  The stability of a cantilevered elastic sheet in a uniform flow has been
studied extensively due to its importance in engineering and its prevalence in
natural structures. Varying the flow speed can give rise to a range of dynamics
including limit cycle behaviour and chaotic motion of the cantilevered sheet.
Recently, the "inverted flag" configuration - a cantilevered elastic sheet
aligned with the flow impinging on its free edge - has been observed to produce
large-amplitude flapping over a finite band of flow speeds. This flapping
phenomenon has been found to be a vortex-induced vibration, and only occurs at
sufficiently large Reynolds numbers. In all cases studied, the inverted flag
has been formed from a cantilevered sheet of rectangular morphology, i.e. the
planform of its elastic sheet is a rectangle. Here, we investigate the effect
of the inverted flag's morphology on its resulting stability and dynamics. We
choose a trapezoidal planform which is explored using experiment and an
analytical theory for the divergence instability of an inverted flag of
arbitrary morphology. Strikingly, for this planform we observe that the flow
speed range over which flapping occurs scales approximately with the flow speed
at which the divergence instability occurs. This provides a means by which to
predict and control flapping. In a biological setting, leaves in a wind can
also align themselves in an inverted flag configuration. Motivated by this
natural occurrence we also study the effect of adding an artificial "petiole"
(a thin elastic stalk that connects the sheet to the clamp) on the inverted
flag's dynamics. We find that the petiole serves to partially decouple fluid
forces from elastic forces and increases the freedom by which the flapping
dynamics can be tuned. These results highlight the intricacies of the flapping
instability and account for some of the varied dynamics of leaves in nature.
