2.1 Wave tank

The experiments were carried out in the wave flume in the hydrodynamic laboratory at the University of Oslo. The wave flume is 25 m long, 0.5 m wide and was filled to a water depth of $h=0.72~\text{m}$ . In one end of the tank there is a hydraulic piston-type wavemaker with motion controlled by a preset voltage time series based on linear wavemaker theory. At the opposite end there is a passive absorbing beach. At the location 10.9 m from the wavemaker, a bottom hinged cylinder, with a diameter $d=6~\text{cm}$ , was placed to obtain the wave-exciting moment and the motion response.

2.2 Wave conditions

A total of six irregular long-crested wave time series based on the JONSWAP spectrum (Hasselmann et al. Reference Hasselmann, Barnett, Bouws, Carlson, Cartwright, Enke, Ewing, Gienapp, Hasselmann, Kruseman, Meerburg, Müller, Olbers, Richter, Sell and Walden1973), each 320 s long, were used in the experiments. The JONSWAP spectrum was chosen to generate an approximately real ocean wave environment. The spectrum as a function of angular frequency $\unicode[STIX]{x1D714}$ is given by

(2.1) $$\begin{eqnarray}S_{J}(\unicode[STIX]{x1D714}_{n})=A_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}_{n}^{-5}\exp \left(-{\textstyle \frac{5}{4}}\unicode[STIX]{x1D714}_{n}^{-4}\right)\unicode[STIX]{x1D6FE}^{\exp (-(1/2\unicode[STIX]{x1D70E}^{2})(\unicode[STIX]{x1D714}_{n}-1)^{2})},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{n}=\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{P}$ and $\unicode[STIX]{x1D6FC}=(5/16)\unicode[STIX]{x1D714}_{P}^{-1}H_{S}^{2}$ . The peak wave frequency is denoted by $\unicode[STIX]{x1D714}_{P}=2\unicode[STIX]{x03C0}/T_{P}$ and the significant wave height by $H_{S}=4\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D702}}$ , where $T_{P}$ is the peak wave period and $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D702}}$ is the standard deviation of the measured surface elevation. The peak shape parameter is $\unicode[STIX]{x1D6FE}=3.3$ , the spectral width parameter is $\unicode[STIX]{x1D70E}=0.07$ for $\unicode[STIX]{x1D714}\leqslant \unicode[STIX]{x1D714}_{P}$ and $\unicode[STIX]{x1D70E}=0.09$ for $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{P}$ . Further, $A_{\unicode[STIX]{x1D6FE}}=1-0.287\ln (\unicode[STIX]{x1D6FE})$ is a normalization factor.

To relate the wave-exciting moment and motion response to undisturbed wave parameters, the surface elevation was measured with the cylinder removed, using ultra-sound wave sensors (UltraLab ULS Advanced Ultrasound, USS02/HFP with 250 Hz sampling rate). The waves were measured at the location for the cylinder, in addition to 0.12 m and 4.9 m upstream, and 4.4 m downstream.

The governing wave parameters of the six time series, given by $H_{S}$ and $T_{P}$ at the location of the cylinder, are listed in table 1. Here the peak wavenumber $k_{P}$ is found from the linear dispersion relation $\unicode[STIX]{x1D714}_{P}^{2}=gk_{P}\tanh (k_{P}h)$ , where $g$ is the acceleration of gravity and $h$ is the water depth. The normalized water depth, $k_{P}h$ , is in the range 2.2–3.3 which is considered to represent deep water waves, and the spectral wave slope, $\unicode[STIX]{x1D716}_{P}=0.5H_{S}k_{P}$ , is in the range 0.10–0.14 which is considered to represent moderately steep waves. The normalized wavenumber, $k_{P}r$ , where $r$ is the radius of the cylinder, is in the range 0.09–0.14 which is considered to be outside of the diffraction regime. The corresponding non-dimensional peak wave period is $T_{P}\sqrt{g/d}\sim 12{-}15$ where $d=2r$ .

Figure 1.

Wave energy density spectrum. Measured surface elevation and JONSWAP spectrum with $\unicode[STIX]{x1D6FE}=3.3$ for (a) all of the six time series and (b) time series c, where $H_{S}=6.24~\text{cm}$ and $T_{P}=1.044~\text{s}$ , measured at the location of the cylinder, in addition to 4.9 m upstream and 4.4 m downstream.

Table 1.

Sea state parameters. Significant wave height $H_{S}$ , peak wave period $T_{P}$ , normalized peak wave period $T_{P}\sqrt{g/d}$ , normalized wavenumber $k_{P}r$ , normalized water depth $k_{P}h$ , spectral wave slope $\unicode[STIX]{x1D716}_{P}=0.5H_{S}k_{P}$ and resonance frequency ratio $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}$ as obtained from the oscillating cylinder.

