Force Element Analysis in Vortex-Induced Vibrations of Side-by-Side Dual Cylinders: A Numerical Study
Abstract
:1. Introduction
2. Control Equations and Numerical Methods
2.1. Flow Field Setting
2.2. Control Equations
2.3. Vibration Response System Model
2.4. Numerical Verification
2.4.1. Flow Meshing
2.4.2. Independence Verification
2.4.3. Model and Method Validation
3. Results and Discussion
3.1. Frequency and Amplitude Response
3.1.1. Frequency Response
3.1.2. Amplitude Response
3.2. Lift and Drag Coefficients
3.3. Motion Trajectory
3.3.1. Trajectory Modes
3.3.2. Trajectory Patterns Analyses
3.4. Wake-Stream Pattern
4. Conclusions
- (1)
- The VIV phenomenon for side-by-side dual cylinders exhibits three vibration regions with different spacing ratios, similar to those observed for a single cylinder, namely the initial branch (IB), the lower branch (LB), and the desynchronization branch (DB). When the spacing ratio L/D is greater than or equal to five, the range of reduced velocities Ur corresponding to the lower branch is wider and more closely resembles that of a single cylinder. For side-by-side cylinders at different spacing ratios, the maximum displacements in the streamwise and cross-flow directions occur at a reduced velocity Ur = 5. Cross-flow displacements are comparable to those of a single cylinder. In the streamwise direction, the amplitude decreases gradually with increasing spacing ratio, approaching the behavior of a single cylinder. Notably, there is no sudden increase in streamwise amplitude at a Ur = 8, unlike the case for a single cylinder.
- (2)
- During the VIV phenomenon, the lift and drag forces acting on side-by-side cylinders are primarily due to the volume vorticity forces associated with the cylinders themselves, combined with surface vortices (including friction) forces. When entering the “lock-in” zone, as cylinder vibrations intensify increase, the acceleration forces on the cylinder surface also increase. The volume vorticity lift force phase opposes the total lift force, thus inhibiting (negatively contributing) to the lift force. As the reduced velocity increases and the cylinder amplitude reaches its maximum, the surface acceleration forces begin to suppress changes in lift force, decreasing with further increases in reduced velocity. At lower reduced velocities, side-by-side cylinders are mainly influenced by the surface vorticity forces from the surface of the other cylinder. As reduced velocity increases, the influence of inter-cylinder surface acceleration forces grows, leading to increased amplitudes. The effect of surface vorticity forces in lift direction stabilizes over time. As the spacing ratio between cylinders increases, the influence between cylinders diminishes, and the force variation trends approach those of a single cylinder.
- (3)
- Once in the “lock-in” zone and experiencing significant vibrations, the wake vortices of side-by-side cylinders transition from a 2S mode to a C(2S) mode. The smaller the spacing ratio, the more pronounced this transformation becomes. At the onset and end of the wake mode transition, the displacement trajectories of the cylinders exhibit irregular “bounded” motions. During this period, the lift and drag coefficients display a “multi-frequency” behavior. Simultaneously, the vibration trajectories of side-by-side cylinders form “bounded elliptical”, and “bounded teardrop” shapes.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Re | Reynolds number |
U∞ | Uniform incoming fluid velocity |
D | Diameter of the cylinder |
ρ | Density of the fluid |
P | Non-dimensional pressure |
P* | Pressure |
Ur | Reduced velocity |
St | Strouhal number |
fs | Frequency of the vortex shedding |
fn | Natural frequency |
f * | Frequency ratio |
k | Spring constant |
c | Damping coefficient |
ζ | Damping ratio |
m* | Mass ratio |
FD | Drag force |
FL | Lift force |
Cd | Drag force coefficient |
Cl | Lift force coefficient |
Time-averaged drag force coefficient | |
Time-averaged lift force coefficient | |
Cda | Surface acceleration drag force coefficient |
Cdv | Volume vorticity drag force coefficient |
Cds | Surface vorticity drag force coefficient |
Cla | Surface acceleration lift force coefficient |
Clv | Volume vorticity lift force coefficient |
Cls | Surface vorticity lift force coefficient |
In-line amplitude | |
Cross-flow amplitude | |
h* | Nondimensional mesh spacing |
hfirst | Near-wall first grid height |
L | Center-to-center spacing of cylinders |
L/D | Spacing ratio |
Ctotal | Total hydrodynamic force coefficient |
Ftotal | Total hydrodynamic force |
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Mesh | Cell Number | h* | ∆tU∞/D | ||||
---|---|---|---|---|---|---|---|
M1 | 56,490 | 0.01 | 0.0025 | 1.901 | 0.191 | 0.069 | 0.514 |
M2 | 80,550 | 0.005 | 0.0025 | 1.872 (1.5%) | 0.186 (2.7%) | 0.066 (4.5%) | 0.511 (0.6%) |
M3 | 146,400 | 0.0025 | 0.0025 | 1.863 (0.5%) | 0.184 (1.1%) | 0.065 (1.5%) | 0.503 (1.6%) |
M2 | 80,550 | 0.005 | 0.00125 | 1.860 (0.6%) | 0.187 (0.5%) | 0.066 (0%) | 0.508 (0.6%) |
Chen et al. [24] | - | - | - | 1.828 | 0.187 | 0.060 | 0.536 |
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Song, M.; Guo, S.; Xu, H.; Tao, W.; Lei, J.; Chang, C.-C. Force Element Analysis in Vortex-Induced Vibrations of Side-by-Side Dual Cylinders: A Numerical Study. J. Mar. Sci. Eng. 2024, 12, 1529. https://doi.org/10.3390/jmse12091529
Song M, Guo S, Xu H, Tao W, Lei J, Chang C-C. Force Element Analysis in Vortex-Induced Vibrations of Side-by-Side Dual Cylinders: A Numerical Study. Journal of Marine Science and Engineering. 2024; 12(9):1529. https://doi.org/10.3390/jmse12091529
Chicago/Turabian StyleSong, Mengtian, Suxiang Guo, Hailong Xu, Weijian Tao, Jiechao Lei, and Chien-Cheng Chang. 2024. "Force Element Analysis in Vortex-Induced Vibrations of Side-by-Side Dual Cylinders: A Numerical Study" Journal of Marine Science and Engineering 12, no. 9: 1529. https://doi.org/10.3390/jmse12091529