A Semi-Analytical Potential Solution for Wave Resonance in Gap Between Floating Box and Vertical Wall

Based on potential flow theory, a dissipative semi-analytical solution is developed for the wave resonance in the narrow gap between a fixed floating box and a vertical wall by using velocity potential decompositions and matched eigenfunction expansions. The energy dissipation near the box is modelled in the potential flow solution by introducing a quadratic pressure loss condition on the gap entrance. Such a treatment is inspired by the classical local head loss formula for the sudden change of cross section in channel flow, where the energy dissipation is assumed to be proportional to the square of local velocity for high Reynolds number flows. The dimensionless energy loss coefficient is calibrated based on experimental data. And it is found to be insensitive to the incident wave height and wave frequency. With the calibrated energy loss coefficient, the resonant wave height in gap and the reflection coefficient are calculated by the present dissipative semi-analytical solution. The predictions are in good agreement with experimental data. Case studies suggest that the maximum relative energy dissipation occurs near the resonant frequency, which leads to the minimum reflection coefficient. The horizontal wave forces on the box and the vertical wall attain also maximum values near the resonant frequency, while the vertical wave force on the box decreases abruptly there to a small value.


Introduction
Ocean waves can induce fluid resonances in confined spaces, such as the narrow gap between closely spaced vessels and that between vessel and wharf, and moonpool of floating platform. In practice, the fluid resonance in narrow gaps needs careful considerations since it can result in violent wave motions inside and very large wave forces on the bodies, and significantly affects the safe operations of maritime structures. The gap resonance problem has been extensively studied in the past decades (Molin, 2001a;Saitoh et al., 2003;Faltinsen et al., 2007;Yeung and Seah, 2007;Lu et al., 2010;Moradi et al., 2016;Zhao et al., 2017).
Wave resonance between a floating box and a vertical wall, for example, a vessel berthing in front of a terminal or wharf, has been studied by a number of researchers. With the matched eigenfunction expansion method, analytical solutions for wave interaction with a floating rectangular box with a leeside vertical wall were developed in Hsu and Wu (1997) and Zheng et al. (2004). By employing the same analytical method, the effect of step bottom on the wave scattering by a floating box with a sidewall was further investigated in Bhattacharjee and Guedes Soares (2011). Moreover, the radiation and diffraction problems were also examined for a floating rectangular tanker with a bottom opening near a vertical wall (Zhang and Zhou, 2013). All of these analytical studies above were based on linear potential flow theory in which the wave energy dissipation near the structures was ignored. Numerical and experimental studies (Kristiansen and Faltinsen, 2009) were carried out for the resonant water motion between a fixed ship and a bottom-mounted terminal. The numerical examinations were based on a potential numerical wave tank without considering the energy loss. And thus the predicted wave height in gap was found to be much larger than the experimental observations around the resonant frequency. Furthermore, the gap resonance between a forced oscillating barge and a bottom-standing terminal was investigated in Kristiansen and Faltinsen (2010), and the wave energy dissipation was mainly attributed to the flow separation from the barge edge. Recently, a series of experimental tests were conducted to measure the resonant wave height in the gap between a fixed floating box and a vertical wall, as well as the reflection coefficient of the box-wall system (Tan et al., 2014).
In order to overcome the over-prediction of resonant response in narrow gap, various methods have been proposed by introducing damping term into the potential flow model. These methods include: adding a lid on the free surface (Buchner et al., 2001;Newman, 2004;Molin et al., 2009), introducing a linear damping free surface condition (Chen, 2004;Lu et al., 2011;Chen et al., 2015), adding a pressure loss (jump) surface inside the flow field , and setting an energy dissipative domain inside the fluid region (Liu and Li, 2014). Recently, Faltinsen and Timokha (2015) proposed an empirical pressure loss formulation for the problem of fluid resonance in moonpool. The pressure loss on the moonpool entrance was then transformed to the modified dynamic free-surface condition inside the moonpool. Cummins and Dias (2017) devised an artificial pressure loss surface near the edges of an oscillating wave energy converter, which allowed the viscous dissipation being modelled in the potential flow solution. Tan et al. (2019) proposed a combined linear and quadratic damping model for predicting the response of gap resonance in the frame of potential flow theory. It was understood that the linear damping may play an important role when the dissipation is mainly attributed to the shear stress (friction) on the solid wall, for example, the low Reynolds number flows in the gap between twin boxes with round corners. It was also expected that damping effect would be closely related to the local flow pattern, depending on not only the Reynolds number but also the Keulegan-Carpenter (KC) number. The aforementioned studies have suggested that with the introduction of suitable damping terms the potential flow model can produce reasonable predictions of the hydrodynamic quantities near the resonant frequency.
In this study, we will develop an analytical solution for the wave resonance in the gap between a fixed floating rectangular box and a vertical wall. It is assumed that the quadratic damping is dominant for the present case with sharp edges of the floating boxes. A quadratic pressure loss condition is imposed on the gap entrance in the fluid domain, rather than on the free surface. Then, the energy loss due to the local dissipative effects, physically happening near the gap entrance, can be modelled in the potential flow solution. Such that the unphysical over-prediction of resonant wave amplitude in gap predicted by the conventional potential solution can be overcome. Efforts were specially made on developing an approach to predict valid damping coeffi-cients for various structural geometries. And the predictions will be validated against experimental tests and viscous numerical simulations.
This paper is organized as follows. In Section 2, the boundary value problem for the gap resonance problem is formulated with introducing a quadratic pressure loss condition inside the fluid domain. And a semi-analytical solution with iterative calculations is developed. In Section 3, the dimensionless energy loss coefficient for the pressure loss boundary condition is devised and validated against experimental data of various incident wave frequencies and wave heights. Modified potential calculations are conducted in Section 4 to examine the effects of gap width, box draft and box breadth on the hydrodynamic quantities. Finally, conclusions are drawn in Section 5.

