### Nomenclature

*ρ*Density (kg/m

^{3})

*m*Mass of Underwater Vehicle (kg)

*I*Moment of Inertia (kg-m

^{2})

*X*

_{0}X Position in Earth-Fixed Frame (m)

*Y*

_{0}Y Position in Earth-Fixed Frame (m)

*ψ*Yaw Angle in Earth-Fixed Frame (rad)

*u*Surge Velocity in Body-Fixed Frame (m/s)

*v*Sway Velocity in Body-Fixed Frame (m/s)

*U*Overall Speed (m/s)

*r*Yaw Rate in Body-Fixed Frame (rad/s)

*β*Drift Angle (rad)

*δ*Rudder Deflection Angle (rad)

*X*Surge Force in Body-Fixed Frame (N)

*Y*Sway Force in Body-Fixed Frame (N)

*N*Yaw Moment in Body-Fixed Frame (N-m)

*L*Length of Underwater Vehicle Body (m)

*x*X coordinate of Center of Gravity (m)

_{g}*y*Y coordinate of Center of Gravity (m)

_{g}### 1. Introduction

### 2. Configuration and Specifications of Underwater Vehicle

### 3. Equations of Motion

### 3.1 Coordinates

##### (1)

### 3.2 Equations of Motion in Horizontal Plane

*X*,

*Y*, and

*N*are the external force and moment on the underwater vehicle, including hydrodynamic, hydrostatic (gravity and buoyancy), and thrust terms. Roll motion can be neglected because there is little snap roll angle when the vehicle turns steadily. The roll angle is controlled by the elevators. In addition, a surge equation is not included since it can be considered as constant, and the speed of the vehicle is uniformly controlled in the sea trial of this study when the vehicle goes straight or turns.

*ϕ*) and pitch angle (

*θ*) are not considered in horizontal plane motion. Hydrodynamic terms (

*Y*and

_{H}*N*) including the interactions between the yaw rate and drift/rudder angle can be expressed as follows:

_{H}*r*is the yaw rate, and

*δ*is the rudder angle.

*ṙ*is the sway acceleration,

*v̇*and

*ṙ*is the yaw acceleration. The subscript “coup” means coupled derivatives.

*i*and

*ṙ*are not important in steady motion, so

*Y*,

_{v̇}*Y*and

_{ṙ}*N*,

_{v̇}*N*can be neglected. In order to solve the equations for steady turning motion, we have to obtain the damping derivatives of the sway and yaw motion, the control derivatives of the rudder, and coupled interaction terms.

_{ṙ}### 4. CFD Analysis

### 4.1 Simulation Method

*v*is the relative velocity in the moving frame,

_{r}*v*is the absolute velocity in the stationary frame,

*u*is the velocity of the moving frame in the stationary frame,

_{r}*v*is the translational frame velocity, and

_{t}*ω*is the angular frame velocity.

*T*is the thrust, and A is the area of the disk (propeller surface), as shown in Fig. 8. In this study, the input value of thrust is defined as the propeller force in self-propulsion conditions of the vehicle speed and was obtained by self-propulsion tests in the towing tank of the Korea Research Institute of the Ship and Ocean Engineering (KRISO).

### 4.2 Governing Equations and Solver Setting

*ω*and

*v*are 0 and

_{t}*v*equals

_{r}*v*.

*k*-

*ω*SST turbulence model was applied, and the Semi-Implicit Method for Pressure Linked Equations – Consistent (SIMPLEC) was used as a pressure-velocity coupling scheme. The operating fluid is sea water, of which the density is 1025.87 kg/m

^{3}, and the viscosity is 0.00122 kg/m·s. Sea water is considered as an incompressible fluid. All CFD simulations in this study were performed in Ansys Fluent, which is well known and verified commercial CFD software.

### 4.3 Simulation Conditions

*Y*′ means

_{v}*∂Y*′/

*∂v*′, and

*N*

_{r}_{|}

_{r}_{|}′ means

*∂N*′/

*∂*(

*r*′|

*r*′|).

*U*is the overall speed of the underwater vehicle and is set to 10 kn (≒ 5.1 m/s). In the case of the rotating arm simulations, the turning radius changes as the yaw rate changes, and the overall speed is constant.

### 5. Results of the CFD Simulations

### 5.1 Results of the Static Drift/Rudder Simulations

*r*′ = 0). Since the vehicle is symmetrical, simulations for only positive

*v*′ and

*δ*were conducted, and then the simulation results were compared to VPMM test results from the towing tank of KRISO.

*v*′), the simulation results follow the tendency of the experiments well and have feasibility for predicting to the hydrodynamic derivatives of the underwater vehicle. According to the operating concept of the underwater vehicle, it is very rare that the drift angle is larger than 10 degrees (

*v*′ = 0.1736) during all flight times.

