Solid waveguide immersed in a perfect fluid

This study is to validate the SAFE method for waveguides immersed in perfect fluids. A 1mm radius steel cylinder bar immersed in water is used, the results of which have already been studied by DISPERSE and shown in Sec. 2.3.4. The geometry of the system is shown in Fig. 2.9. The steel bar is 2mm in diameter and the surrounding water is modeled by a 4mm thick ring having an inner diameter of 2mm. The absorbing region is modeled by a 5mm thick ring having an inner diameter of 10mm. The material properties are chosen to be the same as they were used in DISPERSE. The whole geometry is meshed by 7563 triangular elements of 1st order (each element has 3 nodes), which are automatically generated by the software used [49]. The number of degrees of freedom is 14912. A typical calculation of one SAFE model presented here takes approximately half a minute on a Pentium

4 PC with 1 Gbyte memory.

The system is solved using the SAFE method to find values of the wavenumber k at different frequencies. For each frequency, several solutions are obtained. For each solution, the amplitude of normal stress in the radial direction T_rr is calculated at each nodal position in the solid domain and the pressure p is calculated at each nodal position in the fluid domain and in the absorbing region. These quantities are equal at the border between the solid and fluid according to the imposed boundary condition. Solutions which have higher values of T_rr in the solid domain than −p in the fluid domain generally represent modes guided along the bar and radiating in the water, while other solutions represent resonances of the whole system and are unwanted.

Fig. 2.10 shows SAFE solutions at 500 kHz. There are three propagating modes existing at this frequency: the L(0,1) mode which is shown in Fig. 2.10(a), the T(0,1) mode which is shown in Fig. 2.10(b), and the F(1,1) mode which is shown in Fig. 2.10(c). Fig. 2.10(d) shows an unwanted solution that corresponds to a resonance of the absorbing region. From the figure, it can be seen in the longitudinal mode that the radial normal stress is concentrated in the center of the bar and some energy is radiating to the water; in the torsional mode the radial normal stress is almost zero (theoretically it should be zero, but there is a very small value due to the numerical approximation); in the flexural mode the radial normal stress is symmetric with respect to a diameter of the bar and energy is radiating to the water. It can be seen that the mode shapes of three fundamental modes agree with the DISPERSE prediction shown in Fig. 2.7.

In order to compare the dispersion curves over a range of frequencies, the system is then solved for 71 frequencies from 100 kHz to 1500 kHz, and solutions which repre-sent the propagating modes are sought according to the above rule. The mode shape information (displacements in each direction) is recorded at each nodal position for each sought eigensolution. By comparing these mode shapes, all the solutions can be classified into the different modes. Fig. 2.11 presents the dispersion curves of wave modes propagating along the steel bar and eventually radiating energy in the

(a)

Figure 2.10: Cross-section distribution of normal stress in solid and pressure in fluid at 500 kHz for example modal results: (a) L(0,1) mode (b) T(0,1) mode and (c) F(1,1) mode; (d) mode resonating in the absorbing region.

infinite water from 100 to 1500 kHz, showing the phase velocity, real wave number, group velocity and attenuation. The real wave number and the attenuation can be obtained from the eigensolutions directly, while the phase velocity can be calculated by C_ph = ω/k⁰ and the group velocity is obtained by doing a numerical derivation C_gr = dω/dk⁰. Plain lines are predictions made with the DISPERSE software, while circles represent the SAFE solutions obtained with the model.

(a) (b)

(c) (d)

(a)

Figure 2.11: Dispersion curves of phase velocity (a), wavenumber (b), group velocity (c) and attenuation (d) of 1mm radius circular steel cylinder bar immersed in water, predicted by the SAFE method (◦) and DISPERSE (—).

From the figure it can be seen that the SAFE predictions have good agreement with DISPERSE results at most of the frequencies. The only disagreement appears at 250 kHz - 400 kHz of the F(1,1) mode on the group velocity and attenuation curves, which is a result of inefficiency of the absorbing region at these frequencies. As it has been introduced in Sec. 2.4.5, an efficient length of the absorbing region is dependent on the longest wavelength projecting to the radial direction. In water, only the bulk wave can be radiated, thus only the angle of the radiation decides the

maximum radiated wavelength in the radial direction. This can be derived using:

λrad = λwater

cosθ (2.34)

Where λ_water is the radiation wavelength in water and λ_rad is its projection in the radial direction. θ is the radiation angle illustrated in Fig. 2.4 which has been decided by Eq. (2.18). If the phase velocity of the guided mode is lower than the bulk velocity of water, there will be no radiation.

From Fig. 2.11(a), it can be seen that the phase velocity of the F(1,1) mode at 250 kHz to 400 kHz is just above the bulk velocity of water, therefore leaky waves have large angles of radiation θ, so that λ_rad is very large, thus the absorbing region does not perform well in such cases according to the previous studies [50]. By increasing the length of the absorbing region, the inaccurate frequency range can be reduced, however it will be much more time consuming to solve the model.