Based on the establishment of morphological models and kinematic equations with high biosimilarity, a numerical calculation method of bionic group hydrodynamics based on the immersed boundary method is proposed, and a numerical study of group hydrodynamics of manta rays in tandem, positive triangle, and inverted triangle formations is carried out. The results show that in a tandem formation, manta rays at the head of the line gain hydrodynamic benefits only when the row spacing is small, and that the propulsive performance of manta rays in the middle of the line is the best, and that it directly determines the propulsive performance of the group system. In the two triangular formations, the hydrodynamic performance is both somewhat similar and somewhat unique. The similarities are that the propulsive performance of leader manta rays decreases significantly and the followers' propulsive performance fluctuates according to the spacing in both formations, while the uniqueness is that the followers' propulsive efficiency increases in the inverted triangle formation. The conclusions can provide recommendations for formation and spacing decisions for multi-bionic underwater vehicles performing cluster missions in the same plane.

1.
M.
Ballerini
,
N.
Cabibbo
,
R.
Candelier
,
A.
Cavagna
,
E.
Cisbani
,
I.
Giardina
et al, “
Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study
,”
Proc. Natl. Acad. Sci. U. S. A.
105
(
4
),
1232
1237
(
2008
).
2.
A.
Cavagna
,
A.
Cimarelli
,
I.
Giardina
,
G.
Parisi
,
R.
Santagati
,
F.
Stefanini
, and
M.
Viale
, “
Scale-free correlations in starling flocks
,”
Proc. Natl. Acad. Sci. U. S. A.
107
(
26
),
11865
11870
(
2010
).
3.
M.
Nagy
,
Z.
Ákos
,
D.
Biro
, and
T.
Vicsek
, “
Hierarchical group dynamics in pigeon flocks
,”
Nature
464
(
7290
),
890
893
(
2010
).
4.
J.
Gautrais
,
C.
Jost
,
M.
Soria
,
A.
Campo
,
S.
Motsch
,
R.
Fournier
et al, “
Analyzing fish movement as a persistent turning walker
,”
J. Math. Biol.
58
,
429
445
(
2009
).
5.
V. H.
Sridhar
,
L.
Li
,
D.
Gorbonos
,
M.
Nagy
,
B. R.
Schell
,
T.
Sorochkin
et al, “
The geometry of decision-making in individuals and collectives
,”
Proc. Natl. Acad. Sci. U. S. A.
118
(
50
),
e2102157118
(
2021
).
6.
G.
Li
,
I.
Ashraf
,
B.
François
,
D.
Kolomenskiy
,
F.
Lechenault
,
R.
Godoy-Diana
, and
B.
Thiria
, “
Burst-and-coast swimmers optimize gait by adapting unique intrinsic cycle
,”
Commun. Biol.
4
(
1
),
40
(
2021
).
7.
D.
Helbing
and
P.
Molnar
, “
Social force model for pedestrian dynamics
,”
Phys. Rev. E
51
(
5
),
4282
(
1995
).
8.
D.
Helbing
,
I.
Farkas
, and
T.
Vicsek
, “
Simulating dynamical features of escape panic
,”
Nature
407
(
6803
),
487
490
(
2000
).
9.
M.
Moussaïd
,
D.
Helbing
, and
G.
Theraulaz
, “
How simple rules determine pedestrian behavior and crowd disasters
,”
Proc. Natl. Acad. Sci. U. S. A.
108
(
17
),
6884
6888
(
2011
).
10.
J. K.
Parrish
and
L.
Edelstein-Keshet
, “
Complexity, pattern, and evolutionary trade-offs in animal aggregation
,”
Science
284
(
5411
),
99
101
(
1999
).
11.
D. J.
Sumpter
, “
The principles of collective animal behaviour
,”
Philos. Trans. R. Soc. B
361
(
1465
),
5
22
(
2006
).
12.
B. L.
Partridge
, “
The structure and function of fish schools
,”
Sci. Am.
246
(
6
),
114
123
(
1982
).
13.
I. L.
Bajec
and
F. H.
Heppner
, “
Organized flight in birds
,”
Anim. Behav.
78
(
4
),
777
789
(
2009
).
14.
D.
Weihs
, “
Hydromechanics of fish schooling
,”
Nature
241
(
5387
),
290
291
(
1973
).
