Numerical investigation of the full-scale resistance of a zero-emission fast catamaran in shallow water

Maritime Safety Research Centre (MSRC), Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, G4 0LZ, United Kingdom. Computational Fluid Dynamics Department, Hamburg Ship Model Basin (HSVA), Hamburg, Germany. Ship Design Laboratory, National Technical University of Athens (NTUA), Athens, Greece. Uber Boat by Thames Clippers, London, United Kingdom. Email address for correspondence: evangelos.boulougouris @strath.ac.uk.


Introduction
Low-carbon, environmentally-friendly maritime transport is playing an important role in reducing the emission of greenhouse gases and building a sustainable future. The need for technological innovations in the design of zero-emission ships is posing challenges for the maritime industry in the coming decades.
The research presented herein was conducted in the European Commission (EC) funded research project TrAM (Transport: Advanced and Modular, https://tramproject.eu/), which aims at designing and manufacturing battery-powered fast catamarans operating in coastal areas and inland waterways by implementing modular design and production methods. Given the significant lower specific energy content of batteries compared to conventional fuels [1], the design of zero-emission high-speed marine vehicles poses unique challenges and limitations which are tackled within the TrAM project. These include the selection of the appropriate battery technology and specification, safety considerations and of course, multi-objective hull form optimisation in the presence of shallow water effects [1][2][3][4]. The present study is focused on the battery-driven, zero-emission 'TrAM London Demonstrator', designed for The Thames River. It examines the hydrodynamic performance of the preliminary design of this high-speed catamaran in shallow water as it affects directly the rate by which the vessel consumes the stored energy. Therefore, it verifies and validates the computational methods employed in the hydrodynamic optimisation of the hull form.
Catamarans, due to their favourable performance in efficiency and stability at high speeds, have been widely studied experimentally, theoretically and numerically over the past decades [5][6][7]. A series of model tests were carried out by Insel and Molland [8] and Molland et al. [9] investigating the calm water resistance of fast catamarans with symmetrical demihulls, whereas Zaraphonitis et al. [10] have studied asymmetrical demihulls. Their studies emphasised the effects of demihull dimensions and separation distance on the resistances and motions of the catamarans over a wide range of Froude numbers (0.2 ≤ Fn could increase the total resistance by up to 30%. The interference effects were less strong at very low and very high Froude numbers (Fn < 0.3 or Fn > 0.7). Zaraphonitis et.al. [17] studied the optimisation of the hull shape with regards to powering and wash for a high speed catamaran. Souto-Iglesias et al. [18] also investigated experimentally the interference phenomenon of a catamaran and compared the wave systems created by the catamaran and the corresponding monohull. They concluded that the non-centred inner wave cuts are also important evidence for the analysis of wave interference. Later, Souto-Iglesias et al. [19] further studied the influence of demihull separation and testing condition on the interference resistance of a Series 60 catamaran and found that the free sinkage-trim condition enhanced both the favourable and unfavourable interference effects compared with fixed condition cases. Danışman [20] found that the wave interference resistance between the demihulls could be considerably reduced by placing an optimised Centrebulb, which led to a favourable secondary wave interaction.
With the fast development of computer science and numerical methods, computational fluid dynamics (CFD) has become a feasible approach with sufficient accuracy to investigate ship hydrodynamics [21].
Various CFD solvers have been applied to examine the calm water resistance and seakeeping of both monohulls [22][23][24] and multihulls [25][26][27][28][29]. A combined experimental and numerical study was carried out by Zaghi et al. [30] to analyse the interference effects between the demihulls and the dependency on the separation of a high-speed catamaran. Two humps were found in the total resistance coefficient curves and the second one was much higher, corresponding to a stronger interference. Besides, a smaller separation distance led to a stronger interaction and a larger speed where the peak occurred. Broglia et al. [31] conducted a numerical analysis on the interference phenomena between the demihulls of the catamaran with emphasis on the validation of the CFD code and the Reynolds number effect. It was found that the numerical results agreed very well with the experiment in terms