Viscous Effects on Wave Forces on A Submerged Horizontal Circular Cylinder

Numerical simulations are carried out for wave action on a submerged horizontal circular cylinder by means of a viscous fluid model, and it is focused on the examination of the discrepancies between the viscous fluid results and the potential flow solutions. It is found that the lift force resulted from rotational flow on the circular cylinder is always in anti-phase with the inertia force and induces the discrepancies between the results. The influence factors on the magnitude of the lift force, especially the correlation between the stagnation-point position and the wave amplitude, and the effect of the vortex shedding are investigated by further examination on the flow fields around the cylinder. The viscous numerical calculations at different wave frequencies showed that the wave frequency has also significant influence on the wave forces. Under higher frequency and larger amplitude wave action, vortex shedding from the circular cylinder will appear and influence the wave forces on the cylinder substantially.


Introduction
In ocean engineering, it is an important issue to accurately predict the wave forces on the cylinder structures for safety research. The wave forces on a submerged horizontal circular cylinder beneath waves have attracted extensive attention. The previous related studies were almost based on the potential flow theory. Dean (1948) proposed a linear solution to the diffraction problem of monochromatic waves over a submerged horizontal circular cylinder, and found that under the deep water condition the reflection coefficient is zero. Ursell (1950) confirmed this conclusion by applying a multipole expansion method, which was more reasonable for computation. Ogilvie et al. (1963) firstly computed the first order wave force and the second order mean drift force on the horizontal cylinder by extending Ursell's (1950) method. It is found that the first order horizontal force is equal to the vertical force on the cylinder, and the horizontal second order mean drift force is zero under the deep water condition. To investigate higher order wave action, Vada (1987) applied an integral equation method to develop a second order frequency-domain model, and computed the first and second harmonics of the wave forces on a submerged circular cylinder. Recently, full nonlinear poten-tial-based numerical wave tank (NWT) models have also been used to examine the higher harmonics of wave forces on a horizontal submerged circular cylinder, such as Guerber et al. (2010Guerber et al. ( , 2012 and Bai et al. (2014).
Owing to the inherent limitations, the potential flow model cannot produce reasonable hydrodynamic predictions in the case of viscous effects playing important roles. Thus, some experimental and viscous numerical investigations were conducted. Chaplin (1981Chaplin ( , 1984aChaplin ( , 1984bChaplin ( , 2001 carried out a series of experimental and numerical studies on nonlinear wave interactions with a submerged horizontal circular cylinder. The laboratory tests by Chaplin (1984b) showed that the first harmonics of the wave forces predicted by the potential flow model are significantly overestimated. Based on the experimental observations, the lift force caused by the nonzero circulation and the boundary layer separation due to viscous effects lead to the differences between the experimental measurements and the potential flow solutions. However, the direct support from the flow visualizations was not easy to be obtained in the experiments. Contento and Codiglia (2001) conducted laboratory tests and measured the wave forces on a submerged horizontal circular cylinder under different submergence at low Keulegan-Carpenter (Kc) numbers. The experimental data showed that the high harmonics of the wave forces have evident dependence on the submergence. Tavassoli and Kim (2001) developed a 2D viscous NWT to simulate the wave forces on a submerged horizontal circular cylinder in the laminar flow regime with low Reynolds (Re) number. The numerical results showed notable differences in terms of the first harmonics of wave forces and the mean vertical forces with respect to the experimental observations by Chaplin (1984b). The experimental study of Longuet-Higgins (1977) on the mean forces exerted by waves on a submerged horizontal circular cylinder reported the existence of negative mean horizontal forces, and suggested the negative mean horizontal forces being attributed to wave breaking.
Furthermore, some viscous fluid numerical studies on the vortex shedding phenomenon in flow past circular cylinders have been carried out in recent years. Ohya et al. (2013) simulated the near wake of a horizontal circular cylinder in linear stratified flows, and found that the stratification parameters have significant influence on the vortex formation and shedding. Tong et al. (2015) conducted numerical simulations on the steady flow past two circular cylinders at low Re number. The research found and identified four distinct vortex shedding regimes in the staggered arrangements of the cylinders. Jiang et al. (2016) researched the wake transitions of a circular cylinder at extremely low Re number by 3D direct numerical simulations. It is found that the variation ranges of the location of the saddle point for the two different wake transition modes are obviously different. Li et al. (2017) investigated the vortex shedding behavior of flow past a submerged horizontal circular cylinder with considering the influence of the free surface at different submergence. The numerical results showed that the submergence depth, the Froude number and the Re number all have influence on the vortex shedding. The focus of these studies is on the characteristics of the vortex formation and shedding in flow past circular cylinders. However, there are few researches on the vortex shedding from a submerged horizontal circular cylinder under the action of pure wave or the vortex effect on the wave force on the cylinder.
As the fluid viscosity is believed to have important influence on hydrodynamics, a NWT model based on the viscous fluid theory is used in this study. The numerical model is established based on the OpenFOAM package. The viscous NWT model is equipped with a renormalization group (RNG) model for turbulent closure, a volume of fluid (VOF) technique for interface capture, a no-reflection wave generator and a wave-damping function for long-time simulations. The accuracy of the NWT model in predicting largeamplitude nonlinear wave propagation and hydrodynamic forces on the structures is validated against available analytical solutions and experimental data. The comparisons of the flow fields predicted by the viscous fluid model and the potential flow model are conducted to show the details of the viscous effects on the wave forces, especially the direction and magnitude of the lift force, the dependence of the stagnation-point position on the wave amplitude and the vortex effect on the wave force. The influence of wave frequency on the wave forces is also studied. The comparisons between the viscous numerical results and the linear potential flow solutions under various wave frequencies are presented in order to understand the viscous effects on the wave forces. The relationship between the vortex shedding and the parameters of wave amplitude, Kc number, Re number and wave steepness is analyzed.