All of the six measured wave spectra show good agreement with the JONSWAP spectrum, as seen in figure 1(a). In figure 1(b), the spectrum from series c is shown at the cylinder location in addition to 4.9 m upstream and 4.4 m downstream, showing only minor modification in the spectral shape. Between the upstream and downstream locations, the rate of decrease in  $H_{S}$ is found to be, on average for the six time series, 0.01 per peak wavelength $\unicode[STIX]{x1D706}_{P}=2\unicode[STIX]{x03C0}/k_{P}$ . Measurements from the two wave sensors with a distance of 0.12 m have been used to estimate the reflection from the beach. For the governing wave frequencies, $0.9<\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{P}<1.5$ , the reflection coefficient, in terms of the amplitude, as outlined by Goda & Suzuki (Reference Goda and Suzuki1976), is found to be less than 0.06.

2.3 Local wave properties and statistics

The surface elevation at a fixed position in the wave tank is a function of time. It is convenient to define a single wave event by its crest elevation, $\unicode[STIX]{x1D702}_{C}$ , and its trough-to-trough period, $T_{TT}$ , see figure 2(a). Altogether, the six time series consist of in total 2166 single wave events. A scatter plot of $\unicode[STIX]{x1D702}_{C}$ and $T_{TT}$ , measured at the location of the cylinder, is shown in figure 2(b).

Figure 2.

Local wave properties. (a) Definition of crest height $\unicode[STIX]{x1D702}_{C}$ and trough-to-trough wave period $T_{TT}$ of a wave event and (b) wave scatter plot including all of the 2166 measured waves ( $+$ ).

For a group of events the empirical probability of exceedance is given by

(2.2) $$\begin{eqnarray}P_{ex}(x_{i})=1-P(X\leqslant x_{i})=1-\text{i}/(N+1),\end{eqnarray}$$

where $x_{i}$ for $\text{i}=1,2,\ldots ,N$ indicates the events in ascending order, and $N$ is the total number of the events. In figure 3(a) the crest height exceedance probability $P_{ex}(\unicode[STIX]{x1D702}_{C}/Hs)$ , as found from series c, is presented and compared to the linear Rayleigh and the second-order Forristall crest distribution (Forristall Reference Forristall2000). It is observed that the measurements contain somewhat larger crest heights than expected based on the second-order distribution. This is further visualized by comparing the largest crest elevations from all of the series with the corresponding Forristall distribution. If $\unicode[STIX]{x1D702}_{F}$ denotes the Forristall crest height estimate, the largest crest height observed with regards to significant wave height, $\unicode[STIX]{x1D702}_{C}/H_{S}=1.5$ , has $\unicode[STIX]{x1D702}_{C}/\unicode[STIX]{x1D702}_{F}=1.6$ (found in series d and seen in figure 2 b). Except for this extreme crest event, a plot of $\unicode[STIX]{x1D702}_{C}/\unicode[STIX]{x1D702}_{F}$ versus its probability shows $0.97<\unicode[STIX]{x1D702}_{C}/\unicode[STIX]{x1D702}_{F}<1.28$ for the 10 % largest waves, for all of the six series, see figure 3(b).

Figure 3.

Crest height exceedance probability for (a) series c, $H_{S}=6.24~\text{cm}$ and $T_{P}=1.044~\text{s}$ ( $+$ ), linear Rayleigh (– –) and second-order Forristall (——) and (b) measured crest heights, normalized with the corresponding Forristall estimate, the 10 % largest crest heights from each series, series a (▵), b (▫), c ( $+$ ), d (♢), e ( $\times$ ) and f (○).

For later purposes (§§ 3.2 and 3.5), and following Grue et al. (Reference Grue, Clamond, Huseby and Jensen2003), using a variant of the Stokes’ third-order approximation, the measured $\unicode[STIX]{x1D702}_{C}$ and $T_{TT}$ are used to define a local wavenumber, $k_{TT}$ , and local wave slope, $\unicode[STIX]{x1D716}$ , of the event by

(2.3a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{TT}^{2}=gk_{TT}(1+\unicode[STIX]{x1D716}^{2})\quad \text{and}\quad k_{TT}\unicode[STIX]{x1D702}_{C}=\unicode[STIX]{x1D716}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D716}^{2}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D716}^{3},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{TT}=2\unicode[STIX]{x03C0}/T_{TT}$ , $\unicode[STIX]{x1D716}=ak_{TT}$ and $a$ is the approximated underlying linear amplitude. This enables a wave parametrization of each of the single wave events in the irregular wave time series. From the same approach a maximum horizontal particle velocity below the crest is estimated by $u_{C}=\unicode[STIX]{x1D716}\sqrt{g/k_{TT}}\exp (k_{TT}\unicode[STIX]{x1D702}_{C})$ . The estimation of $\unicode[STIX]{x1D716}$ , $k_{TT}$ and $u_{C}$ in irregular waves have been further tested by Stansberg, Gudmestad & Haver (Reference Stansberg, Gudmestad and Haver2008) and Grue & Jensen (Reference Grue and Jensen2012), showing good agreement with experimental results.

