette flow. Phys. Rev. E 81 (1), 015301(R).Google Scholar
Schneider, T. M. & Eckhardt, B. 2009 Edge states intermediate between laminar and turbulent dynamics in pipe flow. Phil. Trans. R. Soc. Lond. A 367 (1888), 577–587.Google Scholar
Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.Google Scholar
Schneider, T. M., Gibson, J. F., Lagha, M., De Lillo, F. & Eckhardt, B. 2008 Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E 78 (3), 037301.Google Scholar
Schneider, T. M., Marinc, D. & Eckhardt, B. 2010b Localized edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646, 441–451.Google Scholar
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.Google Scholar
Strazisar, A. & Reshotko, E. 1978 Stability of heated laminar boundary layers in water with nonuniform surface temperature. Phys. Fluids 21 (5), 727–735.Google Scholar
Strazisar, A. J., Reshotko, E. & Prahl, J. M. 1977 Experimental study of the stability of heated laminar boundary layers in water. J. Fluid Mech. 83 (2), 225–247.Google Scholar
Toh, S. & Itano, T. 2003 A periodic-like solution in channel flow. J. Fluid Mech. 481, 67–76.Google Scholar
Waleffe, F. 1995 Transition in shear flows. Nonlinear normality versus non-normal linearity. Phys. Fluids 7 (12), 3060–3066.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.Google Scholar
Wall, D. P. & Wilson, S. K. 1996 The linear stability of channel flow of fluid with temperature-dependent viscosity. J. Fluid Mech. 323, 107–132.Google Scholar
Wall, D. P. & Wilson, S. K. 1997 The linear stability of flat-plate boundary-layer flow of fluid with temperature-dependent viscosity. Phys. Fluids 9 (10), 2885–2898.Google Scholar
Xi, L. & Graham, M. D. 2012 Dynamics on the laminar-turbulent boundary and the origin of the maximum drag reduction asymptote. Phys. Rev. Lett. 108 (2), 028301.Google Scholar
Zammert, S. & Eckhardt, B. 2014a Periodically bursting edge states in plane Poiseuille flow. Fluid Dyn. Res. 46 (4), 041419.Google Scholar
Zammert, S. & Eckhardt, B. 2014b Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech. 761, 348–359.Google Scholar
Zonta, F., Marchioli, C. & Soldati, A. 2012 Modulation of turbulence in forced convection by temperature-dependent viscosity. J. Fluid Mech. 697, 150–174.Google Scholar