Semi-analytical solution
A sketch definition of the present semi-analytical model is shown in Fig. 1. A rectangular box with breadth B (= 2b) and draft d is fixed in front of the vertical wall. The gap width between the box and the vertical wall is B g . The spacing between the box bottom and the flat seabed is s = h -d with the water depth h. A Cartesian coordinate is defined with its origin located at the still water level and the z-axis pointing upwards colliding with the vertical wall. The box and the wall are subject to incident waves propagating along the positive x-direction. We adopt potential flow theory to examine the present problem, where the fluid flow is assumed to be incompressible and irrotational. Thus, we can use the velocity potential Φ (x, z, t) to describe the fluid motion. The dynamic pressure is denoted by P (x, z, t). By considering the time-dependent harmonic small-amplitude incident waves with angular frequency ω, we have ; (1) where Re(*) denotes the real part of the argument, , and are the complex spatial velocity potential and dynamic pressure, respectively.
The velocity potential and the dynamic pressure can be 748 LIU Yong et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 747-759 related by using the linearized Bernoulli equation where ρ is the fluid density. The complex spatial velocity potential satisfies the Laplace equation and the following boundary conditions associated with the free surface, seabed, solid walls and far field: where g is the acceleration due to gravity, and k 0 and ϕ 0 are, respectively, the wave number and the velocity potential of the incident waves. Eqs. (4)-(10) formulate a complete boundary value problem, which can be solved either analytically or numerically. It is noted that the mechanical energy loss/dissipation involved in the real flows cannot be modelled in the potential flow solution. The limitation of potential flow model consequently leads to notable overestimations of the hydrodynamic quantities around the resonant frequency, for example, the surface elevation in gap and the wave forces on structures. In order to improve the accuracy of the conventional potential flow model, an appropriate damping term is expected. Such that the ignored dissipative effects can be approximately modelled.

Introduction of energy dissipation
Based on the idea of fairly perfect fluids with potential flow assumption but introducing damping effects,  proposed a linear dynamic pressure jump (energy loss) across the surface near body, where significant dissipative vortical flows may develop. According to previous studies (Molin et al., 2009;Kristiansen and Faltinsen, 2010;Tan et al., 2019), the energy dissipation in the narrow gap or in the similar moonpool are mainly resulted from the flow separation and vortex shedding from the sharp corners of structures. It was known that the resultant pressure loss is generally proportional to the squared local flow velocity. Thus, with a close analogy with the classical local head loss formula for the suddenly changed cross section of channel flow (Idelchik, 1996), the quadratic pressure loss condition is introduced for the present gap resonance problem in the near region of the gap entrance where U is the normal fluid velocity across the gap entrance, and ζ is a dimensionless energy loss coefficient. Eq. (11) is essentially consistent with the quadratic pressure loss condition proposed for a perforated thin plate (Molin, 1992). It is also noted that Eq. (11) is not only valid to linear harmonic waves, but also transient nonlinear waves. For the linear time-harmonic waves, Eq. (11) can be simplified. By substituting Eqs. (1)-(3) into Eq. (11), we have Applying Lorentz linearization to the time-dependence in Eq. (12) (Molin, 1992;Mei, 1989) Eq. (13) is the core of the present semi-analytical model. With the introduced local quadratic energy loss condition, the preceding modified potential flow problem is ready to be solved.

