Numerical Analysis of the Dynamic Effects of Ballast Tank Water Discharge on Submarine Motion Stability During Emergency Rising
Article information
Abstract
Submarine emergency surfacing involves complex dynamic behaviors because of the shift in the center of gravity (CG) and changes in the moment of inertia caused by ballast tank water discharge. This study proposes a numerical analysis model based on six-degree-of-freedom (6DOF) equations of motion to investigate the dynamic behavior of submarines during emergency surfacing. A grid-based Volume Cell model was developed to simulate the water flow dynamics within the ballast tank, incorporating the effects of mass displacement and inertia changes into the equations of motion. The simulation results indicated that linear motions (surge, sway, and heave) are minimally affected by the discharge of ballast tank water. Nevertheless, rotational motions (pitch and yaw) are significantly influenced by the discharge, particularly due to the reduction in inertia. In pitch motion, the sensitivity to external moments increases, resulting in a noticeable increase in angular deviation. The proposed model showed high accuracy through comparisons with theoretical calculations and CAD modeling results, providing a robust foundation for optimizing the design and placement of ballast tanks. This paper presents a novel methodology to enhance the reliability and efficiency of emergency surfacing simulations, offering valuable insights and design guidelines for improving submarine stability and performance during critical operations.
1. Introduction
The design and operational analysis of submarines are considered highly complex and challenging tasks because of the special requirements of conducting operations in above-water and underwater environments. Such designs must harmoniously consider the fluid dynamic characteristics and operational purposes and precisely reflect the speed, maneuverability, and special-purpose performance of submarines. On the other hand, submarine design relies on limited model test data and numerical analysis data, unlike the extensive data available for existing surface ship designs. In particular, the lack of predictive capability for special operations such as emergencies also acts as a limitation. Submarine emergency rising is one of the operational scenarios that demonstrates the importance of such predictive models. Emergency rising involves a complex process of rapidly discharging water from the ballast tank using high-pressure compressed air and increasing buoyancy to bring the submarine swiftly to the surface. This process is influenced by various physical factors such as the interactions between air and water, pressure loss, and changes in inertia, which also significantly affect the attitude and stability of a submarine during rising. During emergency rising, the weight of the hull changes rapidly, resulting in a corresponding change in inertia.
Research related to submarine emergency rising is steadily advancing based on simulation technology and mathematical modeling. Bettle et al. (2009) analyzed the roll instability that occurs during emergency rising using a six-degree-of-freedom motion equation based on the Reynolds-averaged Navier–Stokes (RANS). This study highlights the influence of the initial roll angle on attitude instability during emergence and presents the combined effect of the roll and yaw angles through simulations. In addition, the study further analyzed roll instability during emergency rising through computational fluid dynamics (CFD) simulations that reflected the abnormal fluid dynamic effects. This study identified the sail on the upper part of the submarine and the resulting flow asymmetry as the leading causes of roll instability, and it detailed the sensitive impact of the initial roll angle on the roll angle during surfacing. Font et al. (2010) mathematically modeled the blowing and venting processes of the ballast tank and proposed an optimized control algorithm. This study explored the potential of using blowing and venting operations in emergencies and as an auxiliary means of motion control for submarines through buoyancy adjustment. Zhang et al. (2019) analyzed the strong coupling effect between roll and yaw motions during emergency rising through RANS-based CFD simulations and model tests. They reported that path-keeping performance can significantly enhance roll stability and identified excessive yaw bias as a major factor contributing to roll instability. Mahdaviniaki et al. (2022) used StarCCM+ software to model the interactions between air and water within the ballast tank, quantitatively analyzing the impact of the key parameters on motion stability during emergency rising. This study provided specific guidelines for design optimization based on parameter sensitivity analysis.
Jung et al. (2024) quantitatively analyzed the water discharge process of the ballast tank during emergency rising based on the ballast tank dynamics study by Font et al. (2010) and proposed a mathematical model to explain it. This study, based on the complex interactions between high-pressure compressed air and ballast water, examined the impact of pressure loss on the discharge flow rate and developed a new formula that can be adjusted dynamically according to the depth and pressure conditions. The proposed model significantly improved the accuracy of flow rate calculations compared to the conventional method using fixed coefficients, suggesting the potential for more precise predictions of attitude changes during the emergency rise of a submarine.
