Nomenclature
AL:
Longitudinal projected area of underwater part (m2)
B:
Breadth of vessel (m)
CP:
Specific heat coefficient (−)
d:
Draft of vessel (m)
D:
Depth of vessel (m)
fβ*:
Free – shear modification factor (−)
fi:
Body force component (N/m3)
fβ:
Vortex – stretching modification factor (−)
FrL:
Froude number by vessel length (−)
oxyz:
Local coordinate system
k:
Turbulence kinetic energy (m2/s2)
LOA:
Vessel overall length (m)
L, LWL:
Vessel length at waterline (m)
N:
Yaw moment (N·m)
Averaged pressure (Pa)
Production term (kg/(m·s2))
User-specified source terms (kg/(m·s2))
t:
Time (s)
U:
Speed of current (m/s)
UFS:
Relative error of dRE and the fine solution (%)
u:
Surge velocity (m/s)
Averaged velocity component (m/s)
v:
Sway velocity (m/s)
X:
Surge force (N)
xL:
Longitudinal position of the area center of AL
Y:
Sway force (N)
y+:
Nondimensional distance from wall to first prism cell (−)
α:
Current angle (°)
αi:
Volume ratio of phase i in a cell (−)
αwater:
Volume ratio of water in a cell (−)
αair:
Volume ratio of air in a cell (−)
δRE:
Error of the fine solution by Richardson extrapolation (*)
λ:
Scale ratio (−)
μ:
Viscosity (Pa·s)
μt:
Turbulent eddy viscosity (Pa·s)
ρ:
Density (kg/m3)
σk,:
σω Model coefficients (−)
ω:
Specific dissipation rate (1/s)
1. Introduction
Offshore activity plays a significant role in the global energy landscape. Currently, the average number of operational oil rigs is approximately 2,000, which is comparable to the levels observed in the 1990s (Statista, 2025). In addition, offshore wind energy has gained momentum, marked by an increase in offshore turbine capacity and the number of new installations in recent years (Global Wind Energy Council, 2025). The demand for various dynamic positioning (DP) vessels for offshore operations such as transportation, drilling, and pipe laying is increasing. Although these vessels have specific missions and hull designs, the DP system is crucial for maintaining the position of a vessel against environmental factors, such as waves, wind, and currents. It also facilitates slow maneuvering near offshore installations. The platform supply vessel (PSV) can operate in DP mode for up to 50% of its operational time (Pham & Hoang, 2020). A set of current loads in all horizontal directions is necessary to develop an effective DP system.
The current exerts loads—hydrodynamic forces and moments—on a vessel, which can be understood as a combination of viscous and pressure components, or potential and wave-resistance components. In practical terms, Remery and Oortmerssen (1973) considered current flow to be a steady phenomenon because it does not vary significantly over time. Consequently, the loads are assumed to be steady and can typically be determined experimentally, by empirical formulae, or computational fluid dynamics (CFD). Experimental measurement is the most traditional method, conducted in wind tunnels or water towing tanks. In wind-tunnel tests, however, only the underwater portion of the vessel is utilized. The development of the boundary layer near the tunnel wall can alter the flow upstream, leading to less reliable results (Kim et al., 2009). Using a towing tank is generally preferred, particularly for scaled models (Kim et al., 2009; Koop, 2020; Piaggio et al., 2020; Aydin et al., 2022). Nevertheless, the number of water towing tanks is limited, and adjusting the design of physical vessel models for testing can be challenging. As a result, experimental measurements are not always feasible. Empirical formulae offer the advantage of rapid predictions but are strongly dependent on the type of vessel hull, such as tankers (Remery & Oortmerssen, 1973), drill ships (American Petroleum Institute, 2005), or DP vessels (Det Norske Veritas, 2018). When a vessel hull differs from the sample used or when more specific results are required, the reliability of empirical formulae diminishes. The third option is CFD, which benefits from advances in computational resources and algorithm development. CFD is renowned for its adaptability to various vessel geometries, providing significant advantages during the design phase. Previous CFD-based studies have examined the prediction of current load forces acting on OSVs, primarily using the RANS algorithm. Koop (2020) examined the current load on a liquefied natural gas carrier and a shuttle tanker, focusing on scale and shielding effects. The author used a CFD approach, assuming a steady