On the behaviour of a fluid-loaded cylindrical shell with mean flow

Abstract

The unsteady behaviour of an infinitely long fluid-loaded elastic plate which is driven by a single-frequency point-force excitation in the presence of mean flow is known to exhibit a number of unexpected features, including absolute instability when the normalized flow speed, U, lies above some critical speed U 0 , and certain unusual propagation effects for U < U 0 . In the latter respect Crighton & Oswell (1991) have demonstrated most significantly that for a particular frequency range there exists an anomalous neutral (negative energy) mode which has group velocity pointing towards the driver, in violation of the usual radiation condition of outgoing waves at infinity. They show that the rate of working of the driver can be negative, due to the presence of other negative-energy waves, and can also become infinite at a critical frequency corresponding to a real modal coalescence. In this paper we attempt to extend these results by including, as is usually the case in a practical situation, plate curvature in the transverse direction, by considering a fluid-loaded cylinder with axial mean flow. In the limit of infinite normalized cylinder radius, a, Crighton & Oswell's results are regained, but for finite a very significant modifications are found. In particular, we demonstrate that the additional stiffness introduced by the curvature typically moves the absolute-instability boundary to a much higher flow speed than for the flat-plate case. Below this boundary we show that Crighton & Oswell's anomalous neutral mode can only occur for a > a 1 (U), but in practical situations it turns out that a 1 (U) is exceedingly large, and indeed seems much larger than radii of curvature achievable in engineering practice. Other negative-energy waves are seen to exist down to a smaller, but still very large, critical radius a 2 (U), while the existence of a real modal coalescence point, leading to a divergence in the driver admittance, occurs down to a slightly smaller critical radius a 3 (U). The transition through these various flow regimes as U and a vary is fully described by numerical investigation of the dispersion relation and by asymptotic analysis in the (realistic) limit of small U. The inclusion of plate dissipation is also considered, and, in common with Abrahams & Wickham (1994) for the flat plate, we show how the flow then becomes absolutely unstable at all flow speeds provided that a > a 2 (U).

Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

“…For this we follow the method, originally developed by Briggs (1964) and Bers (1983) for plasma physics applications, but subsequently widely applied for fluid mechanical and aeroacoustical applications (see Huerre & Monkewitz 1985;Peake 1997;Lingwood & Peake 1999;Peake 2002;Brambley & Peake 2006.…”

“…It has discontinuities along (−i∞, 0) and (0, i∞), which correspond with the branch cuts of the square roots. The notation α is very common in this kind of problems (see Lingwood & Peake 1999;Peake 1997Peake , 2002), but of course should not be confused with the complex modulus of α. However, in this paper the complex modulus does not occur.…”

The origin of hysteresis in the flag instability

“…These results show that U * c is extremely sensitive to curvature. As it was shown by Peake (1997), the plate curvature can be taken into account in the linear regime by adding a spring foundation term to the equation of motion (3.1), with spring rate per unit surface, Ehκ 2 , with E = 1.5 GPa the plate Young's modulus, h = 280 µm the plate thickness, and κ ≈ 0.1 m −1 the curvature. In dimensionless terms, this corresponds to adding a constant term of order δ 2 /h 2 to the dispersion relation (3.16), where δ is the typical deflection due to curvature.…”

On transient oscillations of plates in moving fluids

“…In the last case, we could consider including this by a lateral restoring force in the plate. In fact, Peake [15] discussed such a model: axisymmetric disturbances on a circular cylindrical elastic shell immersed in a fluid which flows uniformly in the axial direction. He found, again, little qualitative difference between his results and those of Crighton and Oswell with regard to absolute instability, but quantitatively its onset occurs at much higher flow speeds when the nondimensionalised shell radius is of order unity.…”