Vortex features and vortex-induced vibration behavior under varying flow conditions

2.1 Vortex Shedding

Vortex-induced vibration (VIV) is a critical issue in the design and operation of slender offshore structures such as risers, umbilicals, and submarine power cables because periodic vortex shedding can generate cyclic hydrodynamic forces that lead to fatigue damage. When an external flow exceeds a critical velocity, flow separation occurs because of the strong adverse pressure gradient on the downstream side of the structure, resulting in the formation of alternating vortices. The interaction between these unsteady vortices and the structural motion induces oscillatory responses, commonly referred to as VIV [10].

Submarine power cables are continuously exposed to complex marine environments throughout their service life and are subjected to combined external excitations resulting from ocean waves and currents. Steady or slowly varying tidal currents, together with the excitation transmitted from floating platforms or upper supporting structures, can trigger vortex shedding and subsequently induce VIV. Such flow-induced vibrations are widely recognized as some of the dominant mechanisms responsible for fatigue accumulation and long-term structural degradation in submarine cable systems. Vortex shedding occurs at a characteristic frequency, known as the vortex shedding frequency f_s, which is primarily governed by the incoming flow velocity U and characteristic diameter of the structure D. This frequency is commonly expressed in a dimensionless form using the Strouhal number (S_t), which is defined as

The Strouhal number provides a fundamental relationship linking the flow conditions to the frequency of the unsteady hydrodynamic loading acting on slender marine structures.

2.2 Reynolds Number Dependent Vortex Shedding

The Strouhal number is closely related to Re and remains approximately constant within specific Re ranges, whereas it gradually varies as the wake structure and turbulence characteristics evolve with increasing Re. At relatively low Re values, laminar vortex shedding develops in the wake of the circular cylinder, producing highly regular and periodic fluid-force fluctuations. Vortex shedding is typically initiated when a pair of initially stable vortices becomes unstable under small disturbances, which occurs at Re values greater than 40. In this regime, the boundary layer formed on the surface of the cylinder separates owing to an adverse pressure gradient on the rear side, resulting in alternating vortex formation in the wake. Under low-current conditions, Re generally falls within the range of 40–200, which corresponds to laminar wake flow regimes. As Re increases toward ≈ 1000, the wake flow enters a transitional regime, where the vortex street gradually evolves from laminar to turbulent behavior. In this range, vortices tend to form closer to the cylinder surface, and both the shedding frequency and amplitude characteristics are modified. When Re > 1000, the wake region behind the cylinder becomes fully turbulent, significantly altering the vortex shedding patterns and the associated unsteady hydrodynamic forces.

These Reynolds-number-dependent changes in wake dynamics directly affect the magnitude and frequency of the fluid forces acting on the structure, thereby affecting the vibration amplitude, response mode, and frequency-lock-in behavior associated with VIV. Consequently, a systematic understanding of vortex shedding mechanisms across different Re regimes is essential to accurately predicting VIV responses and assessing fatigue damage in submarine cable systems subjected to tidal current loading.

2.1 Numerical Conditions

A VIV analysis of a structure was performed to investigate the influence of various fluid conditions. The characteristic vortex features and hydrodynamic parameters were examined to validate the numerical methodology. In this context, as shown in Figure 1, D denotes the diameter of the cylindrical structure, whereas L, a, and b represent characteristic geometric lengths associated with the wake structure, and θ denotes the vortex separation angle. These parameters enabled quantitative evaluation of vortex characteristics, including the normalized L/D and a/D and b/D. The numerical predictions were systematically compared with benchmark data reported in the literature to verify the reliability and accuracy of the adopted analytical approach.

Figure 1:

Vortex features for Re=40

Numerical simulations were performed using the commercial computational fluid dynamics software ANSYS Fluent (2024 R2). An unsteady flow analysis was performed, and pressure–velocity coupling was handled using the pressure-implicit with splitting of operators algorithm to ensure numerical stability and accuracy in the transient simulations [11]. The shear stress transport (k–ω SST) turbulence model was employed, as it provides reliable predictions of flow separation and wake development, which are essential for accurately capturing vortex shedding behavior and VIV characteristics.

3.1 Validation of Numerical Method

To validate the developed numerical approach for VIV analysis, we quantitatively compared the calculated vortex features with reference data reported in previous experimental and numerical studies for steady flow past a circular cylinder at Re = 40. The validity of the proposed methodology was assessed by examining the agreement between the present numerical results and the established benchmark values.

The vortex feature results obtained using the proposed numerical approach are shown in Figure 2. Key geometric and hydrodynamic parameters characterizing the vortex structures were employed as validation metrics, including the normalized L/D, a/D and b/D, θ, and Cd. The predicted L/D was 2.35, which corresponded to a deviation of approximately 4.44% from the average reference value of 2.25, indicating that the wake extent behind the cylinder was accurately captured.

