**5. Results and Discussion**

*5.1. Side-Hulls Inhibition of Porpoising Instability in High-Speed Crafts*

To verify that the release of side-hulls can inhibit porpoising instability in high-speed crafts, simulations for the initial designed TFS (Table 1) advancing in calm water at *F*r = 2.73

(condition two) and 2.52 (condition three) were performed, and the oscillation curves with time of *τ* and *R*<sup>T</sup> were attained. Based on the previous computations of the MFS, the oscillation comparisons of the *τ* and *R*<sup>T</sup> in the MFS and TFS were conducted; the results under condition two are presented in Figure 16.

The *τ* and *R*<sup>T</sup> of the TFS exhibited minimal oscillation, but that of the MFS was significant, which indicates that the side-hulls could effectively inhibit porpoising instability.

To ascertain why porpoising occurs, the pressure distributions of the hull bottom in the MFS and the TFS were compared, and results are shown in Figure 17. A weak pressure area arose at the fore of hull bottom in the MFS, indicating that the fore moment *M*<sup>f</sup> (Figure 18) was relatively less than that of the rear moment *M*r, and due to the high trim angle (Figure 16a) and low lift coefficient, porpoising occurred, the specific cause of this are explained in Figure 19.

**Figure 16.** Oscillation comparisons when *L*cg/Lm = 0.38: (**a**) trim angle; (**b**) total resistance.

**Figure 17.** Comparisons of the hull bottom pressure in the MFS and TFS when *L*cg/Lm = 0.38.

**Figure 18.** Forces and moment acted on the hull surface in the TFS.

When porpoising occurred, releasing the twin side-hulls into water caused them to acquire more lift and longitudinal moment *M*<sup>d</sup> to the CG, which is equivalent to increasing the lift of hull bottom and the rear longitudinal moment *M*<sup>r</sup> to the CG; thus, the higher trim angle was decreased (Figure 16a). A strong pressure area arises at the fore of the hull bottom, so the porpoising was inhibited, the detailed explanation is shown in Figure 20.

When porpoising occurs in the MFS and TFS, the pressure from stern to bow on the keel line was acquired by using probe technology; the results are summarized in Figure 21. For the MFS, the maximum pressure mainly concentrated near the CG; when the location ratio *l*/Lm exceeded 0.5, the pressure at the fore area reduced sharply, which led to porpoising. Conversely, for the TFS, most of the maximum pressure was located in the fore area (*l*/Lm

> 0.5), and the pressure at the CG and rear was relatively lower, which further explains the inhibition of the side-hulls on porpoising.

**Figure 19.** Why porpoising occurs.

**Figure 20.** How side-hulls inhibit porpoising.

**Figure 21.** Pressure distribution from stern to bow on keel line in the MFS and TFS.

#### *5.2. Influence of Side-Hulls on Sailing Attitudes and Hydrodynamic Performance at Different Speeds*

In this section, adopting the same CFD setup and the whisker spray equation of Savitsky [27], computations for the initial designed TFS (Table 1) sailing at different speeds were carried out, and the influence of the side-hulls on sailing attitudes and hydrodynamic performance were analyzed. The dimensionless resistance (*C*<sup>T</sup> = *R*T/Δ) and sailing attitudes at different speeds in the MFS and TFS when *L*cg/Lm = 0.38 are shown in Figure 22.

**Figure 22.** Change in total resistance and sailing attitudes with speed when *L*cg/Lm = 0.38: (**a**) total resistance; (**b**) sinkage and trim angle.

Figure 22a shows that the *R*<sup>T</sup> of the boat in the TFS was larger when crossing the resistance peak, and during the high-speed planing stage, the *R*<sup>T</sup> in the TFS was relatively larger than that in the MFS at an equal speed, indicating releasing the side-hulls into the water could bring more resistance when porpoising. For the main hull resistance *R*<sup>M</sup> in the TFS, the changing trend with increasing speed was similar to the MFS, and when *F*r > 1.89, the *R*<sup>M</sup> even surpassed its previous resistance peak value.

