A numerical investigation of interactions between extreme waves and a vertical cylinder ＊

Qun-bin Chen, Yu-xiang Ma, Guo-hai Dong, Xiao-zhou Ma, Bing Tai, Xu-yang Niu

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024,China

Abstract: In this paper, the interactions between extreme waves and a vertical cylinder are investigated through a 3-D two-phase flow model. The numerical model is verified and validated by experimental data. Then, two factors are considered, the global wave steepness and the frequency bandwidth of the wave groups, in the studies of the in-line wave forces and the wave run-up around a cylinder. It is found that both the in-line wave forces and the wave run-up are remarkably increased with the increase of the global wave steepness, whereas the effect of the frequency bandwidth on the in-line wave forces is relatively weak in comparison with its effect on the wave run-up. The minimum and maximum wave run-ups are located in the directions of 22.5° and 180° with respect to the direction of the incident waves, respectively. Additionally, a new empirical formula is proposed for predicting the in-line wave forces by using only the free surface elevations around the cylinder. The results of the formula agree well with the simulation results.

Key words: Extreme waves, vertical cylinder, in-line wave force, wave run-up

Introduction

With the growing demand for energy, many countries pay a special attention to renewable energy in recent years. Among the kinds of renewable energy,the wind energy is clean, free and available in any place where there are relatively strong winds.Compared with the inland areas, the offshore regions have more strong and stable wind resources.Therefore, offshore wind turbines are widely applied in converting kinetic energy in wind into mechanical power and then further into electric energy. However,these structures are always exposed to ocean waves in offshore regions and are typically supported by vertical cylinders. As a result, a clear understanding of wave loads and run-up on vertical cylinders is an important issue[1].

Extreme waves are large waves with extraordinary large crests, significantly higher than the background waves. They can damage structures with the loss of their structural functionality. Due to the potential hazard caused by extreme waves, their impact is an active research subject in ocean and coastal engineering[2].

Generally, the main concern regarding extreme waves acting on a vertical cylinder is the wave forces and the wave run-up. In laboratory experiments, the method of dispersion focusing is often used to generate isolated large waves because it is relatively easy to control the location and the time of the appearance of a large wave. Several physical experiments on focused wave interactions with an isolated circular cylinder were carried out to explore various important phenomena within this process[3],including the ringing responses[4]and the secondary loading cycles[5]. It is well known that a ringing response induced by extreme waves is very dangerous for offshore structures, and it is usually associated with the presence of a secondary loading cycle impacting the structure[5-6]. Recently, the secondary loading cycle process, which is interpreted by a return flow at the back of the vertical cylinder, was successfully simulated by Paulsen et al.[7]with the open source computational fluid dynamics (CFD)-toolbox OpenFOAM?; and their results indicated that the CFD model has the ability to reproduce the strong nonlinear process[8]. As the waves impacted on a vertical cylinder, the components of higher harmonic forces played important role[9]. Under certain wave conditions, high-order harmonics can contribute 60%of the wave forces[10].

On the other hand, the wave run-up on a cylinder can likewise cause unexpected damage to marine structures and electronic equipment on ships. Hence,this process also needs to be considered.Hallermeier[11]proposed an important formula to predict the wave run-up on a cylinder, which says that the maximum wave run-up around a cylinder is equal toηmaxplus20.5βu/gin regular waves, whereis the maximum crest,uis the maximum horizontal water particle velocity at the wave crest andβis an empirical coefficient. Afterwards, many modified formulas under regular waves and irregular waves were proposed, mainly to correct the coefficientβand using the high-order wave theory to calculate the velocityu[12]. All these formulas regarding the wave run-up on a cylinder are based on the calculation of the wave kinematics through an appropriate wave theory. In 2015, a formula[13]was proposed in terms of scaling arguments combined with the available experimental datasets. Based on the second-order diffraction theory, Kriebel[14]proposed an analytical method to predict the wave run-up on a cylinder. However, it was found that the analytical formula significantly underestimated the wave run-up of steep waves. As can be seen, most of the previous studies regarding the wave run-up are mainly for regular or irregular waves, but the wave run-up on a cylinder under extreme waves is still not very well understood[15]. Therefore, a reliable wave run-up formula for extreme waves acting on a vertical cylinder is still to be established.

