Numerical Investigation on the Hydrodynamic Characteristics of Submarine Power Cables for Offshore Wind Turbines Under Combined Wave–Current Loading

Abstract

A 2D numerical model for viscous flow is established in OpenFOAM version 10 to analyze the hydrodynamic response of submarine power cables for offshore wind turbines under combined wave–current conditions. It focuses on analyzing the effect of the cable suspension ratio e/D and the current-to-wave velocity ratio U_c/U_m on the Morison coefficient of the suspended cable. The results indicate that for the cable suspension ratio e/D of less than 0.5, the strength of the dependence of both the drag coefficient C_d and inertia coefficient C_M on the cable suspension ratio e/D is significantly influenced by the current-wave-ratio U_c/U_m, while this dependence becomes less pronounced for e/D greater than 0.5. And the inertia force coefficient C_M decreases monotonically with the current-to-wave velocity ratio U_c/U_m, while the drag force coefficient C_d demonstrates a more complex, non-monotonic relationship with it. Based on the simulation results in this paper, a quantitative relationship between C_d, C_M, and the key governing parameters is established using a two-layer feedforward neural network model, providing a method for predicting wave–current forces on subsea suspended cables.

1. Introduction

As an integral component of marine renewable energy systems, submarine cables connect onshore and offshore facilities, playing a pivotal role in the transmission of electric power harnessed from wind, wave, and tidal energy. This role is particularly critical in China, given its national strategic emphasis on energy transition and its ambitious “Dual Carbon” goals, which have catalyzed the rapid expansion of offshore wind turbines along the coastline. However, complex seabed topography or seabed scouring due to ocean currents can result in suspended spans of submarine cables. And the on-bottom stability and structural safety of subsea cables are governed by the hydrodynamic loads on the suspended spans [1,2]. Therefore, conducting research on the hydrodynamic characteristics of suspended submarine cables for offshore wind turbines holds significant engineering importance.

The suspended span of a submarine cable may be formed due to local scour that extends axially along the cable. Although the scouring process exhibits typical three-dimensional characteristics, as it gradually expands along the cable to a reasonable stage, the scouring and force characteristics at the cable cross-section can be considered to be a two-dimensional process. Liang et al. [3] compared the results of two-dimensional numerical simulations of local scouring with experimental data from three-dimensional physical models, indicating that the two-dimensional model can effectively simulate the development process of local scouring, and the suspended height of the cylinder remains basically at the order of 1.0D, where D represents the cylinder diameter. Cheng et al. [4] conducted a three-dimensional scour experiment, measuring the local scour depth beneath the cylinder using a specially developed conductivity scour probe. A prediction formula for scour depth was derived from the results. Bearman et al. [5] studied the flow around cylinders at different heights above a plane boundary through wind tunnel experiments and found that the vortex-shedding frequency is inhibited by the plane boundary, and the vibration response of the cylinder is, consequently, affected. Lei et al. [6] conducted a numerical study on the drag coefficient of a circular cylinder in unidirectional flow, investigating Reynolds numbers of 80–1000 and clearance ratios of 0.1–3.0. Here, the Reynolds number Re under unidirectional flow conditions is defined as Re = U_cD/υ, where U_c is the unidirectional flow velocity, υ is the kinematic viscosity of the water, and the clearance ratio is defined as e/D, where e is the clearance between the cylinder bottom and the wall. Similarly, Ong et al. [7] studied the vortex-shedding mechanism of a near-wall circular cylinder at a Reynolds number Re = 1.31 × 10⁴ and clearance ratios of 0.1 ≤ e/D ≤ 1.0 based on a two-dimensional k-ε turbulence model. The study showed that vortex shedding is suppressed at e/D ≤ 0.3, resulting in a decrease in the root mean square coefficient of the cylinder. According to Lei et al. [8], the critical clearance ratio for vortex suppression ranges from 0.2 to 0.3 and varies inversely with the boundary layer thickness δ. The drag coefficient’s dependence on the clearance ratio is dominated by the cylinder’s position relative to the boundary layer. It remains nearly constant outside the layer, increases with e/δ within the layer, and becomes negligible for e/D > 2.0. Buresti and Lanciotti [9] reported that the mean lift coefficient decreases with the clearance ratio, and the magnitude of this reduction is directly proportional to the boundary layer thickness, making the drop more dramatic in thicker layers. However, the average drag coefficient varies non-monotonically with the clearance ratio, a trend that is highly sensitive to both the Reynolds number and the boundary layer thickness. For oscillatory flow conditions, An et al. [10] conducted 2D and 3D simulations within the parameter range of Re = 2000 and KC = 1~26.2 (under oscillatory flow conditions, Re = U_mD/υ and KC = U_mT/D, where U_m and T are defined as the velocity amplitude and period of the oscillatory flow). The computational results indicated that the vortex structures obtained from the 2D and 3D models under the same conditions were essentially consistent, and the predicted differences in the Morison force coefficient were within 18%. Munir et al. [11] investigated the influence of the plane boundary on the vortex-induced vibration (VIV) using a two-degree-of-freedom (2DOF) VIV model. The results showed that the reduced velocity V_r (V_r = U_m/(f_nD), where f_n is the natural frequency of the cylinder) affects the vortex shedding significantly, and the vortices are shed only from the bottom side of the cylinder due to the effect of the shear layer on the plane boundary.

