2. Hull Models and Methodology
Stepped hulls, a prevalent design feature in boats, have the potential to diminish resistance and wetted surfaces through flow separation at the bottom of the vessel. Nevertheless, the effectiveness of a stepped hull is contingent upon factors like step type, height, and location. This study employs numerical simulations to investigate the influence of steps on the performance of one- and two-stepped planing hulls in calm water, building upon previous research in this domain.
To scrutinize the planing hull’s behavior and the impact of steps, the parent hull (C1) of the Naples Systematic Series (NSS) was chosen as the foundational model. The C1 underwent design and testing at the naval division of the Department of Industrial Engineering of the
Università degli Studi di Napoli “
Federico II”.
Figure 1 illustrates the C1 model, and the key parameters of the hull are presented in
Table 1 [
31].
We selected this model for analyzing the effect of step height and position on performance prediction due to the availability of towing tank test results for the non-stepped hull. Additionally, the results of the interceptor’s impact on this hull are accessible, enabling readers to easily compare the results of the interceptor with the added step.
This hull is symmetrical along the Y-axis from the centerline which is why half hull is used to perform CFD simulations to save computational time.
Addition of Steps
The C1 hull served as the base model, and steps were incorporated into the hull. Previous studies indicate that steps can offer an effective solution for controlling the dynamic trim angle, mitigating longitudinal instability phenomena such as the porpoising effect [
23].
In this research, single and double steps were introduced by varying their locations, shapes, and sizes. Initially, nine single-stepped hulls were created, each with distinct locations, shapes, and sizes. Subsequently, the second step was added to the hull after fixing the first step, and both steps were adjusted by varying their locations and sizes, resulting in 15 stepped hulls. To facilitate a systematic comparison with the base model, a constant draft was maintained while adjusting hull displacement. This methodological choice aims to isolate the influence of displacement variations on overall performance, with the resistance coefficient employed as a standardized metric for comparative analysis. The resistance coefficient, being a normalized measure, accommodates the diverse hull configurations, ensuring a robust basis for evaluating and comparing the performance of distinct hull designs.
For better comprehension, the longitudinal view of all hulls is presented in
Figure 2 and
Figure 3. Initially, steps were placed in the center of gravity (COG), and then additional steps were introduced in the forward and aft sections of the hull. The variation in step height was also analyzed.
Table 2 provides the technical specifications of the stepped hull. The “Code” column contains different names for the modified hulls, while subsequent columns detail the hull mass and the location of the COG.
In
Figure 2, the first nine single-stepped hulls with varying locations, sizes, and shapes are shown. The design SS9-MD1 has different steps, as can be seen in the figure. Notably, its middle step has a slightly lower height than the sides. To facilitate ventilation and airflow from the steps, the edges are curved.
The double-stepped planing hull has a significant advantage in stability and resistance reduction. Therefore, modifications are considered to analyze the behavior of the hull. The hulls from DSS10-MD1 to DSS15-V3 are all double-stepped hulls with different details, which are presented in
Figure 3.
The CFD was performed in free motion and using the Dynamic Fluid Body Interaction (DFBI) approach to capture the hull motions. Specifically, two motions were allowed: dynamic trim and heave. Four different speeds are simulated (3.05 m/s, 6.0 m/s, 7.0 m/s, and 8.0 m/s).
4. Results and Discussions
The CFD analysis was performed on the single and double-stepped hull and the results were compared with the experimental results. Dimensionless parameters were considered for the analysis such as resistance, wetted surface, sinkage, and dynamic trim. A total of 15 stepped hulls were created from the parent hull of NSS and simulations were performed at four different speeds, 3.05 m/s, 6.0 m/s, 7.0 m/s, and 8.0 m/s, correspondent to = 1.13, 2.22, 2.59, and 2.96, respectively.
In the code column, the last letter of the code such as M, F, B, etc. shows the location and depth of the steps.
The “M” represents that step is placed in the mid of the hull.
The “F” represents that the step is placed in the forward region of the hull.
The “B” represents that the step is placed in the backward region of the hull.
The “D” represents that the step has more depth, and the location is shown by the attached letter “FD” which means the step is in the forward section but has more depth than the previous step in the forward part of the hull.
“BH” is for the base hull.
The results of the dynamic trim are plotted in
Figure 7. In STAR CCM+, the dynamic trim angle is determined using Dynamic Fluid Body Interaction (DFBI). This involves simulating how the hull interacts with fluid flow, allowing rotation about the y-axis to calculate the trim angle accurately. It can be observed that as the speed increased, the dynamic trim also increased in the semi-planing region and reached its maximum at
< 2.22, except for “DSS10-MD1” and “DSS13-V1”, which both show a different trend. As the beam Froude number increased, the planing region was reached, and dynamic trim decreased for all the stepped hulls. The depth and location of the steps play very important roles in step design. Some steps do not have a perfect placement, which raises the dynamic trim as compared to the base hull. Most designs reduce dynamic trim at low speeds except “S8-M1”, “S3-F”, “S2-M”, “S4-B”, and “S1-M” and rose in the semi-planing region which was further reduced as the Froude number increased. “SS5-M”, “SS7-FD”, and “S8-M1” performed well in dynamics trim reduction.
