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Langmuir-type vortices in wall-bounded flows driven by a criss-cross wavy wall topography

Published online by Cambridge University Press:  07 August 2020

Andreas H. Akselsen Open the ORCID record for Andreas H. Akselsen [Opens in a new window]  and
Simen Å. Ellingsen Open the ORCID record for Simen Å. Ellingsen [Opens in a new window]
Andreas H. Akselsen*
Affiliation:
Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491Trondheim, Norway
Simen Å. Ellingsen
Affiliation:
Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491Trondheim, Norway
*
†Email address for correspondence: andreas.h.akselsen@ntnu.no
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Abstract

We investigate a mechanism to manipulate wall-bounded flows whereby wave-like undulations of the wall topography drives the creation of bespoke longitudinal vortices. A resonant interaction between the ambient vorticity of the undisturbed shear flow and the undulation of streamlines enforced by the wall topography serves to slightly rotate the spanwise vorticity of the mean flow into the streamwise direction, creating a swirling motion, in the form of regular streamwise rolls. The process is kinematic and essentially identical to the ‘direct drive’ CL1 mechanism for Langmuir circulation (LC) proposed by Craik (J. Fluid Mech., vol. 41, issue 4, 1970, pp. 801–821). Wall shear is modelled by selecting suitable primary flow profiles. A simple, easily integrable expression for the cross-plane streamfunction is found in two asymptotic regimes: the resonant onset of the essentially inviscid instability at early times, and the fully developed steady-state viscous flow. Linear-order solutions for flow over undulating boundaries are obtained, fully analytical in the special case of a power-law profile. These solutions allow us to quickly map out the circulation response to boundary design parameters. The study is supplemented with direct numerical simulations which verify the manifestation of boundary induced Langmuir vortices in laminar flows with no-slip boundaries. Simulations show good qualitative agreement with theory. Quantitatively, the comparisons rest on a displacement length closure parameter adopted in the perturbation theory. While wall-driven LC appear to become unstable in turbulent flows, we propose that the mechanism can promote swirling motion in boundary layers, a flow feature which has been reported to reduce drag in some situations.

