In the present study, the effects of surface slip on the hydrodynamics and flow around a two-dimensional National Advisory Committee for Aeronautics 0012 hydrofoil are systematically investigated by numerical methods. The objective is to fully understand the effects of surface slip on the streamlined body. Three slip positions (both surfaces, the upper surface, the lower surface) and eight slip lengths (in a wide range from 1 to 500 μm) under 0°–10° angles of attack are fully investigated at a moderate Reynolds number of 1.0  × 106. Surface slip has been found to increase lift and reduce drag by postponing the flow transition, laminar separation bubble, and flow separation on the hydrofoil surface under both surfaces and the upper surface slip conditions. Slip has also been found to induce upshift of the mean velocity profile, decrease the displacement thickness, and mitigate the turbulent kinetic energy in the flow field. However, counterintuitive phenomenon occurs under the lower surface slip condition, where the total drag of the hydrofoil is increased compared to that under the no slip condition. Total drag increase is found mainly due to the increase in the pressure drag under small slip lengths and relatively large angles of attack. Flow maps demonstrating the complex interaction between different surface slip conditions and the flow field are further presented. The results suggest that surface slip can not only reduce drag, but also increase the drag of the streamlined body, which shall provide valuable insights for practical applications of slippery materials.

1.
H.
Choi
,
W.-P.
Jeon
, and
J.
Kim
, “
Control of flow over a bluff body
,”
Annu. Rev. Fluid Mech.
40
,
113
139
(
2008
).
2.
H.
Choi
,
J.
Lee
, and
H.
Park
, “
Aerodynamics of heavy vehicles
,”
Annu. Rev. Fluid Mech.
46
,
441
468
(
2014
).
3.
G.
Liu
,
Z.
Yuan
,
Z.
Qiu
,
S.
Feng
,
Y.
Xie
,
D.
Leng
, and
X.
Tian
, “
A brief review of bio-inspired surface technology and application toward underwater drag reduction
,”
Ocean Eng.
199
,
106962
(
2020
).
4.
D.
Pan
,
X.
Xu
,
B.
Liu
,
H.
Xu
, and
X.
Wang
, “
A review on drag reduction technology: Focusing on amphibious vehicles
,”
Ocean Eng.
280
,
114618
(
2023
).
5.
M.
Zhu
and
L.
Ma
, “
A review of recent advances in the effects of surface and interface properties on marine propellers
,”
Friction
12
(
2
),
185
214
(
2024
).
6.
V. I.
Kornilov
, “
Current state and prospects of researches on the control of turbulent boundary layer by air blowing
,”
Prog. Aeosp. Sci.
76
,
1
23
(
2015
).
7.
C.
Vignon
,
J.
Rabault
, and
R.
Vinuesa
, “
Recent advances in applying deep reinforcement learning for flow control: Perspectives and future directions
,”
Phys. Fluids
35
(
3
),
031301
(
2023
).
8.
K.
Fukagata
,
K.
Iwamoto
, and
Y.
Hasegawa
, “
Turbulent drag reduction by streamwise traveling waves of wall-normal forcing
,”
Annu. Rev. Fluid Mech.
56
,
69
90
(
2024
).
9.
C.
Lee
,
C. H.
Choi
, and
C. J.
Kim
, “
Superhydrophobic drag reduction in laminar flows: A critical review
,”
Exp. Fluids
57
(
12
),
176
(
2016
).
10.
H.
Park
,
C. H.
Choi
, and
C. J.
Kim
, “
Superhydrophobic drag reduction in turbulent flows: A critical review
,”
Exp. Fluids
62
(
11
),
229
(
2021
).
11.
M.
Liu
and
L.
Ma
, “
Drag reduction methods at solid-liquid interfaces
,”
Friction
10
(
4
),
491
515
(
2022
).
12.
W.
Barthlott
and
C.
Neinhuis
, “
Purity of the sacred lotus, or escape from contamination in biological surfaces
,”
Planta
202
(
1
),
1
8
(
1997
).
13.
J. P.
Rothstein
, “
Slip on superhydrophobic surfaces
,”
Annu. Rev. Fluid Mech.
42
,
89
109
(
2010
).
