1. Introduction
The use of towed cable systems is prevalent in subsea exploration, marine surveys, environmental monitoring, and military defense operations, where precise positioning is essential for efficient and safe operations. Accurately estimating the hydrodynamic forces on each cable element poses a significant challenge. Several researchers have investigated the hydrodynamic forces on bare cables and the influence of the Reynolds number on drag and lift coefficients. At low Reynolds numbers (Re < 103), frictional forces contribute predominantly to the drag force, resulting in elevated drag values. For Reynolds numbers (Re) of 103< Re < 4 × 105, the drag coefficient remains nearly constant, approximately CD = 1, with a laminar boundary layer present until separation. The drag coefficient decreased significantly to CD = 0.3 at higher Reynolds numbers (Re > 4 × 105) in turbulent flow (Cao et al., 2014).
Nevertheless, for a flexible underwater towed cable system, oscillatory motions caused by the interaction between fluid flow and submerged flexible structures lead to vortices, causing the cable to vibrate. The strumming effect can dramatically elevate the drag coefficients, with values reaching 3.5 (Griffin et al., 1981; Griffin, 1985).
Various techniques are used to mitigate the effects of vortex-induced vibrations (VIV) on underwater towed cables, including hydrodynamic devices such as hairy fairings. These devices feature hair-like flexible structures extending from a jacket covering the cable and have been shown to significantly reduce hydrodynamic drag on underwater towing cables by 40% to 60%. They help lower the drag coefficients from > 2.0 for bare cables to 0.7–1.5, enhancing the towing efficiency and performance (Every et al., 1982).
In practical applications, researchers often use constant drag and lift force coefficients when calculating the hydrodynamic forces using the Morison equation. In the context of towed cable systems, the inclined angle of each cable element varies along the length of the cable system, affecting the drag coefficient. The angle of attack is another critical factor affecting the drag coefficient of a cylindrical structure (Franzini et al., 2009; Kebede et al., 2020; Li et al., 2024; Vakil and Green, 2009). The drag coefficients of towing cables must be validated based on local inclined angles to establish appropriate hydrodynamic force coefficients for underwater towing system design. Moreover, the shape of hairy fairing cables significantly affects the hydrodynamic drag and lift forces. Modeling the shape of the cable presents its challenges. Therefore, this study modeled two cable shapes to determine how modeling affects the calculation results.
This study examined the towed cable posture with a hairy fairing cable. The hydrodynamic forces acting on the hairy fairing cable element at various inclined angles and operating speeds were calculated using the computational fluid dynamics (CFD) simulation method. In addition, a mathematical model was developed based on the equilibrium of internal and external forces to simulate the steady state of the towed cable system. Simulations of the cable posture, using the finite element method and based on the obtained hydrodynamic forces, validated the accuracy of this approach and compared the results with sea trial data.
2. Mathematical Model
2.1 Assumption
For analytical simplicity, the dynamics of the system were confined to the vertical plane (oxz), with lateral movement being disregarded. The towed cable system experienced consistent forward motion from the towing ship at a constant speed. Fig. 1 presents a diagram of the towed cable system. A boundary condition is specified where the towfish is connected to the towed cable.
2.2 Steady Equation
In this study, the towing cable used to pull the towfish is a hairy fairing cable with a density greater than that of water (heavy towed cable). The released length refers to the length of the heavy towed cable deployed between the towing vessel and the towfish, as shown in Fig. 1. Fig. 2 shows the forces acting on the cable element. The internal forces include tension, while external forces comprise gravity, buoyancy, and hydrodynamic forces. This ensemble is considered the entire force acting on the towed cable element. The released cable length was divided into n small, rigid, straight elements to establish equilibrium in the underwater towed cable system under steady-state conditions. The size of each cable element was also divided finely enough to ensure calculation accuracy. Therefore, it was assumed that T and dT lie on the same straight line, with θ and ds being sufficiently small. An equilibrium equation was proposed in the normal and tangential directions of the cable element (Berteaux, 1976; Zhang et al., 2023)
In the tangential direction of the towed cable element,
In the normal direction of the towed cable element,
where P(N) denotes the weight of the towed cable element in water; Fn (N/m) is the hydrodynamic force acting in the normal direction; Ft (N/m) is the hydrodynamic forces acting on the cable element in the tangential direction; T(N) is the tension acting on both ends of the cable element; θ is the angle between the horizontal baseline and the deflection angle of the towed cable element. dT, dθ, and ds are small increments of tension, inclination angle, and length, respectively.
3. Drag and Lift Coefficients
3.1 Definition
A long, flexible towed cable is connected at one end to the towing vessel and at the other end to the towfish. The hairy fairing cable is equipped with attached hairs. Fig. 3 presents a detailed configuration of the towed cable.
CFD simulations were conducted to determine the drag and lift coefficients of a towed cable element at various inclined angles and operating speeds. Two simplification models presented in Fig. 4 were used to evaluate the impact of this approach on hydrodynamic drag and lift because of the complexity of modeling the hairy fairing cable with its many movable flexible hairs. For cable shape S1, the hairy fairing cable was modeled as a straight tube attached to the main cable. In contrast, cable shape S2 was segmented into numerous small strands, and grooves were added to the main cable to resemble the sample cable.
