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Dynamics of laminar and transitional flows over slip surfaces: effects on the laminar-turbulent separatrix. (English) Zbl 07198784
Summary: 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 [J. S. Park and M. D. Graham, ibid. 782, 430–454 (2015; Zbl 1381.76097)]. 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 \(\Lambda\)-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.
76F06 Transition to turbulence
76F20 Dynamical systems approach to turbulence
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[1] Abdulbari, H. A., Yunus, R. M., Abdurahman, N. H. & Charles, A.2013Going against the flow-a review of non-additive means of drag reduction. J. Ind. Engng Chem.19, 27-36.
[2] Allhoff, K. T. & Eckhardt, B.2012Directed percolation model for turbulence transition in shear flows. Fluid Dyn. Res.44 (3), 031201. · Zbl 1309.76104
[3] Avila, K., Moxey, D., De Lozar, A., Avila, M., Barkley, D. & Hof, B.2011The onset of turbulence in pipe flow. Science333 (6039), 192-196.
[4] Avila, M., Mellibovsky, F., Roland, N. & Hof, B.2013Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett.110 (22), 224502.
[5] Barkley, D.2011Simplifying the complexity of pipe flow. Phys. Rev. E84 (1), 016309.
[6] Barkley, D.2016Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech.803, P1. · Zbl 1454.76047
[7] Barkley, D., Song, B., Mukund, V., Lemoult, G., Avila, M. & Hof, B.2015The rise of fully turbulent flow. Nature526 (7574), 550-553.
[8] Bocquet, L. & Lauga, E.2011A smooth future?Nat. Mater.10 (5), 334-337.
[9] Bottin, S. & Chaté, H.1998Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B6 (1), 143-155.
[10] Brand, E. & Gibson, J. F.2014A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech.750, R3.
[11] Carlson, D. R., Widnall, S. E. & Peeters, M. F.1982A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech.121, 487-505.
[12] Castagna, M., Mazellier, N. & Kourta, A.2018Wake of super-hydrophobic falling spheres: influence of the air layer deformation. J. Fluid Mech.850, 646-673.
[13] Chai, C. & Song, B.2019Stability of slip channel flow revisited. Phys. Fluids31 (8), 084105.
[14] Chang, J., Jung, T., Choi, H. & Kim, J.2019Predictions 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. · Zbl 1419.76305
[15] Choi, C. H. & Kim, C. J.2006Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett.96 (6), 066001.
[16] Chu, A. K. H.2004Instability of Navier slip flow of liquids. C. R. Méc.332 (11), 895-900. · Zbl 1386.76063
[17] Davies, J., Maynes, D., Webb, B. W. & Woolford, B.2006Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys. Fluids18 (8), 087110.
[18] Duguet, Y., Willis, A. P. & Kerswell, R. R.2008Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech.613, 255-274.
[19] Eckhardt, B., Faisst, H., Schmiegel, A. & Schneider, T. M.2007aDynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. Lond. A366 (1868), 1297-1315.
[20] Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J.2007bTurbulence transition in pipe flow. Annu. Rev. Fluid Mech.39 (1), 447-468.
[21] Fairhall, C. T., Abderrahaman-Elena, N. & García-Mayoral, R.2019The effect of slip and surface texture on turbulence over superhydrophobic surfaces. J. Fluid Mech.861, 88-118. · Zbl 1415.76334
[22] Fukagata, K., Kasagi, N. & Koumoutsakos, P.2006A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces. Phys. Fluids18 (5), 051703.
[23] De Gennes, P.-G.2002On fluid/wall slippage. Langmuir18 (9), 3413-3414.
[24] Gibson, J. F.2012 Channelflow: a spectral Navier-Stokes simulator in \(C++\). Tech. Rep. U. New Hampshire. Available at: Channelflow.org.
[25] Gibson, J. F. & Brand, E.2014Spanwise-localized solutions of planar shear flows. J. Fluid Mech.745, 25-61.
