The penguin is the fastest underwater swimmer among the wing-propelled diving birds. To figure out the mechanism for its excellent swimming, the hydrodynamic performance of a penguin wing is numerically investigated using an immersed boundary method with the incompressible flow solver. This study examines the effects of feathering, flapping, and Strouhal number (St) under preset motion. Results indicate that feathering is the primary contributor to thrust generation. The change in angle of attack (AoA) can qualitatively reflect the change in lift but not thrust. Therefore, a new variable, angle of thrust (AoT, αT), is introduced to effectively reflect the change of thrust across different kinematic parameters. Optimal feathering amplitude balances the decrease in AoA and the increase in feathering angle to achieve the highest AoT and thrust. Excessive feathering amplitude degrades the leading-edge vortex to shear layers, transforms the pressure side to the suction side, and ultimately causes negative thrust (drag). Spatial analysis of the thrust shows that the outer three-fifths of the wing are the primary source of thrust, contributing 85.4% of thrust generation at optimal feathering amplitude. Flapping amplitude has little impact on the optimal feathering amplitude. The optimal feathering amplitude increases linearly with the St number in the scope of examination, leading to larger thrust but lower swimming efficiency. Thus, a dimensionless number, Stm, is introduced to describe the optimal wing motion. This work provides new insights into the propulsion mechanism of aquatic swimmers with flapping–feathering wings and helps design novel bio-inspired aquatic vehicles.

1.
K.
Sato
,
K.
Shiomi
,
Y.
Watanabe
,
Y.
Watanuki
,
A.
Takahashi
, and
P. J.
Ponganis
, “
Scaling of swim speed and stroke frequency in geometrically similar penguins: They swim optimally to minimize cost of transport
,”
Proc. R. Soc. B
277
,
707
714
(
2010
).
2.
Y.
Ropert-Coudert
,
A.
Kato
,
R. P.
Wilson
, and
B.
Cannell
, “
Foraging strategies and prey encounter rate of free-ranging little penguins
,”
Mar. Biol.
149
,
139
(
2006
).
3.
J. F.
Schreer
,
K. M.
Kovacs
, and
R. J. O.
Hines
, “
Comparative diving patterns of pinnipeds and seabirds
,”
Ecol. Monogr.
71
,
137
(
2001
).
4.
Y. Y.
Watanabe
,
K.
Sato
,
Y.
Watanuki
,
A.
Takahashi
,
Y.
Mitani
,
M.
Amano
,
K.
Aoki
,
T.
Narazaki
,
T.
Iwata
,
S.
Minamikawa
, and
N.
Miyazaki
, “
Scaling of swim speed in breath-hold divers: Scaling of swim speed
,”
J. Anim. Ecol.
80
,
57
68
(
2011
).
5.
J.
Rafferty
,
Gentoo Penguin
(
Encyclopedia Britannica Inc.
,
2021
).
6.
K.
Sato
,
Y.
Naito
,
A.
Kato
, and
Y.
Niizuma
, “
Buoyancy and maximal diving depth in penguins: Do they control inhaling air volume?
,”
J. Exp. Biol.
205
,
1189
1197
(
2002
).
7.
D.
Weihs
, “
Dynamics of dolphin porpoising revisited
,”
Integr. Comp. Biol.
42
,
1071
1078
(
2002
).
8.
C. A.
Hui
, “
Penguin swimming. I. Hydrodynamics
,”
Physiol. Zool.
61
,
333
343
(
1988
).
9.
M. H.
Masud
,
M.
La Mantia
, and
P.
Dabnichki
, “
Estimate of Strouhal and Reynolds numbers for swimming penguins
,”
J. Avian Biol.
2022
,
e02886
.
10.
D. N.
Beal
and
P. R.
Bandyopadhyay
, “
A harmonic model of hydrodynamic forces produced by a flapping fin
,”
Exp. Fluids
43
,
675
682
(
2007
).
11.
R. W.
Blake
, “
The mechanics of Labriform locomotion: II. An analysis of the recovery stroke and the overall fin-beat cycle propulsive efficiency in the Angelfish
,”
J. Exp. Biol.
85
,
337
342
(
1980
).
12.
P. R.
Bandyopadhyay
, “
Swimming and flying in nature—The route toward applications: The Freeman scholar lecture
,”
J. Fluids Eng.
131
,
031801
(
2009
).
13.
N.
Kato
, “
Locomotion by mechanical pectoral fins
,”
J. Mar. Sci. Technol.
3
,
113
121
(
1998
).
14.
N.
Kato
and
T.
Inaba
, “
Control performance of fish robot with pectoral fins in horizontal plane
,” in
Proceedings of the International Symposium on Underwater Technology
(
IEEE
,
1998
), pp.
357
362
.
15.
R.
Bannasch
, “
Hydrodynamics of penguins-an experimental approach
,”
The Penguins; Ecology and Management
(
Surrey Beatty & Sons Pty Limited
,
1995
), pp.
141
176
.
16.
B. D.
Clark
and
W.
Bemis
, “
Kinematics of swimming of penguins at the Detroit Zoo
,”
J. Zool.
188
,
411
428
(
1979
).