Series solutions of velocity potentials
By aiming at developing series solutions of the velocity potential, the fluid domain is divided into four regions. As shown in Fig. 1, we define Region 1 with x ≤ -(B g +B) and -h ≤ z ≤ 0, Region 2 with -(B g +B) ≤ x ≤ -B g and -h ≤ z ≤ -d, Region 3 with -B g ≤ x ≤ 0 and -h ≤ z ≤d, and Region 4 with -B g ≤ x ≤ 0 and -d ≤ z ≤ 0. We use herein ϕ j (j = 1, 2, 3, 4) to denote the velocity potential in the j-th region. In Region 1, the velocity potential satisfies Eqs. (4) - (6) and (10), and it can be written as: where H 0 is the incident wave height, R m denote unknown expansion coefficients, and k m denote positive real roots of the wave dispersion relation In Region 2, the velocity potential satisfies Eqs. (4), (6) and (9) and has where A m and B m are unknown complex expansion coefficients, and the eigenvalues μ m are given by For Regions 3 and 4, it is not easy to directly develop the series solutions of velocity potentials. Hence the velocity decomposition method (Lee, 1995;Liu and Li, 2011) is employed to deal with the difficulty. The velocity potentials are decomposed as follows: (20) We further make the decomposed potentials satisfy the boundary conditions: The decomposed velocity potentials for Regions 3 and 4, satisfying the Laplace equation and the boundary conditions in Eqs. (22)-(26), can be written as: (32b) where C m , D m , E m and F m are unknown complex expansion coefficients, and the eigenvalues α m read The eigenvalues λ m are the positive roots of the following dispersion relation (34) We note that λ 0 is in fact the usual wave number for linear wave propagation with constant water depth d. Similar expressions to Eqs. (27)-(30) can also be found in Molin (2001b) for wave motion over a submerged horizontal perforated thin plate attached to a vertical wall.
To this point, it is necessary to determine the unknowns associated with the velocity potentials by using the velocity and pressure matching conditions among different regions. Attributed to the nonlinearities of boundary condition in Eq. (13), an iterative procedure is required. The method used in this work is similar to those in An and Faltinsen (2012) and Molin and Remy (2013).

Iterative procedure for determining expansion coefficients
The velocity and pressure matching conditions among different regions can be summarized as follows: where the former Eqs. (13), (20), (21), (23) and (25) have been incorporated. By inserting the velocity potentials of Eqs. (14) and (17) into Eq. (35), multiplying both sides by Z m (z), and integrating with respect to z from -h to 0, it yields , , δ mn = 1 (m = n) and δ mn = 0 (m ≠ n). By using the similar procedure, we transform Eqs.
Eqs. (42)-(48) are simultaneously solved in this work using an iterative method by truncating m and n after M terms. The procedure of iteration is given as follows.
(1) Assuming that the values of C n (n = 0, 1, …, M) in ħ(x) are known, the values of are determined, and Eq. (48) becomes a linear system. The values of C n in ħ(x) are temporally set to be zero for the first iteration.
(3) Evaluate the difference between the previous and the present C n , and stop the iteration as the difference smaller than a prescribed value 10 -4 adopted in this study. Otherwise, the averaged value of the assumed and updated C n is used to obtain the new , and Step 2 is repeated. The numerical tests of this work show that the convergent solution can be normally achieved within 15 iterations, and a truncated number of M = 25 for m and n is sufficient to make sure the convergences of hydrodynamic quantities. Furthermore, we have also numerically solved the present problem using an iterative multi-domain boundary element method (Liu and Li, 2017). And the present semi-analytical solutions are confirmed to be in good agreement with the numerical results of the boundary element model.

