CFD-assisted linearized frequency-domain analysis of motion and structural loads for floating structures with damping plates

Damping plates are widely used on floating offshore structures to reduce the dynamic responses in waves. Linearization of the nonlinear drag loads are needed to solve the floater motions by frequency-domain methods, which, to date, still dominate the analysis of wave-frequency responses of floating structures in engineering practice. While existing studies focus on damping effects on heave motions, there is a lack of understanding on how the drag loads on damping plates should be modeled to predict the angular motions and structural loads of the floaters. Two different configurations of Morison elements are investigated in this paper. The first considers the entire damping plate as a whole, while the second locates Morison elements at the estimated ‘points of acting’ of drag loads for different parts of the damping plate. Computational Fluid Dynamics (CFD) analysis is utilized to obtain the required drag coefficients and the ‘points of acting’ as function of Keulegan–Carpenter number. In comparison to full CFD analysis of the same floater in regular waves, a hybrid approach is shown to give satisfactory wave-frequency results, not only for motion responses, but also bending moments and shear forces on the damping plate. However, non-negligible second-harmonic structural loads that cannot be properly accounted for by the existing Morison-drag formulation are also identified in CFD analysis, calling for special attention to structural assessment under extreme wave conditions. This study also demonstrates that locating the Morison elements at ‘points of acting’ improves the prediction of motion responses and structural loads at resonance.


Introduction
Damping plates, also called heave plates, are widely used on offshore floating structures, e.g. cylindrical FPSOs (Shao et al., 2016), pontoon on floating bridge , Spar platforms (Tao and Cai, 2004) and floating offshore wind turbines (Robertson et al., 2017), to provide significant viscous damping to reduce the motions at resonance (Xiang et al., 2017;Lopez-Pavon and Souto-Iglesias, 2015). The presence of the damping plates also increases the natural periods of heave, roll and pitch motions, thus possibly shifting the natural periods of those Degrees of Freedom (DOFs) out of the wave-frequency range. It is worth noting that a similar mechanism is also present in perforated plates, where flow separation through holes can induce pressure discharge (Molin, 2011;Mackay et al., 2021;Liang et al., 2022).
Although the responses of most large-volume offshore structures are dominated by inertia effects, the influence of viscous drag loads on the motion is non-negligible. Examples are the resonant heave motions of floating structures with small water-plane area and the roll motion * Corresponding author.
of ships, where the radiation damping might be much smaller than the viscous damping at the natural periods of those DOFs. To model the floating structures with a damping plate, high-fidelity CFD models based on Navier-Stokes equations and proper turbulence modeling are the most sophisticated. However, the required computational cost is still limiting wide application of CFD solvers in engineering analysis, particularly in the early design and optimization phase. Therefore, linear or second-order frequency-domain models based on radiationdiffraction theory for potential flows (Newman, 1977;Faltinsen, 1993) have been developed and implemented in most of the state-of-the-art engineering tools in marine hydrodynamic.
In early design phases, a common approach, referred as hybrid method hereafter, is applied to empirically account for the viscous drag loads in the frequency-domain potential-flow analysis Ishihara and Zhang, 2019;Shao et al., 2016Shao et al., , 2019. For a moving structure in water, the viscous drag loads consist of a frictional and a form contribution, where the friction is caused by viscous effects https://doi.org/10. 1016/j.oceaneng.2023.114924 Received 22 February 2023 Received in revised form 30 April 2023; Accepted 21 May 2023 in the boundary layer near the walls, and form drag is related to fluid separation and vortex shedding (Tao and Thiagarajan, 2003a). (Zhang and Ishihara, 2020) performed forced oscillation analysis using CFD to study the hydrodynamic coefficients of a heave plate. The results show that the drag contribution increases significantly near the edge, demonstrating good correlations between vortex generation and drag force. Numerous investigations (Tao and Thiagarajan, 2003b;Tao et al., 2007;Zhu and Lim, 2017;Shao et al., 2019) have shown that the drag coefficient on damping plates are more dependent on the Keulegan-Carpenter number ( ) rather than the Reynolds number ( ), suggesting that scale effects are of less importance for hydrodynamic loads on damping plates, while it also depends on many other factors, such as non-dimensional plate thickness and distances to seafloor or free surface (Garrido Garrido-Mendoza et al., 2015).
Even though the hybrid method has been proven to be very efficient and powerful in the design and analysis of offshore structures, there still exist challenges that have not been properly addressed. The first challenge is how to accurately estimate the drag coefficients and use them in the linearized frequency-domain analysis. Either model tests  or CFD can be employed to obtain a database of for different values. The number is unknown a priori because it depends on the motions of structures, whose solutions need as an input in the linearized motion equations. Thus iterations are required to determine and for a floating structure with a damping plate. Another challenge is related to the viscous-drag induced moments and structural loads. In the calibration of coefficients against model tests, there is often limited information on the drag loads distribution over the damping plate. Consequently, there exists no rational way to estimate the drag-induced roll and pitch moments, which in return may lead to inaccurate prediction of angular responses. The same challenge applies for the structural loads, for instance, the shear force and bending moment at the roots of a bilge box (intersection between the damping plate and the main hull of the floater), which are typically weak points of the whole structure design.
To the best of the authors' knowledge, the majority of existing studies in the literature have focused only on the heave motion of the structures with damping plate, and very few, if not none, have specifically investigated how the roll/pitch moments and structural loads due to the drag loads can be properly modeled. This study aims to provide a practical approach to more accurately model the drag loads (forces, moments and structural loads) in the state-of-the-art frequencydomain hydrodynamic analysis. CFD analysis is utilized to estimate the drag coefficients and 'point of acting' for the drag loads as function of number for different parts of the damping plate. This enables us to design more reasonable configuration of Morison elements in a frequency-domain hydrodynamic model, which runs in iterations and consistently select coefficients in accordance with the actual relative velocity amplitudes. To simplify the analysis but without losing generosity, this study will, as an example, consider a 2D cross section comprising a vertical hull and a bottom plate. Two rigidly connected 2D cross sections are also studied as a second example. Two configurations of the Morison elements are compared. One allocates the Morison elements at the 'point of acting' of the drag loads estimated from dedicate CFD analysis, while in the other configuration, the Morison elements are located close to half radius of the damping plate. The hybrid model is systematically verified and validated through comparison against full CFD analysis of the same floating structures in waves. The limitation of the Morison-drag formulation will also be discussed when it comes to the modeling of nonlinear structural loads.
The rest of the paper is organized as follows: Section 2 presents the theoretical formulation. The hybrid model and the CFD model will be introduced, verified and validated in Section 3. In Section 4, the drag coefficients are obtained from dedicated CFD analysis with forced harmonic motions of the structures. The hybrid numerical model is validated in Section 5 by comparing with full CFD results of floating structures in waves. The scale effects and limitation of the present hybrid model are addressed in Section 6. Finally, conclusions will be summarized in Section 7.  Fig. 1 defines three right-handed Cartesian coordinate systems for description of the motions of the structure, namely two inertial coordinate system and an Earth-fixed coordinate system. The Earth-fixed coordinate system is used to describe the waves. The coordinate system does not follow the dynamic motion of the structure, and the -plane coincides with the calm-water surface. The coordinate system is body-fixed but with its origin at the center of gravity (CoG) of the structure. This coordinate system is used when describing the dynamic motions of the structure.

