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Reynolds-number dependence of turbulent skin-friction drag reduction induced by spanwise forcing

Published online by Cambridge University Press:  05 August 2016

Davide Gatti
Institute of Fluid Mechanics, Karlsruhe Institute of Technology, Kaiserstraße 10, 76131 Karlsruhe, Germany
Maurizio Quadrio*
Department of Aerospace Science and Technology, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
Email address for correspondence:


This paper examines how increasing the value of the Reynolds number $Re$ affects the ability of spanwise-forcing techniques to yield turbulent skin-friction drag reduction. The considered forcing is based on the streamwise-travelling waves of spanwise-wall velocity (Quadrio et al., J. Fluid Mech., vol. 627, 2009, pp. 161–178). The study builds upon an extensive drag-reduction database created via direct numerical simulation of a turbulent channel flow for two fivefold separated values of $Re$ , namely $Re_{\unicode[STIX]{x1D70F}}=200$ and $Re_{\unicode[STIX]{x1D70F}}=1000$ . The sheer size of the database, which for the first time systematically addresses the amplitude of the forcing, allows a comprehensive view of the drag-reducing characteristics of the travelling waves, and enables a detailed description of the changes occurring when $Re$ increases. The effect of using a viscous scaling based on the friction velocity of either the non-controlled flow or the drag-reduced flow is described. In analogy with other wall-based drag-reduction techniques, like riblets for example, the performance of the travelling waves is well described by a vertical shift of the logarithmic portion of the mean streamwise velocity profile. Except when $Re$ is very low, this shift remains constant with $Re$ , at odds with the percentage reduction of the friction coefficient, which is known to present a mild, logarithmic decline. Our new data agree with the available literature, which is however mostly based on low- $Re$ information and hence predicts a quick drop of maximum drag reduction with $Re$ . The present study supports a more optimistic scenario, where for an airplane at flight Reynolds numbers a drag reduction of nearly 30 % would still be possible thanks to the travelling waves.

