The hull–propeller–rudder interaction of the Korea Research Institute of Ships & Ocean Engineering Container Ship is studied using a combined experimental fluid dynamic (EFD) and computational fluid dynamics (CFD) method with an innovative approach employed for the analysis of steady state circular motions. The force and moment balances are analyzed by decomposing into contributions from the bare hull, rudder, and propulsor. Detailed investigation of the computed local flow fields is performed including the hull vortices, surface pressure and streamlines, and propeller and rudder inflows. The force and moment balances mostly have a similar trend for both the EFD and CFD. The propeller inflow in port and starboard turning shows different trends, and the propeller is more heavily loaded with reduced efficiency as compared to the straight-ahead condition. The port side shows larger magnitudes of the hull vortices, more propeller load, lower propeller efficiency, larger drift angle, and smaller circle radius than the starboard side turning. These differences are explained via the hull–propeller–rudder force and moment balances with the aid of transformed circular motion equations of centrifugal force. The surge (X) force is hardly changed, but the lateral (Y) force is reduced (largely due to the rudder force) for the port side turning, which induces a larger drift angle and more speed loss. The overall conclusion is that the circular motion induces the centrifugal force and drift angle, which induce the hull vortices, off-design propeller inflow, reduced propeller efficiency, increased propeller thrust, and speed loss in addition to the propeller rudder interactions.

## I. INTRODUCTION

Maneuvering in both calm water and waves is required for many ship operations such as navigation in confined water and/or littoral zones, ship–ship/platform interactions, and replenishment at sea. Furthermore, ships should have enough maneuverability in adverse weather conditions to ensure the safety of the ship, its crew, and payload. Commercial and military ships must meet International Maritime Organization (IMO) Guidelines and North Atlantic Treaty Organization (NATO) Standardization Agreements. Commercial interest is also increasing due to recent IMO Guidelines for energy efficiency and minimum power requirements. The analysis requires free-running conditions, which challenges both experimental and prediction method capability in comparison to the status for captive condition resistance and propulsion, seakeeping, and maneuvering.

The physics of ship maneuvering are less understood, and prediction capability is lacking in comparison to resistance and propulsion and seakeeping, as evidenced by the International Towing Tank Conference (ITTC) Quality Manual Procedures (QM 75-02-02-014, QM 75-02-07-021, QM 75-02-06001) and, most recent, computational fluid dynamics (CFD)^{3} and SIMMAN (http://www.simman2019.kr/) workshops and Stern *et al.*'s^{18} CFD state-of-the-art assessment. Similarly, the measurement capability is reduced as evidenced by recent facility bias/scale effect studies for Korea Research Institute of Ships & Ocean Engineering Container Ship (KCS) added resistance/power^{11,12} vs maneuvering in calm water and waves.^{13} Resistance and propulsion scaling laws including viscous effects are well established as are those for seakeeping at least for head waves. Works on scaling laws to address similar scale effects are also in progress for maneuvering.^{8,19–21} Oblique seakeeping and especially maneuvering are subject to large viscous effects for horizontal motions, which lack scaling laws and challenge both system based and CFD prediction methods. Errors and scatter for CFD (and prediction capability in general) increase with the complexity of the simulations, increasing from resistance and propulsion to seakeeping to maneuvering, as is also the case with experiments especially in progressing from captive to free-running conditions.

In the present study, the hull–propeller–rudder interaction of the KCS for steady turning circles (TCs) in calm water is investigated using a combined experimental and CFD method. Several previous studies have noted the differences for port vs starboard maneuvering such as Schot and Eggars,^{14} Kuiper *et al.*,^{6} and Schulten *et al.*^{15} but have not provided a detailed explanation. Although system based maneuvering prediction methods can predict the correct trends for port vs starboard turning circles, e.g., Simonsen *et al.*^{16,17} using the whole ship model and Yasukawa and Yoshimura^{22} using the Maneuvering Modeling Group (MMG) model, the physical understanding of the flow field mechanisms of the actual hydrodynamic interactions is still not fully understood. CFD studies have shown the capability of predicting maneuvering for tanker,^{9} container,^{7} and surface combatant^{10} hull forms; however, the current research has not done both port and starboard turning circles and have primarily focused on verification and validation vs detailed study of the maneuvering physics. Note that IMO does not require the capability of the accurate prediction of the differences for port vs starboard turning circles or zig-zag (ZZ) maneuvers, but only that the average values meet the requirements. Nonetheless, the study of such differences as will be shown herein enables a better understanding of the overall physics of maneuvering.

The objective of the present study is to provide physical explanation for the hull–propeller–rudder interaction and differences for port vs starboard turning circles for the KCS. An innovative approach is used, which combines the results from free-running experiments and CFD for both global and local flow analysis. First (1), experimental results from two facilities [University of Iowa, Iowa Institute of Hydraulic Research (IIHR) and Hiroshima University (HU)] are supplemented with MMG system based maneuvering model results^{2} for evaluation of the three degrees of freedom (3DoF) equations of motion for steady state straight ahead course keeping self-propulsion (hereinafter, referred to as self-propulsion in short) and turning circle maneuvering Coriolis terms, hull, propulsor, and rudder force and moment balances. Second (2), CFDShip-Iowa results from free-running simulations^{5} [supplemented by additional Froude number Fr =$\u2009UgL$= 0.157 (where U is the approach speed and L is the ship length) results presented herein] are validated via comparisons with the experimental results and additionally provide the propeller side force and yaw moment information. Third (3), the CFD local flow results are analyzed to assess the hull–propeller–rudder interaction. Fourth (4), the circular motion equations are used to show the relationship between the centrifugal force and drift angle and the force and moment and local flow analysis to fully explain the physics of the differences between the port and starboard turning circles. Directly using the equations of motion Coriolis terms and force and moment balances via the analysis of the relative contributions of the various terms is considered a valuable new approach for physical understanding and CFD validation for maneuvering.

The present paper focuses on “Addressing Challenges in CFD Simulations” for applications in ocean engineering. Thus, the physics of fluids are at the industrial vs the fundamental level, and for the general reader some guidance may be helpful. The following sections of our paper describe the equations of motion inertia and force and moment balances for the experiments and CFD; local flow analysis explicating the physics of the hull vortices and surface pressure and streamlines, propeller inflow, propeller performance and propeller–hull interaction, and rudder inflow/outflow and propeller–hull–rudder interaction; evaluation of the physics using cylindrical coordinates and both the global and local flow analysis; and conclusions which fully explicate the physics of hull–propeller–rudder interactions for steady turning circles. Brief comments are provided at the beginning of most sections describing the content as a guide for the general reader.

## II. 3DoF EQUATIONS OF MOTION FOR STEADY-STATE SELF-PROPULSION AND TURNING CIRCLE MANEUVERING

In this section, the coordinate system and equations of motion used for both the experimental and CFD analysis of the physics of the turning circle maneuvering are presented along with the definitions of the terms in the equations.

Figure 1 shows the Cartesian coordinate systems used for the experiments and computations including ship-fixed (*x*, *y*, *z*) and the earth-fixed (*x*_{0}, *y*_{0}, *z*_{0}). Note that right-handed coordinate systems are used with downward directed vertical coordinates). The earth-fixed coordinate system is assumed to be an inertial reference frame. The ship-fixed coordinate system origin is at the center of gravity G. The experiments use measurement systems in both coordinate systems, which are combined to provide the overall results.

The CFD fluid flow Reynolds-averaged Navier–Stokes equations are solved, and the forces and moments are initially computed using the earth-fixed coordinates. The location and orientation of the ships with respect to the inertial earth-fixed system are described by linear translations and Euler angles. The rigid-body equations for ship motions are solved in the ship-fixed coordinate systems. The forces and moments ($X,Y,Z,K,M,N$) are projected into the ship coordinate system. In Eqs. (1), initial translation and angular velocity values of ($u,v,w,p,q,r$) are specified at the beginning the computations. The accelerations ($u\u0307,v\u0307,w\u0307,p\u0307,q\u0307,r\u0307$) are obtained from solving the motion equations (1) iteratively first, and then the velocities $u,v,w,p,q,r$ and the translation distances and rotation angles ($x1,x2,x3,\varphi ,\theta ,\psi $) are updated and converted back to the earth system accordingly.

Both the experimental measurements and CFD simulations were for six degrees of freedom (6DoF) motions and forces and moments; however, for the present purposes, a 3DoF analysis of the overall results is sufficient. The 3DoF equations of motion for self-propulsion and turning circle maneuvering in the ship-fixed system are

The equations are solved for the three unknowns, $u,v$, and $r$. The forces and moment are decomposed into contributions from the bare hull, rudder, and propulsor. The H, R, and P subscript terms in Eqs. (1) correspond to the bare hull, rudder, and propeller terms, respectively. For steady state conditions, $u\u0307\u2212wq\u2245v\u0307\u2212wp\u2245Izr\u0307+(Iy\u2212Ix)pq\u22450$. The KCS model is equipped with a semi-balanced horn rudder. The horn part is treated as part of the hull, in principle; therefore, the force induced by the horn is included in the hull force. The R subscript terms (*X _{R}*,

*Y*, and

_{R}*N*) only include the forces/moment induced by the rudder movable part.

