7 FREQUENCY-DOUBLING EXCITATION OF 3D GAP RESONANCE

Chapter 7: Frequency-doubling excitation of 3D gap resonance between two fixed vessels with round corners

7-1

7 FREQUENCY-DOUBLING EXCITATION

OF 3D GAP RESONANCE BETWEEN TWO

FIXED VESSELS WITH ROUND CORNERS

7.1 Introduction

The natural frequencies of offshore structures are generally substantially above the peak

frequency of the wave spectrum. However, it is known that forces on offshore structures

may still occur at the natural frequencies. For example, the contributions of second- and

third-harmonic wave loads may induce high-stress levels for offshore structures, known

as ‘springing’ and ‘ringing’ (Faltinsen et al., 1995) respectively. Side-by-side offloading

from an FLNG to an LNG carrier will only be attempted in mild sea states. Long-period

swell may occur in these sea states, but will not drive significant surface motion linearly

in the gap between the two vessels, because the frequency is too low to couple to the

lowest gap mode. However, Zhao et al. (2017) show experimentally that frequency

doubling could excite the gap resonances and this can be substantially larger than the

linear gap response under the excitation of an incident wave group with appropriate

frequency content. Therefore, accurate prediction of the higher harmonic responses is

very important.

In this chapter, the experiments with frequency doubling excitation of 3D gap resonant

responses (including the underlying harmonic structures for the free surface) are

successfully reproduced in the NWT established in Chapter 5. The boxes with round

bilges in a parallel configuration are excited by incident wave groups with peak frequency

equal to half of the first modal frequency for the side-by-side system. Using the validated

numerical results, the velocity and forces are successfully extracted. The evolution of the

local flow field around the body at first, second and third order is presented and discussed.

7.2 Reproducing experiments

With the same experimental setup described in Chapter 6 but halving the peak frequency

of the incident wave groups to 𝑓𝑃=0.508 Hz, Zhao et al. (2017) investigated the effect of

higher harmonics on 3D gap resonance. The natural frequency for the first mode 𝑓𝑚=1 is

twice the peak frequency of the incident wave groups, hence this case is referred to as

‘frequency-doubling excitation’ in the following context. In the experiments four

NewWave-type incident wave groups incident from the beam with the same paddle signal

and maximum wave amplitude of 𝛼=50 mm, but with each component shifted by a

relative phase 𝜃 of 0°, 90°, 180° and 270° respectively were approximately focused at the

middle of the gap (𝑥=0 m) to excite resonant fluid motions in the gap. These wave groups

facilitate separation of higher harmonics using the four-phase harmonic extraction

method which will be described in Section 7.3.

Considering the peak wavelength of the incident wave groups is increased to 𝜆𝑝=6.0 m a

wider numerical wave tank is required since it takes less time for the reflected waves from

side-walls to travel back into the gap and waves with larger wavelength dissipate more

slowly. Thus, the dimensions of the NWT were chosen as (𝐿𝑤, 𝐵𝑤22, 𝐻𝑤) (12.0, 19.998,

4.0) m and the water depth is chosen as ℎ 3.5 m which satisfies the deep-water condition.

Given the incident wave groups are weakly nonlinear and there is not much spectral

‘smearing’ for different harmonics of gap responses, only two wave groups are used here

for harmonic separation to reduce computational effort, i.e. a crest-focused wave group

(𝜃=0°) and a corresponding trough-focused wave group (𝜃=180°).

Two meshes B1 and B2, with near-wall cell widths of Δ𝑥2 2.0 mm and Δ𝑥2 0.1 mm

respectively, were used to simulate 3D gap resonance. The vertical cell heights near the

free surface Δ𝑧1 were chosen as 5.0 mm, 2.0 mm for B1 and B2, respectively, to keep an

appropriate cell aspect ratio. The other mesh dimensions are the same for the two cases

with (Δ𝑥1, Δ𝑦1) (30.0, 15.8) mm and the total number of cells being 33.0 million and

51.0 million for B1 and B2 respectively. Given the experimental maximum gap response

level for the frequency-doubling excitation is smaller than the maximum gap response for

the linear excitation modelled in Chapter 6, the resonant fluid motions in the gap for these

two cases should reside in the same flow regime. According to the mesh independence

study discussed in Chapter 6, B1 is a converged case which can resolve the boundary


7-3

(a) time series

(b) amplitude spectrum

(c) phase spectrum

Figure 7.1 Numerically recreated linear incident wave group compared to experimental

results.

(a) time series


(c) phase spectrum

Figure 7.2 Numerically recreated crest-focused total incident wave group compared to

experimental results.


7-5

(a) time series


(c) phase spectrum

Figure 7.3 Numerically recreated trough-focused total incident wave group compared

to experimental results.

Figure 7.4 Reconstruction of the second-order harmonic signal using the linear signal

for incident wave group.

layer properly while B2 serves as a comparison for the purpose of emphasizing the

importance of resolving the near-wall boundary layer. All the other numerical

configurations are the same as the linear excitation case.

The incident wave groups were recreated using the linear iterative method which

iteratively corrects the incident wave signal based on the linear wave signal extracted

using the phase-inversion harmonic extraction method (see Section 7.3). The obtained

numerical results of the linear signal, crest- and trough-focused total signals are compared

to the experimental measurements in Figure 7.1 to Figure 7.3 respectively. The agreement

between the numerical and experimental results are excellent after only four iterations

(the results are better than in Chapter 6, because the waves are less steep, resulting in less

spectral evolution and a need for fewer iterations). Assuming the incident wave group has

a Stokes form, the second-order bound waves can be reconstructed using the linear signal

𝜂2 = 𝑆(𝜂12 − 𝜂1𝐻

2 ) (7.1)

where 𝑆 is related to the standard Stokes coefficients (see Walker et al., 2004), 𝜂1 and

𝜂1𝐻 are the linear signal and its Hilbert transform, and 𝜂2 is the second-order bound wave

signal. The numerical and reconstructed second-order signals are compared in Figure 7.4.

The main wave packet of the numerical result agrees very well with the reconstructed

signal but there is a second pulse in the numerical result (at 𝑡 > 3 s) which is absent for

the reconstructed signal. This pulse represents error waves, which occur because the

control signal applied at the numerical wave generation boundary is linear while the

waves generated are inherently nonlinear. Nevertheless, the error waves are


7-7

(a) Crest-focused total gap response

(b) Trough-focused total gap response

Figure 7.5 Gap responses measured at WG 4 for mesh B1 and B2.

small and should not have a substantial effect on the numerical results. This is because

the error waves are free waves, not bound to the rest of the group, so interact with the

boxes in an essentially linear fashion – and if we consider that the amplitude of the peak

response for the linear wave group in Chapter 6, with similar distribution of energy, was

of the same order as the amplitude of the incident group, we expect the response to the

error waves to be less than 1 mm. This error should be eliminated in future studies –

second order generation without error waves is discussed by Schäffer (1996) for example.

With boxes in place the recreated incident wave group has been used to excite gap

motions for meshes B1 and B2. Time series of free surface elevation measured at WG 4

for B1 is compared to B2 in Figure 7.5. As expected B1 with coarse near-wall mesh has

a larger gap response as a result of insufficient resolution to resolve the near-wall velocity

gradient (see Figure 7.6). The total crest- and trough-focused time series of free surface

elevation measured at WG 1-4 are compared to the experimental results shown in Figure

7.7 and Figure 7.8. With much better recreation of the incident wave groups in the NWT,

an excellent agreement has been achieved for both crest and trough gap responses. The

time series exhibit a similar ‘beating pattern’ as the linear excitation case modelled in

(a) B1

(b) B2

Figure 7.6 Near-wall vertical velocity profiles at various phases for B1 and B2. Circle:

theoretical solution; solid line: CFD results.