Numerical method and validation
In this section, the mathematical formulations for a viscous NWT model are presented, and the numerical validation for the large-amplitude wave motion is conducted.

Numerical method
The continuity equation and the Navier−Stokes equation are written as: where u i is the velocity components in the i-th direction; ρ, the fluid density; t, the time; p, the pressure; F i , the body force; , the effective dynamic viscosity with being the fluid viscosity and being the turbulent viscosity. The turbulent viscosity is expressed as: The RNG turbulence closure model is used to model the turbulent effects. The turbulent kinematic energy k and its dissipation rate are formulated as: , and are the model constants, which are derived from the theoretical analysis. α The VOF method is used to capture the free surface. For each computational cell, the volume fraction, denoted by herein, is defined as: TENG Bin et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 245-255 The volume fraction is used to determine the density and viscosity mixture inside each computational cell as: where subscripts 'w' and 'a' represent the liquid phase and the air phase, respectively. The volume fraction is calculated by solving the following advection equation, ∂α ∂t u r i where the last term on the left hand side is an artificial compression term with , a relative compression velocity. The introduction of the artificial compression term is helpful to improve the numerical stability and the interface resolution (Weller et al., 1998).
The static pressure and zero velocity are specified as the initial conditions. The boundaries of the viscous NWT are shown in Fig. 1. The velocity components according to the appropriate wave theory are specified at the wave generation boundary (AC). The no-slip boundary condition ( = 0 and u i = 0) is imposed at the wave-damping boundary (BD) and the bottom boundary (CD), where is the dynamic pressure, h is the vertical distance from a point in the liquid phase to the still-water level. The upper-air phase boundary condition (p * = 0 and = 0) is implemented at the top boundary (AB).
To eliminate undesirable wave reflections, the relaxation method (Jacobsen et al., 2012) is adopted in the numerical model. Two relaxation zones (Relaxation zones I and II) are introduced in the computational domain near the upstream and downstream ends, as shown in Fig. 1. In the relaxation zones, the numerical solutions are modified artificially. The following relaxation function is applied in the relaxation zones, where and are the relaxation functions. The velocity, pressure and volume fraction within the relaxation zones are modified as follows: where subscripts r, c and a represent the target, the calculated and the analytical values, respectively.
The wave force on a circular cylinder includes the pressure force and the viscous shear force , which are calculated by n s τ where and are the unit normal and tangential vectors, and is the shear stress. The shear stress is defined as: u s ∂u s /∂n where is the tangential fluid velocity, is the normal velocity gradient. The horizontal and vertical forces can be expanded into the Fourier series as: ) , where F (m) is the m-th harmonic force; F (0) , the mean force; , the base frequency and , the phase of the m-th harmonic component.

