Hydroelastic responses and interactions of mega floating fuel storage modules 6

CHAPTER SIX Wave Propagation along Channel Formed by Storage Modules 6.1 Overview In this chapter, we investigate the wave propagation along the channel formed by placing two floating fuel storage modules side-by-side As wave propagates through the channel, a wave runup would occur due to waves being radiated, diffracted and squeezed into the channel The wave elevation along the channel might be large especially if the channel spacing is very small This could result in wave being splashed on the module deck (also known as green water on deck) if we not have a sufficient freeboard for the module Green water may adversely affect the stability of the floating module and have to be avoided The problem at hand is thus to determine the behavior of wave propagating through the narrow channel with a view to recommend suitable freeboard (or allowable draft) and channel spacing so as to prevent green water on deck The design of the suitable channel spacing involves a trade-off decision between minimising the hydroelastic response of the empty storage modules and maximising the loading capacities of the storage modules 110 Wave Propagation along Channel Formed by Storage Modules We first conduct an experimental test to validate the numerical model and computer code for determining the wave runup along the channel Upon establishing the correctness of the model and computer code, extensive simulations are performed to study the effects of wave angle, channel spacing, parapet wall, floating breakwater and water depth on the wave elevations along the channel The computed wave elevations will permit one to design the allowable draft and freeboard of the storage modules as well as a suitable channel spacing 6.2 Experimental Test Tests on scaled Perspex models for the floating fuel storage modules were conducted in the wave flume of the Hydraulic Engineering Laboratory, National University of Singapore in order to validate the numerical model, formulation and method of solution for the wave elevations along the channel As the wave flume has a smaller width as compared to the wave basin, we have therefore adopted a length scale ratio of 1:100 as opposed to 1:50 (length scale used for the experimental test in Chapter 5) in modeling the floating fuel storage modules Note that although a different length scale ratio was adopted, we employed the same Froude number in scaling the length and time variables Furthermore, the variables are non-dimensionalised in the formulation so that the computed results will not be affected by the choice of the length scale ratio Based on the adopted length scale, the Perspex model dimensions take on 1.22m × 0.5m × 0.1m Three different draft to height ratios, i.e d/h = 0.07, 0.2 and 0.7, are 111 Wave Propagation along Channel Formed by Storage Modules considered for comparison purposes Four channel spacings, i.e s = 0.15m, 0.20m, 0.30m and 0.40m, are treated The width of the wave flume is 1m and the water depth is set at 0.15m The wavelengths λ/L range from 0.32 to 0.55 As the 1-m width of the wave flume was too small to accommodate two floating modules side-by-side, the experimental setup was modified by exploiting the geometric symmetry as proposed by Molin et al (2002) The experiment was performed by placing one floating storage module in the wave flume with the side walls of the wave flume serving as a plane of symmetry (see Fig 6.1) Hence the experiment setup is equivalent to an array of floating storage modules placed sideby-side As we are only concerned with the wave propagation along the channel with spacing of s (see Fig 6.1), five wave probes were installed along the channel closed to the side wall in order to measure the wave elevations as shown in Figs 6.1 and 6.2 The maximum wave elevations measured by the wave probes were used to check their counterparts obtained from the numerical model Note that this modified geometric symmetrical experimental setup is only valid for the investigation of wave elevations along the channel when the wave propagates in a head sea direction and that both the floating