This paper, following Jung et al. (2024), proposes a grid-based numerical analysis model that calculates the center of mass and moment of inertia based on the changes in water volume during the water discharge process in a ballast tank. The proposed model was validated for its appropriateness by comparing it with existing formulae. In addition, a numerical analysis model was developed to reproduce the phenomenon of water tilting inside the tank caused by excessive pitch motion during emergency rising. This model was compared and reviewed against the CAD modeling results. The numerical analysis model, which considers the changes in the longitudinal center of gravity (LCG) and moment of inertia of the ballast tank, was applied to the extended six-degree-of-freedom (6DOF) motion equations based on the Karasuno model proposed by Lee and Ahn (2024). Through this, the flow changes of water within the ballast tank and the dynamic behavior of the hull caused by discharge were simulated, and the changes in linear and rotational motions occurring during emergency rising were analyzed precisely. The analysis demonstrated that linear motions (surge, sway, and heave) are barely affected by the dynamic changes in the ballast tank. In rotational motion (pitch and yaw), however, the moment of inertia decreased due to the water discharge from the ballast tank, resulting in increased sensitivity to external moments and a significant increase in the range of angular variations. During pitch motion, the movement of the center of mass and changes in inertia have a significant impact, which has been confirmed to be an important variable in the stability and motion characteristics of submarines.
This paper presents a numerical analysis model that effectively analyzes the dynamic changes in the center of mass and moment of inertia resulting from water discharge from the ballast tank during the emergency rise of a submarine. Through this, the attitude and motion characteristics of the submarine were quantitatively evaluated. These results are expected to provide important design guidelines for submarines and the design and operational optimization of similar marine structures.
In the following chapter, the grid-based numerical modeling process for analyzing flow changes within the ballast tank and the simulation results of emergency surfacing will be described in detail. The primary study findings and future directions are discussed.
2. Numerical Simulation Modelling for Blowing Operation of Ballast Tank in Submarine
2.1 Main Idea of Numerical Simulation Modelling
A typical submarine is composed of multiple ballast tanks, as shown in Fig. 1. This study selected the locations of two ballast tanks (red boxes) for modeling and numerical calculations.
Configuration of ballast tanks in a submarine
This study modeled a simplified cylindrical shape with a scale ratio of 1/20, based on the linear shape of a MARIN BB2 full-scale submarine (Fig. 2(a)) and referencing the research data of Jung et al. (2024) (Fig. 2(b)).
Information on a BB2 submarine and the model of a ballast tank
This simple cylindrical shape has its center of mass located at the center when the tank is full of water, and the moment of inertia can be calculated as in Eq. (1), as shown in Fig. 3(a):
Conceptual diagram of a cylindrical chamber
where mb is the weight of the water, r is the radius of the cylinder, and B is the width of the cylinder.
When the submarine discharges water from the tank for emergency rising, it becomes impossible to calculate the center of mass and moment of inertia as a cylindrical shape with the top removed, as shown in Fig. 3(b). To determine the center of mass and moment of inertia of the shape after draining the water, the entire volume of the cylindrical shape can be finely divided in the width and height directions, treating it as a collection of small cubes or volume cells, as shown in Fig. 4.
Main Idea for calculating the center of mass and moment of inertia
The center of mass and moment of inertia calculated in this manner can numerically represent the characteristics of each axis, reflecting the mass distribution of the object. The center of mass of a volume cell composed of a collection of small cells is calculated for each axis using Eq. (2):
where mi is the mass of each cell; xi, yi, and zi are the coordinates of each cell. mc is the mass of one cell. Through this equation, the centers of mass, CMx, CMy, and CMz for each axis, represent the center of mass of the object. The moment of inertia is a value that quantifies the degree of resistance to rotation based on how far the mass is separated from each axis, and it is defined by Eq. (3):
where Distx, Disty and Distz are the distances from the center of mass to each cell. They can be represented by Eq. (4):
The above distance equations extend the Euclidean distance between two points in three-dimensional space to calculate the distance for specific axes, allowing for independent distance values for each axis. Using this concept, it shows the simulation technique used to calculate the center of mass (CM) and moment of inertia (I) during the water discharge process in the ballast tank, as shown in Fig. 5. The simulation divides the volume of the ballast tank into small cubic elements to represent the mass distribution of the water accurately. This grid-based volume cell partitioning technique enables the precise analysis of the mass distribution that occurs as water is discharged.