7-9

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 7.7 Crest-focused total responses at various locations along the gap for B2

compared to experiments.

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 7.8 Trough-focused total responses at various locations along the gap for B2

compared to experiments.


7-11

Chapter 6. At the beginning the free surface in the gap oscillates with the same frequency

as the incident waves. When the incident waves have passed the gap the free surface in

the gap starts to oscillate at higher frequencies, which are a superposition of multiple

resonant modes.

7.3 Harmonic extraction method

For weakly nonlinear waves and wave-structure interaction, a regular incident wave will

induce a surface elevation of the form (written here to the fourth order in linear component

𝐴):

𝜂(𝜃) = 𝜂11𝐴 cos(𝜃) + 𝜂20𝐴2 + 𝜂22𝐴

2 cos(2𝜃)

+ 𝜂31𝐴3 cos(𝜃) + 𝜂33𝐴

3 cos(3𝜃)

+ 𝜂40𝐴4 + 𝜂42𝐴

4 cos(2𝜃) + 𝜂44𝐴4 cos(4𝜃) + 𝑂(𝐴5)

(7.2)

where 𝜃 is the phase function of the incident wave, and the coefficients 𝜂𝑖𝑗 represent

transfer functions between the incident surface elevation and response surface elevation

(with 𝑖 and 𝑗 standing for the power of amplitude and the frequency harmonic,

respectively). Separation of higher harmonics for response surface elevation induced by

a regular wave is straightforward using digital filtering, however, it is more complicated

for response surface elevation induced by wave groups where there is often overlap

between different harmonic contents.

To separate the odd and even harmonics of the response free surface elevation arising

from focused incident wave groups, the phase-inversion method applied to wave

kinematics described by Hunt et al. (2002) can be used. This method requires two time

series of a given hydrodynamic quantity induced by two incident waves that are 180° out

of phase. According to Eq. (7.2), when the phase function is shifted from 𝜃 to 𝜃 + 𝜋 the

signs of the odd harmonics will be inverted while the signs of the even harmonics will

remain the same. Therefore, the odd and even harmonics are separated by taking the

averaged difference and sum of the free surface elevation signals, respectively,

𝜂0 − 𝜂1802

= (𝜂11𝐴 + 𝜂31𝐴3) cos(𝜃) + 𝜂33𝐴

3 cos(3𝜃) + 𝑂(𝐴5) (7.3)

and

𝜂0 + 𝜂1802

= 𝜂20𝐴2 + 𝜂40𝐴

4 + (𝜂22𝐴2 + 𝜂42𝐴

4) cos(2𝜃)

+𝜂44𝐴4 cos(4𝜃) + 𝑂(𝐴6)

(7.4)

where 𝜂0 and 𝜂180 refers to the crest-focused and troughed-focused wave groups with

phase 𝜃=0° and 𝜃=180° at the focus point. Then the harmonics of the time series of the

total nonlinear free surface elevation can be extracted by frequency filtering.

To more effectively separate the harmonics, a similar four-phase harmonic extraction

method can be used (Fitzgerald et al., 2014). In this situation four different incident wave

groups are generated using the same paddle signal, but with each component shifted by a

relative phase 𝜃 of 0° , 90°, 180° and 270° which correspond to the ‘crest-focused’,

‘downcrossing’, ‘trough-focused’, and ‘upcrossing’ wave group respectively. A linear

combination of the four wave groups with their Hilbert transforms will result in the

separation of the harmonics as

𝜂0 − 𝜂90𝐻 − 𝜂180 + 𝜂270

𝐻

4= (𝜂11𝐴 + 𝜂31𝐴

3) cos(𝜃) + 𝑂(𝐴5)

𝜂0 − 𝜂90 + 𝜂180 − 𝜂2704

= (𝜂22𝐴2 + 𝜂42𝐴

4) cos(2𝜃) + 𝑂(𝐴6)

𝜂0 + 𝜂90𝐻 − 𝜂180 − 𝜂270

𝐻

4= 𝜂33𝐴

3 cos(3𝜃) + 𝑂(𝐴5)

𝜂0 + 𝜂90 + 𝜂180 + 𝜂2704

= 𝜂20𝐴2 + 𝜂40𝐴

4 + 𝜂44𝐴4 cos(4𝜃) + 𝑂(𝐴6)

(7.5)

where the superscript 𝐻 refers to the Hilbert transform and the subscript 0, 90, 180 and

270 represent the shifted phase of the signals. A more complicated harmonic extraction

method is possible to extract the harmonics more effectively by using a set of 12 wave

groups with phase ranging from 0° to 330° with an interval of 30 ° (see Hann et al., 2014).

7.4 Harmonics of free surface

The high-order harmonic components are separated by applying the ‘phase-inversion’

method to the measured gap responses for the crest-focused wave group and the same

wave group inverted. The spectra of the separated odd and even harmonics for B2 are

shown in Figure 7.9. It is found that the phase inversion method works very well


7-13

Figure 7.9 Separated odd and even harmonic spectra of gap responses measured at WG

4 for B2

for this case and the harmonics are clearly separated in the spectra. The odd harmonics

have obvious peaks in the spectrum at approximately 𝑓𝑝 and 3 𝑓𝑝 while the even

harmonics at 2𝑓𝑝 and 4𝑓𝑝 are also evident, as expected. It is remarkable that the second

harmonic is larger than the first harmonic. A clearer version of odd and even harmonics

is shown in Figure 7.10, in which the experimental results are also included in the figure

for comparison. The agreement between the numerical and experimental results are quite

good for the zeroth to the fourth harmonics. To separate the harmonics, window-based

low-pass and band-pass digital filters with linear phase response are applied to extract the

zeroth harmonic and the first to the fourth harmonic components respectively, and the

corresponding magnitude frequency responses for the filters are illustrated in Figure 7.10.

The extracted long-wave difference component (zeroth), first, second, third and fourth

sum harmonics, measured at WG 1-4 along the gap, are compared to the experimental

results shown in Figure 7.11 to Figure 7.15, respectively. It is found the zeroth harmonic

is extremely small. The numerical and experimental results for the first-, second- and

third-order harmonics are in good agreement, while the agreement for the fourth harmonic

is satisfactory.

(a)

(b)

Figure 7.10 The frequency response of filters used for harmonic separation for gap

responses measured at WG 4. Left and right vertical axes represent the power spectral

density and frequency response of the filters, respectively.


7-15

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 7.11 Separated zeroth harmonic component separated at various locations along

the gap for B2 compared to experiments.

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 7.12 Separated first harmonic component at various locations along the gap for

B2 compared to experiments.


7-17

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 7.13 Separated second harmonic component at various locations along the gap

for B2 compared to experiments.

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 7.14 Separated third harmonic component at various locations along the gap for

B2 compared to experiments.


7-19

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 7.15 Separated fourth harmonic component of gap responses at various locations

along the gap for B2 compared to experiments.

Figure 7.16 Snapshots of wave field for the first, second and third harmonics. The wave

field is normalized using the maximum value of each harmonic wave field.

