The effect of rudder deflections on hydrodynamic performance of underwater vehicles during dynamic maneuver

Abstract

A computational study is conducted for the flow over the submarine under the rudder deflection condition, considering the complicated flow characteristics of a submersible under maneuvering conditions. The STAR-CCM + is used to simulate the rudder force test over an extensive variety of rudder angles using the submarine model Defense Advanced Research Projects Agency (DARPA) SUBOFF. However, in the present research, the force and moment-related parameters (also known as the hydrodynamic parameters) and their coefficients are calculated with the deflection of control surfaces from − 15 to + 15° with a step increase of 3° while dynamically rotating the complete body at multiple drift angles. Dynamic yaw maneuvers with time-step increments were applied to the body in the current study. Furthermore, the control surface was actuated at a specific angle; the actuation was rigidly controlled. The hydrodynamic coefficients of DARPA SUBOFF were calculated using this configuration, and it was discovered that these coefficients’ trends change systematically while the yaw maneuver is executed. At multiple angles of control surfaces, the coefficient of drag (Cd), side slip coefficient (Cy), and yaw moment coefficient (Cn) noticed for different drift angles of a body ranged from 0.005 to 0.0415, − 0.0675 to 0.0625, and − 0.022 to 0.017, respectively. At different speeds ranging from 3.05 to 6.10 m/s, the estimated findings obtained for resistance and hydrodynamic coefficients are compared with experimental data.

Introduction

One of the key aspects of a submarine is maneuverability which affects both mission capability and navigation safety. The assessment of maneuverability has evolved into a crucial step in the submarine design procedure, and in the past few years, there has been a lot of focus on the study of the submarine’s maneuvering capabilities. It is commonly recognized that the hydrodynamic forces applied on the submarine’s hull and appendages, which are closely linked to the local flow areas, govern the submarine’s maneuvering properties. Nevertheless, the submarine’s surrounding flow fields are quite complicated when it is navigating, and a variety of vortex forms emerge from its main body and appendages [16]. Large-scale flow separation would be most noticeable around the rudder when performing tight maneuvers. This could lead to rudder stall, which could negatively impact the submarine’s ability to maneuver. To provide additional information on the assessment of the submarine maneuverability, it is therefore necessary to investigate the flow around the submarine under maneuvering conditions, particularly for the rudder deflection with a large angle. Particle Image Velocimetry (PIV) studies of the flow around a submarine model at various side slip angles were made by Kumar et al. [8]. Manshadi et al. [14] used the oil-flow visualization approach in a wind tunnel to study the impact of the vortex generator on the flow surrounding a submarine model. An effective experimental protocol was developed by Lin et al. [10] to examine the horizontal plane maneuvering derivatives of a submersible model. A submersible model was subjected to a series of captive model experiments at a towing tank by Efremov et al. [5], who also discussed the impact of free surface and Froude number on the hydrodynamic parameters.

Dubbioso et al. [4] performed free-running maneuvering simulations for a submersible design with various rudder configurations at infinite depth and near the free surface. They investigated whether the X-rudder configuration has a better turning capability than the cruciform one. In a confirmation and validation investigation for a submersible design under various maneuvering situations, Takahashi and Sahoo [18] examined how well the curvature adjustment strategy improved predictability. In order to investigate the coherent structures shedding from the tip and root of the appendages, Broglia et al. [2] computed the flow over a submersible model under the operating parameters of straight-ahead, drift, and pitch. In the straight-line towing assessment, Lin and Li [9] examined how well the slide mesh prototype captured the flow around a submarine model. Their findings offered a workable method for simulating the unstable motions. A full-scale self-propulsion simulation for a submarine model was reported by Liu et al. [12], who also discussed how the scale effect affected the specifics of the flow field surrounding the downstream propeller. In their investigation of the formation of a submarine’s wakes and vortex patterns when traveling in a straight line, Cao et al. [3] suggested a vortex tuning technique to control the vorticity field. Paredes et al. [15] examined the flow around a submersible model in deep water and close to the surface, talking about how the free surface affects local flow properties and opposition. Liu et al. [13] examined the flow fields surrounding a submarine model in high Reynolds number conditions, evaluating the predictive power of the numerical method in surface curvature impacts on turbulent boundary layers. It is evident from the research that the hybrid Reynolds-Averaged Navier–Stokes/Large Eddies Simulation (RANS/LES) method requires significantly less computational power than the LES method while still being able to foresee the complex flow separation surrounding a submarine.