storage modules have the same dimensions The results based on this modified experimental setup were found to be in good agreement with the analytical solution (Molin et al., 2001, 2002) 112 Wave Propagation along Channel Formed by Storage Modules Fig 6.1 Experimental setup in wave flume showing mirror plane of symmetry Fig 6.2 Experimental setup in wave flume of Hydraulic Engineering Laboratory, National University of Singapore 113 Wave Propagation along Channel Formed by Storage Modules 6.3 Numerical Model 6.3.1 Equivalent Solid Plate The experimental box-like model is simplified as an equivalent solid Mindlin plate in order to reduce the computational time in the hydroelastic analysis The procedures in obtaining the equivalent solid plate were described in Section 5.4.1 The dimensions of the equivalent plate are kept the same as the actual experimental model (i.e 1.22m × 0.5m × 0.1m), but its Young’s modulus E is tweaked so as to match the vibration modes and natural periods of the experimental model The Young modulus E for the equivalent solid plate is found to be 20MPa For waves propagating in a head sea direction, we have also simplified the enclosed type box-like floating breakwater (see Fig 5.3b) to an equivalent solid plate strip that fronts the storage modules (see Fig 6.3a) We have shown in Fig 5.7 that by using such a simplified model for the breakwater, a high accuracy of computational results could still be achieved provided that the wave comes in a head sea direction However, the enclosed type box-like floating breakwater has to be modelled if the waves come from directions apart from the head sea Figure 6.3 shows the plan view of two floating storage modules under head sea condition and the positive x and y directions In order to facilitate discussion of the wave propagation through the system, we have divided the fluid surface domain into regions These regions are designated by RI, RII and RIII RI region captures the incident and reflected wave elevations RIII region captures the transmitted wave 114 Wave Propagation along Channel Formed by Storage Modules elevations Investigation will focus on the wave elevations at RII region which is the channel formed by the two floating storage modules It is to be noted here that, we only consider two floating storage modules placed side-by-side in our numerical model (and not an array of adjacently placed storage modules as shown in Fig 6.1) by assuming that the reflected and radiated waves by modules 3, 4,…, N (where N is the total number of modules placed side-by-side due to the symmetrical setup in Fig 6.1) are shielded away by the presence of modules and 2, hence does not affect the wave propagation along the channel significantly Besides that, as the responses of the modules under wavelength of λ / L = 0.32 to 0.55 are very small (lesser than wmax / A = 0.25, see Fig.5.15), the effect of radiated wave due to the interaction of multiple floating modules placed side-by-side is hence minimal towards the wave propagation along the channel (a) With simplified breakwater (b) Without breakwater Fig 6.3 Regions for wave field surrounding two floating storage modules (a) with simplified breakwater (b) without breakwater 115 Wave Propagation along Channel Formed by Storage Modules 6.3.2 Convergence Tests Convergence tests for the numerical model were already carried out in Section 5.4.3 It has been found that a minimum mesh size of λ / 30 will suffice for converged results, where λ is the wavelength 6.4 Validation of Wave Elevations along Channel with Experimental Tests The wave elevations along the channel formed by the two 1.22m × 0.5m × 0.1m solid plates are computed The computed wave elevations are compared with the experimental test results Figures 6.4 to 6.6 show the variations of the maximum wave elevations measured along the centerline of the channel for three different draft to height ratios, i.e d/h = 0.07, 0.2 and 0.7 Four different channel spacings, i.e s = 0.15, 0.20, 0.30 and 0.40m, are considered to investigate the effect of channel spacing Three different wavelengths λ/L ranging from 0.32 to 0.55 are treated for each channel spacing