Simulation of calculating the center of mass and moment of inertia for the ballast tank
Fig. 5(a) shows the ballast tank filled with water. The composition of each cell can be seen by zooming in from the right. Fig. 5(b) depicts the shape of the water being discharged over time, and the center of mass shifts due to the uneven distribution of the remaining water. This movement significantly affects the moment of inertia, and simulations were conducted to dynamically reflect these changes, precisely calculating the center of mass and moment of inertia at each time step during the discharge process.
First, the numerical analysis results obtained when the ballast tank was filled with water were compared with the theoretical formula in Table 1, as shown in Fig. 5(a).
Results of simulation of calculating the center of mass and moment of inertia
The center of mass calculated for the X-axis and Y-axis remains close to 0, as listed in Table 1, as expected because of the symmetric geometric structure of the tank. The numerical simulation results were in close agreement with the theoretical predictions, with negligible errors. For the moment of inertia, a small difference was observed between the value obtained using the formula and the numerical results, which was attributed to the error from modeling the cubic cell as a circle on a grid.
2.2 Rotation of Ship Motion for Numerical Simulation
This study designed a numerical analysis algorithm to analyze the distribution of ballast water in emergencies by performing three-dimensional rotation transformations on volume cells. This algorithm uses the roll, pitch, and yaw rotation angles as input, performs axis-specific rotations in the ZYX order, and returns the sorted results. First, the roll, pitch, and yaw angles are input in degrees (°) and converted to radians. The conversion formulae can be represented by Eq. (5):
The rotation transformation for the three axes was calculated by performing matrix multiplication in the ZYX order to obtain the final rotation matrix R. The rotation matrix is given by Eq. (6):
The center coordinates of each volume cell (CMx, CMy, CMz) are converted using this rotation matrix R. The converted coordinates are given by Eq. (7):
After applying this process to all volume cells, the converted cells are sorted in descending order along the Z-axis. This is a post-processing step to reflect the change in the height of the ballast water caused by water discharge during emergency rising.
The ballast tank is filled with water in a cylindrical shape. When the tank is full, its shape does not change with the attitude of the vessel, but it transforms into a (semi)cylindrical shape when some water is drained. The ballast tank, being cylindrical, appears rectangular from the side, as shown in the upper blue part of Fig. 6. It appears rectangular from the top view in Fig. 6. Considering the rapidly changing pitch angle during emergency rising, the flow of water within the tank becomes tilted as shown in the red color of Fig. 6. The blue color in Fig. 6 shows the water in a (semi)cylindrical shape that does not reflect the pitch angle and does not rotate. This phenomenon becomes more pronounced as the pitch angle increases. Figs. 6(a) and 6(b) show pitch angles of 4.6° and 34.3°, respectively. The lower image of Fig. 6 is shown in 3D form under the same conditions as the upper image. The shape of the waterline differs significantly from its original form as the pitch angle increases, indicating the need to recalculate the center of mass and moment of inertia.
Volume cell diagrams calculated with rotation applied
In conclusion, this algorithm can accurately convert the 3D coordinates of volume cells according to roll, pitch, and yaw, and it is expected to precisely reflect the dynamic changes in ballast water during the emergency rising of a submarine.
2.3 Validation of Numerical Simulation
2.3.1 Verification of cubic shape
The grid-based volume cell method described in sections 2.1 and 2.2 was initially evaluated through a three-step verification process, using a cubic shape as the basis for easy calculations. As shown in the conceptual diagram of Fig. 7, the initial state verification in Step 1 was performed based on a cube with dimensions of 0.2 m in width (X-axis), height (Y-axis), and depth (Z-axis). In the second stage, the volume of the top part of the cube was removed to simulate the state where approximately 1.5 kg of water was discharged. Owing to the removal of the upper mass, the effect on the center of mass, decreasing along the Z-axis (height), and the changes in the moment of inertia were examined. In the last step, the modified cubic shape created in Step 2 was rotated at a pitch angle of 30° from the initial origin, and the results were compared with the CAD modeling shape. During the verification process, the theoretical formulae, numerical simulation results, and CAD modeling data were compared to assess the accuracy of the model under various conditions.