© 2016 Cambridge University Press 

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Auteri, F., Baron, A., Belan, M., Campanardi, G. & Quadrio, M. 2010 Experimental assessment of drag reduction by traveling waves in a turbulent pipe flow. Phys. Fluids 22 (11), 115103.CrossRefGoogle Scholar
Baron, A. & Quadrio, M. 1996 Turbulent drag reduction by spanwise wall oscillations. Appl. Sci. Res. 55, 311326.CrossRefGoogle Scholar
Bechert, D. W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105209.CrossRefGoogle Scholar
Belan, M. & Quadrio, M. 2013 A perturbative model for predicting the high-Reynolds-number behaviour of the streamwise travelling waves technique in turbulent drag reduction. Z. Angew. Math. Mech. 93 (12), 944962.CrossRefGoogle Scholar
Bernardini, M., Pirozzoli, S. & Orlandi, P. 2014 Velocity statistics in turbulent channel flow up to Re 𝜏 = 4000. J. Fluid Mech. 742, 171191.CrossRefGoogle Scholar
Brown, G. L. & Thomas, A. S. W. 1977 Large structure in a turbulent boundary layer. Phys. Fluids 20 (10), S243S252.CrossRefGoogle Scholar
Choi, J.-I., Xu, C.-X. & Sung, H. J. 2002 Drag reduction by spanwise wall oscillation in wall-bounded turbulent flows. AIAA J. 40 (5), 842850.CrossRefGoogle Scholar
Choi, K.-S. 1989 Near-wall structure of a turbulent boundary layer with riblets. J. Fluid Mech. 208, 417458.CrossRefGoogle Scholar
Choi, K.-S., DeBisschop, J. R. & Clayton, B. R. 1998 Turbulent boundary-layer control by means of spanwise-wall oscillation. AIAA J. 36 (7), 11571162.CrossRefGoogle Scholar
Choi, K.-S. & Graham, M. 1998 Drag reduction of turbulent pipe flows by circular-wall oscillation. Phys. Fluids 10 (1), 79.CrossRefGoogle Scholar
Clauser, F. H. 1956 The turbulent boundary layer. Adv. Appl. Mech. 4, 151.CrossRefGoogle Scholar
Del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135144.CrossRefGoogle Scholar
Du, Y., Symeonidis, V. & Karniadakis, G. E. 2002 Drag reduction in wall-bounded turbulence via a transverse travelling wave. J. Fluid Mech. 457, 134.CrossRefGoogle Scholar
Duque-Daza, C. A., Baig, M. F., Lockerby, D. A., Chernyshenko, S. I. & Davies, C. 2012 Modelling turbulent skin-friction control using linearised Navier–Stokes equations. J. Fluid Mech. 702, 403414.CrossRefGoogle Scholar
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22 (7), 071704.CrossRefGoogle Scholar
Fukagata, K., Iwamoto, K. & Kasagi, N. 2002 Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Phys. Fluids 14 (11), L73L76.CrossRefGoogle Scholar
Ganapathisubramani, B., Hutchins, N., Monty, J. P. & Marusic, I. 2012 Amplitude and frequency modulation in wall turbulence. J. Fluid Mech. 712, 6191.CrossRefGoogle Scholar
Ganapathisubramani, B., Longmire, E. & Marusic, I. 2003 Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 3546.CrossRefGoogle Scholar
García-Mayoral, R. & Jiménez, J. 2011 Drag reduction by riblets. Phil. Trans. R. Soc. A 369 (1940), 14121427.CrossRefGoogle ScholarPubMed
Gatti, D., Güttler, A., Frohnapfel, B. & Tropea, C. 2015 Experimental assessment of spanwise-oscillating dielectric electroactive surfaces for turbulent drag reduction in an air channel flow. Exp. Fluids 56 (5), 115.CrossRefGoogle Scholar
Gatti, D. & Quadrio, M. 2013 Performance losses of drag-reducing spanwise forcing at moderate values of the Reynolds number. Phys. Fluids 25, 125109.CrossRefGoogle Scholar
Gouder, K., Potter, M. & Morrison, J. F. 2013 Turbulent friction drag reduction using electroactive polymer and electromagnetically driven surfaces. Exp. Fluids 54 (1), 1441.CrossRefGoogle Scholar
Guala, M., Hommema, S. E. & Adrian, R. J. 2006 Large-scale and very-large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521542.CrossRefGoogle Scholar
Hasegawa, Y., Quadrio, M. & Frohnapfel, B. 2014 Numerical simulation of turbulent duct flows at constant power input. J. Fluid Mech. 750, 191209.CrossRefGoogle Scholar
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20, 101511.CrossRefGoogle Scholar
Hurst, E., Yang, Q. & Chung, Y. M. 2014 The effect of Reynolds number on turbulent drag reduction by streamwise travelling waves. J. Fluid Mech. 759, 2855.CrossRefGoogle Scholar
Iwamoto, K., Fukagata, K., Kasagi, N. & Suzuki, Y. 2005 Friction drag reduction achievable with near-wall manipulation at high Reynolds numbers. Phys. Fluids 17, 011702.CrossRefGoogle Scholar
Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36, 173196.CrossRefGoogle Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213240.CrossRefGoogle Scholar
Jung, W. J., Mangiavacchi, N. & Akhavan, R. 1992 Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations. Phys. Fluids A 4 (8), 16051607.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.CrossRefGoogle Scholar
Lardeau, S. & Leschziner, M. 2013 The streamwise drag-reduction response of a boundary layer subjected to a sudden imposition of transverse oscillatory wall motion. Phys. Fluids 25, 075109.CrossRefGoogle Scholar
