Submerged fluid-filled cylindrical shell subjected to a shock wave: Fluid–structure interaction effects

A submerged fluid-filled circular cylindrical shell subjected to a shock wave propagating in the external fluid is considered. The study focuses on a number of acoustic and structural effects taking place during the interaction. Specifically, the influence of the acoustic phenomena in the fluid on the stress–strain state of the shell is analysed using two different visualization techniques. The effect that the parameters of the shell have on the internal acoustic field is addressed as well, and the ‘shock transparency’ of various shells is discussed. Special attention is paid to the analysis of the contribution of the terms in the shell equations representing bending stiffness, and the limits of applicability of the membrane theory of thin shells are discussed in the fluid–structure interaction context. The possibility of cavitation in the internal fluid is investigated, and the effect that cavitation could have on the structural dynamics of the shell is discussed. The present paper is a follow-up of the author's earlier study of the interaction between fluid-filled cylindrical shells and external shock waves.     

A Refined Asymptotic Model of Fluid-Structure Interaction in Scattering by Elastic Shells

A refined asymptotic model of fluid-structure interaction in scattering by elastic shells is proposed. The model takes into consideration transverse compression of a shell by a fluid and some other phenomena. As an illustration, scattering of a plane acoustic wave by a circular cylindrical shell is considered. Comparison of numerical data corresponding various approximate approaches is provided. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interaction between a submerged evacuated cylindrical shell and a shock wave—Part II: Numerical aspects of the solution

The interaction between a submerged elastic circular cylindrical shell and an external shock wave is addressed. A linear, two-dimensional formulation of the problem is considered. A semi-analytical solution is obtained using a combination of the classical analytical approach based on the use of the Laplace transform and separation of variables, and finite difference methodology. The study consists of two parts. Part I focuses on the simulation and analysis of the acoustic fields induced during the interaction. Both the diffraction (absolutely rigid cylinder) and complete diffraction–radiation (elastic shell) are considered. Special attention is paid to the lower-magnitude shell-induced waves representing radiation by the elastic waves circumnavigating the shell. The focus of Part II is on the numerical analysis of the solution. The convergence of the series solution and finite-difference scheme is analysed. The computation of the response functions of the problem is discussed as well, as is the effect of the bending stiffness on the acoustic field. The membrane model of the shell is considered to analyse such effect, which, in combination with the models addressed in Part I, allows for the analysis of the evolution of the acoustic field around the structure as its elastic properties change from an absolutely rigid cylinder to a membrane. The results of the numerical simulations are compared to available experimental data, and a good agreement is observed.     

Interaction between a submerged evacuated cylindrical shell and a shock wave—Part I: Diffraction–radiation analysis

The interaction between a submerged elastic circular cylindrical shell and an external shock wave is addressed. A linear, two-dimensional formulation of the problem is considered. A semi-analytical solution is obtained using a combination of the classical analytical approach based on the use of the Laplace transform and separation of variables, and finite difference methodology. The study consists of two parts. Part I focuses on the simulation and analysis of the acoustic fields induced during the interaction. Both the diffraction (absolutely rigid cylinder) and complete diffraction–radiation (elastic shell) are considered. Special attention is paid to the lower-magnitude shell-induced waves representing radiation by the elastic waves circumnavigating the shell. The focus of Part II is on the numerical analysis of the solution. The convergence of the series solution and finite-difference scheme is analysed. The computation of the response functions of the problem is discussed as well, as is the effect of the bending stiffness on the acoustic field. The membrane model of the shell is considered to analyse such an effect, which, in combination with the models addressed in Part I, allows for the analysis of the evolution of the acoustic field around the structure as its elastic properties change from an absolutely rigid cylinder to a membrane. The results of the numerical simulations are compared to available experimental data, and a good agreement is observed.     

Transient response of a moving spherical shell in an acoustic medium

A ring-stiffened spherical shell is submerged in an acoustic medium. The shell is thin and elastic. The acoustic medium is inviscid, irrotational and compressible. The center of mass of the shell is subjected to a translational acceleration which is an arbitrary function of time. The absolute displacements of the shell are expressed in terms of the relative displacements and the displacement of the base of the shell, base being defined as the rigid ring placed at the equator. The motion of the acoustic medium is governed by the wave equation. The transient response of the shell is investigated numerically. The results are compared with the results of the in-vacuo response. The effects of the plane wave approximation and the base velocity on the transient response of the shell are studied. The numerical results show that the plane wave approximation accurately predicts the response of the shell in the acoustic medium for short times after excitation. The displacements of the shell in fluid are larger than those in vacuo. But when the base of the shell is restrained from translating, the displacements in fluid are smaller than those in vacuo. Therefore, base translation has a very significant effect on the transient response of the shells submerged in an acoustic medium.     

