Conditions and phase shift of fluid resonance in narrow gaps of bottom mounted caissons

This paper studies the viscid and inviscid fluid resonance in gaps of bottom mounted caissons on the basis of the plane wave hypothesis and full wave model. The theoretical analysis and the numerical results demonstrate that the condition for the appearance of fluid resonance in narrow gaps is kh=(2n+1)π (n=0, 1, 2, 3, …), rather than kh=nπ (n=0, 1, 2, 3, …); the transmission peaks in viscid fluid are related to the resonance peaks in the gaps. k and h stand for the wave number and the gap length. The combination of the plane wave hypothesis or the full wave model with the local viscosity model can accurately determine the heights and the locations of the resonance peaks. The upper bound for the appearance of fluid resonance in gaps is 2b/L<1 (2b, grating constant; L, wave length) and the lower bound is h/b≤1. The main reason for the phase shift of the resonance peaks is the inductive factors. The number of resonance peaks in the spectrum curve is dependent on the ratio of the gap length to the grating constant. The heights and the positions of the resonance peaks predicted by the present models agree well with the experimental data.


Introduction
The coastal and marine structure may be composed of large number of caissons, which are vertically situated in water of uniform depth. The water wave interaction with an array of bottom mounted caissons is studied by several researchers. Dalrymple and Martin (1990) have investigated the wave diffractions of offshore detached breakwaters. Porter and Evans (1996) obtained two singular integral equations for the pressure and the velocity through a gap. Abul-Azm and Williams (1997) examined the oblique wave interaction with offshore breakwaters. Fernyhough and Evans (1995) studied the case of a periodic array of rectangular blocks. Peter and Meylan (2007) investigated wave scattering by a semi-infinite periodic array of arbitrary bodies.
In the last two decades, following the development of the offshore technology, the assembly of large caissons with small gaps has been commonly used in oil and gas off-loading facilities and oil storage. Side-by-side operations adopted in production must now consider environmental conditions, operation procedures and so on. Amongst them, the wave resonant phenomenon in a narrow gap is one of the most important topics. The first analytical and experimental results of wave resonance in gaps of bottom mounted and floating caissons were probably obtained by Miao et al. (2000Miao et al. ( , 2001 using an asymptotic matching technique and Saitoh et al. (2002) and Iwata et al. (2007) by using laboratory tests.
Because the floating structures are widely used in actual production, the fluid resonance in a narrow gap of the floating caissons has received substantial attention. In general, numerical methods based on the linear potential theory are widely used to study fluid resonance in narrow gaps of multiple floating bodies (e.g., Hong et al., 2005Hong et al., , 2013Lewandowski, 2008;Sun et al., 2010). However, discrepancies between linear potential results and measured free-surface elevations are significant. To overcome the potential theory being unable to model the viscous effects (skin friction and flow separation on the gaps), an external damping factor was introduced in the gaps. Thus, a number of very effective and powerful numerical simulation methods are de-veloped. Buchner et al. (2001) used the damping lid technique. Li et al. (2005) developed a modified scaled boundary element method. Pauw et al. (2007) applied the damping lid method for the investigation of resonant effects. Faltinsen (2009, 2010) compared their experimental data with results from a two-dimensional numerical analysis using a vortex tracking method. Sun et al. (2010) utilized a 3D program DIFFRACT to simulate firstand second-order resonant waves between adjacent barges. Lu et al. (2010) investigated the effect of adding an artificial damping to the momentum equations. More recently, Lu et al. (2011aLu et al. ( , 2011b have modelled the fluid resonance in narrow gaps and fluid forces on multi-bodies. Kristiansen and Faltinsen (2012) analysed the gap resonance by a new domain-decomposition method combining potential and viscous flow. Liu and Li (2014) obtained a semi-analytical solution for gap resonance by adding the artificial resistance force on the gap-free surface and the boundary between the dissipative domain and the non-dissipative domain. Yeung and Seah (2007) studied Helmholtz and higher-order resonance in the gaps between twin floating bodies. Most recently, Sun et al. (2015) have investigated wave driven free surface motion in the gap. Pessoa et al. (2015) investigated the coupled motion responses in waves of sideby-side LNG floating systems by numerical study. Perić and Swan (2015) conducted an experimental study of the wave excitation in the gap. Moradi et al. (2015) presented the effect of inlet configuration on wave resonance in the narrow gap of two fixed bodies. Watai et al. (2015) introduced an external damping factor and improved the numerical convergence by ranking time-domain method simulations. Feng and Bai (2015) described fully nonlinear waves by separating the contributions from incident and scattered waves.
The terminal gap problem has analogies to the moonpool problem, which has been treated in the frame of linear potential theory by Molin (2001) for infinite water depth and Faltinsen et al. (2007) for finite water depth.
For the appearance conditions of fluid resonance in narrow gaps, theoretical analysis and physical model tests in the gaps of bottom mounted and floating caissons conducted by Miao et al. (2001), Saitoh et al. (2002Saitoh et al. ( , 2008, and Iwata et al. (2007) show that the condition of fluid resonance in the narrow gap can be approximated by sin(kh)=0, kh=nπ (n=1, 2, 3, …), fundamental mode near kh=2.9 or khtanh(kd)=1 (Iwata et al., 2007), where d was the water depth.
The majority of the results mentioned above have concentrated on the prediction of free surface elevations in the gaps between floating bodies, and the results obtained mainly by the numerical method are very rich and effective. However, there is no a priori method of determining the coefficient of the damping term unless being calibrated by experimental tests. Further investigation of the multi-body hydrodynamics within the narrow gaps is still necessary. It is concluded that the following questions remain to be solved.
(1) The previous work suggests that the ability of linear diffraction can accurately predict the height and location of the resonance peaks without doubt and a monochromatic wave field is surely not a good representation of the ocean surface.
Therefore, a suitable wave model needs to be used.
(2) Theoretical arguments on the appearance conditions of fluid resonance in narrow gaps are not sufficient and need to be further investigated. On the other hand, the number of the resonance peaks in the spectrum curve and the reason (and value) of the phase shift of the response peak are still not clear. What is the influence of the geometrical parameters (e.g., the width of a caisson and a gap, length of a caisson or a gap) on the fluid resonance?
(3) It is very important to identify a good viscosity model. A reasonable viscosity model can explain many physical phenomena for the fluid resonance in gaps, for example: What is the relationship between the maximum height and the position of the resonance peak in a spectrum curve? What causes lead to the phase shift of the response peak? What is the influence of the phase shift on the height of the resonance peak?
(4) Whether there is a difference in the resonance peak number calculated by the viscosity model and by the potential flow model?
This paper presents more details on the theoretical derivation method and the numerical results for the inviscid and viscid fluid resonance in the gaps of bottom mounted caissons on the basis of the plane wave hypothesis and the full wave model.

