1. Introduction
The submerged floating tunnel (SFT) is considered to be the most effective method for crossing deeper and wider seas. An anchored SFT typically consists of a tube, end constraints, and the on-way support system. It suspends at a specific water depth through buoyancy force to balance the weight of the tunnel, with a mass ratio
m* < 1 and natural frequency ratio
Rf < 1 [
1,
2,
3,
4]. Compared to traditional water crossing structures, anchored SFTs offer several advantages, including flexible on-way configurations and sections, minimal impact on navigation and the ecological environment, better adaptability to water depth and topography, improved spanning capacity, and lower construction costs. As a result, SFTs have garnered extensive attention from both academic and engineering communities.
When exposed to the ocean environment, the flow-induced vibration (FIV) of SFTs is generated by the alternating vortex shedding in wake flow under current. When the response frequency and vortex shedding frequency closely match the natural frequency of the SFT under critical flow velocity, the responses of the SFT can be significantly amplified. This phenomenon is commonly known as lock-in [
5,
6,
7,
8]. Frequency lock-in occurs within a certain reduced velocity range. Large-amplitude vibration in this lock-in regime can lead to fatigue damage and significantly affect the safety, operation, and remaining fatigue life of the SFT. Hence, this issue should be given careful attention.
Directive experiments on SFTs are lacking due to the high expenses. However, bluff bodies, which are similar to the tubes in SFTs, have been studied experimentally for a long time. Numerous experiments [
9,
10,
11,
12] have demonstrated that the FIV response of a bluff body is influenced by both model parameters, such as mass ratio
m*, damping ratio
ξ, and natural frequency ratio
Rf, etc., as well as flow parameters, including the Reynolds number Re, reduced velocity
Urw, etc. The non-dimensional vibration amplitude and frequency of a bluff body can be expressed as a function of a series of normalized factors.
Early experiments were conducted on a single elastically mounted rigid cylinder, focusing mainly on large-amplitude vibration only in the cross-flow direction (one-degree-of-freedom, 1-DOF) for a high mass ratio
m* =
O(100) [
13]. These experiments indicated that only two branches (the initial excitation branch and the lower branch) for cross-flow vibration amplitude existed under various reduced velocities, as shown in
Figure 1a. The jump from the initial branch to the lower branch corresponded to a wake mode change from 2S (two single vortices shedding per vibration cycle) to 2P (two pairs of vortices shedding per vibration cycle) [
14]. The vibration frequency was close to the natural frequency in the lock-in regime and followed the Strouhal relationship outside, as shown in
Figure 1b.
A decrease in the mass ratio (
m* =
O(10)) can significantly enhance the fluid–cylinder interaction and expand the cross-flow lock-in regime. This leads to the generation of a new large-amplitude branch, known as the upper branch. The vortex shedding mode is 2P, with different vortex intensities for each pair of vortices [
15,
18]. During lock-in, the vibration frequency is larger than the natural frequency (
f* ≈ 1.4), a phenomenon known as soft lock-in [
16].
When the mass ratio decreases to a critical value (i.e.,
mcrit* = 0.54 ± 0.02 for a low mass-damping cylinder), the lower branch in the cross-flow direction disappears. This means that the vibration of the cylinder will have a large amplitude and be continuous with the increase in reduced velocity. Additionally, the vibration frequency may approach or even exceed the vortex shedding frequency [
16,
19].
By confining the vibration of the cylinder only in the cross-flow direction, the true dynamic response and wake pattern may be altered. There, further experiments investigated the in-line vibration of an elastically mounted cylinder and its effect on the cross-flow vibration and wake pattern. These experiments revealed that in-line vibration for a low-mass-ratio cylinder (
m* < 4) occurred more easily at low reduced velocities and significantly enhanced the cross-flow vibration with the increase in reduced velocity. A new large-amplitude branch was generated, known as the super-upper branch (
m* = 2.6). The wake mode observed was 2T, with two sets of triple vortices shedding per vibration cycle [
17].
It is important to note that for the aforementioned experiments, the masses and natural frequencies of the cylinder in the in-line and cross-flow directions are the same.
