3.1. Simulation Parameters
In this section, the relationships between motions of the trimaran and hydrodynamic forces were investigated. The influence from entry velocity and acceleration were studied by the method of variable separation.
For the constant velocity water entering cases, the specific velocities used in the simulation are shown in
Table 1 (in which V is −1.7 m/s).
For the constant acceleration water entering cases, five different accelerations were selected to investigate the effect from acceleration. The positive direction was set to be upward. The velocity at the time instant of contacting the water was kept the same, which was 2 V, i.e., −3.4 m/s. The specific parameters were set as shown in
Table 2 (in which g is −9.81 m/s
2).
All the simulations in this paper are run on the computer with Intel(R) Core(TM) i7-8700 (duo 3.2 GHz) CPU, RAM 32 GB. In terms of computational cost, for a typical case, the computational time was about 10 CPU hours.
3.2. Pressure Analysis
In this section, the pressure under the main hull and wet-deck were analyzed. The pressure time history under the abovementioned different velocity and acceleration conditions were plotted against the corresponding penetration depth, as in
Figure 5 and
Figure 6. It is worth mentioning that the two peaks in
Figure 5a and
Figure 6a correspond to the initial slamming of the main hull and the subsequent wet-deck slamming, respectively. It can be seen from the time history of pressures that the characteristics of the pressure were strongly correlated to the penetration depth, regardless of the time history of different types of penetration. More specifically, the increasing/decreasing patterns and the positions corresponding to the peak value were all the same under different entry velocities and accelerations, respectively. This is consistent with the commonly used slamming prediction theory, such as Wagner’s model [
3] or the MLM model [
18], as follows:
in which
is the penetration depth, and
and
have different definitions in Wagner or MLM models:
where
.
As can be seen from Equation (1), the pressure with any particular Y coordinate can be divided into velocity and acceleration dependents parts, in which the two parts do not affect each other and the corresponding coefficient for each part is only dependent on penetration depth.
For the velocity dependent part, the prediction by Equation (1) implies that with the increase of the speed, the pressure peak at each monitoring point increases quadratically, since the acceleration related parts were expected to be zero under the constant entry velocity condition. This is exactly what occurred for all the pressure monitoring points, as can be seen from
Figure 5b,d,f,h,j. This should be expected for the main hull area since its peaks occurred before the side hull entry, which means the main hull slamming would be the same as the slamming of a single hull, and this is exactly the scenario that these analytical models are based on. However, the flow pattern corresponding to the wet-deck area is quite different from the single hull water entry, i.e., the air gaps gradually close until the instant of the wet-deck slamming peak. The interesting point is that the pressure peak on the side of main hull (i.e., pressure monitor p2) and under the wet-deck (i.e., pressure monitor p3~p5) also followed the patterns predicted by the analytical models. This indicates that if the air is assumed to be evacuated freely (which is guaranteed in this study by carefully selecting the length in the X direction of the trimaran hull in
Section 2.2), the hydrodynamic mechanism behind these two types of slamming events is to some extent quite similar, which would need more in-depth analysis in a future study.
For the case of water entry with constant acceleration, as shown in
Figure 6a, the acceleration had almost no effect on the pressure peak (or indeed the time history before and after this instant) under the main hull (i.e., pressure monitor p1), which means the coefficient for the acceleration dependent part in Equation (1) is negligible (as will also be shown in
Section 3.3). In the area under the wet-deck, in order to further isolate the influence from acceleration on pressure, the velocity dependent parts, which was interpolated from the results in
Figure 5 at corresponding velocity for a particular pressure monitor, were extracted from the total pressure value. This manipulation was based on the analytical models as in Equation (1), in which the velocity and acceleration dependent pressure can be separated. As shown in
Figure 6g–j, the acceleration and pressure peak showed an almost linear dependency, and the upward acceleration (i.e., g and 2 g) would generate negative pressure distribution. These again are consistent with the model predictions in Equation (1). It should be noted that, in
Figure 5 and
Figure 6, the value of the pressure is very large, especially for the peak pressure, which is even more than 1000 kPa. This is partially due to the relatively high impact velocity of the hull, but more importantly, it is due to the process of the air gap collapsing under the wet-deck, which makes the slamming effect much stronger for trimaran structure compared to single blunt body.
