Excitation and evolution of waves on an inhomogeneous flexible wall in a mean flow

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Abstract

The excitation of neutrally stable small-amplitude waves in a system comprising uniform flow over a flexible wall is studied. Linear theory and numerical simulation are used. A feature of wave excitation in such systems is that distinct amplitude ratios exist when more than one wave is present. An amplitude-ratio equation is first derived for the two waves excited directly by oscillatory excitation applied to a spatially homogeneous plate-spring flexible wall. Thereafter, a second amplitude-ratio equation is derived for waves on a wall with spatially varying foundation-spring stiffness coefficient. In this case, waves beyond a region of changing flexibility are continuously excited by waves propagating through the region of change. Alternatively, it can be considered that an incoming wave evolves into the outgoing wave that emerges from the region of changing wall flexibility. Both energy-flux and WKB methods are used to derive the final amplitude-ratio equation. The theory holds for both upstream- and downstream-propagating waves. It is shown, inter alia, that the waves on either side of the region of slowly changing wall properties are well-described by the dispersion equation based upon local wall properties. Finally, the case of waves incident upon a region of rapidly changing wall properties is simulated and discussed.

Introduction

The hydroelastic behaviour of small-amplitude waves has, most often, been studied theoretically using a boundary-value approach with a normal-mode prescription of system waves. Such an approach necessarily assumes a spatially homogeneous system of infinite streamwise extent. It is therefore unable to address the important matter of how disturbances come into being and the subsequent relationship between the excitation that initiates wave motions and the amplitude of the waves that emanate from such forcing. In more recent years, a body of work, for example Crighton and Oswell (1991), Peake (1997), Lucey (1998) and Abrahams and Wickham (2001), has emerged in which the initial-value problem has been solved for the case of oscillatory line excitation in the two-dimensional problem of uniform flow over a thin elastic plate or shell. Such studies show spatial dependence of the response in that particular wave solutions of the dispersion relation are predicted to appear either upstream or downstream of the point of excitation. The present paper takes up this theme and extends the aforementioned studies by tackling the propagation of waves over a flexible boundary with spatially varying mechanical properties.

The problems addressed in this paper are illustrated by the schematic of Fig. 1. Oscillatory line excitation generates upstream- and downstream-propagating waves of different amplitudes. Our first interest lies in determining the amplitude ratio of these waves; it is shown that this problem is closely related to the main wave-evolution problem due to a flexible wall with spatially varying properties. After its initial excitation, the downstream wave passes through a region of changing wall flexibility. As it does so it evolves until it emerges with wavenumber and amplitude modified to match the local conditions in the region downstream of the change. An important goal of the present work is to find the amplitude change of the wave in terms of its local wave properties upstream and downstream of the region of changing wall stiffness. This problem may also be considered as one of wave excitation in that an output wave with frequency ω, wavenumber k2 and amplitude η02 is generated by an incoming disturbance. In this case, the incoming disturbance is a wave with properties ω, k1 and η01 propagating through an upstream region of different wall flexibility. A subsidiary objective is to confirm that waves in different spatial regions of the overall wall-flow system can be accurately described in terms of the local properties of the system. Although Fig. 1 only depicts the evolution of downstream-propagating waves, we also investigate corresponding phenomena for upstream-propagating disturbances. In this paper we mainly concern ourselves with slow variation in the region of wall-property change. However, we briefly consider the limits of our analysis based upon this assumption. Clearly, a combination of wave reflection and transmission can occur if a rapid change in wall properties is prescribed.