_{R}## III. EVALUATION OF THE EQUATIONS OF MOTION USING EXPERIMENTAL DATA SUPPLEMENTED WITH THE MMG MODEL

In this section, the accuracy of the experiments is assessed based on results for multiple facilities, and the accuracy of the CFD is assessed based on comparisons with the experimental data. Then the balance of the terms in the equations of motion is evaluated using the experimental data supplemented with the MMG system based maneuvering model results.

The recent facility bias/scale effects study for KCS maneuvering in calm water and waves^{13} included port and starboard 35° turning circles from six facilities for Fr = 0.26 and from three facilities for Fr = 0.157, as shown in Fig. 2 along with results from CFDShip-Iowa. Sanada *et al.*^{13} and its references provide more details about the experimental methods, including feature engineering, dimensional analysis, and machine learning for the identification of the primary assessment variables; however, herein the focus is specifically on the calm water port and starboard turning circle data. Note that identification of the primary assessment variables (Table I and Fig. 3) is required for maneuvering as the current lists for both turning circles and zig-zag maneuvers cover about twenty variables which are both dimensional and nondimensional and often correlated. The MARIN and HU experimental data were also used as test cases for the SIMMAN 2021 workshop.

No. . | Variables . |
---|---|

1 | Tactical diameter ($TD/L$) |

2 | Advance ($AD/L$) |

3 | Turning radius ($R/L$) |

4 | Roll angle in the constant part of the turn ($\varphi S$) |

5 | Maximum outward roll angle ($\varphi mo$) |

6 | Speed in the steady part of the turn ($US$) |

7 | Drift angle in the constant part of the turn ($\beta S$) |

8 | Average yaw rate in the constant part of the turn ($rS$) (deg/s) |

No. . | Variables . |
---|---|

1 | Tactical diameter ($TD/L$) |

2 | Advance ($AD/L$) |

3 | Turning radius ($R/L$) |

4 | Roll angle in the constant part of the turn ($\varphi S$) |

5 | Maximum outward roll angle ($\varphi mo$) |

6 | Speed in the steady part of the turn ($US$) |

7 | Drift angle in the constant part of the turn ($\beta S$) |

8 | Average yaw rate in the constant part of the turn ($rS$) (deg/s) |

The trends for Fr = 0.26 were as follows. The mean facility (M) starboard tactical diameter (TD) and advance (AD) = 2.9L and the turning radius (R) = 1.24L with standard deviation (SD) = 5.9, 3.2 and 4.1%M, respectively. The M port TD = 2.6L, AD =2.9L, and R = 1.17L with SD = 5.9, 3.2, and 2%M, respectively. The port R is 6%M smaller than starboard, and the port speed loss u = 0.447U is larger than starboard u = 0.465U. The mid-ship port drift angle is −20.1°, whereas the starboard drift angle is 15.6°. The SD for all variables was 10.2 and 15.4%M for port and starboard turning.

The results for Fr = 0.157 are similar. The M starboard TD = 3.2L, AD = 2.9L, and R = 1.3L with SD = 5.9, 3.2 and 4.1%M, respectively. The M port TD = AD = 2.9L and R = 1.2L with SD = 1.8, 3.3, and 1.1%M, respectively. The port R is 8%M smaller than starboard, and the port speed loss u = 0.455U is larger than starboard u = 0.488U. The mid-ship port drift angle is −18.7°, whereas the starboard drift angle is 17.7°. The SD for all variables was 7.4 and 8.2%M for port and starboard turning.

### A. Experimental/MMG model evaluation terms in equations of motion

The IIHR wave basin (40 × 20 × 3 m^{3}) experiments used a 2.7 m free-running model with measurements of trajectories and linear/angular velocities ($x,y,\psi ,u,v,r$) and propeller revolutions, thrust, and torque (n, T, and Q). The HU experiments were performed at the National Research Institute of Fisheries Engineering (NRIFE) wave basin (60 × 25 × 3.2 m^{3}) and used a 3.2 m free-running model with measurements of ($x,y,\psi ,u,v,r$), $n,T,Q$, and rudder normal force F_{N}. Table II lists the principal particulars of the HU and IIHR models.

. | Full scale . | HU . | IIHR . | |
---|---|---|---|---|

Facility dimension | ⋯ | 60 × 25 × 3.2 m^{3} (NRIFE) | 40 × 20 × 3 m^{3} (IIHR wave basin) | |

Main particular | ||||

Scale | 1 | 75.2 | 85.2 | |

L (m) | 230 | 3.057 | 2.700 | |

B (m) | 32.2 | 0.428 | 0.378 | |

T (m) | 10.8 | 0.1435 | 0.1268 | |

∇ (m^{3}) | 52030 | 0.1222 | 0.0842 | |

S (m^{3}) | 9539 | 1.6851 | 1.3145 | |

GM (m) | 0.6 | 0.008 | 0.007 | |

GM/L | 0.002 61 | 0.002 62 | 0.002 59 | |

KG/L | 0.0623 | 0.0623 | 0.0623 | |

k_{xx}/B | 0.4 | ⋯ | 0.39 | |

k_{yy}/L | 0.25 | 0.25 | 0.25 | |

U_{0} (m/s) | Fr: 0.26 | 12.35 [24(kt)] | ⋯ | 1.34 |

Fr: 0.157 | 7.46 [14.5(kt)] | 0.86 | 0.81 | |

Propeller | ||||

Geometry type | KP505 | KP505 | ||

Diameter (D_{p})(m) | 7.9 | 0.105 | 0.093 | |

(P/D_{p}) mean | 0.95 | 0.95 | ||

A_{e}/A_{o} | 0.8 | 0.8 | ||

Hub ratio | 0.180 | 0.180 | ||

Rudder | ||||

Geometry type | Original | Simplified geometry | Original | |

H (m) _{R} | 9.90 | 0.132 | 0.116 | |

A (m_{R}^{2}) | 54.45 | 0.0096 | 0.0075 |

. | Full scale . | HU . | IIHR . | |
---|---|---|---|---|

Facility dimension | ⋯ | 60 × 25 × 3.2 m^{3} (NRIFE) | 40 × 20 × 3 m^{3} (IIHR wave basin) | |

Main particular | ||||

Scale | 1 | 75.2 | 85.2 | |

L (m) | 230 | 3.057 | 2.700 | |

B (m) | 32.2 | 0.428 | 0.378 | |

T (m) | 10.8 | 0.1435 | 0.1268 | |

∇ (m^{3}) | 52030 | 0.1222 | 0.0842 | |

S (m^{3}) | 9539 | 1.6851 | 1.3145 | |

GM (m) | 0.6 | 0.008 | 0.007 | |

GM/L | 0.002 61 | 0.002 62 | 0.002 59 | |

KG/L | 0.0623 | 0.0623 | 0.0623 | |

k_{xx}/B | 0.4 | ⋯ | 0.39 | |

k_{yy}/L | 0.25 | 0.25 | 0.25 | |

U_{0} (m/s) | Fr: 0.26 | 12.35 [24(kt)] | ⋯ | 1.34 |

Fr: 0.157 | 7.46 [14.5(kt)] | 0.86 | 0.81 | |

Propeller | ||||

Geometry type | KP505 | KP505 | ||

Diameter (D_{p})(m) | 7.9 | 0.105 | 0.093 | |

(P/D_{p}) mean | 0.95 | 0.95 | ||

A_{e}/A_{o} | 0.8 | 0.8 | ||

Hub ratio | 0.180 | 0.180 | ||

Rudder | ||||

Geometry type | Original | Simplified geometry | Original | |

H (m) _{R} | 9.90 | 0.132 | 0.116 | |

A (m_{R}^{2}) | 54.45 | 0.0096 | 0.0075 |

Using these measurement variables obtained from the turning tests supplemented with the hydrodynamic derivatives obtained by captive tests in calm water and the MMG model for forces/moment by the hull/rudder, the force balances in steady-state turnings are estimated by the following procedure.

The terms in Eq. (2) are nondimensional using $\rho U2Ld/2$ for the forces (X and Y) and $\rho U2L2d/2$ for the moment (N) where U is the ship approach speed, L is the length between perpendiculars, and d is the draft. Note that for steady state, the left-hand sides are zero as are the second equality right hand sides such that the second equality right hand sides ($\Delta X,\u2009\Delta Y,$ and $\Delta N$) represent the error in the sum of the Coriolis terms, forces, and moments. The H subscript and $R0$ terms correspond to the bare hull terms in Eq. (1). The R and P subscript terms correspond to the rudder and propeller terms in Eqs. (1), and the remaining terms in the middle equality correspond to the left-hand side inertia terms of Eqs. (1) for steady-state conditions. For comparison purposes with CFD results, the hull forces are reformulated as $XH=XH+myvr\u2212R0$ and $YH=YH\u2212mxur$.

The rudder terms (X_{R}, Y_{R}, and N_{R}) are estimated by the following equations:

The steering resistance deduction factor ($tR$), rudder force increase factor ($aH$), and the position of rudder force increase ($xH$) are estimated by the Kijima^{4} model.

For HU, the measured F_{N} is used in Eq. (3) and the IIHR rudder normal force is estimated by the MMG rudder model as per the following equation:

*H*_{R} and *A*_{R} are defined in Table II, and L is the rudder aspect ratio (=*H*_{R}/*B*_{R} = 1.80, where mean chord length *H*_{R} = 5.50, 0.073 and 0.065 m for full scale, HU and IIHR, respectively). *I*_{R}′ and *Y*_{R} are the nondimensional effective longitudinal coordinates of the rudder position and flow straightening coefficient, respectively, and they were tuned for the best fit using the HU experimental data. All other terms are either self-explanatory or Yasukawa and Yoshimura^{22} can be consulted.

