Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T01:36:42.293Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of laminar and transitional flows over slip surfaces: effects on the laminar–turbulent separatrix

Published online by Cambridge University Press:  04 May 2020

Ethan A. Davis Open the ORCID record for Ethan A. Davis [Opens in a new window]  and
Jae Sung Park Open the ORCID record for Jae Sung Park [Opens in a new window]
Ethan A. Davis
Affiliation:
Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0526, USA
Jae Sung Park*
Affiliation:
Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0526, USA
*
† Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of slip surfaces on the laminar–turbulent separatrix of plane Poiseuille flow is studied by direct numerical simulation. In laminar flows, the inclusion of the slip surfaces results in a drag reduction of over 10 %, which is in good agreement with previous studies and the theory of laminar slip flows. Turbulence lifetimes, the likelihood that turbulence is sustained, is investigated for transitional flows with various slip lengths. We show that slip surfaces decrease the likelihood of sustained turbulence compared to the no-slip case, and the likelihood is further decreased as slip length is increased. A more deterministic analysis of the effects of slip surfaces on a transition to turbulence is performed by using nonlinear travelling-wave solutions to the Navier–Stokes equations, also known as exact coherent solutions. Two solution families, dubbed P3 and P4, are used since their lower-branch solutions are embedded on the boundary of the basin of attraction of laminar and turbulent flows (Park & Graham, J. Fluid Mech., vol. 782, 2015, pp. 430–454). Additionally, they exhibit distinct flow structures – the P3 and P4 are denoted as core mode and critical-layer mode, respectively. Distinct effects of slip surfaces on the solutions are observed by the skin-friction evolution, linear growth rate and phase-space projection of transitional trajectories. The slip surface appears to modify the transition dynamics very little for the core mode, but quite considerably for the critical-layer mode. Most importantly, the slip surface promotes different transition dynamics – an early and bypass-like transition for the core mode and a delayed and H- or K-type-like transition for the critical-layer mode. We explain these distinct transition dynamics based on spatio-temporal and quadrant analyses. It is found that slip surfaces promote the prevalence of strong wall-toward motions (sweep-like events) near vortex cores close to the channel centre, inducing an early transition, while long sustained ejection events are present in the region of the $\unicode[STIX]{x1D6EC}$-shaped vortex cores close to the critical layer, resulting in a delayed transition. This should motivate flow control strategies to fully exploit these distinct transition dynamics for transition to turbulence.

JFM classification

Instability: Transition to turbulence Flow Control: Drag reduction Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 894 , 10 July 2020 , A16
Copyright
© The Author(s), 2020. Published by 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

Abdulbari, H. A., Yunus, R. M., Abdurahman, N. H. & Charles, A. 2013 Going against the flow-a review of non-additive means of drag reduction. J. Ind. Engng Chem. 19, 27–36.CrossRefGoogle Scholar
Allhoff, K. T. & Eckhardt, B. 2012 Directed percolation model for turbulence transition in shear flows. Fluid Dyn. Res. 44 (3), 031201.Google Scholar
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.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle ScholarPubMed
Barkley, D. 2011 Simplifying the complexity of pipe flow. Phys. Rev. E 84 (1), 016309.Google ScholarPubMed
Barkley, D. 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.CrossRefGoogle Scholar
Barkley, D., Song, B., Mukund, V., Lemoult, G., Avila, M. & Hof, B. 2015 The rise of fully turbulent flow. Nature 526 (7574), 550–553.CrossRefGoogle ScholarPubMed
Bocquet, L. & Lauga, E. 2011 A smooth future? Nat. Mater. 10 (5), 334–337.CrossRefGoogle ScholarPubMed