Theoretical formulae of fluid resonance in gaps
The structures consist of equally spaced caissons. The width and thickness of each caisson are A and h; the gap width between two adjacent caissons is 2a. The caissons are vertically located in water of a uniform depth d, and Cartesian coordinates (x, y, and z) are employed with the origin located on the still-water level at the centre of the gap. The positive x-direction coincides with the wave direction, the yaxis is parallel with the seaward side of the caissons, and the z-axis is directed vertically upward, as denoted in Fig. 1.
The row of the caissons is normally subjected to plane ZHU Da-tong et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 724-735 725 waves with the wave height H and frequency ω. The whole fluid domain is divided into three regions: the sea field outside the row of the caissons (x≤0), the sea field inside the row of the caissons (x≥h), and in the gap (0≤x≤h) where there is the influence of the fluid viscosity. We use "viscid fluid" or the "viscosity of the fluid" to denote the local effects of the fluid viscosity on wave transmission and fluid resonance. Because the viscous decay length is assumed to be much smaller than the wave length, the viscous effects are confined to a thin region at the two ends of the gaps. Our analysis will proceed under the assumptions that the fluid is incompressible, inviscid, and the motion is irrotational away from the gaps of the caissons. We further assume that the boundary conditions on the free surface can be linearized. The fluid motion can then be expressed in terms of velocity potential. In the gap region between adjacent caissons, these velocities should match across the gap; and the pressures on each side should be equal in the gap to ensure a continuous pressure across the gap.