Theoretical description
Two numerical models will be utilized to study floating structures with a damping plate (or called damping box if the thickness of the structure is not small) • Hybrid model: It solves the motion equations in the frequency domain, while the linearized drag loads are also considered. All potential-flow hydrodynamic coefficients are pre-calculated from linear diffraction-radiation analysis. • CFD model: It solves the incompressible Navier-Stokes equations, which is more comprehensive but less efficient than the Hybrid model.
In Sections 2.1 and 2.2, the theoretical description of the Hybrid model and the CFD model will be briefly presented, respectively.

Equations of motion with empirical corrections
The viscous effect is ignored in potential flow theory. However, it is of vital importance for the structure with damping plate due to the vortex shedding from the sharp edges. Thus, the empirical correction of the viscous loads is needed to obtain more realistic results for engineering applications. For floating bodies in waves, considering linear diffraction and radiation, as well as linearized drag loads, the equations of motion can be formally written as: Here, we have assumed that the complex solutions have a common time-harmonic factor of e i . is the wave frequency. is wave heading, and  is mass matrix of floating structure. ( ) and ( ) are the matrices of added mass and damping calculated from a linear potential-flow solver, respectively.  is the hydrostatic restoring matrix and ( , ) is the complex amplitude of the wave excitation due to potential flow. is the complex amplitudes of the motions. Extra added mass, damping and restoring matrices, denoted by  ,  and  respectively, are also included in the above motion equations. Those extra terms offer possibilities to empirically account for other mechanisms that are not directly included by the linear hydrodynamic coefficients, such as mooring system, viscous drag on damping plate. A correction to the linear potential-flow excitation force , has also been added on the right-hand side of Eq. (1) for completeness.