𝜑/𝜑𝑚𝑎𝑥

(a) First harmonic at 𝑡=14.88s

(b) Second harmonic at 𝑡=14.88s

(c) Third harmonic at 𝑡=12.64s


7-21

Figure 7.17 Frequencies 𝑓 (Hz) and damping coefficients 𝜉 × 103, for mode 1, 3 and

5 of gap resonant free surface elevation for frequency-doubling excitation (results for

linear excitation are included in grey for comparison). ‘Expt.’ refers to ‘Experimental’.

Red dashed lines are modal frequencies predicted using potential flow software

DIFFRACT.

One of the advantages of the CFD method lies in the ability to reproduce the entire flow

field with high spatial and temporal resolution and hence the wave field of different

harmonics are extracted. Figure 7.16 (a) and (b) show the wave field for the first and

second harmonics at a time in the decay stage (for this case the excitation stage is defined

as the time window before 𝑡= 6 s and the decay stage is defined otherwise), respectively.

It can be seen that after the incident wave group has passed the gap the second harmonic

components still oscillate in the gap while the first harmonic is almost absent in the gap.

This clearly illustrates that the gap resonant response in the decay stage is driven by a

local quadratic wave-wave-structure coupling from the incident half-frequency wave

group, rather than the first harmonic components from the incident wave group which are

too long to produce a large persistent gap response.

Similar to the linear excitation case the damping in the decay stage for this frequency-

doubling excitation case is linear. Therefore, the second harmonic of the gap response is

a combination of different resonant modes and the damping coefficients for each mode

can be obtained using the method of Kumaresan and Tufts. As shown in Figure 7.17 both

Figure 7.18 Mode shape illustration of gap resonance driven by second harmonics

(mode 1 to mode 7) and third harmonics (mode 9 to mode 15). Lines: numerical results;

round symbols: experimental results.


7-23

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 7.19 Envelope of the fourth harmonic approximated using the scaled squared

envelope of the second harmonic for the gap responses measured at WG 1-4 for B2.

Figure 7.20 Time series of the crest- and trough-focused vertical velocities measured

at the gap centre (WG 4).

Figure 7.21 Separated odd and even harmonic spectra of the vertical velocity measured

at WG 4 for B2.


7-25

Figure 7.22 Separated zeroth to fourth harmonic component of the vertical velocity

measured at WG 4 for B2.

the numerical results obtained from CFD and potential flow models give an excellent

prediction of the resonant frequencies for each mode. In terms of damping coefficients,

B1 with coarse near-wall mesh clearly underpredicts the experiment. When the near-wall

mesh is refined, it can be seen the damping coefficients for B2 agree well with the

experimental results although there is still some underprediction. These results are

consistent with the linear excitation case.

Similar to the second harmonic wave field, obvious standing waves are found in the gap

for the third harmonic wave field (see the normalized third harmonic in the decay stage

in Figure 7.16 (c)). The first to the fifteenth mode shapes with corresponding resonant

frequencies are shown in Figure 7.18, in which the first to the seventh mode shapes are

identified from the second harmonic while the ninth to the fifteenth mode shapes from

the third harmonic. It is found the numerical results give excellent agreement with the

experimental results.

Given there is a generalized Stokes type perturbation expansion for the gap responses, it

is possible to approximate the envelope of the 𝑛th harmonic component by raising the

fundamental envelope (calculated as in Section 4.2.1) to the power 𝑛 and then fitting the

measured envelope of the 𝑛th harmonic component by a least-square method. Figure 7.19

shows the envelope of the fourth harmonic component approximated using the envelope

of the second harmonic component which is calculated by using the Hilbert transform. It

is found that the extracted fourth harmonic fits the scaled squared envelopes of the second

harmonic quite well. This means the numerical results are self-consistent in that the fourth

harmonic component is bound to the second harmonic component, although some

discrepancies between the numerical and experimental results are found for the fourth

harmonic.

To summarize, these numerical results for the free surface elevations in the gap give a

very good prediction of the experiment. This gives us the confidence to analyse other

quantities in detail, such as the velocity and hydrodynamic force on the boxes which will

be discussed in the following sections.

7.5 Harmonics of velocity

The time series of the crest- and trough-focused vertical velocities measured at the gap

centre in the mid-ship plane are shown in Figure 7.20. The sampling point is located at

0.08 m vertically below the mean free surface which always stays wet. It can be identified


7-27

from the time series that the odd harmonics dominate the excitation stage while the even

harmonics dominate the decay stage. Using the phase-inversion method, harmonic

structures up to the sixth can be identified in the power spectra shown in Figure 7.21. It

is remarkable that the second harmonic component is much larger than the first harmonic

component.

By digital filtering, the time series of the zeroth to the fourth harmonic are shown in

Figure 7.22. The fifth and sixth harmonics (with a maximum amplitude smaller than 2.0

mm/s and 1.4 mm/s) are very small and are not shown here. From the results it is seen

that for the vertical velocity the maximum amplitude of the second harmonic component

is 1.85 times that of the first harmonic component, whereas the ratio for the free surface

elevation is approximately 1. The envelopes for the fourth harmonic component are

obtained as the scaled squared envelopes for the second harmonic component, through a

least squares numerical fit method. The fitted envelope matches the time series of the

fourth harmonic component remarkably well, showing how the fourth harmonic ‘follows’

the second harmonic.

7.6 Harmonics of hydrodynamic force

Wave loading on offloading vessels is the major external force to be considered in the

design stage of offloading operability. Hydrodynamic forces induced by external waves

have a critical effect on the relative motions of the offloading vessels. Using the validated

results from B2 the hydrodynamic forces are obtained by integrating the pressure over

the time-varying wet surface of the boxes. Due to the symmetry condition in the mid-ship

plane only horizontal forces along the wave propagation direction 𝐹𝑥 and vertical forces

𝐹𝑧 are present. Figure 7.23 shows the time histories of the total horizontal and vertical

forces on the up- and down-wave boxes for a crest-focused wave group and the same

wave group inverted. It is evident that the up-wave box ‘shields’ the down-wave box in

the excitation stage – the maximum horizontal and vertical forces for the up-wave box

are 47% and 58% larger than the counterparts for the down-wave box respectively. The

vertical forces are at least two times larger than the horizontal forces.

By applying the phase-inversion method using the crest- and trough- focused forces, the

odd and even harmonics are obtained. The power spectra of the odd and even harmonics

for the horizontal and vertical forces are shown in Figure 7.24, where the harmonics are

(a) Horizontal force on the up-wave box

(b) Horizontal force on the down-wave box

(c) Vertical force on the up-wave box

(d) Vertical force on the down-wave box

Figure 7.23 Time series of crest- and trough-focused total horizontal and vertical force

signals for up- and down-wave boxes.


7-29

(a) Horizontal force

(b) Vertical force

Figure 7.24 Separated odd and even harmonic spectra of horizontal and vertical forces

on the up-wave box.

Figure 7.25 Separated zeroth to fourth harmonic component of horizontal force signals

on up- and down-wave boxes.


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Figure 7.26 Separated zeroth to fourth harmonic component of vertical force signals on

up- and down-wave boxes.

Figure 7.27 Separated odd and even harmonic spectra of pressure measured at 𝑆1 on

the up-wave box.

Figure 7.28 Separated second and fourth harmonic component of pressure measured at

𝑆1 on the up-wave box and corresponding envelopes.


7-33

well separated. The zeroth to the fourth harmonic components for horizontal and vertical

forces are shown in Figure 7.25 and Figure 7.26 respectively, where the forces on the up-

and down-wave boxes are compared in each subfigure. The shielding effect from the up-

wave box is most significant in the excitation stage due to the large incident wave loading.

In this stage the first harmonic is dominant, and the second harmonic is much smaller.