Free surface elevation in gaps between the caissons
The wave potential in a gap (0≤x≤h) can simplistically be denoted as: where A g and B g are unknown constants. The free surface elevation in the gaps can be represented as: ] . (2) In the following, we will neglect the time factor e -jωt for the sake of convenience.
The wave pressure and the x-directed velocity in the gap are: Hence, the wave pressure and the velocity at two ends of the gap, x=0 and h, can be written as: ; where u a (0) and u a (h) are the non-dimensional velocities to be determined. The unknown constants A g and B g are easily obtained from Eqs. (9) and (10): A g and B g are substituted into Eq.
(2) and the free surface elevation in the gaps of the caissons can be denoted as: The free surface elevation at the midpoint location of the gaps of the caissons is: It has been confirmed from Eqs. (11) and (12) that the non-dimensional velocities u a (0) and u a (h) at both ends of the gaps are the sources of disturbance of the free surface motion in the gaps. They can be obtained from a different wave theory and a viscous model. Based on the plane wave hypothesis and the full wave model in this paper, four cases can exist: the inviscid and viscid fluid velocities for plane wave hypothesis and inviscid and viscid fluid velocities for the full wave model on the two ends of the gap.

Non-dimensional velocities under plane wave hypothesis
We first discuss the non-dimensional velocities under plane wave approximation. To obtain the non-dimensional velocities, u a (0) and u a (h) at both ends of the gaps, we must use the wave field for seaward and landward regions of the caissons and matching conditions at the two ends of the gaps.
The potential of the incident and reflected waves in the seaward region can be denoted as (Dean and Dalrymple, 1991): The wave pressure and the x-directed velocity in the field on the left-hand side of the row of caissons have the following formulas: .

(15)
The free surface elevation in the sea field on the lefthand side of the row of caissons is: where R 0 is the reflection coefficient; ρ and c are the fluid density and the wave celerity, respectively. The wave potential in the landward region of the row of caissons (x≥h) is: in which T 0 is the transmission coefficient. The wave pressure and the x-directed velocity in the field on the right-hand side of the row of the caissons have the following formulas: The free surface elevation in the landward region of the row of the caissons is denoted as: . (20)

Matching conditions and non-dimensional velocities on both ends of a gap in an inviscid fluid for plane wave approximation
The continuity of x-directed velocities of fluid particles across each end of the gap requires: where, the porosity is a ratio of the gap width (2a) to the grating constant (2b=2a+A). The continual conditions of wave pressures on both ends of the gap are denoted in an inviscid fluid as: Substituting Eqs. (7) and (15) into Eq. (21) gives the reflection coefficient: Inserting Eqs. (7) and (19) into Eq. (22) yields the transmission coefficient: Eqs. (5), (14) and (25) are substituted into Eq. (23) to obtain the following formula:  (27) and (28), we can obtain non-dimensional velocities u a (0) and u a (h) on both ends of a gap in an inviscid fluid under plane wave approximation, respectively.

Matching conditions and non-dimensional velocities at both ends of a gap in viscid fluid under plane wave approximation
It is no doubt that the skin-friction, flow separation, and vortex shedding around the gap and corners caused by the viscid fluid will dissipate a large amount of wave energy, thereby reducing the height of the free surface and the severity of the fluid resonance in the gaps.
The continual condition of the x-directed velocities of fluid particles across each end of the gap still requires Eqs. (21) and (22).
The continual conditions of pressure on both ends of the gaps in viscid fluid can be represented as: where ∆P 0 and ∆P h represent the loss of fluid pressure across the two ends of the gaps.
We use the local viscosity model, which is analogous to damping lid method in the numerical simulation, and the effects of viscosity are only located around the gaps. The influence of the additional mass on the fluid resonance in the gaps is also considered in the local viscosity model.
The losses of pressure ∆P 0 and ∆P h can be denoted as (Zhu, 2011): where μ, ζ and l′ are the dynamic viscosity, the local resistance coefficient, and the modified length of gap (Zhu, 2013), respectively; δ=0-0.5 is the inductance coefficient, which is also referred to as the resonance participation coefficient. The first terms in Eqs. (35) and (36) are an inductive reactance. It does not dissipate energy, but can store energy, and lead the phase shift. The second term is the energy loss caused by the skin-friction; and the third term is a nonlinear energy dissipation.
Another formula can be derived from Eq. (38) by employing the same procedure as above.