With the advancement in computer performance, some researchers have begun using computational fluid dynamics (CFD) methods to numerically simulate the vibration of elastically mounted cylinders and the resulting vortex patterns.
Singh and Mittal [
20] conducted simulations of the two-degree-of-freedom vibration of a cylinder with a low mass ratio (
m* = 10) and low Reynolds number (
Re ≤ 500). The finding indicated that the Reynolds number had a significant impact on the vibration of the cylinder and the mode of vortex shedding. Kang et al. [
21] established a two-degree-of-freedom numerical model for a cylinder (
m* = 2.6) using the Reynolds-averaged Navier–Stokes (RANS) method with a modified shear stress transport (SST) turbulence model. They simulated and compared three initial conditions, including increasing velocity, decreasing velocity, and constant velocity. Their work revealed that the two-degree-of-freedom vibration response of cylinder and hydrodynamic loads were significantly dependent on the initial conditions, and the super-upper branch of the cross-flow amplitude and 2T vortex shedding mode can be approximately reproduced only under increasing velocity. Zhao et al. [
22] conducted simulations and analyzed the impact of the natural frequency ratio (
Rf = 1–4) on the two-degree-of-freedom vibration of a cylinder (
m* = 2.0) with a low Reynolds number (
Re = 200). The findings revealed that the vibration of the cylinder and the resulting hydrodynamic loads were complex at
Rf = 2.5 or 3, with two or three wake modes (P+S, 2P, N) present in the lock-in regime. These studies focus on the mass ratios of cylinders that are greater than 1.
Liu et al. [
23] simulated the cross-flow vibration of a cylinder with
m* = 0.7 and compared the results to those of a cylinder with
m* = 2.4. They found that no lower branch of cross-flow vibration amplitude was present for
m* = 0.7. Additionally, the vibration frequency remained consistent with the vortex shedding frequency under reduced velocities ranging from 1 to 20. However, it is worth noting that the cylinder is constrained to vibrate only in the cross-flow direction.
Yu et al. [
24] conducted a numerical investigation into the two-degree-of-freedom maximum vibration amplitude of a cylinder as a function of the Reynolds number (
Re = 75–175) under a limiting condition (
m* = 0 and
ξ = 0) and compared it to the one- or two-degree-of-freedom vibrations of a cylinder with
m* = 1. The results indicated that the maximum cross-flow amplitude increased with the rise in the Reynolds number in the laminar flow regime. Notably, this finding differed from the conclusion proposed by Williamson and Govardhan [
25] for one-degree-of-freedom vibration. Additionally, the maximum cross-flow amplitude decreased with an increase in mass ratio, and the maximum amplitude of two-degree-of-freedom vibration decreased at a faster rate than that of one-degree-of-freedom vibration. For one- and two-degree-of-freedom vibrations of the cylinder, the critical mass ratios were 0.117 and 0.106, respectively, signifying that the vibration of the cylinder remained stable below the critical mass ratio at high reduced velocity.
So far, previous studies on the two-dimensional vibration of elastically mounted cylinders and wake patterns have primarily focused on low Reynolds numbers, even within the laminar flow regime, a mass ratio
m* > 1, and a natural frequency ratio
Rf > 1. Additionally, most experiments and numerical simulations have been conducted under low damping to investigate the large-amplitude vibration in the cross-flow direction [
22]. However, the numerical studies on the 2-DOF vibration of the cylinder with mass ratio
m* < 1 and natural frequency ratio
Rf ≤ 1, i.e., an anchored SFT, are limited and should be addressed.
In this study, a two-dimensional numerical model was established for the two-degree-of-freedom FIV of an anchored SFT, using the Reynolds-averaged Navier–Stokes (RANS) method combined with the fourth-order Runge–Kutta method. The specific parameters for the SFT were m* = 0.84, Rf > 0.54, and ξ = 0.05. This model was subjected to current and was verified using the literature results. Our work addressed the effects of two-degree-of-freedom coupled vibration and structural parameters, such as mass ratio and natural frequency ratio, on SFT response and wake pattern. The numerical investigation covered a wide range of reduced velocities, ranging from 0.6 to 10.