Drag loads and their linearization
The viscous drag loads will be empirically accounted for using drag formulation in the Morison equation: where is the drag coefficient. is the relative velocity between the ambient flow and the structure.
is the projection area against the flow direction. In practice, a cross-flow principle is applied, and = Re{ i } is understood as the relative velocity component in the normal direction (e.g. perpendicular to the length of a slender member or a heaving plate).
is the complex amplitude of the relative velocity. Even though the above drag force formulation is empirical and can be questioned, no better empirical model is available in the literature, and thus the Morison-drag formulation is still the most popular approach in engineering practice to incorporate the viscous effects in the early design phases of offshore structures.
To include the viscous drag loads in the frequency-domain analysis, the drag loads have to be linearized, which means that Eq. (2) is approximated as Since = − is the relative velocity where and are the ambient flow velocity and the velocity of the floating structure, the above drag load can be divided into two parts: Here and − are understood as the linearized viscous excitation and viscous-damping forces, respectively.
There are two ways to linearize the drag loads in Eq.
(2), namely the regular-wave linearization and stochastic linearization (DNV, 2017;Shao et al., 2016), which will be introduced shortly. In the regularwave linearization, the equivalent damping is found by requiring that the average work done by the drag loads is equal to that by the linearized load: leading to The overline in Eq. (5) means time average, and | | is the amplitude of relative velocity . In principle, the regular-wave linearization is suitable for motion prediction in regular waves. For irregular waves, the following linearized damping can be applied (Borgman, 1967;Shao et al., 2016;DNV, 2017): Here is the standard deviation of the relative velocity.

Governing equations for the CFD model
In the hybrid method described in Section 2.1, a key to obtain a good estimation of the linearized drag loads is the drag coefficient , which in this study will be obtained from dedicated CFD analysis. For completeness of the paper, this section briefly presents the governing equations for the CFD solver, i.e. the continuity equation and momentum equations where is the control volume bounded by the closed surface . and are the fluid and grid velocities, respectively; is the unit vector normal to and pointing positively outwards; represents time; is the pressure, and is the fluid density; and are the unit vectors in the th and th directions, respectively; is the gravity acceleration; is the viscous stress tensor. denotes the source term which is used to generate and absorb waves.
These equations, when combined with appropriate boundary conditions, can describe the behavior of the fluid flow. For hydrodynamic problem of rigid body motion, it can be assumed that the density of fluid is constant. When the case of two-phase flow is involved in the solver, the interface between water and air needs to be considered, and the volume of fluid (VOF) method is used to capture the free surface. The VOF multiphase model assumes that the immiscible fluids within the control volume share the same velocity, pressure, and temperature fields, so that a solution similar to that of single-phase flow can be employed. At the interface between the water and air, a physical property , e.g. velocity or pressure, is approximated using the volume fraction ∈ [0, 1], so that the mixture of the two fluids is treated as an equivalent fluid: Here = 0 presents air, = 1 presents water, and = 0.5 is chosen to represent the free surface. Subscripts 'water' and 'air' denote the corresponding phases. The conservation equation is used to describe the variation of volume fraction :

Physical and numerical models
This section describes the physical model that will be extensively studied in later sections, followed by the setup of two numerical models, i.e. the hybrid model and the CFD model.

Physical model
A 2D cross section in model scale, comprising a central column and a bottom box, will be considered as a reference structure in this paper. The cross section is sketched in Fig. 2. Here is the diameter of the bottom box, the diameter of the central column, and the thickness of the bottom box.
is the draft, and ℎ is the height of the entire structure. The following model parameters are considered: = 0.48 m, = 0.24 m, = 0.12 m and draft = 0.4 m. A side-by-side configuration of the two identical cross sections will also be studied in our later analysis. The wetted surface of the bottom box consists two parts, which are defined as follows • Bottom flange is the part exceeding the central column, defined by 0.5 < ≤ 0.5 , where is the horizontal distance to the vertical axis of the section.
• Middle bottom is the middle part of the bottom disk with ≤ 0.5 .
In our later analysis, the drag loads on those two parts will be accounted for separately in both the hybrid model and the CFD model.

Hybrid model
In the hybrid model, the potential-flow added mass matrix ( ), radiation-damping matrix ( ) and wave excitation coefficients ( , ) for the considered cross section are obtained from a state-of-theart diffraction-radiation solver WADAM (DNV, 2017). Since WADAM is a three-dimensional (3D) program, we use an elongated barge with the same cross section as defined in Fig. 2 in beam sea to approximate the 2D scenario. Sensitivity analysis has been made using different lengths ( ) of the barge, and a sufficiently large length is selected so that the 3D W. Lei et al.  Similar to previous studies by Shao et al. (2016Shao et al. ( , 2019, the viscous drag loads will be modeled by a set of Morison elements at the bottom box. The drag coefficients for the Morison elements are selected so that the total drag force on the Morison elements is equal to that given by Eq. (3) for the bottom box. Compared with the approaches that directly add the total drag loads in the motion equations, the use of Morison elements have some advantages. Firstly, in addition to the drag coefficients, the locations of Morison elements can be calibrated so that the drag-induced roll and pitch moments also match the model-scale or full-scale measurements, or high-fidelity CFD simulations. In general, it is challenging to measure individual contribution of viscous drag loads from different parts of the floating structures, for instance, the bottom flange and the middle bottom parts. However, in a CFD model, one can easily measure the drag loads on different parts of the structures and even pressure distribution on all wetted surfaces. Secondly, the calibrated Morison model may serve to predict the structural loads, e.g. the bending moments and shear forces at the root of the bottom flange, e.g. intersection between the central column and the bottom box, which has paramount importance for the design of similar structures.
For the cross section illustrated in Fig. 2, two different configurations of the Morison elements shown in Fig. 3(a) and (b) will be considered and compared. In the first configuration, a pair of Morison elements are symmetrically located to the left and right of the vertical central axis of the section, with a horizontal distance to the central axis 0 . The same drag coefficient is applied on the Morison elements. In the second configuration, two pairs of Morison elements are located on two sides of the vertical central axis, with a horizontal distance to the central axis as 1 or 2 . Each Morison element is responsible to account for the vertical force contributed by a certain part of the bottom box. For instance, the two Morison elements in Fig. 2(b) with a distance 2 to the central axis are used to model the vertical drag loads on Table 1 The Boundary conditions for the three types of CFD simulations involved in this study. the bottom flange. The locations of the Morison elements are selected as the 'point of acting' of the drag load on corresponding structure part, so that both the vertical drag force and its induced moments are consistently modeled. As it will be explained in later sections, the drag forces and their 'points of acting' on different parts of the structure can be estimated through dedicated CFD simulation and post-processing of the results.