For both facilities, the same resistance coefficient, hydrodynamic derivatives, and added mass coefficients are used to evaluate R_{0}, X_{H}, Y_{H}, and N_{H}.^{2} X_{P} is estimated by the measured thrust. Added mass/inertia coefficients (m_{x}, m_{y}, J_{z}) are estimated by Motora's empirical charts. The Eq. (2) force and moment balances were done for both Fr = 0.26 and 0.157 with similar results; however, since experimental data are also available from IIHR and HU for port and starboard 35° turning circles in waves and the CFDShip-Iowa studies include additional simulations without the propeller and without both the propeller and rudder for Fr = 0.157, those results are presented herein.

### B. Self-propulsion

The IIHR self-propulsion results are shown in Table III. The X equation shows that X_{P} is balanced by X_{H} with the other terms being small including the error. The Y and N equations are an order of magnitude smaller than the X equation and show that the hull force/moment is balanced by the rudder force/moment due to the neutral rudder angle of 0.864° required for straight ahead course keeping self-propulsion (and Coriolis term for Y). The large errors for Y and N are due to the small values of the measured variables.

X force . | ||||||
---|---|---|---|---|---|---|

. | X_{P}
. | X_{H}
. | X_{R}
. | mvr . | ΔX . | |

IIHR | SP | 1.610 × 10^{–2} | −1.360 × 10^{–2} | −8.730 × 10^{–6} | 1.984 × 10^{–6} | 2.550 × 10^{–3} |

35PSTC | 2.725 × 10^{–2} | −1.280 × 10^{–2} | −6.105 × 10^{–3} | −1.002 × 10^{–2} | −1.667 × 10^{–3} | |

35SBTC | 2.707 × 10^{–2} | −1.238 × 10^{–2} | −5.630 × 10^{–3} | −9.090 × 10^{–3} | −2.689 × 10^{–5} | |

35SBTC–35PSTC | −1.793 × 10^{–4} | 4.161 × 10^{–4} | 4.751 × 10^{–4} | 9.285 × 10^{–4} | 1.640 × 10^{–3} | |

HU | 35PSTC | 3.103 × 10^{–2} | −1.218 × 10^{–2} | −6.766 × 10^{–3} | −9.472 × 10^{–3} | 2.617 × 10^{–3} |

35SBTC | 3.188 × 10^{–2} | −1.233 × 10^{–2} | −7.626 × 10^{–3} | −9.307 × 10^{–3} | 2.606 × 10^{–3} | |

35SBTC–35PSTC | 8.522 × 10^{–4} | −1.551 × 10^{–4} | −8.602 × 10^{–4} | 1.644 × 10^{–4} | −1.112 × 10^{–5} |

X force . | ||||||
---|---|---|---|---|---|---|

. | X_{P}
. | X_{H}
. | X_{R}
. | mvr . | ΔX . | |

IIHR | SP | 1.610 × 10^{–2} | −1.360 × 10^{–2} | −8.730 × 10^{–6} | 1.984 × 10^{–6} | 2.550 × 10^{–3} |

35PSTC | 2.725 × 10^{–2} | −1.280 × 10^{–2} | −6.105 × 10^{–3} | −1.002 × 10^{–2} | −1.667 × 10^{–3} | |

35SBTC | 2.707 × 10^{–2} | −1.238 × 10^{–2} | −5.630 × 10^{–3} | −9.090 × 10^{–3} | −2.689 × 10^{–5} | |

35SBTC–35PSTC | −1.793 × 10^{–4} | 4.161 × 10^{–4} | 4.751 × 10^{–4} | 9.285 × 10^{–4} | 1.640 × 10^{–3} | |

HU | 35PSTC | 3.103 × 10^{–2} | −1.218 × 10^{–2} | −6.766 × 10^{–3} | −9.472 × 10^{–3} | 2.617 × 10^{–3} |

35SBTC | 3.188 × 10^{–2} | −1.233 × 10^{–2} | −7.626 × 10^{–3} | −9.307 × 10^{–3} | 2.606 × 10^{–3} | |

35SBTC–35PSTC | 8.522 × 10^{–4} | −1.551 × 10^{–4} | −8.602 × 10^{–4} | 1.644 × 10^{–4} | −1.112 × 10^{–5} |

Y force . | |||||
---|---|---|---|---|---|

. | Y_{H}
. | Y_{R}
. | −mur . | ΔY . | |

IIHR | SP | 7.120 × 10^{–5} | −7.960 × 10^{–4} | −1.219 × 10^{–3} | −1.940 × 10^{–3} |

35PSTC | −6.453 × 10^{–2} | 1.694 × 10^{–2} | 2.975 × 10^{–2} | −1.784 × 10^{–2} | |

35SBTC | 5.651 × 10^{–2} | −1.560 × 10^{–2} | −3.114 × 10^{–2} | 9.771 × 10^{–3} | |

35SBTC–35PSTC | 1.210 × 10^{–1} | −3.254 × 10^{–2} | −6.089 × 10^{–2} | 2.761 × 10^{–2} | |

HU | 35PSTC | −6.081 × 10^{–2} | 1.876 × 10^{–2} | 2.893 × 10^{–2} | −1.312 × 10^{–2} |

35SBTC | 5.857 × 10^{–2} | −2.115 × 10^{–2} | −3.004 × 10^{–2} | 7.391 × 10^{–3} | |

35SBTC–35PSTC | 1.194 × 10^{–1} | −3.990 × 10^{–2} | −5.897 × 10^{–2} | 2.052 × 10^{–2} |

Y force . | |||||
---|---|---|---|---|---|

. | Y_{H}
. | Y_{R}
. | −mur . | ΔY . | |

IIHR | SP | 7.120 × 10^{–5} | −7.960 × 10^{–4} | −1.219 × 10^{–3} | −1.940 × 10^{–3} |

35PSTC | −6.453 × 10^{–2} | 1.694 × 10^{–2} | 2.975 × 10^{–2} | −1.784 × 10^{–2} | |

35SBTC | 5.651 × 10^{–2} | −1.560 × 10^{–2} | −3.114 × 10^{–2} | 9.771 × 10^{–3} | |

35SBTC–35PSTC | 1.210 × 10^{–1} | −3.254 × 10^{–2} | −6.089 × 10^{–2} | 2.761 × 10^{–2} | |

HU | 35PSTC | −6.081 × 10^{–2} | 1.876 × 10^{–2} | 2.893 × 10^{–2} | −1.312 × 10^{–2} |

35SBTC | 5.857 × 10^{–2} | −2.115 × 10^{–2} | −3.004 × 10^{–2} | 7.391 × 10^{–3} | |

35SBTC–35PSTC | 1.194 × 10^{–1} | −3.990 × 10^{–2} | −5.897 × 10^{–2} | 2.052 × 10^{–2} |

Yaw moment (N) . | ||||
---|---|---|---|---|

. | N_{H}
. | N_{R}
. | ΔN . | |

IIHR | SP | −5.740 × 10^{–4} | 3.420 × 10^{–4} | −2.320 × 10^{–4} |

35PSTC | 1.212 × 10^{–2} | −1.228 × 10^{–2} | −1.660 × 10^{–4} | |

35SBTC | −9.695 × 10^{–3} | 1.132 × 10^{–2} | 1.627 × 10^{–3} | |

35SBTC–35PSTC | −2.181 × 10^{–2} | 2.361 × 10^{–2} | 1.793 × 10^{–3} | |

HU | 35PSTC | 1.167 × 10^{–2} | −1.361 × 10^{–2} | −1.934 × 10^{–3} |

35SBTC | −1.047 × 10^{–2} | 1.534 × 10^{–2} | 4.870 × 10^{–3} | |

35SBTC–35PSTC | −2.214 × 10^{–2} | 2.894 × 10^{–2} | 6.804 × 10^{–3} |

Yaw moment (N) . | ||||
---|---|---|---|---|

. | N_{H}
. | N_{R}
. | ΔN . | |

IIHR | SP | −5.740 × 10^{–4} | 3.420 × 10^{–4} | −2.320 × 10^{–4} |

35PSTC | 1.212 × 10^{–2} | −1.228 × 10^{–2} | −1.660 × 10^{–4} | |

35SBTC | −9.695 × 10^{–3} | 1.132 × 10^{–2} | 1.627 × 10^{–3} | |

35SBTC–35PSTC | −2.181 × 10^{–2} | 2.361 × 10^{–2} | 1.793 × 10^{–3} | |

HU | 35PSTC | 1.167 × 10^{–2} | −1.361 × 10^{–2} | −1.934 × 10^{–3} |

35SBTC | −1.047 × 10^{–2} | 1.534 × 10^{–2} | 4.870 × 10^{–3} | |

35SBTC–35PSTC | −2.214 × 10^{–2} | 2.894 × 10^{–2} | 6.804 × 10^{–3} |

### C. Port and starboard turning circles

Table III shows that the force and moment balances for HU have similar trends for both port and starboard turning. The X_{P} is positive and the largest term and is balanced by the other terms which are all negative. The Y_{H} is the largest term and is balanced by the other terms with opposite signs. The N_{H} is balanced by N_{R}. Thus, the dominant forces are Y_{H} and X_{P} with Y_{H} about one and half times larger than X_{P}.