Bottin, S. & Chaté, H. 1998 Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B 6 (1), 143–155.CrossRefGoogle Scholar
Brand, E. & Gibson, J. F. 2014 A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R3.CrossRefGoogle Scholar
Carlson, D. R., Widnall, S. E. & Peeters, M. F. 1982 A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech. 121, 487–505.CrossRefGoogle Scholar
Castagna, M., Mazellier, N. & Kourta, A. 2018 Wake of super-hydrophobic falling spheres: influence of the air layer deformation. J. Fluid Mech. 850, 646–673.CrossRefGoogle Scholar
Chai, C. & Song, B. 2019 Stability of slip channel flow revisited. Phys. Fluids 31 (8), 084105.Google Scholar
Chang, J., Jung, T., Choi, H. & Kim, J. 2019 Predictions of the effective slip length and drag reduction with a lubricated micro-groove surface in a turbulent channel flow. J. Fluid Mech. 874, 797–820.CrossRefGoogle Scholar
Choi, C. H. & Kim, C. J. 2006 Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett. 96 (6), 066001.CrossRefGoogle Scholar
Chu, A. K. H. 2004 Instability of Navier slip flow of liquids. C. R. Méc. 332 (11), 895–900.CrossRefGoogle Scholar
Davies, J., Maynes, D., Webb, B. W. & Woolford, B. 2006 Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys. Fluids 18 (8), 087110.CrossRefGoogle Scholar
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008 Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255–274.CrossRefGoogle Scholar
Eckhardt, B., Faisst, H., Schmiegel, A. & Schneider, T. M. 2007a Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. Lond. A 366 (1868), 1297–1315.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007b Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (1), 447–468.CrossRefGoogle Scholar
Fairhall, C. T., Abderrahaman-Elena, N. & García-Mayoral, R. 2019 The effect of slip and surface texture on turbulence over superhydrophobic surfaces. J. Fluid Mech. 861, 88–118.CrossRefGoogle Scholar
Fukagata, K., Kasagi, N. & Koumoutsakos, P. 2006 A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces. Phys. Fluids 18 (5), 051703.Google Scholar
de Gennes, P.-G. 2002 On fluid/wall slippage. Langmuir 18 (9), 3413–3414.CrossRefGoogle Scholar
Gibson, J. F.2012 Channelflow: a spectral Navier–Stokes simulator in $C++$. Tech. Rep. U. New Hampshire. Available at: Channelflow.org.Google Scholar
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and travelling-wave solutions of plane couette flow. J. Fluid Mech. 638, 243–266.CrossRefGoogle Scholar
Golovin, K. B., Gose, J., Perlin, M., Ceccio, S. L. & Tuteja, A. 2016 Bioinspired surfaces for turbulent drag reduction. Phil. Trans. R. Soc. Lond. A 374 (2073), 20160189.Google ScholarPubMed
Gose, J. W., Golovin, K., Boban, M., Mabry, J. M., Tuteja, A., Perlin, M. & Ceccio, S. L. 2018 Characterization of superhydrophobic surfaces for drag reduction in turbulent flow. J. Fluid Mech. 845, 560–580.CrossRefGoogle Scholar
Granick, S., Zhu, Y. & Lee, H. 2003 Slippery questions about complex fluids flowing past solids. Nat. Mater. 2 (4), 221.CrossRefGoogle ScholarPubMed
Gruncell, B. R. K., Sandham, N. D. & McHale, G. 2013 Simulations of laminar flow past a superhydrophobic sphere with drag reduction and separation delay. Phys. Fluids 25 (4), 043601.CrossRefGoogle Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305 (5690), 1594–1598.CrossRefGoogle ScholarPubMed
Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 59–62.CrossRefGoogle ScholarPubMed
Ibrahim, J. I., Yang, Q., Doohan, P. & Hwang, Y. 2018 Phase-space dynamics of opposition control in wall-bounded turbulent flows. J. Fluid Mech. 861, 29–54.CrossRefGoogle Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.CrossRefGoogle Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.CrossRefGoogle Scholar
Jung, T., Choi, H. & Kim, J. 2016 Effects of the air layer of an idealized superhydrophobic surface on the slip length and skin-friction drag. J. Fluid Mech. 790, R1.CrossRefGoogle Scholar
Jung, Y. C. & Bhushan, B. 2010 Biomimetic structures for fluid drag reduction in laminar and turbulent flows. J. Phys.: Condens. Matter 22 (3), 35104.Google ScholarPubMed
Kachanov, Y. S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26 (1), 411–482.CrossRefGoogle Scholar