2.4 Cylinder model

A single cylinder with diameter $d=6~\text{cm}$ , in two different set-ups, was used in the experiments. The cylinder was located 10.9 m from the wavemaker and hinged at a horizontal lateral axis at the level of $z=z_{0}=2~\text{cm}$ above the tank bottom, with positive rotation in the wave propagation direction. At a distance of $z_{a}-z_{0}=90~\text{cm}$ above the rotation axis, the cylinder was connected to two load cells (Hottinger Baldwin Messtechnik Type Z6C2 with $10~\text{kg}=2~\text{mV}~\text{V}^{-1}$ and 400 Hz sampling rate), measuring the force from which the overturning moment was determined. In the first set-up the cylinder was fixed and rigidly connected to the load cells, where the wave-exciting force and moment with respect to $z_{0}$ are measured. In the second set-up the cylinder was free to oscillate and springs were used to connect the model and the load cells.

Figure 4.

Oscillating cylinder set-up with angular rotation $\unicode[STIX]{x1D703}(t)$ , cylinder diameter $D=0.06~\text{m}$ , water depth $h=0.72~\text{m}$ , rotation point $z_{0}=0.02~\text{m}$ , distance from tank bottom to load cells $z_{a}=0.92~\text{m}$ , load cells $F_{1}$ and $F_{2}$ , distance to wavemaker $L_{WM}=10.90~\text{m}$ and distance to tank end, $L_{TE}=13.87~\text{m}$ .

A sketch of the second set-up is shown in figure 4. The vertical cylinder is free to rotate with an angle $\unicode[STIX]{x1D703}(t)$ in the pitch mode of motion. Assuming linear motion, the moment due to the pressure forces with respect to $z_{0}$ reads: $M_{wave}(t)-a_{55}\ddot{\unicode[STIX]{x1D703}}-b_{55}\dot{\unicode[STIX]{x1D703}}-c_{55}\unicode[STIX]{x1D703}$ , where $M_{wave}(t)$ , $a_{55}$ , $b_{55}$ and $c_{55}$ denote the wave-exciting moment obtained from the fixed cylinder set-up, added mass, damping and restoring coefficients in the pitch mode of motion, respectively. The moment due to the spring forces reads: $-(z_{a}-z_{0})(F_{2}(t)-F_{1}(t))=-\unicode[STIX]{x1D705}_{0}(z_{a}-z_{0})^{2}\unicode[STIX]{x1D703}$ where $F_{1}$ and $F_{2}$ denote the force recorded at the left and right transducer, respectively, see figure 4, and $\unicode[STIX]{x1D705}_{0}$ the spring constant. Balance of angular momentum gives

(2.4) $$\begin{eqnarray}m_{55}\ddot{\unicode[STIX]{x1D703}}=-a_{55}\ddot{\unicode[STIX]{x1D703}}-b_{55}\dot{\unicode[STIX]{x1D703}}-(\unicode[STIX]{x1D705}_{0}(z_{a}-z_{0})^{2}+c_{55})\unicode[STIX]{x1D703}+M_{wave}(t),\end{eqnarray}$$

where $m_{55}$ denotes the moment of inertia of the cylinder.

The resonance frequency of (2.4) is given by $\unicode[STIX]{x1D714}_{0}^{2}=(c_{55}+\unicode[STIX]{x1D705}_{0}(z_{a}-z_{0})^{2})/(m_{55}+a_{55})$ . Note that the spring force provides the dominant contribution to the restoring force where $c_{55}$ is 0.005 times $\unicode[STIX]{x1D705}_{0}(z_{a}-z_{0})^{2}$ for the actual cylinder. The still water decay tests as well as the irregular wave experiments determine $\unicode[STIX]{x1D714}_{0}=17.7~\text{rad}~\text{s}^{-1}\sim 3\unicode[STIX]{x1D714}_{P}$ (table 1) of the oscillating cylinder. The damping ratio $\unicode[STIX]{x1D701}$ , determined as a fraction of the critical damping, is 0.02 for the cylinder. The small damping ratio implies a very lightly damped oscillating system relevant to offshore wind turbines in extreme conditions (Kallehave et al. Reference Kallehave, Byrne, Thilsted and Mikkelsen2015).

By integration, the pitch angle $\unicode[STIX]{x1D703}(t)$ is obtained as a function of time. For convenience, the response is multiplied by $\unicode[STIX]{x1D705}_{0}(z_{a}-z_{0})^{2}$ , giving the moment of the sum spring force with respect to $z_{0}$ . We denote this quantity by $R(\unicode[STIX]{x1D714}_{0},t)$ where