Fig. 5 shows the structure of the hairy fairing cable as imported into the calculation software. The extension part of the main cable was added to avoid the influences of the two ends of the main cable on the flow and the impact on the main cable. These two parts have the same structure as the main cable and are denoted as the upper and lower parts.
Fig. 6 shows the mesh generation and visualization of the computational domain and boundary conditions of the cable in Ansys Fluent. The regions near the cable and the hairy fairing cable were refined with a smaller mesh size to enhance the mesh quality and ensure appropriate mesh growth. The velocity inlet, pressure outlet, and symmetry settings are applied to the inlet, outlet, and remaining boundaries, respectively. The number of hairy fairings was reduced to minimize the computational burden because of the small and complex shape of the fairings, as shown in Fig. 6(b). Table 1 lists the main numerical setup used in this study.
The position of the towed cable varies in inclined angles ranging from 0° to 90°, depending on factors such as the position and speed of each cable element. Nevertheless, numerical studies often use constant drag coefficients, neglecting the effect of inclined angles. This often results in a significant underestimation of the towed cable position, particularly at high inclined angles. Consequently, two hairy fairing cable models were examined to analyze the drag and lift forces at angles from 0° to 90° at 10° intervals and varying speeds (V1, V2, V3, V4, and V5), where the lowest velocity is denoted as V1. The large eddy simulation (LES) turbulence model, chosen for its established reliability as reported by Hart (2016) in a study on flow over spherical objects, was used to solve the governing equations. Table 2 lists the key dimensions of the hairy fairing cable.
3.2 CFD Results and Hydrodynamic Coefficients
Figs. 7 and 8 show the velocity contours of hairy fairing cable models S1 and S2 at inclined angles of 90°, 70°, 50°, and 30°, respectively. Despite using the LES model to estimate the drag and lift forces, the velocity contour behind the cylinder showed symmetry on both sides of the towed cable, particularly at smaller inclined angles. This symmetry suggests minimal vibration and strumming effects, which were not considered in this study. In addition, the disparity between the front and rear velocities of the cylinder tends to decrease as the inclined angle of the cable element decreases, potentially reducing the hydrodynamic force on the cable element.
The drag force is defined as the force parallel to the flow direction, while the lift force is perpendicular to it. For comparative purposes, the drag and lift from the CFD simulations were converted into coefficients by dividing by the fluid density, cable diameter, and flow velocity around the cable component. The formula also incorporates the inclined angle of the cable to account for its projection onto the plane perpendicular to the flow 0.5 ρsin(θ)DV2. The drag coefficient increased gradually as the inclined angle of the cable increased (Fig. 9(a)). For shape S1, the drag coefficient peaked at 90°. In contrast, the drag coefficient for shape S2 peaked at an inclined angle of approximately 60° and decreased slightly as the angle reached 90°. Fig. 9(b) shows the trend in the lift coefficient for S1 and S2. The lift coefficient was minimal at 0° and 90°, and then increased gradually towards the region between 0° and 90°. The maximum lift coefficient in both cable shapes was observed between 40° and 50°.
Fig. 9 presents a comparative analysis of the drag and lift coefficients on model S1, model S2 hairy fairing, and bare cable elements. Model S1 displays lower drag and lift coefficients than model S2 because of its simplified design, which reduces the meshing complexity and computational time but fails to capture the detailed physical flow properties accurately. Model S2, with its detailed representation of subdivided hairs and added curvature, encounters increased forces, leading to higher overall force magnitudes. Furthermore, Fig. 9 shows a reduction in drag and lift coefficients with increasing operational speeds or Reynolds numbers, ranging from 6.93 × 104 to 5.19 × 105. This finding corroborates previous studies on the flow dynamics over cylindrical objects because of the transition of turbulent flow from the subcritical to the critical regime. In addition, a comparison of the force coefficients on each element of towing cables S1 and S2 with those on the bare cable element showed that the drag and lift coefficients acting on the bare cable element are higher. This was attributed to the vortices generated behind the bare cable element, which induce vibrations. These vibrations significantly increase the drag and lift coefficients. For the hairy cable, the hair-like structures attached to the main cable break up vortex patterns, stabilizing the cable and reducing vibrations. This leads to a decrease in overall drag on the cable.
The force acting on the towfish must be considered to calculate the posture of the towing cable system accurately. Fig. 10 shows the external force components acting on the towfish, which include gravity (GB), buoyancy (BB), drag (FBD), and lift (FBL). The drag and lift forces on the towfish were calculated at various speeds using the CFD method. Fig. 11 shows the lift and drag forces on the towfish at different speeds. The hydrodynamic and hydrostatic forces were combined to establish the boundary conditions for the equilibrium equation. These conditions were then applied to the differential Eq. (3) to determine the state of the cable system. The parameters considered are as follows:
Total force in the z-direction:
Total force in the x-direction:
Initial inclined angle:
Initial Tension:
where, GB and BB are the weight and buoyancy of the towfish, respectively; FBL and FBD are the lift and drag forces acting on the towfish, respectively.
