1. Introduction
Fluid-filled cylinders are components in a variety of systems such as chemical, petrochemical, gas, hydraulic, offshore plants and the human cardiovascular system. The simulation of their dynamic behaviour, together with the diagnosis of their structural health conditions, are important for design, safety, economic, environmental and medical reasons. Numerical prediction of wave propagation in cylindrical structures, both in-vacuo and fluid-filled, provides information that is essential in many applications such as dynamics and shock response, noise transmission, material characterisation, non-destructive testing (NDT) and structural health monitoring (SHM).
The prediction of waves that can propagate over long distances with low attenuation, together with their interaction with defects and inclusions, is fundamental for the implementation of ultrasonic guided wave techniques, which are commonly used for damage detection and long-range inspection of pipes, e.g., [
1]. Characterisation of wave propagation in pipes, including fluid-structure interaction, is also important to detect failure or to predict noise propagation and fluid leakage in pipes. In [
2], for example, theoretical and experimental results for noise and vibration propagation in a polyethylene pipe were presented. When leakage in a pipe occurs, acoustic waves are generated and propagate to both ends of the pipe, and their measurement is used to detect and locate leaks. These techniques are widely used for leak detection in buried pipes and benefit from an understanding of wave propagation characteristics [
3]. In [
4], for example, a computational-fluid-dynamic simulation was used to model wave propagation in gas pipelines for leak detection, while in [
5] the pressure wave velocity in fluid-filled pipes was analysed to detect solid deposits in pipes, which might need to be prevented or removed, especially in oil pipelines. As a further example, the prediction of pressure waves, including fluid-structure interaction, was investigated in [
6] for the diagnosis of cladding failure of a nuclear fuel rod. Fuel rod cladding failure generates a pressure wave in the coolant fluid flowing around the fuel rod, which can be detected by a system of sensors, whose design depends on the precise understanding of wave propagation around the fuel rod.
Simulation of wave propagation in human arteries can also indicate the physiological state of the cardiovascular system [
7]. In particular, pressure wave propagation in the arterial system can indicate cardiovascular risk [
8]. Although simplified models are typically assumed, the construction and dynamic behaviour of arteries is complicated, and refined models are required to both improve diagnostic techniques and develop a new generation of vascular prostheses. In [
9], for example, segments of the aorta were modelled as laminated cylindrical shells, including residual stresses, while in [
10] a 3D fluid-structure interaction model of the arteries was studied and results compared with simpler 1D and rigid wall models.
Since Lamb’s pioneering work [
11], numerous researchers have studied wave propagation in fluid-filled cylinders, as reviewed in the book by Païdoussis [
12]. Although there is an extensive literature, the assumptions generally used to model flexible cylinders with an internal fluid often restrict the analysis to the so-called thin-wall theory and to beam-like motion of the pipe. Developing a continuum model in the framework of elasticity theory, including fluid-structure interaction (FSI), is a challenging task. Therefore, theoretical studies are often based on simplifying assumptions and approximations, resulting in an upper-frequency limit of applicability. In order to properly study elastic and acoustic wave propagation in fluid-filled cylinders, both the flexibility of the wall and the compressibility of the fluid must be taken into account, complicating the model and the analysis further [
13]. In addition, the cylinder may be laminated, which substantially increases numerical and computational challenges. The dynamics of thick, laminated, cylindrical structures with FSI can only be modelled accurately if the through-thickness deformation is included. This typically results in very complex analytical/numerical models, which are, however, often limited to low-frequency applications and beam-type modes. In any case, even for simple cases, the analysis is very complex due to the interdependence of the parameters and the transcendental dispersion equations to which explicit solutions do not exist. Relatively recently, coupled analysis has been conducted using the finite element (FE) method [
14]. This has made it possible to simulate the dynamic motion of an acoustic fluid coupled with a flexible structure with arbitrary geometry. However, standard FE analysis cannot model large structures at high frequency or with short wavelengths: there must be at least six elements per wavelength to accurately model the behaviour, and the model size becomes very large. In the low/low-mid frequency range, computational fluid-structure interaction issues can be handled by applying special reduction techniques [
15] or using boundary element (BEM) and finite element (FEM) discretisation [
16].