### 5.2 Results of Rotating Arm Simulations with Drift/Rudder Angle

*r*′ and the whole range of

*v*′ and

*δ*were conducted because the vehicle is symmetrical, but the relative direction between yaw rate and drift/rudder angle can affect the tendency of the interactions. Figs. 15–16 show the total non-dimensional force

*Y*′ and moment N’ for the yaw rate and drift angle, and Figs. 17–18 show the results for the yaw rate and rudder angle.

*r*′ = 0 are the same as in the static drift and rudder simulations, and rotary derivatives for the yaw rate can be estimated from the results when

*v*′ = 0 and

*δ*= 0. If there is no interaction between them, the shape of the force and moment graphs will coincide regardless of the yaw rate. However, the shape of the graphs changes according to the yaw rate.

*r*′) is positive and the drift angle (

*v*′) is negative (in other words, the bow of the vehicle is heading for the inside of the turning circle), the tangential velocity of the bow becomes slower, and that of the stern becomes faster because the partial turning radius changes. In addition, the flow incidence angle of the bow becomes smaller, and that of the stern becomes bigger, as shown in Fig. 23(b). These phenomena become stronger as the yaw rate increases.

*r*′) and drift angle (

*v*′) are both positive (in other words, the bow of the vehicle is heading for the outside of the turning circle), the tangential velocity of the bow becomes faster, that of the stern becomes slower, the flow incidence angle of the bow becomes bigger, and that of the stern becomes smaller, as shown in Fig. 23(c). However, in this direction, the interaction force and moments are weak compared to the opposite direction.

*r*′) is positive and the rudder angle (

*δ*) is negative, the tangential velocity of the rudder becomes slower, and flow’s incidence angle becomes smaller, as shown in Fig. 24(b). Therefore, reduction of the sway force and yaw moment increases as the yaw rate increases. In contrast, When the yaw rate (

*r*′) and rudder angle (

*δ*) are both positive, the tangential velocity of the rudder becomes faster, and flow’s incidence angle becomes bigger, as shown in Fig. 24(c). Therefore, the additional sway force and yaw moment increase as the yaw rate increases. Fig. 25 shows the velocity distribution of the whole domain in the case of the rotating arm simulations.

### 5.3 Estimation of Hydrodynamic Derivatives

### 6. Results of Turning Motion Simulations and Sea Trial of Underwater Vehicle

*v*) and yaw rate (

*r*) in the body-fixed frame were calculated by solving these equations. The surge velocity (

*u*) was considered as constant. The values in the body-fixed frame were then transformed to the earth-fixed frame using Eq. (3), and the turning radius and drift angle were obtained by deriving the trajectory and attitude of the vehicle. The simulations were carried out using MATLAB.

*Y*′(

_{coup}*r*′,

*v*′) and

*N*′(

_{coup}*r*′,

*v*′)) are applied (Case B). In the third one, all coupled terms (

*Y*′(

_{coup}*r*′,

*v*′)

*, N*′(

_{coup}*r*′,

*v*′)

*, Y*′(

_{coup}*r*′,

*δ*) and

*N*′(

_{coup}*r*′,

*δ*)) are applied (Case C). The initial velocity is 4.15 m/s because this it makes it easy to compare the results with sea trial data. The rudder angle was set to −20 degrees to turn with a positive yaw rate.

*R*=

*U*/

*r*.), and the drift angle can be calculated with the difference between the Euler angle and the flow’s incidence angle from the X velocity and Y velocity in the earth-fixed frame. The advance and tactical diameter are estimated with the Euler angle, X velocity, and Y velocity in the earth-fixed frame.

*r*′ =

*rL*/

*U*) are 0.289–0.377, which is in the range of the CFD analysis (0–0.4). When coupled terms are not included, the turning radius, advance, and tactical diameter are smaller, and the drift angle is bigger than the sea trial data.

*Y*′(

_{coup}*r*′,

*v*′) and

*N*′(

_{coup}*r*′,

*v*′)) are included, the turning radius, advance, and tactical diameter become bigger, and the drift angle becomes smaller because additional force and moment opposite to the rotational direction are produced by the interaction. When coupled terms between the yaw rate and rudder angle (

*Y*′(

_{coup}*r*′,

*δ*) and

*N*′(

_{coup}*r*′,

*δ*)) are included, the turning radius, advance, and tactical diameter become smaller, and the drift angle becomes bigger because additional force and moment in the rotational direction are produced by the interaction.

### 7. Conclusion

The feasibility of the hydrodynamic force and moment from the CFD was verified by comparing the results of the static drift/rudder simulations to those of the VPMM tests.

Rotating arm simulations with the drift and rudder angle were performed, and we confirmed the existence of interactions. Coupled terms from the interactions have strong non-linearities and different tendencies according to the relative direction of the yaw rate and drift/rudder angle.

Turning motion simulations were carried out by solving 2-DOF equations based on hydrodynamic data from CFD. The coupled interaction terms should be considered to estimate the turning performance of the underwater vehicle more accurately.