Wave generation in NWT
In consideration of the present simulations involving large-amplitude nonlinear waves, the accuracy of the viscous numerical model is validated in computing nonlinear wave propagation with a large wave steepness up to nearly 10%. The size of the NWT is 11.71 m×1.05 m and the water depth is d = 0.85 m. The wave number is k = 4.04, the wave amplitude is A = 7.35 cm, and the wave steepness is H/L = 9.45%, where H is the wave height and L is the wave length. 100 and 20 mesh cells are uniformly arranged in the wave length and wave height, respectively.
The time series of the surface elevation at x = 5 m and the wave profile at t = 36T (T is the wave period) predicted by the present numerical model are compared with the fifthorder Stokes analytical solutions (Fenton, 1985), as shown in Fig. 2. The figures show that the numerical results and the analytical solutions are in good agreement. This means that the viscous numerical model works well for simulating the nonlinear waves with a fairly large wave steepness.
In addition, the numerical accuracy of the present viscous numerical model in computing the hydrodynamic forces on the structures will be further validated in Section 3.2.

Wave forces on a submerged horizontal circular cylinder
The wave forces on a submerged horizontal circular cylinder are simulated by using the present viscous NWT model. The comparisons among the present numerical results, experimental data, potential flow analytical solutions and other's numerical results are conducted to show the influence of the viscous effects on the wave forces. Flow fields around TENG Bin et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 245-255 247 the cylinder are also presented to provide clues for explaining the mechanism of the viscous effects on the wave forces.
3.1 Numerical setup Through referring to Chaplin's (1984b) model tests, two groups of deep-water cases were simulated in this study. Fig. 3 is a sketch of the NWT, in which a circular cylinder with a radius of r = 0.051 m and a submergence depth of s is placed at 1.65L away from the wave generation zone and 4L away from the wave damping zone. The lengths of the wave generation zone and damping zone are 2.5L and 3L, respectively.
The generation of nonlinear waves is based on the fifthorder Stokes wave theory of Fenton (1985). The parameters for the numerical simulations are listed in Table 1. For the deep-water cases, the Kc number can be expressed as: where U is the water particle velocity, and D = 2r is the diameter of the cylinder. The Re number can be expressed as: ω υ β where is the circular frequency of wave and is the kinematic viscosity of fluid. The Stokes frequency parameter of can be expressed as: The change of the Kc number is achieved by changing the amplitude of the incident waves. Five wave amplitudes (A/r = 0.12-0.57) are used in the simulations for the low Kc numbers cases (Case A, Kc = 0.25-1.20), and eleven wave amplitudes (A/r = 0.33-1.99) are adopted for the cases with relatively large Kc numbers (Case B, Kc = 0.50-3.00).
High-quality computational meshes are used for the simulations. 100, 20 and 200 mesh cells are uniformly arranged in the wave length, the wave height and around the cylinder surface, respectively.

Wave forces of Case
The first harmonics of the horizontal and the vertical wave forces on the cylinder are plotted as the functions of Kc number in Fig. 4a. By referring to the previous studies, the dimensionless form of the wave force is denoted by F/(ρr 3 ω 2 Kc). The figure shows good agreement between the present viscous numerical results and the experimental measurements of Chaplin (1984b). The present viscous numerical results and the experimental measurements are always smaller than the potential flow solutions, and the discrepancies increase obviously with the increase of Kc number. The viscous numerical results reported by Tavassoli and Kim (2001) are larger than the experimental data. It might be that their computation parameters of = 483 and Re = 242-580 are not consistent with the experimental parameters of = 9120 and Re = 4560-10944. Their laminar flow model with the low Re numbers leads to the over-prediction of the first harmonics of the wave forces. The mean horizontal forces are plotted as the functions of Kc number in Fig. 4b. The drift force from the linear potential flow theory is zero. However, the present mean viscous horizontal forces are negative, and decrease with the increase of the Kc number. The mean horizontal forces obtained by Tavassoli and Kim (2001) are also negative, but different from the present results. Fig. 4c is the change of the mean vertical forces with Kc number. It shows that the current viscous numerical results agree well with the experimental measurements and the potential flow results. The viscous numerical results of Tavassoli and Kim (2001) are notably different from the others. The second and third harmonics of the horizontal and vertical forces are shown in Figs. 4d and 4e, re- spectively. The current viscous numerical results agree well with the experimental measurements, the second-order potential flow results, and the viscous numerical results under the low Re numbers. The horizontal and vertical forces are close to each other for the first, second and third harmonics, which is attributed to the features of the fluid acceleration and velocity of wave in deep-water. The good agreements between the present viscous numerical results and the experimental data indicate that the present viscous fluid model has good numerical accuracy in predicting the hydrodynamic forces on structures.
For the first harmonics of the wave forces, the differences between the viscous fluid results and the potential flow results can be clearly observed. The reason for the over-prediction of the wave forces by the potential flow model has been explained by Chaplin (1984b). Based on the viscous fluid theory, the nonzero circulation appears to generate the lift force in the anti-phase with the conventional inertia force. The comparisons between the present results and the results of Tavassoli and Kim (2001) show that the first harmonics of the wave forces and the mean wave forces are sensitive to the viscous effects under the different Re numbers, while, the high harmonics of wave forces are not sensitive to the viscous effects. According to the study of Longuet-Higgins (1977), the negative mean horizontal forces are due to the wave breaking above the circular cylin-der. However, the present results and the results of Tavassoli and Kim (2001) show that the mean horizontal forces can also be negative when the wave breaking does not occur.