Step-by-step conceptual diagram for verification using a cubic shape
Table 2 lists the verification results using the cubic shape. In Step 1, the center of mass for the X–Y–Z axes was calculated to be close to 0 due to the symmetric structure of the cube, which matched the theoretical calculation results. The moment of inertia was calculated using the theoretical formula presented in the Note of Table 2, showing a high degree of agreement with the numerical analysis results and CAD modeling values. In Step 2, the top portion of the cube was removed to simulate the state where 1.5 kg of water was discharged. Owing to the removal of the mass from the top, the center of mass shifted approximately −18.75 mm in the Z-axis direction, and the moment of inertia decreased, reflecting the distribution of the removed mass. In this step, the numerical analysis results were similar to the theoretical calculations and CAD modeling values. In step 3, the modified cube shape generated in the second step was rotated at a pitch angle of 30° for calculation purposes. The center of gravity shifted approximately −19.12 mm in the Z-axis direction due to the rotation, and the moment of inertia varied according to the rotated shape. The numerical analysis results calculated at this step showed good agreement with the CAD modeling data, further proving the accuracy of the proposed model.
Results of verification using a cubic shape
2.3.2 Verification of the cylindrical shape (ballast tank)
The verification based on the cylindrical shape of the ballast tank was conducted in four stages, as shown in Fig. 8, similar to the method used in the cubic shape verification. In Step 1, the center of mass and moment of inertia were calculated based on the initial state of the circular cylinder filled with water to evaluate the accuracy of the model. In Step 2, the state of discharging approximately 1.5 kg of water from the top was simulated to analyze the shift in the center of mass and the change in the moment of inertia caused by the removal of the top mass. In Step 3, an additional 1.5 kg of water was discharged to examine the dynamic response caused by the larger mass change. In Step 4, a 30° pitch rotation was applied to the shape generated in Step 3, using the origin of the initial shape as the reference point, to assess the changes in the center of mass and moment of inertia resulting from rotation. Table 3 lists the results of this verification process.
Step-by-step conceptual diagram for verification using a ballast tank shape
Results of verification using a cubic shape
In the initial state, the center of mass and moment of inertia perfectly matched the theoretical formulae, numerical simulations, and CAD modeling values due to the symmetrical structure of the circular cylinder, as listed in Table 3. In the upper water discharge stage, the center of mass in the Z-axis direction decreased by approximately −48.75 mm, and the moment of inertia decreased, but all results showed high consistency. In the additional water discharge step, the center of mass along the Z-axis further descended to approximately −96.01 mm. Finally, in the step where a 30° pitch rotation was applied, the center of mass moved to −86.63 mm, reaffirming the reliability of the model along with the moment of inertia. Hence, the proposed model can be used as a reliable analytical tool for complex shapes, such as ballast tanks, and accurately reflects physical changes even under dynamic conditions, including water discharge and rotation.
2.3.3 Analysis of the impact of split size on the volume cell division method
The impact of the split size of the cell volume on the results was analyzed to evaluate the accuracy of the numerical analysis model. Therefore, calculations were performed by varying the split size of each axis based on the three-step verification shape conducted as described in Section 2.3.1 (the state after 1.5 kg of water is discharged from the initial cubic shape and it is rotated at a 30° pitch angle). When performing numerical analysis, the appropriateness of the split size was examined by varying the number of cell divisions, as shown in Fig. 9, and the error in the calculated values was reviewed.
Conceptual diagram for checking the split size of the volume cell division method
As a verification method, the split size was set in seven steps from 2 × 2 × 2 to 100 × 100 × 100. The calculation results of the center of mass and moment of inertia were compared and analyzed for each setting, as shown in Fig. 9. A numerical calculation method was used in the initial stages with cell division counts of 2, 4, 6, and 8, while numerical calculations were difficult for cell division counts of 16 and above. Hence, comparisons were made based on numerical analysis results. Table 4 lists the results.
Results of the check of the split size
According to Table 4, when the split size was set to 16 × 16 × 16 or larger, the error in the calculated center of mass and moment of inertia was reduced to within 1%. In particular, the moment of inertia in the Z-axis direction showed stable values when the split size was 16 or more, and the center of mass error converged to approximately 0 when the split size was 50 or more. Hence, the number of divisions for each axis was selected as x = 50, y = 100, and z = 100 to analyze the flow of ballast water that changes dynamically over time.