To date, there has been limited research on the hydrodynamic problems of cylinders subjected to combined unidirectional and oscillatory flow. Tong et al. [12] used a viscous flow model to study the flow field structure around a forced oscillating cylinder in uniform flow at Re = 100 and explored the influence of different oscillation directions on the flow field structure. Teng et al. [13] utilized a numerical model to calculate the forces acting on a horizontally oscillating cylinder under forced excitation and analyzed the amplitude differences in various harmonic forces through flow field visualization. Zhao et al. [14] employed a 3D direct numerical simulation to study current-to-wave ratio effects on the force acting on the cylinder, finding that the frequency of lift force increases with the current-to-wave ratio. Cao and Li [15] found that the vortex shedding under the combined steady and oscillatory flow is not only dependent on the Reynolds number but also on the KC number and AR number (AR = U_m/U_c). Deng et al. [16] numerically investigated the flow around a transversely oscillating circular cylinder subjected to combined uniform and oscillatory flow. Their study found that the vortex formation length near the cylinder decreases as the amplitude of the oscillatory flow increases. Furthermore, the drag and inertia force coefficients increase when the frequency ratio f/f_s (where f is the oscillation frequency of the cylinder and f_s is the vortex-shedding frequency) approaches the lock-in region. Zhao et al. [17] investigated the vortex-induced vibration of a circular cylinder subjected to combined steady and oscillatory flow and found that the lock-in regime under this condition is wider than that in either pure steady flow or pure oscillatory flow. Chang et al. [18] investigated the flow structure around a vertical cylinder. They found that under a constant KC number, both the strength and lifespan of the main horseshoe vortex formed during each oscillatory cycle increase with the velocity ratio U_c/U_m. Additionally, the phase lag between the force and the oscillatory velocity component decreases with increasing KC number and shows little dependence on U_c/U_m. Teng et al. [19] investigated the hydrodynamic forces on small-diameter cylinders, finding that Fourier coefficients can effectively reconstruct the time-history of hydrodynamic forces, and provided empirical formulas for peak coefficients and Fourier coefficients as functions of KC and U_c/U_m.

However, under combined wave–current loading, the forces on a suspended cable are influenced by multiple control parameters, including KC, e/D, and U_c/U_m. Scientifically quantifying the relationship between the hydrodynamic coefficients of the cylinder and these control parameters is key to predicting the forces on subsea suspended cables. Therefore, this paper investigates the hydrodynamics of subsea suspended cables under combined wave–current conditions, utilizing a viscous flow numerical model. Based on the simulation results, the present study further proposes a method for predicting the forces on cables under combined wave–current loading, providing a reference for related engineering design.