Figure 2:

Vortex feature results obtained using the developed numerical approach

The vortex center locations exhibited deviations of approximately 2.33% for a/D and 8.33% for b/D. Although a relatively large deviation was observed for b/D, the overall symmetry and spatial distribution of the vortex structures remained consistent with the reference studies. The separation angle was predicted to be 53.0°, showing an error of only 0.47% compared with the average reference value of 53.25°, demonstrating that the vortex detachment behavior was reliably reproduced.

Furthermore, the predicted drag coefficient (Cd = 1.61) exhibited a relative error of approximately 0.52% compared with the reference mean value of 1.60, confirming that the integrated hydrodynamic response was accurately captured. As illustrated in Figure 2, identical vortices were clearly observed in the wake region behind the cylindrical surface. A quantitative comparison of the vortex features is presented in Table 1. Overall, the strong agreement between the present numerical results and benchmark data demonstrates the validity and robustness of the developed numerical method for VIV analysis.

Table 1:

Vortices features measurements of a steady flow past a circular cylinder for Re=40

3.2 Evaluation of Lift and Drag Coefficients with Respect to Re

Figure 3 illustrates the time histories of the lift coefficient (CL), Cd, and the corresponding Strouhal frequency obtained from the transient laminar flow simulations at Re = 40 and 100. Table 2 presents a quantitative comparison of the Cd and CL values at Re = 100 and 200 between the present numerical results and representative experimental and numerical data reported in the literature.

Figure 3:

Strouhal frequency with laminar flow of different at Reynolds numbers of Re = 40 and 100, the lift coefficient (CL), drag coefficient (Cd) with Strouhal frequency(a: Re=40, b: Re=100, c: Re=200)

Table 2:

Quantitative comparison of Cd and CL values at Re = 100 and 200

At Re = 40, the flow remained in a stable laminar regime characterized by the formation of symmetric and attached recirculation vortices behind the cylinder. Under these conditions, the drag coefficient converged to a nearly constant value after the initial transient period, whereas the lift coefficient exhibits small-amplitude oscillations near zero. The Strouhal spectrum exhibited a distinct dominant peak, indicating a well-defined vortex-shedding frequency associated with laminar vortex shedding. As Re increased to 100 and 200, the periodic vortex shedding became more pronounced, leading to a clearer oscillatory behavior in the lift coefficient time history. The drag coefficient maintained a relatively stable mean value with small superimposed fluctuations, reflecting a balance between the viscous and inertial effects in the laminar wake regime. The Strouhal number extracted from the frequency analysis remained within the range reported in previous studies, confirming that the dominant shedding frequency is appropriately captured using the present numerical approach.

A detailed quantitative comparison with the reference data is presented in Table 2. For Re = 100, the average drag coefficient reported in the literature is Cd ≈ 1.35, whereas the present numerical result yields Cd = 1.19, corresponding to a deviation of approximately 11.9%. For Re = 200, the average reference value is Cd ≈ 1.31, whereas the present result is Cd = 1.12, resulting in a deviation of about 14.5%. Although the present simulations systematically underestimated the drag coefficient, the predicted values remained within the scatter range of the experimental and numerical results reported in the literature. With respect to the lift coefficient, the average reference value at Re = 200 is CL ≈ 0.59, while the present numerical result gives CL = 0.51, corresponding to a deviation of approximately 13.6%. Considering the sensitivity of the lift force prediction to the mesh resolution, numerical scheme, and flow unsteadiness, this level of discrepancy is considered acceptable. Importantly, the proposed model successfully reproduces the overall magnitude and oscillatory nature of the lift coefficient associated with vortex shedding.

Overall, the comparisons of the Cd, CL, and Strouhal frequencies demonstrate that the present numerical method captures the essential hydrodynamic characteristics of the flow past a circular cylinder over a range of Re. The observed deviations from the reference data are within reasonable bounds and are consistent with the differences reported in previous studies. Therefore, the developed numerical approach is deemed sufficiently accurate and reliable for subsequent VIV analyses of submarine cable structures under current-induced loading conditions.

Author Contributions

Conceptualization, Y. J. Kim, S. W. Choi; Methodology, Y. J. Kim, S. W. Choi; Software, Y. J. Kim, S. W. Choi; Formal Analysis, Y. J. Kim, S. W. Choi; Investigation, Y. J. Kim, S. W. Choi; Resources, S. W. Choi; Data Curation S. W. Choi; Writing-Original Draft Preparation, S. W. Choi; Writing-Review & Editing, S. W. Choi; Visualization, Y. J. Kim, S. W. Choi; Supervision, S. W. Choi; Project Administration, S. W. Choi.