Non-dimensional velocities for a full wave model
The inviscid and viscid fluid non-dimensional velocities at both ends of a gap under full wave model have been studied by Zhu (2013) and Zhu and Xie (2015), and their results are introduced in this section.

Inviscid fluid non-dimensional velocities under full wave model
The inviscid fluid non-dimensional velocities at both ends of the gap under a full wave model can be rewritten as: ( ; .

Viscid fluid non-dimensional velocities under a full wave model
Viscid fluid non-dimensional velocities at both ends of the gap under a full wave model can be obtained by a set of non-linear equations similar to Eqs. (41) and (42), but the parameters contained in these equations are not the same with the plane wave approximation. They have the following forms: ; .

Fluid resonance in gaps 2.4.1 Inviscid fluid resonance in gaps
Eqs. (30) and (31) in plane wave hypothesis are substituted into Eq. (12). The free surface elevation at the midpoint of the gaps of the caissons in an inviscid fluid can finally be denoted as: − je 1 sin (kh) .
h/L=1/2 is the zeroth-order mode, are higherorder modes. kh=mπ (m=1, 2, 3, …) is the appearance condition for the fluid resonance obtained by Miao et al. (2001) and Saitoh et al. (2002Saitoh et al. ( , 2008. For an even number for m, the fluid resonance in narrow gaps does not exist. Hence, Eq. (47) provides an accurate and efficient method for the prediction of the appearance conditions of the fluid resonance in narrow gaps.
The root kh=(2n+1)π is inserted into Eq. (46); both nu-merator and denominator to zero are observed, and its limit value can only be obtained by the L'Hôpital's rule as follows: Eq. (48) shows that the ratio of the amplitude of the resonance wave in narrow gaps with the incident wave height can reach a very large value when the porosity e 1 is smaller than 1.
Eqs. (43) and (44) in the full wave model are substituted into Eq. (12). Free surface elevation at the midpoint of the gaps of the caissons in inviscid fluid can be denoted as:

Viscid fluid resonance in gaps
Owing to a viscid fluid, non-dimensional velocities at both the ends of the gap are found from Eqs. (41) and (42), and they are the two implicit functions. Free surface elevation of viscid fluid at the midpoint of the gaps cannot be represented by explicit functions. Viscid fluid non-dimensional velocities can be directly inserted into Eq. (12) to calculate the free surface elevation at the midpoint of the gaps.

Validation of theoretical formulae
To examine the effectiveness and accuracy of the formulae for viscid and inviscid fluid resonance in the gaps, free surface elevation in gaps predicted by the present formulas are compared with the measured values obtained by Saitoh et al. (2008).
In the experiments of Saitoh et al. (2002Saitoh et al. ( , 2008, the width and the thickness of a square caisson were 2b-2a=77 cm and h=77 cm, respectively. The water depth was d=20 cm. The widths of the gap were 3, 2.6, and 2 cm, and the length was 77 cm.