Boundary conditions
The CFD analysis in this paper involves three types of simulations • Type 1: Forced oscillations in infinite fluid domain without a free surface. • Type 2: Forced oscillations with a free surface.
• Type 3: Freely floating structures in incident waves.
Numerical wave tanks are built up using Star-CCM+ for different simulations. The computational domains in the three types of simulation are similar. Fig. 4 shows an example of the domain and setup for Type-3 simulation, where a structure is located at the middle of the numerical wave tank. The applied boundary conditions for the three types of simulations are summarized in Table 1.
For Type 2 and 3 where free surface is involved, VOF multiphase flow model is activated to capture the free surface. Since only 2D CFD simulations will be considered in this paper, a single layer of mesh in the transverse direction of the wave tank is employed, and the symmetry boundary conditions are applied on the front and back sides of the tank.
Wave forcing is applied at both ends of the numerical wave tank to absorb the diffracted and radiated waves by the structure. More details of wave forcing mechanism can be found in Kim et al. (2012). In this paper, the length of forcing zones is set to 1.5 times the wave length on both ends of the tank. Fifth-order Stokes theory (Fenton, 1988) is used to generate incident waves. The SST − turbulence model with all-+ wall treatment is used for the forced oscillations of the structure. For freely floating structures in regular waves, the standard low-− turbulence model with all-y+ wall treatment is applied, and the initial values of turbulent kinetic energy = 10 −5 J/kg and turbulent dissipation rate = 10 −4 m 2 ∕s 3 are assumed to prevent excessive growth of turbulent viscosity near the free surface. The implicit unsteady segregated solver is employed and a second-order time discretization is set to the second order. In order to obtain accurate and stable results, the maximum convective courant number near the free surface is less than 0.5 in all simulations.

Overset mesh
The overset mesh is employed to accommodate the motion of the structure. Two sets of meshes, namely the background mesh and overset mesh, need to be generated. To reduce the accuracy loss, the background mesh is refined at the overlapping area with the overset mesh. An example of the mesh system is displayed in Fig. 5. Trimmed mesh is employed in the entire domain. Since the sharp edges of the numerical W. Lei et al. Fig. 4. A sketch of the numerical wave tank with the structure at the middle of the tank. model are almost right-angled, almost all cells are square-shaped to ensure the accuracy of the solver.
Since Type-1 analysis does not involve the modeling of the free surface, to capture the flow field around the sharp edge, the meshes are only refined near the sharp edge, while coarser meshes are used in the regions away from the structure. Five cells per heave amplitude are distributed along the vertical direction around the flange. Squared meshes, i.e. with aspect ratio ∕ = 1, are used where possible. Type 2 differs from Type 1 due to the need of modeling the free surface. The mesh aspect ratio near the free surface keeps ∕ = 8. Type 3 simulates the motion response of the structure under regular waves, 15 cells are distributed per wave height. More than 1000 time steps are adopted per oscillatory period.
Verification and validation of the CFD model are presented in Appendix for a rectangular box under forced heave motion on the free surface, as well as a freely floating rectangular box in a regular wave. In both cases, our CFD results are compared with experimental and some existing numerical results. Detailed convergence studies on the mesh size and time step can also be found in Appendix A.1.