In the present study, numerical simulations of interactions between the focused waves and a vertical cylinder are carried out by using the open source hydrodynamic code REEF3D, which has been applied in many different types of flow problems, such as the hydrodynamic characteristics of breakers over impermeable slopes[16], hydrodynamics of oscillating water columns[17]and pier scour under current and waves[18]. More recently, Bihs et al.[19]studied the interaction between focused waves and a vertical cylinder, indicating that the REEF3D also performs well. However, to the best of our knowledge, 3-D CFD simulations of focusing waves interacting with vertical cylinders are still rare. In particular, the influence of the global wave steepness and the frequency bandwidth of the wave groups on the in-line wave forces and the wave run-up remains still unclear. Furthermore, it is unknown whether there is a simple relationship between the wave forces and the wave run-up on a cylinder. These issues are studied in the present paper.

1. Methodology

1.1 Numerical model

In the present study, we solve the 3-D two-phase Navier-Stokes equations by using the open source CFD program REEF3D[20]. The interface between the air and the water is reconstructed by a level set method. The incompressible Reynolds-averaged Navier-Stokes (RANS) equations can be written as:

Chorin’s projection method is employed to solve the RANS equations on a staggered grid. A third-order total variation diminishing (TVD) Runge-Kutta scheme is adopted to discretize the time terms of the Navier-Stokes equations. The weighted essentially non-oscillatory (WENO) scheme is applied to discretize the convection terms. The Poisson equation is solved by the bi-conjugate gradient stabilized(Bi-CGSTAB) solver with a parallel semicoarsening multigrid preconditioning. A ghost cell immersed boundary method[21]is implemented in the REEF3D for simulating complex structures, as shown in Fig.1(a). A close-up view of the 2-D mesh near the cylinder is given in Fig. 1(b). The in-line wave forcesFacting on structures are calculated by integrating the pressurepand the surface normal component of the viscous shear stress tensorτ, i.e.

where Ixis thex-component of the unit normal vector pointing into the fluid,Ωis the surface of the structure. The detailed numerical algorithms can be found in Ref. [20].

Fig. 1 (Color online) The strategy of handling structures: (a) A brief illustration of the ghost cell immersed boundary method; the solid points refer to fluid grid cells, the hollow points refer to solid grid cells. (b) A close-up view of the 2-D mesh near the cylinder in the present numerical simulation

1.2 Experimental setup

The measurements of the free-surface elevations of the waves and the induced in-line wave forces are carried out in a wave flume of 20.00 m long, 0.45 m wide and 0.60 m high at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, China. In the present study, a water depth of 0.30 m is used. A piston-type wavemaker with the mean position atx=0 m is used to generate the dispersive focused waves, and at the other end of the flume, a wave absorber is installed to mitigate the wave reflection. The detailed experimental setup is shown in Fig. 2. A vertical circular cylinder of 0.06 m in diameter and 0.56 m in height is installed in the wave flume, and the center of the cylinder is located inxc=9.100 m andyc=0.225 m . A total of 6 capacitance wave gauges and a two-component force transducer are used to measure the time series of the free-surface elevations and the induced in-line wave forces. Four experimental cases are considered with the frequency range [0.6 Hz,1.4 Hz] and the number of wave componentsN=64, the rest wave parameters are shown in Table 1, whereS= ∑ai kiis the global wave steepness,ai kiis the wave steepness for theith wave component, andAfis the theoretical focusing amplitude. The measurement in each experimental case is repeated three times.

Table 1 Parameters for experimental cases

1.3 Wave generation

The dispersive focusing method is used to generate extreme waves. The theoretical focusing position and time are specified asxfandtf,respectively. Based on the linear potential flow theory,the free surface elevations at an arbitrary point can be expressed as

wherexfandtfare equal to 9.0 m and 14.0 s,ai,kiandωiare the corresponding wave amplitude,wavenumber and radian frequency of theith wave component, respectively. A constant wave steepness spectrum[3]is used to determine theith wave amplitudeai, which can be expressed as

Fig. 2 Sketch of the experimental setup (Not to scale, m)

and each wavenumberkican be obtained from the linear dispersion relation

in whichhandgare the still water depth and the gravitational acceleration. The radian frequencyiωcan be obtained by, where1ωandNωare the minimum and the maximum of the radian frequency,respectively, and can be determined with a predefined frequency band.