When the incident wave group has passed the second harmonic becomes more important.

This is more pronounced for the horizontal force, where the second harmonic can become

larger than the first harmonic over time owing to the very small radiation damping for the

second harmonic. This suggests that in a random sea of this amplitude, the maximum first

and second-order horizontal forces may be comparable in magnitude, since the present

wave group produces a force which is close to a NewWave in shape, while the second-

order force is quite different. It should also be noted that one of the original motivations

for developing ‘lid’ methods was the excessive mean force predictions of potential flow

codes (e.g. Buchner et al., 2001). There is no such issue here, where the low-frequency

forces are quite modest.

For the hydrodynamic forces on the boxes the scaled squared envelopes for the second

harmonic cannot match the time series for the fourth harmonic. This is because the

hydrodynamic forces on the boxes are determined by the flow in the gap and outside the

gap, and also affected by the time-varying free surface elevation. For example, the

contributions for the second harmonic are associated partly with second-order pressure

acting on the mean submerged surface of the boxes, and also with the effect of the first-

order pressure acting on the time-varying boundary at the free surface. To further confirm

this the harmonic structures for the pressure measured at a point on the up-wave box in

the gap 𝑆1 (–𝐵𝑔/2, 0, -0.08) m are checked. The harmonic spectra, and the second and

fourth harmonics with corresponding envelopes are shown in Figure 7.27 and Figure 7.28

respectively. Remarkably good agreement between the scaled squared envelopes of the

second harmonic component and the time series of the fourth harmonic component is

achieved, as expected.

7.7 Conclusions

In this chapter the numerical results of harmonic structures including the zeroth, first,

second and third harmonics for free surface elevation show excellent agreement with

experimental results. Using the validated numerical results, the higher harmonics for the

velocity and the hydrodynamic forces on the boxes are also successfully extracted. This

represents a highly successful application of CFD to a problem with linear viscous

dissipation but nonlinear free surface effects, both of which must be resolved to achieve

satisfactory results. CFD has previously been used to derive second harmonic responses

around non-resonant structures such as cylinders in relatively steep waves (Paulsen et al.,

2014). Here the strength of the second harmonic response is much stronger (so more

challenging to simulate correctly); even in waves which are not steep the second harmonic

responses are as large as the first. Given the success of reproducing higher harmonics

using the CFD method, it is expected that even higher harmonics could be well predicted

if the mesh resolution near the free surface was further refined, though their practical

significance may be limited. However, quadratic excitation of gap resonance does appear

to be a problem of practical relevance.

Significant shielding effect from the up-wave box is observed in the excitation stage, such

that the wave loading on the up-wave box can be up to 58% larger than the down-wave

box. In practical offloading scenarios, it may be preferable to arrange the LNG carrier on

the down-wave side (if it is not achievable to align the vessels parallel to the direction of

the predominant environmental forces) since its motions are more sensitive to large

hydrodynamic forces compared to the FLNG.

Chapter 8: 3D gap resonance between two fixed vessels with square corners

8-1

8 3D GAP RESONANCE BETWEEN TWO

FIXED VESSELS WITH SQUARE CORNERS

8.1 Introduction

According to the discussion in Chapter 4, the flow features and wave responses for 2D

gap resonance are distinct for round and square entrance shapes. For square entrance

shapes, flow separation will occur around the sharp corners. Zhao et al (2017) also

conducted a series of wave basin tests with models with square corners. In this chapter,

the experimental results of 3D gap resonance with square corners under the excitation of

transient groups are first presented, and then the experimental gap responses are

reproduced in the NWT. The numerical and experimental results differ by more than in

the round corner case, and both the resonant frequencies and damping coefficients are

overpredicted by the numerical model. Discussion is given concerning potential causes

of this difference which may prove useful for future study on this topic.

8.2 Physical setup

A series of experiments for 3D gap resonance in between vessels with square corners

were carried out in the State Key Laboratory of Ocean Engineering at Shanghai Jiao Tong

University by Zhao et al (2017). The vessel models used in the experiments are shown in

Figure 8.1, where bespoke strips were stuck on to the round corners of the boxes (which

were used in the previous experiments described in Chapter 5) to form square corners.

Otherwise, the configuration is similar to that for the boxes with round corners. The main

dimensions of the boxes are (𝐿𝑚, 𝐵𝑚, 𝐻𝑚, 𝐷𝑚)= (3.333, 0.767, 0.425, 0.185) m, where

𝐿𝑚, 𝐵𝑚, 𝐻𝑚 and 𝐷𝑚 are the length, width, height and draft of the model, respectively.

(a) Bespoke strip (b) Vessel model with bespoke strips

Figure 8.1 Snapshot of the vessel model in the experiment.

The boxes were fixed in parallel with a gap width of 𝐵𝑔=0.0667 m, which was selected

to scale to 4 m in the prototype. The fluid in the gap was set in motion by transient wave

groups, with peak frequency chosen according to the resonant frequency for the first mode

(𝑓𝑃=0.953 Hz) and a frequency approximately half of that (𝑓𝑃=0.514 Hz). These cases are

referred to as ‘Set C’ (or ‘linear excitation’) and ‘Set D’ (or ‘frequency-doubling

excitation’) in the following text, respectively. Seven wave gauges (referred to as ‘WG

1’ to ‘WG 7’) with the same positions as in the round corner setup were employed along

the gap to measure the gap responses. The sampling frequency of the experiments has

been increased from 25Hz to 40Hz. The free surface elevation measured at WG 4 with

and without the boxes in place are compared in Figure 8.2. Similar to the round corner

setup, the fluid in the gap oscillates with a beating pattern in the absence of the incident

waves, which is more obvious for Set C.

The gap responses for round and square corners measured at WG 4 are compared in

Figure 8.3. The round corner setup has larger maximum gap response, smaller damping

coefficients and larger resonant frequencies (see Table 6.3 and Table 8.1) than the square

corner setup. These results are consistent with the findings in Chapter 4.

8.3 Numerical results

To reproduce the experimental gap responses the incident wave groups used in the


8-3

(a) Set C

(b) Set D

Figure 8.2 Experimental free surface elevation measured at WG 4 with and without

boxes for Set C and D.

(a) Linear excitation

(b) Frequency-doubling excitation

Figure 8.3 Compasion of experimental gap responses measured at WG 4 for round and

square corners.

(a) time series


(c) phase spectrum


experimental results for Set C.


8-5

(a) time series


(c) phase spectrum


experimental results for Set D.

(a) time series


(c) phase spectrum

Figure 8.6 Numerically recreated trough-focused total incident wave group compared

to experimental results for Set D.


8-7

Figure 8.7 Comparison of gap responses measured at WG 4 using different domains

for Set C.

experiment were first recreated in the NWT using the linear iterative method shown in

Figure 8.4 for Set C, and Figure 8.5 and Figure 8.6 for Set D. For Set D the experimental

wave signal used for wave recreation in the NWT is set to zero after 𝑡=15s because the

reflections in the experiments (see Figure 8.2(b) time series without boxes) cannot be

recreated in the NWT. Both numerical results show excellent agreement with the

experimental data.

Following the meshing strategy for the 3D round corner setup, two meshes were used for

the simulations for Set C by changing the cell width of the first near-wall cell, i.e. Δ𝑥2=2

mm for Case C1 and Δ𝑥2=0.4 mm for Case C2. The other mesh dimensions shown in

Figure 5.3 are the same for the two cases with (Δ𝑥1, Δ𝑧1, Δ𝑦1) (7.5, 2.0, 15.8) mm. For

robustness and efficiency, the time step of the simulations was chosen to be runtime

adaptive with maximum Courant number not exceeding 0.5 (chosen based on

experience).