In this paper we consider the excitation and propagation of neutrally stable waves. We therefore restrict ourselves to fluid loading at flow speeds lower than those that would yield hydroelastic instability at any spatial location within the system. Moreover, we choose to model the simplest type of spatially varying flexible wall—an elastic plate of invariant properties supported by a spatially varying distributed spring foundation. Our goal, therefore, is to establish a framework of theoretical and computational methods within which more complete flow and wall models can be developed. Although the present problems are of general scientific interest, specific motivation for their study is provided by the potential of compliant coatings to reduce skin-friction drag in marine vehicles. Carpenter 2000, Carpenter 2001 have proposed that extremely effective laminar-flow control can be achieved by coatings with spatially dependent properties, allowing properties to be tailored to the local Reynolds number. Choi et al. (1997) have experimentally demonstrated that flexible walls can reduce turbulent skin friction. In this application, too, the excitation of fluid-loaded wall waves that might evolve into hydroelastic instability can limit the effectiveness of this drag-reduction strategy. De Langre and Ouvrard (1999) have shown that the open system studied in this paper is closely related to that of plug flow in elastic channels. Clearly, then, further applications for the present work are to be found in biomechanics.

The present paper combines analytical and computational approaches to study the problems illustrated in Fig. 1. Whilst we describe the analytical methods in some detail, the computational method is only briefly summarized because it is a minor development from that formulated and used in Lucey and Carpenter (1992) and Lucey (1998). The simulations that the method permits are, however, an integral part of the present investigation. We therefore divide the remainder of this paper along investigative lines, as opposed to methodologies, in the following way. Section 2 presents the system of equations used to model the system and briefly describes solution methods. In Section 3, we address the problem of wave excitation by an external oscillator. In Section 4, we solve the problem of wave-propagation through a region of spatially varying wall flexibility. Two independent theoretical approaches are developed, one based on energy fluxes, the other at the force level using a WKB method. The predictions of these methods are shown to give good agreement with the results of numerical simulations. Finally, in Section 5, we provide a brief conclusion to the present work.

Section snippets

Governing equations

The small-amplitude motion, of a thin elastic plate, supported by a spatially varying spring foundation K∗(x), in the presence of a fluid flow and subjected to oscillatory line excitation is described byρmh∂2η∂t2+B∂4η∂x4+K∗(x)η=−p(x,0,t)+F∗0exp(iωFt)δ(x−xF)H(t),where η(x,t), ρm, h and B are, respectively, plate's deflection, density, thickness and flexural rigidity, while p(x,y,t) is the unsteady fluid pressure and F∗0, ωF and xF are, respectively, the amplitude, frequency and location of the

Line excitation: wave–amplitude ratios

In this section we consider a spatially homogeneous flexible wall with line excitation at a frequency ωF. Typical simulations for two flow speeds, U=0.02487 and 0.0496, are shown in Fig. 3 for excitation with ωF=0.0119 and amplitude F0=30.03×10−6. The spring-stiffness coefficient, K, is 0.1463×10−3. Eq. (9) gives UD=0.0483 and thus our higher flow speed is marginally in excess of the theoretical divergence-onset critical flow speed based upon a wall of infinite length. In our finite system, the

Wave propagation through a change in wall properties

We now study the passage of a disturbance, commencing as a wave with frequency ω, wavenumber k1=k(x1) and amplitude η01=η0(x1) in region 1, through a region of prescribed change in wall flexibility. It ultimately propagates through region 2 which has uniform wall flexibility (different to that of region 1) as a wave with properties ω, k2=k(x2) and η02=η(x2). In the region of changing wall properties (x1<x<x2, for a downstream-propagating disturbance) both k(x) and η0(x) have continuous

Conclusion

Numerical simulation and theoretical analysis have been combined in a complementary approach to study the excitation of neutrally stable waves on a plate-spring type of flexible wall. A formula has been derived to relate the amplitude of waves downstream of a point of applied oscillatory excitation to those waves that propagate upstream of it. The phenomenon of amplitude difference between such waves is found to be attributable to the selective distribution of energy that originates from the

Acknowledgements

The authors gratefully wish to acknowledge the support of the UK Engineering and Physical Sciences Research Council through the award of a Visiting Research Fellowships that enabled Professor P.K. Sen to visit the University of Warwick during which time much of the work reported above was carried out.

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