1   Introduction

At present, as the demand for faster vessels continues to grow, various high-speed vessels have been designed and constructed for military, recreational, and passenger use. Planing vessels, specifically engineered for high-speed travel, require optimization to improve their hydrodynamic performance and stability during design (Savitsky, 1964). Researchers have proposed various approaches to enhance the hydrodynamic parameters of planing hulls, one of which involves the incorporation of transverse steps on the vessel's bottom. Incorporating these transverse steps helps reduce frictional resistance by inducing flow separation at each step location. Furthermore, this flow separation divides the bottom of the planing hull into multiple parts, thereby distributing hydrodynamic lift along the vessel's length and promoting its longitudinal stability (Martin, 1978; Faltinsen, 2006). According to Clement (2003), approximately 90% of hydrodynamic lift is generated in the frontal region of the hull, with the remaining 10% coming from the aft body, specifically concerning the wetted areas of the bottom. Consequently, the geometric parameters of the step play a crucial role in determining the wetted areas of vessels (Clement and Pope, 2018; Clement and Koelbel, 1991). One major concern with using steps is their unpredictable hydrodynamic performance at remarkably high speeds. To address this issue, researchers such as Clement, Pope and Moore (Clement and Pope, 1961; Moore, 1967) have explored the potential of steps in high-velocity scenarios. Clement and Koelbel (1993) identified specific challenges, including longitudinal instability, that must be addressed in the design of optimal stepped planing hulls. Furthermore, Katayama conducted an extensive investigation into the parameters that contribute to porpoising occurrences (Katayama, 2004). Another important contribution to the field comes from Savitsky and Morabito (2010), who developed a set of formulas related to the parameters of separated flow in stepped planing hulls. In line with the Savitsky method, Svahn (2009) developed a technique for estimating the resistance of stepped planing hulls. Garland and Maki (2012) conducted a two-dimensional numerical simulation to examine the performance of these hulls. They varied the position and height of the steps to analyze their effects on free surface elevation and pressure distribution on the vessel's bottom. Their findings indicated that increasing the height of the steps improved the lift-to-drag ratio, while the position of the steps had a less pronounced effect on this ratio. Taunton et al. (Taunton et al., 2010; 2011) also conducted extensive experiments on stepped planing hulls under calm and wave conditions, revealing that acceleration and pressure were primarily influenced by the position of the steps. Veysi et al. (2015) investigated the effect of transverse steps on the profile of a high-speed vessel through numerical methods. Their findings indicated that incorporating transverse steps reduces the vessel trim, leading to a decrease in wake height. In a 2017 study, De Marco et al. (2017) employed numerical and experimental approaches to examine flow separation and patterns occurring at the step using phenomenological studies. Matveev (2012) applied 2D modeling techniques to investigate stepped planing hulls, revealing that the height of the step and the Fr number notably influenced factors such as free surface generation, the length of wetted areas on the hull's bottom, and the distribution of air pressure. Doustdar and Kazemi (2019) examined the hydrodynamic components of resistance, trim, and rise-up for three types of highspeed vessels: those without steps, with a single step, and those with two steps, using numerical methods. Their results showed that increasing the number of transverse steps improved pressure distribution along the bottom of the vessel and enhanced longitudinal stability. Najafi and Nowruzi (2019) examined the impact of five different types of transverse steps on various performance metrics of planing hulls, including lift-to-drag ratio, resistance, trim angle, and rise-up. They later expanded their research (Najafi et al., 2019b) through experimental methods to analyze the geometric components of the transverse step and proposed an analytical approach for calculating the hydrodynamic components of vessels. Nourghassemi et al. (2018) performed numerical simulations to explore the effects of step height, demonstrating that increasing the height of the transverse step enhanced ventilation. Subsequently, Di Caterino et al. (2018) introduced an optimal design for the dry area behind the step, focusing on drag reduction and improved stability. Ghadimi and Panahi (2019) evaluated the impact of steps on the hydrodynamic forces and moments acting on various stepped planing hulls. Esfandiari et al. (2020) conducted a numerical comparison between the hydrodynamic performance of two-stepped planing hulls and regular wave conditions without a step hull. Their findings revealed that, in waves longer than the vessel's length, the movements and accelerations of the two-stepped planing hulls were notably lower compared to those without a step. Danielsson and Stromquist (2012) proposed a mathematical model based on the Svahn and Savitsky model, revealing uncertainties regarding the applicability of Savitsky and Morabito's formula (Savitsky and Morabito, 2010) for two-stepped planing hulls. Sajedi and Ghadimi (2020) examined the phenomenon of step location variation and its impact on resistance reduction through experimental and numerical methods. Their results indicated that increasing the distance of the step from the stern improved longitudinal stability but also led to an increase in vessel resistance due to a decrease in trim angle. Bilandi et al. (2021) studied a double-stepped design for lifting surfaces and developed a model using a 2D+t framework. Eskandari et al. (2021) presented an appropriate analytical method for studying the hydrodynamic parameters of stepped planing hulls, validating their results against experimental data from a specific stepped planing hull vessel for three step heights of 2%, 4%, and 6% of the width. The results revealed that the developed method outperformed the Savitsky method and showed good agreement with the experimental data. Najafi et al. (2021) investigated the hydrodynamic behavior and bottom wetted surfaces of two-stepped planing hulls by performing towing tank tests, which involves determining the hydrodynamic resistance, reattachment length, trim angle, and bottom wetted surfaces under different geometric variables of the two transverse steps and planing hull speeds. Ghadimi et al. (2022a) conduncted experimental and numerical studies on two different vessels, one with a single step and the other with two steps. They found that as the second step moved away from the transom, resistance increased while trim decreased. They also concluded that the single step and two-step models remained stable at speeds up to 12 m/s. Tran et al. (2022) presented a strategy to optimize planing hull design, beginning with dimensional analysis to establish a resistance objective function based on three dimensionless hull form parameters. They then employed CFD or the Savitsky method to predict resistance for each hull variant formed by altering pairs of hull form parameters in the resistance objective function. Using experimental methods, Vitiello et al. (2022) introduced a systematic series of high-speed vessels equipped with a V-shape step. Their investigation revealed that an appropriately designed V-shaped step could enhance the hydrodynamic performance of high-speed vessels and notably improve stability. In another experimental study, Ghadimi et al. (2022b) investigated the hydrodynamic behavior and stability of a planing vessel with three different step configurations: transverse, aft-wise, and forewise steps. The results indicated that the aft-wise step provided a greater improvement in the hydrodynamic behavior of the vessel compared to the other configurations tested.

