Hovering insects are adapted for high metabolism. Their physiology, morphology, and flight have similar scaling relations. However, there are differences in efficiency. For instance, bumblebees have superior efficiency over orchid bees. Here, we propose that unsteady aerodynamics can explain these differences. Inspired by helical wakes in the bumblebee flight, we study an unsteady viscous model to show that there is intermittently abundant energy in the wake which is available for extraction by flapping wings. We then investigate the transport phenomena in the model and identify the role of kinematic viscosity, nonlinearity, flow topology, and fluid dynamic current to justify the available energy. Wake energetics is applicable to other species such as hummingbirds whose responses rely on environmental conditions. The present work has implications in minimizing power consumption in bio-mimetic locomotion both in air and under water. Interdisciplinary applications include dynamic modeling in superfluid turbulence and magneto-fluid dynamic dynamo theory.

1.
R. K.
Suarez
, “
Energy metabolism during insect flight: Biochemical design and physiological performance
,”
Physiol. Biochem. Zool.
73
,
765
771
(
2000
).
2.
R. B.
Srygley
and
A. L. R.
Thomas
, “
Unconventional lift-generating mechanisms in free-flying butterflies
,”
Nature
420
,
660
664
(
2002
).
3.
T.
Engels
,
D.
Kolomenskiy
,
K.
Schneider
,
F. O.
Lehmann
, and
J.
Sesterhenn
, “
Bumblebee flight in heavy turbulence
,”
Phys. Rev. Lett.
116
,
1
5
(
2016
).
4.
T.
Engels
,
D.
Kolomenskiy
,
K.
Schneider
,
M.
Farge
,
F. O.
Lehmann
, and
J.
Sesterhenn
, “
Helical vortices generated by flapping wings of bumblebees
,”
Fluid Dyn. Res.
50
,
011419
(
2018
).
5.
C. A.
Darveau
,
P. W.
Hochachka
,
D. W.
Roubik
, and
R. K.
Suarez
, “
Allometric scaling of flight energetics in orchid bees: Evolution of flux capacities and flux rates
,”
J. Exp. Biol.
208
,
3593
3602
(
2005
).
6.
C. A.
Darveau
,
P. W.
Hochachka
,
K. C.
Welch
,
D. W.
Roubik
, and
R. K.
Suarez
, “
Allometric scaling of flight energetics in Panamanian orchid bees: A comparative phylogenetic approach
,”
J. Exp. Biol.
208
,
3581
3591
(
2005
).
7.
F.
Billardon
and
C. A.
Darveau
, “
Flight energetics, caste dimorphism and scaling properties in the bumblebee, Bombus impatiens
,”
J. Exp. Biol.
222
,
1
10
(
2019
).
8.
G. N.
Askew
,
R. T.
Tregear
, and
C. R.
Ellington
, “
The scaling of myofibrillar actomyosin ATPase activity in apid bee flight muscle in relation to hovering flight energetics
,”
J. Exp. Biol.
213
,
1195
1206
(
2010
).
9.
V. M.
Ortega-Jimenez
,
R.
Mittal
, and
T. L.
Hedrick
, “
Hawkmoth flight performance in tornado-like whirlwind vortices
,”
Bioinspiration Biomimetics
9
,
025003
(
2014
).
10.
Y. Z.
Lyu
,
H. J.
Zhu
, and
M.
Sun
, “
Aerodynamic forces and vortical structures of a flapping wing at very low Reynolds numbers
,”
Phys. Fluids
31
,
041901
(
2019
).
11.
S.
Bhat
,
J.
Zhao
,
J.
Sheridan
,
K.
Hourigan
, and
M.
Thompson
, “
Aspect ratio studies on insect wings
,”
Phys. Fluids
31
,
121301
(
2019
).
12.
Y.
Zhang
,
J.
Han
, and
G.