CFD data analysis methodology
To obtain drag forces and thus estimate the drag coefficients, the cross section illustrated in Fig. 2 will be forced to oscillate in infinite fluid domain (Type 1 simulation in Section 3.3). We consider a vertical harmonic motion in heave : where is the forced oscillation amplitude, is forced oscillation frequency. For the fully submerged structure in infinite fluid domain, it is assumed that total vertical force consists of two components: wherė( ) and̈( ) are the velocity and acceleration, respectively. ( ) = − 33̈( ) is the added mass force proportional to the acceleration, and 33 is the heave added mass. ( ) = − 1 2̇( )|̇( )| is the drag force. Again, is the project area and is the drag coefficient. If we define the inertia coefficient as where is the displaced fluid volume by the bottom box, and Eq. (13) can be rewritten as Here = is the amplitude of the heave velocity. A Fourier averaging is used to calculate the coefficients (Ommani et al., 2016). The Fourier coefficients are defined as: The inertia and drag coefficients can be obtained as: Similar to the total force, the total moment also consists two contributions, expressed as Here and are the arms of inertia force and drag force, respectively. By applying similar Fourier average to the moment ℎ ( ), we obtain the Fourier coefficient related to the drag-induced moment The arm of the drag force, or alternatively the 'point of acting', is estimated as: The above procedure can be used to estimate the drag coefficient and 'point of acting' of the drag force for any part of the structure. In the next section, this will be done for the bottom flanges and half of the middle bottom of the cross section with bottom damping box (see definition in Fig. 2).

Drag coefficients and 'point of acting' for the cross section with a damping box
To get a database for the drag coefficient as a function of number, we will consider a range of forced heave amplitudes between 0.02 m and 0.014 m, and a fixed oscillatory frequency = 3.0 rad/s. The dimensional and non-dimensional parameters of the simulations are summarized in Table 2 and categorized as Group A. It can be shown by a dimension analysis that the non-dimensional force on a structure in infinite fluid domain only depends on two independent non-dimensional parameters. Thus, in addition to number, also depends, in principle, on Reynolds number or equivalently = 2 ∕( ) which is the ratio between Reynolds number and Froude number. Here is the kinematic viscosity coefficient of the fluid, and = 2 ∕ is the oscillatory period. However, for structures with sharp edges, the dependence of on number is rather weak (Shao, 2019), as the flow separation will always occur at the sharp edges, in contrast to bluff and smooth structures like circular cylinders and spheres. To investigate if the same conclusion holds for the considered cross section with a damping box, some additional simulations are performed, where the heave amplitude is fixed as 0.02 m (thus remains as a constant of 0.262), while the frequency varies between 1.0 rad/s to 7.0 rad/s. Those simulations are summarized in Table 2 and categorized as Group B.
In the CFD analysis, the forces on three parts of the structure have been measured, including: half of the total bottom box, one of the bottom flanges and half of the middle bottom. Consequently, the drag forces and the 'points of acting' on the corresponding parts are obtained by following the procedure described in Section 4.1, which are presented in Fig. 6(a) and 6(b) respectively as function of oscillatory frequency. The results for those three parts are referred as 'total', 'flange' and 'mid_bottom' respectively. The 'points of acting' is defined by the horizontal distance to the vertical central axis. In Fig. 6(b), they are also defined as 0 , 1 and 2 in accordance with the definition in Fig. 3 for the second configuration of Morison elements. All the results in the figure are based on the same = 0.262. As seen from the results, both the drag coefficient and 'points of acting' remain almost the same for different frequencies, indicating that scale effect (or Reynolds dependence) is insignificant for the considered cross section. Therefore, as an estimation, the number is the only important non-dimensional parameter that dominates the results.
The fitted functions are also plotted in Fig. 7. As seen from the results, on the middle bottom, the coefficient remains almost a constant for different numbers. For the bottom flange, decreases with increasing number until ≈ 1.5, after which curve becomes flat.
The 'points of acting' of the drag forces on half of the total cross section, bottom flanges and half of the middle bottom are presented in Fig. 8 for a range of numbers. The above results of drag coefficients

Validation of the hybrid numerical method by CFD
This hybrid numerical method will be applied and validated in this section. The two different configurations of Morison elements defined in Section 3.2 will be considered and compared. For Configuration 1, we use the total drag coefficient from Eq. (23)  We will start with two cases to validate the hybrid numerical method. The first case concerns the added mass and damping coefficients for a forced heaving of the cross section defined in Section 3.1. In the second case, side-by-side configuration of two identical cross sections are free to respond in heave and roll in incident waves. The two cross sections are rigidly connected so that they behave as a single body. To this end, the motion of and structural loads on a single cross section are studied and compared with the full CFD analysis of the floating structure in waves.