Wave forces of Case B
The first harmonics of the horizontal and vertical wave forces are plotted as the functions of the Kc number in Fig.  5. It is seen that the present viscous numerical predictions are generally in good agreement with the experimental data. In the band of low Kc number (Kc = 0.5-2.0), the dimensionless wave forces decrease with the increase of the Kc number, and the horizontal and vertical forces are close to each other. However, in the band of high Kc (Kc = 2.0-3.0), the dimensionless wave forces increase with the increase of Kc number, the horizontal and vertical forces are no longer close to each other, and the experimental data become scattered. Owing to the difficulty in obtaining the direct flow visualizations in the experiments, the viscous effects in the high Kc band were not fully explained.

Flow fields around the cylinder
The horizontal wave forces of Case A4 are chosen as an example. The pressure and the viscous shear contributions to the wave force are shown in Fig. 6. The viscous shear forces are much smaller than the pressure forces. This sug- TENG Bin et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 245-255 gests that the contribution of the viscous shear forces to the total wave forces can be negligible. Thus, the viscous shear forces are not the reason for the discrepancies between the viscous results and the potential flow solutions.
To reveal the mechanism of the viscous effects, the flow fields in the vicinity of the circular cylinder are examined.

Flow fields of Case A
w = D(∂v/∂x − ∂u/∂z)/U The 2D vorticity fields ( , where u and v are the horizontal and vertical velocity components, respectively) of Case A4 (Kc = 1.00) at t = 0, 0.25T, 0.50T and 0.75T, for example, are shown in Fig. 7. It can be seen that vortices are attached to the cylinder without the vortex shedding. Similar phenomena were also found in all the other cases of Case A.
Moreover, the pressure fields and streamline distribution of Case A4 (Kc = 1.00) at several instants in one quarter of a period predicted by the linear potential flow model and the viscous fluid model are shown in Fig. 8. The potential flow fields are predicted by a multipole expansion method (Liu et al., 2012), as shown in Fig. 8a. From the po-tential flow analysis, the directions of the inertia forces, denoted as F i , are nearly 90° clockwise to the flow directions. With the absence of the circulation, there is no lift force on the circular cylinder. The flow fields from the viscous simulations are shown in Fig. 8b. The flow pattern on the opposite sides of the circular cylinder is asymmetric with respect to the direction perpendicular to the main flow direction. The fluid velocity is larger on the positive pressure side. In the viscous solutions, the circulation around the cylinder exists and can be evaluated as: where C is a closed curve enclosing the circumference of the circular cylinder. The circulations are negative for the wave propagating in the x-direction. Thus, a lift force resulted from the rotational flow acts upon the cylinder, denoted as F l . According to the Kutta−Joukowski theorem, when the circulation is in the clockwise direction, the direction of the lift force is nearly 90° counterclockwise to the main flow direction. Thus, the directions of the lift force and the inertia force are nearly opposite. Hence, the lift force cancels a part of the inertia force. It suggests that the lift force plays an important role in reducing the total wave force. According to the Kutta−Joukowski theorem, the magnitude of the lift force can be calculated as follows if the vortex shedding is not involved, When the two stagnation points are close to the cylinder surface without overlapping, the circulation is estimated by where the coefficient depends on the central angle (0°< <180°) determined by the positions of the two stagnation points, with = 1 for = 0° and = 0 for = 180°.
λ At the typical instants that the horizontal forces reach the extreme values, the positions of the stagnation points predicted by the potential flow model keep almost unchanged for different wave amplitudes. The central angle is always 180° and = 0. The dependence of the central angle on the wave amplitude for the viscous flows is examined in Fig. 9, where the '−T Fxmax ' and '+T Fxmax ' represent the instants when the horizontal forces reach the negative and the positive extreme values, respectively. For Case A1 with the