The hull is considered in the planing region when sinkage values increase from negative to positive. When the hull moves out of the water vertically (upward direction on the z-axis), it is considered positive, and vice versa. At a low
, the sinkage is almost the same for all the stepped hulls but as the speed increases, the sinkage value starts rising positively upward in the z-axis as shown in
Figure 8 for higher speeds as well. This shows the relation of interaction between water and hull, as greater upward motion means less resistance.
The “S1-M”, “S2-M”, “S3-F”, and “S4-B” are the step designs that have a greater positive value of sinkage as compared to the base hull. The other designs reduce the sinkage value, as shown in
Figure 8. The “DSS10-MD1” and “DSS13-V1” have irregular behavior because the depth of the step is increased in these designs, which moves the center of pressure in the forward section.
The resistance of the planing hull is a crucial factor for the performance of the planing hull, especially for the stepped hull. The resistance of all stepped hulls and base hulls is plotted in
Figure 9. Total resistance depends on the wetted surface area and the shape of the hull, and water spray also plays its role.
Figure 9 shows the resistance coefficient (Equation (3)) of the stepped hull as compared with the base hull.
At lower beam Froude numbers, all stepped hulls show a lower resistance coefficient compared to the base hull, except for “S1-M”, “S2-M”, “S3-F”, “S4-B”, and “S8-M1”‘ which display higher values at lower Froude numbers. As the speed increases, “SS5-M”, “SS6-MD”, “SS7-FD”, and “SS9-MD1” initially experience a decrease in resistance, but a further speed increment leads to a rise in the resistance coefficient. Notably, “SS9-MD1” demonstrates low resistance compared to the base hull at = 1.13 and 2.22, followed by an increase at = 2.59, maintaining nearly constant values at = 2.96. With its streamlined design, this hull exhibits impressive resistance performance.
The wetted surface of the stepped hull at the four-beam Froude numbers was investigated. Typically, planing hulls move at high speeds and, due to the generation of hydrodynamic lift, the hull comes out of the water, reducing the wetted surface.
The plot for the wetted surface is in
Figure 10. Most stepped hulls reduce wetted surfaces, except for “S1-M”, “S2-M”, “S3-F”, and “SS5-M”. These four designs have slightly higher values or coincide with the base hull at all Froude numbers. The models “DSS10-MD1” and “DSS13-V1” have the lowest wetted surface at
= 1.13 and rise as the Froude number increases to
= 2.22, further increasing in speed
= 2.96 and reducing the values to 2nd or 3rd lowest. The inclusion of steps at significant positions and appropriate step heights will reduce the wetted surface. The augmented spray drag may offset the overall reduction, resulting in an increase in total resistance. In some scenarios, however, the placement and height of the step can lead to an overall increase in resistance.
4.1. Pressure Distribution Contours
The pressure distribution on the bottom surface of the hull plays a crucial role in hydrodynamic studies. The pressure of steps in the hull creates flow separation, which impacts the pressure distribution and results in a low-pressure region at the location of the step and the high-pressure region aft of the step. The pressure difference generates lift, reducing dynamic trim and sinkage while enhancing hull performance. The pressure distribution in
Figure 11 reveals that single-stepped hulls exhibit two pressure regions, while double-stepped hulls have three.
In this scientific context, hydrodynamic pressures are chosen as representative values.
Figure 11 shows the pressure distribution for all stepped hulls and the bare hull. These pressures can have both negative and positive values and are constrained within the range of −3000 to +3000 Pascal. By limiting the pressure range, a more accurate and comprehensive analysis of pressure variations across the entire hull can be conducted, leading to a better understanding of its hydrodynamic behavior. The same pressure restrictions are also applied to the bare hull for comparison and analysis. The steps influence the pressure distribution across the hull’s bottom. It is observed that regions before and after the step have higher pressure and the step itself experiences lower pressure due to the additional lift generated compared to the base hull. Some designs like “DSS10-MD2” at
= 1.13, “DSS10-MD1”, “DSS11-MD2”, “DSS13-V1”, and “SS6-MD” at
= 2.22, “DSS10-MD1” at
= 2.59 and
= 2.96 have more critical pressure distribution as compared to the base hull. The critical pressure distribution refers to a distribution of high or low pressure along the hull’s bottom, where the flow behavior undergoes significant changes and inappropriate pressure can lead to structural damage. Higher or lower pressure on the hull directly affects its structural integrity, potentially leading to stress concentration and deformation. Examining these pressure variations provides valuable insights into how bottom pressure affects structural integrity. Variations in bottom pressure can concentrate localized stresses, which may lead to fatigue, deformation, or even structural collapse over time. Low-pressure areas can cause buckling or collapse, while high-pressure areas can lead to bending of the hull material. Stepped hulls with less depth steps, as seen in the “S3-F” model at all speeds, may not effectively distribute pressure. Conversely, non-stepped hulls at
= 2.96 exhibit higher pressure, but with stepped hulls, the pressure distribution is more uniform and reduced.