Figure 7 and
Figure 8 show the pressure contours and free surface positions of the middle plane of the trimaran (i.e., Y–Z plane) at some typical penetration depths (i.e., 0.089 m, 0.155 m, and 0.178 m) under different entry velocities and accelerations, respectively. At the same depth of penetration, the position and shape of the free surface were almost identical regardless of the velocity or acceleration. Moreover, the patterns of the pressure spatial distribution at the same penetration depth show a high level of similarity as well, although the specific pressure values under higher velocities or accelerations were indeed higher as expected. It worth noticing that the pressure around the tip of the main hull at 0.089 m penetration for entry velocity of V (i.e., 1st row 1st column of
Figure 7) shows a slightly different distribution from the cases at the same penetration depth under different entry velocities, i.e., the pressure hotspot was not as significant as other corresponding cases due to the lower entry velocity.
Overall, this pressure and free surface characteristics are consistent with the above analysis, i.e., the pressure distribution is strongly correlated with the penetration depth regardless of the specific way of entry.
3.3. Force Analysis
The discussion for pressure distribution indicates that the initial slamming of the main hull is dominantly affected by the velocity and the influence from acceleration is negligible, whilst the wet-deck slamming would increase quadratically and linearly with velocity and acceleration, respectively. This pressure pattern fits well with the prediction by analytical models such as Wagner or MLM. In this section, the force characteristic on a thin section centered by the middle plane is further investigated based on those theories. More specifically, according to
Figure 4b,d, it can be seen that the pressure of the monitors 0.025 m before and after the middle plane (i.e., Y–Z plane) was approximately equal, indicating that the flow within this thin layer of the section is appropriate to be considered as two-dimensional. Therefore, the pressure under this area of the section was integrated for the force analysis [
3,
18,
19,
20]. As stated by Korobkin [
18], within the framework of many models that originated from Wagner’s model, by integrating the pressure (as in Equation (1)) along the body surface, the slamming force
can be formulated as the following formula:
where
and
are two functions that only depend on the penetration depth and the shape of the body, and are not affected by the particular penetration time history.
More specifically, the functions and take different forms due to the characters of the body shape and higher orders of pressure term in the Bernoulli equation within different analytical models.
As showed in the work of Seng [
20], CFD can be used to determine the specific forms of functions
and
for different single hull shapes (i.e., wedge and ship section). This would be more effective for cases involving complex body shapes, where analytical models tend to require complicated mathematic operations or are even unable to give direct solutions (such as for the case of trimaran). In order to extract the velocity and acceleration related functions
and
separately, the following procedures [
20] were used:
First, the entry velocity was set to be constant, which means the acceleration related part in Equation (4) is zero. Hence, the velocity related part can be extracted as.
Second, the body was forced to penetrate the water with constant acceleration, and then, by using the velocity related function
obtained in previous step, the acceleration related function
can be calculated by.
The vertical force and the extracted
curve are shown in
Figure 9a,b, respectively. As mentioned in
Section 2.2, the CFD calculated forces were obtained by integrating within the 0.05 m thick layer around middle plane. This force was then converted to the scale of the 0.3 m thick trimaran (i.e., by multiplying with a factor of 6). The same manipulation was used for the analysis with constant acceleration penetration later.
As shown in
Figure 9, the curve extracted from the cases with different entry velocities was almost the same, which shows the suitability of this slamming force decomposition model to a complex shape such as a trimaran for both initial main hull slamming and wet-deck slamming.
The vertical forces and resulting function
are shown in
Figure 10a,b, respectively.
As shown in
Figure 10b, before the wet-deck slamming (i.e., penetration depth at around 0.14 m), the acceleration related functions from different velocity-acceleration combination cases are very consistent, and the value is almost zero. This indicates that before the wet-deck slamming, including the initial water entry of the main hull, the acceleration effect on the slamming force is negligible, which is consistent with the findings in
Section 3.2. In the instant of wet-deck slamming, the function started to oscillate and the pattern from different accelerations was consistent with each (though not as repeatable as the curve before wet-deck slamming), but the value was still significantly smaller than
.
Based on the above analysis, this force decomposition theory is generally applicable for the slamming analysis of trimaran shape body, and the overall effect from velocity on the dynamics of the slamming process is considerably more significant than that from acceleration. This is also consistent from the pressure discussions in
Section 3.2.