Table III also shows that the force and moment balances for the IIHR trends are both qualitatively and quantitatively similar for both port and starboard turning as HU. For both HU and IIHR, the turning circle errors in the balances are reasonably small. Note that the same scales are used for both HU and IIHR and that the magnitudes for the turning circles are orders of magnitude larger than those for the self-propulsion.

Table III also shows a comparison of the equations of motion Coriolis term, force, and moment differences for starboard vs port turning circles ΔD = (starboard − port) for both HU and IIHR. The trends are the same for both HU and IIHR for Y and N equations but are most opposite for X forces.

## IV. EVALUATION OF THE EQUATIONS OF MOTION USING CFD

In this section, the CFD setup is described, and the balance of the terms in the equations of motion is evaluated and compared with the experimental results.

Figure 4 shows the grid layout and distribution for the CFD simulations. The details of the CFD setup for Fr = 0.26 are provided by Kim *et al.*,^{5} and the same approach and grids were used for the Fr = 0.157 simulations. The CFD is for Re = 3.61 × 10^{6} (same as the IIHR experiments), actual propeller, and coarse grid G3 = 12M. The simulations are for 6DoF and exactly mimic the experimental setup and conditions using rudder and propeller rotation speed controllers. The results for G2 = 36M were also obtained; however, the duration of the simulations is less than for G2, and the differences are not that large such that G3 is used for the validation and analysis. The output from the simulations was used to evaluate the equations (1) Coriolis term, force, and moment balances similarly as was done for the experiments.

For Fr = 0.157, two additional simulations were performed to aid in the hull–propeller–rudder interaction analysis after the 6DoF free running conditions. (1) propeller and (2) the propeller and rudder were removed, and captive simulations were performed using the identical trajectories/motions as predicted by the 6DoF free-running simulations.

Sanada *et al.*^{13} provided verification and validation assessment of CFDShip-Iowa results for both Fr = 0.26 and Fr = 0.157 calm water and wave maneuvering. The average errors %M for the CFD were 12.1 and 10.7 for Fr = 0.26 and 12.5 and 10.4 for Fr = 0.157 for port and starboard, respectively, which are comparable to the SD of the experiments.

### A. Self-propulsion

Table IV shows the self-propulsion results using the same format and scales as for the experiments, including propeller side force *Y _{P}* and yaw moment

*N*. The CFD has the same trends as the experiments for

_{P}*X*but shows some differences for

*Y*and

*N*. For the CFD, the errors are small for not only

*X*but also

*Y*and

*N*compared to the experiments. The neutral rudder angle for the CFD is 1.1°, which is comparable to the experimental value.

X force . | ||||
---|---|---|---|---|

X_{P}
. | X_{H}
. | X_{R}
. | mvr . | ΔX . |

2.064 × 10^{–2} | −1.994 × 10^{–2} | −7.051 × 10^{–4} | −9.690 × 10^{–11} | −6.702 × 10^{–6} |

X force . | ||||
---|---|---|---|---|

X_{P}
. | X_{H}
. | X_{R}
. | mvr . | ΔX . |

2.064 × 10^{–2} | −1.994 × 10^{–2} | −7.051 × 10^{–4} | −9.690 × 10^{–11} | −6.702 × 10^{–6} |

Y force . | ||||
---|---|---|---|---|

Y_{P}
. | Y_{H}
. | Y_{R}
. | −mur . | ΔY . |

−5.839 × 10^{–4} | 1.349 × 10^{–3} | −9.402 × 10^{–4} | 1.349 × 10^{–6} | −1.735 × 10^{–4} |

Y force . | ||||
---|---|---|---|---|

Y_{P}
. | Y_{H}
. | Y_{R}
. | −mur . | ΔY . |

−5.839 × 10^{–4} | 1.349 × 10^{–3} | −9.402 × 10^{–4} | 1.349 × 10^{–6} | −1.735 × 10^{–4} |

Yaw moment . | |||
---|---|---|---|

N_{P}
. | N_{H}
. | N_{R}
. | ΔN . |

2.558 × 10^{–4} | −7.062 × 10^{–4} | 4.527 × 10^{–4} | 2.338 × 10^{–6} |

Yaw moment . | |||
---|---|---|---|

N_{P}
. | N_{H}
. | N_{R}
. | ΔN . |

2.558 × 10^{–4} | −7.062 × 10^{–4} | 4.527 × 10^{–4} | 2.338 × 10^{–6} |

Table V shows the CFD self-propulsion force and moment components. In this table, the forces and moments induced by the horn and movable parts of the rudder are shown separately. Figure 5 shows a close-up view of the KCS rudder showing the fixed horn (red) and movable control surface (yellow) parts. Table V shows the nondimensional force and moment balance and each component as a percentage share of the total positive/negative components, respectively. In the X direction, the propeller thrust balances the bare hull resistance. In the lateral force and yaw moment, the force induced by the horn part has the largest magnitude of all the components. In the Y direction, all the force components are making the ship go to the starboard side except the horn force. The yaw moment shows similar trends as the Y force. The moment acting on the rudder horn is making the ship go to the port side, whereas the other components are making the ship go to the starboard side. In the lateral force and yaw moment, the movable part of the rudder cancels out 50% of the magnitude of the force and moment induced by the horn part. Araki *et al.*^{1} have also shown the importance of the effects of the rudder horn during maneuvering.

X force . | |||||
---|---|---|---|---|---|

. | X_{P}
. | X_{H}–X_{R_Horn}
. | X_{R_Horn}
. | X_{R}
. | ΔX . |

Nondim balance | 2.064 × 10^{–2} | −1.949 × 10^{–2} | −4.530 × 10^{–4} | −7.051 × 10^{–4} | −6.702 × 10^{–6} |

Balance in % total | 100.00 | −94.41 | −2.17 | −3.42 | 0.00 |

X force . | |||||
---|---|---|---|---|---|

. | X_{P}
. | X_{H}–X_{R_Horn}
. | X_{R_Horn}
. | X_{R}
. | ΔX . |

Nondim balance | 2.064 × 10^{–2} | −1.949 × 10^{–2} | −4.530 × 10^{–4} | −7.051 × 10^{–4} | −6.702 × 10^{–6} |

Balance in % total | 100.00 | −94.41 | −2.17 | −3.42 | 0.00 |

Y force . | |||||
---|---|---|---|---|---|

. | Y_{P}
. | Y_{H}–Y_{R_Horn}
. | Y_{R_Horn}
. | Y_{R}
. | ΔY . |

Nondim balance | −5.839 × 10^{–4} | −3.667 × 10^{–4} | 1.716 × 10^{–3} | −9.402 × 10^{–4} | −1.735 × 10^{–4} |

Balance in % total | −30.81 | −19.43 | 100.00 | −49.76 | −9.90 |

Y force . | |||||
---|---|---|---|---|---|

. | Y_{P}
. | Y_{H}–Y_{R_Horn}
. | Y_{R_Horn}
. | Y_{R}
. | ΔY . |

Nondim balance | −5.839 × 10^{–4} | −3.667 × 10^{–4} | 1.716 × 10^{–3} | −9.402 × 10^{–4} | −1.735 × 10^{–4} |

Balance in % total | −30.81 | −19.43 | 100.00 | −49.76 | −9.90 |

Yaw moment . | |||||
---|---|---|---|---|---|

. | N_{P}
. | N_{H}–N_{R_Horn}
. | N_{R_Horn}
. | N_{R}
. | ΔN . |

Nondim balance | 2.558 × 10^{–4} | 1.093 × 10^{–4} | −8.155 × 10^{–4} | 4.527 × 10^{–4} | 2.338 × 10^{–6} |

Balance in % total | 31.17 | 13.36 | −100.00 | 55.47 | 0.00 |

Yaw moment . | |||||
---|---|---|---|---|---|

. | N_{P}
. | N_{H}–N_{R_Horn}
. | N_{R_Horn}
. | N_{R}
. | ΔN . |

Nondim balance | 2.558 × 10^{–4} | 1.093 × 10^{–4} | −8.155 × 10^{–4} | 4.527 × 10^{–4} | 2.338 × 10^{–6} |

Balance in % total | 31.17 | 13.36 | −100.00 | 55.47 | 0.00 |

### B. Port and starboard turning circles

Table VI shows that the Coriolis term, force, and moment balance for CFD has similar trends qualitatively and most quantitatively for both port and starboard turning as the experiments. The tables use a similar format and scales as were used for the experimental results. Note that for X_{H} the pressure component is somewhat larger than the friction component, but they are both comparable, whereas for Y_{H} and N_{H} their pressure components are dominant in comparison to their friction component (not shown here). Unlike the results for self-propulsion the rudder horn contribution to the forces and moment is small (not shown here). Interestingly and contrary to many persons' expectations, the role of the propeller side force and yaw moment is relatively small. The turning circle errors in the balances are reasonably small.