An important issue regarding applications involving wave propagation is the ability to develop accurate and computationally efficient models. The computational constraints of a full FE analysis of the structure and the complexity of analytical formulations have inspired the development of semi-analytical and numerical methodologies to analyse wave propagation in both structure and acoustic fluid, using new finite element formulations for high-frequencies. In the context of cylindrical waveguides with FSI, we can cite, for example, the Spectral-Finite-Element method, e.g., [
17,
18], which allows the evaluation of wave propagation using a semi-analytical FE formulation. This method has the advantage of being able to predict wave behaviour up to a very high frequency with high accuracy; however, it requires the formulation of new finite elements that must be realised for each model, resulting in a substantial effort, especially for complicated constructions. Another FE-based approach is the Wave Finite Element (WFE) method, which has seen application to a variety of problems due to its ability to model, in a straightforward way, periodic and homogeneous waveguides with complex characteristics. The unit cell of the structure is taken and discretised using a conventional FE formulation, typically using commercial FE software. Periodicity conditions are then applied to the FE equations of motion as developed by Abdel-Rahman in his PhD thesis [
19], and a quadratic or linear eigenvalue problem is obtained. Solutions to this eigenvalue problem give the dispersion curves (the relation between the wavenumbers and frequency) and the wavemodes (the cross-section deformation caused by the passage of a wave through the structure). These provide information for further analysis, such as forced response, the estimation of the parameters in a Statistical Energy Analysis model, the vibroacoustic response and implementation of SHM and NDT techniques. When both the structure and the fluid are taken into account, the WFE method enables a very significant decrease in the size of the problem and in computation time. The approach has been described in previous studies, which have shown its practical applicability to periodic and continuous structures. Examples include laminated plane panels [
20], cylindrical structures [
21], forced vibration [
22], vibroacoustic response [
23] and metamaterial [
24].
The aim of this paper is to investigate the application of the WFE method to model wave propagation in fluid-filled, laminated cylinders of complex construction, with and without pre-stress conditions. The method can be implemented at a very low computational cost and exploits the versatility of commercial FE software. The method is applied using a 2D formulation for axisymmetric structures such as that described in [
25]. Compared to the 1D formulation, applied for example in [
26], here only an arbitrarily small segment is assumed as the unit cell instead of an entire axial segment of the structure. This results in significant advantages compared to the 1D formulation: the dispersion curves can be evaluated for a specified circumferential mode order separately without requiring any further analysis; helical wave propagation can be also studied; due to the small segment analysed, the use of solid elements instead of shell elements is not an issue—computational cost is very small; by avoiding the use of shell elements, numerical issues related to the use of rotational DOFs are removed while, at the same time, the through-thickness dynamics up to high frequency can be predicted accurately.
In this paper, the WFE method is developed and applied to model wave propagation in cylindrical structures with internal fluid and including pre-stress effects due to internal pressure. In this method a very small segment of the system is discretised using 3D solid elements, through the thickness, and 3D fluid elements. In our numerical examples, we use the FE software ANSYS, although any FE package could be used. Thus, each layer is discretised using one or more solid elements to accurately represent the stress/strain behaviour of the structure. It is shown that the inclusion of pre-stress in the structure is also straightforward. In particular, it is shown that pre-stress conditions can be applied with very little effort using FE software by evaluating the FE static stiffness matrix of the loaded 2D small segment of the structure using the same FE model of the unloaded structure. Residual stresses in the structure are included in the model by evaluating the FE pre-stress stiffness matrix of the loaded segment, which is then added to the segment’s original stiffness matrix.
The paper is organised as follows. In the next section the formulation of the method for the coupled motion of a cylindrical structure and internal acoustic fluid is briefly presented. The resulting eigenvalue problem is formulated such that the real-valued frequency is given and the complex-valued wavenumbers (eigenvalues) and wavemodes (eigenvectors) are found. In
Section 3 numerical examples are presented. The first example concerns a thin-walled steel pipe filled with water, for which results are validated with respect to results obtained using an analytical formulation. The effect of increasing pipe thickness is also studied. The second example shows the application of the method to the more complicated example of a laminated sandwich cylinder containing air. The third example deals with a helical reinforced hard rubber pipe with internal pressure, this being an example of a laminated cylinder used for high-pressure applications.
2. 2D WFE Formulation for Fluid-Filled Cylinders
Consider a fluid-filled cylinder as depicted in
Figure 1a, where cylindrical coordinates (
y,
r,
) and geometrical parameters are defined. The cylinder is of thickness
h and mean radius
R. At frequency
, time-harmonic waves propagate in the structure as helical waves so that
, where
is the complex wave amplitude,
t is time, while
and
are the wavenumber components in the
y and
directions.
The cylinder is homogeneous in the axial and circumferential directions, with an arbitrary period. Exploiting this periodicity, a very small segment of angle
and length
is taken,
Figure 1a. The segment is then meshed by a number of elements as shown in
Figure 1b, using standard FE method. The elements used depend on the nature of the structure: typically solid, 8-noded solid elements might be used to model the structure, with acoustic elements for the fluid.