Forced oscillation of the cross section
The heave added mass and damping coefficients of the cross section are studied for a range of oscillatory frequencies ( = 1 − 10 rad/s) and two numbers ( = 0.131 and 0.262) by using the hybrid numerical method and the CFD method with a free surface (Type 2 analysis defined in Section 3.3).
Figs. 9 and 10 present the non-dimensional added mass and damping results of the hybrid numerical method, denoted by 'Pot.+Vis. config-1' and 'Pot.+Vis.config-2' for Morison elements Configuration 1 and 2 respectively, and the CFD results. A potential-flow results without any viscous drag correction, denoted by 'Pot.' are also included in the comparisons. Since no corrections have been considered for the inertia forces in the hybrid numerical method, the added mass results denoted by 'Pot.+Vis.config-1' and 'Pot.+Vis.config-2' are identical to that of 'Pot.'. For the added mass, the CFD results and the potentialflow results agree well with each other for both = 0.131 and = 0.262, indicating also that the viscous effects have not been secondary for the added mass coefficients of the considered structure and flow parameters. For much larger values, stronger vortex shedding and flow convection occur, and thus larger viscous effects on the added mass are expected.
For the damping coefficients, the potential-flow results without viscous correction clearly under-predict the total heave damping. With the linearized drag forces, the equivalent viscous damping contributes significantly to the total damping, and this contribution is greater for larger value. The latter can be explained by the fact that the linearized equivalent viscous damping is proportional to the motion amplitude, which in our analysis is greater for larger number. The encouraging agreement between the hybrid method and CFD model also indicates that the hybrid model works very well to provide adequate total damping coefficients.

Freely floating side-by-side cross sections in regular waves
Heave and roll motions of two side-by-side identical cross sections are studied by both the hybrid numerical model and the fully nonlinear numerical wave tank based on CFD. The two cross sections are rigidly connected so that they behave as a single rigid body. The center-tocenter distance between the two cross sections is selected as 0 = 2.08 = 1.0 m. In the hybrid model, an elongated barge with length = 60 0 exposed to beam-sea waves is considered to approximate a 2D scenario. Details of the hybrid method has been described in Sections 3.2 and 2.1. Morison elements Configuration 1 and 2 are used to model the linearized drag loads, with the needed drag coefficients on different parts of the structure defined in Eqs. (23), (24) and (25). In the CFD model, only a narrow strip of the barge with length = 0.032 0 is considered. Symmetric conditions are applied at two ends of the strip to represent a 2D case. The boundary conditions in the CFD model have been summarized in Table 1. The parameters used in the hybrid and CFD models are listed in Table 3.
The three DOFs motion responses (sway, heave, roll) induced by regular waves are studied under different regular waves. Fig. 11 shows the heave and roll motion results of the hybrid method and the corresponding CFD results. The same wave height = 0.06 m is considered for all wave lengths. The presented heave RAO is non-dimensionalized as 3 ∕ , and the roll RAO by 4 ∕ , where is the wave amplitude in CFD simulation and is the wave number. For heave and roll motions, without the viscous correction, the responses predicted by the hybrid model near the resonance period is much larger compared to the CFD results. With the linearized drag forces, the numerical results of the hybrid model, either by Morison element Configuration 1 or 2, agree reasonably well with CFD results.
Since the linearized viscous force is proportional to the relative velocity amplitude, the motion RAOs are dependent on the incident wave amplitude. Different wave heights between = 0.06 m and 0.16 m are also considered in the hybrid numerical model. Fig. 12 presents the RAO curves at different wave heights, which are obtained by applying the Configuration 2. In the hybrid model, both heave and roll motions close to resonance decrease with increasing wave heights, attributed to larger linearized viscous damping. Fig. 13 shows the motion results of the side-by-side section for a range of wave heights ( = 0.06 − 0.16 m) and the same wave period ( = 2.7 s) corresponding to ∕( 0 + ) ≈ 7.46.