In addition, Figure 22b shows in any navigation stage, the trim angle in the TFS was smaller than that of the MFS at equal speeds, especially during the draining stage, the discrepancy was more obvious. The sinkage of TFS in the draining stage was larger compared with the MFS; however, during the planing stage, the sinkage of the MFS was relatively larger. This was mainly because most of the volume of the side-hulls was submerged into the water during the draining stage; the side-hulls could acquire more lift and lead to the further lifting of the CG. When entering the planing stage, the sidehulls were gradually lifted out of the water with increasing speed, as shown in Figure 23; the reduction of side-hulls immersion volume made them acquire less lift, which is mainly used to generate more rear moment to adjust the trim angle; only a fraction of the lift is used to raise the CG of craft.

**Figure 23.** Volume fraction of water for the twin side-hulls at different speeds: (**a**) *F*r = 0.84; (**b**) *F*r = 1.26; (**c**) *F*r = 1.68; (**d**) *F*r = 2.1.

Figure 24a presents the change with speed of the side-hulls resistance (*R*d) and the moment of side-hulls to the CG (*M*d), which shows the *R*<sup>d</sup> and *M*<sup>d</sup> gradually increased with the increase of speed during the draining stage, but when entering the planing stage, both the *R*<sup>d</sup> and the *M*<sup>d</sup> had a significant decline due to the reduction of the side-hulls immersion volume; but when *F*r > 1.89 the *M*<sup>d</sup> increased as speed further increased, indicating porpoising could be inhibited by the side-hulls.

**Figure 24.** Forces and moment acted on the hull in the TFS at different speeds: (**a**) resistance and moment of the side-hulls; (**b**) lift of the side-hulls and the main hull.

Moreover, Figure 24b shows that the changing trend of the side-hulls lift (*C*Nd) was consistent with that of the *C*Md. For the main hull lift (*C*Nm) in the TFS, with the increase of speed the *C*Nm reduced during the draining stage; but when entering the planing stage, the *C*Nm exhibited an obvious rise; when *F*r > 1.89, the upward trend gradually slowed.

Figure 25 shows the free surface of the vessel in the TFS when porpoising; we observed that the main hull was lifted very high due to the strong hydrodynamic lift during the high-speed planing stage. At the same time, only the rear part of the side-hulls was slightly immersed in the water, so the interference influence on the side-hulls, caused by the ship traveling wave of the main hull, was lesser, the wave surface change mainly concentrated on the stern wake.

**Figure 25.** The free surface of the vessel in the TFS when porpoising.

Based on the attained *Z*cg and *τ* values, the waterline surfaces (WS) of the two navigation states at different speeds were intercepted and compared utilizing the Creo software; the results are shown in Figure 26. As the speed increased, the root of the WS gradually moved backward to the stern, and the WS became sharper and longer; the area of WS also decreased, causing porpoising. However, the release of side-hulls delayed the backward movement of the root, lengthened the WS, and the stagnation points moved forward even far beyond the MFS, which further enhanced the longitudinal stability during high-speed navigation.

#### *5.3. Influence of Longitudinal and Vertical Side-Hull Locations on Inhibiting Porpoising*

The longitudinal and vertical locations of the side-hulls had a profound impact on porpoising instability. To simulate the location adjustments of twin side-hulls in the two directions, we chose the TFS, whose side-hulls were installed at *a*<sup>t</sup> = 0.23 m, *b*<sup>t</sup> = 0.548 m and *c*t = 0.07 m (No. 3 in Table 8) as the basic research object.