In the present numerical simulation, a Dirichlet inlet boundary condition is used to generate the focused waves on the left side of the numerical wave tank. According to the linear potential flow theory, the velocities on the inlet boundary can be written as

At the end of the numerical wave tank, the relaxation method[22]with a relaxation length of 3.0 m is implemented to absorb the reflection wave.

2. Results and discussions

2.1 Validation

2.1.1Convergence analysis

To validate the accuracy of the numerical model in simulating the focused wave groups, a series of focused waves (without the cylinder) are simulated first by setting different grid sizes to test the convergence of the numerical solution. Note that the grid size in the present version of REEF3D 18.01 is always uniform in the three directions, i.e.,dx= dy=dz, and the high-order temporal and spatial discretization schemes can be more easily implemented. Furthermore, the uniform grid size can also enhance the stability of the numerical computation.The experimental cases with the global wave steepnessS=0.15, 0.30 and the input frequency(f) range [0.6 Hz,1.4 Hz] are used to test the convergence of the numerical results with different grid sizes. As shown in Fig. 3, by gradually reducing the grid size from 0.040 m to 0.005 m, the numerical results gradually converge to a stable solution.Because the simulation results for dx=0.010 m and dx=0.005 mare almost identical, the grid size dx= 0.010 m is used in the following simulations.

Fig. 3 (Color online) The simulation results for the two experimental cases without the cylinder with different grid sizes,and for both,the location of the measured points is at x=9.0 m andy=0.1m(#7 in Fig. 2): (a)S=0.15with the frequency range [0.6Hz,1.4Hz]. (b)S =0.30 with the frequency range [0.6Hz,1.4Hz]

2.1.2Comparison with the experimental data

To further test the numerical model for its performance of predicting the free-surface elevations and the in-line wave forces when the focused waves interact with a structure, numerical simulations for two cases of with grid size dx= 0.010 m are carried out and comparisons are made with the corresponding experimental data. In the simulations, the computational domain is 17.60 m long, 0.45 m wide and 0.60 m high. The total cell number in the computational domain is 4.752×106. Note that this 3-D numerical wave tank setup is also used in the following simulations. Figure 4 shows the comparison results for the free surface elevations. Excellent agreement is observed. The comparison results for the in-line wave forces also show good agreement, as indicated in Fig. 5. The small fluctuations of the time series of the in-line wave forces can also be captured.The focusing positions obtained by the experiments and the numerical simulations deviate slightly with increasing wave steepness, and a similar phenomenon was also found in previous studies[4,23]. Based on the above validations, it can be said that the present numerical model can reproduce the laboratory experiments and can capture the free-surface elevations and the in-line wave forces on the vertical cylinder well.

Fig. 4 (Color online) Comparison of the free-surface elevations without the cylinder between numerical results (─) and experimental data (·): (a) S=0.15 with the frequency range [0.6Hz,1.4Hz]. (b) S=0.30 with the frequency range [0.6Hz,1.4Hz]

After the validation of the numerical model, a comprehensive analysis of the in-line wave forces and the wave run-up is carried out with different wave parameters. Note that all following numerical results are for the cases with the cylinder. Two important wave parameters, i.e., the global wave steepnessSand the frequency bandwidth,are considered for their influences on the in-line wave forces and the wave run-up in the subsequent part. All wave parameters used in the following simulations are shown in Table 2.