solution with the free surface treated as a slip wall. Zan et al. (2020) simulated the current load for a pipelaying crane vessel and validated their results against wind-tunnel experiments. This simulation focused solely on the submerged portion of the hull for the current testing. Aydin et al. (2022) compared empirical formulae with CFD techniques for determining the current load on a PSV, including the free surface in their simulations despite anticipating its negligible impact. Mauro et al. (2023) conducted CFD analyses to generate a dataset of hydrodynamic forces, using a numerical technique based on the double-body assumption and a time step of 0.05 s for the unsteady RANS (URANS) model, and developed a rapid prediction method for the current load on a PSV. The studies mentioned either excluded the free surface to simplify calculations or included it to minimize uncertainty. While including the free surface increased the computational costs, excluding it relied heavily on assumptions. Furthermore, steady and unsteady RANS models were evaluated without considering the free surface. They revealed limitations in method comparisons that undermined confidence in the chosen CFD model. The numerical resolution may have been overly simplistic or too computationally expensive. Regardless of the method used, accurately capturing the hydrodynamic forces due to the current is essential.
This study examined the impact of the free surface on current load calculations using CFD by comparing two modeling approaches. The first approach used the volume of fluid (VOF) model to represent the free surface, while the second treated it as a symmetry plane based on a double body assumption. The former used the URANS model while the latter was analyzed using steady RANS and URANS models. The shear stress transport (SST) k-ω equations were used to resolve the turbulence kinetic energy and dissipation rates. A 1:64 scaled PSV model, corresponding to a full-scale length of 110 m, was selected for the simulation, with a target current speed of 0.193 m/s. The comparison was conducted with three representative current angles and varying current speeds. The experimental data from the Changwon National Towing Tank and values from the Det Norske Veritas (DNV) empirical formula (DNV, 2018) supported the analysis. This study validated an efficient CFD method that balances cost and accuracy for predicting current loads, particularly those affecting DP vessels.
2. Current Load Calculation
2.1 Vessel Specification and Test Condition
The PSV model used for the test is characterized by the main parameters listed in Table 1. Its geometry is illustrated in Fig. 1. Fig. 2 presents the local coordinate system (oxy) along with the sign convention for (α). The origin (o) is positioned at the midpoint of the waterline and along the center plane of the vessel, with the ox axis directed towards the bow and the oy axis oriented towards the starboard. The vessel was held stationary under a uniform current flow during the test. By referring to relative studies (Kim et al., 2009; Koop, 2020; Aydin et al., 2022; Mauro et al., 2023) and the available statical data of buoys deployed in United State offshore oil and gas field (National Data Buoy Center, 2025), the current speed was selected as Free Surface Effect in the Current Load Calculation Based on CFD 125 0.193 m/s for the scaled model, corresponding to 1.543 m/s in full-scale conditions. Under certain conditions, the speed was increased to 0.579 m/s for additional verification. Table 2 provides details of the test conditions. The test was conducted in fresh water at 20 °C.
2.2 Numerical Method
The numerical method was implemented using the commercial CFD software Star-CCM+ version 20.04.008. The fluid flow of an incompressible Newtonian fluid was governed by the Navier–Stokes equations. The CFD simulation used the RANS method to model these equations, as shown in Eqs. (1) – (2). The SST k-ω model was used to capture the flow separation around the hull, with its formulations presented in Eqs. (3) – (4). The free surface distribution was represented using the VOF model, as detailed in Eqs. (5) – (9). Under the steady-state assumption, the time derivative term was set to zero; otherwise, a time step was applied. The y+ value was maintained below 10, with an average of 5 or less. The computational time was recorded on a computer equipped with 32 GB of RAM and a 12th Gen Intel Core i7-12700 CPU.