Transient radiation by a submerged fluid-filled cylindrical shell

The radiation by a submerged fluid-filled cylindrical shell in response to a transient external pressure pulse is considered, and a semi-analytical model based on the Reissner–Mindlin shell theory is employed to simulate the interaction numerically. Two types of radiated waves that have been previously seen in experimental images for a submerged evacuated cylindrical shell are observed in both the external and internal fluids, the symmetric Lamb waves S₀ and the antisymmetric Lamb (or pseudo-Rayleigh) waves A₀. The third type of radiated waves is also observed that has not been explicitly imaged either experimentally or numerically for a submerged evacuated cylindrical shell, and it is demonstrated that these waves are the Scholte–Stoneley waves A. The effect that the complex structure of the radiated field has on the wave phenomena in the internal fluid is analyzed for shells of several different thicknesses, and the results of this analysis are summarized in the form of diagrams suitable for the use at the pre-design stage.     

Interaction of a spherical shockwave and an elastic circular cylindrical shell

The paper studies the interaction of a spherical shock wave with an elastic circular cylindrical shell immersed in an infinite acoustic medium. The shell is assumed infinitely long. The wave source is quite close to the shell, causing deformation of just a small portion of the shell, which makes it possible to represent the solution by a double Fourier series. The method allows the exact determination of the hydrodynamic forces acting on the shell and analysis of its stress state. Some characteristic features of the stress state are described for different distances to the wave source. Formulas are proposed for establishing the safety conditions of the shell.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 94–104, September 2004.     

VIBRATION AND ACOUSTIC RADIATION FROM SUBMERGED STIFFENED SPHERICAL SHELL WITH DECK-TYPE INTERNAL PLATE

Based on the motion differential equations of vibration and acoustic coupling system for thin elastic spherical shell with an elastic plate attached to its internal surface, in which Dirac-δ functions are employed to introduce the moments and forces applied by the attachment on the surface of shell, by means of expanding field quantities as Legendre series, a semi-analytic solution is derived for the vibration and acoustic radiation from a submerged stiffened spherical shell with a deck-type internal plate, which has a satisfactory computational effectiveness and precision for an arbitrary frequency range. It is easy to analyze the effect of the internal plate on the acoustic radiation field by using the formulas obtained by the method proposed. It is concluded that the internal plate can significantly change the mechanical and acoustic characteristics of shell, and give the coupling system a very rich resonance frequency spectrum. Moreover, the method can be used to study the acoustic radiation mechanism in similar structures as the one studied here.     

On the possibility of shock-induced cavitation in submerged cylindrical shell systems

A circular cylindrical shell loaded by one or two fluids and responding to an external shock wave is analyzed in the context of the possible inception of shock-induced cavitation. Several scenarios of fluid contact are considered including a submerged evacuated shell and a submerged fluid-filled shell for three different combinations of the parameters of the internal and external fluids. A semi-analytical shell-shock interaction model is employed in order to predict the regions of the fluids where cavitation is likely to occur, and the respective cavitation development is hypothesized about. The most interesting and practically important finding is that when fluid is present both inside and outside the shell, there exist conditions when cavitation is expected to occur in both the internal and external fluid, resulting in a particularly complex and violent structural re-loading occurring upon the collapse of the respective cavitation regions. The inception of cavitation in the internal fluid alone and in the external fluid alone is also possible. The findings are summarized in a manner that is suitable for use at the pre-design stage as a guide for preliminary assessment of the possibility of shock-induced cavitation in fluid-interacting industrial systems.     

Sound transmission through laminated composite cylindrical shells using analytical model

Composite structures are often used in aircraft because of advantages offered by a high strength to weight ratio. Sound transmission through an infinite laminated composite cylindrical shell is studied in the context of the transmission of airborne sound into aircraft interior. The shell is immersed in an external fluid medium and contains an internal fluid, and airflow in an external fluid medium moves with a constant velocity. The different parameters were used to see how laminate specification affected noise transmission. An exact solution is obtained by solving the vibration equation of laminated composite shell and acoustic wave equations simultaneously. Transmission losses (TLs) obtained from numerical solution are compared with those of other authors. The effects of different source condition, structural properties and flight conditions on TL are studied for a range of values, especially, incident angle of the plane wave, Mach number and flight altitude of aircraft, stack sequences, angle of warp and damping.     