Symmetry boundary conditions were applied in the mid-ship plane to save computational

cost for C1 and C2. To check the effect of this, an additional simulation using an entire

domain without symmetry boundary conditions was carried out and the gap response

measured at the gap centre (WG 4) is compared to C1 shown in Figure 8.7. It is found the

agreement is very good.

The time series of gap responses measured at WG 1 to WG 4 for Case C1 and C2 are

compared to the experimental data for Set C in Figure 8.8. Several points are apparent.

Firstly, mesh size of Δ𝑥2 has insignificant influence on the gap responses that the time

series for C1 and C2 are in very good agreement. Secondly, the agreement of phase

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 8.8 Responses at various locations along the gap for different meshes for Set C.


8-9

Figure 8.9 Comparison of gap response measured at WG 4 using different methods for

Set C. ‘Expt.’ stands for ‘experimental data’, ‘Potential’ stands for ‘potential flow

model’.

𝑓1 𝜉1 𝑓3 𝜉3 𝑓5 𝜉5

C1 0.957 11.8 1.057 12.2 1.180 10.1

Experiment 0.938 11.8 1.045 13.5 1.174 14.1

Molin et al. (2002) 0.933 N/A 1.059 N/A 1.192 N/A

Table 8.1 Frequencies 𝑓 (Hz) and damping coefficients 𝜉 × 103 for each mode for Set

C.

𝑓1 𝜉1 𝑓3 𝜉3 𝑓5 𝜉5

D1 0.960 9.1 1.058 9.7 1.183 11.6

Experiment 0.939 9.2 1.048 11.6 1.176 18.9

Table 8.2 Frequencies 𝑓 (Hz) and damping coefficients 𝜉 × 103 for each mode for Set

D.

between CFD and experimental results is very good for the first pulse in the excitation

stage (𝑡 < 5 s). After that, the CFD results begin to lead in phase, with accumulation over

time. At the end of the time window in the figure, the experimental and CFD results are

approximately 180 ° out of phase. For comparison, the time series of gap response

predicted using a linear potential flow model (DIFFRACT, Sun et al 2015) is included in

Figure 8.9. It is found that the CFD results give a much better prediction compared to the

potential flow results. It is also interesting to note that, in terms of phase lead, ‘Potential

flow model’ > ‘CFD model’ > ‘Experiment’.

Applying the method of Kumaresan and Tufts (Kumaresan & Tufts, 1982) to the linear

time series, the resonant frequency and damping coefficient for each mode are listed in

Table 8.1. It is found the numerical damping coefficients generally agree with the

experimental results well. Specifically, the damping coefficient for the first mode is the

same as the experimental result while those for the third and fifth modes underpredict the

experimental results. The frequency difference for the first mode for the square corner

setup (2%) is larger than the round corner setup (0.1%) but the frequency differences for

the third and the fifth mode are similar. This means that the phase difference is mainly

determined by the frequency difference in the first mode given that the round corner setup

has a much better phase agreement.

A mesh D1 with Δ𝑥2=2 mm was used for the simulations for Set D. The crest-focused

and trough-focused gap responses measured at WG 1-4 are shown in Figure 8.10 and

Figure 8.11. The second harmonic component is extracted using the phase inversion

harmonic extraction method and the results are shown in Figure 8.12. The same trend in

phase between the experimental and the numerical results are found in the time series.

Applying the method of Kumaresan and Tufts to the time series for the second harmonic

component, the resonant frequency and damping coefficient for each mode are obtained

as listed in Table 8.2. It is found the numerical results still over-predict the resonant

frequencies. In terms of damping coefficients, the numerical results give very good

agreement with the experiment for the first and third modes, but the fifth mode is

obviously underpredicted by the CFD model (though the amplitude of the fifth mode is

small and is of less importance). Two caveats of obtaining the damping coefficients are

discussed here. Firstly, the method of Kumaresan and Tufts assumes linear superposition

of modes. If the damping is quadratic in the velocity for the square corner case, this


8-11

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 8.10 Comparison between experimental and numerical crest-focused total gap

responses measured at various locations along the gap for D1.

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 8.11 Comparison between experimental and numerical trough-focused total gap

responses measured at various locations along the gap for D1.


8-13

(a) WG 1

(b) WG 2

(c) WG 3

(d) WG 4

Figure 8.12 Separated second harmonic component separated at various locations along

the gap for D1.

(a) (b)

Figure 8.13 Dependency of the resonant frequency for each mode on (a) gap width (b)

draft using Molin’s method.

analysis will not perform as well as in the round corner (linear) case. Therefore, the

method of Kumaresan and Tufts must be treated with some caution, though the trends in

frequencies and damping are consistent with the time series. Secondly, if the damping is

quadratic it means the drag force is proportional to the square of the amplitude of gap

oscillation, but it enters odd harmonics when expanded into harmonic components (see

Fitzgerald et al., 2014). This breaks the Stokes’ structure assumed to exist for the phase

inversion harmonic extraction method.

8.4 Discussion

From the numerical results for Set C and Set D, the resonant frequencies are always over

predicted by the numerical model. Given the good prediction of the resonant frequencies

(or phase) for the round corner setup, it is quite unexpected that the resonant frequencies

for the square corner setup are not well predicted by the CFD model. This means there

might be some flow features existing in the experiments which are not captured or

resolved in the CFD model. To investigate the differences in more detail, several possible

causes are considered below.


8-15

(a) 𝐴0=0.05m (b) 𝐴0=0.24m

Figure 8.14 Illustration of initial water column height.

Firstly, the square corners of the vessel models used in the experiments were formed by

sticking bespoke strips onto the bilges. This was accompanied by the model being

removed and replaced in the basin. It is possible that this replacement could have resulted

in an experimental error in gap width and/or draft on the order of 1 mm (or 1.5% and

0.5% of the intended gap width and draft, respectively). To roughly estimate how much

change in gap width or draft will induce the difference in resonant frequencies listed in

Table 8.1, the theoretical method proposed by Molin et al. (2002) was used. Under the

so-called ‘single-mode approximation’ which treats the water in the gap as a solid body

when the draft 𝐷𝑚 is large with respect to the gap width 𝐵𝑔, the resonant frequencies can

be obtained as

𝜔𝑛02 ≃ 𝑔𝜆𝑛

1 + 𝐽𝑛0 tanh 𝜆𝑛𝐷𝑚𝐽𝑛0 + tanh𝜆𝑛𝐷𝑚

(8.1)

with

𝐽𝑛0 =2

𝑛𝜋2𝑟{∫

𝑟2

𝑢2√𝑢2 + 𝑟2[1 + 2𝑢 + (𝑢 − 1) cos(𝑛𝜋𝑢)

1

0

−3

𝑛𝜋sin(𝑛𝜋𝑢)] d𝑢 −

1

sin𝜃0+ 1 + 2𝑟ln

1 + cos𝜃01 − cos𝜃0

}

(8.2)

where 𝜆𝑛 = 𝑛𝜋/𝐿𝑚, 𝑟 = 𝐵𝑔/𝐿𝑚, and tan 𝜃0 = 𝑟−1. 𝐵𝑔, 𝐷𝑚, 𝐿𝑚 and 𝑛 are the gap width,

draft of boxes, length of boxes and mode number, respectively. The resonant frequencies

calculated using this method based on (𝐵𝑔, 𝐷𝑚, 𝐿𝑚)=(0.0667, 0.185, 3.333) m are listed