Most studies examining the effect of transverse steps on the behavior of high-speed vessels have focused on geometric parameters, such as the height and location of the transverse step, in relation to hydrodynamic components. However, studies that examine the effect of the shape of the transverse step on the dynamic behavior of the vessel are few. Notably, very few research efforts have resulted in analytical formulas for assessing transverse step components. In most studies, the hydrodynamic behavior of stepped vessels has been evaluated using experimental or numerical methods specific to individual vessels. The existence of analytical formulas for evaluating vessel behavior is crucial during the preliminary and conceptual design stages of high-speed vessel development. However, this aspect has received less attention in existing research. In the present study, the effect of changing the angle of the aft-wise transverse step on the dynamic behavior of the vessel is evaluated using experimental methods. Using the results, the analytical formulas provided by Savitsky and Morabito are refined to calculate the hydrodynamic parameters of high-speed vessels equipped with aft-wise transverse steps. These enhanced formulas can facilitate a faster and more accurate evaluation of hydrodynamic performance during the early design stages.

2   Problem definition

Reducing resistance and improving longitudinal stability are key challenges in the design of high-speed vessels. Various methods are employed to overcome these challenges, with the use of a transverse step being one of the most common approaches. This study explores the effect of changing the angle of the aft-wise step and incorporates these changes into existing analytical formulas, resulting in new formulas specifically for high-speed vessels equipped with aft-wise steps. For this research, the Fridsma highspeed vessel (Fridsma, 1969), characterized by a 20° deadrise angle, has been chosen as the base vessel. The parameters of the vessel examined in this study are presented in Table 1.

In this study, the height and length of the transverse step were set as 0.2 and 1.1 m, respectively, based on findings from Najafi et al. (2021), which indicated acceptable performance in terms of resistance and longitudinal stability. According to the results presented in the research (Vitiello et al., 2022; Ghadimi et al., 2022b; Nourghasemi et al., 2017) and an analysis of the hydrodynamic behavior of the vessel (considering factors such as the center of gravity, trim angle, and wetted surface at different speeds), the angle of the aft-wise transverse step was evaluated at 0°, 9°, 11°, 13°, 15°, and 17° for the aft-wise transverse step (Figure 1). Each mode was evaluated at three speeds of 8, 10, and 12 m/s. Table 2 summarizes the parameters for the transverse steps considered in this research.