of resistance and wave cuts and the dependency on the scale effect was rather weak. He et al. [32] computationally investigated the effects of Froude number and demihull separation distance on the resistance and motion of the catamaran. They found that the resistance coefficient became higher at smaller separation distances, indicating stronger interference between the demihulls. Besides, the strongest demihull interaction occurred when Froude number is between 0.45 and 0.65 (0.45 < Fn < 0.65). When the Froude number is below 0.45 or above 0.65, the variation of the separation distance had a negligible effect on the resistance as well as the sinkage and trim of the catamaran. Haase et al. [33] proposed a novel CFD-based method for the prediction of full-scale ship resistance, which relied on the results of the model test experiment and CFD simulation at both model-scale and full-scale Reynold number. Farkas et al. [34] carried out a numerical study on the interference of resistance components for a Series 60 catamaran at medium Froude numbers, where the interference factor was decomposed into viscous interference and wave interference. They found that the form factor of the catamaran was independent of the Froude number, but decreased to the value of the monohull when the separation distance became larger. It was also observed that the viscous interference factor was independent of the Froude number but relied on the separation ratio of the catamaran.
The shallow water effects must be considered when designing ships for restricted waterways (e.g., inland rivers, canals). Previous studies regarding shallow water effects for monohulls [35][36][37][38][39] revealed that the depth Froude number ( = √ ⁄ , where is the ship speed, is gravity acceleration and is the water depth) is playing a key role in determining the performance of the vessel. A ship moving near the critical depth Froude number ( = 1.0) will experience a surge in total resistance coefficient and drastic changes in motions and wave patterns, which should be taken into account when passing through shallow water areas. In terms of catamarans operating in shallow water, several experimental and numerical studies are also available [40][41][42][43]. Molland et al. [44,45] experimentally investigated the resistance of a series of fast displacement catamarans in shallow water. Similar to monohulls, the catamarans experienced large increases in total resistance and wave elevation, and significant changes in sinkage and trim near the critical depth Froude number. The resistance increase was higher for the smaller water depth. Gourlay [46] theoretically predicted the sinkage and trim of various catamaran configurations in shallow water. It was found that the maximum sinkage and trim occurred at the trans-critical speed range. Lee et al. [47] designed and tested the shallow water behaviours of a small catamaran and further investigated the influence of the separation ratio between the demihulls on the resistance characteristics. The residual resistance coefficient surged near the critical depth Froude number and the sinkage and trim also varied significantly in the critical region. Castiglione et al. [48] studied the interference effects between the demihulls of a high-speed catamaran in shallow water using a CFD method. They concluded that for all separation ratios, the total resistance coefficients were significantly increased due to shallow water effects, with peaks achieved near the critical depth Froude number. However, at extreme subcritical and supercritical speeds, the total resistance coefficients in shallow water became smaller than the values of corresponding deepwater cases. It was also found that the interference factor reached its peak values around the critical speed and increased for smaller separation distances. Moreover, the sinkage and trim were also increased compared with deep water values and maximised at the critical speed.
Despite the extensive studies on the calm water resistance and interference of high-speed catamarans, CFD simulations on full-scale fast catamarans in shallow water are still rare. As aforementioned, the work in the present paper is part of the ongoing TrAM Project (https://tramproject.eu/) and the objectives of the current study are twofold: 1) validate the numerical methods and setups that will be employed in the hull optimisation stage, 2) investigate the shallow water effect on the calm water resistance, sinkage, trim and wave creation of the full-scale London Demonstrator catamaran using a CFD method. The rest of this paper is organized as follows: In Section 2, the geometry of the London Demonstrator and parameters used for analysis are presented. The computational methods are introduced in Section 3 and they are validated in Section 3.4. In the next section, the numerical results are given. The conclusions are drawn in the final section.

