Numerical Investigation of Run-ups on Cylinder in Steep Regular Wave

The run-up on offshore structures induced by the steep regular wave is a highly nonlinear flow with a free surface. This article focuses on the investigation of the steep regular wave run-up on a single vertical cylinder by solving the Navier-Stokes equations. A numerical wave tank is established based on the open-source package to simulate the wave scattering induced by a vertical cylinder. The VOF method is applied to capture the large deformation and breaking of the free surface. The numerical model is validated by experimental results. The relative wave run-ups on the front face and the back face along the centerline of a cylinder are analyzed. The changes of the relative run-ups with the wave steepness, the relative diameter and the relative depth are studied. It is found that the relative run-ups on the front face and the back face of the cylinder depend mainly on the wave steepness and the relative diameter, while the dependence on the relative depth is weak. The empirical formulae are proposed to calculate the relative run-ups in terms of the wave steepness of incident regular waves and the relative diameter of a cylinder.


Introduction
The linear diffraction theory for a vertical cylinder in regular waves was developed by MacCamy and Fuchs (1954), from which the theoretical solution of the run-up on the cylinder can be obtained. Kriebel (1990Kriebel ( , 1992 developed a closed-form solution for the velocity potential resulting from the interaction of second-order Stokes waves with a large vertical circular cylinder and the maximum wave crest run-up on the cylinder from the nonlinear theory is found to exceed that predicted by the linear diffraction theory by up to 50%, which was confirmed by the experiments. Morris- Thomas and Thiagarajan (2004) analyzed the run-up of regular waves on a vertical cylinder up to the third-order solution and argued that the complete wave runup was not well accounted for by the second-order diffraction theory as the wave steepness was large enough. Hallermeier (1976) suggested an estimate for run-up on a cylinder which considers the stagnation head at the wave crest and it is called the velocity stagnation head theory. They used linear wave theory for the crest kinematics calculation. Based on the linear wave theory, Niedzwecki and Huston (1992) proposed a new formula and two coefficients were added. Martin et al. (2001) investigated run-up on columns caused by steep, deep water regular waves. They found that the results of velocity stagnation head theory using a high-order wave theory for crest kinematics coincides better with the measurements.
The wave run-up is a strong nonlinear phenomenon, which is important to the safety of offshore structures including offshore wind farms and platforms in rouge waves and tsunamis. Actually, for the steep regular waves, local deformation of the free surface around the cylinder will be violent and result in breaking of the free surface. Fully nonlinear numerical wave tank was used to study such strong nonlinear flow phenomena. Büchmann et al. (1998) established a numerical wave tank using the second order boundary element method to simulate the wave run-up. Trulsen and Teigen (2002) applied the fully nonlinear potential flow model to study of the wave scattering around a vertical cylinder. Chen et al. (2014) compared the computed wave runup and wave loads on a vertical cylinder in the regular waves and focused waves with the experimental data, conducted in a wave basin, and indicated that OpenFOAM is very capable of accurate modelling of nonlinear wave interaction with offshore structures, with up to the fourth order harmonic correctly captured. Using OpenFOAM, Mohseni et al. (2018) studied the run-up at the front face of the cylinder in detail. It is found that in short waves the run-up at the front of the cylinder will be more obvious and the increase of the wave steepness will also enhance this kind of run-up. Paulsen et al. (2014) investigated the forcing by steep regular water waves on a vertical circular cylinder at finite depth by solving the two-phase incompressible Navier-Stokes equations and discussed the mechanism of the secondary load cycle which refers to a short additional loading in the direction of wave propagation shortly near the negative peak value in time history. Fan et al. (2018) conducted numerical experiments on the secondary load cycle on a vertical cylinder by OpenFOAM and proposed the empirical formulae of evaluating the characteristic parameters of the secondary load cycle in terms of the wave steepness and the relative cylinder diameter. It showed that the secondary load cycle is clearly associated with the re-entering flow near the rear face of the cylinder. For the cases that the second load cycle appears, the violent flows with the breaking free surface will affect the run-up at the back face of the cylinder. However, less quantitative analysis on the run-ups has been conducted.
Based on the open-source package WAVES2FOAM, the numerical wave tank is developed to simulate the run-ups on the front and back faces of a vertical cylinder. The experimental data obtained by Nielsen (2003) are used to validate the numerical model. Numerical experiments with different wave steepness, relative diameters and relative depths are carried out. The empirical relations between the relative run-ups at the front and back faces of a cylinder and the above parameters are proposed.