Maneuverability is an important consideration in the development of unmanned underwater vehicles. This study focuses on the dynamic drift rotation of an underwater vehicle model when the rudders are deflected. The Computational Fluid Dynamics (CFD) software STAR-CCM + is used to simulate the rudders’ deflection test over an extensive range of rudder angles from − 15 to + 15° using a standard submarine model DARPA SUBOFF as the research geometry. During dynamic drift rotation, the results have been collected at every increment of drift angle, i.e., from − 15 to + 15°. This concept enables researchers to cut down on computing time, iterations, reconstructed mesh configurations, and resetting physical factors, which are still not being investigated and used. Furthermore, outcomes contrasted with experimental findings from DTRC publications, as stated in Roddy [17] and Groves, Huang et al. [6], provide a detailed comparison, nevertheless, the data available from their research provide certain or specific data for some specific drift angles with variable control surfaces positions. However, our study provides multiple/array of data/data points at each increment of time step, which are better for designing control algorithms and give further authority on control surface or drift angle controllability/maneuverability. In the resistance test and rudder’s deflection testing, the accuracy of the CFD results is validated with experimental results.

Methods

For a DARPA SUBOFF project in 1989, Groves et al. [6] developed an axially symmetric SUBOFF with symmetrical appendages at the stern. This designed SUBOFF has an overall length of 4.3560 m. The trials were conducted on the DARPA SUBOFF model with different forms such as a fully appended hull, the hull with a sail, the hull with a ring wing, and the hull with no appendages as stated by Liu and Huang [11]. The fully appended form of DARPA SUBOFF’s geometric properties is illustrated in Fig. 1.

Fig. 1 [Images not available. See PDF.]

Geometric properties of fully appended DARPA SUBOFF [19]

Computational domain

The computational domain was split into two sections for this research. One of them is the rotating domain, and the other one is the static domain. The static zone is made up of blocks, while the rotating domain which is in a vertical position is made up of a cylinder and DARPA SUBOFF geometry. For examinations, an automated mesh was developed for the two regions that STAR-CCM + recommends. The automated mesh, a highly advanced function in STAR-CCM +, allows meshing at specific locations, and it is employed in this research. This feature of STAR-CCM + is dependable for creating high-quality meshes, and it speeds up computation when meshing. Figure 2a and b displays the generated mesh of the stationary and rotational zone.

Fig. 2 [Images not available. See PDF.]

Mesh generation of a static domain and b rotating domain

Physical conditions

The following physical parameters are listed to calculate the solution after mesh generation: (a) All y + wall treatment, (b) constant density liquid (water), (c) implicit unsteady, (d) three-dimensional, (e) segregated flow, (f) RANS, (g) turbulent, and (h) K −$\omega$ model. Before starting the simulation, the following boundary conditions have been identified.

Mesh independence study

To determine the effect of the number of cells on the simulation outcome, the drag force was computed for various cell counts during the process of convergence analysis. The simulation results for the drag force are convergent at 450,000 cells which is equivalent to 0.1 m of base size as shown in the convergence analysis in Fig. 3a. The mesh and time independence study for drag for dynamic yaw maneuver in the case of a rudder’s deflection at 3.05 m/s is shown in Fig. 3b and c. The mesh and time independent study for dynamic yaw maneuver has been carried out at multiple base sizes and time steps.

Fig. 3 [Images not available. See PDF.]

Mesh and time independence study. a) Cd (static analysis), b) Cd (mesh), and c) Cd (time)

In current simulation, changing angle of attack dynamically induces variation in flow field around the DARPA SUBOFF causing variable outcome of Cd on transient time frame. CFD simulations, especially with turbulent flows, show that the drag coefficient can change over time due to the transition of the flow from laminar to turbulent or due to instabilities in the wake region. At the initial stages of the simulation, the flow might be stabilizing, and as time progresses, the turbulent eddies and vortices might form in the wake of the object, which increases drag. In such types of simulations, the wake flow may stabilize over time, and vortex shedding patterns can develop or intensify.