The FE equations of motion of the discretised model in
Figure 1b can be obtained using standard FE analysis, and vary according to the assumed field variables. The FE equation of motion in their standard, unsymmetrical formulation for acoustic FSI, when the structural and acoustic variables are the nodal displacements,
, and acoustic pressures,
, is
In Equation (
1), the subscripts
s and
f refer to structure and fluid respectively,
is the fluid density,
,
,
,
are mass and stiffness matrices,
and
are coupling submatrices,
and
are the nodal forces,
is the viscous damping matrix for the structure, while
is a dissipation term added to account for absorption of the acoustical pressure waves. Note that Equation (
1) can alternatively be rewritten using acoustic potential variables for the fluid, e.g., [
15].
Residual radial and hoop stresses in the structure can be found by evaluating the FE pre-stress stiffness matrix, which is then added to the original stiffness matrix. In particular, the effect of internal pressure can be modelled by the pre-stress matrix
, which is obtained simply by solving a FE static analysis for a slice of the cylinder to which the internal pressure is applied. The effect of axial stress can be also considered by evaluating the pre-stress stiffness matrix,
, obtained through a FE static analysis of the segment of the cylinder loaded, for example, by the axial stress
which is applied on the faces parallel to the
plane. The resulting total stiffness matrix is
where
is the stiffness matrix of the unloaded segment.
Assuming time-harmonic behaviour and omitting the time-dependence
, Equation (
1) can be rewritten in the form
where
is the dynamic stiffness matrix,
,
where
T denotes the matrix transpose, and the matrices are those of the coupled system in Equation (
1), typically obtained using FE commercial software.
Following the WFE formulation in [
25,
27], a 4-noded super-element is formed by concatenating all the acoustical and structural degrees of freedom (DOFs) at the four corners of the prismatic segment in
Figure 1c. The total vectors of the DOFs
and nodal forces
are ordered as
The local coordinates must be rotated as described in [
25],
Figure 1d. This is required to model the curvature of the cylinder and can be easily accomplished by calculating the rotation matrix
; the new FE matrices in the global system of reference become
,
and
.
Using periodicity conditions, the relations between the nodal DOFs at the four corners are
where
Equation (
6) then gives
where
denotes the identity matrix. Considering internal equilibrium and periodicity conditions, the relationship between the nodal forces can be written as
Using Equations (
8) and (
9), the WFE reduced equation of motion becomes
and hence
where
,
and
are the WFE reduced stiffness, mass and damping matrices whose size are
,
m being the total number of nodal DOFs
at corner 1,
Figure 1c.
For closed cylindrical structures, the number of wavelengths around the circumference must be integral and therefore the circumferential wavenumber
must be an integer
. It follows that
for a given circumferential mode order
n. Equation (
11) thus becomes an eigenvalue problem in
and
, whose solutions give the WFE predictions of the wavenumbers in the
y direction (through
) and the wavemodes (through
). The form of the eigenproblem depends on the physical problem and on whether the frequency or the wavenumber is assumed to be real [
28].
In this paper, the frequency is assumed to be real, and a quadratic eigenvalue problem in
is obtained as shown in
Appendix A. In particular, the eigenvalue formulation solved in this paper (to obtain the complex dispersion curves and wavemodes) is a linear companion form of the quadratic eigenvalue problem, see Equations (
A3)–(
A5). As a result of the 2D formulation and because of the small number of DOfs, the equation can be solved without numerical difficulties and at a very low computational cost using standard functions and routines; in the examples herein presented the function
eigs in Matlab was used.
4. Conclusions
In this paper the Wave Finite Element method was applied to modelling wave propagation in thick, laminated, cylinders of different construction and filled with different fluids, including pre-tensioning and residual stress effects. Compared to previous studies, the paper focused on these complicated cases, showing an alternative to other methodologies, and to the 1D formulation of the method, for the wave and vibro-acoustic characterisation of fluid-structure interaction and pre-stress effects.
In particular, the formulation proposed in the paper showed that results for any given circumferential mode order can be found without any additional investigation by specifying the circumferential wavenumber. The laminated structures were modelled using 3D solid elements, thus representing the through-thickness dynamics with high accuracy up to high frequency, but at very low computational cost, since only a very small segment of the system needs to be analysed. Therefore, the method shows promise for its implementation in optimisation processes and biomechanics research involving blood vessels, for example.
Numerical examples were presented and discussed. These concerned thick, laminated cylinders filled with water and air and a composite reinforced rubber cylinder pre-stressed by circumferential tension. It was found that an attempt to interpret the dispersion behaviour of the system in terms of the dispersion curves of the corresponding uncoupled subsystems, viz. an empty cylinder, fluid in a rigid duct and fluid in a soft duct, generally leads to incorrect conclusions, in particular when the wavenumber increases and laminated constructions are considered; in these cases, it was found that coupling and mode exchange between acoustic and elastic waves become significant. When complicated constructions with internal reinforcements and pre-stress conditions (for example due to internal pressure) are considered, it was shown that the analysis of dispersion curves could be used to investigate and improve the performance of the pipe at a design stage.