3. Numerical Results
3.1. Two-Degree-of-Freedom FIV
To investigate the effect of coupled vibration for the SFT under current, the two-degree-of-freedom (2-DOF) FIV was simulated numerically in a wide range from 0.6 to 10, and the results were compared to that of one-degree-of-freedom (1-DOF) FIV.
Figure 5 shows the time histories of displacements for the SFT and hydrodynamic coefficients under typical reduced velocities. Note that the spectra of displacements for the SFT are also included. It reveals that, for
= 0.6, the two-degree-of-freedom FIV of the SFT is periodic and dominant at one frequency (narrow peak). The in-line main vibration frequency (i.e., the maximum peak value in the spectrum) is twice that in the cross-flow direction. Phase differences exist between the two-degree-of-freedom vibrations and the hydrodynamic coefficients. For
= 2.5, since the in-line vibration frequency is close to the natural frequency, the amplitude of in-line displacement is larger than that in cross-flow. Cross-flow vibration is irregular and multi-frequency, corresponding to two narrow peaks in the spectrum (i.e., main frequency and its third frequency). For
= 6.3 or 10, the in-line vibration is multi-frequency, corresponding to multiple and wide peaks in the spectrum, and the cross-flow vibration is still in the one-frequency domain.
Figure 6 shows the
XZ-trajectories of the SFT versus reduced velocity. It should be noted that the smaller two-degree-of-freedom responses at
≤ 1.8 are amplified by 10 times synchronously. It reveals that the variation in shape and moving direction for the
XZ-trajectory critically affects the vibration of the SFT (i.e., amplitude, frequency, and phase) and wake pattern. Typical “8”-shaped trajectories are present under most reduced velocities corresponding to the 2:1 in-line to cross-flow dominant frequency ratios. For 0.6 ≤
≤ 1.8, since the phase difference
between the in-line and cross-flow vibration is close to 270°, the
XZ-trajectories bend downstream and appear like a circular arc (“C” shape). The two-degree-of-freedom vibration amplitudes of the SFT are identical. Due to the one-frequency dominant vibration of the SFT, the
XZ-trajectories are periodic and regular. For
= 2.5, since the in-line vibration frequency is close to the natural frequency (as shown in
Figure 5b), the
XZ-trajectory becomes flat. For 3.1 ≤
≤ 3.8, the
XZ-trajectories bend upstream, and the phase difference
is close to 90°. For
= 4.4, the
XZ-trajectory changes to an enclosed loop shape (i.e., the in-line vibration frequency is the same as that in cross-flow). With the further increase in reduced velocity, the cross-flow amplitude of the SFT increases and is larger than the in-line. Due to the in-line multi-frequency vibration, the
XZ-trajectories become chaotic and do not repeat the same path from cycle to cycle (even drift from one mode to another). For
= 8.2, the
XZ-trajectory restores to the standard “8” shape. For
= 10, the trajectory becomes irregular again since the multi-frequency vibration is in the in-line direction.
Figure 7 shows the non-dimensional responses (i.e., amplitude, frequency, and phase) of the SFT versus reduced velocity. Note that the numerical results of one-degree-of-freedom FIV for the SFT under current are also included for comparison purposes. For the multi-peak spectra, only the maximum peak value is shown in
Figure 7.
The comparisons show that the effect of coupled vibration on the SFT is significant.
- (1)
Amplitude
Generally, the 2-DOF vibration amplitudes of the SFT increased with the reduced velocity. For in-line vibration, the maximum vibration amplitude is 0.63D at = 10. For cross-flow vibration, the initial branch and upper branch are present, and the maximum vibration amplitude is in synchronization with the in-line amplitude (1.03D). In addition, the cross-flow amplitudes are significantly larger than the in-line when ≥ 3.1.
A comparison with the 1-DOF results reveals that the coupled vibration has a significant impact on the in-line and cross-flow vibration amplitudes of the SFT at ≥ 4.4 and ≥ 5.7, respectively.