Fig. 5 (Color online) Comparison of the in-line wave forces between numerical results (─) and experimental data (·):(a) S= 0.15 with the frequency range [0 .6 Hz,1.4Hz].(b)S=0.30with the frequency range[0.6Hz,1.4Hz]

Table 2 Wave group parameters

2.2 Influence of global wave steepness

2.2.1In-line wave forces

In this subsection, the in-line wave forces acting on the vertical cylinder by the focused waves with different global wave steepness values are investigated with a fixed input frequency bandf=[0.6 Hz,1.4 Hz]. With the global wave steepnessSincreasing from 0.15 to 0.40, the peak values of the free surface elevations and the in-line wave forces gradually increase, as shown in Fig. 6. Due to the advantage of numerical simulations, the computational results for the free surface elevations and the in-line wave forces are readily synchronous. The time corresponding to the maximum value of the in-line wave forces is slightly ahead of the time of the maximum free surface elevations (Fig. 7). Generally,the in-line wave forces are caused by the pressure difference between the upstream and the downstream of the cylinder, which might lead to the difference of the time between the peak values of the free surface and the in-line wave force. For example, whenS=0.30, the peak value of the in-line wave force occurs att= 13.85s although the maximum wave run-up appears att=14.00 s. Because, att= 13.85s the asymmetry of the velocity field and the wave elevation for the cylinder upstream and downstream is more apparent than that at the timet= 14.00 s , as shown in Figs. 8(a) and 8(b),respectively. This results in the pressure difference for the cylinder upstream and downstream att=13.85s larger than that oft= 14.00s .

Fig. 6 For f =[0.6Hz,1.4Hz], the peak value of the time series with different global wave steepness values: (a)The free surface elevations measured at x=9.1m and y = 0.1m . (b) The in-line wave forces acting on the cylinder

Fig. 7 For a fixed frequency band f =[0.6Hz,1.4Hz], the numerical results with different global wave steepness values: (a) The time series of the free surface elevation measured at the same location, i.e., x=9.1m and y = 0.1m . (b) The time series of the in-line wave force.Note that the vertical dash-dot line represents the theoretical focusing time t=14.00s

Fig. 8 (Color online) 3-D views of the interaction between the focused waves and the vertical cylinder at two different moments with the global wave steepness S=0.30 and the frequency band f =[0.6Hz,1.4Hz].v is the velocity magnitude defined by

On the other hand, the value of the rise timetr(defined as the duration between the peak value of the in-line wave force and its left adjacent trough, as shown in Fig. 9) gradually increases as the global wave steepness increases. However, the increase of the total impact durationtd(which is the time interval between the two troughs on either side of the maximum in-line wave force) is very slow compared with the increase oftr. Thus, the ratio oftrtotdgradually increases (ranging from 0.59 to 0.73) with the increase of the global wave steepness, indicating a horizontal asymmetry of the in-line wave forces.Because the process of the focused waves acting on the cylinder is transient in nature, the wavelet transform may be used to analyze the time-frequency distribution of the forces. In the present study, the Morlet wavelet is chosen[24]as the mother wavelet.The wavelet power spectra of the in-line wave forces for different global wave steepness values are shown in Fig. 10. It can be found that with increasing the initial global wave steepness, the high-frequency components of the in-line wave forces become gradually pronounced in the frequency range from 3 Hz to 8 Hz, suggesting that the nonlinear effects increase with the increase of the global wave steepness.

Fig. 9 (Color online) Numerical solution regarding the rise time with the frequency band f=[0.6Hz,1.4Hz], the total impact duration on the vertical cylinder and the ratio of tr to td

2.2.2Wave run-up

In this subsection, the numerical results of the wave run-up on the cylinder with different global wave steepness values are examined. Nine wave probes with an equal spacing of 22.5° are placed to measure the free surface elevations around the cylinder, and the layout of the wave probes is depicted in Fig. 11. The time series of the free surface elevations at 0° and 180° (i.e., the first and the ninth probes in Fig. 11) for different global wave steepness are illustrated in Fig. 12. As expected, the maximum crest elevation increases with the increase of the global wave steepness. Furthermore, to study the wave run-up around the cylinder, the maximum crest elevations at different positions around the cylinder are extracted and shown in Fig. 13. The maximum value of the wave run-up occurs at 180° in each case.However, the minimum value of the wave run-up does not take place at 0°, mainly due to the flow return as the wave crest passes the cylinder (as shown in Fig. 8).More detailed discussions regarding the flow return were given by Paulsen et al.[7].