Numerical method
The water flow and air flow are computed simultaneously. The governing equations of incompressible viscous fluid flow are recalled as follows: where t is the time, is the Cartesian coordinates, is the velocity, is the density of water, is the kinetic viscosity, is the volume force, and is the pressure. , in which g is the gravity acceleration and stands for the upward vertical coordinate.
In order to simulate the large deformation and breaking of the free surface around a cylinder, the Volume of Fluid method (VOF) is used to track the free surface. Details of the VOF method can be found in Paulsen et al. (2014) and Hirt and Nichols (1981). The coefficient α is defined as the volume fraction coefficient of the water: α The conservation equation of can be formulated as: in which, the compression term takes effect only on the interface due to the existence of , is the fluid velocity, is the velocity field suitable to compress the interface (Weller et al., 1998).
The density and kinematic viscosity can be calculated by: and denote the density of water and air; and denote the kinetic viscosity of water and air. The numerical wave tank is established using the open source package WAVES2FOAM which is developed by Jacobsen et al. (2012). The computational domain can be divided into the wave generation zone, working zone and damping zone. Fig. 1 shows the compositions of the numerical wave tank.
For all computations, the incident waves are the fifth order Stokes waves. Within the generation and damping zones, the relaxation method is applied to update the velocity and the wave surface elevation to generate and damp the incident waves. The relaxation algorithm is formulated as: where ϕ is either U or α, α is the volume fraction. Following Jacobsen et al. (2012) and Guo et al. (2012), the relaxation parameter C is defined as: The definition of X is the relative coordinates so that C is always unit at the interface of the non-relaxed domain and the relaxed domain.

Verification and validation
Nielsen (2003) investigated the wave run-up of the vertical cylinder experimentally at MARINTEK. This study has been used as an ISSC benchmark study with the full-scale model. The present study reproduced the experiment for the model validation.  Table 1 lists the wave parameters of the incident waves. The depth of the wave tank is . Since , the incident wave is the deep-water wave and the wavelength can be obtained with the wave period . The diameter of the circular cylinder is D=16.0 m, and the draft of the cylinder is 24.0 m. As shown in Fig. 2, the total length of computational domain is 5L with a 0.5L length of the wave generating zone and a 3L damping zone. The transverse width of the wave tank is 20D. The cylinder is located at the center of working zone. In the work zone the mesh size in longitudinal direction is , in transverse is , near the free surface is . Around the cylinder the mesh is refined for computation of the wave force and other information accurately. Fig. 3 shows the mesh around the cylinder in detail. The total number of mesh is about 3.3 million. According to the experimental work, eight probes are set to record the wave elevation. The locations of the probes are shown in Table 2 and the positions are illustrated in Fig. 4. Fig. 5 shows the comparison of the computed free sur-face elevation and the experimental data. The good agreement reveals the stability and accuracy of the wave generation method.
The computed maximum wave crest elevations in two rows A1 and A2 are recorded every 20 time steps and the η m A comparisons with the measured data are shown in Fig. 6. is the average maximum elevation of 10 wave period; is the wave height of the incident wave, defined as A=H/2; r is the radial distance of each probe from the cylinder centre and a is the radius of the cylinder. The numerical results coincide well with the measurements. It means that the numer-