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TENG Bin et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 245-255 θ λ smallest wave amplitude, the central angle is nearly 180°a nd the streamline distribution on the left and right sides of the circular cylinder is almost symmetrical. With the increase of the wave amplitude, the streamlines become increasingly asymmetrical, and the two stagnation points gradually move towards the side with low pressure, and the corresponding central angle decreases. This means that the coefficient increases with the increase of the wave amplitude.
By substituting Eq. (21) and the first-order velocity of the wave field in deep water into Eq. (20), the magnitude of the lift force can be expressed as: In the simulations of Case A, the parameters , k and s remain unchanged. This gives rise to that the magnitude of the lift force depends on the parameter and the wave amplitude A. The experimental data of Chaplin (1984b) are fitted as a curve of C m = 2.25−0.58Kc 2 for the dimensionless first harmonics of the wave forces. On this basis, the coefficient for Case A is λ = 0.145 2 (A/r). It means that λ may be nearly in the linear proportion with A. Thus, with the increase of the wave amplitude, the proportion of the lift force in the total wave force increases.

Flow fields of Case B
The vorticity fields at the instants of t = 0, 0.25T, 0.5T and 0.75T for the typical cases of Case B are shown in Fig.  10. For Cases B1-B7 (Kc = 0.50-2.00), the wave amplitudes are relatively small, and no obvious vortex shedding occurs. Thus, the variation of the wave force with Kc number is observed to be similar with that in Case A. However, as the Kc number exceeds 2.00, for Cases B8-B11 (Kc = 2.00-3.00), obvious vortex shedding appears.
The pressure and streamline distribution at the instants that the horizontal forces reach the extreme values for the three typical cases are shown in Fig. 11. Owing to the vortex shedding, the pressure and streamline distribution around the circular cylinder for Case B10 (Kc = 2.75) varies more notably than those observed in Case B7 (Kc = 2.0) and Case B8 (Kc = 2.25).
The pressure distributions on the cylinder surface at the instants when the horizontal forces reach the extreme values are shown in Fig. 12 for Case B, where the circumferen-  TENG Bin et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 245-255 φ tial angle on the cylinder surface is defined in Fig. 11a. It can be seen that the curves of the non-dimensional dynamic pressure are smooth in Case B7 (Kc = 2.0), but with obvious irregular oscillations in Case B10 (Kc = 2.75) where the vortex shedding develops. Moreover, the pressure distributions for the two cases are also different. In Case B10, the pressure at the force direction side is lower than that in Case B7. This phenomenon is due to the appearance of the periodic vortex shedding as the Kc number exceeds the critical value. The vortex shedding dramatically changes the pressure distribution on the cylinder surface, and then influ-ences the wave forces on the cylinder substantially. Because of the instability of the vortex shedding, the first harmonics of the horizontal and vertical forces are no longer close to each other, and the corresponding experimental data look scattered.
In accordance with the changes of the streamline distribution, the positions of the stagnation points and the central angle due to the vortex shedding, the reason for the characteristics of the wave forces can be explained as that the role of the lift force in reducing the wave force decreases under the vortex effect. It should be mentioned that Eq. (22) cannot be used to estimate the lift force if the vortex shedding occurs.

Influence of wave frequency on wave forces
In the above studies, the examinations are only carried out at one specific wave frequency for Cases A and B (see Table 1). In this section, the effect of the wave frequency on the wave force will be investigated. More cases with five wave frequencies and five wave amplitudes are simulated by using the viscous NWT model, referring to Table 2 for details. The largest wave amplitude satisfies the condition that the circular cylinder is fully submerged. The cylinder radius r = 0.051 m, the submergence depth of the cylinder s/r = 2 and the water depth d = 1.785 m remain the same. The corresponding Kc numbers, wave steepness and Re numbers for the examined waves are also listed in Table 2.
In infinite water depth, the first harmonics of the horizontal and vertical forces on a 2D horizontal cylinder are exactly identical from the potential flow theory. It is also found by the viscous numerical results that they are close to each other. Therefore, the comparisons below are only made for the first harmonics of the horizontal wave force. The dimensionless form of the wave force is denoted by .