The experimental results measured on each wave probe are plotted in each figure It can be seen from Figs 6.4 to 6.6 that the numerical results are in good agreement with the experimental tests, thereby validating the numerical model and computational technique As expected, the wave elevation increases as the wave propagates through the channel and then the wave elevation starts to decrease as the wave reaches the centre of the channel These findings are similar to the behavior of a wave propagating along a gap in an ice floe (Molin et al., 2002) However, the numerical results of the wave elevation along the channel with a 116 Wave Propagation along Channel Formed by Storage Modules narrow spacing s = 0.15m and d/h = 0.07 not agree with the experimental test results as shown in Fig 6.4a Under such a condition, it was observed from the experiment that wave slamming of the model occurred and this enhanced the hydroelastic response of the model The slamming generated radiated waves which cause non-linearity in the wave propagating along the channel Therefore, the wave elevation along the channel formed by the empty storage modules (d/h = 0.07) is found to be larger than their fully loaded counterparts (d/h = 0.7) As the numerical scheme is based on frequency domain analysis and linear potential theory, the slamming effect of the storage module and the non-linearity of the wave field were not taken into consideration This explains why the numerical and the experimental wave elevations differ Our numerical model could be furnished by using the modified boundary integral equation proposed by Malenica et al (2005) The proposed boundary integral equation consists of modifying the boundary conditions on the free surface and hull surface by introducing an artificial body on part of the free surface between the two bodies Such modified boundary integral equation is able to predict the resonant behavior, flow separation, wave breaking phenomena that occur at certain frequencies Interested readers are encouraged to refer to the paper for the details The relatively accurate numerical results for d/h = 0.2 (Fig 6.5) indicate that precise wave elevations along the channel can be determined from the numerical model as long as the slamming does not occur Similarly, as slamming does not occur for the floating storage modules with d/h = 0.7 (loaded storage module), wave 117 Wave Propagation along Channel Formed by Storage Modules propagation along the channel behaves linearly due to negligibly radiated wave being generated by the small heaving response of the loaded storage module This explains the strong agreement in wave elevation predicted by the numerical model and by the experimental test (see Fig 6.6) The high accuracy in predicting the wave elevations along the channel formed by the loaded storage modules has significant importance in determining the maximum allowable draft and freeboard for the storage modules 118 λ/L = 0.32 s(m) (a) 0.15 (c) 0.30 (d) 0.40 (cm/cm) ηmax /A (cm/cm) ηmax /A (cm/cm) ηmax /A (cm/cm) 2 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 RI RII RIII ( ) Experimental results -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 y/L (cm/cm) -1.0 -0.5 0.0 0.5 1.0 () Numerical method 2 0 1 2 1 λ/L = 0.55 0 -1.0 -0.5 0.0 0.5 1.0 RI RII -1.0 -0.5 0.0 0.5 y/L (cm/cm) RIII 1.0 RI RII RIII -1.0 -0.5 0.0 0.5 1.0 y /L (cm/cm) 119 Fig 6.4 Maximum wave elevations along channel with spacing (a) s = 0.15m (b) s = 0.20m (c) s = 0.30m (d) s = 0.40m Draft d/h = 0.07 Water depth H = 0.15m Wave Propagation along Channel Formed by Storage Modules (b) 0.20 ηmax /A λ/L = 0.43 z h FB 0.8 0.4 0.0 (a) λ/L = 0.20 0.2 FB/A d/h 1.00 1.50 2.00 2.50 0.90 0.85 0.80 0.75 (wmax+ηmax)/FB (cm/cm) Legend 0.6 0.8 1.0 1.2 0.8 0.4 0.0 (b) λ/L = 0.32 0.2 s/L (cm/cm) d Side elevation 0.4 1.6 1.5 1.0 0.0 (c) λ/L = 0.43 0.2 0.4 0.6 s/L (cm/cm) 0.6 0.8 1.0 0.8 1.0 s/L (cm/cm) 2.0 0.5 0.4 0.8 1.0 2.0 1.5 1.0 0.5 0.0 (d) λ/L = 0.67 0.2 0.4 0.6 s/L (cm/cm) Fig 6.9 Maximum relative displacement (wmax + ηmax)/FB with respect to different channel spacings s/L for wavelength (a) λ/L = 0.20 (b) λ/L = 0.32 (c) λ/L = 0.43 (d) λ/L = 0.67 