Comparison of the free surface elevations in gaps predicted by the present models with the measured values
The viscid fluid free surface elevations in the gaps are predicted utilizing Eqs. (41) and (42) for the plane wave hy-pothesis and using Eqs. (41), (42) and (45) for the full wave model, and the measured values (Saitoh et al., 2008) are together plotted in Fig. 2. Curves and symbols in the figure describe theoretical predictions and test data, respectively.
The solid line in Fig. 2 is to denote the results of the full wave model and the dotted line for the plane wave hypothesis. The theoretical curves calculated by the two models are almost coincident. Fig. 2 indicated that the heights and the positions of the resonance peaks predicted by the present models are in agreement with the experimental ones (Saitoh et al., 2008). On both sides of the resonance peak, the theoretical results are smaller than the experimental data. In Fig.  2b the predicted height of the resonance peak is a little larger than the measured value.
The phase shifts of the response peaks corresponding to Fig. 2 are denoted in Fig. 3; the theoretical values are in good agreement with the experimental data.
Comparisons of the present model with the previously measured results confirmed that the viscosity model based on full wave solution and plane wave hypothesis provides satisfactory results from the viewpoint of practical applications. Especially, the viscosity model can accurately predict the phase shift of the response peak. A reasonable viscosity model can reduce the free surface height in the gaps, move the position of the resonance peak, and lead to the phase shift of the response peak.

Wave transmission through narrow gaps
The transmission of waves through gaps is closely related to the gap length. In consideration of the influences of gap length and porosity, the transmission coefficients under the plane wave hypothesis are derived in Eq. (32) for inviscid fluids and Eqs. (41) and (42) for viscid fluids. The transmission coefficients for the full wave model have been obtained by Zhu (2013) and Zhu and Xie (2015).
The transmission coefficients calculated utilizing two kinds of models are shown in Fig. 4. Fig. 4a is the transmission coefficient for plane wave Fig. 2. Comparison of the predicted free surface elevation with test data (Saitoh et al., 2008). Solid line is from full wave model; dotted line is from plane wave hypothesis; symbols are test data.
ZHU Da-tong et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 724-735 conditions; Fig. 4b is for the full wave model. The horizontal coordinate is the ratio, 2b/L of grating constant (2b= 2a+A) with wavelength. The inductance coefficient or resonance participation coefficient in the viscosity model is taken as δ=0.1, i.e., the phase shift factor is considered. There are two curves in each figure; one curve is for the inviscid fluid, and the other for the viscid fluid. The transmission peaks (dotted line) for plane wave hypothesis can be plotted from infinite; the six peaks are only denoted in Fig. 4a.
In the full wave model, we only observe four transmission peaks (dotted line) in Fig. 4b, and when 2b/L>1, other transmission peaks vanish. It has been shown that the ratio, 2b/L=1 indeed is a critical value because the transmission peaks are truncated at this value. This phenomenon is shown to have a bound, which is the upper limit for the appearance of the transmission peaks. The solid curves in Figs value, , which is the upper limit for the appearance of the transmission peaks. It is perfect that the remaining two numbers are included in the range of 2b/L<1. It is thus reasonable to think that the ratio, 2b/L<1 is the upper limit for the appearance of fluid resonance in the gaps. The comparison of Fig. 4c and Fig. 4d shows that although the plane wave hypothesis can predict the positions and the heights of the transmission peaks, it cannot determine the upper limit for the appearance of the transmission peaks.
The positions of transmission peaks (dotted line) of inviscid fluid in Fig. 4d are a little different from Fig. 4c and are moved to the low frequency side. This phenomenon  shows that the evanescent waves in the full wave model have a phase shift function (Zhu and Xie, 2015), while the plane wave model does not include the evanescent waves. However, the value of the phase shift by the evanescent waves is smaller than the ones obtained by the resonance participation coefficient.
4.2 Conditions for the appearance of the fluid resonance in narrow gaps and the reason of phase shift The condition of fluid resonance in gaps of the caissons is . However, this condition is very rough, and it is not complete. Moreover, the modes (number) of the resonance peaks seem to be infinite in the spectrum curve. The relationships of the resonance peaks with caisson geometry are not clear. In this section, we research the appearance condition of fluid resonance in the gaps in detail.