Two domain types were tested: the first for simulation without a free surface (sim. w/o FS) and the second for simulation with a free surface (sim. w/ FS). Both domains were the same size, but sim. w/ FS included an additional zone for the air phase. Fig. 3 and Table 3 present the dimensions and boundary conditions of the domains. The VOF flat wave functions were used for the velocity inlet and pressure outlet conditions when the free surface was considered. The mesh refinement strategy was optimized for the comparison and flow characteristics. It was similar in the vertical plane (Fig. 4(a)) but adapted for flow separation depending on the current angle in the horizontal plane (Fig. 4(b)). This adjustment helped distribute the small mesh cells to the necessary zones. Furthermore, sim. w/ FS and sim. w/o FS shared the same mesh refinement for the underwater zone. In this study, trimmed cells were used for mesh generation, and the prism mesh was applied for boundary-layer meshing around the vessel hull.
2.3 Experiment Work and DNV Formula
The experiment on a physical PSV model at the same scale ratio was conducted in the Changwon National University Towing Tank, which has dimensions of 20.0 m × 14.0 m × 1.8 m in length, width, and depth, respectively (Fig. 5). The hydrodynamic force was measured by rotating the vessel to the desired angle and then towing it straight (Fig. 6). In addition to the experimental validation data, the DNV formula for a DP vessel was also used to assess the performance of CFD methods, as shown in Eqs. (10) – (12).
3. CFD Verification
The verification process followed the Stern et al. method (ITTC, 2024). Fig. 7 shows the number of mesh cells for a base mesh size of 1. This base size controlled the cell dimensions in the region of interest near the hull. Increasing the base size from 1 to 2 resulted in the trimmed cell size doubling in each Cartesian direction. Tables 4 and 5 list the results of the mesh and time step convergence tests, which were derived from the hydrodynamic forces illustrated in Figs. 8 and 9. Not all cases had a magnitude of R < 1, indicating convergence, but the values still fluctuated within a certain range. This behavior may have been caused by the high turbulence in the disturbed flow around the hull, which is difficult to capture accurately with an averaged model, such as RANS. In some instances, the resulting force showed minimal changes across refinement levels (< 1%), even though the index suggested divergence. For the other divergence cases, the relative error UFS appeared relatively large because the forces calculated with the finest refinement differed noticeably from those obtained with the coarser refinements. Nevertheless, the discrepancies in magnitude remained below 8% of the finest-refinement result. They exhibited the inherent limitation of RANS models, especially at coarser refinements. Despite these challenges, the results remained reliable for calculating the current load.
4. Free Surface Effect
4.1 Current Angle
An investigation of the representative angles of the current load test at FrL = 0.0491 revealed an obvious transition in the lateral force distribution along the longitudinal axis of the PSV from the heading current to the beam current conditions (Fig. 10). A head-on current did not generate a noticeable lateral force. On the other hand, when the current angle changed from 180° to 90°, this distribution became sizable and shifted from a high concentration near the bow to a more even distribution along the hull. The bow and stern zones were not under the Y force impact because the local flow mainly tended to move away from the hull along the ox axis. The comparison showed that three CFD approaches were consistent in predicting the Y force. Although local discrepancies up to 20% were observed between sim. w/o FS (RANS) and the other ones at (α = 120°), the URANS model produced similar distributions regardless of whether the free surface was resolved. The velocity field shown in Fig. 11 showed similar distributions. The difference in downstream flow in the horizontal view at (α = 120°) was combined with the introduced local discrepancies. In addition, the region upstream and near the vessel showed slight variations. This was due to the run-up effect and the flow reflection by the vessel hull when the free surface was included. The wave elevation was minimal, accounting for less than 2% of the vessel draft (Fig. 12(a)). No noticeable waves were observed during the experiment (Fig. 12(b)). The hydrodynamic forces calculated using different CFD methods were compared with experimental data and the DNV formula, as shown in Fig. 13. In general, the forces and moment follow coherently similar trends except for the X force by the DNV formula. When the current angle decreased from 180° to 90°, the Y force increased and reached its maximum state while the N moment rose from zero and fell back to zero. Specifically, CFD provided the current load closer to the experimental data than the DNV formula. Instead of using the empirical formula, using the CFD method increased the accuracy of the force prediction up to almost 100%, especially where a was 120°–150°. Nevertheless, the proposed CFD approaches produced approximately the same differences as the experimental data. Either free-surface resolution by the VOF model or the one-phase simulation following the double body assumption reached the same level of current load accuracy. Nevertheless, the URANS model incurred an extremely high computational cost to handle the slow-propagating flow in the current load test. Furthermore, the VOF model was solved by an unsteady turbulence model. This combination turned out to be more expensive. The recorded computational costs ranged from a hundred to a thousand times those of sim. w/o FS (RANS) (Fig. 14). The steady RANS model showed its robustness in the current load calculation (FrL = 0.0491), where it provided a reliable result with a small cost. The free surface had a negligible effect on the resulting force, but required substantially more computational resources.