3. Application to Six-degree-of-freedom Motion Simulation for Submarine Emergency Rising
3.1 Mathematical Model of Six-degree-of-freedom Equations of Motion for Submarine Emergency Rising
This study applied the 6DOF motion equation for submarine emergency rising based on the extended form of the Karasuno model presented by Lee and Ahn (2024). The mathematical model proposed by Lee and Ahn (2024) was designed to reflect the complex motion characteristics of a submarine hull more accurately, including drift and angle of attack, by extending the physics-based model introduced by Karasuno et al. (1991; 1992). The Karasuno model defines the fluid dynamic forces and moments acting on the submarine hull by dividing them into components, including the ideal fluid force, viscous drag, lateral drag, and lift, and is evaluated to have higher prediction accuracy in oblique motion compared to the existing Taylor series-based model. Lee and Ahn (2024) proposed a methodology to extend the Karasuno model from a 3DOF motion equation to a 6DOF motion equation, enabling more precise predictions of submarine motion even under abnormal conditions such as emergency rising.
Assuming that the submarine is a rigid body and is symmetric about the left and right sides, Fig. 10 shows the coordinate system fixed to the submarine. Based on the coordinate system of Fig. 10, the 6DOF motion equation includes the three linear motions (surge, sway, and heave) and three rotational motions (roll, pitch, and yaw) of a submarine. It can be represented by the following Eq. (8):
Coordinate system
where in the body coordinate system, the velocity along the x-axis is u; and the acceleration is u̇; the velocity along the y-axis is v; the acceleration is y; the velocity along the z-axis is w; the acceleration is ẇ; the roll rotation speed is q; the pitch rotation speed is p; the yaw rotation speed is r; X, Y, and Z are the external forces (thrust, drag, and lift); K, M, and N are the moments related to roll, pitch, and yaw; m is the total mass of the submarine; Ixx, Iyy, and Izz are the principal moments of inertia; XH, YH, and ZH are forces related to crossflow; KH, MH, and NH are moments related to crossflow;XV, YV, and ZV are the viscous forces and lift; KV, MV, and NV are the viscous moments.
This study analyzed the effects of changes in hull motion and changes in the center of mass and inertia caused by ballast water discharge on the motion by applying the extended Karasuno model to the emergency rising scenario of the MARIN BB2 submarine.
3.2 Numerical Simulation of Motion for Submarine Emergency Rising
Based on the 6DOF motion model described in Section 3.1, the simulation of the emergency rising of a submarine was conducted by operating two ballast tanks (Fig. 11) under the conditions outlined in Table 5.
Simplified diagram of the ballast tank positions of a submarine
Principal dimensions of MARIN BB2
In Table 5, the parallel axis theorem must be used to combine the moment of inertia of the ballast tank itself with the moment of inertia of the submarine body. According to the parallel axis theorem, the moment of inertia about one axis was calculated by summing the moment of inertia about the center of mass (center of gravity, CG) and the product of the mass and the square of the distance from the center of mass to the parallel axis. The formula for this is given by Eq. (9):
where I is the moment of inertia for the given axis; ICG는 is the moment of inertia based on the ballast tank CG; d is the distance from CG to the new axis; M is the mass of the object. Based on this equation, if the moment of inertia of a submarine body is Isub and the moment of inertia of a ballast tank is Itank, the total moment of inertia
The numerical analysis of the 6DOF motion of the submarine during emergency rising was designed based on the 6DOF motion equations to reflect the main specifications of the submarine and the dynamic changes in the ballast tank. Each stage was structured, as shown in Fig. 12.
Conceptual diagram of the three-stage structure of the emergency rising simulation
In the first stage, the initial input data for the simulation was set, and the initialization of each motion variable was performed (Fig. 12). In this process, the main specifications of the submarine (length, mass, and moment of inertia), performance data for the propeller and rudder, and the location and characteristics of the ballast tank were input. In addition, the motion states of the hull, such as speed, position, and attitude, were defined in the initial state, and the initial center of mass position and draft state were set. In the second stage, each term was calculated to solve the 6DOF motion equations. First, this study calculated the external forces and moments acting on the hull, which included the external forces from the thrusters and rudders, as well as the changes in the moment of inertia caused by flow changes within the ballast tank. The changes in the center of mass and inertia occurring within the ballast tank were dynamically reflected in the equations of motion. After calculating the external forces, the inertia and acceleration terms of the hull were calculated to interpret the linear motions (surge, sway, and heave) and rotational motions (roll, pitch, and yaw). The analysis of the nonlinear ordinary differential equation was performed using the fourth-order Runge–Kutta method (RK4) for time integration. In the third stage, the simulation results were output and analyzed. The 6DOF data, including the interpreted linear motion variables (speed and position) and rotational motion variables (attitude and angular velocity), were output over time. The simulation results are presented in graphs and visual materials to assess the dynamic behavior of the submarine during emergency rising.
