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Turbulent drag reduction by anisotropic permeable substrates – analysis and direct numerical simulations

  • G. Gómez-de-Segura (a1) and R. García-Mayoral (a1)


We explore the ability of anisotropic permeable substrates to reduce turbulent skin friction, studying the influence that these substrates have on the overlying turbulence. For this, we perform direct numerical simulations of channel flows bounded by permeable substrates. The results confirm theoretical predictions, and the resulting drag curves are similar to those of riblets. For small permeabilities, the drag reduction is proportional to the difference between the streamwise and spanwise permeabilities. This linear regime breaks down for a critical value of the wall-normal permeability, beyond which the performance begins to degrade. We observe that the degradation is associated with the appearance of spanwise-coherent structures, attributed to a Kelvin–Helmholtz-like instability of the mean flow. This feature is common to a variety of obstructed flows, and linear stability analysis can be used to predict it. For large permeabilities, these structures become prevalent in the flow, outweighing the drag-reducing effect of slip and eventually leading to an increase of drag. For the substrate configurations considered, the largest drag reduction observed is ${\approx}$ 20–25 % at a friction Reynolds number $\unicode[STIX]{x1D6FF}^{+}=180$ .


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Abderrahaman-Elena, N., Fairhall, C. T. & García-Mayoral, R. 2019 Modulation of near-wall turbulence in the transitionally rough regime. J. Fluid Mech. 865, 10421071.
Abderrahaman-Elena, N. & García-Mayoral, R. 2017 Analysis of anisotropic permeable surfaces for turbulent drag reduction. Phys. Rev. Fluids 2, 114609.
Auriault, J. L. 2009 On the domain of validity of Brinkman’s equation. Trans. Porous Med. 79 (2), 215223.
Battiato, I. 2012 Self-similarity in coupled Brinkman/Navier–Stokes flows. J. Fluid Mech. 699, 94114.10.1017/jfm.2012.85
Battiato, I. 2014 Effective medium theory for drag-reducing micro-patterned surfaces in turbulent flows. Eur. Phys. J. E 37, 19.
Battiato, I. & Rubol, S. 2014 Single-parameter model of vegetated aquatic flows. Water Resour. Res. 50, 63586369.
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1), 197207.
Bechert, D. W., Bruse, M., Hage, W., Van der Hoeven, J. G. T. & Hoppe, G. 1997 Experiments on drag-reducing surfaces and their optimization with an adjustable geometry. J. Fluid Mech. 338, 5987.
Bottaro, A. 2019 Flow over natural or engineered surfaces: an adjoint homogenization perspective. J. Fluid Mech. (submitted).
Breugem, W. P. & Boersma, B. J. 2005 Direct numerical simulations of turbulent flow over a permeable wall using a direct and a continuum approach. Phys. Fluids 17, 025103.
Breugem, W. P., Boersma, B. J. & Uittenbogaard, R. E. 2006 The influence of wall permeability on turbulent channel flow. J. Fluid Mech. 562, 3572.10.1017/S0022112006000887
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 2734.
Busse, A. & Sandham, N. D. 2012 Influence of an anisotropic slip-length boundary condition on turbulent channel flow. Phys. Fluids 24, 055111.
Clauser, F. H. 1956 The turbulent boundary layer. Adv. Appl. Mech. 4, 151.
Darcy, H. 1856 Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont.
Fairhall, C. T., Abderrahaman-Elena, N. & García-Mayoral, R. 2019 The effects of slip and surface texture on turbulence over superhydrophobic surfaces. J. Fluid Mech. 861, 88118.10.1017/jfm.2018.909