Results and discussion

Resistance validation

Coefficient of drag (Cd) for the fully appended hull form was computed at 3.05 m/s. It was contrasted with the outcomes of the experiment. The Cd of the bare hull form was estimated at 3.04 m/s using an identical methodology as shown in Table 1. Moreover, the difference between the results indicates that the error for the simulation conducted in this work is under 8%, which is way below that of the study already conducted earlier by Heipeng Guo et al. [7] in 2023. They investigated the flow characteristics over the fully appended hull form with rudder deflection and depicted a low level of accuracy towards their obtained results using two different physical models, i.e., Unsteady Reynolds-Averaged Navier–Stokes (URANS) and Delayed Detached-Eddy Simulations (DDES). It is pertinent to mention that in our study, the higher time step was used, i.e., 0.01 s for all simulation cases, as a time-independent study shows that the temporal discretization is carried out tangibly in a similar fashion to that of the lower time step. In addition to this, if the lower time step had been used, the accuracy would have been better than what is currently depicted, but the computational time would have required more time and power. Following this confirmation, additional research was carried out on the appended and bare hull shapes at various velocities, and Cd was calculated. Table 1 lists the experimental readings of Cd for the DARPA SUBOFF of the fully appended hull form and the bare hull form as well as the variance in the CFD results.

Table 1. CFD and EFD Cd values for fully appended hull form and bare hull form of DARPA SUBOFF

Velocity (m/s)	Cd				Error (%)
	EFD (bare hull form)	EFD (fully appended hull form)	CFD (bare hull form)	CFD (fully appended hull form)	CFD (bare hull form)	CFD (fully appended hull form)
3.05	0.00314	0.00347	0.00310	0.00369	− 0.987	6.431
5.14	0.00306	0.00338	0.00280	0.00348	− 8.215	3.154
6.10	0.00300	0.0033	0.00272	0.00342	− 9.013	3.903
7.16	0.00294	0.00324	0.00266	0.00337	− 9.5	4.307
8.23	0.00285	0.00315	0.00260	0.00333	− 8.508	6.026
9.25	0.00272	0.00309	0.00256	0.00330	− 5.669	7.064

Choosing the value of wall y + is the most important stage in CFD simulations. The size of the mesh and the growth rate of the body are changed to keep wall y + values within an acceptable range. All velocities have an average wall y + of between 10 and 100. The boundary grid can be used for simulations because the values at every point are within the required y + range. Figure 4 shows the depiction of the wall y + value of the body.

Fig. 4 [Images not available. See PDF.]

Wall y + depiction at 3.05 m/s. a Side view. b Front view

Lateral control in hydrodynamics (rudders)

In this section, hydrodynamic parameters are calculated at three separate velocities during the dynamic simulation of DARPA SUBOFF. The simulation has been run with control surface (rudder) deflection at various drift angles between − 15° and + 15°. A confirmation assessment of the dynamic simulation of the DARPA SUBOFF with deflection of control surfaces, i.e., rudders from − 15° to + 15°, was rotated about the normal axis from − 30° to + 30°.

From the CFD simulation, the drag force, side slip force, yawing moment, and their coefficients were computed. Tonio, A. S. et al. presented the experimental findings of the bare hull model for yawing moment and drag force in their article [19], respectively. However, the experimental data of the fully appended hull form for side slip coefficient were provided by Haipeng Guo et al. [7] in their research. The results of the experiments and the CFD were compared.

The correlation between the experimental outcomes and the findings of the STAR-CCM + ’s drag, side slip, and moment coefficients, for different drift angles at 3.05 m/s, is shown in Fig. 5a, b, and c.

Fig. 5 [Images not available. See PDF.]

Comparison of hydrodynamic coefficients at 3.05 m/s with experimental results. a) Drag coefficient, b) Side slip coefficient, c) Yawing moment coefficient

When compared to the experimental results at the maximum positive drift angle, the drag coefficient calculation for the bare hull results shows a variance of about − 9.4%, whereas the drag coefficient estimate for the bare hull with sail shows an error of about 20%. This suggests that when it comes to calculating drag coefficients, the bare hull exhibits identical results to the bare hull with sail. In contrast, the side slip coefficient produced by STAR-CCM + is acceptable with the findings of the experimental outcome. The graph displays a consistent trend of experimental data for both bare hulls with and without sails as depicted in Fig. 5b.

Furthermore, the findings of the experimental data are compared to the yaw moment coefficient computed using STAR-CCM + in Fig. 5c. Yaw moment coefficient calculations for the results of the bare hull with sail show an error of − 32%, while the outcomes of the bare hull without sail demonstrate a variation of about − 6.4% when contrasted to the experimental outcomes at the positive highest drift angle. Additionally, there is a considerable correlation between the experimental results and the CFD outcomes, demonstrating high agreement between drag, side slip, and yawing moment coefficients. In addition to this, the forces and moments trends show similar patterns with their respective coefficients.