[26] Gibson, J. F., Halcrow, J. & Cvitanović, P.2008Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech.611, 107-130. · Zbl 1151.76453
[27] Gibson, J. F., Halcrow, J. & Cvitanović, P.2009Equilibrium and travelling-wave solutions of plane couette flow. J. Fluid Mech.638, 243-266. · Zbl 1183.76688
[28] Golovin, K. B., Gose, J., Perlin, M., Ceccio, S. L. & Tuteja, A.2016Bioinspired surfaces for turbulent drag reduction. Phil. Trans. R. Soc. Lond. A374 (2073), 20160189. · Zbl 1404.76125
[29] Gose, J. W., Golovin, K., Boban, M., Mabry, J. M., Tuteja, A., Perlin, M. & Ceccio, S. L.2018Characterization of superhydrophobic surfaces for drag reduction in turbulent flow. J. Fluid Mech.845, 560-580. · Zbl 1404.76125
[30] Granick, S., Zhu, Y. & Lee, H.2003Slippery questions about complex fluids flowing past solids. Nat. Mater.2 (4), 221.
[31] Gruncell, B. R. K., Sandham, N. D. & Mchale, G.2013Simulations of laminar flow past a superhydrophobic sphere with drag reduction and separation delay. Phys. Fluids25 (4), 043601.
[32] Hof, B., Van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F.2004Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science305 (5690), 1594-1598.
[33] Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B.2006Finite lifetime of turbulence in shear flows. Nature443 (7107), 59-62.
[34] Ibrahim, J. I., Yang, Q., Doohan, P. & Hwang, Y.2018Phase-space dynamics of opposition control in wall-bounded turbulent flows. J. Fluid Mech.861, 29-54. · Zbl 1415.76341
[35] Itano, T. & Toh, S.2001The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan70 (3), 703-716.
[36] Jiménez, J. & Moin, P.1991The minimal flow unit in near-wall turbulence. J. Fluid Mech.225, 213-240. · Zbl 0721.76040
[37] Jung, T., Choi, H. & Kim, J.2016Effects of the air layer of an idealized superhydrophobic surface on the slip length and skin-friction drag. J. Fluid Mech.790, R1.
[38] Jung, Y. C. & Bhushan, B.2010Biomimetic structures for fluid drag reduction in laminar and turbulent flows. J. Phys.: Condens. Matter22 (3), 35104.
[39] Kachanov, Y. S.1994Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech.26 (1), 411-482.
[40] Kawahara, G.2005Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Phys. Fluids17 (4), 041702. · Zbl 1187.76260
[41] Kawahara, G., Uhlmann, M. & Van Veen, L.2012The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech.44, 203-225. · Zbl 1352.76031
[42] Kerswell, R. R.2005Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity18 (6), R17.
[43] Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M.1962The three-dimensional nature of boundary-layer instability. J. Fluid Mech.12 (1), 1-34. · Zbl 0131.41901
[44] Lauga, E. & Cossu, C.2005A note on the stability of slip channel flows. Phys. Fluids17 (8), 1-4. · Zbl 1187.76297
[45] Lee, C., Choi, C. H. & Kim, C. J.2016Superhydrophobic drag reduction in laminar flows: a critical review. Exp. Fluids57 (12), 176.
[46] Lee, J., Jelly, T. O. & Zaki, T. A.2015Effect of Reynolds number on turbulent drag reduction by superhydrophobic surface textures. Flow Turbul. Combust.95 (2-3), 277-300.
[47] Lemoult, G., Shi, L., Avila, K., Jalikop, S. V., Avila, M. & Hof, B.2016Directed percolation phase transition to sustained turbulence in couette flow. Nat. Phys.12 (3), 254.
[48] Ling, H., Srinivasan, S., Golovin, K., Mckinley, G. H., Tuteja, A. & Katz, J.2016High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces. J. Fluid Mech.801, 670-703.
[49] Luchini, P., Manzo, F. & Pozzi, A.1991Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech.228, 87-109.