2. Numerical Method

2.1. Governing Equations

The flow is governed by the two-dimensional, incompressible Reynolds-averaged Navier–Stokes and continuity equations:

$${\displaystyle \frac{\partial u_i}{\partial x_i}}=0$$

(1)

$${\displaystyle \frac{\partial u_i}{\partial t}}+u_j{\displaystyle \frac{\partial u_i}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(2\upsilon S_{ij}-\overline{u_i^{\prime }{u}_j^{\prime }}\right)$$

(2)

where x_i represents the coordinate component, and where i = 1 and 2 represent the x-component and y-component, respectively. u_i represents the velocity component corresponding to the coordinate x_i, where u₁ = u and u₂ = v. t is time, ρ is fluid density, p is pressure, and υ is fluid kinematic viscosity. S_ij is the strain rate tensor, defined as S_ij = (∂u_i/∂x_j + ∂u_j/∂x_i)/2. The Reynolds stress $\overline{u_i^{\prime }{u}_j^{\prime }}$ is defined as follows:

$$\overline{u_i^{\prime }{u}_j^{\prime }}=-{\upsilon }_t\left(\partial u_i/\partial x_j+\partial u_j/\partial x_i\right)+{\displaystyle \frac{2}{3}}k{\delta }_{ij}$$

(3)

where k represents turbulent kinetic energy, and δ_ij represents the Kronecker operator.

Due to the unknown Reynolds stress term $\overline{u_i^{\prime }{u}_j^{\prime }}$ in the Navier–Stokes equation, a reasonable turbulence model needs to be further introduced for the closed-form solution of the equation. The SST k-ω two-equation turbulence model [20,21] is adopted in the present study: the standard k-ω model is used near the wall, and it is transformed into the standard k-ε model in the free stream and core region, which can accurately predict boundary layer flows with adverse pressure gradients. The governing equations of the SST k-ω two-equation turbulence model are as follows:

$${\displaystyle \frac{\partial k}{\partial t}}+u_j{\displaystyle \frac{\partial k}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\upsilon +{\sigma }_k{\upsilon }_t\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+P_k-{\beta }^{\ast }\omega k$$

(4)

$${\displaystyle \frac{\partial \omega }{\partial t}}+u_j{\displaystyle \frac{\partial \omega }{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\upsilon +{\sigma }_{\omega }{\upsilon }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+{\displaystyle \frac{\alpha p_k}{{\upsilon }_t}}-\beta {\omega }^2+2\left(1-F_1\right){\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}$$

(5)

where k represents the turbulent kinetic energy, ω represents the turbulent energy dissipation rate, P_k is the product of turbulent kinetic energy, υ_t is the turbulent eddy viscosity, and F₁ is the mixing function, which are expressed as follows:

$$P_k=\min \left[{\upsilon }_t{\displaystyle \frac{\partial u_i}{\partial x_j}}\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right),10{\beta }^{\ast }k\omega \right]$$

(6)

$${\upsilon }_t={\displaystyle \frac{{\alpha }_{1}k}{\max \left({\alpha }_1\omega ,S{F}_2\right)}}$$

(7)

$$F_1=\tanh \left\{\min \left\{\min \left[\max \left({\displaystyle \frac{\sqrt{k}}{{\beta }^{\ast }\omega y^{+}}},{\displaystyle \frac{500\upsilon }{y^{+2}\omega }}\right)\right.\right.\right.,{\left.\left.\left.{\displaystyle \frac{4\rho {\sigma }_{\omega 2}k}{D_{k\omega }{y}^{+2}}}\right],10\right\}\right\}}^4$$