Fig. 10 (Color online)For a fixed frequency band f=[0.6Hz,1.4Hz], the wavelet power spectra of the inline wave forces with different global wave steepness values.Note that each wavelet power spectrum is normalized by the corresponding maximum wavelet energy

Fig. 11 (Color online) Schematic view of the layout of wave probes around the vertical cylinder

Fig. 12 The numerical results of the time series of the free surface elevation around the cylinder with the frequency range [0.6Hz,1.4 Hz]

Fig. 13 For a fixed frequency bandwidth f=[0.6 Hz,1 .4 Hz],variations in the maximum value of the wave run-up around the cylinder with different global wave steepness value s.Note that the numerical results are normalized by the corresponding theoretical focusing amplitude, as shown in Table 2

As shown in Fig. 13, the minimum value of the wave run-up appears at 22.5°, which is slightly different from the experimental results of Li et al.[15],in which the minimum value of the wave run-up takes place at 45°. This difference regarding the position of the minimum wave run-up is mainly induced by the following two factors. One factor is the number of wave probes used in the experiments of Li et al.[15],which is just 5. Therefore, the wave elevations at 22.5°were not recorded in their experiment. Another factor is the focused waves used in Li et al.[15], which are multidirectional, but in the present simulations, the focused waves are unidirectional.

2.3 Influence of the frequency bandwidth

2.3.1In-line wave forces

For a fixed global wave steepnessS=0.30, the theoretical focusing amplitude increases as the input frequency bandwidth is broadened with the same number of wave components. As a result, the maximum crest elevations at the locationx=9.1m gradually increase, except for the input frequency bandwidth Δf=0.8 Hz , as shown in Fig. 14(a).However, the corresponding peak value of the in-line wave forces does not have a significant variation, and the difference between the maximum (2.56 N) and minimum (2.03 N) values in Fig. 14(b) is just 26.1%.On the other hand, the horizontal asymmetry of the trough becomes more pronounced, i.e., the value of the trough on the left side of the peak value gradually decreases, whereas the value of the trough on the right side of the peak value increases with the increase of the input frequency bandwidth (see Fig. 15). By examining the corresponding wavelet power spectra of the in-line wave forces, as shown in Fig. 16, it is shown that, when the input frequency bandwidth is 0.6 Hz, the main energy of the focusing wave groups is contained around the theoretical focusing time.Continuously increasing the input frequency bandwidth from 1.0 Hz to 1.4 Hz, the contour of the wavelet power becomes somewhat disperse;especially in case 9 with the broadest-band spectrum,some parts of the wavelet power arise at the timet= 17.00s. This finding likewise indicates that the actual focusing position might somehow deviate from the theoretical focusing position due to the nonlinear effects of the wave-wave interactions[25-26].

2.3.2Wave run-up

In this subsection, the wave run-up around the cylinder’s surface for different input frequency bandwidths is investigated. The time series of the free surface elevation at 180° and 0° are shown in Fig. 17.With increasing the input frequency bandwidth, the maximum and minimum wave run-ups gradually increase. The other distinctive feature for the free surface elevation at 180° is that the central wave crest becomes higher and narrower, whereas the adjacent wave trough becomes broader and less deep, indicating that the effects of the nonlinear wave-wave interactions increase with the increase in the input frequency bandwidth. In addition, similar to the numerical results for different global wave steepness values, the position of the minimum wave run-up is also located at the second probe (i.e., 22.5°) with increasing the input frequency bandwidth (see Fig. 18).In case 9 with the broadest-band spectrum, the value ofη/Afcan reach 1.4 around the cylinder’s surface.

Fig. 14 For a fixed global wave steepness S=0.30, the peak value of the time series with different frequency bandwidths Δf: (a) The free surface elevations measured at x = 9.1m and y=0.1m. (b) The in-line wave forces acting on the cylinder

2.4 An empirical formula for predicting in-line wave forces

Fig. 15 For a fixed global wave steepness S= 0.30,the numerical results with different frequency bandwidths:(a)The time series of the free surface elevation measured at the same location,i.e., x=9.1m and y = 0.1m . (b) The time series of the in-line wave force

Fig. 16 (Color online) For a fixed global wave steepness S=0.30, the wavelet power spectra of the in-line wave force with different input frequency bandwidths. Note that each wavelet power spectrum is normalized by the corresponding maximum wavelet energy

