4.1. Resistance, Sinkage and Trim
As the present paper aims to validate the numerical methods adopted for simulation, a blind validation study was carried out by MSRC and HSVA, where the commercial solver Star CCM+ 14.06 was used by MSRC, whereas an in-house code FreSCo+ [
51] was employed by HSVA. The number of mesh cells for both simulations was approximately 6 million.
Figure 7 shows the resistances and motions of the full-scale London Demonstrator in deep water, from which we can observe that very good agreement is accomplished between the present results (Strath) and those from HSVA. It is seen from
Figure 7a that the total resistance (
) rises monotonously as the speed of the catamaran increases. The relation between
and
is almost linear when
< 0.4. A continuous change in the slope of the total resistance curve can be observed when 0.4 <
< 0.6, which is also reported by Zaghi et al. [
30] in the same Froude number range and indicates the experience of unfavourable interferences. The frictional component (
) also rises monotonously with increased speed, while the pressure component (
) experiences a peak of
= 0.575. Besides,
is the larger component at lower speeds whilst it becomes smaller when
is greater than 0.65.
The sinkage and trim are demonstrated in
Figure 7b, 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 of
= 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+. The results from both solvers also agree very well with each other for shallow water scenarios. It is interesting to observe from
Figure 8a 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 Froude number is rarely reported in previous studies.
rises monotonously after the hump (when
> 0.35) as the continuous increase of the frictional resistance. An inspection of
and
curves reveal that the hump comes from the pressure component of the resistance, indicating it is the consequence of wave interference between the demihulls. Unlike the total resistance,
declines after the hump and the frictional resistance exceeds
and becomes the larger part of the total resistance when
> 0.55. The existence of such a hump in the
curve should be carefully considered in the design of the catamaran to guarantee that the installed power is sufficient to overcome the hump resistance in the process of accelerating the vessel to the designed speed. It is observed from
Figure 8b that the sinkage and motion of the catamaran change significantly near the critical speed, which agrees with previous studies on high speed catamarans [
45,
48].
Figure 9 compares the resistances and motions of the London Demonstrator in deep and shallow water. Hereafter, only the results computed using Star CCM+ 14.06 are used for further analysis. The total resistance in shallow water is higher than that in deep water at smaller Froude numbers (
< 0.45) because of the hump near the critical speed. When
further increases,
in shallow water becomes lower, due to the reduction of pressure resistance
. The frictional resistances in deep and shallow water are almost the same, i.e., the difference between the total resistance in deep and shallow water results from significantly different wave patterns and interferences, which will be demonstrated in the following sections. By comparing the motions of deep and shallow water cases, it is found that the maximum trim angles accomplished in deep and shallow water are close to each other (≈1.0 degree). However, the maximum of trim in shallow water is reached near the critical speed (
= 0.287), whereas the peak in deep water is achieved at
= 0.575. Similar to deep water cases, the maximum value of shallow water trim is also achieved at the Froude number where the pressure resistance peaks (
= 0.287). The sinkage of the catamaran in shallow water is larger in sub- and trans-critical ranges (
< 0.3), whilst in the supercritical region, the sinkage in shallow water becomes smaller than that in deep water, which leads to a considerable reduction in pressure drag, as observed from
Figure 9a. Furthermore, the sinkage of the catamaran in shallow water is positive at subcritical speeds. With the further increase of Froude number, the catamaran’s centre of mass starts to move upward and when
> 0.35, the change rate of sinkage becomes less significant.
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 with 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. Unlike the hump of the
curve in shallow water, as shown in
Figure 9a, 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 9a), where
reaches its minimum value.
4.2. 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 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 degrees, 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 8a and the remarkable
peak, shown in
Figure 11b. Besides, the critical wave significantly elevates the bow, creating the trim maxima observed from
Figure 8b. Behind the stern of the vessel, divergent waves are generated. As the moving speed increases to the supercritical range, the critical wave disappears, and divergent waves are created near both the bow and stern of the hull. The further increase of the Froude number reduces the angles of the divergent waves. However, the overall wave patterns are not significantly changed. In both deep and 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
< 0.3), multiple crests and troughs exist within the inner region between the two hulls. Enhanced crests and troughs become pronounced when
= 0.345 at the symmetry plane of the catamaran, where the waves meet and strengthen each other. At
= 0.46, another two troughs are generated on each side of the symmetry plane apart from the one created at the central plane, indicating a significant secondary wave interference. At this Froude number, the secondary troughs 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 7b. Moreover, as discussed in
Figure 9a, 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
Figure 12), which leads to a decrease in the trim as the secondary troughs are closer to the hull surface, thereby having a more direct impact on the motion of the demihull. Another observation from the wave pattern at
= 0.805 is that the first crest in the inner region is produced near midship, which results in the reduction of the moment causing the pitch motion, leading to the decrease in trim angle. On the other hand, with the first crest further strengthened and moved near the catamaran’s centre of mass, this crest will lift the entire catamaran instead of the bow. Therefore, the sinkage becomes negative (the hull moves upward) at higher Froude numbers.
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 creating 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 creating the secondary crests, i.e., at higher speeds, both the trim and sinkage in shallow water are smaller, as seen from
Figure 9b.
4.3. 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 (
= 0.896 and 1.12) in shallow water, where the water is elevated at least 0.5
Lpp ahead of the catamaran and reaches the maximum height near the FP. In deep water, both the wave height and wave length increase as the Froude number rises, confirming the observations from
Figure 14. The increase of the wave length leads to a reduction in the number of waves between FP and aft perpendicular (AP). For example, there are approximately three waves between FP and AP when
= 0.23, while the number becomes less than one when
increases to 0.805. It is interesting to observe that at
= 0.575, the wave number between FP and AP is approximately unity and this Froude number corresponds to the maximum value of the pressure component of total resistance (see
Figure 9a). 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 trans-critical 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 9a. 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.