Investigation of Maneuvering Coefficients for the Darpa Suboff Aff1 Model via Computational Fluid Dynamics

Abstract

Keywords: DARPA Suboff; CFD; Maneuvering coefficients; Static drift

1. Introduction

Throughout history, geopolitical competition and the pursuit of power have driven the development of advanced technologies. As over 70% of the Earth’s surface is covered by oceans, maritime dominance has become a key element in global strategic dynamics. Among the most transformative innovations in naval warfare is the submarine, which has significantly reshaped underwater operations. Today, submarines serve a diverse range of purposes, including military operations, scientific exploration, and

even tourism. In the defense sector, they play vital roles such as deterrence, strategic offense, force projection, surveillance, and covert attack missions from beneath the surface to shore based targets. The critical role of submarines in military operations necessitates a range of performance capabilities, among which maneuverability is essential for maintaining operational effectiveness. To successfully conduct reconnaissance and execute underwater missions, submarines must operate at considerable depths with precise control. Additionally, conventional diesel-electric submarines are periodically

Received 17 September 2025; revised 28 December 2025; accepted 6 January 2026. Published online 19 January 2026 E-mail address: hpalavar@turkloydu.org (H. Palavar). https://doi.org/10.29187/2792-0771.1045 2792-0771/© 2026 Published by Yıldız Technical University Press, İstanbul, Türkiye. This is an open access article under the CC BY-NC 4.0 Licence (https://creativecommons.org/licenses/by-nc/4.0/).

required to surface to intake atmospheric oxygen— both for engine combustion and for sustaining onboard life-support systems. Consequently, robust maneuvering performance is vital not only in deep-sea environments but also during near surface operations. During the preliminary design phase of submarines, several maneuvering-related parameters are evaluated, including course-keeping ability, dynamic stability, turning performance, heading-change capability, and stopping behavior. These characteristics are typically assessed using linear motion equations in the early stages of design. While linear methods provide a first order approximation, more complex scenarios—such as those involving propulsion and control systems—require nonlinear modeling techniques. At this stage, maneuvering performance is analyzed using nonlinear mathematical models incorporating surge, sway, and yaw motions. In advanced phases, model testing and Computational Fluid Dynamics (CFD) simulations are employed to enhance the accuracy of maneuverability predictions (Sukas et al., 2017). Both manned and unmanned submarines are employed for a variety of applications, including scientific research, search and rescue missions, and the maintenance or repair of underwater structures. During such operations—particularly when maneuvering in close proximity to the seabed—the hydrodynamic forces and moments acting on the hull can vary significantly due to ground effect phenomena (Sabuncu, 1985). Consequently, submarines must perform accurate and responsive maneuvers under rapidly changing flow conditions. To ensure safe and effective operation, maneuverability analyses should be conducted prior to the construction phase. These evaluations can be performed using empirical or semi-empirical formulations, Computational Fluid Dynamics (CFD) methods, and physical model testing (Sukas et al., 2017). Since submarine design is primarily carried out within the defense sector, access to detailed geometries and performance data is often restricted, limiting academic research in this field. To facilitate broader scientific investigation and ensure repeatability across studies, the development and dissemination of standardized submarine model geometries became essential. In response, various institutions and research centers have undertaken efforts to establish publicly available reference models. In the 1980s, the Defense Advanced Research Projects Agency (DARPA) initiated the SUBOFF project to support the development of submarine technologies. As part of this initiative, a mathematically defined submarine model—available in both appendage and bare hull configurations—was introduced, with its geometric specifications

published (Groves et al., 1989). Huang et al. (1989) and colleagues also documented the experimental setup and instrumentation used to test the DARPA SUBOFF model. The symmetric body DARPA SUBOFF model has since been extensively investigated under various configurations. Resistance tests were conducted in towing tanks, while pressure distributions and wake characteristics were examined in wind tunnel environments. Liu and Huang (1998) further analyzed the effects of geometric variations on wake behavior, static pressure fields, and boundary layer velocity profiles. Their study also provided comprehensive datasets on surface pressure and skin friction measurements. The geometry of the DARPA SUBOFF model used in this study is shown in Fig. 1.