(8)

where S = (2S_ijS_ij)^1/2 represents the invariant measure of strain rate, y⁺ denotes the non-dimensional distance of the grid closest to the wall, and the variables F₂ and D_kω are expressed as follows:

$$F_2=\tanh {\left\{\min \left[\max \left({\displaystyle \frac{2\sqrt{k}}{{\beta }^{\ast }\omega y^{+}}},500{\displaystyle \frac{\upsilon }{y^{+2}\omega }}\right),100\right]\right\}}^2$$

(9)

$$D_{k\omega }=\max \left(2\rho {\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}},{10}^{-10}\right)$$

(10)

This is based on the mixed function F₁, σ_k = F₁σ_k1 + (1 − F₁)σ_k2, σ_ω = F₁σ_ω1 + (1 − F₁)σ_ω2, α = F₁α₁ + (1 − F₁)α₂, and β = F₁β₁ + (1 − F₁)β₂. In addition, the basic theoretical parameters of the SST k-ω turbulence model used in Formulas (4)–(10) are shown in

Table 1. Parameters in the SST k-ω turbulence model.

β*	α1	β1	σk1	σω1	α2	β2	σk2	σω2
0.09	5/9	3/40	0.85	0.5	0.44	0.0828	1.0	0.856

.

2.2. Boundary Conditions

Figure 1 shows the schematic of the 2D numerical model for simulating the hydrodynamics of a suspended cable under combined wave–current loading. The coordinate origin O is located on the seabed, directly below the cable center. To minimize boundary effects on the near-cable flow, the inlet (left) and outlet (right) of the fluid are both set at a distance of 25D from the center of the cable. The center of the cable is located at a height of e + D/2 above the seabed. The specific boundary conditions are set as follows: (1) the left boundary is the inflow boundary, with boundary conditions of u = U_c + U_msin(2πt/T), v = 0, ∂p/∂x = (2πU_m/T)cos(2πt/T), ∂k/∂n = 0, and ∂ω/∂n = 0; (2) the right boundary is the outflow boundary, with boundary conditions of ∂u/∂x = 0, ∂v/∂y = 0, p = 0, ∂k/∂n = 0, and ∂ω/∂n = 0; (3) the upper boundary is the symmetric boundary condition, with ∂u/∂n = 0, v = 0, ∂p/∂y = 0, ∂k/∂n = 0, and ∂ω/∂n = 0; and (4) both the lower boundary and the cable surface are non-slip boundary conditions, with boundary conditions of u = 0, v = 0, ∂p/∂n = 0, and ω = 6υ/0.075Δ². The boundary conditions for turbulent kinetic energy k at the lower boundary and cable surface are ∂k/∂n = 0 and k = 0, respectively. Here, n represents the unit normal vector, and Δ represents the first-layer height of the grid.

3. Validation of Numerical Model

3.1. Grid Convergence Verification of Numerical Model

Prior to performing numerical simulations, grid convergence verification was first conducted. The grids used for numerical calculations in this paper are all structured grids, as shown in Figure 2. Local densification of the grid around the cable was achieved mainly through the following two methods: adjusting the circumferential nodes count and the first-layer height of the grid on the cable surface (An et al. [22]). Next, convergence studies were performed with respect to the circumferential node density and the first-layer grid height. The specific information of the four sets of grids is shown in

Table 2. Grid convergence verification for the height of the first grid layer on the cable.

Grid Number	The Height of the First Grid Layer on the Cable	Cd	CM
G1	0.003 D	1.25	1.60
G2	0.002 D	1.51	2.05
G3	0.001 D	1.42	2.07
G4	0.0005 D	1.41	2.08

. The calculation conditions were set as follows: Reynolds number Re = 0.8 × 10⁵, KC = 10, current wave ratio U_c/U_m = 0, and suspension ratio e/D = 0.5.