Water depth H = 0.15m 129 Wave Propagation along Channel Formed by Storage Modules y 1.2 (wmax+ηmax)/FB (cm/cm) FB = h – d h = 0.1m A = 0.01m Plan view 1.6 (wmax+ηmax)/FB (cm/cm) 0.5m Module s Module (wmax+ηmax)/FB (cm/cm) Head sea L=1.22m 0.8 0.6 0.4 0.2 (a) λ/L = 0.20 0.0 0.2 0.4 0.6 0.8 1.0 (wmax+ηmax)/FB (cm/cm) 1.0 1.2 1.0 0.8 0.6 0.4 0.2 (b) λ/L = 0.32 0.0 0.2 0.4 0.6 1.4 1.2 1.0 0.8 0.6 0.4 0.0 (c) λ/L = 0.43 0.2 0.4 0.6 s/L (cm/cm) 1.0 0.8 1.0 s/L (cm/cm) 0.8 1.0 (wmax+ηmax)/FB (cm/cm) Legend FB/A d/h 1.50 0.90 2.00 0.85 2.50 0.80 3.00 0.75 (wmax+ηmax)/FB (cm/cm) s/L (cm/cm) 0.8 1.4 1.2 1.0 0.8 0.6 0.4 0.0 (d) λ/L = 0.67 0.2 0.4 0.6 s/L (cm/cm) Fig 6.10 Maximum relative displacement (wmax + ηmax)/FB with respect to different channel spacings s/L for wavelength (a) λ/L = 0.20 (b) λ/L = 0.32 (c) λ/L = 0.43 (d) λ/L = 0.67 Water depth H = 0.15m With parapet wall of height hp = 0.005m 130 Wave Propagation along Channel Formed by Storage Modules (wmax+ηmax)/FB (cm/cm) 1.2 0.8 0.6 0.4 0.2 (a) λ/L = 0.20 0.0 0.2 0.4 0.6 0.8 1.0 (wmax+ηmax)/FB (cm/cm) 1.0 1.2 1.0 0.8 0.6 0.4 0.2 (d) λ/L = 0.32 0.0 0.2 0.4 0.6 1.2 1.0 0.8 0.6 0.2 0.0 (c) λ/L = 0.43 0.2 0.4 0.6 s/L (cm/cm) 0.8 1.0 (wmax+ηmax)/FB (cm/cm) (wmax+ηmax)/FB (cm/cm) 1.4 0.4 1.0 0.8 1.0 s/L (cm/cm) s/L (cm/cm) Legend FB/A d/h 1.50 0.90 2.00 0.85 2.50 0.80 3.00 0.75 0.8 1.4 1.2 1.0 0.8 0.6 0.4 0.2 (d) λ/L = 0.67 0.0 0.2 0.4 0.6 s/L (cm/cm) 131 Fig 6.11 Maximum relative displacement (wmax + ηmax)/FB with respect to different channel spacings s/L for wavelength (a) λ/L = 0.20 (b) λ/L = 0.32 (c) λ/L = 0.43 (d) λ/L = 0.67 Water depth H = 0.15m With parapet wall of height hp = 0.005m and breakwater of cross section 0.05m × 0.025m Wave Propagation along Channel Formed by Storage Modules (wmax+ηmax)/FB (cm/cm) 1.2 y/L (cm/cm) y/L (cm/cm) Without floating breakwater With floating breakwater (a) λ/L = 0.32m x/L (cm/cm) (b) λ/L = 0.43m x/L (cm/cm) x/L (cm/cm) (c) λ/L = 0.67m (c) λ/L = 0.89m 132 Fig 6.12 Comparison of maximum wave elevations surrounds floating storage modules with and without presence of floating breakwater Wavelength (a) λ/L = 0.32 (b) λ/L = 0.43 (c) λ/L = 0.67 (d) λ/L = 0.89 Water depth H = 0.15m Wave amplitude A = 0.01m Channel spacing s/L = 0.20 Head sea Wave Propagation along Channel Formed by Storage Modules x/L (cm/cm) Wave Propagation along Channel Formed by Storage Modules 6.6.4 Effect of Water Depth We next study the effect of water depth on the normalised relative displacement (wmax + η max ) / FB with different channel spacings s/L Figures 6.13 to 6.14 show the variations of the normalised (wmax + η max ) / FB with respect to two different water depths i.e H = 0.15m and H = 0.25m when the storage modules are loaded to d/h = 0.85 and 0.90, respectively For wave periods T = 0.4s to 0.6s, Figs 6.13a-c and 6.14a-c show that the water depth has negligible effect on the maximum allowable draft d/h whereas it becomes significant when the wave period T is 0.8s as shown in Figs 6.13d and 6.14d To better understand the influences of water depth towards the normalised (wmax + η max ) / FB , we investigate the effect of water depth on the wave elevation η max and the maximum deflection wmax of the floating storage modules, separately, for the case when d/h = 0.85 We begin by first investigating the effect of water depth on the wave elevation η max along the channel Figure 6.15 shows the variations of the maximum wave elevation η max along the channel with respect to different channel spacings s/L for two different water depths, i.e H = 0.15m and 0.25m Their corresponding wavelengths are given in each figure For wave periods of T = 0.4s to 0.6s, Figs 6.15a-c show that the relative difference in the maximum wave elevation η max for H = 0.15m is negligibly small as compared to their counterpart for H = 0.25m due to the small difference in their corresponding wavelengths On the other hand, this relative difference in maximum wave elevation is prominent for T = 0.8s because of the larger difference in wavelengths This shows 133 Wave Propagation along Channel Formed by Storage Modules that the influence of the water depth on the wave elevation