Figure  1  Transverse step placed on the bottom of the high-speed vessel

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In this research, the hydrodynamic behavior of a vessel equipped with an aft-wise step was investigated using experimental methods. Based on the results, an analytical formula derived from the work of Savitsky and Morabito (2010) has been developed to calculate the parameters of the wetted surface for vessels with an aft-wise step (angled transverse step).

3   Experimental study

3.1   Test procedure

In this research, a high-speed vessel equipped with a transverse step was investigated using experimental methods. The tests were conducted on a mono-hull planing vessel of the Fridsma model with a 20° deadrise angle. These hulls were subjected to towing tests in the towing tank at the National Iranian Marine Laboratory in Iran (https://ittc.info), with the main parameters outlined in Table 3. The tests were performed in two degrees of freedom, pitch, and heave.

To conduct the laboratory studies, the first step involved the construction of a vessel model for experiments. The process of creating and preparing the vessel for testing included the following steps:

• The vessel model was designed using the SolidWorks software environment based on the specifications and formulas provided by Fridsma (1969). The model was created in multiple parts to manage construction costs, allowing for the adjustment of steps in different positions, heights, and angles (Figure 2(a)).

Figure  2  Steps of vessel model production in the Persian Gulf National Laboratory

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• The different parts of the vessel hull were produced using 3D printing technology (Figure 2(b)).

• The various parts of the hull were connected and assembled to create the complete vessel geometry (Figure 2(c)).

• The vessel hull was then painted and marked in accordance with the guidelines set by the International Towing Tank Conference (ITTC, 2014) (Figures 2(d) and (e)).

The geometric similarity of the constructed model with the 3D-CAD design was based on guidelines established by the ITTC (2014). Measurements indicated an error of less than 1 mm, demonstrating the accuracy of the built model.

The equipment installed on the vessel for conducting experimental tests includes a dynamometer, which measures the resistance force acting on the vessel, and a potentiometer, which is used to determine the trim angle and rise-up of the vessel. Figure 3 shows the equipment installed on the vessel.

Figure  3  Vessel installation under the towing carriage

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During the towing tests, several parameters are measured, including the wetted length of the chain and keel, dynamic trim, the extent of water separation from the step, and the wetted surface of the vessel's bottom. Four cameras were strategically placed at the front, side, back, and underwater to calculate the wetted surface area of the vessel. Figure 4 presents a schematic view of the camera placements around the vessel. The images captured by these cameras were analyzed to determine the wetter surface parameters. Additionally, the uncertainty of the experimental conditions and results was assessed in accordance with the recommendations of the ITTC (ITTC, 2014; Sajedi et al., 2021)

Figure  4  Schematic view of the cameras placed around the vessel

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3.2   Determining test outputs

In this research, tests were conducted to examine the effect of the transverse step angle on the behavior of the vessel, as well as to develop analytical formulas for calculating the wetted surface of the vessel. The measured parameters from the laboratory tests were determined and analyzed in line with these objectives. One important study on analytical formulas for calculating the hydrodynamic components of stepped vessels was conducted by Savitsky and Morabito (2010). Using laboratory methods, they analyzed the wake height formed behind the vessel and provided a formula for calculating the wake height relative to the centerline and the plate corresponding to 0.25 times the beam of the vessel, specifically at deadrise angles of 10°, 20°, and 30°. The formulas are presented in Equations (1) and (2):

where n is the constant coefficient based on the deadrise angle, which is equal to 1.5 at the deadrise angle of 10° and equal to 2 at the deadrise angles of 20° and 30°. The parameters g, X_cl, and X_1/4 denote the acceleration of gravity and the distance between the wetted area of the rear hull (behind the transverse step) and the transverse step, which is determined in the center line and 0.25% of the vessel beam, respectively. Additionally, C_v, τ, and L_k represent the speed coefficient (C_v = $\frac{V}{\sqrt{\mathrm{gb}}}$), static trim, and the distance of the wetted surface behind the transverse step to the stern in the center line, respectively. Figure 5 presents a schematic of the parameters related to the wetted surface of the vessel bottom.