Based on the numerical results using the four different sets of grids, the horizontal forces (the inertia coefficient C_M and drag coefficient C_d) can be approximated using the Morison equation (Equation (11)). By fitting using the least squares method, the optimal solutions for C_d and C_M can be obtained, as shown in Table 2:

$$F_D\left(t\right)={\displaystyle \frac{1}{2}}\rho D{C}_d\left|u(t)\right|u(t)+\rho {\displaystyle \frac{\pi D^2}{4}}{C}_M{\displaystyle \frac{du(t)}{dt}}$$

(11)

Table 2 and

Table 3. Grid convergence verification for the number of nodes on the cable surface.

Grid Number	Number of Nodes on the Cable Surface	CD	CL
G5	120	1.18	0.151
G6	140	1.27	0.124
G7	160	1.28	0.115
G8	180	1.29	0.118

present the results for oscillatory flow and steady current, respectively. Comparing the results corresponding to the grids G4 and G3 in Table 2, the force coefficients show a difference below 1%, indicating that the grid G3 meets the convergence requirements. As shown in Table 3, the results for mesh G7 (with 140 nodes) and G8 (with 160 nodes) show an insignificant difference. Therefore, the mesh that features a first-layer height of 0.001 D and 160 nodes on the cable surface was selected to ensure accuracy while maintaining computational efficiency. Figure 3 shows the variation in the force coefficients with dimensionless time. The results for Δt* = 0.005 (Δt* = u₀Δt/D) are in close agreement with those for a finer time step of Δt* = 0.002, with a negligible difference. Consequently, a time step of Δt* = 0.005 was adopted for all subsequent simulations.

3.2. Accuracy Verification of Numerical Model

This section conducted the accuracy verification on the hydrodynamic characteristics of suspended cables. The computational condition is adopted from Oner et al. [23], i.e., Reynolds number Re = 9500 and suspension ratio e/D = 0.2 and 0.3. Figure 4 compares the numerically simulated and experimental [23] distributions of the average horizontal velocity around the cable, showing good agreement between the present results (red line) and the data from Oner et al. [23] (black circles). The model shows its capability to faithfully capture the velocity distribution around the suspended cable.

The model is subsequently validated against the hydrodynamics of a cylinder in oscillatory flow using the experimental data of Sumer et al. [24], i.e., (a) KC = 4, Re = 0.4 × 10⁵, e/D = 0.1, (b) KC = 10, Re = 0.8 × 10⁵, e/D = 0.1, and (c) KC = 10, Re = 0.8 × 10⁵, e/D = 1.0. As shown in Figure 5, the calculated C_d and C_M values are in good agreement with the results of Sumer et al. [24], demonstrating the model’s accuracy for simulating the hydrodynamic forces subjected to cylinders under oscillating flow. Although the flow around a circular cylinder exhibits inherent three-dimensional characteristics, the two-dimensional assumption remains a widely adopted and practical approach for estimating hydrodynamic forces acting on the subsea structures. The reliability of the 2D model is supported by the present study and prior studies [10,25], which demonstrate that it is capable of accurately capturing the dominant characteristics of drag and inertia forces. These force coefficients could be used for on-bottom stability design and buckling analysis. It is noteworthy that the present results are typically workable for a single, endless cross-section, where the effects of span flexibility, the 3D vortex shedding, and the contact at span ends are not considered.

4. Numerical Results and Analysis

This paper conducts research on the hydrodynamic characteristics of underwater suspended cables under the combined wave–current conditions. To simulate the combined effect of wave and current, the oscillatory flow was superimposed onto the steady unidirectional flow, where the oscillatory flow was prescribed by a sine function. The definition of the flow velocity u(t) is as follows:

$$u\left(t\right)=U_c+U_m\sin \left(2\pi t/T\right)$$

(12)

Under the combined wave–current conditions, the current-to-wave ratio U_c/U_m and KC number are the two key parameters. The operating conditions in this article are set as follows: Reynolds number Re = 0.8 × 10⁵, KC number KC = 4, 7, 10, suspension ratio e/D = 0.1–2.0, and the ratio of unidirectional flow velocity to oscillating flow amplitude U_c/U_m = 0.2, 0.5, 0.8, 1.0, and 1.5, totaling 120 calculation conditions. The drag coefficient C_d and inertia coefficient C_M are still solved by fitting the Morrison equation (Equation (11)) using the least squares method.