(2.5) $$\begin{eqnarray}R(\unicode[STIX]{x1D714}_{0},t)=\unicode[STIX]{x1D705}_{0}(z_{a}-z_{0})^{2}\unicode[STIX]{x1D703}(t)=\frac{\unicode[STIX]{x1D714}_{0}^{2}}{\unicode[STIX]{x1D714}_{d}}\int _{0}^{t}M_{wave}(\unicode[STIX]{x1D70F})\text{e}^{-\unicode[STIX]{x1D701}\unicode[STIX]{x1D714}_{0}(t-\unicode[STIX]{x1D70F})}\sin (\unicode[STIX]{x1D714}_{d}(t-\unicode[STIX]{x1D70F}))\,\text{d}\unicode[STIX]{x1D70F},\end{eqnarray}$$

and $\unicode[STIX]{x1D714}_{d}=\unicode[STIX]{x1D714}_{0}\sqrt{1-\unicode[STIX]{x1D701}^{2}}$ . A derivation of (2.5), commonly known as Duhamel’s integral, is given in appendix A. Using (2.5) to obtain the motion response $R(\unicode[STIX]{x1D714}_{0},t)$ , this is fully described by the wave-exciting moment, the resonance frequency and the damping ratio. Use of (2.5) makes possible a response analysis given $M_{wave}(t)$ on the fixed cylinder and varying the resonance frequency $\unicode[STIX]{x1D714}_{0}$ to investigate the response dependency on the ratio $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}$ . We shall find a good correspondence between the measured and calculated response maxima, see § 3.1.

For the calculations of the response, a low pass filter has been applied above the significant wave frequencies at the frequency $\unicode[STIX]{x1D714}=60~\text{rad}~\text{s}^{-1}>9\unicode[STIX]{x1D714}_{P}$ . This is considered as well above the significant wave and load frequencies of interest.

3.1 Single response maxima

Figure 6.

All of the six time series with 2166 events ( $+$ ) where $2.6<\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}<3.3$ . Comparing response maxima by (a) direct comparison and (b) quantiles (sorted values).

The response maxima, obtained from the measured wave-exciting moment on the fixed cylinder, with the resonance calculated by the transfer function (2.5), denoted by $R_{calc}^{max}=\max (R(\unicode[STIX]{x1D714}_{0},t))$ , are compared to the measured response, denoted by $R_{meas}^{max}$ . The data from all of the six time series give a total of 2166 events with a frequency ratio in the range $2.6<\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}<3.3$ . The two quantities show good agreement for the largest response events $R_{meas}^{max}/\unicode[STIX]{x1D70E}_{M}>7$ , where the deviation is up to approximately 5 %, see figure 6(a). Good agreement is also found when looking at the calculated and measured maxima, sorted according to magnitude, denoted by the so-called quantiles, see figure 6(b). This justifies the use of the measured wave-exciting moment in combination with the transfer function, both for estimating the probability levels and for the identification of the extreme events. In what follows, we obtain only the calculated response maxima using the notation $R^{max}=R_{calc}^{max}$ , for the extended frequency range $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=3,4$ and 5.

3.2 Higher harmonic wave forces

The high frequency response is driven by higher harmonic wave force components. We investigate the third, fourth and fifth harmonic forces with regards to the local wave period $T_{TT}$ . The forces are obtained from the wave-exciting moment assuming that they are acting at the still water level. The high frequency forces are considered to act close to the surface (Rainey Reference Rainey1989, Reference Rainey and Eatock Taylor1995a ,Reference Rainey b ; Faltinsen et al. Reference Faltinsen, Newman and Vinje1995; Newman Reference Newman, Grue, Gjevik and Weber1996), which indicates an error of less than $(\unicode[STIX]{x1D702}_{C}/h)$ when using the moment to obtain the forces. For each of the single events, where an event is defined in figure 5(a,b), a window function of 20 s has been applied, from which the high frequency harmonic force components are extracted, see figure 5(c). The maximum amplitude found within the event is defined as the local high frequency harmonic force contribution.

The third harmonic force $F^{(3\unicode[STIX]{x1D714}_{TT})}$ is found using a filter covering $2.5<\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{TT}<3.5$ . Likewise, the fourth harmonic force $F^{(4\unicode[STIX]{x1D714}_{TT})}$ is found for $3.5<\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{TT}<4.5$ , and the fifth harmonic force $F^{(5\unicode[STIX]{x1D714}_{TT})}$ using $4.5<\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{TT}<5.5$ . The forces are expressed for the proxies; the normalized wavenumber $k_{TT}r$ and the wave slope $\unicode[STIX]{x1D716}=ak_{TT}$ , where both are defined in (2.3) and $r$ is the cylinder radius. The obtained results from the irregular waves are compared to previous works with a regular wave input; the leading-order third harmonic FNV solution, $F_{FNV}^{(3)}/\unicode[STIX]{x1D70C}ga^{3}=2\unicode[STIX]{x03C0}(k_{TT}r)^{2}$ , the measurements from Huseby & Grue (Reference Huseby and Grue2000), denoted by H&G and the CFD computations for finite depth by Paulsen et al. (Reference Paulsen, Bredmose, Bingham and Jacobsen2014b ).