15.
D.
Weihs
, “
Optimal fish cruising speed
,”
Nature
245
(
5419
),
48
50
(
1973
).
16.
J.
Deng
and
X. M.
Shao
, “
Hydrodynamics in a diamond-shaped fish school
,”
J. Hydrodyn., Ser. B
18
(
3
),
438
442
(
2006
).
17.
M. H.
Chung
, “
Hydrodynamic performance of two-dimensional undulating foils in triangular formation
,”
J. Mech.
27
(
2
),
177
190
(
2011
).
18.
A. P.
Maertens
,
A.
Gao
, and
M. S.
Triantafyllou
, “
Optimal undulatory swimming for a single fish-like body and for a pair of interacting swimmers
,”
J. Fluid Mech.
813
,
301
345
(
2017
).
19.
M.
Daghooghi
and
I.
Borazjani
, “
The hydrodynamic advantages of synchronized swimming in a rectangular pattern
,”
Bioinspiration Biomimetics
10
(
5
),
056018
(
2015
).
20.
J. L.
Johansen
,
R.
Vaknin
,
J. F.
Steffensen
, and
P.
Domenici
, “
Kinematics and energetic benefits of schooling in the labriform fish, striped surfperch Embiotoca lateralis
,”
Mar. Ecol. Prog. Ser.
420
,
221
229
(
2010
).
21.
Y.
Ma
,
Q.
Huang
,
G.
Pan
, and
P.
Gao
, “
Investigation of the hydrodynamic characteristics of two manta rays tandem gliding
,”
J. Mar. Sci. Eng.
10
(
9
),
1186
(
2022
).
22.
P.
Gao
,
Q.
Huang
,
G.
Pan
,
Y.
Cao
, and
Y.
Luo
, “
Group gliding of three manta rays in multiple formations
,”
Ocean Eng.
278
,
114389
(
2023
).
23.
P.
Gao
,
X.
Tian
,
Q.
Huang
,
G.
Pan
,
Y.
Chu
, and
J.
Bai
, “
Group hydrodynamic analysis of two manta rays under spacing and phase-difference coupling
,”
Phys. Fluids
36
(
9
),
091923
(
2024
).
24.
P.
Gao
,
X.
Tian
,
Q.
Huang
, and
G.
Pan
, “
Research on the swimming performance of two manta rays under staggered propulsion on the same frequency: When the follower is above the leader
,”
Phys. Fluids
36
(
1
),
011902
(
2024
).
25.
Z.
Huang
,
A.
Menzer
,
J.
Guo
, and
H.
Dong
, “
Hydrodynamic analysis of fin–fin interactions in two-manta-ray schooling in the vertical plane
,”
Bioinspiration Biomimetics
19
(
2
),
026004
(
2024
).
26.
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
).
27.
D.
Zhang
and
W. X.
Huang
, “
Hydrodynamics of a swimming batoid fish at Reynolds numbers up to 148 000
,”
J. Fluid Mech.
963
,
A16
(
2023
).
28.
L. M.
Yang
,
C.
Shu
, and
J.
Wu
, “
A three-dimensional explicit sphere function-based gas-kinetic flux solver for simulation of inviscid compressible flows
,”
J. Comput. Phys.
295
,
322
339
(
2015
).
29.
L. M.
Yang
,
C.
Shu
,
Y.
Wang
, and
Y.
Sun
, “
Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows
,”
J. Comput. Phys.
319
,
129
144
(
2016
).
30.
L. M.
Yang
,
C.
Shu
,
W. M.
Yang
,
Y.
Wang
, and
J.
Wu
, “
An immersed boundary-simplified sphere function-based gas kinetic scheme for simulation of 3D incompressible flows
,”
Phys. Fluids
29
(
8
),
083605
(
2017
).
31.
P.
Gao
,
Q.
Huang
,
G.
Pan
,
D.
Song
, and
Y.
Cao
, “
Research on swimming performance of fish in different species
,”
Phys. Fluids
35
(
6
),
061909
(
2023
).
32.
P.
Han
,
Y.
Pan
,
G.
Liu
, and
H.
Dong
, “
Propulsive performance and vortex wakes of multiple tandem foils pitching in-line
,”
J. Fluids Struct.
108
,
103422
(
2022
).
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