5.3.1. Location Adjustments of the Twin Side-Hulls

Figure 27 shows the side-hulls adjustment mode of the longitudinal location. The ratio a/Lm was adjusted and the longitudinal location of side-hulls, the Lcg, and the longitudinal inertia tensor (*I*yt) of the TFS were changed. Then, six different longitudinal locations of the side-hulls are designed and shown in Figure 28a, corresponding to the NOs. 1–6, as listed in Table 8. Besides, for condition two, the *L*cg and *I*yt of the TFS after adjustment was measured by utilizing the Creo software, presented in Table 8.

**Figure 26.** Waterline surface comparisons of the two navigation states when *L*cg/Lm = 0.35.


**Table 8.** Six longitudinal locations of side-hulls in the TFS.

**Figure 27.** Longitudinal adjustment mode of the twin side-hulls.

 Figure 29 shows the vertical adjustment mode of the twin side-hulls, in which the solid and dashed lines separately represent the two positions before and after adjustment, *d* represents the waterline position relative to the Base line (BL) of the main hull, *c* is the distance between main hull BL and demihull BL, Δdh represents the variation of BL position for the main hull before and after adjustment. Six different vertical locations of the side-hulls were obtained, shown in Figure 28b, corresponding to the six draft ratios *D*d/Tm (NOs. 1–6) listed in Table 9, where Sm and Sd represent the waterline (WL) areas of the main hull and twin side-hulls, respectively.

**Figure 28.** Different (**a**) longitudinal locations; (**b**) vertical locations of the side-hulls in the TFS.

**Figure 29.** Vertical adjustment mode of the twin side-hulls.

**Table 9.** Six vertical locations of the side-hulls in the TFS.


5.3.2. The Optimal Range of Side-Hull Locations on Porpoising Instability

To achieve the inhibition of porpoising at the cost of lower resistance, the *R*<sup>T</sup> and *τ* in the TFS with different side-hulls longitudinal and vertical locations were computed, and the positive location ranges are presented in this section. Moreover, to evaluate the inhibition effect on porpoising, the dimensionless oscillation amplitude of trim angle (C<sup>τ</sup> = |*τ*-*τ*av|/τav) and

the average resistance (*C*Ra = *R*av/Δ) were introduced. As shown in Figure 30, for the *C*<sup>τ</sup> prescribing that when the side-hulls were at the optimal location (*C*τ < 2%), the porpoising was completely inhibited; at the weak location (2% < *C*τ < 10%), the craft was regarded as achieving the stable navigation; at the unfit location (*C*τ > 10%), porpoising occurred; for the *C*Ra prescribing when *C*Ra < *C*Ra \_m (*R*av\_m/Δ), the side-hulls were at the optimal location, here the *R*av\_m is the average resistance of the MFS when porpoising; when *C*Ra \_m < *C*Ra < 1.5 *C*Ra \_m the craft suffered more navigation resistance; when *C*Ra > 1.5 *C*Ra \_m, sailing forward in the TFS required too much thrust and was uneconomical.

**Figure 30.** The optimal, weak, and unfit locations: (**a**) trim angle; (**b**) half total resistance.

According to the six longitudinal locations (Nos. 1–6 in Table 8) and the six vertical locations (Nos. 1–6 in Table 9) of side-hulls in the TFS, adopting the spatial sampling method of full factor design [40], simulations for the TFS with 36 different side-hull locations under condition two were performed, and the results are shown in Table 10.

**Table 10.** Influence of side-hull longitudinal and vertical locations on the inhibition of porpoising instability and total resistance.


Table 10 shows with the backward movement of side-hulls, the probability that porpoising could be entirely suppressed increased. However, when the side-hulls were too backward (*a*/Lm = −0.1–0.1), along with an increasing draft, the average resistance *R*av also increased, and when draft ratio *D*d/Tm exceeded 0.442, the craft frequently suffered too much resistance, which inversely consumes more thrust. In addition, when side-hulls were relatively forward (*a*/Lm = 0.2–0.4) and the draft ratio exceeded 0.442, the probability that porpoising would occur, increased significantly. The more forward location (*a*/Lm = 0.4) is not recommended owing to the significant risk of porpoising. Considering the *C*<sup>τ</sup> and CRa, the draft of the side-hulls is not recommended to exceed the ratio (*D*d/Tm) of 0.442.