Chen
, “
Effects of the flapping frequency on the thrust performance for three-dimensional bionic multi-wings in a schooling
,”
Phys. Fluids
31
,
117110
(
2019
).
13.
X.
Cheng
and
M.
Sun
, “
Revisiting the clap-and-fling mechanism in small wasp Encarsia formosa using quantitative measurements of the wing motion
,”
Phys. Fluids
31
,
101903
(
2019
).
14.
W.
Tong
,
Y.
Yang
, and
S.
Wang
, “
Characterizing three-dimensional features of vortex surfaces in the flow past a finite plate
,”
Phys. Fluids
32
,
011903
(
2020
).
15.
M.
Lei
and
C.
Li
, “
The aerodynamic performance of passive wing pitch in hovering flight
,”
Phys. Fluids
32
,
051902
(
2020
).
16.
P.
Chakraborty
,
A.
Roy
, and
S.
Chakraborty
, “
Mechanistic basis of transport in unconfined swirling flows
,”
Phys. Fluids
33
,
053109
(
2021
).
17.
P.
Chakraborty
,
A.
Roy
, and
S.
Chakraborty
, “
Topology and transport in generalized helical flows
,”
Phys. Fluids
33
,
117106
(
2021
).
18.
P.
Chakraborty
and
A.
Roy
, “
Wake aerodynamics of flapping systems in formation flight
,”
Phys. Fluids
34
,
047113
(
2022
).
19.
D. R.
Warrick
,
B. W.
Tobalske
, and
D. R.
Powers
, “
Aerodynamics of the hovering hummingbird
,”
Nature
435
,
1094
1097
(
2005
).
20.
M. H.
Dickinson
,
F.-O.
Lehmann
, and
S. P.
Sane
, “
Wing rotation and the aerodynamic basis of insect flight
,”
Science
284
,
1954
1960
(
1999
).
21.
R. J.
Bomphrey
,
G. K.
Taylor
, and
A. L. R.
Thomas
, “
Smoke visualization of free-flying bumblebees indicates independent leading-edge vortices on each wing pair
,”
Exp. Fluids
46
,
811
821
(
2009
).
22.
N. T.
Jafferis
,
E. F.
Helbling
,
M.
Karpelson
, and
R. J.
Wood
, “
Untethered flight of an insect-sized flapping-wing microscale aerial vehicle
,”
Nature
570
,
491
495
(
2019
).
23.
H.
Liu
,
S.
Ravi
,
D.
Kolomenskiy
, and
H.
Tanaka
, “
Biomechanics and biomimetics in insect-inspired flight systems
,”
Philos. Trans. R. Soc., B
371
,
20150390
(
2016
).
24.
X.
Cheng
and
M.
Sun
, “
Very small insects use novel wing flapping and drag principle to generate the weight-supporting vertical force
,”
J. Fluid Mech.
855
,
646
670
(
2018
).
25.
R. J.
Wood
,
B.
Finio
,
M.
Karpelson
,
K.
Ma
,
N. O.
Pérez-Arancibia
,
P. S.
Sreetharan
,
H.
Tanaka
, and
J. P.
Whitney
, “
Progress on 'pico' air vehicles
,”
Int. J. Rob. Res.
31
,
1292
1302
(
2012
).
26.
M. A.
Badger
,
H.
Wang
, and
R.
Dudley
, “
Avoiding topsy-turvy: How Anna's hummingbirds (Calypte anna) fly through upward gusts
,”
J. Exp. Biol.
222
,
jeb176263
(
2019
).
27.
T.
Dombre
,
U.
Frisch
,
J. M.
Greene
,
M.
Henon
,
A.
Mehr
, and
A. M.
Soward
, “
Chaotic streamlines in the ABC flows
,”
J. Fluid Mech.
167
,
353
391
(
1986
).
28.
C. F.
Barenghi
,
D. C.
Samuels
,
G. H.
Bauer
, and
R. J.
Donnelly
, “
Superfluid vortex lines in a model of turbulent flow
,”
Phys. Fluids
9
,
2631
2643
(
1997
).