Hydrodynamic quantities
Once all the unknowns are obtained, the hydrodynamic quantities can be calculated. The surface elevation is obtained by The dimensionless wave height in the narrow gap is estimated by In the right hand side of Eq. (14), the first and second terms denote the incident and reflected waves, respectively, while the third part denotes a series of evanescent modes, exponentially decaying along the negative x-direction. Thus, the reflection coefficient of the box-wall system can be calculated by The dynamic pressure in the fluid domain is determined by the Bernoulli equation (3). By integrating the dynamic pressure along the solid surfaces, the horizontal wave force on the vertical wall, and the horizontal and vertical wave forces on the floating box are, respectively, estimated by The dimensionless wave forces are defined as: (55) (56)

Experimental tests
The energy loss coefficient ζ introduced in the present semi-analytical solution needs to be obtained with the aiding of experimental data. Laboratory tests were carried out in a wave flume (56.0 m in length and 0.7 m in width) in Dalian University of Technology. A constant water depth h = 0.5 m was adopted throughout the experiments. The breadth of the fixed floating box was the same as the water depth, that is, B = h = 0.5 m. Seven cases with different box drafts and gap widths were tested. Details of the experimental parameters are listed in Table 1. For these seven cases, the incident wave height was kept to be H 0 = 2.4 cm while the wave period ranged from 0.92 s to 2.91 s. It was evaluated that the wave steepness H 0 /L, with L being the wavelength, varied between 0.003 0 and 0.0185. The first four cases are conducted for varied gap width B g = 3, 5, 7 and 9 cm with a constant body draft d = 25.2 cm, while Cases 5-7 are made for different drafts d = 15, 35 and 45 cm with a constant gap width B g = 5 cm. In the experiments the wave height in the narrow gap and the reflection coefficient of the box-wall system were measured. More details on the experimental tests can be found in Tan et al. (2014).
It should be mentioned that in the above reference the experimental results of Cases 2, 3, 4, 6 and 7 were included, while Case 1 and Case 5 were newly conducted for the present study. As for the former Case 7, the present re-examinations show that the seepage flows below the vertical wall may result in lower resonant wave amplitude in gap. And hence the experimental data are rectified in this work. Furthermore, laboratory tests were also conducted for different incident wave heights in the present study, which will be discussed later.  Fig. 2 shows the time histories of the wave elevations in gap for the two newly added Case 1 and Case 5. And the viscous numerical results based on the Navier−Stokes equations are also included for comparisons. The viscous numerical model has been developed in our previous studies for the gap resonance among multiple rectangular boxes (Lu et al., 2010). The comparisons shown in Fig. 2 indicate that the viscous numerical results agree well with the experimental data although constant phase differences are observed. The phase difference is resulted from the ramped function applied in the initial stage of the numerical simulations. Viscous numerical simulations are further conducted 752 LIU Yong et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 747-759 to extend the parameter scope of the laboratory test. The results will be shown later.

Predictive formula of energy loss coefficient
It is expected that if the geometric parameters keep unchanged, a constant energy loss coefficient can be obtained, that is, the energy loss coefficient will not be sensitive to the wave frequency. Our results below will confirm the above assumption. For each tested case with fixed body draft d, body breadth B, water depth h and gap width B g , we calculate the wave heights in gap by increasing the energy loss coefficient ζ from zero to a sufficiently large value with a small increment. Then, the calculated wave heights in gap are compared with the experimental data at different wave frequencies. Thus, the best fitted energy loss coefficient can be obtained by considering the overall agreement between the predicted and measured wave heights in gap. To estimate the overall agreement, the following index of agreement (Willmott et al., 2012) is adopted: where N is the total number of data, p n and o n are the predicted and measured results, respectively, and is the mean value of the measurements. The upper and lower limits of d r are 1 and -1, respectively. The better agreement is identified as d r is closer to 1. By fitting the predicted and measured results, we obtain 7 energy loss coefficients for the different experimental setups. All the energy loss coefficients ζ are listed in the last column of Table 1. The maximum and minimum values of d r for the best fittings are 0.965 and 0.805, respectively, corresponding to Case 5, and Cases 6 and 7. The predicted and measured results of the wave heights in gap for the above three cases are shown in Fig. 3. It can be seen that even for Cases 6 and 7 with the worst d r = 0.805, the predicted and measured results are in fairly good agreement. The comparisons shown in Fig. 3 are made for various wave frequencies. The agreements suggest that a constant energy loss coefficient ζ can be applied for a specific case. It confirms that the loss coefficient is insensitive to the wave frequency.
The comparisons in Fig. 3 have confirmed the applicab-  LIU Yong et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 747-759 753 ility of the quadratic damping in predicting the gap resonant response. In addition to the understanding that the damping effect is proportional to the squared local velocity, the energy loss coefficient (or the damping coefficient) needs to be further formulated. Based on the results in Table 1, we find that, for the present problem of a rectangular box fixed in front of a vertical wall, the energy loss coefficient ζ mainly depends on the ratio of the box entrance width s (≡ h -d) to the gap width B g . Hence a predictive formula for the energy loss coefficient is developed as: The results of Eq. (59) are compared in Fig. 4 with the values of ζ listed in Table 1. It can be seen that Eq. (59) can well predict the energy loss coefficient.