Catamaran geometry and dimensions
The London Demonstrator catamaran investigated in the present work is designed by the Maritime Safety Research Centre (MSRC) at the University of Strathclyde, which is a partner at the ongoing EU funded project TrAM (https://tramproject.eu/). The London Demonstrator is designed for The Thames River as a battery-driven, zero-emission passenger ferry. As the catamaran is still at the initial design stage, the geometry illustrated in Figure 1 is selected as a showcase validating the numerical methods and examining the shallow water effect. Some main dimensions of the London Demonstrator are summarised in Table 1, where is the length between perpendiculars.

Parameters for analysis
In ship hydrodynamics, Froude number is an important non-dimensional parameter measuring the speed of the vessel, which is defined as: where is the ship speed, is gravity acceleration. For ships advancing in shallow waterways, the Froude number defined based on the water depth is playing a significant role in determining the ship hydrodynamics: The hydrodynamic performance of the TrAM London Demonstrator is analysed by examining its total resistance ( ), sinkage ( ) and trim ( ). The total resistance is decomposed into the frictional component Similarly, the frictional and pressure resistance coefficients can be formulated as where is the water density, is the moving speed of the catamaran and ∈ ( , ) is the wetted surface area, where and are the static and dynamic wetted surface areas respectively. The frictional resistance coefficient can also be estimated by the ITTC 1957 correlation line formula 3 Methodology

Flow simulation
The unsteady incompressible turbulent flow in the present study is simulated by solving the unsteady Reynolds-Averaged Navier-Stokes (URANS) equations. The corresponding continuity and momentum equations can be formulated as where is the fluid density, and are the components of the position vector in Cartesian coordinate, ̅ and ̅ are the components of the mean velocity vector, ′ ′ ̅̅̅̅̅̅ is the Reynolds stresses and ̅ is the mean pressure. ̅̅̅ are the components of the mean viscous stress tensor, which can be written as where is the fluid kinematic viscosity.
The − SST turbulence model, which has been widely used for marine hydrodynamics [21,29,32], is employed as the closure for Eq. (6) and (7). Here, the flow governing equations including the turbulent model are solved using a finite volume method implemented in the commercial code Star CCM+ 14.06. 9 The Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) was used as the solution procedure, where the continuity and momentum equations are solved sequentially and then coupled via a predictorcorrector approach. The spatial discretisation was achieved using a second-order scheme while a firstorder scheme was employed for temporal discretisation since we are only focused on the final converged equilibrium state.

Free surface capturing
For marine hydrodynamics, the appropriate capture of the free surface is of great importance in order to accurately predict the wave height. In the present work, the volume of fluid (VOF) method in combination with the High-Resolution Interface Capturing (HRIC) scheme was adopted to calculate the wave elevation induced by the motion of the catamaran. To avoid the wave's reflection at the boundaries of the computational domain, a wave forcing method was used at relevant boundaries to guarantee that the wave is completely damped out when it reaches the domain boundary. The wave forcing length and relevant boundaries are illustrated in Figure 3.

Dynamic trim and sinkage
As the catamaran is advancing in the water, the surface of the hull will interact with the surrounding water, leading to a fluid-body interaction problem. In the present study, only the heave and pitch motions were allowed while the rest degrees of freedom were fixed. The Dynamic Fluid-Body Interaction (DFBI) method provided in Star CCM+ 14.06 package was employed to calculate the sinkage and trim of the catamaran according to the fluid forces and moments acting on the hull surface. As the overset grid strategy was used in the present study, the DFBI method was only applied to the demi-hull and its associated region (see section 3.2).

Coordinate system
In the present simulation, two different coordinate systems are used: an earth-fixed (global) system and a ship-fixed (local) system. The flow simulation was carried out within the earth-fixed coordinate system and the computed forces and moments were then transformed to the ship-fixed coordinate system whose origin was located at the centre of mass of the catamaran. The equations of the motion were solved based on the latest forces and moments using the DFBI method. The new position and velocity of the hull were then converted back to the earth-fixed system as the boundary condition for the flow simulation. After updating the position of the hull, the connectivity between the two sub-domains in the overset grid method was re-calculated accordingly.

Computational domain and boundary conditions
As the catamaran is geometrically symmetrical about its mid-plane, only one demi-hull was used for CFD simulation in order to reduce computational cost. Besides, the overset grid method was employed in the present study, i.e., the entire computational domain was decomposed into two regions: an inner region around the demi-hull (Hull Region) and an outer region forming the virtual tank (Tank Region). Figure 2 shows

Mesh generation
The CFD mesh used in the present study was generated using the automated meshing functionality in Star  To properly resolve the flow boundary layer, prism mesh layers should be used in the vicinity of the hull.
Here, the thickness of the turbulent boundary layer was estimated by where is the Reynolds number based on .
layer. As the wall function was used in the turbulent model, the distance of the first prism layer to the hull surface was targeted at y+ =100. Figure 4 demonstrates the computed y+ distribution on the hull surface and it can be observed that for both Froude numbers the y+ values are within the range (30 < y+ < 300) that the wall function can be appropriately applied.

NPL 4a02 catamaran
The first case used to validate the computational methods used in the present study was the NPL 4a02 catamaran from a series of model tests carried out by Molland et al. [9]. The same computational methods and mesh generation strategies presented in Section 3 were also applied here. The total number of mesh cells used for this validation case was around 4.7 million. Figure 5 demonstrates the total resistance coefficients, sinkage-to-draught ratios and trim angles as functions of the Froude number, from which it is observed that the computed results are in good agreement with the experimental data.

