Self-propulsive performances of the flexible plates undergoing pitching and heaving motions are investigated numerically. The effects of multiple key dimensionless parameters are considered, such as bending stiffness, heaving amplitude, pitching amplitude and flapping frequency. Despite so many influence factors, results indicate that the cruising speed
$U$
(or the cruising Reynolds number
$Re_c$
), the thrust
$T$
and the input power
$P$
can be summarized as some simple scaling laws vs the flapping Reynolds number
$Re_f$
. In the heaving motion, the scaling laws may be not fully independent of bending stiffness because in the motion the role of bending stiffness is more complicated for the thrust generation. Our scaling laws are well supported by biological data on swimming aquatic animals.
Fine fibre immersed in different flows is ubiquitous. For a fibre in shear flows, most motion modes appear in the flow-gradient plane. Here the two-dimensional behaviours of an individual flexible flap in channel flows are studied. The nonlinear coupling of the fluid inertia (
$\textit {Re}$
), flexibility of the flap (
$K$
) and channel width (
$W$
) is discovered. Inside a wide channel (e.g.
$W=4$
), as
$K$
decreases, the flap adopts rigid motion, springy motion, snake turn and complex mode in sequence. It is found that the fluid inertia tends to straighten the flap. Moreover,
$\textit {Re}$
significantly affects the lateral equilibrium location
$y_{eq}$
, therefore affecting the local shear rate and the tumbling period
$T$
. For a rigid flap in a wide channel, when
$\textit {Re}$
exceeds a threshold, the flap stays inclined instead of tumbling. As
$\textit {Re}$
further increases, the flap adopts swinging mode. In addition, there is a scaling law between
$T$
and
$\textit {Re}$
. For the effect of
$K$
, through the analysis of the torque generated by surrounding fluid, we found that a smaller
$K$
slows down the tumbling of the flap even if
$y_{eq}$
is comparable. As
$W$
decreases, the wall confinement effect makes the flap easier to deform and closer to the centreline. The tumbling period would increase and the swinging mode would be more common. When
$W$
further decreases, the flaps are constrained to stay inclined, parabolic-like or one-end bending configurations moving along with the flow. Our study may shed some light on the behaviours of a free fibre in flows.
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