Fig. 17 The numerical results of the time series of the free surface elevation around the cylinder

Fig. 18 For a fixed global wave steepness S=0.30,the variations in the maximum value of the wave run-up around the cylinder with different global wave steepness values. Note that the numerical results are normalized by the corresponding theoretical focusing amplitude, as shown in Table 2

Cylinders are widely used in the coastal and offshore engineering, serving usually as the basic support. Therefore, it is important to monitor efficiently the safety or health of the cylinders by assessing the wave forces acting on the cylinders in real-time. Generally, directly measuring the wave forces is not easy, especially in a way without damaging the cylinders. Thus, it is necessary to explore an indirect and efficient method to monitor the wave forces in real-time. As discussed above, the main concern in this study is focused on the in-line wave forces and the wave run-up around the cylinder,specifically, the possible relationship between the in-line wave forces and the wave run-up, that can be used to predict or estimate the in-line wave forces using only the information of the free surface elevation around the cylinder. In fact, some attempts were made in this respect. For example, Boccotti et al.[27]used the Morison equation to predict the wave forces induced by the random waves. Based on the potential theory, Liu et al.[28]monitored the wave forces using the free surface elevations around a cylinder in an experiment. Because these formulas usually require the velocity potential of the waves as an intermediate step to predict the in-line wave forces,the process is slightly complex. In the present study,an empirical formula using only the original information of the free surface elevations around a cylinder is proposed and can be written in a simple form as:

wheremandM-m-1represent the number of probes arranged upstream and downstream the cylinder. Note that the number of probes in the first summation should be equal to that in the second summation (the value ofmis equal to 4 in the present study), and the probes at the upstream and the downstream of the cylinder should be symmetric with respect to they-axis, as shown in Fig. 11. The basic idea behind Eq. (9) is that the in-line wave forces are caused by the pressure difference between the upstream and the downstream of the cylinder, which means that the in-line wave forcesFare equal to Δptimes the surface areasof the cylinder as the free surface. Furthermore, Δpis equal toρwatergΔη(t) , andscan be obtained as, where Δη(t)/2π represents the net wave elevations around the cylinder. From Fig.19, the results of the in-line wave forces calculated with the empirical formula Eq. (9), agree well with all numerical results. It could be expected that when increasing the number of probes, the performance of Eq. (9) might be improved further.

3. Conclusions

Numerical simulations and laboratory experiments are carried out to investigate the interactions between the focused waves and a vertical cylinder.The numerical model is validated using the experimental data, including both the free surface elevations at six different positions and the in-line wave forces acting on the vertical cylinder. Then, the wave groups with different global wave steepness values and input frequency bandwidths are numerically generated to investigate their effects on the in-line wave forces and the wave run-up. The continuous wavelet transform is adopted to analyze the time-frequency distribution of the in-line wave forces. The main conclusions are as follows:

Fig. 19 (Color online) The comparison of in-line wave forces,the results of the red solid line corresponding to the CFD solutions, and the black solid line corresponding to the results obtained by the present empirical formula (a) For the different global wave steepness values with the frequency range [0.6 Hz,1.4Hz]. (b) For the different input frequency bandwidths with the global wave steepness S=0.30

For a fixed input frequency bandwidth, both the peak value and the high frequency components of the in-line wave forces increase with the increase of the global wave steepness. The value of the rise timetrfor the in-line wave forces gradually increases as the global wave steepness increases, while the increase of the total impact durationtdis relatively slower than the increase oftr, with a gradual increase of the ratio oftrtotd. This behavior indicates that the horizontal asymmetry of the in-line wave force becomes increasingly pronounced with the increase of the global wave steepness. This increase means that the nonlinear wave-wave interactions become increasingly intense. On the other hand, the frequency bandwidth of the wave groups has little influence on the in-line wave forces. The investigation of the wave run-up shows that the position of the maximum wave run-up is located at the weather side of the centerline(180°), however, the position of the minimum wave run-up is located at 22.5°.

Finally, a new and reliable empirical formula for predicting the in-line wave force is proposed by using the time series of the free surface elevation around a vertical cylinder.