Upper and lower limit conditions for the appearance of fluid resonance in gaps
From Section 4.1, a ratio of grating constant with the wavelength 2b/L<1 is the upper bound for the appearance of fluid resonance in the gaps, i.e., the resonance can occur only when the condition 2b/L<1 is satisfied.
Taking 2b=Ƹh gets the following formula: When Ƹ=2, the length of the gaps, h=b is half of the grating constant, can be obtained from Eq. (50), and this value is already lower than the limit of 1/2. Fig. 5b for the full wave model shows that the viscid fluid resonance does not appear in the gaps. However, the viscid fluid resonance peaks predicted by the plane wave model still appear in Fig. 5a. If Ƹ=1.7, , then the fluid resonance occurs in the gaps (Fig. 6). Thus, for the length of the gaps to equal half of the grating constant may be the lower limit for the appearance of the zeroth-order resonance in the gaps. While the resonance peaks calculated by the plane wave model are distributed in all frequency domains, and the upper and lower bound for the appearance of the resonance peaks are not found by the plane wave model. When Ƹ=1/4, the length of the gaps h=4(2b) is four times that of the grating constant, can be derived from Eq. (50). This value is already larger than 7/2. There are four resonance peaks (Fig. 7) at in the spec-h L = 1 2 trum curve. The zeroth-order mode is , and the thirdorder mode is h/L=7/2. Therefore, the upper limit of the resonance peak completely depends on the ratio of the length of the gap to the grating constant; the length of the gap increases, and the upper bound is raised without a fixed value.

Number of resonance peaks in the spectrum curve
The number of resonance peaks depends on the ratio of the gap length to the grating constant. When h≤b, there is no resonance in the gaps. As the length of the gap increases, the number of the resonance peaks also increases. there is only a resonance peak for the zeroth-order mode (Fig. 8). For values such as h=2(2b), 2b/L=h/(2L)<1, h/L<2, h/L can take and two resonance peaks for the zeroth-and first-order modes are displayed on the spectrum  ZHU Da-tong et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 724-735 h L = 1 2 , 3 2 curve in Fig. 9. From Fig. 4d  Again, the length of the gaps, h=3(2b) is three times that of the grating constant, 2b/L=h/(3L)<1, h/L<3; h/L then can select . Fig. 10 shows that, in the spectrum curve   there are three resonance peaks for the zeroth-, first-and second-order modes. Therefore, the index 2b/L<1 determines the number of resonance peaks in the spectrum curve and its upper limit for the appearance of fluid resonance in the gaps.