4.1. Flow Field Analysis

Figure 6, Figure 7 and Figure 8 show the evolution of vorticity contours over one oscillatory cycle for different current-to-oscillatory ratios and suspension ratios. When the oscillatory flow dominates, the vortex oscillates around the cable and eventually decays. In contrast, as the current intensifies, the vortex tends to propagate downstream. Under the effect of oscillatory flow, vortex shedding persists even at small clearance ratios and is not suppressed by the wall. A comparison of Figure 7 and Figure 8 reveals that the bottom wall significantly influences the shed vortex path. At a small gap ratio, the vortex is deflected upward, whereas at a larger ratio, it propagates horizontally downstream.

4.2. The Influence of Suspension Ratio e/D on Force Coefficients

Figure 9 shows the relationship between the drag coefficient C_d and the suspension ratio e/D for a suspended cable under combined wave–current conditions, where (a)–(c) correspond to KC numbers of 4, 7, and 10, respectively. The results show that under the conditions of three KC numbers, when the current-to-wave ratio U_c/U_m ≤ 0.5, the drag coefficient C_d decreases with the suspension ratio, with its influence diminishing markedly for U_c/U_m > 0.5. Across all tested conditions, the effect of suspension ratio on the drag coefficient C_d exhibits a marked reduction for e/D values exceeding 0.5. Taking the results at KC = 4 shown in Figure 9a as an example, it is found that when the component of current-induced velocity is relatively weak, say U_c/U_m = 0.2, the drag coefficient C_d decreases noticeably with the increasing suspension ratio e/D. This behavior is similar to that found under pure wave conditions, namely U_c/U_m = 0, as reported by Teng et al. [26], indicating that the force amplification is due to the effect of wall proximity. However, when the current-induced velocity is strong, say U_c/U_m = 1, C_d is less sensitive to the increase in e/D. This phenomenon agrees well with that found under pure current conditions, as reported by Teng et al. [27], indicating that the force is dominated by the velocity distribution in the boundary layer rather than the effect of wall proximity.

Figure 10 presents the variation in the inertia force coefficient C_M with the suspension ratio e/D for KC numbers of 4, 7, and 10. The results indicate that the inertia force coefficient C_M tends to decrease with e/D. However, when KC = 10 and U_c/U_m ≥ 1, the dependence of C_M on e/D is weak. Across all tested conditions, the influence of suspension ratio on the inertia force coefficient C_M exhibits a marked reduction for e/D values exceeding 0.5.

4.3. The Influence of Current-to-Wave Ratio U_c/U_m on Force Coefficients

The drag and inertia coefficients under different current-to-wave ratios U_c/U_m are shown in Figure 11 and Figure 12, respectively. The results indicate that when U_c/U_m < 1, the variation trend of C_d with U_c/U_m is different under the same KC number but different e/D conditions, and it is also different under the same suspension ratio but different KC number conditions. However, under all KC numbers and e/D conditions, C_M shows a monotonic decreasing trend with the increase in U_c/U_m, as the flow becomes increasingly dominated by the steady current component, which contributes less to the inertial force. For U_c/U_m greater than 1.0, both C_d and C_M remain nearly constant with a further increase in U_c/U_m. The above results indicate that predicting the horizontal force on suspended cables under combined wave–current conditions, especially when U_c/U_m < 1, requires establishing quantitative relationships for the force coefficients C_d, C_M with KC number, current-to-wave ratio U_c/U_m, and the suspension ratio e/D. Taking the results at KC = 7, shown in Figure 11b, as an example, opposite variation trends with the increasing U_c/U_m are found in the regions of U_c/U_m < 0.5 and U_c/U_m > 0.5 when e/D ≤ 0.3. These opposite trends reflect the complicated effect of the co-existence of wave and current. However, the physics responsible for the local minimum drag force at U_c/U_m = 0.5 is not well understood.