Furthermore, the ITTC Manoeuvring Committee (2014) has officially recommended the use of the DARPA SUBOFF models specifically the bare hull (AFF1) and fully appended (AFF8) configurations; as standard geometries for submarine related hydrodynamic studies. Wu et al. (2005) investigated the influence of hydrodynamic forces acting on submarines operating close to the seabed, with particular emphasis on bottom effect phenomena. In their study, the bottom effect was modeled as a linear function of the vertical distance between the submarine and the seabed. In addition, the influence of the free surface was analyzed for scenarios in which the submarine operated near the water surface. Vaz et al. (2010) assessed the maneuvering performance of the DARPA SUBOFF model in both appended and bare hull configurations under varying angles of attack. Their study also examined the influence of turbulence models on hydrodynamic predictions by employing two different viscous flow CFD solvers. The findings indicated that distinct configurations of the SUBOFF model are commonly utilized for evaluating the maneuverability of fully submerged bodies. Moreover, the results emphasized that in addition to bottom and free surface effects, frictional forces significantly influence hydrodynamic behavior, and turbulence model selection plays a critical role in simulation accuracy.

Polis et al. (2013) employed Computational Fluid Dynamics (CFD) techniques to examine the maneuvering coefficients of the DARPA SUBOFF model under free surface effects at varying snorkel heights and speeds. In their study, hydrodynamic coefficients accounting for free surface influences were incorporated into a mathematical model developed for submarine simulations. The resulting predictions were found to be in good agreement with experimental measurements. Duman et al. (2018) investigated the effect of the propeller on maneuvering forces of a submerged body using CFD. Analyses were performed on the DARPA Suboff bare hull form (AFF-1) for different drift angles and Reynolds numbers. Results were validated against experimental data. The study showed that at small drift angles, the propeller significantly affects longitudinal forces but has negligible impact on sway forces and yaw moments. Furthermore, CFD accurately predicts longitudinal forces, sway forces, and yaw moments with high precision. Atik (2022) investigated the hydrodynamic coefficients of the DARPA SUBOFF submarine using static drift test simulations. CFD results at 0–18° drift angles were validated against experimental data. At small drift angles, the Spalart-Allmaras model showed the closest agreement with experiments, while SST k-ω and k- models performed better at higher angles. Since all turbulence models produced similar results for sway forces and yaw moments, significant differences in maneuvering behavior are not expected. Öztürk et al. (2023) numerically investigated the static drift maneuvering forces and moments of the DARPA SUBOFF submarine for AFF-3 and AFF-8 configurations using CFD. The AFF-3 results were validated against experimental data and showed good agreement. For the AFF-8 configuration, where no experimental data were available, the results at 18° drift angle were consistent with the numerical study of Vaz et al. The study showed that with increasing drift angle, the lateral force and yaw moment increase while the axial force decreases, and the presence of the sail significantly affects the maneuvering characteristics. Çıplakkaya and Arıkan Özden (2023) numerically investigated the forces and moments under static drift and resistance for the model and full scale geometries of the Autosub submarine using CFD. The optimum mesh was determined through a mesh independence study, and results were validated against experimental data. The analysis showed that with increasing drift angle, axial force decreases while lateral force and yaw moment increase. In addition, Kahramanoğlu (2023) examined the DARPA SUBOFF model in both horizontal and vertical planes under steady speed conditions. In his study