3.3 Results of Numerical Simulation of Motion for Submarine Emergency Rising
3.3.1 Result of numerical analysis of the water discharge motion inside the ballast tank
Under the conditions listed in Table 5, where two ballast tanks are discharged sequentially, the water from the first tank is discharged from zero to four seconds, and the water from the second tank is discharged from four to eight seconds. This was reviewed (Fig. 13) to verify whether the flow changes in the ballast tanks, as described in Sections 2 and 3.2, were functioning correctly in the simulation.
Simulation results of the ballast tank action
Ballast tanks 1 and 2 sequentially discharged water, resulting in a decrease in weight (Fig. 13(a)). The vertical center of mass of the ballast tank can be calculated, as shown in Fig. 13(b). As a result, the moments of inertia of each axis decreased, as shown in Figs. 13(c), (d), and (e).
3.3.2 Result of numerical analysis of the emergency rising 6DOF motion
This study conducted numerical analysis based on the 6DOF motion equations to evaluate the impact of movements of the center of mass and changes in the moment of inertia caused by water discharge from (a) Mass of water in the Ballast Tank (b) VCG of the Ballast Tank (c) Inertia of the X Axis (d) Inertia of the Y Axis the ballast tank during the emergency rising of a submarine on the hull motion. The simulation was conducted under three conditions. The first condition reflected only the weight change due to water discharge from the ballast tank. The second condition reflected the changes in water weight and positional inertia, and the third condition reflected the effects of changes in water weight, positional inertia, and the own inertia of the ballast tank. These conditions can be expressed based on Eq. (11) as follows:
The numerical analysis of motion involved a model ship with a propeller rotating to maintain a speed of 1 m/s, and under conditions where the submarine is moving straight while submerged at a constant depth. In this state, without any rudder movement and only discharging ballast water, Fig. 14 shows the simulation results for the attitude change of a submarine.
Simulation results of 6-DOF motion
The motion characteristics of linear motion (surge, sway, and heave) were similar among the three conditions, as shown in Figs. 14(a), (b), and (c). This suggests that linear motion is determined governed by the total mass of the hull and its initial motion state and that dynamic changes in the ballast tank have a minimal impact on linear motion. In particular, in the surge (forward) motion, the water discharge from the ballast tank had little to no effect on the initial speed and inertial terms. In rotational motion (pitch, roll, yaw), however, there were some differences depending on the conditions. In the pitch motion (Fig. 14(d)), the center of mass shifts caused by water discharge, and the moment of inertia decreases, leading to increased sensitivity to external moments, resulting in the largest angle variation. These results indicate that the dynamic changes in the ballast tank have a direct impact on pitch stability. In roll motion (Fig. 14(e)), the variation among the three conditions was relatively small, but the change in positional inertia had a slight impact on the stability of roll motion. In yaw motion (Fig. 14(f)), the combined effects of water discharge and inertia reduction showed a clear difference over time, but the overall variation did not show significant differences.
Nevertheless, an analysis of the three conditions showed that the results of the second condition (considering water weight and changes in positional inertia) and the third condition (considering water weight, (e) Inertia of the Z Axis positional inertia, and the inertia of the ballast tank itself) were similar. Hence, the influence of the inertia of the ballast tank on the overall motion was relatively small compared to the change in positional inertia. Therefore, the position inertia plays a more important role in rotational motion stability during emergency rising.