4.4. Prediction Method of Horizontal Force on Suspended Cables Under Combined Wave–Current Conditions

The force acting on a suspended cable under combined wave–current conditions cannot be considered as merely a superposition of the forces induced by unidirectional flow and oscillatory flow alone. As mentioned earlier, the force coefficients of the cable are governed by three key parameters (i.e., KC number, flow wave ratio U_c/U_m, and suspension ratio e/D), and the non-trivial relationships between these parameters make it impractical to define the quantitative influence on the Morrison hydrodynamic coefficient using simple empirical expressions. Therefore, based on a two-layer feedforward neural network model, as shown in Figure 13, the quantitative relationships between the force coefficients C_d, C_M, and {KC, U_c/U_m, e/D} are established, where the hidden layer contains 10 sigmoid neurons and the output layer contains one linear neuron, where ‘w’ and ‘b’ represent weight and bias, respectively. The Levenberg–Marquardt algorithm is used to train the neural network. The hyperparameters were determined by trials, and those that led to the minimum rms deviation were chosen. Out of the 120 simulation results in this article, 84 were used for training, 18 for validation, and 18 for testing. The root mean square deviations of C_d and C_M for model training are less than 1% and 2%, respectively. Note that the applicable range of this model is as follows: 4 ≤ KC ≤ 10, 0 ≤ U_c/U_m ≤ 1.5, 0.1 ≤ e/D ≤ 2.

Figure 14 shows the drag and inertia coefficients under the conditions of KC = 7, U_c/U_m = 0.5, and e/D = 0.1~2. For ease of comparison, the figure also includes results from simulations with KC = 7 and U_c/U_m = 0.5. It should be noted that these simulations have been excluded from any phase of model training or validation. As shown in the figure, the prediction results based on the model proposed in this paper are generally in good agreement with the simulations. The deviation of C_d is relatively larger compared to that of C_M, because the variation trend of the training data C_d is more complicated under the effect of the co-existing wave and current.

5. Conclusions

This study investigates the hydrodynamic characteristics of underwater suspended cables under combined wave–current conditions. A two-dimensional viscous flow calculation and analysis model is established by solving the RANS equations with the SST k-ω turbulence model. The influences of the suspension ratio e/D and the current-to-wave ratio U_c/U_m on the Morrison coefficient of the cable are analyzed within the parameter ranges of 4 ≤ KC ≤ 10, 0 ≤ U_c/U_m ≤ 1.5, and 0.1 ≤ e/D ≤ 2. The low KC values working together with a wide range of U_c/U_m in the present study reflect steady currents in co-existence with mild waves. The primary findings are summarized below:

(1)

For the cable suspension ratio e/D of less than 0.5, the strength of the dependence of both the drag and inertia coefficients on the cable suspension ratio e/D is significantly influenced by the current–wave ratio U_c/U_m, while this dependence becomes less pronounced for e/D that is greater than 0.5.

(2)

When U_c/U_m < 1, the variation trend of C_d with U_c/U_m is different under the same KC number but different e/D conditions, and the variation trend of C_d with U_c/U_m is also different under the same suspension ratio but different KC number conditions. However, under all KC numbers and e/D conditions, C_M shows a monotonically decreasing trend with the increase in U_c/U_m. When U_c/U_m is greater than 1.0, both C_d and C_M almost no longer change with U_c/U_m variation.

(3)

Based on the simulation results in this article, a double-layer feedforward neural network model was used to establish quantitative relationships between C_d, C_M, and {KC, U_c/U_m, e/D}, with root mean square deviations of less than 1% and 2%, respectively.