JFM classification

Vortex Flows: Vortex dynamics Boundary Layers: Boundary layer structure
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A19
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Akselsen, A. H., Ellingsen, S. Å. 2019 a Sheared free-surface flow over three-dimensional obstructions of finite amplitude. J. Fluid Mech. 878, 740–767.CrossRefGoogle Scholar
Akselsen, A. H., Ellingsen, S. Å. 2019 b Weakly nonlinear transient waves on a shear current: ring waves and skewed Langmuir rolls. J. Fluid Mech. 863, 114–149.CrossRefGoogle Scholar
Anderson, W., Barros, J. M., Christensen, Ke. T. & Awasthi, A. 2015 Numerical and experimental study of mechanisms responsible for turbulent secondary flows in boundary layer flows over spanwise heterogeneous roughness. J. Fluid Mech. 768, 316–347.CrossRefGoogle Scholar
Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.CrossRefGoogle Scholar
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
Bakewell, H. P. & Lumley, J. L. 1967 Viscous sublayer and adjacent wall region in turbulent pipe flow. Phys. Fluids 10 (9), 1880–1889.CrossRefGoogle Scholar
Barenblatt, G. I. 1993 Scaling laws for fully developed turbulent shear flows. Part 1. basic hypotheses and analysis. J. Fluid Mech. 248, 513–520.CrossRefGoogle Scholar
Belcher, S. E., Grant, A. L. M., Hanley, K. E., Fox-Kemper, B., Van Roekel, L., Sullivan, P. P., Large, W. G., Brown, A., Hines, A., Calvert, D. et al. . 2012 A global perspective on Langmuir turbulence in the ocean surface boundary layer. Geophys. Res. Lett. 39, L18605.CrossRefGoogle Scholar
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow. Phys. Fluids 3, 656–657.CrossRefGoogle Scholar
Bergstrom, D. J., Tachie, M. F. & Balachandar, R. 2001 Application of power laws to low Reynolds number boundary layers on smooth and rough surfaces. Phys. Fluids 13, 3277–3284.CrossRefGoogle Scholar
Chan, L., MacDonald, M., Chung, D., Hutchins, N. & Ooi, A. 2015 A systematic investigation of roughness height and wavelength in turbulent pipe flow in the transitionally rough regime. J. Fluid Mech. 771, 743–777.CrossRefGoogle Scholar
Chan, L., MacDonald, M., Chung, D., Hutchins, N. & Ooi, A. 2018 Secondary motion in turbulent pipe flow with three-dimensional roughness. J. Fluid Mech. 854, 5–33.CrossRefGoogle Scholar
Chen, C-L 1991 Unified theory on power laws for flow resistance. J. Hydraul. Engng 117, 371–389.CrossRefGoogle Scholar
Cheng, Nian-Sheng 2007 Power-law index for velocity profiles in open channel flows. Adv. Water Resour. 30, 1775–1784.CrossRefGoogle Scholar
Corino, E. R. & Brodkey, Robert S. 1969 A visual investigation of the wall region in turbulent flow. J. Fluid Mech. 37, 1–30.CrossRefGoogle Scholar
Cossu, C. & Brandt, L. 2002 Stabilization of Tollmien–Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer. Phys. Fluids 14, L57–L60.CrossRefGoogle Scholar
Cossu, C. & Brandt, L. 2004 On tollmien–schlichting-like waves in streaky boundary layers. Eur. J. Mech. B/Fluids 23 (6), 815–833.CrossRefGoogle Scholar
Craik, A. D. D. 1970 A wave-interaction model for the generation of windows. J. Fluid Mech. 41 (4), 801–821.CrossRefGoogle Scholar
Craik, A. D. D. 1977 The generation of Langmuir circulations by an instability mechanism. J. Fluid Mech. 81, 209–223.CrossRefGoogle Scholar
Craik, A. D. D. 1982 a The generalized Lagrangian-mean equations and hydrodynamic stability. J. Fluid Mech. 125, 27–35.CrossRefGoogle Scholar
Craik, A. D. D. 1982 b Wave-induced longitudinal-vortex instability in shear flows. J. Fluid Mech. 125, 37–52.CrossRefGoogle Scholar
Craik, A. D. D. & Leibovich, S. 1976 A rational model for Langmuir circulations. J. Fluid Mech. 73, 401–426.CrossRefGoogle Scholar
Dean, B. & Bhushan, B. 2010 Shark-skin surfaces for fluid-drag reduction in turbulent flow: a review. Phil. Trans. R. Soc. Lond. A 368, 4775–4806.CrossRefGoogle ScholarPubMed
Deguchi, K. & Hall, P. 2014 The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99–112.CrossRefGoogle Scholar
Djenidi, L., Dubief, Y. & Antonia, R. A. 1997 Advantages of using a power law in a low r$\theta$ turbulent boundary layer. Exp. Fluids 22, 348–350.CrossRefGoogle Scholar
Dolcetti, G, Horoshenkov, K. V., Krynkin, A. & Tait, S. J. 2016 Frequency-wavenumber spectrum of the free surface of shallow turbulent flows over a rough boundary. Phys. Fluids 28, 105105.CrossRefGoogle Scholar
Du, Y. & Karniadakis, G. E. 2000 Suppressing wall turbulence by means of a transverse traveling wave. Science 288 (5469), 1230–1234.CrossRefGoogle ScholarPubMed
Fransson, J. H. M., Brandt, L., Talamelli, A. & Cossu, C. 2005 Experimental study of the stabilization of Tollmien–Schlichting waves by finite amplitude streaks. Phys. Fluids 17 (5), 054110.CrossRefGoogle Scholar
