Identical tandem flippers of plesiosaurs, which are unique among all animals, have been a source of debate regarding the role of hind flippers in their locomotion. Here, inspired by the kinematics of plesiosaur flippers, the effect of the amplitude ratio on the propulsive performance of in-line tandem pitching foils is investigated through a series of particle image velocimetry experiments. Three leader-to-follower amplitude ratios are considered for the foils pitching over a range of 02π phase difference. For the first time, it is shown that the amplitude ratio can significantly affect the performance of the hind foil at spacing larger than one chord length. It is found that the thrust generation of the hind foil at the optimum phase difference augments by 130% when it is pitching at the twice angular amplitude of the upstream foil. Although the total performance of the rear-biased and equal amplitude models reaches similar values, thrust production of the hind foil in the equal amplitude model increases only by 23%. By contrast, the performance of the forward-biased model decreases drastically for all phase differences due to the destructive wake–foil interaction of the hind foil. Studying the instantaneous wake–foil interactions, it is found that high thrust generation is associated with the formation of a vortex pair on the suction side of the hind foil, which causes stronger trailing edge vortices to shed with a greater total wake spacing. Finally, through scaling analysis, high-thrust configurations of tandem models are ranked based on the total efficiency of the system.

1.
E.
Frey
and
J.
Riess
, “
Considerations concerning plesiosaur locomotion
,”
Neues Jahrb. Geol. Palaeontol., Abh.
164
,
193
194
(
1982
).
2.
M. A.
Taylor
, “
Plesiosaurs-rigging and ballasting
,”
Nature
290
,
628
629
(
1981
).
3.
J. A.
Massare
, “
Swimming capabilities of Mesozoic marine reptiles: Implications for method of predation
,”
Paleobiology
14
,
187
205
(
1988
).
4.
J. M.
Anderson
,
K.
Streitlien
,
D.
Barrett
, and
M. S.
Triantafyllou
, “
Oscillating foils of high propulsive efficiency
,”
J. Fluid Mech.
360
,
41
72
(
1998
).
5.
K. N.
Lucas
,
G. V.
Lauder
, and
E. D.
Tytell
, “
Airfoil-like mechanics generate thrust on the anterior body of swimming fishes
,”
Proc. Natl. Acad. Sci.
117
,
10585
10592
(
2020
).
6.
X.
Wu
,
X.
Zhang
,
X.
Tian
,
X.
Li
, and
W.
Lu
, “
A review on fluid dynamics of flapping foils
,”
Ocean Eng.
195
,
106712
(
2020
).
7.
R.
Knoller
and
O.
Verein
,
Die Gesetze des Luftwiderstandes
(
Verlag des Osterreichischer Flugtechnischen Vereines
,
1909
).
8.
A.
Betz
, “
Ein beitrag zur erklaerung segelfluges
,”
Z. Flugtech. Motorluftschiffahrt
3
,
269
272
(
1922
).
9.
R.
Katzmayr
, “
Effect of periodic changes of angle of attack on behavior of airfoils
,”
Technical Report No. NACA-TM-147
,
1922
.
10.
Y. S.
Baik
,
L. P.
Bernal
,
K.
Granlund
, and
M. V.
Ol
, “
Unsteady force generation and vortex dynamics of pitching and plunging aerofoils
,”
J. Fluid Mech.
709
,
37
68
(
2012
).
11.
M. M.
Koochesfahani
, “
Vortical patterns in the wake of an oscillating airfoil
,”
AIAA J.
27
,
1200
1205
(
1989
).
12.
R.
Godoy-Diana
,
J.-L.
Aider
, and
J. E.
Wesfreid
, “
Transitions in the wake of a flapping foil
,”
Phys. Rev. E
77
,
016308
(
2008
).
13.
D. G.
Bohl
and
M. M.
Koochesfahani
, “
MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency
,”
J. Fluid Mech.
620
,
63
88
(
2009
).
14.
M.
Wolfgang
,
J.
Anderson
,
M.
Grosenbaugh
,
D.
Yue
, and
M.