14.
Y.
Xue
,
P.
Lv
,
H.
Lin
, and
H.
Duan
, “
Underwater superhydrophobicity: Stability, design and regulation, and applications
,”
Appl. Mech. Rev.
68
(
3
),
030803
(
2016
).
15.
M. A.
Samaha
and
M.
Gad-el-Hak
, “
Slippery surfaces: A decade of progress
,”
Phys. Fluids
33
(
7
),
071301
(
2021
).
16.
C. L. M. H.
Navier
, “
Sur les lois du mouvement des fluides
,”
Mem. Acad. R. Sci. Inst. Fr.
6
,
389
440
(
1823
).
17.
V. S. J.
Craig
,
C.
Neto
, and
D. R. M.
Williams
, “
Shear-dependent boundary slip in an aqueous Newtonian liquid
,”
Phys. Rev. Lett.
87
(
5
),
054504
(
2001
).
18.
C. H.
Choi
,
K. J. A.
Westin
, and
K. S.
Breuer
, “
Apparent slip flows in hydrophilic and hydrophobic microchannels
,”
Phys. Fluids
15
(
10
),
2897
2902
(
2003
).
19.
P.
Joseph
and
P.
Tabeling
, “
Direct measurement of the apparent slip length
,”
Phys. Rev. E
71
(
3
),
035303
(
2005
).
20.
R.
Pit
,
H.
Hervet
, and
L.
Léger
, “
Friction and slip of a simple liquid at a solid surface
,”
Tribol. Lett.
7
,
147
152
(
1999
).
21.
R.
Pit
,
H.
Hervet
, and
L.
Leger
, “
Direct experimental evidence of slip in hexadecane: Solid interfaces
,”
Phys. Rev. Lett.
85
(
5
),
980
983
(
2000
).
22.
Y. X.
Zhu
and
S.
Granick
, “
Rate-dependent slip of Newtonian liquid at smooth surfaces
,”
Phys. Rev. Lett.
87
(
9
),
096105
(
2001
).
23.
D. C.
Tretheway
and
C. D.
Meinhart
, “
Apparent fluid slip at hydrophobic microchannel walls
,”
Phys. Fluids
14
(
3
),
L9
L12
(
2002
).
24.
J.
Ou
,
B.
Perot
, and
J. P.
Rothstein
, “
Laminar drag reduction in microchannels using ultrahydrophobic surfaces
,”
Phys. Fluids
16
(
12
),
4635
4643
(
2004
).
25.
C.
Lee
and
C. J.
Kim
, “
Underwater restoration and retention of gases on superhydrophobic surfaces for drag reduction
,”
Phys. Rev. Lett.
106
(
1
),
014502
(
2011
).
26.
C.
Lee
,
C. H.
Choi
, and
C. J.
Kim
, “
Structured surfaces for a giant liquid slip
,”
Phys. Rev. Lett.
101
(
6
),
064501
(
2008
).
27.
C.
Lee
and
C. J.
Kim
, “
Maximizing the giant liquid slip on superhydrophobic microstructures by nanostructuring their sidewalls
,”
Langmuir
25
(
21
),
12812
12818
(
2009
).
28.
S.
Srinivasan
,
W.
Choi
,
K. C.
Park
,
S. S.
Chhatre
,
R. E.
Cohen
, and
G. H.
McKinley
, “
Drag reduction for viscous laminar flow on spray-coated non-wetting surfaces
,”
Soft Matter
9
(
24
),
5691
5702
(
2013
).
29.
D.
Byun
,
J.
Kim
,
H. S.
Ko
, and
H. C.
Park
, “
Direct measurement of slip flows in superhydrophobic microchannels with transverse grooves
,”
Phys. Fluids
20
(
11
),
113601
(
2008
).
30.
W.
Abu Rowin
and
S.
Ghaemi
, “
Streamwise and spanwise slip over a superhydrophobic surface
,”
J. Fluid Mech.
870
,
1127
1157
(
2019
).
31.
R. J.
Daniello
,
N. E.
Waterhouse
, and
J. P.
Rothstein
, “
Drag reduction in turbulent flows over superhydrophobic surfaces
,”
Phys. Fluids
21
(
8
),
085103
(
2009
).