Summing up the above, the positive range of the side-hull location was −0.1 ≤ *a*/Lm ≤ 0.3 and 0.274 ≤ *D*d/Tm ≤ 0.386. As shown in Figure 31, in the preferred area, the craft could bear slightly more resistance and sail stably, while in the optimal area, porpoising was suppressed, and the boat could sail in the sea with lesser resistance.

Figure 32a shows that side-hulls being placed relatively forward (*a*/Lm = 0.4) was not conducive to inhibiting porpoising. For the side-hulls, when backward relative to the CG of the main hull, the deeper the draft in the vertical direction, and the greater the total resistance (Figure 32b), then the side-hulls resistance and the ratio of *R*<sup>d</sup> to *R*<sup>T</sup> also increased when porpoising.

**Figure 31.** Optimization of longitudinal and vertical side-hull locations on porpoising instability inhibition and resistance reduction.

**Figure 32.** *Cont*.

**Figure 32.** Influence of the longitudinal and vertical side-hull locations on (**a**) *C*τ, (**b**) *C*Ra, (**c**) *C*Rd, (**d**) *R*d/*R*T, (**e**) *C*Md, and (**f**) *C*Nd.

Figure 32e shows that when *a*/Lm > 0.3, the *C*Md becomes negative, porpoising occurs at the unfit location (Figure 32a); as the side-hulls move backward, there was an obvious increase in *C*Md, which further proves the previous analysis.

Figure 32f shows the more backward locations of the side-hulls relative to the CG of the main hull was more conducive to the increase of the side-hulls lift due to the planing attitudes; however, the influence of the side-hull draft on its lift was somewhat messy.

#### *5.4. Inhibition of the Side-Hulls on the Porpoising of Real Ship with Scale Effect*

To determine the impact at the scale of a real ship (RS) of the side-hulls being located in the positive range on restraining porpoising with a lesser resistance cost, the RS model (scaling λ = 2.5) in the MFS and TFS (cases 1–4 in Table 11) when porpoising were simulated, and the total resistance *R*T, effective power *P*<sup>w</sup> curves with speed increasing of the RS were also offered in this section.

**Table 11.** The main geometric parameters of the real-scale MFS and cases 1–4 under condition two.


Considering higher solution cost caused by enlarged computational domain and the model, the grid parameters in Section 4.1 were enlarged in proportion to the λ, and boundary layer grids were further refined to ensure the consistency of *y*+ on the hull surface. Then, as in the previous CFD setup, computations for the MFS and cases 1-4 were completed.

The oscillation comparisons of *τ* and *Z*cg are shown in Figure 33, further processing the results, the dimensionless oscillation amplitudes of trim angle and sinkage (*C*<sup>Z</sup> = |*Z*cg − *Z*av|/*Z*av) were attained, and based on the acquired *τ*av, *τ* (maximum), and the whisker spray equation of Savitsky [27], the total resistance (*R*av) including *R*S, the maximum resistance (*R*am) and the percentage of resistance increment (*PRI* = (*R*av − *R*av\_m)/*R*av\_m%) on releasing side-hulls when porpoising were solved and listed in Table 12.

Figure 33 shows that on the scale of a real ship, releasing the side-hulls in the positive range could significantly reduce the pitch and heave oscillation when porpoising, and the oscillation amplitudes of the more forward side-hulls placement (case 4) were larger compared with cases 1–3. Table 12 further shows the more backward side-hulls placement brought a greater resistance increment, especially for case 1, the *PRI* attained was 27.04%, so to avoid increasing too much resistance of RS, longitudinally, it is recommended not to have a ratio less than *a*/Lm = 0.1.