(a) Coarse mesh (b) Fine mesh

Figure 8.15 Illustration of mesh near the gap entrance.

in Table 8.1. It is found that Molin’s method predicts the resonant frequency reasonably

well for the first mode, but over-predicts the resonant frequencies for the third mode and

the fifth mode. According to Eq. (8.1), the dependencies of the resonant frequency for

each mode on gap width (while fixing draft) and draft (while fixing gap width) are shown

in Figure 8.13(a) and (b) respectively, in which the curves can be fitted using a linear

function over the range of interest. It is found that a 1% change in gap width while fixing

draft will result in 0.14%, 0.08%, 0.04% change in the resonant frequencies for the first,

third and fifth modes respectively. In turn, a 1% change in draft while fixing gap width

will result in 0.32%, 0.28% and 0.19% change in the resonant frequencies for the first,

third and fifth modes respectively. This means that the resonant frequencies are more

sensitive to a change in draft than a change in gap width. Taking C2 as an example, the

resonant frequency for the first mode over predicts the experiment by 1.5%, meaning a

10.5% change in gap width or 4.7% change in draft would be required to cause such

frequency difference, which is very unlikely to occur in the experiment.

A second possible cause of the frequency over-prediction is under-resolution of the flow

around the entrance of the gap, with possible effects on the ‘added mass’ or volume of

water oscillating within the gap. To check the effect of the mesh under the boxes (whilst

minimising computational time) a series of 2D decay tests were carried out by releasing

a water column with predefined initial height in a gap formed by two fixed boxes. The

2D configuration was the same as the cross-section in the mid-ship plane for the 3D gap

resonance problem. Two initial water column heights were used as illustrated in Figure

8.14, i.e. 𝐴0=50 mm, which is approximately the level of the maximum gap response for

the 3D experimental results (referred to as ‘standard case’), and 𝐴0=240 mm, which is an

extreme case with an initial elevation set equal to the top of the boxes (referred to as


8-17

(a) 𝐴0=50 mm, ‘standard case’

(b) 𝐴0=240 mm, ‘extreme case’

Figure 8.16 Time series of free surface elevation in the gap with different meshes.

‘extreme case’). Two meshes were used for both cases as shown in Figure 8.15, i.e. a

coarse mesh with near-wall cell width Δ𝑥2=2 mm, and a fine mesh with near-wall cell

width Δ𝑥2=0.1 mm and refined mesh under the boxes. After release, the water column in

the gap carries out decaying oscillatory motions as shown in Figure 8.16. For the standard

case it is apparent that mesh refinement under the boxes appears to have a negligible

effect on damping or phase. Figure 8.17 shows the vortex structures for the standard case

with different meshes. At 𝑡=0.800 s, the vorticity contour for the coarse and the fine

meshes exhibit similar vortex structures – two pairs of symmetric vortices are observed

near the gap entrance. At 𝑡=1.525 s the pattern of vortices is quite different for different

meshes – the fine mesh is able to capture vortices of smaller scale and hence the vortex

structures are more complex. However, the vortices still remain symmetric for different

meshes. At 𝑡=1.850 s the pattern of vortices is distinct for the two meshes – the vortex

structures for the fine mesh become asymmetric while the coarse mesh still has symmetric

vortex structures. As time passes, the contour of vorticity for the two meshes become

more different. However, the disparity in vortex structures appears to have little effect on

the time series of free surface elevation in the gap, which are almost coincident. For the

extreme case, the coarse mesh lags the fine mesh for the first few cycles (which cannot

(a) Coarse mesh, 𝑡=0.800s (b) Fine mesh, 𝑡=0.800s

(c) Coarse mesh, 𝑡=1.525s (d) Fine mesh, 𝑡=1.525s

(e) Coarse mesh, 𝑡=1.850s (f) Fine mesh, 𝑡=1.850s

Figure 8.17 Vortex structures for 2D decay test with an initial water height of 50 mm.

𝜔𝑦


8-19

Figure 8.18 Comparison of time series of free surface elevation in the gap for initial

elevation and depression of 50 mm.

explain the over-prediction of the resonant frequency for the 3D gap resonance problem,

since the trend is opposite) and becomes in phase with the fine mesh near the end of the

time window. This result suggests that mesh refinement beneath the gap does not explain

the discrepancies in phase between the NWT and the laboratory measurements.

For completeness, to check the difference between flow contraction and expansion in the

water column release experiments, an additional case was carried out by depressing the

initial free surface in the gap by 50 mm. The resulting time series for the gap free surface

is inverted and compared to the result for the standard case with an initial elevation of 50

mm shown in Figure 8.18. It is found the agreement of both results is very good, meaning

flow expansion and contraction have similar effects for the decay test.

Finally, to further explore the observed differences in phase, the same CFD model as

above was used to simulate the experimental 2D decay tests conducted by Maisondieu et

al. (2001). As shown in Figure 8.19, in these experiments the pressure above the free

surface in the gap is reduced, resulting in a rise in water level in the gap. Once stabilized,

a hatch on top of the gap is opened to bring the pressure in the gap back to the atmospheric

pressure. In this way, the water column is released and starts to carry out decaying

oscillatory motions in the gap. As shown in the figure, the draft of the model was chosen

as 𝐷𝑚=200 mm and two initial water column heights (𝐴0=50 mm and 𝐴0=100 mm) were

used in the experiments. Using the present CFD model, the time series of free surface

elevation measured at the gap centre is compared to the experimental data shown in

Figure 8.20, where the CFD results obtained by Sauder et al. (2010) are included for

comparison. It is found that the present CFD results are in excellent agreement with

Figure 8.19 Experimental setup of Maisondieu et al. (2001).

Sauder et al’s CFD results for the decay test with 𝐴0=50 mm, both of which lead the

experimental data in phase. For the decay test with 𝐴0=100 mm, the two CFD results

agree well in phase and both lead the experimental data, though some discrepancies occur

near the end of the time window with respect to the amplitude of oscillation. It should be

noted that the decay rate here is underestimated by the CFD models, which is opposite to

the CFD results for Set C. Consequently, the comparison with the 2D decay tests

conducted by Maisondieu et al. (2001) appear to show similar characteristics (in phase)

to that observed in the 3D simulations presented earlier in this chapter.

A turbulence model has not been used to simulate the 3D gap resonance problem with

square corners, since a turbulence model introduces extra dissipation to the system by

inclusion of turbulence eddy viscosity. Without an appropriate model to accurately

predict this viscosity, it is possible that this viscosity may deteriorate the agreement

between the experimental and numerical results. Nevertheless, it is worth noting that

vortex structures shed from the corners of the box during inflow into the gap (see Figure

8.17). These structures interact with the boundary layer, which may lead to locally

turbulent flow which cannot be adequately resolved by the existing CFD model.

8.5 Conclusions

The 3D experimental results for resonant fluid motions in the gap formed by two fixed

boxes with square corners have been presented in this chapter. Following the same

procedures as for the round corner setup, CFD simulations have been carried out for the

square corner setup but the results are not as good as the round corner setup, in that the


8-21

(a) 𝐴0=50 mm (b) 𝐴0=100 mm

Figure 8.20 Time series of free surface elevation measured in the middle of the gap.

resonant frequencies are over-predicted (the disagreement of resonant frequency has also

been reported by Sauder et al., 2010). The agreement of phase between the experimental

and numerical time series is good in the excitation stage but becomes worse in the decay

stage. Hence it seems that this phenomenon only is most apparent in decay tests with

square corners and is not a problem for gap resonance driven by regular waves (since the

frequency is set by the regular incident waves), e.g. the CFD model in Chapter 4 has

demonstrated good performance in predicting the 2D average gap responses under the

excitation of regular waves with square corners. In practice, designs are based on response

curves, and the present model is a vast improvement on potential flow theory. Therefore,

this is less of a problem for the prediction of the more realistic situation.

