A numerical simulation of the interaction of a compliant wall and inviscid flow

Abstract

A method for numerically simulating the hydroelastic behaviour of a passive compliant wall of finite dimensions is presented. Using unsteady potential flow, the perturbation pressures which arise from wall disturbances of arbitrary form are calculated through a specially developed boundary-element method. These pressures may then be coupled to a suitable solution procedure for the wall mechanics to produce an interactive model for the wall/flow system. The method is used to study the two-dimensional disturbances which may occur on a Kramer-type compliant wall of finite length. Finite-difference methods are used to yield wall solutions driven by the fluid pressure after some perturbation from the equilibrium position. Thus, histories of surface deflection and wall energy are obtained. Such a modelling of the physics of the system requires no presupposition of disturbance form.A thorough investigation of divergence instability is carried out. Most of the results presented in this paper concern the response of the compliant wall while (and after) a point pressure pulse, carried in the applied flow, travels over the compliant panel. Above a critical flow speed and once sufficient time has passed, the compliant wall is shown to adopt the particular profile of an unstable mode. After this divergence mode has been established, instability is realized as a slowly travelling downstream wave. These features are in agreement with the findings of experimental studies. The role of wall damping is clarified: damping serves only to reduce the growth rate of the instability, leaving its onset flow speed unchanged. The present predictions provide an improvement upon some of the unrealistic aspects of predictions yielded by travelling-wave and standing-wave treatments of divergence instability.The response of a long compliant panel after a single-point pressure-pulse initiation, applied at its midpoint, is simulated. At flow speeds higher than a critical value, parts of the formerly (at subcritical flow speeds) upstream-travelling wave system change to travel downstream and show amplitude growth. The development of this ‘upstream-incoming’ wave illustrates how divergence instability can occur at locations upstream of the point of initial excitation. Faster flexural waves transmit energy upstream, thereafter these disturbances can evolve into slow downstreamtravelling divergence waves. The spread of the instability to locations both downstream and upstream of the point of initial excitation indicates that divergence is an absolute instability. This behaviour and the effects of wall damping clarified by the present work strongly suggest that divergence is a Class C instability.

Suppression of absolute instabilities in the flow inside a compliant tube

“…The stability of flows over a flat-plate as well as plug flow has been extensively analyzed by Brazier-Smith and Scott [19], Carpenter and Carrad [39], and Lucey and Carpenter [41]. The nature of absolute instability of this kind of flows has been pointed out by Crighton and Oswell [40], Triantafyllou [42] and Peake [43].…”

The Nonlinear Hydroelastic Behaviour of Flexible Walls

“…In evaluating the tangential velocities, the source strength has been 'smeared' over the panel as a linear variation with coefficient G given by a forwarddifference gradient of sources strengths on adjacent panels. This strategy was used by Lucey & Carpenter (1992a) to improve the accuracy of the hydrodynamic-stiffness term in the pressure without recourse to excessively fine discretizations. This approximates a secondorder panel method (Hess 1973) for the hydrodynamic stiffness, whilst evaluations of hydrodynamic damping and inertia remain at first order.…”

“…In wall-flow structure interactions where the fluid inertia THE NONLINEAR HYDROELASTIC BEHAVIOUR OF FLEXIBLE WALLS greatly exceeds the wall inertia, the lumping of inertias is essential for numerical stability of the coupled system when solving for the wall-acceleration in the presence of fluid loading. This was the strategy adopted in the equivalent linear problem of Lucey & Carpenter (1992a) and has found application in the more complex fluid-loading problem studied by Davies & Carpenter (1997).…”

“…These differences led Gad-el-Hak et al (1984) to infer that the action of wall damping was crucial in the mechanism which causes divergence instability. However, Lucey & Carpenter (1992a) have suggested that the viscoelastic wall damping only served to facilitate the formation of the most dangerous mode when a continuous source of excitation was present due to the turbulent boundary layer. It was particulary noteworthy that the nonlinear surface deformations due to divergence instability on the strongly damped wall of Gad-el-Hak et al (1984) differed greatly from the mainly sinusoidal profiles found in experiments on lightly damped flexible surfaces (Dugundji et al 1963;Gad-el-Hak 1986).…”

On the direct determination of the eigenmodes of finite flow–structure systems

“…Numerical simulations such as those of Lucey & Carpenter (1992) and Lucey (1998) are ideal for capturing the transient behaviour of finite flow-structure systems in response to an initial, or continuing, applied excitation. However, they are less suited to identifying the long-time solution in terms of system eigenmodes.…”