29.
D. W.
Murphy
,
D.
Adhikari
,
D. R.
Webster
, and
J.
Yen
, “
Underwater flight by the planktonic sea butterfly
,”
J. Exp. Biol.
219
,
535
543
(
2016
).
30.
B. J.
Gemmell
,
D.
Adhikari
, and
E. K.
Longmire
, “
Volumetric quantification of fluid flow reveals fish's use of hydrodynamic stealth to capture evasive prey
,”
J. R. Soc. Interface
11
,
20130880
(
2014
).
31.
R. M.
Kerr
, “
Histograms of helicity and strain in numerical turbulence
,”
Phys. Rev. Lett.
59
,
783
(
1987
).
32.
C. W.
Hamman
,
J. C.
Klewicki
, and
R. M.
Kirby
, “
On the Lamb vector divergence in Navier-Stokes flows
,”
J. Fluid Mech.
610
,
261
284
(
2008
).
33.
S.
Kida
and
M.
Takaoka
, “
Vortex reconnection
,”
Annu. Rev. Fluid Mech.
26
,
169
189
(
1994
).
34.
A.
Sengupta
,
V. K.
Suman
,
T. K.
Sengupta
, and
S.
Bhaumik
, “
An enstrophy-based linear and nonlinear receptivity theory
,”
Phys. Fluids
30
,
054106
(
2018
).
35.
H. K.
Moffatt
, “
Helicity and singular structures in fluid dynamics
,”
Proc. Natl. Acad. Sci. U. S. A.
111
,
3663
3670
(
2014
).
36.
Y.
Kimura
and
H. K.
Moffatt
, “
Reconnection of skewed vortices
,”
J. Fluid Mech.
751
,
329
345
(
2014
).
37.
J.
Han
,
Z.
Yuan
, and
G.
Chen
, “
Effects of kinematic parameters on three-dimensional flapping wing at low Reynolds number
,”
Phys. Fluids
30
,
081901
(
2018
).
38.
D.
Lentink
,
G. F. V.
Heijst
,
F. T.
Muijres
, and
J. L. V.
Leeuwen
, “
Vortex interactions with flapping wings and fins can be unpredictable
,”
Biol. Lett.
6
,
394
397
(
2010
).
39.
H.
Marmanis
, “
Analogy between the Navier-Stokes equations and Maxwell's equations: Application to turbulence
,”
Phys. Fluids
10
,
1428
1437
(
1998
).
40.
H.
Nagai
,
K.
Isogai
,
T.
Fujimoto
, and
T.
Hayase
, “
Experimental and numerical study of forward flight aerodynamics of insect flapping wing
,”
AIAA J.
47
,
730
742
(
2009
).
41.
J.
Bluman
and
C. K.
Kang
, “
Wing-wake interaction destabilizies hover equilibrium of a flapping insect-scale wing
,”
Bioinspiration Biomimetics
12
,
046004
(
2017
).
42.
D. L.
Altshuler
and
R.
Dudley
, “
Kinematics of hovering hummingbird flight along simulated and natural elevational gradients
,”
J. Exp. Biol.
206
,
3139
3147
(
2003
).
43.
J. P.
Whitney
and
R. J.
Wood
, “
Aeromechanics of passive rotation in flapping flight
,”
J. Fluid Mech.
660
,
197
220
(
2010
).
44.
K. Y.
Ma
,
P.
Chirarattananon
,
S. B.
Fuller
, and
R. J.
Wood
, “
Controlled flight of a biologically inspired, insect-scale robot
,”
Science
340
,
603
607
(
2013
).
45.
A.
Alexakis
, “
Searching for the fastest dynamo: Laminar ABC flows
,”
Phys. Rev. E
84
(
10
),
026321
(
2011
).
You do not currently have access to this content.