For the longest waves, with $0.10\leqslant k_{TT}r\leqslant 0.14$ ( $14.1>T_{TT}^{0}\sqrt{g/d}>11.9$ , where $T_{TT}^{0}$ is the linear estimate) we observe that $F^{(3\unicode[STIX]{x1D714}_{TT})}$ tends towards a constant level close to the FNV result, with the average of the irregular results approximately 11 % below the theory (figure 7 a). The FNV force is evaluated for the middle value of $k_{TT}r$ in each of the $k_{TT}r$ ranges. In the range $0.14\leqslant k_{TT}r\leqslant 0.18$ ( $11.9>T_{TT}^{0}\sqrt{g/d}>10.5$ ), the results from H&G are lower, but within the standard deviation of the present irregular wave results. The average of the irregular wave results are approximately 29 % below the FNV theory, when the waves are steep (figure 7 b). For the shorter waves, with $0.18\leqslant k_{TT}r\leqslant 0.22$ ( $10.5>T_{TT}^{0}\sqrt{g/d}>9.5$ ), the results from H&G are close to the irregular wave results, tending towards the same level, which is ${\sim}44\,\%$ below the theory (figure 7 c). Compared to the results on finite depth by Paulsen et al. (Reference Paulsen, Bredmose, Bingham and Jacobsen2014b ) for $k_{TT}r=0.1$ , a deviation is observed. However, Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2017) point at a substantial difference between the forces in deep water and finite depth.

Figure 7.

Higher harmonic wave force components. Individual events ( $+$ ), ensemble average (—— ▪) and standard deviation (– – ▪), $F_{FNV}^{(3)}$ (- - -), H&G (●) and Paulsen et al. (Reference Paulsen, Bredmose, Bingham and Jacobsen2014b ) for finite water depth (▴). (a) $F^{(3\unicode[STIX]{x1D714}_{TT})}$ for $0.10\leqslant k_{TT}r\leqslant 0.14$ , (b) $F^{(3\unicode[STIX]{x1D714}_{TT})}$ for $0.14\leqslant k_{TT}r\leqslant 0.18$ , (c) $F^{(3\unicode[STIX]{x1D714}_{TT})}$ for $0.18\leqslant k_{TT}r\leqslant 0.22$ , (d) $F^{(4\unicode[STIX]{x1D714}_{TT})}$ for $0.14\leqslant k_{TT}r\leqslant 0.18$ , (e) $F^{(4\unicode[STIX]{x1D714}_{TT})}$ for $0.18\leqslant k_{TT}r\leqslant 0.22$ and (f) $F^{(5\unicode[STIX]{x1D714}_{TT})}$ for $0.18\leqslant k_{TT}r\leqslant 0.22$ .

For the fourth harmonic force, $F^{(4\unicode[STIX]{x1D714}_{TT})}$ , the results show good agreement with H&G (figure 7(d,e). The results for the fifth harmonic force, $F^{(5\unicode[STIX]{x1D714}_{TT})}$ , are similar to those of H&G for $k_{TT}r=0.245$ , where the present results are obtained for the wider range of $0.18<k_{TT}r<0.22$ (figure 7 f). A comparison between the normalized forces for $\unicode[STIX]{x1D716}=0.25$ is provided in table 2.

Table 2.

The ensemble average of the higher harmonic wave force components $F^{(3)}=F^{(3\unicode[STIX]{x1D714}_{TT})}/\unicode[STIX]{x1D70C}ga^{3}$ , $F^{(4)}=F^{(4\unicode[STIX]{x1D714}_{TT})}/\unicode[STIX]{x1D70C}ga^{4}a^{-1}$ and $F^{(5)}=F^{(5\unicode[STIX]{x1D714}_{TT})}/\unicode[STIX]{x1D70C}ga^{5}a^{-2}$ for local wavenumber $0.10<k_{TT}r<0.22$ and wave slope $\unicode[STIX]{x1D716}=ak_{TT}=0.25$ .

The present extracted higher harmonic force components in the irregular waves, for $0.1<k_{TT}r<0.22$ and $0.1<\unicode[STIX]{x1D716}<0.32$ , provide a quite strong generalization of the higher harmonic forces measured by H&G in the regular waves with $0.1<\unicode[STIX]{x1D716}<0.24$ . This in spite of the present results being obtained from the wave-exciting moment, assuming a moment arm equal to the still water level. We note that the present irregular wave results have a significant standard deviation not observed in the regular wave measurements.

As expected for the third harmonic forces, the FNV approximation is found to best fit the longest waves ( $0.1<k_{TT}r<0.14$ ). In general we observe that the force components are tending towards a constant level for the steep waves. The compliance with the previous results in periodic waves indicates that the high frequency contribution originates from nonlinearities and not from shorter linear free waves, since regular waves do not contain energy from linear free waves. Moreover, it illustrates that the local wave slope and the wavenumber defined in (2.3) are useful proxies of the local wave events.

3.3 Wave load mechanisms

In this section we discuss the following different wave load mechanisms driving the response:

1. (i) wave-exciting inertia forces, a function of the fluid acceleration;

2. (ii) wave slamming, due to both non-breaking and breaking wave events;

3. (iii) the secondary load cycle; and

4. (iv) possible drag forces as a function of the fluid velocity.