Hydrodynamic variables obtained at 3.05 m/s, 5.14 m/s, and 6.10 m/s

The following figures illustrate the drag, side slip, and moment coefficients for various drift angles at a speed of 3.05 m/s with control surfaces deflection varying from −15° to +15°. The relationship between drag coefficients and the deflection of the control surfaces (−15° to +15°) at different drift angles is shown in Fig. 6a.

Fig. 6 [Images not available. See PDF.]

Comparison of hydrodynamic parameters at 3.05 m/s with deflection of control surfaces. a) Drag coefficient, b) Side slip coefficient, and c) Yaw moment coefficient

The image shows that as the DARPA SUBOFF turns dynamically from − 30° to + 30°, the value of the drag coefficient drastically rises. In comparison to greater angles of control surfaces, the value of the drag coefficient is numerically smaller and closer to other values of control surfaces for smaller angles of control surfaces. When there is a positive drift angle, the drag coefficient is larger, and when there is a negative drift angle, it is lower. The correlation between side slip coefficients and the deflection of the control surfaces (− 15 to + 15°) at a velocity of 3.05 m/s at various drift angles is shown in Fig. 6b.

The graphic demonstrates how the side slip coefficient significantly increases as the submersible vehicle rotates dynamically from a −30° to +30° drift angle. At negative drift angles, the side slip force coefficient is negative, whereas at positive drift angles its value is positive.

Figure 6c demonstrates a strong connection between the moment coefficient and the deflection of the control surfaces (−15° to +15°) at a velocity of 3.05 m/s at varying drift angles. The diagram shows how the moment coefficient dramatically rises as the submersible vehicle rotates dynamically from −30° to +30°. The moment coefficient is negative at negative drift angles, whereas it is positive at positive drift angles. All hydrodynamic variables corresponding to other velocities such as 5.14 m/s and 6.10 m/s show the same pattern as in the case of a velocity of 3.05 m/s, even though the values will increase dramatically as the velocity increases. Figure 7a and b demonstrates cases of 5.14 m/s and 6.10 m/s, respectively. The values of drag coefficients, side slip coefficients, and yawing moment coefficients at smaller angles of control surfaces (rudders) are numerically smaller and more comparable to other values of control surfaces. It should be noted that the hydrodynamic coefficient has a lower value when the drift angle is negative and a greater value when the drift angle is positive, and the numerical values of the hydrodynamic coefficients rise in accordance with the velocity. Error was also calculated using root-mean-square error analysis using Eq. 1 to quantify the difference between the experimental value and that of simulation one. For this purpose, data of 3.0 m/s velocity was taken to find the frequency of RMSE variation which is demonstrated using Fig. 8. Equation 1 was used for this purpose.

1

$$\mathrm{RMSE}=\sqrt{\frac{{\sum \left({\mathrm{x}}_{\mathrm{efd}},-,{\mathrm{x}}_{\mathrm{cfd}}\right)}^2}{\mathrm{N}}}$$

Fig. 7 [Images not available. See PDF.]

Hydrodynamic coefficients with deflection of control surfaces: a) Cd for 5.14 m/s velocity: a1) Cy for 5.14 m/s and a2) Cn for 5.14 m/s. b) Cd for 6.10 m/s: b1) Cy for 6.10 m/s and b2) Cn for 6.10 m/s

Fig. 8 [Images not available. See PDF.]

RMSE values differentiated using experimental values and computational fluid dynamic values, where Cn is for the coefficient of yawing moment, Cy is for the sideslip coefficient, and Cd is for the coefficient of drag

Where RMSE is root-mean-square error, ${\mathrm{x}}_{\mathrm{efd}}$ is an experimental value, ${\mathrm{x}}_{\mathrm{cfd}}$ is a CFD value, and N is the number of counts.

Figure 8 demonstrates the RMSE values for Cd and Cn are very much closer to zero showing very little deviation from experimentally obtained values. However, values for Cy are deviating more, which is mostly seen at higher positive angles of attack, as it causes the flow to distort due to boundary layer detachment increases as the AoA rises. The rise in AoA induces a decrease in wet surface area.