32.
T.
Min
and
J.
Kim
, “
Effects of hydrophobic surface on skin-friction drag
,”
Phys. Fluids
16
(
7
),
L55
L58
(
2004
).
33.
T.
Min
and
J.
Kim
, “
Effects of hydrophobic surface on stability and transition
,”
Phys. Fluids
17
(
10
),
108106
(
2005
).
34.
A.
Busse
and
N. D.
Sandham
, “
Influence of an anisotropic slip-length boundary condition on turbulent channel flow
,”
Phys. Fluids
24
(
5
),
055111
(
2012
).
35.
F.
Picella
,
J. C.
Robinet
, and
S.
Cherubini
, “
Laminar-turbulent transition in channel flow with superhydrophobic surfaces modelled as a partial slip wall
,”
J. Fluid Mech.
881
,
462
497
(
2019
).
36.
Y.
Xiao
,
L. S.
Zhang
, and
J. J.
Tao
, “
Slip boundary effect on the critical Reynolds number of subcritical transition in channel flow
,”
Theor. Appl. Mech. Lett.
13
(
2
),
100431
(
2023
).
37.
M.
Yoon
and
H. J.
Sung
, “
Wall-attached structures in a drag-reduced turbulent channel flow
,”
J. Fluid Mech.
943
,
A14
(
2022
).
38.
E. A.
Davis
and
J. S.
Park
, “
Dynamics of laminar and transitional flows over slip surfaces: Effects on the laminar-turbulent separatrix
,”
J. Fluid Mech.
894
,
A16
(
2020
).
39.
A.
Jouin
,
S.
Cherubini
, and
J. C.
Robinet
, “
Turbulent transition in a channel with superhydrophobic walls: Anisotropic slip and shear misalignment effects
,”
J. Fluid Mech.
980
,
A49
(
2024
).
40.
E.
Lauga
and
H. A.
Stone
, “
Effective slip in pressure-driven Stokes flow
,”
J. Fluid Mech.
489
,
55
77
(
2003
).
41.
J.
Davies
,
D.
Maynes
,
B. W.
Webb
, and
B.
Woolford
, “
Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs
,”
Phys. Fluids
18
(
8
),
087110
(
2006
).
42.
M. B.
Martell
,
J. B.
Perot
, and
J. P.
Rothstein
, “
Direct numerical simulations of turbulent flows over superhydrophobic surfaces
,”
J. Fluid Mech.
620
,
31
41
(
2009
).
43.
M. B.
Martell
,
J. P.
Rothstein
, and
J. B.
Perot
, “
An analysis of superhydrophobic turbulent drag reduction mechanisms using direct numerical simulation
,”
Phys. Fluids
22
(
6
),
065102
(
2010
).
44.
H.
Park
,
H.
Park
, and
J.
Kim
, “
A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow
,”
Phys. Fluids
25
(
11
),
110815
(
2013
).
45.
R.
Costantini
,
J. P.
Mollicone
, and
F.
Battista
, “
Drag reduction induced by superhydrophobic surfaces in turbulent pipe flow
,”
Phys. Fluids
30
(
2
),
025102
(
2018
).
46.
C.
Schönecker
,
T.
Baier
, and
S.
Hardt
, “
Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state
,”
J. Fluid Mech.
740
,
168
195
(
2014
).
47.
P.
Dey
,
S. K.
Saha
, and
S.
Chakraborty
, “
Microgroove geometry dictates slippery hydrodynamics on superhydrophobic substrates
,”
Phys. Fluids
30
(
12
),
122007
(
2018
).
48.
F.
Picella
,
J. C.
Robinet
, and
S.
Cherubini
, “
On the influence of the modelling of superhydrophobic surfaces on laminar-turbulent transition
,”
J. Fluid Mech.
901
,
A15
(
2020
).
49.
J.
Seo
and
A.
Mani
, “
On the scaling of the slip velocity in turbulent flows over superhydrophobic surfaces
,”
Phys. Fluids
28
(
2
),
025110
(
2016
).
50.