Stavanger demonstrator
The computational methods presented in Section 3 were further validated against the experimental data of the Stavanger demonstrator [2,3] measured in the Hamburg Ship Model Basin (HSVA). The geometry of the Stavanger demonstrator is illustrated in Figure 6 (a) and the mesh used for simulation is demonstrated in Figure 6 (b), which was consisted of about 11.4 million cells. The computational domain, boundary conditions and mesh system were generated in similar manners to those presented in Section 3. Table 2 compares the total resistance coefficients of the Stavanger demonstrator obtained from CFD simulations with that from physical model tests. It is seen that for the four speeds considered here, the difference between the present numerical result and the experimental data is within 1.5%.   The sinkage and trim are demonstrated in Figure 7 (b) and it is observed that the trim angle of the catamaran is always positive, i.e., the stern goes down for all speeds considered here. At lower speeds ( < 0.4), the trim angle of the London Demonstrator remains almost zero. When becomes higher than 0.4, it rises significantly and reaches its peak at = 0.575 where also achieves its maximum value.
In terms of the sinkage of the catamaran, it keeps positive (the hull moves downwards) until the is higher than 0.7. The largest sinkage is experienced at = 0.517, which is slightly smaller than the Froude number where the trim maximum is accomplished. It should also be noted that the significant changes in trim and sinkage occur when 0.4 < < 0.6, corresponding to the range where the total resistance curve varies. It will be shown in the following sections that these behaviours of resistance and motion are closely associated with the position and strength of the crests and troughs at the central plane of the catamaran.

Figure 8 compares the resistances and motions of the London Demonstrator in shallow water obtained
from the present calculation with those computed by HSVA using FreSCo+. It can be seen that the results from both solvers also agree very well with each other for shallow water scenarios. It is interesting to observe from Figure 8 (a) that experiences a hump at = 0.287, corresponding to a depth Froude number ( = 1.12) around the critical value. It has been widely acknowledged that fast catamarans will experience a dramatic surge in total resistance coefficient near the critical speed in shallow water [45,48].
However, the existence of such a hump in total resistance rather than the coefficient near the critical depth speed catamarans [45,48]. the catamaran's centre of mass starts to move upward and when > 0.35, the change rate of sinkage becomes less significant.

Figure 9 Comparison of resistances (a) and motions (b) of London Demonstrator in deep and shallow water (H = 2.15 m).
The resistance coefficients of the London Demonstrator in deep and shallow water are illustrated in Figure   10 and Figure 11 respectively. The total resistance coefficients ( ) are normalised using both static and dynamic areas and the differences are small for both deep and shallow water cases. Generally, the coefficients calculated based on the dynamic wetted area are slightly smaller and the difference only becomes noticeable for the highest speed ( ≈ 0.8). The frictional resistance coefficients ( ) of the catamaran in both deep and shallow water agree well with those predicted using the ITTC 1957 correlation line formula, indicating the frictional resistance is not significantly affected by shallow water. Moreover, for deep water cases shown in Figure 10, and experience multiple peaks as the increase of Froude number. The peaks at lower Froude numbers ( < 0.4) are higher than that at = 0.46. The total resistance coefficient drops significantly as the further increase of the advance speed. The present curve differs from those observed in some previous studies, where the humps at smaller Froude numbers were usually lower [9,13,30]. This may be associated with the exact hull form and configuration of the catamaran, which leads to a different wave interference between the demihulls. For the shallow water scenario (Figure 11), the resistance coefficient of the catamaran reaches its peak value around the critical depth Froude number and then declines dramatically as the moving speed increases. The maximum value in shallow water is approximately 2.4 times higher than that created in deep water. This ratio is smaller than the value obtained by Castiglione et al. [48] for a similar catamaran configuration, where the peak in shallow water is about 4.2 times larger than that in deep water.
Different from the hump of the curve in shallow water, as shown in Figure 9 (a), which is not commonly seen in previous papers, the dramatic increase of near the critical speed has been widely observed in both model tests and numerical simulations [45,48]. It is worth noting that the maximum total resistance coefficient does not correspond to the maxima of the total resistance, according to which the propulsion power should be installed. For the London Demonstrator examined here, the maximum total resistance is accomplished at the highest speed considered here (see Figure 9 (a)), where reaches its minimum value.