Results and discussion
When the incident steep wave crest approaches a cylinder, the wave elevation on the front face of the cylinder increases. The maximum value of the vertical displacement of the waterline on the surface of the cylinder is the run-up, which could be determined on the front and back faces in the centerline of it. Owing to the increase of the wave elevation near the front face, two sheets of water move from the front region and the rear region around the cylinder. As the two sheets of water meet near the rear face of the cylinder, the violent flows with the large deformation of the free surface appear, which will generate larger run-up on the back face of the cylinder.
The run-ups on the front face and the back face of a cylinder are of high nonlinearity in the steep wave condition as reported by Chen et al. (2014) and Paulsen et al. (2014). In regular waves, the relative run-up for the case of small KC number can be represented as a function: η m H m where is the maximum wave elevation in one wave period, k is the wave number, is the limiting wave height which can be calculated by:  . Five maximum elevations for five wave periods are extracted and the mean value can be calculated as the diemensionless run-up R. To study the run-up more clearly, the theoretical solution computed by the linear wave diffraction theory is also presented for the comparison.
Nine cases are simulated to study the effect of relative depth on and , where is the relative run-up at the front face of the cylinder and is the relative run-up at the back face of the cylinder. The wave steepness and the relative diameter are and . The range of the relative depth is . As shown in Fig. 8, the values of and are close to a constant with the change of relative depth. It means that the effect of relative depth on and is weak. The computed and are larger than those evaluated by the linear diffraction theory, which confirms again that the linear diffraction theory is under-predicted for the run-up in steep wave condition.
Ten cases are simulated to study the effect of wave steepness on and . The relative diameter and the relative depth are and . The range of the wave steepness is . Based on the results, the empirical formulae of and , which are shown in Fig. 9, can be presented as: The formula shows that increases with the increase of firstly and then decreases. The peak occurs at approximately. However, approaches a constant as goes to 0.60 from 0.76. The comparison between the numerical results and the theoretical data also shows that exact prediction of the run-ups on the back face

604
FAN Xiang et al. China Ocean Eng., 2019, Vol. 33, No. 5, P. 601-607 of the cylinder is beyond the linear diffraction theory and the weakly nonlinear theoretical solutions. kD The effect of relative diameter on and is studied through numerical experiments. Eleven cases in terms of the relative diameter are carried out. The wave steepness and the relative depth are and . The relative diameter varies within the range of . By fitting the computed data, the empirical formulae of and , as shown in Fig. 10 increases with the increase of .
increases with the increase of first and then decreases. The peak occurs at . The linear diffraction theory fails in predicting the run-ups on a cylinder in a steep regular wave.
Based on the analysis above, the empirical formulae of and can be given as: in which and are the coefficients to be determined.
The computed results for the cases of different wave steepness and different relative diameters are plotted in Fig. 9,   Fig. 8. Effect of the relative depth on the run-ups.  FAN Xiang et al. China Ocean Eng., 2019, Vol. 33, No. 5, P.  which can be used to determine and . In Fig. 11a, the x-axis is and the yaxis is . A fitting line is given by twenty points and the slope is . Similarly, the slope of the fitting line in Fig. 11b is . Finally, the empirical formulae of and can be presented as:  Martin et al. (2001) studied the run-up on the front face of a cylinder induced by steep, deep water regular waves. In their experiment, the working depth is 0.7 m and the diameter of the cylinder is 0.11 m. Parts of their measurements, which , are plotted in Fig. 12 to compare with the empirical formula. Besides, three kinds of velocity stagnation head theories and numerical results are also added in Fig. 12. It can be found that the measurements (Martin et al., 2001) and the numerical results are co-incided with the formula lines well. The results of the linear velocity stagnation head theory (Hallermeier, 1976) and the high order velocity stagnation head theory (de Vos et al., 2007) both underestimate the run-up. While the results of the formula proposed by Niedzwecki and Huston (1992) overestimate the run-up. Numerical data are also used to test the accuracy of the empirical formula about the run-up on the back face of a cylinder. The parameters of the test cases are shown in Table 3. The computational points are close to the formula line in Fig. 13. Based on the comparison above, it is clear that the empirical formulae can estimate the runup on both front and back face of a cylinder well.

Concluding remarks
The wave run-ups on the front face and the back face along the centerline of a vertical cylinder in steep regular waves are investigated numerically. A numerical wave tank is established based on the OpenFOAM and validated by the experimental data available. The effects of the wave steepness, the relative diameter and the relative depth on the rel-   ative run-ups are discussed. Numerical results confirm again that the linear diffraction theory is under-predicted for runup in steep wave condition and, additionally, show that the maximum run-up on the back face of the cylinder in a steep regular wave is hard to be predicted by the weakly nonlinear wave diffraction solutions for the cases associated with the secondary load cycle. The relative run-ups on the front and back faces of a cylinder depend mainly on the wave steepness and the relative diameter, while the dependence on the relative depth is weak. With fitting the computed values of the maximum run-ups, empirical formulae are proposed to calculate the relative run-ups in terms of the wave steepness of the incident wave and the relative diameter of the cylinder.