4.2 Froude Number
The DP vessel was large and combined with a low current speed to produce a small Froude number in the current load test. Under head-current conditions, a test using the double-body assumption could underestimate the force because the Froude number was substantially large due to the lack of the wave-making component. Nevertheless, a current coming into the side of the vessel (e.g., α = 90°) may lead to an unknown phenomenon. Therefore, higher current speeds were studied to reinforce the previous findings. The current speed for the scaled model ranged from 0.193 m/s (FrL = 0.0491) to 0.579 m/s (FrL = 0.1473). This corresponds to the range of (1.543–4.630) m/s in the full-scale vessel. As shown in Fig. 15, the case of (α = 180°) had a higher X by sim. w/ FS when the speed was increased. This agreed with the wave-making theory. By contrast, the case of (α = 90°) indicated that the double body assumption provided a slightly higher Y. This was attributed to the assumption of the free surface as a symmetry plane, where this mirror-like plane reflected the disturbed flow tending to rise. Consequently, the flow produced a larger pressure on the upstream side of the hull, which mostly focused on the zone around the middle of the vessel and near calm waterline (Fig. 16). The Y, N of (α = 180°) and the X, N of (α = 90°) were relatively small in value compared to the force exerted along the current direction. Although discrepancies in force distribution were observed, the sim. w/o FS produced the primary force exerted by heading current or beam current less than 4% compared to the corresponding sim. w/ FS’s result. The difference between the two CFD-based methods became more pronounced as the Froude number increased, particularly for (FrL > 0.1). Nevertheless, the investigated current speed for a full-scale DP vessel was typically below 1.543 m/s (FrL = 0.0491). Therefore, the current load test at (FrL = 0.0491) could be conducted using the steady RANS model for sim. w/o FS. It helped reduce the computational costs while providing a reliable result.
5. Conclusion
This study examined the effect of the free surface on the current load calculations for a DP vessel using CFD. Three representative current angles were tested at the desired current speed. The effect of the Froude number on the current load was also examined for beam current (α = 90°) and head current (α = 180°). Two types of domains were compared under each condition: one modeled the free surface, while the other assumed it to be a symmetry plane. The findings were as follows:
(1) The CFD result agreed well with the experimental data and provided better value than the empirical formula approach.
(2) The double-body assumption with the steady RANS model yielded a fast and accurate method for the current load calculation, especially when the FrL was approximately 0.0491.
The study suggests that similar current load tests for large surface vessels (conventional monohull types at FrL ≈ 0.05) can adopt the proposed CFD approach based on the steady RANS model without resolving the free surface. This strategy can reduce the computational cost by at least a hundredfold while maintaining the same accuracy in force prediction as the simulations with free-surface resolution. On the other hand, a validation regarding the free-surface effect is recommended when significantly different vessel hull forms or higher Froude numbers are considered. The procedure proposed in this study can serve as a reference framework, involving an examination of the representative current angles and a wide range of current speeds. The flow typically propagates slowly. Therefore, the simulation time should be sufficiently long to ensure converged results, especially in unsteady simulations. This approach enables the examiner to decide whether the free surface can be safely neglected. Future work will extend the study to additional vessel types and flow conditions to further assess the free-surface effects and enhance the general applicability of the present findings to current load predictions. Moreover, scale effects and uncertainty in experimental model tests will also be valuable topics for further investigation to improve the reliability of the present findings.
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