- (2)
Frequency
For ≤ 3.8, the 2-DOF vibration frequencies follow the Strouhal relationship (i.e., the in-line frequency is twice that in cross-flow). For 4.4 ≤ ≤ 7.5, the in-line vibrations become multi-frequency and dominated by lower frequency. The in-line main frequencies jump between 0.2 and 1.0. For 8.2 ≤ ≤ 9.4, the in-line vibrations recover one frequency, and the non-dimensional vibration frequency (≈2.5) is less than 2. The in-line vibration becomes multi-frequency again at = 10, and the non-dimensional vibration frequency (≈0.3) is about 0.2. For cross-flow vibration, the frequency locking presents at 5.7 ≤ ≤ 10 (≈0.7).
A comparison with the 1-DOF results reveals that the coupled vibration has a significant impact on the in-line and cross-flow vibration frequencies of the SFT at ≥ 4.4 and ≥ 5.7, respectively. Multi-frequency competition and conversion of in-line vibration appear at larger reduced velocities, and the main frequency significantly decreases. The lock-in frequency of cross-flow vibration is less than the natural frequency.
- (3)
Phase difference
Variation in phase can affect the vortex pattern. For 0.6 ≤ ≤ 1.8, the phase difference is nearly 270°, and the hydrodynamic load is in phase with the vibration of the SFT in the same direction. For 2.5 ≤ ≤ 5.7, the phase difference jumps to 90° and then varies irregularly due to the in-line multi-frequency vibration. For 10 > ≥ 8.2, the phase difference becomes quasi-steady at 200° and increases again at = 10.
A comparison with the 1-DOF results reveals that the coupled vibration has a significant impact on the phase difference (between the in-line vibration of the SFT and drag force) at ≥ 4.4) and is limited by the phase difference .
Figure 8 shows the transient vorticity magnitude of the SFT at typical reduced velocities. Note that the numerical results of one-degree-of-freedom FIV for the SFT are also included for comparison purposes. It indicates that, for lower reduced velocities, a single vortex sheds from each side of the SFT during one cycle (2S mode). With the increase in reduced velocity, the wake pattern of two-degree-of-freedom FIV for the SFT becomes irregular, and the vortex length becomes larger. The coupled vibration has a significant impact on the wake pattern at
≥ 4.4.
3.2. Effect of Mass Ratio on SFT Vibration
To investigate the effect of mass ratio on the two-degree-of-freedom FIV of the SFT under current, four mass ratios were selected for simulation.
Figure 9 shows the comparison of the
XZ-trajectories of SFTs with various mass ratios at typical reduced velocities.
Figure 10 shows the spectra of in-line and cross-flow displacements under the same conditions. Comparisons show that the two-degree-of-freedom vibrations of the SFT are significantly suppressed with the increase in mass ratio at
≥ 4.4, especially for the in-line vibration. Due to the in-line multi-frequency effect at larger velocities, the
XZ-trajectories become regular and periodic for the SFT with a larger mass ratio. Furthermore, due to the variations in vibration frequency and phase, the pattern of
XZ-trajectories and the width of the spectra also vary with mass ratio.
Figure 11 shows the comparison of the non-dimensional responses (i.e., amplitude, frequency) of the SFT versus reduced velocity under various mass ratios. It shows that the two-degree-of-freedom vibration amplitudes of the SFT increase with the increase in reduced velocity. The increase in mass ratio significantly suppresses SFT vibration due to the multi-frequency vibration and early frequency locking for the SFT with lower mass ratios. For
= 0.5, the in-line and cross-flow amplitudes of the SFT are equal to 1.5
D at
= 10. By contrast, they decrease 92% and 57% when
= 2.0, respectively. The in-line and cross-flow vibration frequencies of the SFT when
= 2.0 follow the Strouhal relationship at
≤ 8.8, and one frequency dominates the vibration with a narrow peak in the spectra.
Figure 12 shows the comparison of the transient vorticity magnitudes of the SFT at typical reduced velocities under various mass ratios. It shows that, for
≤ 4.4, the effect of mass ratio on vortex pattern and length is limited. With the increase in reduced velocity, the vortex pattern of the SFT with a lower mass ratio becomes unstable, corresponding to the irregular
XZ-trajectories. Due to the multi-frequency vibration and early frequency locking, vortex mode switching may occur during the vibration.