Figure  5  Parameters related to the wetted surface of the vessel bottom (Savitsky and Morabito, 2010)

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The wetted area behind the transverse step and toward the stern of the vessel (X_cl, and X_1/4) is a crucial parameter that changes depending on the shape of the transverse step and the vessel speed (V_s). In this study, these parameters were measured at different angles of the transverse step for different vessel speeds. Based on the results, a correction factor was added to Equations (1) and (2) to enable their use in vessels equipped with aft-wise transverse steps. By calculating the parameters X_cl, and X_1/4, the wetted surface of the high-speed vessel equipped with an aft-wise transverse step can be calculated. The analytical formulas for calculating the wetted surface of stepped vessels are determined based on the two parameters, as referenced in Najafi et al. (2019a).

4   Results and Discussions

In this section, the hydrodynamic behavior of the highspeed vessel at different angles of the aft-wise transverse step has been investigated and analyzed. First, the hydrodynamic components of the vessel, including resistance, trim angle, and wetted surface, were evaluated. Second, based on the results of the experimental tests, the analytical formulas developed by Savitsky and Morabito (2010) were adapted to calculate the wetted surface of a vessel equipped with an aft-wise transverse step.

4.1   Investigation of vessel hydrodynamic components

In this section, the effect of changing the angle of the aft-wise transverse step on the hydrodynamic components of resistance and trim angle has been analyzed. The angles examined include 0°, 9°, 11°, 13°, 15°, and 17°, with calculations performed at three speeds of 8, 10, and 12 m/s. Figure 6 shows the effect of these angle changes on vessel resistance across different speeds. The results indicate that changing the angle of the aft-wise transverse step leads to a reduction in vessel resistance, attributed to the increased ventilation generated behind the aft-wise transverse step. Notably, increasing the angle of the aft-wise transverse step to 11° results in a decrease in resistance. However, beyond 11°, resistance begins to rise due to the diminishing effect of the step angle of the aft-wise transverse step on the wetted surface of the vessel. Thus, from the viewpoint of resistance reduction, the aft-wise transverse step with an angle of 11° demonstrates the most effective performance, resulting in an average resistance reduction of 7.8% compared to a vessel without a transverse step.

Figure  6  Effect of changing the angle of the aft-wise transverse steps on the vessel resistance at different speeds

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In the conducted experiments, the dynamic behavior of the vessel was examined in two degrees of freedom. Figure 7 shows the changes in the vessel trim angle at different angles of the aft-wise transverse step for speeds of 8, 10, and 12 m/s. The results indicate that introducing an angle to the transverse step reduces the trim angle of the vessel. This reduction is attributed to the stagnation line moving closer to the stern of the vessel and the increased moment at the stern due to the creation of an angle in the aft-wise transverse step. Therefore, increasing the step angle leads to a decrease in the vessel trim angle, enhancing longitudinal stability and mitigating the risk of porpoising instability. Notably, the trim reduction for the aft-wise transverse step at an angle of 11° (where resistance was also minimized) revealed an average of 12.8% reduction compared to the vessel with a transverse step.

Figure  7  Effect of changing the angle of the aft-wise transverse steps on the vessel trim angle at different speeds

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4.2   Determining the parameters of the vessel wetted surface

In this research, the wetted surface parameters of the vessel were determined using images recorded by the cameras installed on the carriage, along with image processing techniques (Najafi et al., 2019a). Figure 8 shows views of the images recorded by the underwater camera for the vessel equipped with a transverse step at angles of 0° and 11° (the optimal angle). In the figure, the red area represents the wetted surface of the vessel behind the transverse step. Notably, creating an angle in the transverse step increases ventilation in the bottom of the vessel (Figure 8(a)), which leads to a reduction in the wetted surface and vessel resistance (Figure 8(b)).

Figure  8  Wetted surface of the bottom of the planing hull

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In this research, the wetted surface parameters of the vessel were calculated using image processing techniques. Two key parameters were assessed: the wetted surface area of the front hull (in front of the transverse step) and the rear hull (behind the transverse step). These calculations were performed at three speeds of 8, 10, and 12° m/s for transverse step angles of 0°, 9°, 11°, 13°, 15°, and 17°. Figure 9 shows the area of the wetted surface of the front and rear of the vessel hull in the aforementioned states. Notably, the wetted surface of the vessel decreased with increasing speed. Furthermore, as the angle of the aft-wise transverse step increased up to 11°, the wetted surface began to increase again. This change can be attributed to the enhanced ventilation in the bottom of the vessel at a step angle of 11° (Figure 8(b)).