Consider the wave-exciting moment in figure 5(b) where the maximum occurs for $t/T_{P}=0$ . This is simultaneous to the maximum wave crest and means that the orbital velocity is approximately horizontal and at maximum. The force at this instant is associated with wave breaking, slamming and possible viscous drag forces (Paulsen et al. Reference Paulsen, Bredmose, Bingham and Jacobsen2014b ; Kristiansen & Faltinsen Reference Kristiansen and Faltinsen2017) which are included in the categories (ii) and (iv) above.

Consider then the negative response maximum in figure 5(d), of absolute value $R^{max}\approx 8\unicode[STIX]{x1D70E}_{M}$ and occurring at $t/T_{P}\approx 0.25$ . This is approximately at the same time as the wave elevation has a zero down-crossing, corresponding to a maximum horizontal particle acceleration at the surface. The acceleration is associated with an inertia force and is in accordance with category (i). Returning to the load history in figure 5(b), the secondary load cycle (Grue et al. Reference Grue, Bjørshol and Strand1993) occurs slightly before the time of the negative response extreme.

The dynamic part of the response, $R(t)-M(t)$ , further highlights the effects of the different load mechanisms (i)–(iv). In figure 5(e) we observe that the large wave crest produces a significant change of the response amplitude and its phase for $t/T_{P}>0$ . Between the two local response peaks at $t/T_{P}\approx -0.1$ and $t/T_{P}\approx 0.1$ the dynamic part experiences a local oscillation of duration equal to half of the resonance period. The modification of the response is due to a strongly nonlinear impulse type of loading, originating from the slamming event. As a result, the dynamic contribution attains a value of $R(t)-M(t)\approx 4\unicode[STIX]{x1D70E}_{M}$ at $t/T_{P}\approx 0.1$ . The response is further increased to $R(t)-M(t)\approx 5\unicode[STIX]{x1D70E}_{M}$ around the wave zero down-crossing, at $t/T_{P}\approx 0.25$ , with load contributions from the large inertia force and the secondary load cycle. Another effect that adds to the large negative response peak is the restoring force of the cylinder. Even with no wave-exciting forces, this would cause a negative response peak after the positive build-up. The timing of this, relative to the wave forces, is governed by $\unicode[STIX]{x1D714}_{0}$ , the natural frequency. A result of the different load contributions working together, is that the maximum response occurs after and in the opposite direction of the maximum wave-exciting moment, approximately at the same time as the wave elevation has a zero down-crossing.

We have now discussed the load event in figure 5. Further, we consider the load histories of the 21 largest response events which are listed in table 3. More specifically, these events are obtained with regards to $R^{max}/\unicode[STIX]{x1D70E}_{M}$ for $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=3$ . We observe that different wave load and response mechanisms contribute to the response level, including:

1. (i) a large nonlinear inertia force before the wave crest has passed, $F_{I,front}$ , which is observed for 15 of the 21 events. The inertia force is characterized by the front of the wave being steep with $\unicode[STIX]{x0394}\unicode[STIX]{x1D702}/\unicode[STIX]{x0394}t>5H_{S}/T_{P}$ ;

2. (ii) a large nonlinear inertia force after the wave crest has passed, $F_{I,back}$ , observed for 16 of 21 events. This is characterized by the back of the wave being steep with $\unicode[STIX]{x0394}\unicode[STIX]{x1D702}/\unicode[STIX]{x0394}t<-5H_{S}/T_{P}$ ;

3. (iii) wave slamming, $F_{slam}$ , observed for 8 of 21 events, including the 4 largest events. Here slamming is characterized by a coinciding wave crest and a maximum wave-exciting moment; and

4. (iv) the secondary load cycle, $F_{II}$ , observed for 17 of 21 events. The $F_{II}$ values are found to coincide with a steep crest back and occur close to the wave zero down-crossing.

Further we note:

1. (i) an opposite direction of the maximum response, $R_{opp}$ , characterized by the maximum response occurring after and in the opposite direction to the maximum wave-exciting moment and simultaneous to the wave zero down-crossing. This is observed for 16 out of 21 events;

2. (ii) an effect of a preceding wave, $\unicode[STIX]{x1D706}_{prec}$ , where the response is affected by the inertia of the moving pile. The oscillations are significant before the wave event appears. This is observed for 7 out of 21 events, where 3 among the 7 events are strongly dominated by the effect.

Table 3.

Wave parameters and observed wave load and response characteristics, for the 21 largest response events when $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=3$ , listed in decreasing order with respect to $R^{max}/\unicode[STIX]{x1D70E}_{M}$ . The characteristics are confirmed with Y $=$  Yes, N $=$  No or Y(S) $=$  Yes, strongly dominated. Parameters in the table: event number, series index, time of occurrence and the rest of the parameters are defined in the text.

We observe that slamming plays a dominant role for the largest response events. Apart from one of the preceding wave cases, large nonlinear inertia forces are present for all of the events, where either the front or back of the wave, or both, are observed to be steep. However, the large inertia force and the resulting response rather occurs for a large elevation gradient in the back of the wave while what happens in the wave front is less important. Apart from one preceding wave case, the secondary load cycle is found to coincide with the steep wave gradient in the back of the wave.