Acoustic scattering from baffled membranes that are backed by elastic cavities

An elastic membrane backed by a fluid-filled cavity in an elastic body is set into an infinite plane baffle. A time harmonic wave propagating in the acoustic fluid in the upper half-space is incident on the plane. It is assumed that the densities of this fluid and the fluid inside the cavity are small compared with the densities of the membrane and of the elastic walls of the cavity, thus defining a small parameter . Asymptotic expansions of the solution of this scattering problem as →0, that are uniform in the wave number k of the incident wave, are obtained using the method of matched asymptotic expansions. When the frequency of the incident wave is bounded away from the resonant frequencies of the membrane, the cavity fluid, and the elastic body, the resultant wave is a small perturbation (the “outer expansion”) of the specularly reflected wave from a completely rigid plane. However, when the incident wave frequency is near a resonant frequency (the “inner expansion”) then the scattered wave results from the interaction of the acoustic fluid with the membrane, the membrane with the cavity fluid, and finally the cavity fluid with the elastic body, and the resulting scattered field may be “large”. The cavity backed membrane (CBM) was previously analyzed for a rigid cavity wall. In this paper, we study the effects of the elastic cavity walls on modifying the response of the CBM. For incident frequencies near the membrane resonant frequencies, the elasticity of the cavity gives only a higher order (in ) correction to the scattered field. However, near a cavity fluid resonant frequency, and, of course, near an elastic body resonant frequency the elasticity contributes to the scattered field. The method is applied to the two dimensional problem of an infinite strip membrane backed by an infinitely long rectangular cavity. The cavity is formed by two infinitely long rectangular elastic solids. We speculate on the possible significance of the results with respect to viscoelastic membranes and viscoelastic instead of elastic cavity walls for surface sound absorbers.     

Sound wave scattering by circular cylindrical shells

The symmetric (S₀) and antisymmetric (A₀) Lamb-type waves generated in a thin elastic cylindrical shell by normal incidence of an acoustic wave have been considered. The typical frequency dependencies (FD) of the backscattered acoustic pressure at on observetion point in the far field are presented. The spectra of the S₀ and A₀ waves are marked on them. It was found that if the A₀ wave is excited in the shell, then its phase velocity is greater than the sound velocity in the fluid surrounding the shell. The parameter which defines the center of the strong bending domain (SBD) is defined. It is shown that in this domain the A₀ wave is practically non-dispersive. Phase velocity data for the A₀ wave are given. Spectra and dispersion curves of the S₀ wave for shells which have different relative thickness s and which are made of different materials have been examined.     

矩形弹性壳液耦合系统中的重力波分析

根据非线性动力学理论，建立了矩形壳液耦合系统的非线性振动方程组，通过数值求解，发现当激振频率为壳体固有频率与重力波频率之和，且激振力足够大时，会产生大幅低频重力波，通过实验验证，发现了壳液耦合系统中存在的大幅低频重力波现象，实验结果与理论结果基本吻合。     

基于二阶双渐近法的双层圆柱壳在水下爆炸作用下的鞭状运动

针对潜艇在水下爆炸载荷下的鞭状运动,从波动方程出发,推导了二阶双渐近法后期近似,并结合声固耦合法初步解决了双层圆柱壳的内域问题,然后将其与显式有限元耦合形成了圆柱壳结构水下爆炸流固耦合分析方法。通过简单算例验证了本文分析方法的有效性和精度。最后,基于此方法分析了双层圆柱壳结构在水下爆炸载荷下的总体响应特性以及周期比和爆距比对其影响规律。     

A Godunov-type finite-volume scheme for unified solid-liquid elastodynamics on arbitrary two-dimensional grids

A Godunov-type finite-volume scheme is presented for the elastodynamic equations written in terms of displacement velocities and stresses. The mathematical model universally includes elastic solids and liquids resulting in a method highly suitable for studies of acoustic wave scattering at solid-liquid interfaces. The scheme is written for a generic grid of control volumes in two spatial dimensions. The governing equations in an integral conservation form are applied to the control volumes. The exact one-dimensional Riemann problem solution at the volumes borders is used to evaluate fluxes. The spatial accuracy of the scheme is enhanced to second order using linear extrapolation of variables to the volumes faces where the Riemann solver is applied. A predictor-corrector technique is used to improve the accuracy in time. The scheme is then considered in a triangular unstructured grid environment with a dual grid of node-centered control volumes. A dynamic reversible refinement/derefinement procedure is described to enhance resolution locally at sharp variations in the solution or in subdomains of special interest. Implementation of the external and internal boundary conditions is discussed. A demonstrative application to acoustic pulse interaction with a solid shell in liquid is presented.Received: 13 February 2003, Accepted: 29 July 2003, Published online: 2 September 2003     