Several possible causes for the phenomenon have been explored. From the experimental

point of view, error in gap width or draft is not expected to result in the frequency

difference. From the numerical point of view, it is found that mesh refinement does not

improve the phase difference between the numerical and the experimental results. In

addition, both the numerical results obtained from the present CFD model and that from

Sauder et al. (2010) over-predict the resonant frequency for the 2D experimental decay

tests conducted by Maisondieu et al. (2001). This means such over-prediction of resonant

frequency appears to be a general problem – there must be some phenomenon in the

experiment that is not well captured by the CFD models. A turbulence model has not been

used to simulate the gap problem with square corners, which may be a viable option but

requires a proper modelling of turbulence eddy viscosity.

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Chapter 9: Conclusions

9-1

9 CONCLUSIONS

9.1 Introduction

This thesis is concerned with using CFD to estimate the resonant water motions in the

gap between two fixed boxes arranged in a parallel configuration at lab-scale, which is

representative of a side-by-side offloading scenario. By solving the Navier-Stokes

equations combined with the free surface captured by VOF method, both 2D and 3D

numerical simulations under the excitation of regular waves and transient focused wave

groups respectively, have been carried out to successfully reproduce the physical

experiments. Flow field information in high spatial and temporal resolution has been

utilized to explain the phenomena found in the experiments. Unlike most previous

numerical studies on this topic which are concerned with predicting the time-averaged

responses in the gap, the reproduction of the much more demanding time series of gap

responses included in this thesis has made the study very ambitious and challenging,

especially for a 3D configuration where multiple spatial modal structures are present and

the flow-structure interactions are more complex. The mesh size in the boundary layer

can be 𝑂(104) times smaller than the wavelength scale, which has led to significant

computational effort being required for the simulations as well as the post-processing. In

this chapter, the main conclusions of the work are summarised according to the aims set

out in Chapter 2 and possible future research is suggested.

9.2 Conclusions

9.2.1 Two-dimensional gap resonance between two fixed vessels

• Mesh convergence studies for 2D gap resonance under the excitation of regular

waves with round and sharp corners have been carried out. It is found that the

time-averaged gap responses for sharp corners are not affected significantly by

the near-wall cell width normal to the wall, which is consistent with the finding

by Kristiansen & Faltinsen (2012). On the contrary, the gap responses for round

corners are highly sensitive to the near-wall cell width normal to the wall. This

implies wall friction is more important for round corners than sharp corners. The

response curve of gap responses with square corners is in good agreement with

the experimental results obtained by Saitoh et al. (2006).

• With respect to the amplitude of resonant gap responses with different entrance

shapes, it has been shown that the round entrance results in the largest response,

the bilge keel the smallest and the square entrance an intermediate response. This

is consistent with the results for round and square corners found by Moradi et al.

(2015). Since the response is damped by entrance and exit energy losses of flow

in the gap, these rankings are analogous to fluid flowing from/to a reservoir

into/from a pipe with round, protruded and square entrance shape. By utilizing an

equivalent linearized viscous damping model (Tan et al., 2018), the damping

coefficients are found to be smallest for the round entrance shape while largest for

the bilge keel entrance shape, which explains the difference in resonant amplitude

between the gap responses for different entrance shapes. The resonant frequencies

for square and bilge keel entrance shapes are close to each other, while that for

the round entrance shape is larger. This is because there is little change in ‘added

mass’ when the entrance shape is altered from square corner to bilge keel, whereas

the trapped volume of fluid in the gap is reduced when the entrance shape is

modified from square corner to round corner.

• With increasing incident wave amplitude (or equivalently amplitude of gap

oscillation), the resonant wave elevation in the gap will increase while the peak

of response curves will decrease for all entrance shapes, which is related to the

increasing damping coefficients in the system. When the amplitude of gap

oscillation increases, the resonant frequency exhibits different trends for different

entrance shapes. This can be explained by the nonlinear damping effect and the

Duffing-like behaviour with a hardening spring associated with the nonlinear free

surface conditions. The former will result in a decrease in resonance frequency

while the latter will cause the opposite trend.

• Instead of running a series of long-time regular wave tests, simulations of a few

cycles of decay following release of an initial water column in the gap can be used

to predict the resonant frequency of gap responses. The results of this method have


9-3

shown very good agreement with the resonant frequencies identified from the

response curves for different entrance shapes, amplitudes of gap oscillation and

water depths.

• Scaling effects are investigated approximately by varying fluid viscosity in the

numerical simulations. It is found the scaling effects (or effects of varying

viscosity) lead to greater changes in response amplitude for round corners than

square corners. Though the present model may not be able to resolve all fluid

structures in the large-scale (low viscosity) simulations, this does suggest one

should be careful when interpreting results in the laboratory to predict results in

the field, especially for vessels with round corners.

9.2.2 Development of a numerical wave tank to simulate 3D gap

resonance between two fixed vessels with round corners

• A 3D NWT for gap resonance with round corners has been successfully

established by systematically optimizing the mesh topology and extent, together

with appropriate boundary conditions for the purpose of replicating the physical

experiments conducted by Zhao et al. (2017). The meshing strategy is based on

physical insights to resolve both the wave and boundary layer scales. Within the

computational domain, the boundary layer thickness will be the smallest length

scale and require the finest mesh. To improve efficiency the key is to have the

highest possible aspect ratio for this fine mesh, so that upstream of the boxes the

propagation of the incident waves can be simulated reliably without overly fine

vertical resolution.

• Various iterative methods have been employed to recreate the experimental

incident transient wave groups. The linear iterative method shows the best result,

by adjusting the amplitude and phase information based on the difference between

the recorded linear wave signal in the NWT and the desired linear wave signal.

9.2.3 Linear excitation of 3D gap resonance between two fixed vessels

with round corners

• Without any parametric tuning, the 3D numerical time series of gap responses,

resonant amplitudes and frequencies, and mode shapes have shown very good

agreement with the experimental data obtained by Zhao et al. (2017). It is

demonstrated that the near-wall cell width normal to the wall must be sufficiently

small to resolve the boundary layer to accurately reproduce the experimental

results. As the near-wall mesh becomes finer, the time series of gap responses

become closer to the experimental results.

• Interrogation of flow field information finds that nearly all the viscous dissipation

is confined to a thin layer surrounding the boxes, meaning the contribution from

flow separation is negligible and wall friction is the main source of damping.

Using the validated numerical data, the near-wall boundary layer can be well

predicted by solving the Stokes equation using the time-varying pressure

measured at a position near the wall in the gap but still outside the boundary layer.

The near-wall boundary layer has been decomposed into modal components

which gives excellent agreement with the theoretical Stokes boundary layer

solution. Collectively, these pieces of numerical evidence strongly support the

postulation by Zhao et al. (2017) that the viscous damping beyond the radiation

damping affecting the decaying gap oscillations in their experiments was linear.

• The viscous dissipation in the boundary layer has been computed, separated into

modes and added to the modal radiation damping contribution to produce modal

global damping coefficients in reasonable agreement with those estimated from

the decay of the free surface in the simulation. Both sets of damping coefficients

show modestly good agreement with the experimental values.