3.4 Nonlinear versus linear exceedance probability

The empirical exceedance probabilities of the nonlinear wave-exciting moment and motion response are found using (2.2). Each of the series is considered separately, where the frequency ratio is varied with $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=3,4$ or 5. The load and responses are presented for increasing nonlinearity ( $0.10<\unicode[STIX]{x1D716}_{P}<0.14$ ) and for long and moderately long waves ( $0.09<k_{P}r<0.14$ ).

Estimates for the linear wave-exciting moment and motion response were carried out for reference purposes. In order to estimate the underlying linear wave spectrum, the measured surface elevation was linearized as proposed by Johannessen (Reference Johannessen2010, Reference Johannessen2012). The second-order contribution was calculated from the measured wave time series, using the total surface elevation. Subsequently, the linear surface elevation was found by subtracting the second-order contribution from the measured surface elevation. Further, irregular waves were created from each of the estimated linear wave spectra. The MacCamy & Fuchs solution (Reference MacCamy and Fuchs1954) was used to obtain the wave-exciting moment for a fixed cylinder. More details are found in appendix B.

As expected, the linear and nonlinear analyses agree well for $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=3$ when the waves are long and have small amplitude ( $\unicode[STIX]{x1D716}_{P}=0.10$ and $k_{P}r=0.09$ , figure 8 a). The same is true when the wave slope is moderate and the waves are long ( $\unicode[STIX]{x1D716}_{P}=0.12$ and $k_{P}r=0.09$ , figure 8 b). In these cases the estimated linear response provides a good representation of the nonlinear probability.

Figure 8.

The empirical exceedance probability for the linear wave-exciting moment (– –), nonlinear wave-exciting moment ( $+$ ), linear motion response (——) and nonlinear motion response (○) where $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=3$ for (a) series a, $(\unicode[STIX]{x1D716}_{P},k_{P}r)=(0.10,0.09)$ ,(b) b, $(0.12,0.09)$ , (c) c, $(0.12,0.11)$ , (d) d, $(0.12,0.14)$ , (e) e, $(0.14,0.09)$ and (f) f, $(0.14,0.14)$ .

Figure 9.

The empirical exceedance probability for the linear wave-exciting moment (– –), nonlinear wave-exciting moment ( $+$ ), linear motion response (——) and nonlinear motion response (○) where $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=4$ for (a) series a, $(\unicode[STIX]{x1D716}_{P},k_{P}r)=(0.10,0.09)$ , (b) b, $(0.12,0.09)$ , (c) c, $(0.12,0.11)$ , (d) d, $(0.12,0.14)$ , (e) e, $(0.14,0.09)$ and (f) f, $(0.14,0.14)$ .

For moderate wave slope and moderately long waves ( $\unicode[STIX]{x1D716}_{P}=0.12$ and $k_{P}r=0.11$ , figure 8 c), the linear estimate gives a good representation of the distribution of the response for $P_{ex}(R^{max}/\unicode[STIX]{x1D70E}_{M})>0.06$ . However, for $P_{ex}(R^{max}/\unicode[STIX]{x1D70E}_{M})<0.06$ the nonlinear contribution becomes significant, showing a deviation of ${\sim}50\,\%$ for the largest nonlinear response events (marked by circles) when compared to the corresponding linear estimates. For steeper and shorter waves, we observe that the deviation appears at an earlier stage: for $\unicode[STIX]{x1D716}_{P}=0.12$ and $k_{P}r=0.14\,P_{ex}(R^{max}/\unicode[STIX]{x1D70E}_{M})\sim 0.1$ (figure 8 d), for $\unicode[STIX]{x1D716}_{P}=0.14$ and $k_{P}r=0.09\,P_{ex}(R^{max}/\unicode[STIX]{x1D70E}_{M})\sim 0.2$ (figure 8 e) and for $\unicode[STIX]{x1D716}_{P}=0.14$ and $k_{P}r=0.14\,P_{ex}(R^{max}/\unicode[STIX]{x1D70E}_{M})\sim 0.3$ (figure 8 f). For the steepest waves ( $\unicode[STIX]{x1D716}_{P}=0.14$ ) the nonlinear force deviates earlier from the linear force for longer waves (compare figure 8 e,f). We note that while the wave-exciting moment is dominated by the energy around the governing wave frequency, the response is governed by the nonlinear high frequency forces.

For $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=4$ the response for small wave slopes and long waves ( $\unicode[STIX]{x1D716}_{P}=0.10$ and $k_{P}r=0.09$ , figure 9 a) follows the linear results, but shows a deviating trend for small exceedance probability. For moderate wave slope and long waves ( $\unicode[STIX]{x1D716}_{P}=0.12$ and $k_{P}r=0.09$ , figure 9 b) the deviation between the nonlinear and linear results becomes evident for $P_{ex}\approx 0.1$ . In moderate and steep waves ( $\unicode[STIX]{x1D716}_{P}>0.12$ , figure 9 b–f) a significant deviation between the nonlinear and linear results is found for $P_{ex}<0.2$ . For $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=5$ the same tendencies as for $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=4$ are observed. When the frequency is increased, the response tends towards the wave-exciting moment, which is equivalent with the quasi-static level.

