Abstract
A nonlinear fluid-elastic continuum model for the dynamics of a slender cantilevered plate subjected to axial flow directed from the free end to the clamped end, also known as the inverted flag problem, is proposed. The extension of elongated body theory to large-amplitude rotations of the plate mid-plane along with Bollay's nonlinear wing theory is employed in order to express the fluid-related forces acting on the plate, while retaining all time-dependent terms in both modelling and numerical simulations; the unsteady fluid forces due to vortex shedding are not included. Euler-Bernoulli beam theory with exact kinematics and inextensibility is employed to derive the nonlinear partial integro-differential equation governing the dynamics of the plate. Discretization in space is carried out via a conventional Galerkin scheme using the linear mode-shapes of a cantilevered beam in vacuum. The pseudo-arclength continuation technique is adapted to construct bifurcation diagrams in terms of the flow velocity, in order to gain insight into the stability and post-critical behaviour of the system. Integration in time is conducted using Gear's backward differentiation formula. The sensitivity of the nonlinear response of the system to different parameters such as the aspect ratio, mass ratio, initial inclination of the flag, and viscous drag coefficient is investigated through extensive numerical simulations. It is shown that for flags of small aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of sufficiently large aspect ratio, however, the first instability encountered is a supercritical Hopf bifurcation giving rise to flapping motion around the undeflected static equilibrium; increasing the flow velocity further, the flag then displays flapping motions around deflected static equilibria, which later lead to fully-deflected static states at even higher flow velocities. The results exposed in this study help understand the dynamics of the inverted-flag problem in the limit of inviscid flow theory.
Highlights
The dynamics of cantilevered plates in incompressible reverse axial flows, otherwise known as inverted flags, is of considerable interest, as an abstract problem with rich dynamics, and for engineering applications such as small scale energy harvesting systems [19, 28] and heat transfer enhancement in heat exchangers [35, 20]
The present paper aims at producing such a model in the limit case of slender flexible plates, and testing its capability to predict the dynamical behaviour of heavy slender inverted flags in axial flow, such as the onset of instabilities, frequency and amplitude of flapping, as well as elucidating the physical mechanisms involved in the transition between different dynamical states
While comparisons above provide partial agreement between the sequence of flow regimes shown in Figure 8 of the present paper with [24, Figure 3], as well as the computational study of Tang et al.[29], we have noted a strong discrepancy between the structure of the bifurcation diagram predicted by the proposed model and that predicted via 2D simulations by Goza et al [11] for infinitely wide inverted flags
Summary
The dynamics of cantilevered plates in incompressible reverse axial flows, otherwise known as inverted flags, is of considerable interest, as an abstract problem with rich dynamics, and for engineering applications such as small scale energy harvesting systems [19, 28] and heat transfer enhancement in heat exchangers [35, 20]. The present paper aims at producing such a model in the limit case of slender flexible plates, and testing its capability to predict the dynamical behaviour of heavy slender inverted flags in axial flow, such as the onset of instabilities, frequency and amplitude of flapping, as well as elucidating the physical mechanisms involved in the transition between different dynamical states To this end, continuum representations of fluid-flow forces are detailed on the basis of the large amplitude elongated body theory [17] to formulate the reactive force and Bollay’s model to express the nonlinear vortex lift mechanism, with no account for the vortex shedding mechanism from the leading edge. The results are summarized in the form of bifurcation diagrams to identify transitions between various regimes
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