Triantafyllou
, “
Near-body flow dynamics in swimming fish
,”
J. Exp. Biol.
202
,
2303
2327
(
1999
).
15.
E. G.
Drucker
and
G. V.
Lauder
, “
Locomotor function of the dorsal fin in teleost fishes: Experimental analysis of wake forces in sunfish
,”
J. Exp. Biol.
204
,
2943
2958
(
2001
).
16.
M.
Triantafyllou
,
G.
Triantafyllou
, and
R.
Gopalkrishnan
, “
Wake mechanics for thrust generation in oscillating foils
,”
Phys. Fluids A
3
,
2835
2837
(
1991
).
17.
G. K.
Taylor
,
R. L.
Nudds
, and
A. L.
Thomas
, “
Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency
,”
Nature
425
,
707
711
(
2003
).
18.
D. A.
Read
,
F.
Hover
, and
M.
Triantafyllou
, “
Forces on oscillating foils for propulsion and maneuvering
,”
J. Fluids Struct.
17
,
163
183
(
2003
).
19.
K.
Jones
,
C.
Dohring
, and
M.
Platzer
, “
Experimental and computational investigation of the Knoller–Betz effect
,”
AIAA J.
36
,
1240
1246
(
1998
).
20.
W.
Birnbaum
, “
Das ebene problem des schlagenden flügels
,”
ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech.
4
,
277
292
(
1924
).
21.
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
).
22.
E.
Salami
,
T. A.
Ward
,
E.
Montazer
, and
N. N. N.
Ghazali
, “
A review of aerodynamic studies on dragonfly flight
,”
Proc. Inst. Mech. Eng., Part C
233
,
6519
6537
(
2019
).
23.
I.
Akhtar
,
R.
Mittal
,
G. V.
Lauder
, and
E.
Drucker
, “
Hydrodynamics of a biologically inspired tandem flapping foil configuration
,”
Theor. Comput. Fluid Dyn.
21
,
155
170
(
2007
).
24.
D.
Rival
,
G.
Hass
, and
C.
Tropea
, “
Recovery of energy from leading-and trailing-edge vortices in tandem-airfoil configurations
,”
J. Aircr.
48
,
203
211
(
2011
).
25.
T. M.
Broering
,
Y.
Lian
, and
W.
Henshaw
, “
Numerical investigation of energy extraction in a tandem flapping wing configuration
,”
AIAA J.
50
,
2295
2307
(
2012
).
26.
D.
Weihs
, “
Hydromechanics of fish schooling
,”
Nature
241
,
290
291
(
1973
).
27.
D.
Weihs
and
T. Y. T.
Wu
,
Swimming and Flying in Nature
(
Springer
,
1975
).
28.
H.
Yu
,
X.-Y.
Lu
, and
H.
Huang
, “
Collective locomotion of two uncoordinated undulatory self-propelled foils
,”
Phys. Fluids
33
,
011904
(
2021
).
29.
A.
Gungor
,
M. S. U.
Khalid
, and
A.
Hemmati
, “
How does switching synchronization of pitching parallel foils from out-of-phase to in-phase change their wake dynamics?
,”
Phys. Fluids
33
,
081901
(
2021
).
30.
P. B.
Lissaman
and
C. A.
Shollenberger
, “
Formation flight of birds
,”
Science
168
,
1003
1005
(
1970
).
31.
D.
Hummel
, “
Aerodynamic aspects of formation flight in birds
,”
J. Theor. Biol.
104
,
321
347
(
1983
).
32.
B. M.
Boschitsch
,
P. A.
Dewey
, and
A. J.
Smits
, “
Propulsive performance of unsteady tandem hydrofoils in an in-line configuration
,”
Phys. Fluids
26
,
051901
(
2014
).
33.
M.
Kurt
and
K. W.
Moored
, “
Flow interactions of two-and three-dimensional networked bio-inspired control elements in an in-line arrangement
,”
Bioinspiration Biomimetics
13
,
045002
(
2018
).
34.
T. M.
Broering
and
Y.-S.