Wave patterns
The wave patterns created by the London Demonstrator at various speeds in deep water are demonstrated in Figure 12. The catamaran generates typical Kelvin wave patterns at lower speeds, which comprise of both transverse and divergent waves. As the increase of the Froude number, the amplitude and length of the induced wave also increase while the Kelvin wave angle becomes smaller. Besides, the divergent waves become dominant in the wave pattern at = 0.805. Figure 13 demonstrates the wave elevations of the catamaran in shallow water, which are profoundly different from those shown in Figure 12. As expected, when the depth Froude number is near its critical value ( = 1.0), the Kelvin wave angle is close to 90 degree and the critical wave is created at = 1.12, which is located right in front of the catamaran. The critical wave is normal to the advance direction of the vessel and its attitude is significantly elevated, which leads to the hump observed in the curve in Figure 8 (a) and the remarkable peak shown in Figure 11 (b). Besides, the critical wave significantly elevates the bow, creating the trim maxima observed from Figure 8  shallow water, the decrease of the Kelvin wave angle leads the intersection point of the bow waves created by the two demihulls to move astern, which will be more clearly observed from Figure 14 and Figure 15 as well as the wave cuts demonstrated in the next section.    The behaviours of the resistance, trim and sinkage discussed in the previous section can be better understood by analysing the interaction between the wave systems generated by the demihulls. Figure 14 shows a closer inspection of the wave interference between the demihulls in deep water. We can observe that at smaller Froude numbers (e.g. when Fn=0.575 Fn=0.805 are located slightly behind midship. As the Froude number increases to 0.575, the crest and troughs between the demihulls are moved further downstream, which has also been reported in previous studies [13,30]. In particular, the secondary wave troughs are generated near the stern with higher amplitudes, which leads to a larger sinkage at the stern, thereby creating the peak of trim as shown in Figure 7 (b).
Moreover, as discussed in Figure 9 (a), the pressure resistance reaches its maximum value at = 0.575, implying the wave interference is the strongest at this Froude number. When = 0.805, the wave troughs created due to the secondary wave interaction are moved behind the aft of the catamaran (see The wave interferences between demihulls in shallow water are demonstrated in Figure 15. Several significant differences from those in deep water can be observed. First, at trans-critical speeds ( = 0.23 and 0.287), wave interactions between the demihulls seem to be suppressed due to the creation of the critical wave in front of the catamaran (see Figure 13), i.e., the phenomenon of existing multiple crests and troughs within the inner region disappears. At supercritical speeds ( > 0.345), the three troughs observed in deep water (e.g., in Figure 14 when = 0.46) are not seen in shallow water cases. Instead, another two secondary crests are generated apart from the primary one at the catamaran's central plane.
As the Froude number increases, the wave crests are stretched and moved towards the stern. As previously discussed, both trim and sinkage will be decreased with the first crest moving midship. This trend will be further enhanced due to the creation of the secondary crests, i.e., at higher speeds, both the trim and sinkage in shallow water are smaller, as seen from Figure 9 (b).

Figure 15
The wave interaction between demihulls in shallow water.

Longitudinal wave cuts
The wave propagation within the inner region can be better understood by analysing the longitudinal wave cuts at the central plane of the catamaran as demonstrated in Figure 16. It is seen that the wave starts to come into being at the forward perpendicular (FP) for all cases except those at trans-critical speeds (   Figure 9 (a)). In shallow water, the first wave crest behind the bow is always higher than that created in deep water, especially near the critical speed. The difference is considered small only when the Froude number is greater than 0.575. Moreover, no noteworthy wave troughs are generated between FP and AP in shallow water, which significantly differs from those in deep water. Furthermore, the catamaran generates higher wave crests behind the stern in deep water while creating deeper wave troughs in shallow water.
As observed from previous wave patterns in Figure 14 and Figure 15, the catamaran generates a remarkable trough right behind the stern of the demihull. The magnitude of this trough can be more clearly demonstrated by the longitudinal wave cuts at the mid-plane of the demihull, as shown in Figure 17. In deep water, the magnitude of the trough reaches its maximum value at = 0.575, where the water level difference between FP and AP is also maximised. In shallow water, the trough's magnitudes at transcritical speeds are significantly larger than those in deep water. The maximum amplitude is achieved at = 0.287, where the critical wave is also created in front of the bow, resulting in a remarkably large difference between the water levels at the FP and AF of the catamaran. It is worth emphasising that = 0.287 and 0.575 correspond to the speeds where the maximum pressure resistance is produced in shallow and deep water respectively, as seen from Figure 9 (a). At supercritical speeds, the trough's amplitude in shallow water becomes smaller than that in deep water, which can be attributed to smaller sinkage and trim created in shallow water.