X force . | |||||
---|---|---|---|---|---|

. | X_{P}
. | X_{H}
. | X_{R}
. | mvr . | ΔX . |

35PSTC | 3.171 × 10^{–2} | −1.408 × 10^{–2} | −8.448 × 10^{–3} | −1.042 × 10^{–2} | −1.239 × 10^{–3} |

35SBTC | 2.963 × 10^{–2} | −1.346 × 10^{–2} | −7.240 × 10^{–3} | 9.759 × 10^{–3} | −8.281 × 10^{–4} |

35SBTC–35PSTC | −2.082 × 10^{–3} | 6.225 × 10^{–4} | 1.208 × 10^{–3} | 6.618 × 10^{–4} | 4.110 × 10^{–4} |

X force . | |||||
---|---|---|---|---|---|

. | X_{P}
. | X_{H}
. | X_{R}
. | mvr . | ΔX . |

35PSTC | 3.171 × 10^{–2} | −1.408 × 10^{–2} | −8.448 × 10^{–3} | −1.042 × 10^{–2} | −1.239 × 10^{–3} |

35SBTC | 2.963 × 10^{–2} | −1.346 × 10^{–2} | −7.240 × 10^{–3} | 9.759 × 10^{–3} | −8.281 × 10^{–4} |

35SBTC–35PSTC | −2.082 × 10^{–3} | 6.225 × 10^{–4} | 1.208 × 10^{–3} | 6.618 × 10^{–4} | 4.110 × 10^{–4} |

Y force . | |||||
---|---|---|---|---|---|

. | Y_{P}
. | Y_{H}
. | Y_{R}
. | −mur . | ΔY . |

35PSTC | −1.553 × 10^{–3} | −3.908 × 10^{–2} | 1.054 × 10^{–2} | 3.309 × 10^{–2} | 3.006 × 10^{–3} |

35SBTC | 6.153 × 10^{–4} | 4.125 × 10^{–2} | −9.434 × 10^{–3} | −3.574 × 10^{–2} | −3.315 × 10^{–3} |

35SBTC–35PSTC | 2.168 × 10^{–3} | 8.033 × 10^{–2} | −1.998 × 10^{–2} | −6.884 × 10^{–2} | −6.320 × 10^{–3} |

Y force . | |||||
---|---|---|---|---|---|

. | Y_{P}
. | Y_{H}
. | Y_{R}
. | −mur . | ΔY . |

35PSTC | −1.553 × 10^{–3} | −3.908 × 10^{–2} | 1.054 × 10^{–2} | 3.309 × 10^{–2} | 3.006 × 10^{–3} |

35SBTC | 6.153 × 10^{–4} | 4.125 × 10^{–2} | −9.434 × 10^{–3} | −3.574 × 10^{–2} | −3.315 × 10^{–3} |

35SBTC–35PSTC | 2.168 × 10^{–3} | 8.033 × 10^{–2} | −1.998 × 10^{–2} | −6.884 × 10^{–2} | −6.320 × 10^{–3} |

Yaw moment . | ||||
---|---|---|---|---|

. | N_{P}
. | N_{H}
. | N_{R}
. | ΔN . |

35PSTC | 7.461 × 10^{–4} | 4.382 × 10^{–3} | −5.124 × 10^{–3} | 3.547 × 10^{–6} |

35SBTC | −2.632 × 10^{–4} | −4.300 × 10^{–3} | 4.560 × 10^{–3} | −3.497 × 10^{–6} |

35SBTC–35PSTC | −1.009 × 10^{–3} | −8.682 × 10^{–3} | 9.684 × 10^{–3} | −7.044 × 10^{–6} |

Yaw moment . | ||||
---|---|---|---|---|

. | N_{P}
. | N_{H}
. | N_{R}
. | ΔN . |

35PSTC | 7.461 × 10^{–4} | 4.382 × 10^{–3} | −5.124 × 10^{–3} | 3.547 × 10^{–6} |

35SBTC | −2.632 × 10^{–4} | −4.300 × 10^{–3} | 4.560 × 10^{–3} | −3.497 × 10^{–6} |

35SBTC–35PSTC | −1.009 × 10^{–3} | −8.682 × 10^{–3} | 9.684 × 10^{–3} | −7.044 × 10^{–6} |

Table VI also shows a comparison of the equations of motion Coriolis term, force, and moment differences for starboard vs port turning circles ΔD = (starboard – port) for the CFD. The trends for the Y and N are the same as both the experiments, whereas the trend for the X only matches IIHR results.

In addition to condition (1) 6DoF free running with the propeller, results are also obtained for imposed 6DoF motions (referred to as towed) conditions (2) without the propeller and (3) without the propeller and rudder as shown in Tables VII and VIII. Table VII shows that the propeller weakly affects the hull forces but has substantial effect on the rudder forces and moment such that X_{R}, Y_{R}, and N_{R} are negligible. Results for bare hull shown in Table VIII confirm that the rudder has less effect on the hull forces without the propeller.