Considerations of water depth and incident wave height
It should be addressed here that Eq. (59) and Fig. 4 are obtained based on the present experimental tests with constant water depth h = 0.5 m. The present experimental data support the validity of Eq. (59) within the scope of 1 ≤ s/B g < 8.27. For a fixed body with draft d, the continuous increase of water depth h will finally lead to extremely large values of s. As s is sufficiently large, it can be expected that the energy loss coefficient would not depend on the dimensionless parameter s/B g since B g is generally small in practice. It hence deserves determining the critical water depth h, for which the wave response in gap and the damping effect are free from the water depth. Useful clues can be found in the previous study (Moradi et al., 2016), where viscous numerical simulations have been conducted to investigate the influence of water depth on the resonant response in the narrow gap between two identical fixed floating rectangular boxes. The case studies in Moradi et al. (2016) are much similar to the present work. The two boxes with narrow separation there have the same breadth B = 0.5 m and gap width B g = 5 cm as those used in this work. The viscous numerical studies have shown that if the water depth reaches up to a critical value of 10 times the body draft, namely, h/d ≥ 10, it would be possible to neglect the effect of water depth on the resonant response in gap. For a typical case in Moradi et al. (2016) with parameters of d = 0.15 m, B = 0.5 m and B g = 5 cm (similar to Case 5 of this work), the critical value of h / d = 10 will result in s/B g = 27, which is much larger than the present upper limit of s/B g = 8.27 for Eq. (59). By considering the condition with s/B g = 27, the required water depth is h = 1.5 m, which is not available for our experimental facility. Concerning the present Eq. (59), if the water depth changes largely, Eq. (15) needs to be revised. However, the idea proposed in this work for finding such a predictive formula can still be applied. For a specific body draft d, the critical water depth can be identified based on either laboratory tests in a deep-water basin or viscous numerical simulations as Moradi et al. (2016). Another aspect to be addressed here is that the present study aims at the gap resonance of a fixed floating body is closely arranged in front of a vertical wall. It can be thought of as a barge berthing in front of a wharf. For this situation it is generally not likely to encounter the condition with very large values of s/B g .
It is worthy to further examine whether the energy loss coefficient is sensitive to the incident wave height or not. For this purpose, we calculated the wave responses in the narrow gap for all the tested cases in Table 1 but with different incident wave heights. The calculations are based on the present modified potential flow model with the energy loss coefficient predicted by Eq. (59). The modified potential solutions are then compared with measured resonant wave heights in gap subjected to incident waves with varied wave height. In the experiments (Tan et al., 2014), nine incident wave heights were adopted for Cases 1−5 which are: H 0 = 1.0, 1.5, 2.0, 2.4, 2.8, 3.2, 3.6, 4.0 and 4.5 cm. For Case 6, seven incident wave heights were used with H 0 = 1.0, 1.5, 2.0, 2.4, 3.0, 3.5 and 4.0 cm, while eight incident wave heights were tested for Case 7 with H 0 = 1.2, 1.8, 2.2, 2.4, 2.7, 3.2, 3.4 and 4.0 cm. The comparisons between the calculated and measured dimensionless wave height C H are shown in Fig. 5. It can be seen that the overall agreement is fairly good. The largest discrepancy with a mean relative error (about 11%) is observed for Case 6 with d/h=0.7. The comparisons show underestimates of the wave heights in gap by the analytical solutions, indicating an over-prediction of the energy loss coefficient ζ for this special case. The overall acceptable agreements in Fig. 5 suggest that, although Eq. (59) is developed based on the experimental data with a constant incident wave height H 0 = 2.4 cm, it can still work well over a wide range of wave height. In other words, the energy loss coefficient ζ seems little sensitive to the incident wave height, at least for the examined cases in the present study. Such an advantage is mainly attributed to the quadratic pressure loss condition introduced into the potential flow model. It is also found in Fig. 5 that the relative resonant wave height C H generally decreases with the increase of the incident wave height, suggesting a nonlinear dependence of the resonant response on the incident wave height. And the decrease rate decreases gradually with the increase of the incident wave height, which might be partially due to the more significant wave reflection for the larger incident wave height.