Cross flow fields
With the wave interference between the demihulls, the flow field created by the demihull becomes nonsymmetrical against its mid-plane, which will cause a transverse pressure gradient. This can further lead to a cross flow under the keel of the demihull, which is believed to be one of the main causes of the increase in total resistance [31]. The cross flow fields of the London Demonstrator are plotted in Figure   18, where the positive and negative velocities indicate that the flow moves to the outer and inner regions respectively. In deep water, the location and strength of the cross flow are closely associated with the wave interaction between the demihulls. At lower Froude numbers, multiple changes of the cross flow direction under the keel can be observed, which corresponds to the existence of multiple waves between the demihulls (see Figure 14 and Figure 16). With the increase of the Froude number, the strength and extension of the cross flow are significantly enhanced and the locations where the cross flow occurs is also moved towards the stern. This phenomenon was also observed by Zaghi et al. [30] and Farkas et al. [34]. Besides, the number of changes in the cross flow direction is also reduced with the increase of the speed. At higher Froude numbers, significant cross flows are also generated behind the stern. For shallow water scenarios, similar to the deep water cases, the strength of the cross flow is considerably enhanced and the location where the maximum cross flow occurs is also moved towards the stern with the increase of the speed. However, the cross flows created in shallow water are remarkably stronger than the corresponding cases in deep water. Moreover, the phenomenon of multiple changes in cross flow direction observed at lower Froude numbers no longer exists, and for all speeds in shallow water, the cross flow moves from the inner side of the demihull to the outer region.

Conclusion
In the present work, the hydrodynamics of a full scale, zero-emission, high-speed catamaran (London demonstrator) in both deep and shallow water was numerically investigated. The numerical methods used in the current study were validated against experimental data of the NPL 4a02 model [9] and the Stavanger demonstrator [2]. For numerical simulations on the London Demonstrator, a blind validation was also carried out in collaboration with HSVA and good agreement was accomplished. The resistance, sinkage and trim of the London Demonstrator as functions of Froude number (ranged from 0.2 to 0.8) in deep and shallow water were firstly analysed. The total resistance in deep water increased continuously while in shallow water, a hump was experienced at = 0.287 ( = 1.12). Besides, the total resistance in shallow water was higher when < 0.45 and became smaller at larger speeds. As the frictional resistance was almost the same in deep and shallow water, i.e., the difference in total resistance was mainly caused by the pressure component. The variations of the pressure resistance were closely related to the behaviours of trim and sinkage. In particular, the maximum trim was accomplished at the Froude number where the pressure resistance was maximised ( = 0.287 and 0.575 for shallow and deep water respectively). The largest sinkage in shallow water occurred at the lowest speed whereas in deep water the sinkage reaches its maxima at a Froude number ( = 0.517) slightly lower than the one where the maximum trim occurred. Furthermore, the total resistance coefficient curve in deep water showed multiple humps while only one significant peak near the critical speed was produced in shallow water.
The computed wave patterns, longitudinal wave cuts and cross flow fields were also analysed and correlated with the behaviours of the resistance and motion of the catamaran. In general, for both deep and shallow water scenarios, the crests and troughs generated within the inner region were strengthened and moved astern with the increase of Froude number. In deep water, the maximum pressure resistance was related to the creation of a secondary trough near the stern of the demihull. In contrast, the mechanism involved in shallow water was due to the generation of a critical wave in front of the catamaran and normal to the moving direction. Moreover, the creation of maximum pressure resistance was also correlated with the largest water level difference between the forward and aft perpendiculars. Cross flows occurred in both deep and shallow water scenarios due to the asymmetrical flow fields between the inner and outer regions. Compared with deep water cases, the cross flows created in shallow water were much stronger.
Moreover, the cross flow in shallow water moved towards the outer region for all speeds considered here, whereas, in deep water, changes in cross flow directions were observed.
the views of DNV and Royal Caribbean Group. Results of CFD simulations run by the University of Strathclyde were obtained using ARCHIE-WeSt High Performance Computer (www.archie-west.ac.uk).