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a single flexibility in order to explore the possible roles of the wing deformation in
the propulsive performance. Considering our previous studies on the effect of flex-
ibility with a two wing flyer (Thiria and Godoy-Diana
2010
; Ramananarivo et al.
2011
), where the trailing-edge-leading-edge phase lag
was shown to be a crucial
parameter to determine performance, we attempted here to monitor modifications in
ʳ
ʳ
as a function of
˕
(see the analysis of wing kinematics in Figs.
6
and
7
). We picked
a range of
) where a clear deterioration of the flying
performance is observed in Fig.
2
as the hindwing starts leading the forewing. The
observations on the trailing-edge-leading-edge phase lag
˕
(decreasing from 2
ˀ
to 1
.
3
ˀ
ʳ
are, however, not conclu-
sive. A slight decrease of
, which
at these frequencies could explain a diminishing thrust performance, but the effect
is too weak and the present results do not permit to give a thorough and quantitative
confirmation of the effect if it exists.
ʳ
is indeed observed, while
˕
goes from 2
ˀ
to 1
.
3
ˀ
5 Conclusions
We have shown that a four-winged flapping flyer in a cruising regime does present
different optimal forewing-hindwing phase lags depending on whether one would
want to tune the system for maximum cruising speed or minimum energy expendi-
ture. These results are in accordance with previous studies in hovering configurations
(Wang and Russell
2007
; Usherwood and Lehmann
2008
), hinting that the mech-
anisms described here should be robust elements to consider in any aerodynamic
model: on the one hand, wing inertia is a major player in the power expenditure,
while, on the other hand, the thrust production is ruled by the aerodynamics around
the flexible wings.
A full flow field reconstruction around the wings is certainly desirable to compare
different points in the parameter space with different performances and clearly iden-
tify the aerodynamic mechanisms at play. Experimentally this can be challenging,
and numerical simulations such as a 3D version of the fluid-structure simulations of
Kolomenskiy et al. (
2013
) can be an interesting option to define robust aerodynamic
models.
Other issue that remains not fully understood is the effect of the forewing-
hindwing separation
d
as a parameter independent of the forewing-hindwing phase
lag which will have a different role depending on whether the system is hovering or
cruising. This point will be of particular importance when going beyond the “steady”
regimes of hovering or cruising and into the study of transient regimes. These bring
indeed a vast set of open questions related to the unsteadiness—like for instance the
multi-body dynamics of manoeuvring—and where the wings are to be analysed as
part of a full system accelerating, performing sharp turns (Bergou et al.
2010
)or
taking off (Bimbard et al.
2013
).
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