17.
Y.
Shen
,
N.
Harada
,
S.
Katagiri
, and
H.
Tanaka
, “
Biomimetic realization of a robotic penguin wing: Design and thrust characteristics
,”
IEEE/ASME Trans. Mechatron.
26
,
2350
2361
(
2021
).
18.
N.
Harada
,
T.
Oura
,
M.
Maeda
,
Y.
Shen
,
D. M.
Kikuchi
, and
H.
Tanaka
, “
Kinematics and hydrodynamics analyses of swimming penguins: Wing bending improves propulsion performance
,”
J. Exp. Biol.
224
,
jeb242140
(
2021
).
19.
M.
Maeda
,
N.
Harada
, and
H.
Tanaka
, “
Hydrodynamics of gliding penguin flipper suggests the adjustment of sweepback with swimming speeds
,” bioRxiv (
2021
).
20.
Y.
Shen
and
H.
Tanaka
, “
Experimental analysis of the sweepback angle effect on the thrust generation of a robotic penguin wing
,”
Bioinspiration Biomimetics
18
,
026007
(
2023
).
21.
X.
Lin
,
J.
Wu
, and
T.
Zhang
, “
Effect of torsional spring and shape on the performance of bioinspired caudal fin
,”
Phys. Fluids
33
,
071903
(
2021
).
22.
L.
Yao
,
C.
Hefler
,
W.
Shyy
, and
H. H.
Qiu
, “
Effects of gradual flexibility and trailing edge shape on propulsive performance of pitching fins
,”
Phys. Fluids
33
,
071910
(
2021
).
23.
X.
Lang
,
B.
Song
, and
W.
Yang
, “
Effect of spanwise folding on the aerodynamic performance of three dimensional flapping flat wing
,”
Phys. Fluids
34
,
021906
(
2022
).
24.
R. R.
Harbig
,
J.
Sheridan
, and
M. C.
Thompson
, “
The role of advance ratio and aspect ratio in determining leading-edge vortex stability for flapping flight
,”
J. Fluid Mech.
751
,
71
105
(
2014
).
25.
C. P.
Ellington
,
C.
van den Berg
,
A. P.
Willmott
, and
A. L. R.
Thomas
, “
Leading-edge vortices in insect flight
,”
Nature
384
,
626
630
(
1996
).
26.
F. T.
Muijres
,
L. C.
Johansson
, and
A.
Hedenström
, “
Leading edge vortex in a slow-flying passerine
,”
Biol. Lett.
8
,
554
557
(
2012
).
27.
D.
Lentink
and
M. H.
Dickinson
, “
Rotational accelerations stabilize leading edge vortices on revolving fly wings
,”
J. Exp. Biol.
212
,
2705
2719
(
2009
).
28.
G.
Liu
,
Y.
Ren
,
H.
Dong
,
O.
Akanyeti
,
J. C.
Liao
, and
G. V.
Lauder
, “
Computational analysis of vortex dynamics and performance enhancement due to body–fin and fin–fin interactions in fish-like locomotion
,”
J. Fluid Mech.
829
,
65
88
(
2017
).
29.
L. C.
Johansson
and
R. Å.
Norberg
, “
Delta-wing function of webbed feet gives hydrodynamic lift for swimming propulsion in birds
,”
Nature
424
,
65
68
(
2003
).
30.
J.
Wang
,
Y.
Ren
,
C.
Li
, and
H.
Dong
, “
Computational investigation of wing-body interaction and its lift enhancement effect in hummingbird forward flight
,”
Bioinspiration Biomimetics
14
,
046010
(
2019
).
31.
I.
Borazjani
and
M.
Daghooghi
, “
The fish tail motion forms an attached leading edge vortex
,”
Proc. R. Soc. B
280
,
20122071
(
2013
).
32.
P.
Han
,
G. V.
Lauder
, and
H.
Dong
, “
Hydrodynamics of median-fin interactions in fish-like locomotion: Effects of fin shape and movement
,”
Phys. Fluids
32
,
011902
(
2020
).
33.
D.
Zhang
,
Q.-G.
Huang
,
G.
Pan
,
L.-M.
Yang
, and
W.-X.
Huang
, “
Vortex dynamics and hydrodynamic performance enhancement mechanism in batoid fish oscillatory swimming
,”
J. Fluid Mech.
930
,
A28
(
2022
).
34.
Y.
Gong
,
J.
Wang
,
W.
Zhang
,
J. J.
Socha
, and
H.
Dong
, “
Computational analysis of vortex dynamics and aerodynamic performance in flying-snake-like gliding flight with horizontal undulation
,”
Phys. Fluids
34
,
121907
(
2022
).
35.
C.
Li
,
H.
Dong
, and
B.
Cheng
, “
Tip vortices formation and evolution of rotating wings at low Reynolds numbers
,”
Phys. Fluids
32
,
021905
(
2020
).
36.
C.
Li
and
H.
Dong
, “
Three-dimensional wake topology and propulsive performance of low-aspect-ratio pitching-rolling plates
,”
Phys. Fluids
28
,
071901
(
2016
).
37.
H.