Results and discussions
In this section, results of calculations are presented to show the effects of geometry parameters on the wave response in gap, wave reflection and wave forces on the box and vertical wall. In the calculations, the water depth h = 0.5 m and the incident wave height H 0 = 2.4 cm are the same as those used in the experimental tests. The energy loss coefficients are estimated by Eq. (59).

Effect of box draft
The calculated dimensionless wave height in gap C H and the reflection coefficient C R at different relative drafts d/h are shown in Fig. 6, respectively. The corresponding experimental results are also included in these figures. Note that the reflection coefficients for the case with d/h = 0.3 were absent in the experiments. Alternatively, the numerical results based on the viscous fluid model are used for comparisons. The accuracy of the viscous numerical model has been validated against experimental data, as shown in Fig. 6a, in terms of the variations of wave height in gap with wave frequency for various drafts. The results of Fig. 6a suggest that the relative resonant wave heights in gap may attain 6-9 times the incident wave height. With the increase of the draft, all the experimental data, CFD results, and the present analytical solutions with damping term show that the reson-  LIU Yong et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 747-759 ant wave amplitude increases, while the resonant frequency decreases. The observed dependences on the draft are in consistence with the previous studies (Saitoh et al., 2003;Kristiansen and Faltinsen, 2009;Lu et al., 2010Lu et al., , 2011 of the gap resonance between multiple floating barges.
For comparison, the potential flow solutions without damping term are given in Fig. 6b. As shown in this figure, the conventional potential model over-predicts much the resonant amplitudes. However, the overall trends of the variations of resonant wave height and resonant frequency on the draft can be correctly predicted by the conventional potential flow model, namely, the larger draft the larger resonant amplitude and the lower resonant frequency.
Comparisons of reflection coefficients are shown in Fig. 6c. It suggests that reflection coefficients predicted by the present modified potential model are in agreement with the experimental data and CFD results. Examinations confirm that the minimum reflection coefficients appear at the resonant conditions. As 1−C R 2 can be used to evaluate the relative amount of dissipation, hence the most significant dissipation happens at resonance with the occurrence of the largest wave response in gap. Fig. 7 presents the calculated results of the dimensionless wave forces, including the horizontal wave force C W on the vertical wall, horizontal wave force C Bx on the box and the vertical component C Bz on the box. The calculation conditions are the same as those used in Fig. 6. It is seen from Figs. 7a and 7b that both of the horizontal wave forces (C W and C Bx ) attain their large peak values near the resonant frequencies, and the resonant horizontal wave forces at resonance notably increase with the increase of the relative box draft d/h. However, Fig. 7c suggests that the dimensionless vertical wave force C Bz on the box abruptly decrease to small values near the resonant wave frequency. It is also observed from Fig. 7 that as the wave frequency approaches zero, C W and C Bz at different d/h all converge to a unity value, and C Bx at different d/h decreases to zero. These observations are consistent with the well-established theoretical understandings.

Effect of gap width
The modified analytical solutions and experimental measurements of the wave height in gap C H and reflection coefficient C R at different relative gap widths B g /h are presented in Fig. 8. It can be seen from Fig. 8a that with the increase of gap width, the resonant frequency decreases monotonically, and the resonant wave height attains a peak value at a moderate gap width B g /h = 0.10, indicating a nonlinear dependence on the gap width. For the purpose of comparison, we also calculated the wave response in gap by using the conventional linear potential flow model without damping term. The conventional potential solutions in Fig. 8b show a monotonic dependence of the resonant amplitude on gap width, i.e., the smaller the gap width, the larger resonant wave amplitude in gap. Such a dependence is distinct from the experimental observations and the calculations by the modified potential model with damping term. It is expected that the damping effect is nonlinearly dependent on the gap width, which consequently results in the nonmonotonic behavior. The nonlinear dependence of the relative resonant wave height on the gap width was also observed in the experimental tests (Saitoh et al., 2003) and viscous numerical simulations (Lu et al., 2010), among others. Fig. 8c shows again that the minimum reflection coefficients appear at the resonant wave frequencies. It is seen that the narrowest gap B g /h = 0.06 with the highest resonant frequency leads to the largest value of C R = 0.55.
The calculated results of the dimensionless wave forces C W , C Bx and C Bz at different B g /h are shown in Fig. 9. The calculation conditions are the same as those used for Fig. 8. It is observed again that when the wave resonance occurs in the narrow gap, the horizontal wave forces C W and C Bx attain their maximum values while the vertical wave force C Bz decreases abruptly. In addition, with the increase of the relative gap width B g /h, the resonant wave forces C W and C Bx first increase, attain their maximum values and finally decrease. This is consistent with the observations in Fig. 8a for the resonant wave amplitude in gap.