depends very much on their corresponding wavelength λ/L The effect of water depth on the maximum wave elevation η max is negligibly small if the difference of the corresponding wavelength λ/L is less than 0.1 We next investigate the effect of water depth on the maximum deflection wmax of the plate Figures 6.16a-c compare the maximum plate deflection and hydrodynamic force on the floating storage modules when subjected to wave periods T = 0.5s, 0.6s and 0.8s The deflections for T = 0.4s are not shown as the values are rather small Two different water depths, i.e H = 0.15m and 0.25m are also considered for each wave period Results show that the deflection trend follows a similar pattern as the hydrodynamic force This implies that the plate deflections are significantly affected by the hydrodynamic forces induced on the floating modules as shown in Figs 6.16 However, the computed maximum deflections wmax are relatively small as compared to the maximum wave elevations η max presented in Fig 6.15 This shows that the influence of η max is more dominant on the normalised (wmax + η max ) / FB , and therefore, the normalised (wmax + η max ) / FB presented in Figs 6.13 (for d/h = 0.85) follows a similar trend of η max as presented in Fig 6.15 The effect of water depth H on (wmax + η max ) / FB may be neglected if the difference in wavelength λ/L is less than 0.1 as one can observe from Figs 6.13 and 6.14 134 0.60 0.50 (a) T = 0.4s 0.0 0.1 0.2 1.10 0.3 0.4 s/L (cm/cm) H = 0.15, λ/L = 0.4340 H = 0.25, λ/L = 0.4570 1.00 0.90 0.80 (c) T = 0.6s 0.0 0.1 0.2 0.3 s/L (cm/cm) 0.4 (wmax+ηmax)/FB (cm/cm) 0.70 (wmax+ηmax)/FB (cm/cm) (wmax+ηmax)/FB (cm/cm) 0.80 H=0.15, λ/L=0.2045 H=0.25, λ/L=0.2045 0.90 0.80 0.70 0.60 H = 0.15, λ/L = 0.3149 H = 0.25, λ/L = 0.3194 0.50 (b) T = 0.5s 0.0 0.1 0.2 1.10 1.00 0.3 0.4 s/L (cm/cm) H = 0.15m, λ/L = 0.6699 H = 0.25m, λ/L = 0.7639 0.90 0.80 (d) T = 0.8s 0.0 0.1 0.2 0.3 s/L (cm/cm) 0.4 Fig 6.13 Maximum relative displacement (wmax + ηmax)/FB along channel with draft d/h = 0.85 for wave period (a) T = 0.4s (b) T = 0.5s (c) T = 0.6s (d) T = 0.8s With parapet wall of height hp = 0.005m Wave amplitude A = 0.01m 135 Wave Propagation along Channel Formed by Storage Modules (wmax+ηmax)/FB (cm/cm) 0.90 0.60 (a) T = 0.4s 0.0 0.1 0.2 0.3 0.4 s/L (cm/cm) 1.40 H = 0.15m, λ/L = 0.4347 H = 0.25m, λ/L = 0.4570 1.20 1.00 0.80 (c) T = 0.6s 0.0 0.1 0.2 0.3 s/L (cm/cm) 0.4 (wmax+ηmax)/FB (cm/cm) 0.80 (wmax+ηmax)/FB (cm/cm) (wmax+ηmax)/FB (cm/cm) 1.00 H=0.15m, λ/L=0.2045 H=0.25m, λ/L=0.2045 1.20 1.00 H=0.15m, λ/L=0.3149 H=0.25m, λ/L=0.3194 0.80 0.60 (b) T = 0.5s 0.0 0.1 0.2 0.3 0.4 s/L (cm/cm) 1.40 H=0.15m, λ/L=0.6699 H=0.25m, λ/L=0.7639 1.20 1.00 0.80 (d) T = 0.8s 0.0 0.2 0.4 0.6 s/L (cm/cm) 0.8 1.0 Fig 6.14 Maximum relative displacement (wmax + ηmax)/FB along channel with draft d/h = 0.90 for wave period (a) T = 0.4s (b) T = 0.5s (c) T = 0.6s (d) T = 0.8s With parapet wall of height hp = 0.005m Wave amplitude A = 0.01m 136 Wave Propagation along Channel Formed by Storage Modules (wmax+ηmax)/FB (cm/cm) 1.20 0.60 0.50 (a) T = 0.4s 0.0 0.1 0.2 0.3 0.4 0.80 (ηmax)/FB (cm/cm) 0.70 H = 0.15m, λ/L = 0.2045 H = 0.25m, λ/L = 0.2045 0.70 0.60 0.50 (b) T = 0.5s 0.0 0.1 0.2 0.80 0.70 0.3 s/L (cm/cm) 0.4 0.90 (ηmax)/FB (cm/cm) (ηmax)/FB (cm/cm) H = 0.15m, λ /L = 0.4347 H = 0.25m, λ/L = 0.4570 0.60 (c) T = 0.6s 0.0 0.1 0.2 0.3 0.4 s/L (cm/cm) s/L (cm/cm) 0.90 H = 0.15m, λ/L = 0.3149 H = 0.25m, λ/L = 0.3194 0.80 H = 0.15m, λ/L = 0.6699 H = 0.25m, λ/L = 0.7639 0.70 0.60 (d) T = 0.8s 0.0 0.1 0.2 0.3 0.4 s/L (cm/cm) Fig 6.15 Maximum wave elevations along channel with draft d/h = 0.85 for wave period (a) T = 0.4s (b) T = 0.5s (c) T = 0.6s (d) T = 0.8s With parapet wall of height hp = 0.005m Wave amplitude A = 0.01m 137 Wave Propagation along Channel Formed by Storage Modules (ηmax)/FB (cm/cm) 0.80 2.00 1.00 0.00 0.0 0.1 0.2 0.3 0.4 3.00 2.00 3.00 H = 0.15m, λ/L = 0.4347 H = 0.25m, λ/L = 0.4570 4.00 1.00 0.00 0.0 5.00 F/ρgL (kN/kN) H = 0.25m, λ/L = 0.3194 F/ρgL (kN/kN) F/ρgL (kN/kN) 4.00 5.00 H = 0.15m, λ/L = 0.3149 0.1 0.2 0.3 0.4 4.00 3.00 2.00 1.00 0.00 0.0 0.2 0.3 0.4 s/L (cm/cm) 0.30 0.25 0.25 0.25 0.20 0.15 0.10 0.0 0.1 