3.3.2. Vortex-Induced Vibration

The cable of the mooring system can be abstracted as a flexible riser in the water with the upper end subjected to vertical tension and the lower end fixed. The current, when flowing past the cylinder, generates a vortex alternately, therefore, periodic pressure changes occur around the cylinder [30]. When the cylinder is a flexible structure, the periodic vibration of the cylinder structure is caused by the alternating pressure, namely vortex-induced vibration [30]. This can be used to account for the vibration caused by internal waves flowing past the cable system.

As far as we know, little previous research has investigated the VIV induced by ISWs. However, VIV produced by other kinds of currents around a cylinder has been widely documented. In 1911, Karman conducted a pioneering study on vortex shedding around the flow based on the non-viscous method. When a viscous fluid passes through a cylindrical

structure, its wake will undergo flow separation and form two rows of vortices alternately, which is called a Karman vortex street [31]. When the frequency of the vortex is close to the natural frequency of the structure, structural resonance or VIV occurs. Kassen [32], Sarpkaya [33] and Wu [34] provided detailed explanations of VIV, including generation mechanism, models, wake form, self-locking phenomenon and so on.

Although there have been many studies of VIV on underwater flexible risers in the laboratory, there have been few underwater experimental studies owing to the high cost of full-scale field experiments. Large-scale model tests of tensioning risers (90 m in length and 3 cm in diameter) were carried out in a lake on Norway's west coast in 1997 [35]. Lie used modal analysis to determine that the VIV was irregular in the non-lock-incase, and that the vibration frequency was composed of the natural frequency of the structure plus the frequency of vortex shedding along the length of the structure [35]. Recent studies have shown that the VIV of the flexible riser had multi-order harmonics. Wu determined that higher harmonics could be present with flexible beam VIV [36]. Trim analyzed the second harmonic frequency component of the VIV from a flexible riser (38 m in length and 2.7 cm in diameter) in the Marintek Ocean Basin in Trondheim [37]. When Wang analyzed the multi-order modes and harmonics of the cable-coupled vibration response, he found that, the closer to the midpoint of the cable, the higher the excitation degree of the first two modes, and the farther away from the midpoint of the cable, the more average the excitation degree of the first six modes. [38]. The uniform harmonic component was evidence of internal wave noise. This was also consistent with the results of our analysis in Figure 6. It was considered that the oscillations of the marine cables system would cause noise in the sonar system, reducing the performance of other environmental sensors [30].

When uniform turbulence flows past a rigid, fixed riser of diameter *d*, the time-varying motion of the riser can be made up of a series of mode-shapes if the dynamical process of the riser vibration is approximately linear [35].

$$\mathbf{x}(t,z) = \sum\_{n=1}^{\infty} w\_n(t)\boldsymbol{\varrho}\_n(z), \; z \in [0,L], \tag{7}$$

where *L* is the length of the riser, *z* is the vertical coordinate, and *t* is the time. The horizontal displacement of the riser *x(t,z)* is determinded by the mode-shape *ϕn*(*z*) and the modal weight *wn*(*t*), *n* = 1, 2, 3 . . . [35].

The vortex shedding frequency is

$$f\_{VIV} = \text{St} \cdot \text{u} / d,\tag{8}$$

where the *u* is the free-current velocity. When the cable diameter is no more than 0.1 m, the maximum Reynolds number for the mooring system is usually less than 10<sup>5</sup> , and during a wide range of the Reynolds number (Re), 10<sup>2</sup> < Re < 10<sup>5</sup> , the Strouhal number (*St*) varies little and has a value around 0.2 [30,35].

We knew the diameter of the cable (1 cm) and the velocity at the depth of the hydrophones at different times. The vortex shedding frequency was consistent with the structural natural frequency [30]. The shedding frequency was calculated using Equation (8). As shown in Table 3, shedding frequency was positively correlated with the ISW velocity. The velocity increased as the internal wave approached the hydrophone and decreased as the internal wave moved away from the hydrophone. Therefore, the frequency of the noise appearred to increase first and then decrease, appearing as a frequency fluctuation in the time–frequency spectra.


**Table 3.** ISW velocities and calculated frequencies of the hydrophones at different times.

The fundamental frequencies (Table 2) of the noise with maximum spectra at 18:09, 18:06 and 06:00 UTC+8 were 12.72 Hz, 9.16 Hz and 8.18 Hz, respectively. The actual frequency was higher than the calculated frequency. The vibration of the cable depended not only on the velocity of the internal waves, but also on the mass of the cable, the added mass, the elasticity and the damping coefficient [35]. Locking and synchronization may occur if the natural resonant frequency of the cable approximates the Strouhal frequency [30]. In this case, the vortex fell off at the actual frequency, not the frequency calculated by Equation (8), because the formation and shedding process of the vortex changed the added mass of the cable [30,35]. The change in added mass may be positive or negative, causing an increase or decrease in shedding frequency [35]. In addition, the velocity of ISW varied greatly with time and space, further complicating the vortex shedding frequency [38].

Together, the characteristics and frequency components of the VIV were consistent with our analysis of flow noise. The results of the experiment found clear support for low frequency noise induced by VIV when ISWs flowed past the marine cable system.