J. Y.
Zhu
,
F. L.
Zhu
,
W. D.
Su
,
S. F.
Zou
,
L. Q.
Liu
,
Y. P.
Shi
, and
J. Z.
Wu
, “
A vorticity dynamics view of “effective slip boundary” with application to foil-flow control
,”
Phys. Fluids
26
(
12
),
123602
(
2014
).
51.
D.
You
and
P.
Moin
, “
Effects of hydrophobic surfaces on the drag and lift of a circular cylinder
,”
Phys. Fluids
19
(
8
),
081701
(
2007
).
52.
D.
Legendre
,
E.
Lauga
, and
J.
Magnaudet
, “
Influence of slip on the dynamics of two-dimensional wakes
,”
J. Fluid Mech.
633
,
437
447
(
2009
).
53.
P.
Muralidhar
,
N.
Ferrer
,
R.
Daniello
, and
J. P.
Rothstein
, “
Influence of slip on the flow past superhydrophobic circular cylinders
,”
J. Fluid Mech.
680
,
459
476
(
2011
).
54.
R.
Daniello
,
P.
Muralidhar
,
N.
Carron
,
M.
Greene
, and
J. P.
Rothstein
, “
Influence of slip on vortex-induced motion of a superhydrophobic cylinder
,”
J. Fluids Struct.
42
,
358
368
(
2013
).
55.
J. C.
Brennan
,
D. J.
Fairhurst
,
R. H.
Morris
,
G.
McHale
, and
M. I.
Newton
, “
Investigation of the drag reducing effect of hydrophobized sand on cylinders
,”
J. Phys. D
47
(
20
),
205302
(
2014
).
56.
N.
Kim
,
H.
Kim
, and
H.
Park
, “
An experimental study on the effects of rough hydrophobic surfaces on the flow around a circular cylinder
,”
Phys. Fluids
27
(
8
),
085113
(
2015
).
57.
P.
Sooraj
,
M. S.
Ramagya
,
M. H.
Khan
,
A.
Sharma
, and
A.
Agrawal
, “
Effect of superhydrophobicity on the flow past a circular cylinder in various flow regimes
,”
J. Fluid Mech.
897
,
A21
(
2020
).
58.
D. D.
Li
,
S. C.
Li
,
Y. H.
Xue
,
Y. T.
Yang
,
W. D.
Su
,
Z. H.
Xia
,
Y. P.
Shi
,
H.
Lin
, and
H. L.
Duan
, “
The effect of slip distribution on flow past a circular cylinder
,”
J. Fluids Struct.
51
,
211
224
(
2014
).
59.
Y.
Yu
,
L. M.
Yang
,
Y.
Gao
, and
G. X.
Hou
, “
Lattice Boltzmann investigation of the influence of slip distributions on the flow past a diamond cylinder at low-Reynolds-number
,”
Phys. Fluids
33
(
7
),
073611
(
2021
).
60.
Y.
Yu
,
L. M.
Yang
,
Y.
Gao
, and
G. X.
Hou
, “
Lattice Boltzmann study of low-Reynolds-number flow past square cylinders with varying slip distributions
,”
Ocean Eng.
236
,
109425
(
2021
).
61.
A. K.
Balasubramanian
,
A. C.
Miller
, and
O. K.
Rediniotis
, “
Microstructured hydrophobic skin for hydrodynamic drag reduction
,”
AIAA J.
42
(
2
),
411
414
(
2004
).
62.
S.
Gogte
,
P.
Vorobieff
,
R.
Truesdell
,
A.
Mammoli
,
F.
van Swol
,
P.
Shah
, and
C. J.
Brinker
, “
Effective slip on textured superhydrophobic surfaces
,”
Phys. Fluids
17
(
5
),
051701
(
2005
).
63.
J.
Lee
,
H.
Kim
, and
H.
Park
, “
Effects of superhydrophobic surfaces on the flow around an NACA0012 hydrofoil at low Reynolds numbers
,”
Exp. Fluids
59
(
7
),
111
(
2018
).
64.
P.
Sooraj
,
S.
Jain
, and
A.
Agrawal
, “
Flow over hydrofoils with varying hydrophobicity
,”
Exp. Therm. Fluid Sci.