0.2 0.3 s/L (cm/cm) 0.4 wmax /FB (cm/cm) 0.30 wmax /FB (cm/cm) 0.30 wmax /FB (cm/cm) 0.1 s/L (cm/cm) s/L (cm/cm) deflections H = 0.15m, λ/L = 0.6699 H = 0.25m, λ/L = 0.7639 0.20 0.15 0.10 0.0 0.1 0.2 0.3 s/L (cm/cm) 0.4 0.20 0.15 0.10 0.0 0.1 0.2 0.3 0.4 s/L (cm/cm) 138 (a) T = 0.5s (c) T = 0.8s (b) T = 0.6s Fig 6.16 Maximum hydrodynamic force and deflection of storage modules with draft d/h = 0.85 at wave period (a) T = 0.5s (b) T = 0.6s (c) T = 0.8s With parapet wall of height hp = 0.005m Wave amplitude A = 0.01m Wave Propagation along Channel Formed by Storage Modules Hydrodynamic forces 5.00 Wave Propagation along Channel Formed by Storage Modules 6.7 Selection of Suitable Channel Spacing We will now determine a suitable channel spacing based on the results obtained in Figs 5.15 and 6.11 From the parametric studies, we found that the channel spacing does not significantly affect the design of the floating storage modules as long as the freeboard is at least twice the incident wave amplitude We consider the case given in Fig 6.11 where the storage modules are loaded up to d/h = 0.90 leaving a freeboard of 1.5A (FB < 2A, A = 0.01m) The channel spacing between the storage modules should be greater than 0.23 in order to prevent a wave runup when the wavelength λ/L < 0.67 However, the sea space and the number of mooring dolphins have to be increased if the storage modules are spaced farther apart Besides that, we have shown in Chapter that the undesirable interactions between the empty floating storage modules may occur at certain expanded channel spacings Hence, the choice for a suitable channel spacing involves a trade-off between the limited sea space, maximum loading capacity and minimum hydroelastic response of the floating storage modules Figure 5.15 shows that in order to mitigate undesirable interactions between empty floating storage modules, the channel spacing s/L should be in the range of 0.10 to 0.25 As s/L is proposed to be greater than 0.23 for d/h = 0.90 in Fig 6.11, the suitable channel spacing of the storage module, that minimises the response of the empty structure as well as avoids green water on deck when loaded to d/h = 0.90, should be s/L = 0.23 to 0.25 It is important to note that: • The proposed channel spacings are only meant for box-like floating storage modules sited in a narrow waterway or near shore under regular waves 139 Wave Propagation along Channel Formed by Storage Modules • The proposed channel spacing s/L = 0.23 to 0.25 is meant only for the case when the floating storage modules are loaded to d/h = 0.90 (FB = 1.5A) If the minimum allowable FB is at least 2A, green water would not occur and s/L can have a wider range of 0.10 to 0.25 as shown in Fig 5.15 6.8 Concluding Remarks We have investigated the behavior of the wave propagation through a channel formed by adjacently placed floating storage modules The numerical results agree well with the experimental test results as long as slamming of the storage modules does not occur The numerical model and computational technique are thus validated In the investigation of the normalised (wmax + η max ) / FB with respect to different wave angles, it is found that the worst-case scenario for the floating storage modules (spaced at s/L < 0.5) sited in a narrow waterway or near shore is that of the head sea condition The minimum freeboard should be at least 2A in order to avoid a wave runup on deck In contrast, the worst-case scenario for the storage modules sited in an open sea might occur under a beam sea condition and the minimum FB should be increased to 3A The positioning of the floating storage modules in narrow waterway allows for a larger d/h because the storage modules are shielded from wave approaching at angles θ < π / Based on the parametric studies carried out on the floating storage modules sited in a narrow waterway under a head sea condition (the worst-case scenario), it 140 Wave Propagation along Channel Formed by Storage Modules is found that the wave elevations along the channel are affected by the channel spacing, floating breakwater and water depth The wave elevation along the channel was found to increase