Effect of box breadth
In the preceding analysis, the box breadth is fixed at B = h = 0.5 m. For the cases with various box breadths, the wave height in gap, the reflection coefficient and the wave forces are calculated, and the results are presented in Figs. 10 and 11, respectively. It can be seen from these figures that when the relative box breadth B/h increases, the resonant wave frequency and the corresponding minimum reflection coefficients decrease, while the resonant wave height C H and the resonant wave forces C W and C Bx increase. In addition, when the incident wave frequency is lower than the resonant wave frequency, the vertical wave force C Bz on the box increases significantly with the increase of the relative box breadth B/h. Overall, notable influence of the box breadth on the wave resonance in gap can be observed.

Concluding remarks
This study presents a dissipative semi-analytical potential solution for predicting the wave resonance in the narrow gap between a fixed floating rectangular box and a bottom-mounted vertical wall. A quadratic pressure loss condition with a dimensionless energy loss coefficient was introduced on the gap entrance. The introduced dissipative term can correctly account for the damping/dissipative effects near the floating body in the context of potential flow theory. The velocity potential decomposition method and the LIU Yong et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 747-759 757 separation of variables are used to develop the series solutions of the velocity potentials in different fluid regions. The unknown expansion coefficients in the velocity potentials are determined by matching boundary conditions among different fluid regions. And the iterative calculations are adopted to deal with the difficulties resulted from the introduced quadratic pressure loss condition. Based on the dissipative semi-analytical solution, the calculations of the wave height in gap, the reflection coefficient and the wave forces are conducted. Based on available experimental data, a formula for predicting the energy loss coefficient for the quadratic pressure loss condition was developed. It was confirmed that the proposed energy loss coefficient mainly depends on the ratio of the box entrance width to the gap width, namely s/B g . And the energy loss coefficient seems not sensitive to the incident wave height and the wave frequency. The less dependence on the incident wave amplitude is mainly attributed to the quadratic pressure loss condition introduced into the potential flow model. After incorporating the predictive formula of the energy loss coefficient into the present dissipative semi-analytical solution, the predicted wave heights in gap and reflection coefficients agree well with experimental data and viscous numerical results, indicating that the damping effects associated with the gap resonance are well modelled in the present modified potential model.
The results suggest that the geometric parameters of gap width, box draft and box breadth have remarkable effects on the wave resonance in gap. Increasing each of these parameters leads to the decrease of the resonant frequency. When the wave resonance occurs in the narrow gap, the reflection coefficient attains its minimum value, both horizontal wave forces on the box and vertical wall attain their maximum values, and the vertical wave force on the box decreases to a small value.
The present study is mainly focused on the problem of a ship section closely and firmly moored in front of a wharf. As for the situation of floating barges in side-by-side arrangement in open sea is concerned, several aspects are worthy of further attention. First, the parameter s/B g may hold very large value in deep water. The predictive formula of the quadratic damping coefficient may need a more appropriate parameter instead. As the motions of the floating bodies and the three-dimensional effects are involved, their influences should be carefully evaluated. Finally, the present modified potential model is based on linear assumptions, namely, the linearized free surface boundary conditions. For nonlinear waves and more complex breaking waves it remains great challenges though the present dissipative term is incorporated inside the fluid domain rather than the free surface. Although attempts of the present study provided an approach to develop a quadratic damping model for the dissipative effects in gap resonance, a large number of experimental data or field measurements are much necessary in order to devise a more general damping model and the damping coefficient.