[50] Lustro, J. R. T., Kawahara, G., Van Veen, L., Shimizu, M. & Kokubu, H.2019The onset of transient turbulence in minimal plane Couette flow. J. Fluid Mech.862, R2. · Zbl 1415.76278
[51] Maynes, D., Jeffs, K., Woolford, B. & Webb, B. W.2007Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction. Phys. Fluids19 (9), 93603. · Zbl 1182.76502
[52] Min, T. & Kim, J.2004Effects of hydrophobic surface on skin-friction drag. Phys. Fluids16 (7), 55. · Zbl 1186.76377
[53] Min, T. & Kim, J.2005Effects of hydrophobic surface on stability and transition. Phys. Fluids17 (10), 108106. · Zbl 1188.76097
[54] Morkovin, M. V.1985 Bypass transition to turbulence and research desiderata. NASA Conference Publication, pp. 161-211.
[55] Nagata, M. & Deguchi, K.2013Mirror-symmetric exact coherent states in plane Poiseuille flow. J. Fluid Mech.735, R4. · Zbl 1294.76115
[56] Navier, C. L. M. H.1823Mémoire sur les lois du mouvement des fluides. Mém. Acad. Sci. Inst. Fr.6 (1823), 389-440.
[57] Neto, C., Evans, D. R., Bonaccurso, E., Butt, H.-J. & Craig, V. S. J.2005Boundary slip in Newtonian liquids: a review of experimental studies. Rep. Prog. Phys.68 (12), 2859.
[58] Nishi, M., Ünsal, B., Durst, F. & Biswas, G.2008Laminar-to-turbulent transition of pipe flows through puffs and slugs. J. Fluid Mech.614, 425-446. · Zbl 1178.76041
[59] Ou, J., Perot, B. & Rothstein, J. P.2004Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids16 (12), 4635-4643. · Zbl 1187.76393
[60] Park, H., Park, H. & Kim, J.2013A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow. Phys. Fluids25 (11), 110815.
[61] Park, H., Sun, G. & Kim, C. J.2014Superhydrophobic turbulent drag reduction as a function of surface grating parameters. J. Fluid Mech.747, 722-734.
[62] Park, J. S. & Graham, M. D.2015Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech.782, 430-454. · Zbl 1381.76097
[63] Park, J. S., Shekar, A. & Graham, M. D.2018Bursting and critical layer frequencies in minimal turbulent dynamics and connections to exact coherent states. Phys. Rev. F3 (1), 014611.
[64] Picella, F., Robinet, J.-C. & Cherubini, S.2019Laminar-turbulent transition in channel flow with superhydrophobic surfaces modelled as a partial slip wall. J. Fluid Mech.881, 462-497. · Zbl 1430.76225
[65] Pomeau, Y.1986Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D23 (1-3), 3-11.
[66] Pope, S. B.2000Turbulent Flows. Cambridge University Press.
[67] Quéré, D.2005Non-sticking drops. Rep. Prog. Phys.68 (11), 2495.
[68] Reynolds, O.1883Xxix. 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. A174, 935-982. · JFM 16.0845.02
[69] Rothstein, J. P.2010Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech.42 (1), 89-109.
[70] Sano, M. & Tamai, K.2016A universal transition to turbulence in channel flow. Nat. Phys.12 (3), 249.
[71] Saric, W. S., Reed, H. L. & Kerschen, E. J.2002Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech.34 (1), 291-319. · Zbl 1006.76029
[72] Sayadi, T., Hamman, C. W. & Moin, P.2013Direct numerical simulation of complete H-type and K-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech.724, 480. · Zbl 1287.76138
[73] Schmid, P. J. & Henningson, D. S.2012Stability and Transition in Shear Flows. Springer Science & Business Media.
[74] Schmiegel, A. & Eckhardt, B.1997Fractal stability border in plane Couette flow. Phys. Rev. Lett.79, 5250-5253.
[75] Schneider, T. M. & Eckhardt, B.2008Lifetime statistics in transitional pipe flow. Phys. Rev. E78 (4), 1-10.
[76] Schneider, T. M., Eckhardt, B. & Yorke, J. A.2007Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett.99 (3), 034502.
[77] Seo, J., García-Mayoral, R. & Mani, A.2018Turbulent flows over superhydrophobic surfaces: flow-induced capillary waves, and robustness of air-water interfaces. J. Fluid Mech.835, 45-85. · Zbl 1421.76120
[78] Seo, J. & Mani, A.2018Effect of texture randomization on the slip and interfacial robustness in turbulent flows over superhydrophobic surfaces. Phys. Rev. F3 (4), 44601.
