It has been known over several decades that canonical wall-bounded internal flows of a pipe and channel share flow similarities, in particular, close to the wall due to the negligible curvature effect. In the present study, direct numerical simulations of fully developed turbulent pipe and channel flows are performed to investigate the influence of the superhydrophobic surfaces (SHSs) on the turbulence dynamics and the resultant drag reduction (DR) of the flows under similar conditions. SHSs at the wall are modeled in spanwise-alternating longitudinal regions with a boundary with no-slip and shear-free conditions, and the two parameters of the spanwise periodicity (P/δ) and SHS fraction (GF) within a pitch are considered. It is shown, in agreement with previous investigations in channels, that the turbulent drag for the pipe and channel flows over SHSs is continuously decreased with increases in P/δ and GF. However, the DR rate in the pipe flows is greater than that in the channel flows with an accompanying reduction of the Reynolds stress. The enhanced performance of the DR for the pipe flow is attributed to the increased streamwise slip and weakened Reynolds shear stress contributions. In addition, a mathematical analysis of the spanwise mean vorticity equation suggests that the presence of a strong secondary flow due to the increased spanwise slip of the pipe flows makes a greater negative contribution of advective vorticity transport than the channel flows, resulting in a higher DR value. Finally, an inspection of the origin of the mean secondary flow in turbulent flows over SHSs based on the spatial gradients of the turbulent kinetic energy demonstrates that the secondary flow is both driven and sustained by spatial gradients in the Reynolds stress components, i.e., Prandtl’s secondary flow of the second kind.
REFERENCES
1.
Adrian
, R. J.
, “Hairpin vortex organization in wall turbulence
,” Phys. Fluids
19
, 041301
(2007
).2.
Adrian
, R. J.
, Meinhart
, C. D.
, and Tomkins
, C. D.
, “Vortex organization in the outer region of the turbulent boundary layer
,” J. Fluid Mech.
422
, 1
–54
(2000
).3.
Aghdam
, S. K.
and Ricco
, P.
, “Laminar and turbulent flows over hydrophobic surfaces with shear-dependent slip length
,” Phys. Fluids
28
, 035109
(2016
).4.
Akselvoll
, K.
and Moin
, P.
, Report No. TF-63, Thermoscience Division, Department of Mechanical Engineering, Stanford University
, 1995
.5.
Aljallis
, E.
, Sarshar
, M. A.
, Datla
, R.
, Sikka
, V.
, Jones
, A.
, and Choi
, C.-H.
, “Experimental study of skin friction drag reduction on superhydrophobic flat plates in high Reynolds number boundary layer flow
,” Phys. Fluids
25
, 025103
(2013
).6.
Anderson
, W.
, Barros
, J. M.
, Christensen
, K. T.
, and Awasthi
, A.
, “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
(2015
).7.
Belyaev
, A. V.
and Vinogradova
, O. I.
, “Effective slip in pressure-driven flow past superhydrophobic stripes
,” J. Fluid Mech.
652
, 489
–499
(2010
).8.
Brundrett
, E.
and Baines
, W. D.
, “The production and diffusion of vorticity in duct flow
,” J. Fluid Mech.
19
(03
), 375
–394
(1964
).9.
Busse
, A.
and Sandham
, N. D.
, “Influence of an anisotropic slip-length boundary condition on turbulent channel flow
,” Phys. Fluids
24
, 055111
(2012
).10.
Busse
, A.
, Sandham
, N. D.
, Mchale
, G.
, and Newton
, M. I.
, “Change in drag, apparent slip and optimum air layer thickness for laminar flow over an idealized superhydrophobic surface
,” J. Fluid Mech.
727
, 488
–508
(2013
).11.
Cassie
, A. B. D.
and Baxter
, S.
, “Wettability of porous surfaces
,” Trans. Faraday Soc.
40
, 546
–551
(1944
).12.
Chin
, C.
, Philip
, J.
, Klewicki
, J.
, Ooi
, A.
, and Marusic
, I.
, “Reynolds-number-dependent turbulent inertia and onset of log region in pipe flows
,” J. Fluid Mech.
757
, 747
–769
(2014
).13.
Choi
, H.
, Moin
, P.
, and Kim
, J.
, “Direct numerical simulation of turbulent flow over riblets
,” J. Fluid Mech.
255
, 503
–539
(1993
).14.
Daniello
, R. J.
, Waterhouse
, N. E.
, and Rothstein
, J. P.
, “Drag reduction in turbulent flows over superhydrophobic surfaces
,” Phys. Fluids
21
, 085103
(2009
).15.
Deck
, S.
, Renard
, N.
, Laraufie
, R.
, and Weiss
, P. É.
, “Large-scale contribution to mean wall shear stress in high-Reynolds-number flat-plate boundary layers up to 13 650
,” J. Fluid Mech.