Fransson, J. H. M., Talamelli, A., Brandt, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96, 064501.CrossRefGoogle ScholarPubMed
George, W. K. & Castillo, L. 1997 Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689–729.CrossRefGoogle Scholar
Gong, W., Taylor, P. A. & Dörnbrack, A. 1996 Turbulent boundary-layer flow over fixed aerodynamically rough two-dimensional sinusoidal waves. J. Fluid Mech. 312, 1–37.CrossRefGoogle Scholar
Head, M. R. & Bandyopadhyay, P. 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338.CrossRefGoogle Scholar
Iuso, G., Onorato, M., Spazzini, P. G. & Di Cicca, G. M. 2002 Wall turbulence manipulation by large-scale streamwise vortices. J. Fluid Mech. 473, 23–58.CrossRefGoogle Scholar
Kevin, K., Monty, J. P., Bai, H. L., Pathikonda, G., Nugroho, B., Barros, J. M., Christensen, K. T. & Hutchins, N. 2017 Cross-stream stereoscopic particle image velocimetry of a modified turbulent boundary layer over directional surface pattern. J. Fluid Mech. 813, 412–435.CrossRefGoogle Scholar
Kühnen, J., Song, B., Scarselli, D., Budanur, N. B., Riedl, M., Willis, A. P., Avila, M. & Hof, B. 2018 Destabilizing turbulence in pipe flow. Nat. Phys. 14 (4), 386–390.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge university press.Google Scholar
Langmuir, L. 1938 Surface motion of water induced by wind. Science 87, 119–123.CrossRefGoogle ScholarPubMed
Leibovich, S. 1977 On the evolution of the system of wind drift currents and Langmuir circulations in the ocean. Part 1. theory and averaged current. J. Fluid Mech. 79, 715–743.CrossRefGoogle Scholar
Leibovich, S. 1980 On wave-current interaction theories of Langmuir circulations. J. Fluid Mech. 99, 715–724.CrossRefGoogle Scholar
Leibovich, S. 1983 The form and dynamics of Langmuir circulations. Annu. Rev. Fluid Mech. 15, 391–427.CrossRefGoogle Scholar
Li, M., Zahariev, K. & Garrett, C. 1995 Role of Langmuir circulation in the deepening of the ocean surface mixed layer. Science 270 (5244), 1955–1957.CrossRefGoogle Scholar
Lieu, B. K., Moarref, R. & Jovanović, M. R. 2010 Controlling the onset of turbulence by streamwise travelling waves. Part 2. direct numerical simulation. J. Fluid Mech. 663, 100–119.CrossRefGoogle Scholar
Melville, W. K., Shear, R. & Veron, F. 1998 Laboratory measurements of the generation and evolution of Langmuir circulations. J. Fluid Mech. 364, 31–58.CrossRefGoogle Scholar
Offen, G. R. & Kline, S. J. 1974 Combined dye-streak and hydrogen-bubble visual observations of a turbulent boundary layer. J. Fluid Mech. 62, 223–239.CrossRefGoogle Scholar
Phillips, W. R. C. 1998 Finite-amplitude rotational waves in viscous shear flows. Stud. Appl. Maths 101, 23–47.CrossRefGoogle Scholar
Phillips, W. R. C. 2005 On the spacing of Langmuir circulation in strong shear. J. Fluid Mech. 525, 215–236.CrossRefGoogle ScholarPubMed
Phillips, W. R. C. & Dai, A. 2014 On Langmuir circulation in shallow waters. J. Fluid Mech. 743, 141–169.CrossRefGoogle Scholar
Phillips, W. R. C. & Shen, Q. 1996 A family of wave-mean shear interactions and their instability to longitudinal vortex form. Stud. Appl. Maths 96, 143–161.CrossRefGoogle Scholar
Phillips, W. R. C. & Wu, Z. 1994 On the instability of wave-catalysed longitudinal vortices in strong shear. J. Fluid Mech. 272, 235–254.CrossRefGoogle Scholar
Phillips, W. R. C., Wu, Z. & Lumley, J. L. 1996 On the formation of longitudinal vortices in a turbulent boundary layer over wavy terrain. J. Fluid Mech. 326, 321–341.CrossRefGoogle Scholar
Pujals, G., Cossu, C. & Depardon, S. 2010 a Forcing large-scale coherent streaks in a zero-pressure-gradient turbulent boundary layer. J. Turbul. 11, N25.CrossRefGoogle Scholar
Pujals, G., Depardon, S. & Cossu, C. 2010 b Drag reduction of a 3d bluff body using coherent streamwise streaks. Exp. Fluids 49, 1085–1094.CrossRefGoogle Scholar
Sirovich, L. & Karlsson, S. 1997 Turbulent drag reduction by passive mechanisms. Nature 388 (6644), 753–;755.CrossRefGoogle Scholar
Thorpe, S. A. 2004 Langmuir circulation. Annu. Rev. Fluid Mech. 36, 55–79.CrossRefGoogle Scholar
Vanderwel, C. & Ganapathisubramani, B. 2015 Effects of spanwise spacing on large-scale secondary flows in rough-wall turbulent boundary layers. J. Fluid Mech. 774, R2.CrossRefGoogle Scholar
Willingham, D., Anderson, W., Christensen, K. T. & Barros, J. M. 2014 Turbulent boundary layer flow over transverse aerodynamic roughness transitions: induced mixing and flow characterization. Phys. Fluids 26, 025111.CrossRefGoogle Scholar
Yang, J. & Anderson, W. 2018 Numerical study of turbulent channel flow over surfaces with variable spanwise heterogeneities: topographically-driven secondary flows affect outer-layer similarity of turbulent length scales. Flow Turbul. Combust. 100, 1–17.CrossRefGoogle Scholar
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