[79] Shekar, A. & Graham, M. D.2018Exact coherent states with hairpin-like vortex structure in channel flow. J. Fluid Mech.849, 76-89. · Zbl 1415.76137
[80] Shih, H.-Y., Hsieh, T.-L. & Goldenfeld, N.2016Ecological collapse and the emergence of travelling waves at the onset of shear turbulence. Nat. Phys.12 (3), 245-248.
[81] Sipos, M. & Goldenfeld, N.2011Directed percolation describes lifetime and growth of turbulent puffs and slugs. Phys. Rev. E84 (3), 035304.
[82] Skufca, J. D., Yorke, J. A. & Eckhardt, B.2006Edge of chaos in a parallel shear flow. Phys. Rev. Lett.96 (17), 174101.
[83] Spille, A., Rauh, A. & Buehring, H.2000Critical curves of plane Poiseuille flow with slip boundary conditions. Nonlinear Phenom. Complex Syst.3 (2), 171-173.
[84] Squires, T. M. & Quake, S. R.2005Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys.77 (3), 977.
[85] Suri, B., Tithof, J., Grigoriev, R. O. & Schatz, M. F.2017Forecasting fluid flows using the geometry of turbulence. Phy. Rev. Lett.118 (11), 114501.
[86] Tithof, J., Suri, B., Pallantla, R. K., Grigoriev, R. O. & Schatz, M. F.2017Bifurcations in a quasi-two-dimensional Kolmogorov-like flow. J. Fluid Mech.828, 837-866. · Zbl 07135912
[87] Truesdell, R., Mammoli, A., Vorobieff, P., Van Swol, F. & Brinker, C. J.2006Drag reduction on a patterned superhydrophobic surface. Phys. Rev. Lett.97, 044504.
[88] Tuckerman, L. S., Kreilos, T., Schrobsdorff, H., Schneider, T. M. & Gibson, J. F.2014Turbulent-laminar patterns in plane Poiseuille flow. Phys. Fluids26 (11), 114103.
[89] Viswanath, D.2009The critical layer in pipe flow at high Reynolds number. Phil. Trans. R. Soc. Lond. A367 (1888), 561-576. · Zbl 1221.76098
[90] Waleffe, F.2001Exact coherent structures in channel flow. J. Fluid Mech.435, 93-102. · Zbl 0987.76034
[91] Wang, J., Gibson, J. & Waleffe, F.2007Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett.98 (20), 204501.
[92] Watanabe, K., Okido, K. & Mizunuma, H.1996Drag reduction in flow through square and rectangular ducts with highly water-repellent walls. Trans. Japan Soc. Mech. Engng B62, 3330-3334.
[93] Wu, X. & Moin, P.2009Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech.630, 5-41. · Zbl 1181.76084
[94] Wygnanski, I. J. & Champagne, F. H.1973On 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.
[95] Ybert, C., Barentin, C., Cottin-Bizonne, C., Joseph, P. & Bocquet, L.2007Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries. Phys. Fluids19 (12), 123601. · Zbl 1182.76848
[96] You, X. Y., Zheng, J. R. & Jing, Q.2007Effects of boundary slip and apparent viscosity on the stability of microchannel flow. Forsch. Ing. Engng Res.71 (2), 99-106.
[97] Yu, K. H., Teo, C. J. & Khoo, B. C.2016Linear stability of pressure-driven flow over longitudinal superhydrophobic grooves. Phys. Fluids28, 22001.
[98] Zammert, S. & Eckhardt, B.2014Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech.761, 348-359.
[99] Zhang, J., Tian, H., Yao, Z., Hao, P. & Jiang, N.2015Mechanisms of drag reduction of superhydrophobic surfaces in a turbulent boundary layer flow. Exp. Fluids56 (9), 179.
[100] Zhou, J., Adrian, R. J., Balachandar, S. & Kendall, T. M.1999Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech.387, 353-396. · Zbl 0946.76030
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