Flow field contours around the submarine

Figure 9a, however, shows the top view of a 5° deflection of the control surface at a 30° drift angle. On the other hand, Fig. 9b shows a drift angle of 29.4° and a deflection angle of 5°. Moreover, the wake generation is clearly depicted over the snorkel due to a high drift angle as shown in Fig. 9b. The deviation in the flow profile caused by the drift angle is prominent, with the blue region being formed asymmetrically to the axisymmetric path. In addition to this, Fig. 10b shows heavy flow vortices behind the snorkel due to the drift angle. In line with this, the heavy boundary attachment is seen on the rudder portion of the DARPA, demonstrating that the DARPA rudder efficacy is good at high drift angles, and that rudders have a maximum wetted area, providing less boundary layer detachment during drift profiles.

Fig. 9 [Images not available. See PDF.]

Five-degree deflection angles at a 30° angle of drift; a) Wall y + and b) Streamlines

Fig. 10 [Images not available. See PDF.]

Top view velocity depiction with different angles of drift with + 5-degree control surfaces (Rudders) actuation: a) − 30°, b) − 14°, c) − 1°, d) 5°, e 16°, f) 30°

Figure 10 shows the top view of the dynamic rotation of DARPA SUBOFF’s velocity contours with streamlines from −30° (upward) to +30° (downward) as the rudders are deflected at a +5-degree angle. The following figures demonstrate how the wake is generated as the body changes its drift angle. Maximum wake generation occurs when the body is rotated at its maximum angles either positive or negative. Moreover, the forces on the nose of the vessel become very high as well at these drift angles. However, at lower angles, nose pressure levels are not so dominant.

The residual output obtained at the end of each run is utilized to demonstrate the accuracy of the predicted outcomes and the characteristics of the flow fields that are examined in a variety of dynamic simulations at multiple drift angles. It is significant to highlight that the residuals of flow, continuity, momentum, and other turbulence components should be addressed below the 10⁻³ level to obtain higher approximations of results. Apart from validation and convergence, the factor of skewness angle and the cell quality of the complete computational domain were checked to verify the quality of the cells generated during mesh development. The elements are proven to have an impact on both mesh development and quality. Additionally, in each set of dynamic simulations, the model is rotated from one field of study to another, ensuring that it satisfies the previous one’s convergence requirements before moving on to the next. It is important to note that the frequency of the skewness factor for both the current model and the full domain was determined to be well within the 90-degree skewness angle [1]. To verify the correctness of the mesh development, mesh cell quality was evaluated in addition to skewness. The frequency of the cell quality for the created mesh has been found to be well within the 0.9 ranges [1]. The present study offers a variety of data points that are preferable for developing control algorithms and offers more authority on the control surface controllability/maneuverability.

Conclusions

Dynamic models are crucial for simulating complex underwater motions prior to the launching of the UVs. A key component of this dynamic simulation of UVs is the estimation of the force and moment coefficients. The goal of the present research was to calculate hydrodynamic variables at various drift angles, with deflection of control surfaces ranging from − 15° to + 15° for rudders for lateral stability of DARPA SUBOFF. DARPA SUBOFF’s bare and appended hull form resistance and its coefficient were analyzed.

Our study, however, offers a variety of data/data points that are better for constructing control algorithms and that provide additional authority on the control surface controllability/maneuverability. The accuracy of the CFD outcomes is tested using experimental data in the resistance test and rudder deflection test. Finally, the depiction of the boundary layer at the tail section at high drift angle provides a detailed insight on the effectiveness of the rudder and its geometrical aspect during high maneuver actions.

Acknowledgements

Not applicable.

Authors’ contributions

IM, first author, manuscript writing, execution, and simulation work. AL, second author, manuscript writing, editing, and supervision of the work. SSHZ, third author and editing and reviewing. AK, fourth author and support for HPC. MA, fifth author and formatting and reviewing

Funding

There is no funding associated with this research.

Data availability

The datasets generated and/or analyzed during the current study are available in the Encyclopedia repository, Darpa Suboff Yawing with (5deg)Rudder | Encyclopedia MDPI.

Declarations

Competing interests

The authors declare that they have no competing interests.

Abbreviations

UVs

Underwater Vehicles

DARPA

Defense Advanced Research Projects Agency

Cd

Coefficient of Drag

Cy

Side Slip Coefficient

Cn

Yaw Moment Coefficient

CFD

Computational Fluid Dynamics

PIV

Particle Image Velocimetry

RANS/LES

Reynolds-Averaged Navier–Stokes/Large Eddy Simulation

URANS

Unsteady Reynolds-Averaged Navier–Stokes

DDES

Delayed Detached-Eddy Simulations

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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