Lee, M. & Moser, R. 2015 Direct numerical simulation of turbulent channel flow up to Re 𝜏 ≈ 5200. J. Fluid Mech. 774, 395415.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014a Effect of the computational domain on direct simulations of turbulent channels up to Re 𝜏 = 4200. Phys. Fluids 26, 011702.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014b Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432471.CrossRefGoogle Scholar
Luchini, P. 1996 Reducing the turbulent skin friction. In Computational Methods in Applied Sciences ’96, Proceedings of the Third ECCOMAS Computational Fluid Dynamics Conference (ed. Desideri, J.-A., Hirsch, C. & Tallec, P.). Wiley.Google Scholar
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87109.Google Scholar
Luchini, P. & Quadrio, M. 2006 A low-cost parallel implementation of direct numerical simulation of wall turbulence. J. Comput. Phys. 211 (2), 551571.CrossRefGoogle Scholar
Mishra, M. & Skote, M. 2015 Drag reduction in turbulent boundary layers with half wave wall oscillations. Math. Prob. Engng 2015, 253249.Google Scholar
Moarref, R. & Jovanović, M. R. 2012 Model-based design of transverse wall oscillations for turbulent drag reduction. J. Fluid Mech. 707, 205240.CrossRefGoogle Scholar
Mockett, C., Knacke, T. & Thiele, F. 2010 Detection of initial transient and estimation of statistical error in time-resolved turbulent flow data. In Proceedings of the 8th ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM8), pp. 911. ERCOFTAC.Google Scholar
Oliver, T. A., Malaya, N., Ulerich, R. & Moser, R. D. 2014 Estimating uncertainties in statistics computed from direct numerical simulation. Phys. Fluids 26, 035101.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Quadrio, M. 2011 Drag reduction in turbulent boundary layers by in-plane wall motion. Phil. Trans. R. Soc. A 369 (1940), 14281442.CrossRefGoogle ScholarPubMed
Quadrio, M., Frohnapfel, B. & Hasegawa, Y. 2016 Does the choice of the forcing term affect flow statistics in DNS of turbulent channel flow? Eur. J. Mech. (B/Fluids) 55, 286293.CrossRefGoogle Scholar
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent drag reduction through spanwise wall oscillation. J. Fluid Mech. 521, 251271.CrossRefGoogle Scholar
Quadrio, M. & Ricco, P. 2011 The laminar generalized Stokes layer and turbulent drag reduction. J. Fluid Mech. 667, 135157.CrossRefGoogle Scholar
Quadrio, M., Ricco, P. & Viotti, C. 2009 Streamwise-traveling waves of spanwise wall velocity for turbulent drag reduction. J. Fluid Mech. 627, 161178.CrossRefGoogle Scholar
Ricco, P., Ottonelli, C., Hasegawa, Y. & Quadrio, M. 2012 Changes in turbulent dissipation in a channel flow with oscillating walls. J. Fluid Mech. 700, 77104.CrossRefGoogle Scholar
Ricco, P. & Quadrio, M. 2008 Wall-oscillation conditions for drag reduction in turbulent channel flow. Intl J. Heat Fluid Flow 29, 601612.CrossRefGoogle Scholar
Ricco, P. & Wu, S. 2004 On the effects of lateral wall oscillations on a turbulent boundary layer. Exp. Therm. Fluid Sci. 29 (1), 4152.CrossRefGoogle Scholar
Schmeiser, B. W. 1982 Batch size effects in the analysis of simulation output. Oper. Res. 30 (3), 556568.CrossRefGoogle Scholar
Skote, M. 2011 Turbulent boundary layer flow subject to streamwise oscillation of spanwise wall-velocity. Phys. Fluids 23, 081703.CrossRefGoogle Scholar
Skote, M. 2013 Comparison between spatial and temporal wall oscillations in turbulent boundary layer flows. J. Fluid Mech. 730, 273294.CrossRefGoogle Scholar
Skote, M. 2014 Scaling of the velocity profile in strongly drag reduced turbulent flows over an oscillating wall. Intl J. Heat Fluid Flow 50, 352358.CrossRefGoogle Scholar
Spalart, P. R. & McLean, J. D. 2011 Drag reduction: enticing turbulence, and then an industry. Phil. Trans. R. Soc. A 369 (1940), 15561569.CrossRefGoogle ScholarPubMed
Touber, E. & Leschziner, M. A. 2012 Near-wall streak modification by spanwise oscillatory wall motion and drag-reduction mechanisms. J. Fluid Mech. 693, 150200.CrossRefGoogle Scholar
Walsh, M. J. 1980 Drag characteristics of V-groove and transverse curvature riblets. In Viscous Drag Reduction (ed. Hough, G. R.). American Institute of Aeronautics and Astronautics.Google Scholar
Walsh, M. J., Sellers, L. W. & McGinley, C. B. 1989 Riblet drag at flight conditions. J. Aircraft 26 (6), 570575.CrossRefGoogle Scholar
Xie, W.2014 Turbulence skin-friction reduction by traveling waves: a DNS study. PhD thesis, Politecnico di Milano.Google Scholar
Yakeno, A., Hasegawa, Y. & Kasagi, N. 2014 Modification of quasi-streamwise vortical structure in a drag-reduced turbulent channel flow with spanwise wall oscillation. Phys. Fluids 26, 085109.CrossRefGoogle Scholar
Yudhistira, I. & Skote, M. 2011 Direct numerical simulation of a turbulent boundary layer over an oscillating wall. J. Turbul. 12 (9), 117.CrossRefGoogle Scholar
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