Kawahara, G. 2005 Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Phys. Fluids 17 (4), 041702.CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & Van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.CrossRefGoogle Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (6), R17.CrossRefGoogle Scholar
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12 (1), 1–34.CrossRefGoogle Scholar
Lauga, E. & Cossu, C. 2005 A note on the stability of slip channel flows. Phys. Fluids 17 (8), 1–4.CrossRefGoogle Scholar
Lee, C., Choi, C. H. & Kim, C. J. 2016 Superhydrophobic drag reduction in laminar flows: a critical review. Exp. Fluids 57 (12), 176.CrossRefGoogle Scholar
Lee, J., Jelly, T. O. & Zaki, T. A. 2015 Effect of Reynolds number on turbulent drag reduction by superhydrophobic surface textures. Flow Turbul. Combust. 95 (2-3), 277–300.CrossRefGoogle Scholar
Lemoult, G., Shi, L., Avila, K., Jalikop, S. V., Avila, M. & Hof, B. 2016 Directed percolation phase transition to sustained turbulence in couette flow. Nat. Phys. 12 (3), 254.CrossRefGoogle Scholar
Ling, H., Srinivasan, S., Golovin, K., McKinley, G. H., Tuteja, A. & Katz, J. 2016 High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces. J. Fluid Mech. 801, 670–703.CrossRefGoogle Scholar
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87–109.Google Scholar
Lustro, J. R. T., Kawahara, G., van Veen, L., Shimizu, M. & Kokubu, H. 2019 The onset of transient turbulence in minimal plane Couette flow. J. Fluid Mech. 862, R2.CrossRefGoogle Scholar
Maynes, D., Jeffs, K., Woolford, B. & Webb, B. W. 2007 Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction. Phys. Fluids 19 (9), 93603.CrossRefGoogle Scholar
Min, T. & Kim, J. 2004 Effects of hydrophobic surface on skin-friction drag. Phys. Fluids 16 (7), 55.CrossRefGoogle Scholar
Min, T. & Kim, J. 2005 Effects of hydrophobic surface on stability and transition. Phys. Fluids 17 (10), 108106.CrossRefGoogle Scholar
Morkovin, M. V.1985 Bypass transition to turbulence and research desiderata. NASA Conference Publication, pp. 161–211.Google Scholar
Nagata, M. & Deguchi, K. 2013 Mirror-symmetric exact coherent states in plane Poiseuille flow. J. Fluid Mech. 735, R4.CrossRefGoogle Scholar
Navier, C. L. M. H. 1823 Mémoire sur les lois du mouvement des fluides. Mém. Acad. Sci. Inst. Fr. 6 (1823), 389–440.Google Scholar
Neto, C., Evans, D. R., Bonaccurso, E., Butt, H.-J. & Craig, V. S. J. 2005 Boundary slip in Newtonian liquids: a review of experimental studies. Rep. Prog. Phys. 68 (12), 2859.CrossRefGoogle Scholar
Nishi, M., Ünsal, B., Durst, F. & Biswas, G. 2008 Laminar-to-turbulent transition of pipe flows through puffs and slugs. J. Fluid Mech. 614, 425–446.CrossRefGoogle Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16 (12), 4635–4643.CrossRefGoogle Scholar
Park, H., Park, H. & Kim, J. 2013 A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow. Phys. Fluids 25 (11), 110815.CrossRefGoogle Scholar
Park, H., Sun, G. & Kim, C. J. 2014 Superhydrophobic turbulent drag reduction as a function of surface grating parameters. J. Fluid Mech. 747, 722–734.CrossRefGoogle 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.CrossRefGoogle Scholar
Park, J. S., Shekar, A. & Graham, M. D. 2018 Bursting and critical layer frequencies in minimal turbulent dynamics and connections to exact coherent states. Phys. Rev. F 3 (1), 014611.Google Scholar
Picella, F., Robinet, J.-C. & Cherubini, S. 2019 Laminar-turbulent transition in channel flow with superhydrophobic surfaces modelled as a partial slip wall. J. Fluid Mech. 881, 462–497.CrossRefGoogle Scholar
Pomeau, Y. 1986 Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23 (1-3), 3–11.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Quéré, D. 2005 Non-sticking drops. Rep. Prog. Phys. 68 (11), 2495.CrossRefGoogle Scholar
Reynolds, O. 1883 Xxix. an experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. A 174, 935–982.Google Scholar
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42 (1), 89–109.CrossRefGoogle Scholar