X force . | |||||
---|---|---|---|---|---|

. | X_{T}
. | X_{H}
. | X_{R}
. | mvr . | ΔX . |

35PSTC | 2.388 × 10^{–2} | −1.225 × 10^{–2} | −1.216 × 10^{–3} | −1.042 × 10^{–2} | 0.000 |

35SBTC | 2.444 × 10^{–2} | −1.309 × 10^{–2} | −1.596 × 10^{–3} | −9.754 × 10^{–3} | 0.000 |

35SBTC–35PSTC | 5.579 × 10^{–4} | −8.397 × 10^{–4} | −3.796 × 10^{–4} | 6.614 × 10^{–4} | 0.000 |

X force . | |||||
---|---|---|---|---|---|

. | X_{T}
. | X_{H}
. | X_{R}
. | mvr . | ΔX . |

35PSTC | 2.388 × 10^{–2} | −1.225 × 10^{–2} | −1.216 × 10^{–3} | −1.042 × 10^{–2} | 0.000 |

35SBTC | 2.444 × 10^{–2} | −1.309 × 10^{–2} | −1.596 × 10^{–3} | −9.754 × 10^{–3} | 0.000 |

35SBTC–35PSTC | 5.579 × 10^{–4} | −8.397 × 10^{–4} | −3.796 × 10^{–4} | 6.614 × 10^{–4} | 0.000 |

Y force . | |||||
---|---|---|---|---|---|

. | Y_{T}
. | Y_{H}
. | Y_{R}
. | −mur . | ΔY . |

35PSTC | 8.837 × 10^{–3} | −4.314 × 10^{–2} | 1.225 × 10^{–3} | 3.308 × 10^{–2} | 0.000 |

35SBTC | −4.677 × 10^{–3} | 4.209 × 10^{–2} | −1.693 × 10^{–3} | −3.572 × 10^{–2} | 0.000 |

35SBTC–35PSTC | −1.351 × 10^{–2} | 8.523 × 10^{–2} | −2.918 × 10^{–3} | −6.880 × 10^{–2} | 0.000 |

Y force . | |||||
---|---|---|---|---|---|

. | Y_{T}
. | Y_{H}
. | Y_{R}
. | −mur . | ΔY . |

35PSTC | 8.837 × 10^{–3} | −4.314 × 10^{–2} | 1.225 × 10^{–3} | 3.308 × 10^{–2} | 0.000 |

35SBTC | −4.677 × 10^{–3} | 4.209 × 10^{–2} | −1.693 × 10^{–3} | −3.572 × 10^{–2} | 0.000 |

35SBTC–35PSTC | −1.351 × 10^{–2} | 8.523 × 10^{–2} | −2.918 × 10^{–3} | −6.880 × 10^{–2} | 0.000 |

Yaw moment . | ||||
---|---|---|---|---|

. | N_{T}
. | N_{H}
. | N_{R}
. | ΔN . |

35PSTC | −5.100 × 10^{–3} | 5.696 × 10^{–3} | −5.960 × 10^{–4} | 0.000 |

35SBTC | 3.616 × 10^{–3} | −4.439 × 10^{–3} | 8.225 × 10^{–4} | 0.000 |

35SBTC–35PSTC | 8.716 × 10^{–3} | −1.013 × 10^{–2} | 1.419 × 10^{–3} | 0.000 |

Yaw moment . | ||||
---|---|---|---|---|

. | N_{T}
. | N_{H}
. | N_{R}
. | ΔN . |

35PSTC | −5.100 × 10^{–3} | 5.696 × 10^{–3} | −5.960 × 10^{–4} | 0.000 |

35SBTC | 3.616 × 10^{–3} | −4.439 × 10^{–3} | 8.225 × 10^{–4} | 0.000 |

35SBTC–35PSTC | 8.716 × 10^{–3} | −1.013 × 10^{–2} | 1.419 × 10^{–3} | 0.000 |

X force . | ||||
---|---|---|---|---|

. | X_{T}
. | X_{H}
. | mvr . | ΔX . |

35PSTC | 2.267 × 10^{–2} | −1.226 × 10^{–2} | −1.041 × 10^{–2} | 0.000 |

35SBTC | 2.296 × 10^{–2} | −1.321 × 10^{–2} | −9.751 × 10^{–3} | 0.000 |

35SBTC–35PSTC | 2.868 × 10^{–4} | −9.481 × 10^{–4} | 6.613 × 10^{–4} | 0.000 |

X force . | ||||
---|---|---|---|---|

. | X_{T}
. | X_{H}
. | mvr . | ΔX . |

35PSTC | 2.267 × 10^{–2} | −1.226 × 10^{–2} | −1.041 × 10^{–2} | 0.000 |

35SBTC | 2.296 × 10^{–2} | −1.321 × 10^{–2} | −9.751 × 10^{–3} | 0.000 |

35SBTC–35PSTC | 2.868 × 10^{–4} | −9.481 × 10^{–4} | 6.613 × 10^{–4} | 0.000 |

Y force . | ||||
---|---|---|---|---|

. | Y_{T}
. | Y_{H}
. | −mur . | ΔY . |

35PSTC | 1.076 × 10^{–2} | −4.383 × 10^{–2} | 3.307 × 10^{–2} | 0.000 |

35SBTC | −7.587 × 10^{–3} | 4.330 × 10^{–2} | −3.572 × 10^{–2} | 0.000 |

35SBTC–35PSTC | −1.835 × 10^{–2} | 8.713 × 10^{–2} | −6.878 × 10^{–2} | 0.000 |

Y force . | ||||
---|---|---|---|---|

. | Y_{T}
. | Y_{H}
. | −mur . | ΔY . |

35PSTC | 1.076 × 10^{–2} | −4.383 × 10^{–2} | 3.307 × 10^{–2} | 0.000 |

35SBTC | −7.587 × 10^{–3} | 4.330 × 10^{–2} | −3.572 × 10^{–2} | 0.000 |

35SBTC–35PSTC | −1.835 × 10^{–2} | 8.713 × 10^{–2} | −6.878 × 10^{–2} | 0.000 |

Yaw moment . | |||
---|---|---|---|

. | N_{T}
. | N_{H}
. | ΔN . |

35PSTC | −5.929 × 10^{–3} | 5.929 × 10^{–3} | 0.000 |

35SBTC | 4.854 × 10^{–3} | −4.854 × 10^{–3} | 0.000 |

35SBTC–35PSTC | 1.078 × 10^{–2} | −1.078 × 10^{–2} | 0.000 |

Yaw moment . | |||
---|---|---|---|

. | N_{T}
. | N_{H}
. | ΔN . |

35PSTC | −5.929 × 10^{–3} | 5.929 × 10^{–3} | 0.000 |

35SBTC | 4.854 × 10^{–3} | −4.854 × 10^{–3} | 0.000 |

35SBTC–35PSTC | 1.078 × 10^{–2} | −1.078 × 10^{–2} | 0.000 |

These results along with the previous verification and validation studies provide sufficient validation of the CFD for the evaluation of the hull–propeller–rudder interaction for the port vs starboard turning using the local flow CFD analysis as follows.

## V. EVALUATION OF HULL–PROPELLER–RUDDER INTERACTION FOR STEADY STATE SELF-PROPULSION AND TURNING CIRCLE MANEUVERING USING CFD LOCAL FLOW ANALYSIS

In this section, the CFD local flow is analyzed. The drift angle induced hull vortices are shown to alter the propeller inflow with differences for port vs starboard turning since the drift angles of opposite sign induce opposite directions for the vortical motions. The vortices degrade the propeller performance since they induce off-design blade section angles of attack. The off-design propeller slipstream degrades the rudder performance. The overall hull–propeller–interaction explains the previously discussed balances of the equations of motion.

The CFD local flow is analyzed using nondimensional variables for which the characteristic length $L$, approach velocity $U=FrgL$, and time $L/U$ scales are used.

Figure 6 shows the wave pattern for the self-propulsion and port and starboard turning circles for Fr = 0.157. The amplitudes are about three to four times smaller compared to the Fr = 0.26 results.

### A. Hull vortices and surface pressure and streamlines

Figure 7 shows the overall vortex structures based on the Q criterion for the steady-state self-propulsion and port and starboard turning circles. The drift angle is positive on the starboard and negative on the port sides; therefore, when observing from the stern, the hull vortices into the propeller are counterclockwise for the starboard turning circle and clockwise for the port turning circles vs weak anti-symmetric for self-propulsion, whereas the propeller rotation is clockwise in all cases.

Figure 8 shows the surface pressure and streamlines for the self-propulsion and starboard and port turning circles, respectively. The results shown with the propeller are averaged over 14 propeller rotations. For the self-propulsion condition, the port vs starboard surface pressure and streamlines are nearly symmetric with limited stern region cross flow separation; thus, the side forces and yaw moments are small as per Tables IV and V, whereas for the turning circle conditions, the port vs starboard surface pressure and streamlines are not symmetric with extensive cross flow separations; thus, the side forces and yaw moments are large as per Table V for condition (1). The crossflow separation flow patterns correlate with the Q criterion shown in Fig. 9. Comparisons of condition (1) and (2) show that the hull–propeller interaction affects the surface pressure and streamlines locally in the stern region, whereas comparisons of condition (2) and (3) show that the rudder has limited effect on the surface pressure and streamlines.

### B. Propeller inflow

Figures 9–11 show the propeller inflow at x/L = 0.975 (see Fig. 4) using wide angle views from the stern for self-propulsion and starboard and port turning circles, respectively, including the condition (1)–(3) results.

For self-propulsion, the hull anti-symmetric “bilge” vortices are weak, and the hull–propeller–rudder interactions are minimal, although relatively small hull–propeller interaction is evident. The starboard and port turning circles show strong drift angle hull-induced vortices with counterclockwise and clockwise rotation, respectively, and strong hull–propeller interaction and weak hull–propeller–rudder interaction. Note the strong hull-induced vortex shaft wake flow. The hull-induced vorticity magnitudes are larger for the port than for the starboard turning circle. The maximum/minimum vorticity values for the starboard/port turning circles for the conditions (1), (2), and (3) are 77/−84, 82/−83, and 83/−88.

Figures 12–14 show the propeller inflow at x/L = 0.975 close-up propeller disk views for self-propulsion and starboard and port turning circles, respectively, including the results for conditions (1)–(3).

The propeller inflow highlights the wide view trends. For self-propulsion, the maximum/minimum vorticity values for the conditions (1), (2), and (3) are 354/–375, 128/−128, and 154/−157. For the turning circles, the without propeller hull-induced vortex shaft wake flow becomes jet flow due to propeller suction. The port turning circle without propeller hull-induced vortex shaft wake flow has lower velocity due to its stronger vortex; therefore, higher velocity jet flow occurs due to the stronger propeller suction. For the starboard turning circle, the maximum/minimum vorticity values for the conditions (1), (2), and (3) are 427/−650, 294/−545, and 344/−564. For the port turning circle, the maximum/minimum vorticity values for the conditions (1), (2), and (3) are 648/−462, 559/−313, and 610/−366.

Table IX provides the nondimensional average velocity (1-total wake fraction) for the propeller inflow. Note that the average velocity is based on the total axial velocity U integrated over the propeller disk, which includes the propeller induced velocities for the condition with propeller. The average velocity is reduced substantially without the propeller and slightly reduced by the rudder. For the without propeller turning circles, the outboard average velocity is greater than the inboard for both side turning circles, and the starboard average velocity is greater than the port due to the hull-vortex induced shaft wake, which is larger on the port side due to the larger axial vorticity. For the with propeller condition, the outboard starboard average velocity is larger than the port as before, but the inboard port average velocity is larger than the starboard due to the propeller suction, having a greater increase in the lower momentum shaft wake on the port side, which induces a higher velocity jet flow. The effects of the rudder are relatively small, i.e., about 5%. The upper vs lower average velocity is smaller for self-propulsion and larger for turning.