on the horizontal plane, he numerically modeled sway and yaw motions at different scales to investigate the scale effects on maneuvering derivatives. Similarly, Kahramanoğlu et al. (2024) conducted a study on the vertical plane, they analyzed heave and pitch motions using CFD and identified the associated scale effects. However, since these investigations were conducted at constant speed, the number of studies addressing results obtained at varying speeds remains quite limited in the literature. The objective of this study is to determine the maneuvering coefficients of the DARPA SUBOFF form in its bare hull (AFF1) configuration under static drift conditions using CFD. STAR-CCM+ software was employed to perform the simulations, and the results were validated against experimental data from the literature. The CFD results were found to be in good agreement with the experiment results, remaining within acceptable accuracy margins. To further ensure result reliability, a grid independence study and uncertainty analysis were conducted. This methodological approach ensured that the findings were not influenced by mesh sensitivity and achieved a balance between computational efficiency and accuracy. The simulations were repeated for multiple speeds, and nondimensional force and moment values were used to derive maneuvering coefficients. Overall, this study not only provides a validated maneuvering analysis for a widely used submarine geometry but also demonstrates the effectiveness of CFD based methods in capturing hydrodynamic behavior. By comparing the derived coefficients with previous literature, the study contributes to improving the reliability and applicability of computational tools in underwater vehicle design and analysis. Moreover the maneuvering coefficients obtained in this study can be used as reference values in future research and analyses of different DARPA Suboff configurations, thereby providing a new contribution to the existing validated and shareable CFD based maneuvering data in the literature.

2. Methodology

To develop a mathematical model for ship maneuvering behavior, it is essential to define the reference coordinate systems. Two primary coordinate systems are commonly used: the earth-fixed system (o0x0y0z0) and the body-fixed system (o-xyz). In the earth-fixed system, the origin is typically assigned to the point where the maneuver begins, while in the body-fixed system, the origin is placed at either the ship’s midship section or its longitudinal center of gravity (LCG) (Sukas et al., 2017). According to Yasukawa and Yoshimura (2014), positioning the origin

at the midship is more appropriate, as the center of gravity may vary with different loading conditions. The earth-fixed and body-fixed coordinate systems adopted in this study are illustrated in Fig. 2.

motion. In most hydrodynamic studies, submarine maneuvering is modeled using a three degrees of freedom (3DOF) framework that focuses on motion in the horizontal plane.

In order to model the horizontal plane motion of a ship, it is necessary to determine the linear and nonlinear maneuvering derivatives associated with velocity and acceleration terms. Several approaches are commonly employed to estimate these derivatives (Çakici & Kahramanoğlu, 2017): • Physical maneuvering experiments (e.g., planar motion mechanism (PMM) tests in yaw and sway, rotating arm tests, and static drift tests) • Statistical methods such as multiple regression analysis, using data from similar ship models • Computational Fluid Dynamics (CFD), typically based on the finite volume method Given the high cost and time requirements of experimental testing, CFD has become a preferred method in both academic and industrial applications. CFD is extensively used for estimating maneuvering derivatives, particularly during the early design stages. Alternatively, regression based approaches using empirical data from comparable hull forms may also be applied.

6. Yaw – Rotation about the vertical axis

The six degrees of freedom motions of a submarine are schematically shown in Fig. 3. Among these, heave, roll, and pitch are primarily governed by hydrostatic restoring forces, while surge, sway, and yaw are considered as the key components of maneuvering

In the earth-fixed coordinate system, external forces and moments acting on the vessel are defined as X0 in the longitudinal (x) direction, Y0 in the lateral (y) direction, and N as the yaw moment about the vertical (z) axis. According to Newton’s Second Law, the governing equations are expressed as: X0 = mẍG

In the equations, ẍG represents the acceleration in the x-direction, ÿG represents the acceleration in the y-direction, and ṙG denotes the angular acceleration around the z-axis. Here, xG and yG represent the coordinates of the ship’s center of gravity, while ψ refers to the yaw angle, which is the angle between the ship-fixed coordinate system and the earth-fixed coordinate system. The parameter m denotes the mass of the ship, while IZ represents the moment of inertia about the z-axis, taken around the center of gravity. Roddy (1990) conducted comprehensive experiments on submarine motion under six degrees of freedom and published foundational data that remain widely referenced in maneuvering studies. His experimental work, typically analyzed in the earth-fixed coordinate system, has enabled a more detailed understanding of vessel dynamics in multiple directions. Roddy’s contributions continue to serve as a critical benchmark for validating numerical and empirical models in submarine maneuvering analysis. • Surge: 
m u̇ − vr + wq − xG q2 + r 2 + yG (pq − ṙ)  +zG (pr + q̇) = X