Fairhall, C. T. & García-Mayoral, R. 2018 Spectral analysis of slip-length model for turbulence over textured superhydrophobic surfaces. Flow Turb. Combust. 100 (4), 961978.
Forchheimer, P. 1901 Wasserbewegung durch boden. Z. Ver. Deutsch. Ing. 45, 17821788.
García-Mayoral, R., Gómez-de-Segura, G. & Fairhall, C. T. 2019 The control of near-wall turbulence through surface texturing. Fluid Dyn. Res. 51 (1), 011410.
García-Mayoral, R. & Jiménez, J. 2011 Drag reduction by riblets. Phil. Trans. R. Soc. A 369, 14121427.
García-Mayoral, R. & Jiménez, J. 2011 Hydrodynamic stability and breakdown of the viscous regime over riblets. J. Fluid Mech. 678, 317347.
García-Mayoral, R. & Jiménez, J. 2012 Scaling of turbulent structures in riblet channels up to Re 𝜏 ≈ 550. Phys. Fluids 24, 105101.
Gatti, D. & Quadrio, M. 2016 Reynolds-number dependence of turbulent skin-friction drag reduction induced by spanwise forcing. J. Fluid Mech. 802, 553582.
Ghisalberti, M. 2009 Obstructed shear flows: similarities across systems and scales. J. Fluid Mech. 641, 5161.
Gómez-de-Segura, G., Fairhall, C. T., MacDonald, M., Chung, D. & García-Mayoral, R. 2018a Manipulation of near-wall turbulence by surface slip and permeability. J. Phys.: Conf. Ser. 1001, 012011.
Gómez-de-Segura, G., Sharma, A. & García-Mayoral, R. 2018b Turbulent drag reduction using anisotropic permeable substrates. Flow Turb. Combust. 100 (4), 9951014.
Hahn, S., Je, J. & Choi, H. 2002 Direct numerical simulation of turbulent channel flow with permeable walls. J. Fluid Mech. 450, 259285.
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re 𝜏 = 2003. Phys. Fluids 18 (1), 011702.
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20, 101511.
Itoh, M., Tamano, S., Iguchi, R., Yokota, K., Akino, N., Hino, R. & Kubo, S. 2006 Turbulent drag reduction by the seal fur surface. Phys. Fluids 18, 065102.
James, D. F. & Davis, A. M. J. 2001 Flow at the interface of a model fibrous porous medium. J. Fluid Mech. 426, 4772.
Jiménez, J. 1994 On the structure and control of near wall turbulence. Phys. Fluids 6 (2), 944953.
Jiménez, J., Uhlmann, M., Pinelli, A. & Kawahara, G. 2001 Turbulent shear flow over active and passive porous surfaces. J. Fluid Mech. 442, 89117.
Joseph, D. D., Nield, D. A. & Papanicolaou, G. 1982 Nonlinear equation governing flow in a saturated porous medium. Water Resour. Res. 18 (4), 10491052.10.1029/WR018i004p01049
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59 (2), 308323.
Kuwata, Y. & Suga, K. 2016 Lattice Boltzmann direct numerical simulation of interface turbulence over porous and rough walls. Intl J. Heat Fluid Flow 61 (A), 145157.
Kuwata, Y. & Suga, K. 2017 Direct numerical simulation of turbulence over anisotropic porous media. J. Fluid Mech. 831, 4171.
Lācis, U. & Bagheri, S. 2017 A framework for computing effective boundary conditions at the interface between free fluid and a porous medium. J. Fluid Mech. 812, 866889.
Le, H. & Moin, P. 1991 An improvement of fractional step methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 92, 369379.
Le Bars, M. & Worster, M. G. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149173.
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to Re 𝜏 = 5200. J. Fluid Mech. 774 (1), 395415.
Lévy, T. 1983 Fluid flow through an array of fixed particles. Intl J. Engng Sci. 21 (1), 1123.
Lozano-Durán, A. & Jiménez, J. 2014 Effect of the computational domain on direct numerical simulations of turbulent channels up to Re 𝜏 = 4200. Phys. Fluids 26, 011702.