Sano, M. & Tamai, K. 2016 A universal transition to turbulence in channel flow. Nat. Phys. 12 (3), 249.CrossRefGoogle Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34 (1), 291–319.CrossRefGoogle 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.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2012 Stability and Transition in Shear Flows. Springer Science & Business Media.Google Scholar
Schmiegel, A. & Eckhardt, B. 1997 Fractal stability border in plane Couette flow. Phys. Rev. Lett. 79, 5250–5253.CrossRefGoogle Scholar
Schneider, T. M. & Eckhardt, B. 2008 Lifetime statistics in transitional pipe flow. Phys. Rev. E 78 (4), 1–10.Google ScholarPubMed
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.CrossRefGoogle ScholarPubMed
Seo, J., García-Mayoral, R. & Mani, A. 2018 Turbulent flows over superhydrophobic surfaces: flow-induced capillary waves, and robustness of air-water interfaces. J. Fluid Mech. 835, 45–85.CrossRefGoogle Scholar
Seo, J. & Mani, A. 2018 Effect of texture randomization on the slip and interfacial robustness in turbulent flows over superhydrophobic surfaces. Phys. Rev. F 3 (4), 44601.Google Scholar
Shekar, A. & Graham, M. D. 2018 Exact coherent states with hairpin-like vortex structure in channel flow. J. Fluid Mech. 849, 76–89.CrossRefGoogle Scholar
Shih, H.-Y., Hsieh, T.-L. & Goldenfeld, N. 2016 Ecological collapse and the emergence of travelling waves at the onset of shear turbulence. Nat. Phys. 12 (3), 245–248.CrossRefGoogle Scholar
Sipos, M. & Goldenfeld, N. 2011 Directed percolation describes lifetime and growth of turbulent puffs and slugs. Phys. Rev. E 84 (3), 035304.Google ScholarPubMed
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.CrossRefGoogle Scholar
Spille, A., Rauh, A. & Buehring, H. 2000 Critical curves of plane Poiseuille flow with slip boundary conditions. Nonlinear Phenom. Complex Syst. 3 (2), 171–173.Google Scholar
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77 (3), 977.CrossRefGoogle Scholar
Suri, B., Tithof, J., Grigoriev, R. O. & Schatz, M. F. 2017 Forecasting fluid flows using the geometry of turbulence. Phy. Rev. Lett. 118 (11), 114501.CrossRefGoogle ScholarPubMed
Tithof, J., Suri, B., Pallantla, R. K., Grigoriev, R. O. & Schatz, M. F. 2017 Bifurcations in a quasi-two-dimensional Kolmogorov-like flow. J. Fluid Mech. 828, 837–866.CrossRefGoogle Scholar
Truesdell, R., Mammoli, A., Vorobieff, P., van Swol, F. & Brinker, C. J. 2006 Drag reduction on a patterned superhydrophobic surface. Phys. Rev. Lett. 97, 044504.CrossRefGoogle ScholarPubMed
Tuckerman, L. S., Kreilos, T., Schrobsdorff, H., Schneider, T. M. & Gibson, J. F. 2014 Turbulent-laminar patterns in plane Poiseuille flow. Phys. Fluids 26 (11), 114103.CrossRefGoogle Scholar
Viswanath, D. 2009 The critical layer in pipe flow at high Reynolds number. Phil. Trans. R. Soc. Lond. A 367 (1888), 561–576.CrossRefGoogle ScholarPubMed
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.CrossRefGoogle Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.CrossRefGoogle ScholarPubMed
Watanabe, K., Okido, K. & Mizunuma, H. 1996 Drag reduction in flow through square and rectangular ducts with highly water-repellent walls. Trans. Japan Soc. Mech. Engng B 62, 3330–3334.CrossRefGoogle Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 5–41.CrossRefGoogle Scholar
Wygnanski, I. J. & Champagne, F. H. 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59 (2), 281–335.CrossRefGoogle Scholar
Ybert, C., Barentin, C., Cottin-Bizonne, C., Joseph, P. & Bocquet, L. 2007 Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries. Phys. Fluids 19 (12), 123601.CrossRefGoogle Scholar
You, X. Y., Zheng, J. R. & Jing, Q. 2007 Effects of boundary slip and apparent viscosity on the stability of microchannel flow. Forsch. Ing. Engng Res. 71 (2), 99–106.CrossRefGoogle Scholar
Yu, K. H., Teo, C. J. & Khoo, B. C. 2016 Linear stability of pressure-driven flow over longitudinal superhydrophobic grooves. Phys. Fluids 28, 22001.CrossRefGoogle Scholar
Zammert, S. & Eckhardt, B. 2014 Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech. 761, 348–359.CrossRefGoogle Scholar
Zhang, J., Tian, H., Yao, Z., Hao, P. & Jiang, N. 2015 Mechanisms of drag reduction of superhydrophobic surfaces in a turbulent boundary layer flow. Exp. Fluids 56 (9), 179.CrossRefGoogle 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.CrossRefGoogle Scholar