• Sway: 
m v̇ − wp + ur − yG p2 + r 2 + zG (qr − ṗ)
 +xG gp + ṙ = Y

• Heave: 
m ẇ − uq + vp − zG q2 + p2 + xG (r p − q̇)  +yG (rq + ṗ) = Z

DARPA Suboff form in a static drift condition using the CFD method is aimed. The study was conducted using Star CCM+ software, and experimental data from the literature were utilized as a reference to assess the accuracy of the results. The analysis results indicate that the obtained data exhibit consistency with the reference experimental results and fall within acceptable accuracy limits.

3. Darpa suboff numerical analyses

In this study, the determination of maneuvering coefficients for the bare hull (AFF1) model of the

Fig. 5. The side view of the DARPA Suboff AFF1 form. Table 1. Main dimension. Length Over All

Length Between Perpendicular Maximum Diameter Longitudinal Center of Buoyancy (fore)

4. Calculation of maneuvering coefficients

The maneuvering coefficients were obtained using the static drift method. For this, the results from analyses conducted at three different angles of attack and four different speeds were used. The obtained force and moment values were then nondimensional. Nondimensional is a method used in hydrodynamic analyses and engineering calculations to make physical quantities independent of units. This process facilitates the comparison of systems and conditions at different scales. By eliminating these differences, nondimensional allows data to be interpreted in a more universal manner. In the study conducted on the DARPA Suboff submarine form, nondimensional allowed the analysis results at different speeds and angles of attack to be examined within a common reference framework. The nondimensional formula for the force in the X direction is given in Eq. (19), for the force in the Y direction in Eq. (20), for the moment around the Zaxis in Eq. (21), and for speed in Eq. (22).

X 2U 2 ρl 2 Y Y0 = 1 2 2 ρl U 2 N N0 = 1 3 2 ρl U 2 V Usinβ v0 = = = sinβ U U X0 = 1

He nondimensional X and Y forces, as well as the N moment, calculated using the nondimensional formulas, are presented for different speeds in Tables 10 to 13. To clearly illustrate the differences between the values, the Y’-v’ graph in Fig. 13, and the N’-v’ graph in Fig. 14. To calculate the maneuvering derivatives of the Y-direction force (sway) and the moment around the Z-axis (yaw), a linear equation (ax + b) is fitted to the nondimensional values. The least squares method is used for all cases (Yoon et al., 2015). The slope of the Y’-v’ graph gives the value of Yv0 . This derivative represents the nondimensional form of the Yv maneuvering coefficient. Similarly, the slope of the N’ - v’ graph gives the value of Nv0 , which represents the nondimensional form of the Nv maneuvering coefficient. Y 0 = Yv0 v0

Table 10. Nondimensional X, Y, and N values at a velocity of 1,029 m/s. Drift Angle

Table 11. Nondimensional X, Y, and N values at a velocity of 2,058 m/s. Drift Angle

Table 12. Nondimensional X, Y, and N values at a velocity of 3,344m/s. Drift Angle

Table 13. Non-dimensional X, Y, and N values at a velocity of 4,373 m/s. Drift Angle

Fig. 14. N’-v’ graph. Table 14. Determination of the nondimensional derivative Yv0 from the Y’–v’ graph.

Relative difference with respect to the design velocity of 6.5 kn

Table 15. Determination of the nondimensional derivative Nv0 from the N’–v’ graph.

