Influence of Wave Variability on Ship Response During Deterministically Repeated Seakeeping Tests at Forward Speed

Abstract

In order to validate numerical results or verify design choices using experiments, knowledge about the experimental variability is essential. This variability was evaluated for seakeeping tests at forward speed with a model in a steep wave condition over the long axis of a basin and in a less steep oblique wave condition, in a commonly applied test procedure. The incoming wave and response variability was evaluated using deterministic repeat tests. The results for incoming waves at some distance before the model have been published already; the present study discusses the model responses. Overall time trace similarity as well as the amplitude and timing variability of individual wave crests and response peaks were studied, after assessing the input uncertainties. The response variability increases with distance from the wave generator for the wave crest height and (relative) ship motion peaks. The variability of the impact loads on a deck structure is large with a lot of scatter. Small wave-induced currents may build up differences in wave propagation speed between the repeat runs, which means that the seakeeping variability partly depends on previous wave conditions. A proportional relation could be identified between most response peaks and the corresponding incoming wave peaks. The timing variability of the response peaks follows from that of the incoming wave crests. Unfortunately, there is no direct relation between the response amplitude variability and that of the corresponding wave crest. The presented results can be used as reference for the typical variability of free-sailing seakeeping experiments.

Appendices

Appendix A: Sensor Locations and Test Times

The sensor locations and sample frequency can be found in Table 3 and the measurement accuracy of these sensors in Table 4. Figure 18 shows the times of the day that the tests were done.

Table 3. Locations of relevant sensors for the KCS in waveA and the plate in waveB (w.r.t. midship, where FS = sample frequency, WG = wave generator)

Table 4. Measurement uncertainties (where X = measured value)

Uncertainties scaled linearly with the RMS of the WG flap motion over the RMS of the response for the results in Sect. 4.1.

Fig. 18.

Timing of original and repeat tests (2018), and other preceding tests [21]

Appendix B: Details of Variability Formulations

Surface Similarity Parameter

As explained in Sect. 3, [16] proposed surface similarity parameter Q, which is a measure for the similarity of two functions f₁ and f₂ (Eq. 1, where F₁ is the Fourier transform of f₁). This parameter considers both the amplitude and the phasing of the time trace. The lower Q, the better the match (Q = 0 for a perfect match and Q = 1 for signals perfectly out of phase). [16] provides some values of Q for analytical functions and measured wave time traces.

$$ Q\left( {f_{1} ,f_{2} } \right) = \frac{{\left( {\smallint \left| {F_{1} \left( \omega \right) - F_{2} \left( \omega \right)} \right|^{2} d\omega } \right)^{1/2} }}{{\left( {\smallint \left| {F_{1} \left( \omega \right)} \right|^{2} d\omega } \right)^{1/2} + \left( {\smallint \left| {F_{2} \left( \omega \right)} \right|^{2} d\omega } \right)^{1/2} }} $$

(1)

Criteria for Variability of Individual Crests and Their Timing

As mentioned in Sect. 3, zero up-crossing analysis was applied to identify peaks in the wave and response time traces (Fig. 19). The force impulse is used as a measure for the shape of the event (Fig. 19). Risetime could also be an important indicator for resonance of a structure, but this is presently disregarded.

Fig. 19.

Wave or response definitions

The variability V(C_m) of wave crest m is defined as the ratio between the standard deviation of all repeats (σ) and the mean of all repeats (μ) for that crest, where C is the height of the crest and N is the number of repeats (Eq. 2). The standard deviation of the timing of each peak over all repeats with respect to the timing of that peak in a reference run χ(C_m) of crest m is defined by Eq. 3 and Eq. 4. In this formulation, t(C_m,n) is the timing of crest m in run n. Similarly, C can be substituted by impulse I.

$$ V\left( {C_{m} } \right) = \frac{{\sigma \left( {C_{m,0} ,C_{m,1,} C_{m,2} , \ldots , C_{m,N} } \right)}}{{\mu \left( {C_{m,0} ,C_{m,1,} C_{m,2} , \ldots , C_{m,N} } \right)}} $$

(2)

$$ \tau \left( {C_{m,n} } \right) = t\left( {C_{m,n} } \right) - t\left( {C_{m,0} } \right),{\text{for }}\,\,\,n\, = \, 1\,\,\,{\text{to}}\,\,N $$

(3)

$$ \chi \left( {C_{m} } \right) = \sigma \left[ {\tau \left( {C_{m,1} } \right),\tau \left( {C_{m,2} } \right), \ldots ,\tau \left( {C_{m,N} } \right)} \right] $$

(4)

Matching Method Peaks in Original and Repeat Runs

As the individual crests in the repeat runs do not occur exactly at the same time, a matching method was applied. A space interval of ±0.3 m was applied in [21] to relate crests above a threshold in the repeat run to crests in the original run. The same method was applied here to the response peaks.

Appendix C: Matching Response Peaks to Wave Peaks

Each response peak in the original run gets attributed its variability based on the repeat runs as explained in Sect. 3. Then it is coupled to the peaks in W1. As explained in Sect. 3, time window starting a few seconds after the wave peak was applied in order to match incoming wave peaks to response peaks. The maximum response peak within this window is identified as the corresponding response peak. All peaks that cannot be coupled this way are omitted. The intervals are based on the response ‘location’ and linear wave propagation, combined with patterns in the data such as the time difference for the best time trace correlation.

The peaks in for instance W1 and relbow for the KCS (see example in Fig. 20) show typical time differences of 2–3 s. Assuming linear dispersion, a wave with a period of 1.58 s would travel from W1 to midship in approximately 2 s. If the local vertical motion plays a large role in relbow, the observed time difference seems reasonable. The figure shows reasonable relations between coupled W1 and relbow peaks. Similarly, time intervals and matching results were obtained for the other responses of the KCS and the plate.

Fig. 20.

Example: matched peaks in incoming wave W1 and KCS relative wave elevation at the bow in waveA, with an indication of the coupled and omitted peaks

The resulting matched peak value and timing results are shown in Fig. 21 for waveA and in Fig. 22 for waveB, with the associated peak variability V (Eq. 2) and timing variability χ (Eq. 4). Some conclusions based on these figures were drawn in Sect. 5.3 and Sect. 5.4.

Fig. 21.

WaveA: values of matched peaks in incoming wave and KCS responses (left), their variability in peak value V (middle) and in timing χ (right)

Fig. 22.

WaveB: values of matched peaks in incoming wave and relative wave elevation along the plate (left), their variability in peak value V (middle) and in timing χ (right, using F1 as reference)

Appendix D: Variability Plots

See Figs. 23 and 24.

Fig. 23.

KCS highest peaks in incoming wave W1 (>0.08 m), heave (>0.015 m), pitch (>0.8°), relbow (>0.15 m), FX and impulses FX (>0.01 kN): repeat runs vs. original run and variability as a function of the original values

Fig. 24.

Plate highest peaks in incoming wave W1 (>0.08 m), relbow (>0.1 m, left), relmid (>0.1 m, mid) and relaft (>0.12 m, right), repeat runs vs. original run and variability as a function of the original values