# Wave Propagation in Laminated Cylinders with Internal Fluid and Residual Stress
## 3 Numerical Examples
### Water-Filled Steel Pipe  
As a first example, a water-filled, isotropic, cylindrical pipe with \(h/R=0.05\) is considered. The case is presented to validate the procedure by comparison with analytical results obtained by solving the Donnell-Mushtari equations for a thin cylindrical structure coupled with internal fluid as in [30]. At the same time, the effect of the FSI on the waves propagating in the structure is investigated. The assumptions and material properties are those used in [30]: [PERSON]'s modulus \(E=192\) GPa, [PERSON]'s ratio \(\
u=0.3\), density \(\rho=7800\) kg m\({}^{-3}\) for steel; density \(\rho=1000\) kg m\({}^{-3}\) and free wave speed \(c=1481\) m s\({}^{-1}\) for water. The FE model for this system is realised using 1 solid structural brick element (SOLID45 in ANSYS) and 20 acoustic fluid elements (FLUID30 in ANSYS), resulting in 27 DOFs after the WFE reduction. Although a single shell element could be used in this case, one solid element is preferred to avoid numerical issues related to the presence of rotational degrees of freedom. Figure 2 shows the dispersion curves for free waves propagating in the fluid-filled pipe together with the dispersion curves for the empty cylinder, and acoustic modes in a hard-walled duct and in a soft-walled duct. In the graphs, the frequency is non-dimensionalised with respect to the ring frequency of the _in vacuo_ shell, \(\Omega_{ring}=1/R\sqrt{E/[\rho(1-v^{2})]}\), while the wavenumber is non-dimensionalised with respect to \(R\). The WFE and the analytical formulation results show good agreement. Differences are explained considering that the analytical dispersion relation for axisymmetric and nonaxisymmetric waves was obtained using Donnell-Mushtari equations of motion for the shell. Figure 2 shows that the dispersion curves of the system are modified by the fluid-structure interaction: the presence of the fluid affects the structure's behaviour and therefore 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, leads to incorrect conclusions. It can be noted that branches represent waves which are predominantly either structural-borne or sound-borne, with a behaviour intermediate between structural wave motion in the empty cylinder and fluid wave motion in a rigid or in a soft waveguide. It can also be noted that veering phenomena occur [31]: structural wavemodes become quasi-fluid wave modes and vice versa, giving information regarding the frequency and branches for which wave mode exchange occurs. For \(n=0\), Figure 2a, branch 1 cuts-off at \(\omega=0\) as an acoustical wave in a hard-walled duct and changes to a structural flexural branch as the frequency increases. The second curve represents an almost pure shear wave (fluid viscosity has been ignored and tangential motion is uncoupled from the fluid). The third curve starts propagating as a structural longitudinal wave and then becomes a fluid wave for \(\omega\approx\Omega_{ring}\). For \(\omega\geq\Omega_{ring}\), due to the increase in the radial displacement of the structure, the movements of the shell and the fluid become coupled and mode veering take place. The fourth curve starts propagating as a fluid wave; at \(\omega\approx\Omega_{ring}\) it veers to an extensional structural wave before veering again to a fluid wave. It can be seen that higher order branches exhibit similar behaviour: close to points where the first extensional branch and acoustic branches cross, veering occurs through Poisson contraction effects and coupling between fluid and structure radial motion can be seen. Figure 2b shows the results for the \(n=1\) circumferential mode. A pure torsional mode uncoupled from the fluid is absent since axial, radial and tangential displacements are coupled.