As expected, the linear and nonlinear analyses agree well when the waves are long and have small amplitude. The nonlinearity of the wave-exciting moment becomes more important in the steeper and shorter waves. The nonlinearity also becomes more prominent when the frequency ratio $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}$ is increased from 3 to 4.

3.5 Extreme response events

Figure 10.

Wave scatter plot including all the 2166 events ( $+$ ), events with $P_{ex}(R^{max}/\unicode[STIX]{x1D70E}_{M})<0.01$ (●), $Fr=0.4$ (——) and $KC=5$ (– –) for (a) $(\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P},\unicode[STIX]{x1D701})=(3,0.02)$ , (b) $(3,0.06)$ , (c) $(4,0.02)$ , (d) $(4,0.06)$ , (e) $(5,0.02)$ and (f) $(5,0.06)$ .

The most extreme responses are critical for design. It is of interest to further investigate the wave effects driving these responses. The 1 % largest events (satisfying $P_{ex}(R^{max}/\unicode[STIX]{x1D70E}_{M})<10^{-2}$ ) are presented in scatter diagrams according to the wave proxies $T_{TT}\sqrt{g/d}$ and $k_{TT}\unicode[STIX]{x1D702}_{C}$ . The resonance frequencies are $\unicode[STIX]{x1D714}_{0}/\unicode[STIX]{x1D714}_{P}=3,4$ and 5. The majority of these events occur for waves with period $T_{TT}\sqrt{g/d}$ between 9 and 16 and wave slope $k_{TT}\unicode[STIX]{x1D702}_{C}$ larger than 0.2 (figure 10 a,c,e).

In order to discuss the history effect in the response, i.e. the contribution from the inertia of the moving pile due to successive large wave events, we present calculations with a damping of both $\unicode[STIX]{x1D701}=0.02$ and $\unicode[STIX]{x1D701}=0.06$ , where in the latter the effect of preceding waves, for the local response, is smaller compared to the former. The extreme response events obtained with $\unicode[STIX]{x1D701}=0.06$ all gather in a common region (figure 10 b,d,f).

The nonlinear high frequency forces driving the response originates either from flow separation, free surface gravity waves or a combination. Flow separation is governed by the Keulegan–Carpenter number $KC=u_{C}T_{TT}/d$ , while gravity wave effects are governed by the Froude number $Fr=u_{C}/\sqrt{gd}$ . We have in figure 10 indicated the line corresponding to $KC=5$ . Almost all of the extreme events are found for $KC>5$ and $k_{TT}\unicode[STIX]{x1D702}_{C}>0.15$ . Note that a $KC$ -number larger than 2 is commonly associated with flow separation (Sarpkaya Reference Sarpkaya1986).

A contribution of flow separation to the higher harmonic wave forces has recently been suggested by Paulsen et al. (Reference Paulsen, Bredmose, Bingham and Jacobsen2014b ), Kristiansen & Faltinsen (Reference Kristiansen and Faltinsen2017). We remark that the details of the flow separation depend on the Reynolds number ( $Re$ ) and are different in the model scale compared to the large scale, where $Re$ is approximately 1000 times larger. The possible flow separation may contain more three-dimensional effects in the large scale compared to the model scale.

We have also indicated the line $Fr=0.4$ , where $Fr=u_{C}/\sqrt{gd}$ . The extreme response events found for waves with $Fr<0.4$ are all affected by preceding waves. The Froude number indicates a gravity wave effect at the scale of the cylinder diameter, where $Fr=0.4$ corresponds to a local wavelength $\unicode[STIX]{x1D706}_{C}=2\unicode[STIX]{x03C0}u_{C}^{2}/g$ of the local crest velocity $u_{C}$ with $\unicode[STIX]{x1D706}_{C}\approx d$ contributing to a particular wave–body interaction, as proposed by Grue et al. (Reference Grue, Bjørshol and Strand1993, Reference Grue, Bjørshol, Strand and Ohkusu1994).

The present data of the obtained responses show that the extreme events occur for $KC>5$ . The regime $KC>5$ , is significantly above the separation limit ( $KC>2$ , Sarpkaya Reference Sarpkaya1986), indicating a flow separation effect and contribution from possible drag forces, even though the flow separation itself has not been measured. The findings supports $KC$ and flow separation as a more relevant criterion for the extreme response events. However, we choose to include the Froude criterion, as the results are not considered sufficient in order to exclude the surface gravity wave effects. The more visible local free surface wave, suggested at the cylinder diameter scale, is for comparison governed by $Fr>0.4$ . The free surface effect is very clear in the experiments. The wave may co-interact with the flow separation effects, suggesting that both effects and criteria are relevant.