21st Australasian Fluid Mechanics Conference Adelaide, Australia 10-13 December 2018 Modelling of a Cantilevered Flexible Plate undergoing Large-Amplitude Oscillations due to a High Reynolds-Number Axial Flow. R. O. G. Evetts, R. M. Howell and A. D. Lucey Fluid Dynamics Research Group, Department of Mechanical Engineering, Curtin University of Technology, GPO Box U1987, Perth, Western Australia 6845 Abstract We present development of a model of the non-linear fluid- structure interaction of a cantilevered flexible plate with an ideal flow that can account for the effect of boundary-layer separa- tion from the plate surface upstream of its trailing edge. The model also allows for the wake to be formed solely from the trailing edge, an assumption used in previous studies of the sys- tem that also constrain the path of its lumped vorticity thereby precluding roll-up. Short plates are studied herein for which the behaviour is dominated by low-order structural modes. When the wake is forced to form from the trailing edge the typical sequence of amplitude growth to non-linearly saturated oscil- lations at flow speeds above that of the onset of linear insta- bility is found. However, if separation is included the system evidences the same sequence at a flow speed for which the sys- tem is neutrally stable to linear disturbances. This suggests that flow separation may be the cause of the sub-critical instability found in experimental studies of the system. The mechanism for this effect is briefly discussed though a consideration of the wake dynamics. The reduced complexity of our model relative to others allows us then to offer further insights into the origins of the sub-critical instability. Introduction Extending our work of [4] we further develop our model of the non-linear fluid-structure interaction (FSI) of a cantilevered flexible plate of length L in uniform axial flow of velocity U ∞ as depicted in figure 1. Inviscid flow is assumed and therefore the FSI model approximates the very high Reynolds number (Re) flows that predominate in engineering applications. How- ever, viscous effects are implicitly incorporated either through the imposition of the Kutta condition at the plate’s trailing edge or through boundary-layer separation (as drawn in figure 1) that can occur in an adverse pressure gradient upstream of the trail- ing edge. Previous approaches, for example [11, 12], have mod- elled this FSI system using the former whereby the wake forms from the trailing edge and in [12] is assumed to follow a sinu- soidal path following the spatio-temporal characteristics of the plate motion. Therefore, the main purpose of this paper is to de- termine the effect of flow separation on the non-linear stability of the FSI system by comparing its results with those in which the boundary-layer vorticity is assumed to remain attached on both sides for the full length of the flexible plate. The latest experiments on this FSI are provided by [14], who also present a comparison of their results with those of the model presented in [12] which showed excellent agreement for the limit of linear stability. However, the theoretical model pre- dicted a supercritical bifurcation while their experiment demon- strated a subcritical bifurcation creating a hysteresis loop. With regards to the origin of hysteresis in this FSI, they reiterated the explanation posed in [3, 12]: that large aspect ratio plates suffer more from spanwise deformations which have a psuedo- stiffening effect leading to higher critical velocities. As yet numerical modelling has been unable to capture the hys- teresis phenomenon because modelling is still restricted to val- ues of Re that are too low for separation to occur. Further- more, the type of model employed is usually very complex e.g. direct-numerical simulation, leading to difficulty in exploring the underlying physics with regards to the specific interactions involved in the hysteresis phenomenon. The most complete nu- merical study to date of a flag in viscous flow is provided by [2] reaching Re = 10 3 . They coupled a finite-element model to a solver for thin membrane dynamics of arbitrarily large mo- tion. They identified three distinct regimes of instability: (I) fixed-point stability, in which the flag settles into a stable non- oscillatory straight form; (II) limit-cycle flapping, where the body enters steady oscillations of constant amplitude and fre- quency; and (III) chaotic flapping, where the flag undergoes ir- regular non-periodic flutter. Herein it is shown that the simple high-Re model developed is able to capture hysteresis and that the reduced complexity of our model relative to others allows us to more deeply investigate its origins. Method The present solution of the Laplace equation utilises a non- linear boundary-element flow solution similar to that developed in [7] and is an extension of the second-order linear boundary- element method detailed in [5] so as to capture finite-amplitude effects. The flexible plate is discretised into N panels each of length δs = L/N and the vector of non-linear vortex strengths, γ, for the N panels is found by imposing the no-flux condition giving {γ} =[I N ] −1 {U ∞ sin θ + ˙ η cos θ − ˙ x sin θ + u T b sin θ − u N b cos θ}, (1) where θ is the panel angle relative to y = 0 and ˙ x and ˙ η are respectively the velocities of each panel control point in the x- and y-directions. u N b and u Tb are respectively the normal and tangential velocities induced at the panel control points by the discrete vortices of the wake. [I N ] comprises the normal influ- ence coefficients. The non-linear version of the Euler-Bernoulli Figure 1: Schematic of the fluid-structure system studied and the approach taken to model separation.