Freely floating single cross section in regular waves
The hybrid numerical method and the fully nonlinear CFD-based wave tank were also used to study the heave and roll motions of a single cross section. The hybrid model and CFD model are similar to side-by-side cross section, but we have to lower the CoG to ensure the hydrostatic stability. Consequently, the gyration radius has also been adapted. The parameters used in the hybrid and CFD models are list in Table 4. Fig. 14 shows the heave and roll motion results of the hybrid and the CFD models. The same wave height = 0.06 m is considered for different wave periods ( = 1.3 − 2.8 s). For heave motion, with the linearized drag loads, the numerical results of the hybrid model, either by Morison element Configuration 1 or 2, agree reasonably well with CFD results. However, Configuration 2 is seen to predict better roll motion than Configuration 1 at the roll resonance. This is expected as the drag forces has been applied on the 'point of acting'.
We have also studied the structural loads in terms of bending moment and shear force at the root of Bottom flange based on the hybrid method and the CFD method. The linearized viscous-drag induced shear Here is the hydrodynamic pressure. is the wetted surface of the bottom flange. 2 and 3 are components of the normal vector = ( 1 , 2 , 3 ). is the -coordinate of a point on the flange. Fig. 15 shows the total non-dimensional shear force ( , + , ) and bending moment ( , + , ) at the root of the flange.
Here the presented total force and moment results include only the wave-frequency components. Since the CFD model is fully nonlinear, higher-order wave loads, e.g. second-and third-harmonic loads can also be predicted. To make a fair comparison with the hybrid model, only the first-harmonic component of the CFD results are included in Fig. 15. The corresponding higher-order shear force and bending moment will be presented later in this section.
As seen in Fig. 15, without the viscous correction, the shear force and bending moment predicted by the hybrid model near the resonance is much larger compared to the CFD results. This is due to nonphysically large resonant response in heave and roll. With the linearized drag forces either by Morison element Configuration 1 or 2, the numerical results of the hybrid model agree reasonably well with CFD results. It is worth mentioning that the predicted shear forces by the two configurations are almost the same, but differ in bending moment, in W. Lei et al.   particular near the heave and roll natural periods. In general, Configuration 2 predicts better bending moment than Configuration 1, which is as expected since the Morison elements in Configuration 2 have been located at the estimated 'points of acting'. It should be noted that the pressure-induced and drag-induced loads have a phase difference as shown in Fig. 18, and thus the total force or moment is not a simply linear superposition of the two components. In the frequency-domain analysis, not only the motions but also the pressure-and drag-induced structural loads are complex variables, from which the amplitudes and the phase angles can be obtained, respectively. The presented phase difference in Fig. 18 is defined as = − , where and are the phase angles for the pressure-induced and drag-induced structural loads, respectively. The phase angles are defined relative to the wave elevation at the origin of the coordinate system. The phase angles and are presented in Fig. 19. For longer waves with ∕ > 20, remains as a constant of −90 deg, while increases towards 180 deg for increasing ∕ . To understand the higher-order contribution in the shear force and bending moment, the amplitudes of first, second and third harmonics are obtained by a Fourier analysis of the CFD results, and compared with the CFD model in Fig. 20. Non-negligible second-harmonic shear force and bending moments are observed near the roll resonance (with ∕ ≈ 7.5). The third-order contribution is relatively smaller for most wave lengths, except at the heave resonance (with ∕ ≈ 16) where third harmonic amplitude is larger than that of the second harmonics. For larger wave heights than the considered = 0.06 m in this study, the relative importance of the higher-order effects is expected to be greater.
In this paper, we have only presented wave-frequency (or the linearized) results of the hybrid model. In principle, the quadratic formulation of the drag loads in Eq. (2) also contains higher-order contribution, which can be extracted through a post-processing of results from the hybrid model. This can be done by using Eq. (2) again to generate a time series of the total drag loads, where the required relative velocity in the equation is readily known once the linearized motion equations are solved. To this end, a Fourier analysis of the time series based on Eq. (2) can be performed to obtain different harmonics of the drag loads. However, as it can be easily shown that the assumed drag loads formulation in Eq. (2) can only give non-zero contribution to the odd harmonics, while the even harmonics, including the second harmonics, are always zeros. This represents a theoretical limitation of the Morison-drag type of formulation when it comes to higherharmonic structural loads on damping plates under storms conditions where higher-order nonlinearities are expected to be significant. In such cases, a combination of hybrid model and CFD model might be helpful from an engineering point of view. The hybrid model with a proper drag linearization could be used for the motion prediction, followed by a screening process based on the hybrid model to design critical wave episodes, and subsequently a dedicated CFD analysis with either forced oscillation or freely-floating structure in waves for the designed waves.

Scale effects and limitations
In addition to the limitation in the prediction of sum-frequency structural loads as demonstrated in Section 5, there are other limitations of the present the hybrid method that one must keep in mind in real applications. The same as any other numerical methods based on a linear or weakly-nonlinear theory, the hybrid method which utilizes linear frequency-domain hydrodynamic coefficients as inputs, cannot be applied to accurately predict violent wave-structure problems where the intrinsic assumptions are violated. On the other hand, discussions in this paper are limited to regular waves, while future dedicated studies are needed on how to linearize the drag loads and properly include them for floaters exposed to irregular waves.
Another important topic that has not been a focus of the present study is the scale effects. Even though only model scale is considered in this study, it is expected that the results are still relevant for fullscale applications of similar structures. Shao et al. (2019) studied a similar 2D cross section with a bottom flange and a vertical central column, in relation to the pontoons of a floating bridge, where the drag coefficients based on model-scale and full-scale simulations were compared. Only small differences have been observed, indicating that the Reynolds-scale effects are secondary and that Froude scaling can be used in practice for this type of structures. The study by Tao and Cai (2004), where a disk attached to a vertical cylinder was considered, has also shown that the friction forces are much smaller compared with the form drag. Most of the damping forces are geometry related form damping due to flow separation and vortex shedding.