4. Simulation Results
Steady simulations of the towed cable system were conducted at various speeds (VS1, VS2, VS3) and released lengths to validate the hydrodynamic forces on the hairy fairing cable calculated using the CFD method, as listed in Table 3. These simulations involved solving differential Eq. (3) within the spatial domain using the explicit Runge–Kutta method, allowing for a determination of the position, tension, and inclined angle of each cable element. The boundary conditions are based on the initial tension and initial inclination angle of the towfish.
Fig. 12 compares the depth of the sea trial, hairy fairing cable shapes S1, and hairy fairing cable shapes S2, highlighting the disparities between the simplified and complex models of the hairy fairing cable. In particular, simulations using the complex modeling of the hairy fairing cable aligned closely with the sea trial results, contrasting with the oversimplification in the simplified model. This disparity results from the reduced drag and lift forces in the simplified model and the lack of consideration for the true complexities of the physical phenomena acting on the hairy fairing cable element.
Table 4 lists the difference percentages in depth between shapes S1 and S2 compared to sea trial results, indicating significant improvements in model shape S2 over shape S1. On the other hand, a thorough investigation of the hydrodynamic drag and lift on the towed cable element remains critical. Although the disparity between the simulated and actual sea trial depths was reduced, the simulated depths still exceeded the expected values. This discrepancy highlights the need for larger drag coefficients to depict the behavior of the towed cable system accurately under real-world conditions.
Detailed modeling produced satisfactory results compared to simplistic approaches for the towed cable element. Therefore, a simulation was conducted to evaluate the effect of pitch angle on drag and lift coefficients using model shape S2. Fig. 13 shows the drag and lift coefficients for the towed cable model S2 at five different speeds and inclination angles ranging from 0° to 90°. For ease of calculation and comparison, the drag and lift can be transformed into coefficients of force in the normal and tangential directions. Unlike the conversion formula for drag and lift, the flow direction is perpendicular to the cable, termed the normal force, and parallel to the cable, called the tangential force. Thus, the projection for the normal force is D and πD for the tangential force. Consequently, the nondimensional factor for normal and tangential force is
0.5 ρ π D V n 2 0.5 ρ π D V t 2
The transformation formula from the lift and drag to normal and tangential forces is as follows, as shown in Fig. 2:
Fig. 14 shows the drag and lift coefficients acting on the cable element in relation to the inclined angle of the fairing cable shape S2. The normal force coefficient tended to decrease as the angle of water flow incidence increased to 90°, as shown in Fig. 14(a). At an angle of 0°, the water flow ran parallel to the cable element, resulting in no force acting on the cable element in the normal direction. Conversely, the force acting in the tangential direction of the cable element increased as the inclined angle increased to 90°, as shown in Fig. 14(b). At a 90° angle, there was no force along the cable element in the tangential direction. The determination of hydrodynamic coefficients for the normal forces at an inclined angle of approximately 0° and for the tangential forces at approximately 90° presents challenges, requiring meticulous investigation to ensure accurate calculations for the state of the towed cable system.
Fig. 15 compares the cable posture between simulations using the coefficients of various inclined angles and a constant coefficient across different speeds. The dashed line indicates the posture of the towing cable system using a constant dynamic coefficient, while the solid line shows the cable posture that considers the influence of the inclined angle on the forces acting on the cable element. The horizontal line denotes the sea trial value of depth. With a constant coefficient, the force coefficient in the longitudinal direction along the cable element was used when the cable was in the 0° position, while the force coefficient perpendicular to the cable element corresponded to the 90° position. Fig. 15 shows that the cable system calculated with the constant coefficient showed a greater depth than the system using coefficients dependent on the inclined angles. The cable elements in a towed cable system can take any inclined angle from 0° to 90°. The normal and tangential force coefficients varied with different inclined angles (Fig. 14). Using the constant coefficient, usually assumed to be the smallest, results in underestimations of the tangential and normal forces compared to the actual values, corroborating the simulation results depicted in Fig. 15. In addition, the simulation results suggest that applying a dynamic coefficient that accounts for the inclined angle of the cable component yields outcomes more aligned with the sea trial results.
Fig. 16 compares the cable tension under simulation using the coefficients for various inclined angles and a constant coefficient at different speeds. The tension in the cable system is generally lower when using a constant hydrodynamic force coefficient. This can be explained by the first equation in Eq. (3), showing that a reduced tangential force decreases the tension.
5. Conclusions
This study evaluated the underwater position and tension in a towing system, considering the effects of the drag and lift coefficients as modified by inclined angles and operational speed. These results were compared with the data from sea trials. In terms of accurate cable element modeling, the cable elements must be modeled as close as possible to the actual cable configuration to achieve more precise calculations than those obtained with oversimplified models. Using a constant dynamic coefficient without considering the effects of inclined angles on cable elements can cause simulation inaccuracies for hydrodynamic coefficient considerations. Angles of 0° and 90° significantly influence the normal and tangential forces. Although some improvements have been observed compared to the sea trial results, ongoing refinement is necessary to enhance the accuracy in calculating the forces acting on the cable element.
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