Luchini, P. 1996 Reducing the turbulent skin friction. In Computational Methods in Applied Sciences, Proceedings 3rd ECCOMAS CFD Conference, pp. 466470. Wiley.
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87109.
MacDonald, M., Chan, L., Chung, D., Hutchins, N. & Ooi, A. 2016 Turbulent flow over transitionally rough surfaces with varying roughness densities. J. Fluid Mech. 804, 130161.
Min, T. & Kim, J. 2004 Effects of hydrophobic surface on skin-friction drag. Phys. Fluids 16 (7), L55.
Neale, G. & Nader, W. 1974 Practical significance of Brinkman’s extension of Darcy’s Law. Can. J. Chem. Engng 52, 475478.
Ochoa-Tapia, J. A. & Whitaker, S. 1995a Momentum transfer at the boundary between a porous medium and a homogeneous fluid – I. Theoretical development. Intl J. Heat Mass Transfer 38 (14), 26352646.
Ochoa-Tapia, J. A. & Whitaker, S. 1995b Momentum transfer at the boundary between a porous medium and a homogeneous fluid – II. Comparison with experiment. Intl J. Heat Mass Transfer 38 (14), 26472655.
Orlandi, P. & Leonardi, S. 2006 DNS of turbulent channel flows with two- and three-dimensional roughness. J. Turb. 7, N73.
Perot, J. B. 1993 An analysis of the fractional step method. J. Comput. Phys. 108, 5158.
Perot, J. B. 1995 Comments on the fractional step method. J. Comput. Phys. 121, 190.
Rosti, M. E., Brandt, L. & Pinelli, A. 2018 Turbulent channel flow over an anisotropic porous wall – drag increase and reduction. J. Fluid Mech. 842, 381394.
Rosti, M. E., Cortelezzi, L. & Quadrio, M. 2015 Direct numerical simulation of turbulent channel flow over porous walls. J. Fluid Mech. 784, 396442.
Rubol, S., Ling, B. & Battiato, I. 2018 Universal scaling-law for flow resistance over canopies with complex morphology. Sci. Rep. 8, 4430.
Gómez-de Segura, G.2019 Turbulent drag reduction by anisotropic permeable substrates. PhD thesis, University of Cambridge.
Seo, J., Garcia-Mayoral, R. & Mani, A. 2018 Turbulent flows over superhydrophobic surfaces: flow-induced capillary waves, and robustness of air–water interfaces. J. Fluid Mech. 835, 4585.
Spalart, P. R. & McLean, J. D. 2011 Drag reduction: enticing turbulence, and then an industry. Phil. Trans. R. Soc. A 369, 15561569.
Suga, K., Matsumura, Y., Ashitaka, Y., Tominaga, S. & Kaneda, M. 2010 Effects of wall permeability on turbulence. Intl J. Heat Fluid Flow 31 (6), 121.
Suga, K., Nakagawa, Y. & Kaneda, M. 2017 Spanwise turbulence structure over permeable walls. J. Fluid Mech. 822, 186201.
Suga, K., Okazaki, Y., Ho, U. & Kuwata, Y. 2018 Anisotropic wall permeability effects on turbulent channel flows. J. Fluid Mech. 855, 9831016.
Tam, C. K. W. 1969 The drag on a cloud of spherical particles in low Reynolds number flow. J. Fluid Mech. 38 (3), 537546.
Tilton, N. & Cortelezzi, L. 2008 Linear stability analysis of pressure-driven flows in channels with porous walls. J. Fluid Mech. 604, 411445.
Vafai, K. & Kim, S. J. 1990 Fluid mechanics of the interface region between a porous medium and a fluid layer – an exact solution. Intl J. Heat Fluid Flow 11 (3), 254256.10.1016/0142-727X(90)90045-D
Whitaker, S. 1996 The Forchheimer equation: a theoretical development. Trans. Porous Med. 25, 2761.
Zampogna, G. A. & Bottaro, A. 2016 Fluid flow over and through a regular bundle of rigid fibres. J. Fluid Mech. 792, 535.
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Turbulent drag reduction by anisotropic permeable substrates – analysis and direct numerical simulations

  • G. Gómez-de-Segura (a1) and R. García-Mayoral (a1)


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