SP . | |||
---|---|---|---|

With propeller and rudder | 0.758 | ||

SB | PS | SB/PS | |

0.762 | 0.753 | 1.012 | |

Top | Bottom | Top/bottom | |

0.689 | 0.827 | 0.833 | |

With rudder no propeller | 0.586 | ||

SB | PS | SB/PS | |

0.585 | 0.587 | 0.997 | |

Top | Bottom | Top/bottom | |

0.52 | 0.652 | 0.798 | |

Bare hull | 0.606 | ||

SB | PS | SB/PS | |

0.608 | 0.603 | 1.008 | |

Top | Bottom | Top/bottom | |

0.537 | 0.674 | 0.797 | |

35PSTC | |||

With propeller and rudder | 0.563 | ||

Inboard | Outboard | In/out | |

0.549 | 0.577 | 0.951 | |

Top | Bottom | Top/bottom | |

0.554 | 0.572 | 0.969 | |

With rudder no propeller | 0.396 | ||

Inboard | Outboard | In/out | |

0.352 | 0.441 | 0.798 | |

Top | Bottom | Top/bottom | |

0.424 | 0.369 | 1.149 | |

Bare hull | 0.418 | ||

Inboard | Outboard | In/out | |

0.383 | 0.453 | 0.845 | |

Top | Bottom | Top/bottom | |

0.442 | 0.395 | 1.119 | |

35SBTC | |||

With propeller and rudder | 0.583 | ||

Inboard | Outboard | In/out | |

0.529 | 0.638 | 0.829 | |

Top | Bottom | Top/bottom | |

0.563 | 0.603 | 0.934 | |

With rudder no propeller | 0.444 | ||

Inboard | Outboard | In/out | |

0.397 | 0.491 | 0.809 | |

Top | Bottom | Top/bottom | |

0.465 | 0.422 | 1.102 | |

Bare hull | 0.468 | ||

Inboard | Outboard | In/out | |

0.429 | 0.506 | 0.848 | |

Top | Bottom | Top/bottom | |

0.489 | 0.447 | 1.094 |

SP . | |||
---|---|---|---|

With propeller and rudder | 0.758 | ||

SB | PS | SB/PS | |

0.762 | 0.753 | 1.012 | |

Top | Bottom | Top/bottom | |

0.689 | 0.827 | 0.833 | |

With rudder no propeller | 0.586 | ||

SB | PS | SB/PS | |

0.585 | 0.587 | 0.997 | |

Top | Bottom | Top/bottom | |

0.52 | 0.652 | 0.798 | |

Bare hull | 0.606 | ||

SB | PS | SB/PS | |

0.608 | 0.603 | 1.008 | |

Top | Bottom | Top/bottom | |

0.537 | 0.674 | 0.797 | |

35PSTC | |||

With propeller and rudder | 0.563 | ||

Inboard | Outboard | In/out | |

0.549 | 0.577 | 0.951 | |

Top | Bottom | Top/bottom | |

0.554 | 0.572 | 0.969 | |

With rudder no propeller | 0.396 | ||

Inboard | Outboard | In/out | |

0.352 | 0.441 | 0.798 | |

Top | Bottom | Top/bottom | |

0.424 | 0.369 | 1.149 | |

Bare hull | 0.418 | ||

Inboard | Outboard | In/out | |

0.383 | 0.453 | 0.845 | |

Top | Bottom | Top/bottom | |

0.442 | 0.395 | 1.119 | |

35SBTC | |||

With propeller and rudder | 0.583 | ||

Inboard | Outboard | In/out | |

0.529 | 0.638 | 0.829 | |

Top | Bottom | Top/bottom | |

0.563 | 0.603 | 0.934 | |

With rudder no propeller | 0.444 | ||

Inboard | Outboard | In/out | |

0.397 | 0.491 | 0.809 | |

Top | Bottom | Top/bottom | |

0.465 | 0.422 | 1.102 | |

Bare hull | 0.468 | ||

Inboard | Outboard | In/out | |

0.429 | 0.506 | 0.848 | |

Top | Bottom | Top/bottom | |

0.489 | 0.447 | 1.094 |

### C. Propeller performance and propeller–hull interaction

Figure 15 shows the KCS KP505 propeller geometry and open water curves for IIHR, HU, and the CFD. The current self-propulsion J ≈0.8 is less than optimum since at the model self-propulsion point, and the propeller was designed for the full-scale self-propulsion point which is J = 0.93. The agreement between the different facilities and CFD is satisfactory.

Table X provides the propeller performance variables for the experiments and CFD, which shows that *J = U*/*nD* is reduced for turning circles, especially for port turning; thus, *K _{T}* =

*T*/

*ρn*

^{2}

*D*

^{4}and

*K*=

_{Q}*Q*/

*ρn*

^{2}

*D*

^{5}are increased and the efficiency

*η*=

*JK*/2

_{T}*πK*is reduced (where

_{Q}*ρ*is water density,

*n*is the propeller revolutions,

*D*is the propeller diameter,

*T*is the thrust, and

*Q*is the torque). The agreement between the different facilities and CFD is satisfactory.

. | . | HU (EFD) . | IIHR (EFD) . | CFDShip-Iowa V4.5 . | E%D_{IIHR}
. |
---|---|---|---|---|---|

Re | 2.6 × 10^{6} | 2.2 × 10^{6} | ⋯ | ||

Fr | 0.157 | ⋯ | |||

K_{T} | SP | 0.249 | 0.249 | 0.243 | −2.410 |

SB | 0.390 | 0.378 | 0.347 | −8.170 | |

PS | 0.379 | 0.380 | 0.371 | −2.498 | |

10 K_{Q} | SP | 0.432 | 0.412 | 0.411 | −0.243 |

SB | 0.639 | 0.584 | 0.545 | −6.610 | |

PS | 0.618 | 0.585 | 0.571 | −2.325 | |

η | SP | 0.722 | 0.744 | 0.723 | −2.913 |

SB | 0.362 | 0.381 | 0.422 | 10.799 | |

PS | 0.343 | 0.348 | 0.386 | 10.955 | |

K_{T}/J^{2} | SP | 0.402 | 0.416 | 0.412 | −0.879 |

SB | 2.815 | 2.773 | 2.006 | −27.666 | |

PS | 3.071 | 3.365 | 2.657 | −21.043 | |

J | SP | 0.787 | 0.774 | 0.768 | −0.775 |

SB | 0.372 | 0.369 | 0.416 | 12.673 | |

PS | 0.351 | 0.336 | 0.374 | 11.124 | |

Mean =(|J_{SB}| + |J_{PS}|)/2 | 0.362 | 0.353 | 0.395 | 11.933 | |

ΔJ = |J_{SB}| − |J_{PS}| | 0.021 | 0.033 | 0.043 | 28.399 | |

ΔJ% mean | 5.81 | 9.38 | 10.76 | 14.712 |

. | . | HU (EFD) . | IIHR (EFD) . | CFDShip-Iowa V4.5 . | E%D_{IIHR}
. |
---|---|---|---|---|---|

Re | 2.6 × 10^{6} | 2.2 × 10^{6} | ⋯ | ||

Fr | 0.157 | ⋯ | |||

K_{T} | SP | 0.249 | 0.249 | 0.243 | −2.410 |

SB | 0.390 | 0.378 | 0.347 | −8.170 | |

PS | 0.379 | 0.380 | 0.371 | −2.498 | |

10 K_{Q} | SP | 0.432 | 0.412 | 0.411 | −0.243 |

SB | 0.639 | 0.584 | 0.545 | −6.610 | |

PS | 0.618 | 0.585 | 0.571 | −2.325 | |

η | SP | 0.722 | 0.744 | 0.723 | −2.913 |

SB | 0.362 | 0.381 | 0.422 | 10.799 | |

PS | 0.343 | 0.348 | 0.386 | 10.955 | |

K_{T}/J^{2} | SP | 0.402 | 0.416 | 0.412 | −0.879 |

SB | 2.815 | 2.773 | 2.006 | −27.666 | |

PS | 3.071 | 3.365 | 2.657 | −21.043 | |

J | SP | 0.787 | 0.774 | 0.768 | −0.775 |

SB | 0.372 | 0.369 | 0.416 | 12.673 | |

PS | 0.351 | 0.336 | 0.374 | 11.124 | |

Mean =(|J_{SB}| + |J_{PS}|)/2 | 0.362 | 0.353 | 0.395 | 11.933 | |

ΔJ = |J_{SB}| − |J_{PS}| | 0.021 | 0.033 | 0.043 | 28.399 | |

ΔJ% mean | 5.81 | 9.38 | 10.76 | 14.712 |

Figure 16 shows the propeller thrust coefficient *K _{T}*, which highlights the results in Table X. Sanada

*et al.*

^{12}showed that the Adachi and Sugai (1978) propeller

*t*′ =

*K*/

_{T}*J*

^{2}scaling (also shown in Table IV) provides a good correlation for model size for both self-propulsion and added power in head and oblique waves. Appendix A shows that this scaling also collapses the data for maneuvering and adds additional explanation of the loss in efficiency for turning circles vs self-propulsion and the larger loss for the port vs the starboard turning circle. Note the phase differences for maximum thrust for self-propulsion vs starboard and port turning circles, i.e., about 0°, 150°, and 250°, as shown in Fig. 16.

The KP505 propeller uses NACA66 blade sections, which for Re ≅10^{6} have the maximum C_{L}/C_{D} when the angle of attack is 3.25°, which is assumed to be the design value for full-scale operation. Figure 17 shows the blade section angle of attack at the 70% propeller radius vs propeller blade angle for the self-propulsion and starboard and port turning. The average angle of attack is 4.65° for self-propulsion, 8.60° for port turning, and 7.84° for starboard turning. The larger value for self-propulsion than the design value is due to the model vs ship self-propulsion point, whereas the larger values for turning are due to the hull induced vortices and hull–propeller interaction, especially for the port turning.

Figure 18 shows the propeller blade section angle of attack at the 40% and 70% propeller radius vs propeller blade angle for the self-propulsion and starboard and port turning, including the K_{T} values for comparison. For the self-propulsion condition, the angle of attack at the r/R = 0.7 shows good correlation with K_{T}, whereas for the turning circles even the average of the r/R = 0.7 and 0.4 angles of attack shows insufficient correlation such that the entire blade flow is required to correlate the angle of attack with K_{T}.

### D. Rudder inflow/outflow and propeller–hull–rudder interaction

Figure 19 shows a close-up side view of the propeller and rudder showing the locations of the vertical inflow x/L = 0.988, outflow x/L = 1.025, and horizontal planes for evaluating the local rudder flow.

Figures 20–22 show close-up views of the rudder vertical inflow at x/L = 0.988 and outflow at x/L = 1.025 axial velocity contours with cross plane vectors for the self-propulsion and starboard and port turning circles, respectively, including the results for conditions (1)–(3). The results shown are averaged over 14 propeller rotations. The results in comparison to Figs. 12–14 show the development of the flow as it progresses from V1 to V2 to V4 for all three conditions (1), (2), and (3). Figures 23 and 24 show the close-up view of the rudder horizontal plane axial velocity contours and horizontal plane vectors for self-propulsion and starboard and port turning circles for the with and without propeller conditions, respectively. Figure 25 shows the rudder angle of attack at the rudder leading-edge location. Clearly, the rudder flow is complex and needs more study for system based maneuvering prediction methods modeling, which is a topic for future study.

## VI. EVALUATION OF THE PHYSICS OF HULL–PROPELLER–RUDDER INTERACTION FOR TURNING CIRCLES BASED ON CYLINDRICAL COORDINATES AND LOCAL FLOW ANALYSIS

The experimental and CFD results analyzed thus far can be better interpreted from a different perspective using a cylindrical (*ρ*, *φ*, *z*_{0}) coordinate system in a noninertial frame, as shown in Fig. 26.