Figure  9  Wetted surface of the bottom of the planing hull

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Two components X_cl and X_1/4, were also calculated using the image processing method (Najafi et al., 2019a) for the performed tests. Table 4 shows the calculation results of two parameters of the vessel wetted surface (X_cl and X_1/4), which were compared using the experimental method and the method provided by Savitsky and Morabito (2010) (Equations (1) and (2)). Notably, the lowest amount of wetted surface parameters in the front and rear hull is related to the aft-wise transverse step with an angle of 11°. The values of these parameters have also experienced a downward trend up to an angle of 11° and then an upward trend due to changes in the number of vessel trim (Figure 7) and its effect on the amount of wetted surface in the front and rear hull. Furthermore, the results revealed notable errors using the Savitsky and Morabito (2010) method, demonstrating increased errors with the angle of the aftwise transverse step. The analytical formulas of Savitsky and Morabito (2010) demonstrate an average error of 12.4% and 21.5% for the two parameters X_cl and X_1/4, respectively, when compared to experimental results. Consequently, these formulas are not suitable for vessels equipped with an aftwise transverse step, lacking the necessary accuracy for reliable predictions.

4.3   Modification of analytical method for calculation of wetted surface parameters

In this section of the current research, analytical formulas have been developed to calculate the wetted surface parameters of vessels equipped with an aft-wise transverse step. Based on the analysis of the vessel's behavior and he two wetted surface parameters (X_cl and X_1/4), the analytical formulas provided by Savitsky and Morabito (2010) (Equations (1) and (2)) have been modified for use with aftwise transverse steps. This modification involves incorporating two correction coefficients, F₁(α, V_s) and F₂(α, V_s), into the Savitsky and Morabito (2010) formulas, leading to the revised Equations (3) and (4).

Based on the calculated parameters (X_cl and X_1/4) and using the polynomial regression method (Edwards, 2002; Theil, 1992), correction coefficients F₁(α, V_s) and F₂(α, V_s) have beenwere determined according to equationsbased on Equations (5) and (6).

CorrectionThe correction coefficients F₁(α, V_s) and F₂(α, V_s) are calculated based on the transverse step angle and vessel speed, which are expressed in equations 3 and 4 with symbols α and V_s respectively. The correctiveThese coefficients provided by consideringare designed to account for the constraints mentioned in Table 5 and are applicable for the calculation ofcalculating the parameters of the vessel wetted surface parameters of vessels equipped with an aft-wise transverse step.

In Table 6, the components of the wetted surface (X_cl and X_1/4) derived from laboratory calculations for the vessel equipped with an aft-wise transverse step are compared with the results obtained from the modified formula.

The results indicate that by incorporating the two correction coefficients into the equations of Savitsky and Morabito (2010), the wetted surface parameters for the vessel equipped with an aft-wise transverse step have been calculated with acceptable accuracy. The modified analytical formulas of Savitsky and Morabito (2010) exhibit average errors of 3.6% and 1.8% for the parameters X_cl and X_1/4, respectively, when compared to the experimental methods. Thus, the addition of these correction coefficients has notably improved the accuracy of the calculations for determining the wetted surface of the vessel.

5   Conclusions and future work

In this research, the effect of the aft-wise transverse step on the hydrodynamic behavior and longitudinal stability of a high-speed vessel was comprehensively evaluated. The hydrodynamic components of the vessel were calculated for six angles of the aft-wise transverse step (angles 0°, 9°, 11°, 13°, 15°, and 17°) at three speeds (8, 10, and 12 m/s) using experimental methods. Based on the findings, analytical formulas were developed to calculate the wetted surface parameters for the high-speed vessel equipped with an aft-wise transverse step. The key results from this study are as follows:

1) Creating an angle in the transverse step notably improved the hydrodynamic components of resistance and trim angle. Therefore, at the optimal angle of the aft-wise transverse step (11°), compared to the transverse step, the resistance and trim angle of the vessel are reduced by 7.8% and 12.8%, respectively.

2) The vessel equipped with an aft-wise transverse step had a lower wetted surface than that with a transverse step due to the enhanced ventilation created behind the step.

3) Creating an angle in the transverse step can increase longitudinal stability and prevent porpoising instability in the vessel by minimizing vessel movements.

4) The correction coefficients derived from laboratory tests enhanced the accuracy of existing analytical formulas, making them valuable tools for the preliminary and conceptual design phases of high-speed vessels.

For future work, the investigation of the effects of forewise and aft-wise transverse steps at different deadrise angles will be conducted using experimental and numerical methods. Based on the results, analytical formulas that can enhance the accuracy and speed of initial and conceptual designs for high-speed vessels will be developed.