Luo
,
H.
Dai
,
P. J.
Ferreira de Sousa
, and
B.
Yin
, “
On the numerical oscillation of the direct-forcing immersed-boundary method for moving boundaries
,”
Comput. Fluids
56
,
61
76
(
2012
).
38.
S.
Huang
,
D.
Guo
,
Y.
Wang
,
G.
Yang
, and
B.
Yin
, “
Hydrodynamics of morphology for thunniform swimmers: Effects of the posterior body shape
,”
Ocean Eng.
272
,
113866
(
2023
).
39.
G.
Liu
,
H.
Dong
, and
C.
Li
, “
Vortex dynamics and new lift enhancement mechanism of wing–body interaction in insect forward flight
,”
J. Fluid Mech.
795
,
634
651
(
2016
).
40.
R.
Mittal
,
H.
Dong
,
M.
Bozkurttas
,
F.
Najjar
,
A.
Vargas
, and
A.
von Loebbecke
, “
A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries
,”
J. Comput. Phys.
227
,
4825
4852
(
2008
).
41.
R. T.
Jantzen
,
K.
Taira
,
K. O.
Granlund
, and
M. V.
Ol
, “
Vortex dynamics around pitching plates
,”
Phys. Fluids
26
,
053606
(
2014
).
42.
M.
Bozkurttas
,
R.
Mittal
,
H.
Dong
,
G. V.
Lauder
, and
P.
Madden
, “
Low-dimensional models and performance scaling of a highly deformable fish pectoral fin
,”
J. Fluid Mech.
631
,
311
342
(
2009
).
43.
J. M.
Anderson
,
K.
Streitlien
,
D. S.
Barrett
, and
M. S.
Triantafyllou
, “
Oscillating foils of high propulsive efficiency
,”
J. Fluid Mech.
360
,
41
72
(
1998
).
44.
D.
Zhang
,
G.
Pan
,
L.
Chao
, and
Y.
Zhang
, “
Effects of Reynolds number and thickness on an undulatory self-propelled foil
,”
Phys. Fluids
30
,
071902
(
2018
).
45.
J.
Eldredge
,
C.
Wang
, and
M.
Ol
, “
A Computational Study of a Canonical Pitch-Up, Pitch-Down Wing Maneuver
,” AIAA Paper No. AIAA 2009-3687,
2009
.
46.
J.
Song
,
B. W.
Tobalske
,
D. R.
Powers
,
T. L.
Hedrick
, and
H.
Luo
, “
Three-dimensional simulation for fast forward flight of a calliope hummingbird
,”
R. Soc. Open Sci.
3
,
160230
(
2016
).
47.
R. G.
Bottom
II
,
I.
Borazjani
,
E. L.
Blevins
, and
G. V.
Lauder
, “
Hydrodynamics of swimming in stingrays: Numerical simulations and the role of the leading-edge vortex
,”
J. Fluid Mech.
788
,
407
443
(
2016
).
48.
I.
Borazjani
and
F.
Sotiropoulos
, “
On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming
,”
J. Exp. Biol.
213
,
89
107
(
2010
).
49.
P. R.
Bandyopadhyay
,
D. N.
Beal
, and
A.
Menozzi
, “
Biorobotic insights into how animals swim
,”
J. Exp. Biol.
211
,
206
214
(
2008
).
50.
A. K.
De
and
S.
Sarkar
, “
Dependence of wake structure on pitching frequency behind a thin panel at Re = 1000
,”
J. Fluid Mech.
924
,
A33
(
2021
).
51.
K.
Zhang
,
S.
Hayostek
,
M.
Amitay
,
W.
He
,
V.
Theofilis
, and
K.
Taira
, “
On the formation of three-dimensional separated flows over wings under tip effects
,”
J. Fluid Mech.
895
,
A9
(
2020
).
52.
T.
Maxworthy
, “
The fluid dynamics of insect flight
,”
Annu. Rev. Fluid Mech.
13
,
329
350
(
1981
).
53.
D.
Zhang
,
J.-D.
Zhang
, and
W.-X.
Huang
, “
Physical models and vortex dynamics of swimming and flying: A review
,”
Acta Mech.
233
,
1249
1288
(
2022
).
54.
K.
Ohmi
,
M.
Coutanceau
,
T. P.
Loc
, and
A.
Dulieu
, “
Vortex formation around an oscillating and translating airfoil at large incidences
,”
J. Fluid Mech.
211
,
37
60
(
1990
).
55.
L.
Chen
,
L.
Wang
,
C.
Zhou
,
J.
Wu
, and
B.
Cheng
, “
Effects of Reynolds number on leading-edge vortex formation dynamics and stability in revolving wings
,”
J. Fluid Mech.
931
,
A13
(
2022
).
56.
M. S.
Triantafyllou
and
G. S.
Triantafyllou
, “
An efficient swimming machine
,”
Sci. Am.
272
,
64
70
(
1995
).
57.
M.
Triantafyllou
,
A.
Techet
, and
F.
Hover
, “
Review of experimental work in biomimetic foils
,”
IEEE J. Oceanic Eng.
29
,
585
594
(
2004
).
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