Relative difference with respect to the design velocity of 6.5 kn

The reference design speed is 6,5 knots. R2 represents the accuracy of the fitted curve. R2 value close to 1 indicates that the curve accurately represents the data. The non-dimensional maneuvering coefficients calculated as a function of velocity are given in Figs. 15 and 16. When examining the graphs of the obtained nondimensional maneuvering coefficients, the effects of speed changes on the DARPA Suboff submarine form are clearly observed. Specifically, the Yv0 coefficient shows significant variation at low speeds, while the Nv0 coefficient is not significantly affected by changes in speed. This suggests that the yaw moment coeffi-

cient exhibits a more stable behavior in response to speed variations. The change in the Yv0 coefficient indicates that lateral forces have a more pronounced effect at low speeds, while the more stable trend of the Nv0 coefficient, independent of speed, demonstrates that the forces related to the yaw moment are less influenced by speed. In the analyses, these coefficients were calculated using the static drift method and CFD techniques, highlighting the effects of speed on the coefficients. These findings provide an important foundation for assessing submarine maneuvering performance and conducting speed based maneuvering analyses.

5. Conclusions

In this study, the hydrodynamic behavior of the DARPA Suboff (AFF1) bare hull under various speeds and angles of attack was investigated using a CFD based static drift method. A mesh independence study was performed at the design speed of 3.343 m/s and an angle of attack of 8 degrees using three different mesh densities, and the optimum mesh structure was identified. The analyses were then repeated for different angles of attack and speeds using the optimum mesh configuration. To compare the numerical results with experimental data, static drift analyses were conducted at the design speed of 3.343 m/s for angles of attack of 3, 6, and 8 degrees. The forces in the X and Y directions, as well as the moment about the Z-axis, were obtained. When the computed force and moment values were compared with the experimental results reported by Roddy (1990), average deviations of approximately 3.5% for the X-direction force (longitudinal force), 9.5% for the Y-direction force (sway force), and 5% for the yaw moment were observed. These deviation levels are consistent with similar studies in the literature and indicate that the results can provide reliable references for engineering applications. Furthermore, as expected, increasing the angle of attack resulted in a pronounced increase in the magnitude of forces and moments acting on the hull. To compute the maneuvering coefficients, analyses were repeated at speeds of 1.029, 2.058, 3.344, and 4.373 m/s for angles of attack of 3, 6, and 8 degrees. As expected, the force in the X direction (longitudinal force) increased with speed, indicating that the forward resistance of the submarine rises proportionally with velocity. No significant variation in the X-direction force was observed with changes in angle of attack. In contrast, increases in both speed and angle of attack led to substantial increases in the Y-direction force (sway force) and the moment about the Z-axis (yaw moment). This behavior can be attributed to the expansion of separation zones, the enhancement of pressure asymmetry, and the increasingly irregular distribution of streamlines around the hull at high angles of attack. The calculated force and moment values were nondimensionalized. To determine the maneuvering derivatives of the Y-direction force (sway) and the moment about the Z-axis (yaw), a linear equation (ax + b) was fitted to the nondimensional values. Examination of the resulting maneuvering coefficients 0 showed that the derivative Yv varied significantly within the speed range of 1.029–3.344 m/s, whereas 0 the derivative Nv exhibited a stable behavior with

respect to speed variations. Therefore, it can be stated that the yaw moment coefficient shows a more consistent response to changes in speed. The validation of the numerical results against experimental data and the successful completion of the mesh independence study demonstrate the reliability of the present work. Additionally, the nondimensional maneuvering coefficients obtained in the study provide reference values for future investigations at different speeds and angles of attack. The maneuvering coefficients and analysis results presented herein are planned to be utilized in future studies involving different experimental configurations (AFF1–AFF8).

Acknowledgments

This paper is derived from the author’s undergraduate graduation project conducted at Bandırma Onyedi Eylül University, Department of Naval Architecture and Marine Engineering. The author would like to express sincere gratitude to Assoc. Prof. Dr. Emre Kahramanoğlu for his supervision and guidance, and to Dr. Utku Cem Karabulut for his continuous support and encouragement.

Ethics

There are no ethical issues with the publication of this manuscript.

Conflict of interest

Financial disclosure The author declared that this study has received no financial support.

Authors contribution Conceptualization, methodology, writing, investigation, verification and software: Haydar Palavar.

Data availability statement

The published publication includes all graphics and data collected or developed during the study.

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