• The vertical and spanwise velocities can be regarded as a superposition of various

mode shapes as well. The mode shapes for the vertical velocity follow the free

surface elevation, while the mode shapes for the spanwise velocity are inverse.

The vertical velocity is found to be dominant in the gap, meaning the

simplification of a 3D gap resonance model to its 2D counterpart is somewhat

possible.

9.2.4 Frequency-doubling excitation of 3D gap resonance between two

fixed vessels with round corners

• The experimental results of frequency doubling excitation including first, second

and third harmonics of free surface elevation have been successfully reproduced

in the NWT. The numerical fourth harmonic component is self-consistent in a way


9-5

that the scaled squared envelopes of the second harmonic component match the

time series of the fourth harmonic component remarkably well. This represents a

highly successful application of CFD to a problem with linear viscous dissipation

but nonlinear free surface effects, both of which must be resolved to achieve

satisfactory results. This is expected to have important application to swell

excitation of gap resonance.

• Mode shapes for the free surface are successfully captured up to the fifteenth and

agree well with the experimental data. The first to the seventh mode shapes are

identified from the second harmonic component, while the ninth to the fifteenth

mode shapes are identified from the third harmonic component.

• The harmonic structures for the velocity in the gap and forces on the boxes have

been extracted successfully using a phase-inversion harmonic extraction method.

Significant shielding from the up-wave box is observed in the excitation stage

such that the wave loading on the up-wave box is much larger than that on the

down-wave box.

9.2.5 3D gap resonance between two fixed vessels with square corners

• 3D gap resonance with square corners excited by transient wave groups has been

simulated in the NWT and the results are compared to the available experimental

data obtained by Zhao et al. (2017). It is found that the CFD model generally

overpredicts the resonant frequencies and damping coefficients, with the

discrepancy between numerical results and experiment substantially greater than

for the round corner cases. The overprediction of resonant frequencies will result

in a phase lead accumulating over time. Several possible causes for the

overprediction of resonant frequencies have been investigated. It is found that the

error in gap width or draft in the experiment is very unlikely to cause such

frequency difference and refining the mesh in the CFD model does not improve

the results.

• 2D decay tests of a water column with predefined initial heights in a gap with

square corners have been simulated using the present CFD model. The results are

compared to the experimental data by Maisondieu et al. (2001) and also the

numerical results by Sauder et al. (2010). It is found the present CFD results agree

well with those obtained from Sauder et al. (2010), both of which lead the

experimental data over time. This means the phenomenon is not a coincidence and

may generally exist for decay tests in a gap with square corners. The results imply

that there are flow behaviours which exist in the laboratory but are not captured

by the CFD models used for comparison.

9.3 Future research

9.3.1 Gap resonance at larger scales

This thesis primarily contributes to understanding resonant fluid behaviours in gaps at

lab-scale. Ultimately the gap resonance problem has industrial relevance at a substantially

larger scale and conclusions drawn from lab-scale in this thesis may need to be revisited

at larger scales. For gaps with round corners at lab-scale, it is expected the viscous

damping is dominated by laminar boundary layer damping, though flow separation may

sometimes occur. As the scale is increased, the boundary layer will inevitably become

turbulent and flow separation will contribute more of the damping. For square (sharp)

corners flow separation will occur in the lab as well as in the field.

In this thesis, the scaling effect is touched on via varying the fluid velocity in 2D

numerical simulations. The numerical results suggest that the scaling effect is more

important for round corners than square corners since the gap responses for round corners

increase dramatically when the scale is increased towards a prototype scale. However, it

should be noted that the same mesh (converged at lab-scale) has been used to study the

2D gap resonance problem at different scales (or with different viscosity). This is not a

problem for scales smaller than the lab-scale, but the mesh is not expected to resolve all

aspects of the flows at larger scales. Therefore, further study should be carried out to

consider gap resonance at larger scales with a more appropriate numerical setup

(turbulence models may be needed) to verify these results.

It is favourable to investigate scaling effects on 2D gap resonance in the first place,

considering it is much less computationally costly than its 3D counterpart. For 3D gap

resonance at larger scales, it would be tempting to start simulations with square corners

first, if the gap responses at larger scales are not affected much by the mesh in the gap (it

is reasonable to have such expectation given the 2D time-averaged gap responses with

square corners excited by regular waves have smaller scaling effects and are almost

independent of the mesh in the gap at lab-scale). This may allow us to use a relatively

coarse mesh to simulate the problem, which is at least 20 times faster than a fine mesh,


9-7

based on the experience gained here. It can be much more difficult to simulate 3D gap

resonance with round corners at larger scales since the mesh in the gap should be

adequately fine to resolve turbulent boundary layers and flow separation at a point

dependent on the Reynolds number. Ultimately field data collected from a full-scale

facility would be invaluable and serve to shed some light on these fundamental issues

associated with scaling.

9.3.2 3D Gap resonance with different headings

Only beam sea conditions are considered in this thesis. In real sea states the offloading

vessels are exposed to external waves incident from different headings. For headings

other than beam sea condition both odd (symmetric) and even (antisymmetric) mode

shapes are present along the gap. To enable CFD simulations in a NWT for different

headings, several points are important. Firstly, oblique sea conditions break down the

symmetry in geometry (as in beam sea conditions), preventing the use of symmetry

boundary conditions to save computational cost. Secondly, considering wave reflection

caused by incident waves interacting with the vessels and radiation from the first mode

(of most concern) can have larger influence on the responses in the gap for head and

oblique sea conditions, wave absorption at the side-walls are required. Finally,

unstructured mesh may be needed for oblique sea conditions and the performance of

resolving, e.g. wave as well as boundary layer scales, should be examined.

9.3.3 Gap resonance with moving vessels

This thesis is concerned with the gap resonance problem with two fixed vessels. For

industrial applications the amplitude of the resonance during operations may be important

in its own right, or due to its coupling with vessel motion. For a moving body there is an

addition nonlinearity caused by the change of gap width in time, which is not investigated

in this study. Therefore, it is desirable to extend the scope of the work to consider gap

resonance with moving vessels. To this end, dynamic mesh should be used. To realise

motions of rigid bodies within the framework of OpenFOAM, many researchers choose

to extend the two-phase solver ‘interDyMFoam’ with dynamic mesh handling coupled

with a motion solver. Higuera et al. (2015) and Dao et al. (2018) modelled physical

movements of piston wave paddles and flap paddles, respectively, to realise wave

generation in a NWT. Schmitt and Elsaesser (2015) simulated the rotational motion of a

flap-type oscillating wave surge converter in waves by making extensions to the

‘interDyMFoam’ libraries in terms of equations of motion and mesh motion algorithm.

The numerical model reproduces the experimental data within the bounds of experimental

accuracy. Devolder et al. (2017) simulated free decay tests and regular wave tests for

heaving wave energy converters, and the numerical results show reasonable agreement

with the experimental data.

Despite the above applications of modelling rigid body motions using OpenFoam, it

should be admitted that dynamic mesh is more difficult to handle than static mesh and

sometimes numerical stability is a problem. For the gap resonance problem, simulations

with moving vessels are more challenging than for a single body case in that the mesh

deformation can be problematic due to the narrow gap width and should be treated with

caution. Hence, development of a reliable solver as well as mesh algorithm is required

for the successful simulation of gap resonance with moving vessels. To break down the

complexity, a step-by-step procedure should be carried out, by considering the motion of

one vessel first and releasing the degrees of freedom one by one.

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Documents

7 FREQUENCY-DOUBLING EXCITATION OF 3D GAP RESONANCE