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TENG Bin et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 245-255 ω The first harmonics of the wave forces predicted by the linear potential flow model and the viscous fluid model are plotted against kr at several wave amplitudes in Fig. 13. It can be seen that, for all the simulated cases, the wave forces predicted by the viscous fluid model are definitely smaller than those predicted by the potential flow model. For the case with the smallest wave amplitude of A/r = 0.12, the discrepancy between the viscous result and the potential flow result increases gradually with the increase of the dimensionless wave number kr. This is because the magnitude of the lift force (F l ) is more closely related to the frequency ( ) as indicated in Eq. (21). For small kr (kr = 0.1-0.3), the smaller wave amplitude A/r leads to the larger dimensionless wave forces. This is because the magnitude of the lift force (F l ) is more closely related to the wave amplitude (A). λ However, for large kr (kr = 0.5-0.8), the dimensionless wave forces seem less dependent on the wave amplitude. This is because the magnitude of the lift force (F l ) is not only related to the wave amplitude (A), but also related to the coefficient .
To show the variation of the first harmonic wave forces with the wave amplitude at different wave number kr, they are plotted against the wave amplitude (or Kc number) at kr = 0.1, 0.2, 0.3, 0.4 and 0.5 in Fig. 14. It can be seen that for the cases with kr = 0.1, 0.2 and 0.3, the dimensionless wave forces predicted by the viscous fluid model decrease gradually with the increase of the wave amplitude, leading to a clear departure from the potential flow solutions. For the cases with kr = 0.4 and 0.5, the dimensionless wave forces predicted by the viscous fluid model decrease with the increase of the wave amplitude when the wave amplitude is small, but increase with the wave amplitude when the wave amplitude is large. It means that the viscous effect plays different roles in the total wave forces.
Further examinations on the viscous flow fields for the cases of kr = 0.1-0.3, and those for the cases of kr = 0.4 and 0.5 with small wave amplitudes show that no vortex shedding appears in the vicinity of the cylinder under wave action. The lift force is in the anti-phase with the inertia force and plays an important role in reducing the wave force. However, for the cases of kr=0.4 and 0.5 with large wave amp-  TENG Bin et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 245-255 litudes, viscous numerical results show obvious vortex shedding from the circular cylinder, which will induce a larger pressure force from the pressure distribution around the cylinder, and decrease the effect of the lift force, thereby increase the total wave forces. The corresponding vorticity fields for kr =0.4 and 0.5 are shown in Fig. 15. The vortices are attached to the cylinder surface without the shedding in the small wave amplitude cases (e.g., kr = 0.4, A/r = 0.36, 0.47 and kr = 0.5, A/r = 0.36). Noticeable vortex shedding occurs when the wave amplitude increases to a certain value (e.g., kr = 0.4, A/r = 0.57 and kr = 0.5, A/r = 0.47, 0.57). This shows that the vortex effect has an important influence on the wave forces for the high-frequency and large-amplitude cases. In addition, according to the parameters in Table 2, for the cases considered in this study, the occurrence of the vortex shedding may have no direct correlation with the Kc number or the Re number, but may be related to the wave steepness.

Conclusions
The objective of this research is to study the mechanism of the viscous effects on the wave forces on a submerged horizontal circular cylinder. Numerical simulations were carried out on a 2D horizontal cylinder under the action of waves with different amplitudes and frequencies.
The examinations on the flow fields around the cylinder show that when the vortex shedding does not occur, the lift force is in the anti-phase with the inertia force and plays an important role in reducing the total wave forces compared λ with the potential flow solutions; the contribution of the lift force to the wave force increases with the increase of the wave amplitude, depending on the wave amplitude and the positions of the stagnation points (the coefficient ); the role of the lift force in reducing the wave forces decreases as the vortex shedding occurs. The investigations on the influence of wave frequency on wave force show that the viscous effects play different roles at different wave frequencies. The influence of wave frequency on the wave forces can be attributed to the role of lift force on the wave force for the  cases with relatively low frequency and small amplitude, and the vortex effect on the wave force for the cases with relatively high frequency and large amplitude. For the cases considered in this study, compared with the Kc number and the Re number, the wave steepness is more relevant to the vortex shedding.