For the steady-state circular motion of a ship with a constant speed *U* and radius *R*, Eqs. (1) can be transformed into the cylindrical system,

where $U=u2+v2$ and $R=U/r$. The single force equation expresses that the centrifugal force is balanced by the hydrodynamic forces exerted on the whole ship system (hull, propeller, and rudder). The drift angle is due to the ratio of the surge and sway forces. The moment equation remains the same and indicates the total moment for the ship system is zero. The speed loss is a direct result of the centrifugal force balance, i.e.,

Comparison of the radius, speed, drift angle, and forces for port and starboard side turning is given in Table XI. As compared to the starboard side turning, the radius *R* and speed *U* decrease and the drift angle β increases in magnitude for both experimentally measured values and CFD simulation results. The calculated speed *U _{x_y}* and drift angle

*β*using the

_{x_y}*X*and

*Y*forces via the above equations are also shown in Table XI. For both experiments, the calculated results show different trends from the measured values. This is probably due to the inaccurate estimation of the forces using the models, especially for rudder, and Y

_{P}is neglected. As for the CFD, the calculated speed and drift angle are in close agreement with the simulation results, which indicates the consistency of the forces, speeds, and drift angles in the simulations. As a result, the CFD results are used for the analysis in the following.

. | HU . | IIHR . | CFD . | |||
---|---|---|---|---|---|---|

35SBTC . | 35PSTC . | 35SBTC . | 35PSTC . | 35SBTC . | 35PSTC . | |

R (m) | 4.0291 | 3.7479 | 3.7665 | 3.3183 | 4.3038 | 3.7557 |

U (m/s) | 0.4093 | 0.3860 | 0.3959 | 0.3628 | 0.4379 | 0.3935 |

β (°) | 16.676 | −17.427 | 15.751 | −18.194 | 14.707 | −16.676 |

X (N) _{H} | −2.000 | −1.975 | −1.381 | −1.427 | −1.498 | −1.499 |

X (N) _{R} | −1.237 | −1.097 | −0.628 | −0.681 | −0.815 | −1.019 |

X (N) _{P} | 5.171 | 5.032 | 3.020 | 3.040 | 3.311 | 3.544 |

X (N) | 1.934 | 1.960 | 1.011 | 0.932 | 0.998 | 1.026 |

Y (N) _{H} | 9.499 | −9.862 | 6.304 | −7.199 | 4.488 | −4.683 |

Y (N) _{R} | −3.430 | 3.042 | −1.740 | 1.890 | −0.933 | 1.494 |

Y (N) _{P} | ⋯ | ⋯ | ⋯ | ⋯ | 0.069 | −0.174 |

Y (N) | 6.6090 | −6.820 | 4.564 | −5.309 | 3.624 | −3.363 |

U (m/s) _{x_y} | 0.4581 | 0.4664 | 0.4572 | 0.4608 | 0.4382 | 0.3959 |

β_{x_y} (°) | 17.675 | −16.034 | 12.491 | −9.952 | 15.397 | −16.970 |

. | HU . | IIHR . | CFD . | |||
---|---|---|---|---|---|---|

35SBTC . | 35PSTC . | 35SBTC . | 35PSTC . | 35SBTC . | 35PSTC . | |

R (m) | 4.0291 | 3.7479 | 3.7665 | 3.3183 | 4.3038 | 3.7557 |

U (m/s) | 0.4093 | 0.3860 | 0.3959 | 0.3628 | 0.4379 | 0.3935 |

β (°) | 16.676 | −17.427 | 15.751 | −18.194 | 14.707 | −16.676 |

X (N) _{H} | −2.000 | −1.975 | −1.381 | −1.427 | −1.498 | −1.499 |

X (N) _{R} | −1.237 | −1.097 | −0.628 | −0.681 | −0.815 | −1.019 |

X (N) _{P} | 5.171 | 5.032 | 3.020 | 3.040 | 3.311 | 3.544 |

X (N) | 1.934 | 1.960 | 1.011 | 0.932 | 0.998 | 1.026 |

Y (N) _{H} | 9.499 | −9.862 | 6.304 | −7.199 | 4.488 | −4.683 |

Y (N) _{R} | −3.430 | 3.042 | −1.740 | 1.890 | −0.933 | 1.494 |

Y (N) _{P} | ⋯ | ⋯ | ⋯ | ⋯ | 0.069 | −0.174 |

Y (N) | 6.6090 | −6.820 | 4.564 | −5.309 | 3.624 | −3.363 |

U (m/s) _{x_y} | 0.4581 | 0.4664 | 0.4572 | 0.4608 | 0.4382 | 0.3959 |

β_{x_y} (°) | 17.675 | −16.034 | 12.491 | −9.952 | 15.397 | −16.970 |

As shown in the table, X slightly increases and Y decreases for the port side turning with decreased circle radius. As shown in in the table, X_{H} is almost the same for port vs starboard side turning, and both X_{R} and X_{P} increase in magnitude but with opposite signs such that X slightly increases. Both Y_{H} and Y_{P} increase in magnitude with the same sign but Y_{R} has opposite sign and increases more such that Y is reduced.

The ratio explains the drift angle trends, i.e., increased drift angle for port turning is due to slightly increased *X* and decreased *Y* which is largely due to *Y _{R}*. The centrifugal force equation explains that the speed loss is due to reduced

*Y*along with the reduced turning radius

*R*, which is largely due to

*Y*. Thus, the difference between the port and starboard turning is due to the interactions between the hull, propeller, and rudder forces:

_{R}*X*force is hardly changed but the

*Y*force is reduced since the hull and propeller forces combine with the same sign but are reduced by the rudder force.

## VII. CONCLUSIONS AND FUTURE RESEARCH

Experiments and CFD are combined to study the hull–propeller–rudder interactions of the KCS with the aim at providing a physical explanation for the differences between the port and starboard turning circles. An innovative approach is used, which combines the results from free-running experiments and CFD for both global and local flow analysis. The experimental results were supplemented with the MMG system based maneuvering model results for the evaluation of the hull, propulsor and rudder force, and moment balances. The CFD results are validated via comparisons with the experimental results and additionally provide the propeller side force and yaw moment, and local flow information. The circular motion equations are used to show the relationship between the centrifugal force and drift angle and the force and moment and local flow analysis to fully explain the physics of the differences between the port and starboard turning circles.

The CFD shows the mostly similar force and moment balance as the experiments. As compared to the steady state straight ahead condition, the propeller inflow is different, and the propeller is more heavily loaded for turning with a loss in efficiency, especially for port turning. The circular motion equation results explain the difference between the port and starboard turning circles: the increased drift angle for port turning is due to slightly increased *X* and decreased *Y* which is largely due to *Y _{R}*, and speed loss is due to reduced

*Y*along with the reduced turning radius

*R*. The overall conclusion is that the circular motion (solution to the 6DoF equations) induces the centrifugal force (balanced by longitudinal and lateral forces) and drift angle (ratio of longitudinal and lateral force), which induce the hull vortices, off-design propeller inflow, reduced propeller efficiency, increased propeller thrust, and speed loss in addition to the propeller rudder interactions. The findings in the present study are based on the steady state circular turning conditions and may differ from other general circle turning cases with various rudder angles; however, the overall approach should be applicable and useful for other turning circles and other maneuvers including transient types such as zigzag, which will be explored in the future work. It should be mentioned that current ship design in some cases includes energy saving devices such as stern bulbs, propeller ducts, guide fins, and advanced propeller and rudder designs alone or in combination, which can also be of benefit for maneuvering.

## ACKNOWLEDGMENTS

The research at IIHR was supported by the Office of Naval Research Grant Nos. N00014-17-1-2083 and N00014-17-1-2084 under the administration of Dr. Thomas Fu, Dr. Woei-Min Lin, and Dr. Ki-Han Kim. The simulations were performed using DoD, Navy DSRC HPCMP resources including 18 Pathfinder program allocations. The SIMMAN 2020 Organizing Committee and Dr. Thad Michael, Dr. Serge Toxopeous, and Dr. Matteo Diez provided helpful discussions.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: PROPELLER LOAD SCALING FOR MANEUVERING

The usual propeller thrust K_{T} and torque *K*_{Q} scaling used for open water performance shows large scatter, whereas the Adachi and Sugai (1978) q′ (t′) = K_{Q}/J^{3} and n′ (t′) = t/2πq′ scaling, where t′ = K_{T}/J^{2} shows good correlation for added power studies of facility bias/scale effects.^{11} The reason is that the t′ scaling depends on the hull size with D^{2} replace by ∇^{2/3} where D is the propeller diameter and ∇ is the hull displacement volume. The Adachi and Sugai (1978) scaling is also used by ITTC 1978 ITTC Performance Prediction Method to predict speed–power–rpm relationship in waves, where the resistance and thrust identity method or direct powering method is applied.^{23}

Figure 27 extends the q′, n′, and t′ correlation for KCS maneuvering in calm water (also for waves, but results not shown), including comparisons with the previous added power studies. The results show that t′ has a similar correlation as added power/course keeping (head and oblique waves), but much larger range and scatter. Different size models move up/down the t′ correlation curve depending on propeller loading during the turning circle (TC) and zig-zag (ZZ) and both calm water and waves maneuvers (ZZ and waves results not shown). Model size scale effects need more study, as more complex than added power/course keeping. System based propeller models should use t′ correlation vs current propeller open water curve and J = U(1 − w)/nD similar to CFD noninteractive axisymmetric body force model.

## References

*, September 8–11*(