Lian
, “
The effect of phase angle and wing spacing on tandem flapping wings
,”
Acta Mech. Sin.
28
,
1557
1571
(
2012
).
35.
L.
Muscutt
,
G.
Weymouth
, and
B.
Ganapathisubramani
, “
Performance augmentation mechanism of in-line tandem flapping foils
,”
J. Fluid Mech.
827
,
484
505
(
2017
).
36.
G.
Xu
,
W.
Duan
, and
W.
Xu
, “
The propulsion of two flapping foils with tandem configuration and vortex interactions
,”
Phys. Fluids
29
,
097102
(
2017
).
37.
V.
Joshi
and
R. C.
Mysa
, “
Mechanism of wake-induced flow dynamics in tandem flapping foils: Effect of the chord and gap ratios on propulsion
,”
Phys. Fluids
33
,
087104
(
2021
).
38.
L.
Cong
,
B.
Teng
, and
L.
Cheng
, “
Hydrodynamic behavior of two-dimensional tandem-arranged flapping flexible foils in uniform flow
,”
Phys. Fluids
32
,
021903
(
2020
).
39.
M.
Kurt
,
A.
Mivehchi
, and
K.
Moored
, “
High-efficiency can be achieved for non-uniformly flexible pitching hydrofoils via tailored collective interactions
,”
Fluids
6
,
233
(
2021
).
40.
S.
Tarsitano
and
J.
Riess
, “
Plesiosaur locomotion-underwater flight versus rowing
,”
Neues Jahbr. Geol. Palaeontol., Abh.
164
,
188
192
(
1982
).
41.
L. E.
Muscutt
,
G.
Dyke
,
G. D.
Weymouth
,
D.
Naish
,
C.
Palmer
, and
B.
Ganapathisubramani
, “
The four-flipper swimming method of plesiosaurs enabled efficient and effective locomotion
,”
Proc. R. Soc. B
284
,
20170951
(
2017
).
42.
S.
Liu
,
A. S.
Smith
,
Y.
Gu
,
J.
Tan
,
C. K.
Liu
, and
G.
Turk
, “
Computer simulations imply forelimb-dominated underwater flight in plesiosaurs
,”
PLoS Comput. Biol.
11
,
e1004605
(
2015
).
43.
K.
Carpenter
,
F.
Sanders
,
B.
Reed
,
J.
Reed
, and
P.
Larson
, “
Plesiosaur swimming as interpreted from skeletal analysis and experimental results
,”
Trans. Kansas Acad. Sci.
113
,
1
34
(
2010
).
44.
F. R.
O'Keefe
, “
The evolution of plesiosaur and pliosaur morphotypes in the plesiosauria (Reptilia: Sauropterygia)
,”
Paleobiology
28
,
101
112
(
2002
).
45.
F. R.
O'Keefe
and
M. T.
Carrano
, “
Correlated trends in the evolution of the plesiosaur locomotor system
,”
Paleobiology
31
,
656
675
(
2005
).
46.
T. L.
Hilderman
, “
Measurement, modelling, and stochastic simulation of concentration fluctuations in a shear flow
,” Ph.D. thesis (
Department of Mechanical Engineering, University of Alberta
,
2004
).
47.
L.
Muscutt
, “
The hydrodynamics of plesiosaurs
,” Ph.D. thesis (
University of Southampton
,
2017
).
48.
M.
Raffel
,
C. E.
Willert
, and
J.
Kompenhans
,
Particle Image Velocimetry: A Practical Guide
, 3rd ed. (
Springer International Publishing AG
,
Cham, Switzerland
,
2018
).
49.
U.
Senturk
and
A. J.
Smits
, “
Reynolds number scaling of the propulsive performance of a pitching airfoil
,”
AIAA J.
57
,
2663
2669
(
2019
).
50.
S.
Heathcote
and
I.
Gursul
, “
Flexible flapping airfoil propulsion at low Reynolds numbers
,”
AIAA J.
45
,
1066
1079
(
2007
).

Supplementary Material

You do not currently have access to this content.