3.3. Effect of Natural Frequency Ratio on SFT Vibration
To investigate the effect of the natural frequency ratio on the two-degree-of-freedom FIV of the SFT under current, three natural frequency ratios were selected for simulation.
Figure 13 shows the comparison of the
XZ-trajectories of the SFT under various natural frequency ratios at typical reduced velocities.
Figure 14 shows the spectra of in-line and cross-flow displacements under the same conditions. Comparisons show that the two-degree-of-freedom vibration amplitudes of the SFT vary significantly with the increase in natural frequency ratio at
≥ 2.5. The pattern of
XZ-trajectories and the width of spectra also vary with the natural frequency ratio.
Figure 15 shows the non-dimensional response (i.e., amplitude, frequency) of the SFT versus reduced velocity under various natural frequency ratios. It shows that peak points of the two-degree-of-freedom vibration amplitudes exist with the increase in reduced velocity. The increase in the natural frequency ratio significantly affects the peak point and lock-in width of SFT vibration since the frequency-locking relationship varies. For
Rf = 1.0 and
=
=
, the maximum in-line vibration amplitude is 0.8
D at
= 6.9, and the maximum cross-flow amplitude is 1.4
D. For
Rf = 1.85, the maximum in-line vibration amplitude is 0.7
D at
= 4.1, and the maximum cross-flow amplitude is 1.4
D. In addition, the in-line and cross-flow vibration frequencies of the SFT with various natural frequency ratios follow the Strouhal relationship at
≤ 3.4.
Figure 16 shows the comparison of the transient vorticity magnitudes of the SFT at typical reduced velocities under natural frequency ratios. It shows that, for
≥ 2.5, the effect of the natural frequency ratio on vortex pattern and length can be observed. With the increase in reduced velocity, the vortex pattern of the SFT with a larger natural frequency ratio becomes more unstable, corresponding to the irregular
XZ-trajectories and various vortex modes.
4. Conclusions
Based on the Reynolds-averaged Navier–Stokes (RANS) method, a two-dimensional numerical model was developed to investigate the two-degree-of-freedom FIV of an anchored SFT under current. The numerical results were verified by the literature results, and the effects of coupled vibration and structural parameters, such as mass ratio and natural frequency ratio, on SFT response and wake pattern were addressed across a wide range of reduced velocities. Based on the numerical results, the following conclusions can be drawn:
- (1)
Compared to the 1-DOF numerical results, coupled vibration has a significant impact on SFT response at ≥ 4.4.
- (2)
A decrease in mass ratio (m* < 1) significantly amplified the 2-DOF vibration amplitudes of the SFT at ≥ 4.4, especially for the in-line vibration. Frequency lock-in was observed in advance for ≤ 1.0, leading to enhanced in-line multi-frequency vibration, and the vortex pattern became more unstable. For = 2.0, the cross-flow vibration of the SFT is dominant.
- (3)
Decreasing the natural frequency ratio (Rf < 1) significantly suppressed the in-line vibration of the SFT at ≥ 2.5. The peak points and lock-in width of 2-DOF vibration amplitudes versus reduced velocity showed significant variations. For Rf ≤ 1.0, frequency jumping was observed, leading to smaller in-line vibration amplitude, and the vortex pattern became more steady.
In summary, for SFT design, the effect of mass ratio and natural frequency ratio on the in-line vibration of the SFT should be given more attention.
It should be noted that to quickly investigate the dynamic response of the SFT and to compare it to the physical model test results in the next stage, two-dimensional numerical simulations with the same Reynolds number range as the physical model test [
29] were addressed in this study. Additionally, the damping effects, specifically the discussion of hydrodynamic damping, were not covered in this study. We plan to investigate this aspect in more detail in our future work.
For an actual SFT, a higher Reynolds number (i.e., Re = O(107)) is more representative of turbulent flow conditions encountered in practice. The factors, i.e., three-dimensional vortex instability or non-uniform elastic deformation of the SFT in the spanwise direction, can affect the dynamic response of the SFT. The effect will be further investigated based on the three-dimensional fluid–structure coupling simulation, and the difference with the two-dimensional results will be addressed. Furthermore, we will also consider the design of motion response control devices or damping devices, as they have the potential to improve the performance of SFTs under current.