4.2. Volume of Fraction Contours
The figure depicts the volume fraction of water for various stepped hulls. The value “1” corresponds to water, while “0” represents air. Values between “1” and “0” indicate an air–water mixture. The red portion represents water, the blue portion represents air, and the green portion is the air–water mixture. Potential overestimations of the ventilated part related to numerical ventilation issues are contained using the artificial VOF suppression scheme detailed in De Luca et al. [
32] and Mancini et al. [
37].
Under low speeds, there is minimal change in the water’s surface. However, as the speed increases, the wake becomes more distinct. Steps play a crucial role in reducing the wetted surface area. Compared to the bare hull, the addition of steps noticeably reduces the wetted surface area at all speeds.
Figure 12 presents the distribution of volume fraction along the bottom of the hulls, considering both stepped configurations and the bare hull, across a range of beam Froude numbers. At
= 2.96, the wetted surface of hulls “DSS11-MD2” and “DSS12-MD3” appears “W”-shaped. This shape is characterized by three triangles, with one in the middle and two on the port and starboard sides. The middle triangle represents the wetted surface caused by the solid wake from the forebody, while the other two triangles result from the spray, which increases hydrodynamic resistance, as noted by Savitsky and Morabito [
13]. Hence, optimal step design involves positioning it strategically to minimize the impact of the spray area, ensuring that the stagnation line avoids intersecting the steps, particularly in the forward planing surface.
5. Conclusions
The aim of the present study is to investigate the behavior of stepped hull shapes, height, and position by modifying the base hull of the NSS (C1 hull). For this purpose, the CFD code SIEMENS PLM STAR CCM+ was utilized, and its accuracy was confirmed against experimental data. Simulations were conducted using different beam Froude numbers, including = 1.13, 2.22, 2.56, and 2.96. A total of 15 stepped hull designs were developed, consisting of 9 single-step hulls and 6 double-step hulls, to explore the effects of step addition, step height, and step position. Control outputs such as resistance coefficient, wetted surface, sinkage, and dynamic trim were considered to evaluate the impact of the proposed modifications. To account for the hull motions (heave and pitch), Dynamic Fluid Body Interaction (DFBI) was employed in conjunction with the overset mesh approach. The main outcomes of the present study can be summarized as follows.
It was observed from the simulations that the single-stepped hulls “SS5-M”, “SS6-MD”, “SS7-FD”, and “SS9-MD1” had the lowest resistance coefficients at low Froude numbers compared to the base hull.
The single-stepped hull “SS9-MD1” showed a reduction in resistance at = 1.13 and = 2.22 and had nearly the same value at = 2.96.
The primary goal of using steps in the hull is to attain high speed by minimizing resistance and controlling trim. The design of these steps requires careful consideration to meet this objective. Notably, the double-stepped hulls “DSS10-MD1” and “DSS12-MD3” exhibited the highest resistance values and started trimming by the bow due to the shift of the center of gravity in the forward section, and these types of designs are advisable to be avoided. Dynamic trim is an important factor for planing hulls at high speeds. The rise in trim and slamming can affect the structure and comfort of the vessel. The single-stepped hulls “SS5-M”, “SS7-FD”, and “S8-M1” were able to control the dynamic trim angle at all Froude numbers.
Most double-stepped hulls have greater dynamic trim angles at high speed due to the placement of the second step near the transom. At low speeds, the double-step designs have lower trim angles as compared to the base hull. A positive value for sinkage means the hull is coming out of the water, which directly reduces the wetted surface and resistance. The single-stepped hulls “S1-M”, “S2-M”, “S3-F”, and “S4-B” had larger (positive) values of sinkage compared to the base hull.
When planing a hull, as speed increases, the wetted surface usually decreases due to the generation of hydrodynamic lift, which pushes the hull out of the water. Adding steps also reduces the wetted surface and changes the pressure distribution on the bottom of the hull. In this study, all added steps reduced the wetted surface area except for the “S3-F” model. The placement of the steps close to the transom reduces trim at high speeds but also increases the resistance and wetted surface of the single-step hull. On the contrary, for a single-stepped hull, having a step at the front will reduce the trim angle at low step heights but increase the trim angle at high step heights. Regarding sinkage, hull configurations with a single step exhibit a similar or slightly higher positive sinkage compared to the base hull, unlike what occurs with hulls featuring double-step configurations.