102
,
479
492
(
2019
).
65.
H.
Choi
,
J.
Lee
, and
H.
Park
, “
Wake structures behind a rotor with superhydrophobic-coated blades at low Reynolds number
,”
Phys. Fluids
31
(
1
),
015102
(
2019
).
66.
S. R.
Mallah
,
P.
Sooraj
,
A.
Sharma
, and
A.
Agrawal
, “
Effect of superhydrophobicity on the wake of a pitching foil across various Strouhal numbers
,”
Phys. Fluids
33
(
11
),
111905
(
2021
).
67.
K.
Wang
,
L. M.
Yang
,
Y.
Yu
, and
G. X.
Hou
, “
Influence of slip boundary on the hydrofoil with a curved slip boundary condition for the lattice Boltzmann method
,”
Phys. Fluids
30
(
12
),
123601
(
2018
).
68.
E. T.
Katsuno
,
J. L. D.
Dantas
, and
E. C. N.
Silva
, “
Low-friction fluid flow surface design using topology optimization
,”
Struct. Multidiscip. Optim.
62
(
6
),
2915
2933
(
2020
).
69.
A.
Shahsavari
,
A.
Nejat
,
E.
Climent
, and
S. F.
Chini
, “
Using genetic algorithm to find the optimum piecewise superhydrophobic pattern maximizing the lift to drag ratio on a SD 7003 foil at different working conditions
,”
Ocean Eng.
278
,
114438
(
2023
).
70.
E. T.
Katsuno
,
J. L. D.
Dantas
, and
E. C. N.
Silva
, “
Topology optimization of low-friction painting distribution on a marine propeller
,”
Struct. Multidiscip. Optim.
65
(
9
),
269
(
2022
).
71.
H.
Li
,
S.
Ji
,
X.
Tan
,
Z.
Li
,
Y.
Xiang
,
P.
Lv
, and
H.
Duan
, “
Effect of Reynolds number on drag reduction in turbulent boundary layer flow over liquid-gas interface
,”
Phys. Fluids
32
(
12
),
122111
(
2020
).
72.
Z.
Feng
and
Q.
Ye
, “
Turbulent boundary layer over porous media with wall-normal permeability
,”
Phys. Fluids
35
(
9
),
095111
(
2023
).
73.
J.
Hu
and
Z.
Yao
, “
Drag reduction of turbulent boundary layer over sawtooth riblet surface with superhydrophobic coat
,”
Phys. Fluids
35
(
1
),
015104
(
2023
).
74.
B.
Liu
and
Y.
Zhang
, “
A numerical study on the natural transition locations in the flat-plate boundary layers on superhydrophobic surfaces
,”
Phys. Fluids
32
(
12
),
124103
(
2020
).
75.
B.
Liu
,
J.
Liu
, and
Y.
Zhang
, “
Numerical investigation of the natural transition in boundary layers on underwater axisymmetric bodies with superhydrophobic surfaces
,”
Phys. Fluids
36
(
1
),
014124
(
2024
).
76.
P. B. S.
Lissaman
, “
Low-Reynolds-number airfoils
,”
Annu. Rev. Fluid Mech.
15
,
223
239
(
1983
).
77.
T. J.
Mueller
and
J. D.
DeLaurier
, “
Aerodynamics of small vehicles
,”
Annu. Rev. Fluid Mech.
35
,
89
111
(
2003
).
78.
S.
Wang
,
Y.
Zhou
,
M. M.
Alam
, and
H.
Yang
, “
Turbulent intensity and Reynolds number effects on an airfoil at low Reynolds numbers
,”
Phys. Fluids
26
(
11
),
115107
(
2014
).
79.
R.
Bai
,
J.
Li
,
F.
Zeng
, and
C.
Yan
, “
Mechanism and performance differences between the SSG/LRR-ω and SST turbulence models in separated flows
,”
Aerospace
9
(
1
),
20
(
2021
).
80.
S.
Rodriguez
,
Applied Computational Fluid Dynamics and Turbulence Modeling: Practical Tools, Tips and Techniques
(
Springer
,
Berlin/Heidelberg, Germany
,
2019
).