3.3.3 Analysis of the numerical analysis results
Based on the numerical analysis results in Fig. 14, Eq. (12) was introduced to quantitatively analyze the impact of the position and the inertial changes of the ballast tank caused by water discharge on the hull motion during an emergency rising, using the fundamental formula of rotational motion:
where T is the external moment; I is the moment of inertia; a is the angular acceleration. This equation was used to calculate the effect of the position and the inertia change of the ballast tank caused by water discharge on the rotational motion of the hull by applying the parallel axis theorem. Table 6 lists the inertial values of the submarine hull and ballast tank presented in Table 5, as well as the maximum inertial value combined by equation (10).
Matrix of the inertia moment value
Table 6 lists the changes in the overall moment of inertia due to the position of the ballast tank and the water discharge scenario. According to the Tank Inertia and Ship couple conditions, the moment of inertia (Isub) can affect the overall ship inertia (Itotal) by approximately 4.1% (Ixx) and 7.9% (Iyy, Izz) depending on the position of the tank. This ratio indicates that the ballast tank has a minimal impact on linear motion but exhibits significant variation in rotational motion, particularly in pitch and yaw. Tanks placed far from the center of the hull showed a significant reduction in the moment of inertia, which has a greater impact on rotational motion (pitch and yaw). On the other hand, tanks closer to the center of the hull exhibited relatively small changes in inertia, but the effects of a shift in the center of mass decreased, resulting in increased stability. In addition, although the effect of the inertia of the ballast tank is minimal, the moment of inertia, depending on its position, affects the rotational motion. Therefore, the position of the ballast tank can be an important factor in the behavior during emergency rising. In conclusion, the inertia analysis results in Table 6 emphasize that the position of the tank is an important design variable for the stability of hull motion. Tank placement closer to the center of the hull tends to enhance rotational motion stability, whereas tanks closer to the stern or bow act as a cause for the increased sensitivity of rotational motion due to the reduction in moment of inertia.
The increase in sensitivity of such rotational motion can be interpreted through the fundamental equation of rotational motion in Eq. (11). According to the formula, if the moment of inertia I decreases, the angular acceleration a will increase for the same external moment T, which directly causes a larger angle change in pitch and yaw motions. Therefore, it is necessary to meticulously adjust the placement and design of the tanks to optimize the stability and motion characteristics during emergency rising. This paper provides foundational data for optimizing such designs, and further research will involve additional simulations that reflect various environmental conditions.
4. Conclusion
This study quantitatively analyzed the impact of the movement of the longitudinal center of gravity (LCG) and changes in the moment of inertia caused by water discharge from the ballast tank during submarine emergency rising on hull motion stability. To this end, numerical analysis was conducted using a 6DOF motion equation and a grid-based volume cell model, systematically evaluating the impact of the position and water discharge scenarios of the ballast tank on hull motion.
The analysis showed that linear motions (surge, sway, and heave) are relatively less affected by the dynamic changes in the ballast tank. This difference was attributed to linear motion, which depends primarily on the total mass of the hull and its initial state of motion. On the other hand, in rotational motion (pitch, roll, and yaw), the reduction in moment of inertia and the shift in center of mass caused by the discharge of water from the ballast tank had a significant impact. Specifically, in pitch motion, the changes in inertia and the movement of the center of mass made the system much more sensitive to outside forces, leading to a much larger range of angle variations.
According to the analysis in Table 6, the moment of inertia of the ballast tank accounts for up to 7.9% of the total hull inertia, which significantly affects motion stability depending on the position of the tank. Ballast tanks located near the center of the hull have relatively small changes in inertia and a reduced center of mass movement, which enhances the stability of pitch and yaw motions. In contrast, tanks closer to the stern or bow contribute to increased sensitivity to rotational motion because of the reduction in moment of inertia.
This study proposes a numerical analysis model to quantitatively assess the impact of dynamic changes caused by water discharge from the ballast tank on hull motion during emergency rising. These findings showed that the arrangement and design of the tank are the key design variables for optimizing the stability and performance of submarines. In addition, foundational data for design optimization was obtained by analyzing the interaction between the moment of inertia and the movement of the center of mass. In future research, additional simulations will be conducted to reflect various environmental conditions and changes in the ballast tank shape.
Notes
No potential conflict of interest relevant to this article was reported.
This study was conducted by the “Deep Ocean Engineering Basin-based standard ocean structure performance evaluation technology development (4/5)”, one of the major projects of KRISO (PES5140) and this research was supported by “Regional innovation Strategy (RIS)” through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2023RIS-007).