Value and reason of phase shift of resonance peaks in spectrum curve
The previous analysis shows that there are two kinds of phase shift phenomena, i.e., the phase shift caused by the evanescent waves in the full wave model and the phase shift caused by the resonance participation coefficient in the viscous model. The former influence is little; the latter effect is obvious. The main task of this section is to discuss the influence of resonance participation coefficient on the phase shift. Fig. 2 describes the theoretical predictions and test data, and the results for full wave model and for plane wave hypothesis are almost coincident. The theoretical height and location of the resonance peak for the zeroth-order mode agree well with the experimental data.
In Fig. 11, we also plotted the curves of influence of the gap width on the resonance. Fig. 11a is the phase shift to equal zero, the position of the resonance peak is h/L=1/2, i.e., kh=π, when the ratio of the gap width to the gap length is 2a/h<0.1, the viscosity of the fluid plays a very important role, and the resonance peak is obviously reduced; 2a/h≥ 0.1, the heights of the resonance peak predicted by using viscid fluid models are the same as that with the inviscid fluid model. In Fig. 11b, δ=0.1. The values of the phase shift increase and heights of the resonance peaks decrease with an increase of the gap width. When the gap width is larger than 10% of the gap length, the heights of the resonance peaks are smaller than 5 times the incidence wave height. The average values at three locations of the resonance peaks for the zeroth-order mode in Fig. 3 (Saitoh et al., 2008)).
Figs. 7 and 10 also show that the phase shifts increase gradually with an increase of the resonance order.
The relation of the phase shift of the resonance peaks for the zeroth-order mode with the resonance participation coefficient is denoted in Fig. 12. Four resonance curves for zeroth-order mode correspond to four resonance participation coefficients, δ=0, 0.1, 0.3 and 0.5. When δ=0, there is no phase shift, kh=3.1416 (the upper limit for zeroth-order mode); for δ=0.5, the phase shift is larger; the locations of the resonance peaks for the zeroth-order mode are moved to the lower frequency side. Fig. 12 shows that the heights of the resonance peaks decrease with an increase in the resonance participation coefficients, δ. The first term in Eqs. (35) and (36) for the viscosity model is an inductive reactance. It does not dissipate energy, but can store energy and lead to the phase shift. According to the study for the perforated wall caisson breakwater (Zhu and Zhu, 2010), the first term in the viscosity model can be used to change the position of the minimum reflection coefficient on the curve of the reflection coefficient via the relative width of the caisson, to move the minimum reflection coefficient to the low frequency side. The function of this term for the fluid resonance in the gap is similar to the perforated wall breakwater. The volume and mass of the resonance water are relatively easy to be determined in the perforated wall caisson breakwater, but the water volume and mass for the fluid resonance in the gap is difficult to be identified. So the resonance participation coefficient δ is used for approximation.
The fluid energy in the gaps can be divided into kinetic energy and potential energy. Kinetic energy is the energy  ZHU Da-tong et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 724-735 carried by the water body (including add mass) inflow and outflow at both ends of the gap; potential energy can be expressed with the free surface elevation above the still water level. When the width of the gap is fixed, increasing the coefficient δ yields to an increase in the volume of water involved in the resonance, the phase shift of the resonance peaks and the kinetic energy to reduce the potential energy and the height of the resonance peaks. To fix the coefficient δ, increasing the width of gaps also obtains the same results (Fig. 11). Therefore, a reasonable selection of the coefficient δ is the key to determine the heights and the locations of the resonance peaks.

Conclusions
This paper presents details of the theoretical analysis and the numerical results for the inviscid and viscid fluid resonance in the gaps of bottom mounted caissons by using a plane wave hypothesis and a full wave model. The main focus is to determine the upper and lower bounds for the appearance of fluid resonance in the gaps, the heights, the number and the phase shift of the resonance peaks in the spectrum curve. On the basis of the comparisons of the present model with the existing theoretical and experimental results, the following conclusions can be drawn: (1) The theoretical analysis and the numerical results all show that the general conditions for the appearance of fluid resonance in narrow gaps are kh=(2n+1)π (n=0, 1, 2, 3, …), i.e., , rather than kh=nπ (n=1, 2, 3, …).
(2) The plane wave hypothesis and the full wave model can accurately determine the heights and the locations of the resonance peaks in the viscid fluid, and reasonably explain the value and the reason for the phase shift of the resonance peaks in the spectrum curve. However, the plane wave hypothesis does not obtain the upper and lower limit for the appearance of fluid resonance in gaps and the number of the resonance peaks in the spectrum curve.
(3) The upper bound, 2b/L<1 for the appearance fluid resonance in the gaps can be obtained by the full wave model in an inviscid fluid, and lower bound, h/b≤1 can be obtained by the full wave model in viscid fluids.
(4) The number of resonance peaks in the spectrum curve is completely dependent on the ratio of the length of the gaps to the grating constant. Increasing the length of the gaps yields a higher number of resonance peaks.
(5) The main reason for the phase shift of the resonance peaks is the inductive factors (resonance participation coefficient). With an increase in the phase shift, the potential energy contained in the gap fluid is gradually converted to kinetic energy, thereby reducing the resonance peak height and moving the resonance peak to the low-frequency side. The range 0.0≤δ≤0.15 of the coefficient may be reasonable. The viscosity model can accurately predict the phase shift of the response peak. However, a suitable coefficient δ still needs to be chosen.
The influence of the evanescent waves in the full wave model on the phase shift is smaller than the resonance participation coefficient.
(6) The gap width is larger than 10% of the gap length, and the heights of resonance peaks are smaller than 5 times the incidence wave height.
(8) It should be further studied whether the content of this paper can be applied to the fluid resonance in the gaps of multiple floating bodies.