743
, 202
–248
(2014
).16.
Fukagata
, K.
, Iwamoto
, K.
, and Kasagi
, N.
, “Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows
,” Phys. Fluids
14
, L73
–L76
(2002
).17.
Fukagata
, K.
, Kasagi
, N.
, and Koumoutsakos
, P.
, “A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces
,” Phys. Fluids
18
(5
), 051703
(2006
).18.
Goldstein
, D.
and Tuan
, T. C.
, “Secondary flow induced by riblets
,” J. Fluid Mech.
363
, 115
–151
(1998
).19.
Goldstein
, D.
, Handler
, R.
, and Sirovich
, L.
, “Direct numerical simulation of turbulent flow over a modelled riblet covered surface
,” J. Fluid Mech.
302
, 333
–376
(1995
).20.
Hinze
, J. O.
, “Secondary currents in wall turbulence
,” Phys. Fluids
10
, S122
–S125
(1967
).21.
Jang
, S. J.
, Sung
, H. J.
, and Krogstad
, P. Å.
, “Effects of an axisymmetric contraction on a turbulent pipe flow
,” J. Fluid Mech.
687
, 376
(2011
).22.
Jelly
, T. O.
, Jung
, S. Y.
, and Zaki
, T. A.
, “Turbulence and skin friction modification in channel flow with streamwise-aligned superhydrophobic surface texture
,” Phys. Fluids
26
, 095102
(2014
).23.
Jimenez
, J.
and Pinelli
, A.
, “The autonomous cycle of near-wall turbulence
,” J. Fluid Mech.
389
, 335
–359
(1999
).24.
Jung
, T.
, Choi
, H.
, and Kim
, J.
, “Effects of the air layer of an idealized superhydrophobic surface on the slip length and skin-friction drag
,” J. Fluid Mech.
790
, R1
(2016
).25.
Kametani
, Y.
and Fukagata
, K.
, “Direct numerical simulation of spatially developing turbulent boundary layers with uniform blowing or suction
,” J. Fluid Mech.
681
, 154
–172
(2011
).26.
Kim
, K.
, Baek
, S.-J.
, and Sung
, H. J.
, “An implicit velocity decoupling procedure for the incompressible Navier-Stokes equations
,” Int. J. Numer. Methods Fluids
38
, 125
–138
(2002
).28.
Klewicki
, J. C.
, Murray
, J. A.
, and Falco
, R. E.
, “Vortical motion contributions to stress transport in turbulent boundary layers
,” Phys. Fluids
6
(1
), 277
–286
(1994
).29.
Lauga
, E.
and Stone
, H. A.
, “Effective slip in pressure-driven Stokes flow
,” J. Fluid Mech.
489
, 55
–77
(2003
).30.
Lee
, J.
, Jelly
, T. O.
, and Zaki
, T. A.
, “Effect of Reynolds number on turbulent drag reduction by superhydrophobic surface textures
,” Flow, Turbul. Combust.
95
(2-3
), 277
–300
(2015
).31.
Lumley
, J. L.
, “Drag reduction in turbulent flow by polymer additives
,” J. Polym. Sci. D: Macromol. Rev.
7
, 263
(1973
).32.
Martell
, M. B.
, Perot
, J. B.
, and Rothstein
, J. P.
, “Direct numerical simulations of turbulent flows over superhydrophobic surfaces
,” J. Fluid Mech.
620
, 31
–41
(2009
).33.
Min
, T.
and Kim
, J.
, “Effect of superhydrophobic surfaces on skin-friction drag
,” Phys. Fluids
16
, L55
(2004
).34.
Monty
, J. P.
, Stewart
, J. A.
, Williams
, R. C.
, and Chong
, M. S.
, “Large-scale features in turbulent pipe and channel flows
,” J. Fluid Mech.
589
, 147
–156
(2007
).35.
Monty
, J. P.
, Hutchins
, N.
, Ng
, H. C. H.
, Marusic
, I.
, and Chong
, M. S.
, “A comparison of turbulent pipe, channel and boundary layer flows
,” J. Fluid Mech.
632
, 431
–442
(2009
).36.
Ng
, T. S.
, Lawrence
, C. J.
, and Hewitt
, G. F.
, “Interface shapes for two-phase laminar stratified flow in a circular pipe
,” Int. J. Multiphase Flow
27
(7
), 1301
–1311
(2001
).37.
Park
, H.
, Park
, H.
, and Kim
, J.
, “A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow
,” Phys. Fluids
25
, 110815
(2013
).38.
Park
, H.
, Sun
, G.
, and Kim
, C.-J.
, “Superhydrophobic turbulent drag reduction as a function of surface grating parameters
,” J. Fluid Mech.
747
, 722
–734
(2014
).39.
Peguero
, C.
and Breuer
, K.