**Figure 33.** Oscillation comparisons of (**a**) *τ* and (**b**) *Z*cg when porpoising under condition 2.

**Table 12.** The comparisons of the dimensionless oscillation amplitude of trim angle (Cτ), dimensionless oscillation amplitudes of sinkage (*C*Z), total average resistance (*R*av), maximum resistance (*R*am), and percentage of resistance increment (*PRI*) under condition two.


In addition, through the deliberation to *C*τ, *R*av, and *R*am in Table 12, case 2 was chosen as the better side-hulls placement. And utilizing the verified CFD method and the whisker spray equation of Savitsky [27], *R*<sup>T</sup> and *P*<sup>w</sup> of the RS in the MFS and TFS (cases 2) at different speeds were computed, the *R*<sup>T</sup> and *P*<sup>w</sup> curves with speed increasing under conditions two and three, are shown in Figure 34, which can provide a sufficient reference for the overall design of planing boats with this concept.

**Figure 34.** The *R*T, power (*P*W) curves of the real ship in the MFS and TFS under (**a**) condition two, (**b**) condition 3.

### **6. Conclusions**

In this study, the hydrodynamics of twin side-hulls and their inhibiting effect on the porpoising of planing boats were analyzed. Based on the CFD method, the whisker spray equation of Savitsky [27], and the test, a comparative analysis was conducted on the hydrodynamic performance of the vessel in the MFS and TFS. The optimal location range of the side-hulls for porpoising inhibition and lesser resistance was also provided.

The CFD setup can accurately forecast the sailing attitudes of the model during the planing regime, but the forecastability of the total resistance during the high-speed planing stage was weaker. The application of the whisker spray equation of Savitsky [27] allowed the deviations of amendatory resistance to be controlled within 7%, which indicates the calculation method utilized in this research could effectively forecast the total resistance, including spray resistance and sailing attitude of the vessel during the high-speed planing stage.

The weak pressure area at the fore of hull bottom in the MFS causes the fore moment to be lesser than the rear moment, coupled with the existence of high trim angle and the low lift coefficient, porpoising occurs. Releasing the side-hulls in the water increased the total lift of the hull bottom and the rear longitudinal moment; thus, the high trim angle could be decreased, and a strong pressure area at the fore inhibited porpoising instability.

The comparisons of the MFS and TFS show that releasing the side-hulls into the water was conducive to inhibiting porpoising, but sailing in the TFS yields more navigation resistance when crossing the resistance peak and during the high-speed planing stage. In addition, at any stage, the trim angle of TFS was smaller compared with the MFS at equal speeds, and the sinkage of the TFS in the draining stage was larger, but during the planing stage, that of the MFS is relatively larger.

Side-hull longitudinal locations exceeding the ratio of *a*/Lm = 0.3 are not recommended, and vertically, the draft ratio (*D*d/Tm) is suggested not to exceed 0.442. The positive range of side-hull locations is −0.1 ≤ *a*/Lm ≤ 0.3 and 0.274 ≤ *D*d/Tm ≤ 0.386, and in the optimal area, porpoising could be suppressed; the boat can also then sail with lesser resistance. In addition, at the scale of a real ship, longitudinally, the side-hull locations are recommended not to be less than the ratio *a*/Lm = 0.1 to avoid increasing resistance.

**Author Contributions:** Conceptualization, Y.S. and J.W.; methodology, J.W.; software, J.W. and X.B.; validation, X.B., J.W. and Y.S.; formal analysis, J.W.; investigation, J.W.; resources, J.Z.; data curation, X.B.; writing—original draft preparation, J.W.; writing—review and editing, J.W. and J.Z.; visualization, J.Z.; supervision, Y.S.; project administration, Y.S.; funding acquisition, J.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research reported here was supported by the National Natural Science Foundation of China (Grant No. 52071100). The authors would like to express their gratitude to all the test participants for their suggestions and observations that helped in improving the present research.

**Data Availability Statement:** Data is contained within the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