Conclusions
This paper proposes an improved hybrid model to improve the soundness and accuracy of the linearized frequency-domain hydrodynamic analysis of floater structures with damping plates. Both wavefrequency motion of the floating structure and structural loads on the damping plates are investigated. Dedicated CFD models in infinite fluid domain are used to generate a database for the drag coefficients and 'points of acting' of the drag forces on different parts of the damping plate. In the hybrid model, the nonlinear drag loads are linearized, and motion equations are solved by iterations, so that the drag coefficients are precisely selected based on the estimated Keulegan-Carpenter (KC) number from each iteration.
Taking a floating 2D cross section with a central column and a damping box at the bottom as an example, the hybrid model is validated against full CFD analysis, with satisfactory wave-frequency results for not only the rigid-body motions but also structural loads (shear force and bending moment) at the root of the bilge. For the considered structure, it is found that the pressure-induced structural loads have a larger contribution, while the viscous drag also has nonnegligible contribution. Placing the Morison elements located at the estimated 'points of acting' is shown to improve the prediction of rotational motions and the structural loads. Strong second-harmonic nonlinear structural loads are identified in the full CFD analysis for a freely floating structure with a damping plate, indicating a theoretical limitation of the applied drag-load formulation when structural loads under extreme wave conditions are concerned. Unless an alternative formulation can be developed to overcome this limitation, a combined usage of the hybrid model (for motion prediction) and CFD analysis (for structural loads) in a smart way may be the way forward in engineering applications.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
Data will be made available on request.

Acknowledgment
Jikang Chen acknowledges the support from the Postdoctoral Scientific Research Developmental Fund (Grant No. LBH-Q20088).

A.1. Rectangular box under forced heave motion on the free surface
To verify and validate the numerical models, we consider a rectangular box under forced heave motion at the free surface, that has been experimentally studied by Vugts (1968). The box has a breath = 0.4 m, a height ℎ = 0.4 m, and a draft = 0.2 m (see Fig. A.21). Mesh refinement in the vicinity of the sharp edge is performed. Convergence study is carried out for the CFD model to obtain reliable numerical results. Three different mesh resolutions are considered, which have = 0.008 m, 0.004 m, 0.002 m near the sharp corners of the box, respectively. Three different time steps = ∕2000, ∕1000, and ∕500 are also considered to check the convergence of the temporal discretization. The time series of vertical force on the heaving box is shown in Fig. A.22 for different mesh density and time steps.
Based on results of the convergence study in Fig. A.22, it is decided to use a minimum grid cell = 0.002 m and a time step ≤ ∕1000 considering the computational costs and the required accuracy. The added mass and damping coefficients for Keulegan-Carpenter number = 2 = 0.157 (heave amplitude = 0.01 m) and = 0.314 ( = 0.02 m) are shown in Fig. A.23 and Fig. A.24, respectively. The present CFD results are compared with the experiments and two potential-flow results. The results denoted by 'CFD' are based on the present CFD model described in Sections 2.2 and 3.3, while those denoted by 'Pot.3D' are obtained from WADAM by considering an elongated barge with its cross section the same as the 2D rectangular W. Lei et al.   box. The barge length = 150 = 60 m was used in WADAM, which is sufficiently large to avoid the 3D effects influencing the nondimensional 2D added mass and damping coefficients. The presented 2D added mass is non-dimensionalized as 33 ∕ , and the damping by

∕
, where is the submerged area of the 2D section ( × ). In the figures, 'Pot.2D' denotes the linear potential-flow results given by Wang et al. (2021). The agreement between the two potential-flow models also confirms that the elongated barge can be used to approximate the 2D scenario. Both the CFD results and the potential-flow results agree reasonable well with the experimental results (Vugts, 1968). For the smaller number, the viscous effects are less significant for the considered frequency range, while the potential-flow results underpredict the damping for larger number indicating the increased viscous effects.

A.2. Freely floating rectangular box in a regular wave
A freely floating rectangular box in regular waves that was experimentally in He et al. (2016) is studied in this section to validate the 2D NWT described in Section 3. Sway, heave and roll motions are considered. Dimensions and the inertia properties of the structure are given in Table A.5.
The considered regular wave has a wave height = 0.06 m and wave period = 1.2 s. The water depth is ℎ 0 = 0.4 m. Fig. A.25 shows the present 2D CFD results, in comparison with the experimental results (He et al., 2016), OpenFOAM results (Rivera Arreba, 2017), and a 3D CFD results based on Star-CCM+ (Yao et al., 2020) where the narrow gap between the 3D box and the front and back walls of the wave flume were also modeled in accordance with the model test.
The present numerical results of heave motion, normalized by the wave height, are similar to the experimental, OpenFOAM, and the 3D STAR-CCM+ results. The present roll response is on average smaller than the experimental and OpenFOAM CFD results, but agrees better with the 3D CFD result in Yao et al. (2020). With respect to sway motion, all numerical results seem to diverge from the experimental data. This is due to the homogeneous part of the time-domain solution of the motion equation that is dependent on the initial conditions. The wavefrequency response, obtained by applying a high-pass filter to the surge time series, agrees very well with the other two numerical results and the experiment. In general, the present 2D numerical wave tank numerical has shown to provide reasonably good results for the freely floating box, in comparison with both experimental and other CFD studies.