, “On drag reduction in turbulent channel flow over superhydrophobic surfaces
,” Advances in Turbulence XII
(Springer Berlin Heidelberg
, 2009
), pp. 233
–236
.40.
Perkins
, R. M.
, “PIV measurements of turbulent flow in a rectangular channel over superhydrophobic surfaces with riblets
,” All Theses and Dissertations, 5547, Brigham Young University
, 2014
.41.
Philip
, J. R.
, “Flows satisfying mixed no-slip and no-shear conditions
,” Z. Angew. Math. Phys.
23
, 353
–372
(1972
).42.
43.
Rastegari
, A.
and Akhavan
, R.
, “On the mechanism of turbulent drag reduction with superhydrophobic surfaces
,” J. Fluid Mech.
773
, R4
(2015
).44.
Reynolds
, W. C.
and Hussain
, A. K. M. F.
, “The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments
,” J. Fluid Mech.
54
(02
), 263
–288
(1972
).45.
Robinson
, S. K.
, “Coherent motions in the turbulent boundary layer
,” Annu. Rev. Fluid Mech.
23
, 601
–39
(1991
).46.
Seo
, J.
, Garcia-Mayoral
, R.
, and Mani
, A.
, “Pressure fluctuations and interfacial robustness in turbulent flows over superhydrophobic surfaces
,” J. Fluid Mech.
783
, 448
–473
(2015
).47.
Seo
, J.
and Mani
, A.
, “On the scaling of the slip velocity in turbulent flows over superhydrophobic surfaces
,” Phys. Fluids
28
, 025110
(2016
).48.
Stroh
, A.
, Hasegawa
, Y.
, Kriegseis
, J.
, and Frohnapfel
, B.
, “Secondary vortices over surfaces with spanwise varying drag
,” J. Turbul.
17
(12
), 1142
–1158
(2016
).49.
Tennekes
, H.
and Lumley
, J. L.
, A First Course in Turbulence
(MIT Press
, 1972
).50.
Tomkins
, C. D.
and Adrian
, R. J.
, “Spanwise structure and scale growth in turbulent boundary layers
,” J. Fluid Mech.
490
, 37
–74
(2003
).51.
Türk
, S.
, Daschiel
, G.
, Stroh
, A.
, Hasegawa
, Y.
, and Frohnapfel
, B.
, “Turbulent flow over superhydrophobic surfaces with streamwise grooves
,” J. Fluid Mech.
747
, 186
–217
(2014
).52.
Watanabe
, K.
, Udagawa
Y., and Udagawa
, H.
, “Drag reduction of Newtonian fluid in a circular pipe with a highly water-repellent wall
,” J. Fluid Mech.
381
, 225
–238
(1999
).53.
Watanabe
, S.
, Mamori
, H.
, and Fukagata
, K.
, “Drag-reducing performance of obliquely aligned superhydrophobic surface in turbulent channel flow
,” Fluid Dyn. Res.
49
(2
), 025501
(2017
).54.
Wei
, T.
and Willmarth
, W. W.
, “Modifying turbulent structure with drag-reducing polymer additives in turbulent channel flows
,” J. Fluid Mech.
245
, 619
–641
(1992
).55.
Wei
, T.
, Fife
, P.
, Klewicki
, J.
, and McMurtry
, P.
, “Properties of the mean momentum balance in turbulent boundary layer, pipe, and channel flows
,” J. Fluid Mech.
522
, 303
–327
(2005
).56.
Wu
, X.
, Baltzer
, J. R.
, and Adrian
, R. J.
, “Direct numerical simulation of a 30R long turbulent pipe flow at R+ = 685: Large- and very large-scale motions
,” J. Fluid Mech.
698
, 235
(2012
).57.
Yoon
, M.
, Ahn
, J.
, Hwang
, J.
, and Sung
, H. J.
, “Contribution of velocity-vorticity correlations to the frictional drag in wall-bounded turbulent flows
,” Phys. Fluids
28
(8
), 081702
(2016
).58.
Zhang
, J.
, Tian
, H.
, Yao
, Z.
, Hao
, P.
, and Jiang
, N.
, “Evolutions of hairpin vortexes over a superhydrophobic surface in turbulent boundary layer flow
,” Phys. Fluids
28
, 095106
(2016
).59.
Zhao
, J.
, Du
, X.
, and Shi
, X.
, “Experimental research on friction-reduction with super-hydrophobic surfaces
,” J. Mar. Sci. Appl.
6
, 58
–61
(2007
).60.
Zhou
, J.
, Adrian
, R. J.
, Balachandar
, S.
, and Kendall
, T. M.
, “Mechanisms for generating coherent packets of hairpin vortices
,” J. Fluid Mech.
387
, 353
–396
(1999
).© 2017 Author(s).
2017
Author(s)
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