with decreasing channel spacings Under normal condition, i.e without a parapet wall and the presence of floating breakwater, the maximum allowable d/h is about 0.80 leaving a freeboard FB/h of 0.20 This FB/h is equivalent to twice the incident wave amplitude (assuming that A = 0.01m, h = 0.1m) By having a parapet wall around the deck of the floating storage modules, the freeboard FB is easily enhanced and thereby the maximum allowable d/h can be increased correspondingly With the presence of floating breakwater protecting the floating storage modules, the wave elevations along the channel decrease significantly especially when the wavelength λ/L is smaller than 0.67 As a rule of thumb, the minimum freeboard FB of the floating storage modules placed in a narrow waterway or near shore has to be at least twice the incident wave amplitude (2A) to avoid green water on deck For floating storage modules located in an open sea, a wave runup would not occur as long as the minimum freeboard FB is at least 3A Should FB be designed smaller than the minimum value, the wave runup on the deck could be prevented by adopting a suitable channel spacing This channel spacing should be chosen such that the normalised (wmax + η max ) / FB is less than 1.0 in order to avoid wave runup on deck The selected channel spacings should be able to minimise the undesirable response of the structure when empty as well as to avoid a wave runup on deck when loaded to the maximum draft d/h By considering the case given in Fig 6.11 where the maximum d/h = 0.90, the channel spacing that 141 Wave Propagation along Channel Formed by Storage Modules allows for a trade-off between minimum wave response and prevention of green water on deck is s/L = 0.23 to 0.25 However, if the minimum freeboard is at least 2A, the channel spacing could take a wider range of s/L = 0.10 to 0.25 as shown in Fig 5.15 The effect of water depth on the wave runup along the channel was also investigated As the design of the allowable freeboard depends on the wave elevation and deflection of the floating storage modules, we investigated the effects of water depth on the maximum plate deflection and wave elevation separately When subjected to wave periods smaller than 0.6s, we found that the design of the allowable freeboard is dominated by the maximum wave elevation and not the module deflection since the hydrodynamic response of the loaded floating modules is very small We observed that the water depth has negligible effect on the normalised (wmax + η max ) / FB of the floating storage modules as long as the difference in the corresponding wavelength λ/L is smaller than 0.1, that is when T < 0.6s In sum, the developed numerical model and computational technique enable engineers and naval architects to investigate the behavior of the wave elevations along the channel formed by the floating modules placed side-by-side The computed wave elevations are needed for designing the allowable freeboard, maximum draft and suitable channel spacing The computational model can also handle the presence of floating breakwaters 142 Wave Propagation along Channel Formed by Storage Modules Upon determining the suitable channel spacing, the steady drift forces acting on the adjacently placed floating storage modules are then computed based on the pressure distribution method These steady drift forces are necessary for designing the mooring system that restraints the floating modules from horizontal displacements This study on drift forces will be conducted in the next chapter 143 ... Section 6. 6 The choice for the channel spacing involves a trade-off between minimising the hydroelastic response of empty floating modules and maximising the loading capacity of the floating modules. .. floating storage modules It is to be noted here that, we only consider two floating storage modules placed side-by-side in our numerical model (and not an array of adjacently placed storage modules. .. purposes, the floating storage modules are assumed to be located in an open sea and the wave can arrive from any direction From Figs 6. 7 and 6. 8, the largest (wmax + η max ) / FB of the floating modules