81.
C.
Yan
,
F.
Qu
,
Y.
Zhao
,
J.
Yu
,
C.
Wu
, and
S.
Zhang
, “
Review of development and challenges for physical modeling and numerical scheme of CFD in aeronautics and astronautics
,”
Acta Aerodyn. Sin.
38
(
5
),
829
857
(
2020
).
82.
R.
Wang
and
Z. L.
Xiao
, “
Transition effects on flow characteristics around a static two-dimensional airfoil
,”
Phys. Fluids
32
(
3
),
035113
(
2020
).
83.
Y.
Liu
,
P. F.
Li
, and
K. Y.
Jiang
, “
Comparative assessment of transitional turbulence models for airfoil aerodynamics in the low Reynolds number range
,”
J. Wind Eng. Ind. Aerodyn.
217
,
104726
(
2021
).
84.
F. R.
Menter
,
P. E.
Smirnov
,
T.
Liu
, and
R.
Avancha
, “
A one-equation local correlation-based transition model
,”
Flow. Turbul. Combust.
95
(
4
),
583
619
(
2015
).
85.
R. E.
Sheldahl
and
P. C.
Klimas
,
Report No. SAND80-2114
(
Sandia National Laboratories
,
1981
).
86.
I. H.
Abbott
and
A. E.
Von Doenhoeff
,
Theory of Wing Sections: Including a Summary of Airfoil Data
(
Dover
,
New York
,
1959
).
87.
S.
Gupta
,
J.
Zhao
,
A.
Sharma
,
A.
Agrawal
,
K.
Hourigan
, and
M. C. C.
Thompson
, “
Two- and three-dimensional wake transitions of a NACA0012 airfoil
,”
J. Fluid Mech.
954
,
A26
(
2023
).
88.
R. A.
Piziali
,
Report No. USAATCOM TR-94-A-011
(
Ames Research Center
,
1994
).
89.
J.
Szydlowski
and
M.
Costes
, “
Simulation of flow around a static and oscillating in pitch NACA0015 airfoil using URANS and DES
,” in
ASME 2004 Heat Transfer/Fluids Engineering Summer Conference
,
Charlotte, NC
(
2004
).
90.
E.
Bakhtiari
, “
Super-hydrophobicity effects on performance of a dynamic wind turbine blade element under yaw loads
,”
Renewable Energy
140
,
539
551
(
2019
).
91.
E.
Bakhtiari
,
K.
Gharali
, and
F.
Chini
, “
Corrigendum to “Super-hydrophobicity effects on performance of a dynamic wind turbine blade element under yaw load” [Renew. Energy 140 (2019) 539–551]
,”
Renewable Energy
147
,
2528
2528
(
2020
).
92.
B.
Mele
and
R.
Tognaccini
, “
Slip length-based boundary condition for modeling drag reduction devices
,”
AIAA J.
56
(
9
),
3478
3490
(
2018
).
93.
B.
Mele
,
R.
Tognaccini
,
P.
Catalano
, and
D.
de Rosa
, “
Effect of body shape on riblets performance
,”
Phys. Rev. Fluids
5
(
12
),
124609
(
2020
).
94.
T.
Knopp
,
N.
Reuther
,
M.
Novara
,
D.
Schanz
,
E.
Schülein
,
A.
Schröder
, and
C. J.
Kähler
, “
Experimental analysis of the log law at adverse pressure gradient
,”
J. Fluid Mech.
918
,
A17
(
2021
).
95.
J.
Jiménez
, “
Turbulent flows over rough walls
,”
Annu. Rev. Fluid Mech.
36
,
173
196
(
2004
).
96.
G. Z.
Ma
,
C. X.
Xu
,
H. J.
Sung
, and
W. X.
Huang
, “
Scaling of rough-wall turbulence by the roughness height and steepness
,”
J. Fluid Mech.
900
,
R7
(
2020
).
97.
G. Z.
Ma
,
C. X.
Xu
,
H. J.
Sung
, and
W. X.
Huang
, “
Scaling of rough-wall turbulence in a transitionally rough regime
,”
Phys. Fluids
34
(
3
),
031701
(
2022
).

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