Analytical model of sound transmission through laminated composite cylindrical shells considering transverse shear deformation

Composite structures are often used in the aerospace industry due to the advantages offered by a high strength to weight ratio. Sound transmission through an infinite laminated composite cylindrical shell is studied in the context of the transmission of airborne sound into the aircraft interior. The shell is immersed in an external fluid medium and contains internal fluid. Airflow in the external fluid medium moves with a constant velocity. An exact solution is obtained by simultaneously solving the first-order shear deformation theory （FSDT） of a laminated composite shell and the acoustic wave equations. Transmission losses （TL） obtained from numerical solutions are compared with those of other authors. The effects of structural properties and flight conditions on TL are studied for a range of values, especially, the Mach number, stack sequences, and the angle of warp. Additionally, comparisons of the transmission losses are made between the classical thin shell theory （CST） and FSDT for laminated composite and isotropic cylindrical shells.     

Isolation of the resonant component in acoustic scattering from fluid-loaded cylindrical shells

For the problem of scattering of a plane acoustic wave by a cylindrical elastic shell immersed in a fluid, we demonstrate that a recently proposed “intermediate background” formalism serves to separate the resonant component from the non-resonant background in the scattering amplitude. Numerical calculations are performed for the case of an air-filled aluminium shell in water.     

Reflection and radiation of a wave system at the open end of a submerged elastic pipe

The reflection and radiation of a wave system at the open end of a submerged semi-infinite elastic pipe are studied. This wave system consists of a flexural wave in the pipe, an acoustic surface wave in the fluid exterior to the pipe and an acoustic wave in the pipe’s interior. Fourier transform techniques are used to formulate this semi-infinite geometry problem rigorously as a Wiener-Hopf type equation. An approximate solution is obtained by using a perturbation method in which the ratio of the massdensities of the fluid and the pipe material is regarded as a small parameter. The calculation of the reflection coefficient is emphasized, and the polar plots of the radiation coefficient are also presented.     

Acoustic Radiation from a Finite Length Cylindrical Shell Excited by an Internal Acoustic Source: Solution Based on a Boundary Element Method and a Matched Asymptotic Expansion

This paper deals with acoustic radiation by a thin elastic shell, closed by two perfectly rigid discs, immersed in water and filled with air. The system is driven by an internal acoustic source. The shell has a length L, is clamped along one of its boundaries and is freely supported along the other boundary. Using the infinite domain Green's function, the radiated acoustic pressure is modeled by a hybrid layer potential (linear combination with nonreal coefficient of a simple layer and a double layer). Using Green's tensor of the in vacuo shell operator, the shell displacement is expressed as the sum of the field generated by the acoustic pressures and that due to boundary sources. Finally, the Green's function of the interior Neumann problem is used to express the acoustic pressure inside the shell in terms of the acoustic source and shell normal displacement: this representation fails for any frequency equal to one of the resonance frequencies of the shell interior. To overcome this, a light fluid approximation, which is allowed because the inner fluid is a gas, is adopted. Around each resonance frequency, an inner approximation is defined which matches the classical outer approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Wave propagation in an elastic tube: a numerical study

In this paper, a fluid–wall interaction model, called the elastic tube model, is introduced to investigate wave propagation in an elastic tube and the effects of different parameters. The unsteady flow was assumed to be laminar, Newtonian and incompressible, and the vessel wall to be linear-elastic, isotropic and incompressible. A fluid–wall interaction scheme is constructed using a finite element method. The results demonstrate that the elastic tube plays an important role in wave propagation. It is shown that there is a time delay between the velocity waveforms at two different locations and that the peak velocity increases while the low velocity decreases in the elastic tube model, contrary to the rigid tube model where velocity waveforms overlap each other. Compared with the elastic tube model, the increase of the wall thickness makes wave propagation faster and the time delay cannot be observed clearly, however, the velocity amplitude is reduced slightly due to the decrease of the internal radius. The fluid–wall interaction model simulates wave propagation successfully and can be extended to study other mechanical properties considering complicated geometrical and material factors.