Effect of bilge keels on maneuverability of a fine ship

Abstract

This paper discusses the effect of bilge keels on maneuverability of a fine ship experimentally and theoretically. To capture the effect, first, free-running model tests were conducted using a KCS container ship model with detachable bilge keels. We found that bilge keels improve the turning performance and enlarge overshoot angles in zig-zag maneuvers, and that this tendency becomes more pronounced with increasing ship speed. Next, captive model tests were conducted to capture the hydrodynamic forces’ effects on maneuvering to clarify the reason for the difference in maneuvering in the presence of bilge keels. By attaching bilge keels, the absolute values of $Y_v'$, $N_v'$, and $K_v'$ (the linear derivatives of lateral force, yaw moment, and roll moment acting on the ship with respect to lateral velocity v, respectively) are increased and $Y_{\phi }'$ (the derivative of the lateral force with respect to roll $\phi$) is also increased. It can be explained theoretically that the change of the hydrodynamic derivatives leads to course instability according to the course stability criterion presented by Yasukawa and Yoshimura [3]. The special feature of the criterion is to include the roll-coupling effect. Thus, the roll-coupling significantly affects ship maneuverability through hydrodynamic derivative changes by attaching bilge keels to fine ships with small metacentric height $\overline{GM}$.

Appendices

Appendix A: captive model test results in heeled conditions

Figures 18, 19, 20, and 21 show the test results for different roll angles $\phi =0$, $-5$, $-10$, and $-15^\circ$. The results exclude the inertia force components with respect to the ship model. In each figure, the longitudinal force coefficient ($X'$), lateral force coefficient ($Y'$), yaw moment coefficient ($N'$) and roll moment coefficient ($K'$) are shown versus hull drift angle $\beta$ for different non-dimensional yaw rates $r'$. The dotted line plotted in the figure is the fitting line obtained using Eq. 1.

Fig. 18

Comparison of hydrodynamic force coefficients for $\phi = 0^\circ$ (left w/o BK, right with BK)

Fig. 19

Comparison of hydrodynamic force coefficients for $\phi = -5^\circ$ (left w/o BK, right with BK)

Fig. 20

Comparison of hydrodynamic force coefficients for $\phi = -10^\circ$ (left w/o BK, right with BK)

Fig. 21

Comparison of hydrodynamic force coefficients for $\phi = -15^\circ$ (left w/o BK, right with BK)

Appendix B: correction of propeller and rudder effects on linear derivatives

The following formulas were used for the correction of propeller and rudder effects on linear derivatives in the framework of the MMG model [9]:

$$\begin{aligned} \left. \begin{array}{lll} Y_v^{*'} &{}=&{} Y_v' - k_Y \gamma _{R} u_{R}' \\ N_v^{*'} &{}=&{} N_v' + k_N \gamma _{R} u_{R}' \\ K_v^{*'} &{}=&{} K_v' + k_Y \gamma _{R} u_R'z_{R}' \\ Y_r^{*'} &{}=&{} Y_r' - k_Y \gamma _{R} u_R'\ell _{R}' \\ N_r^{*'} &{}=&{} N_r' + k_N \gamma _{R} u_R'\ell _{R}' \\ \end{array} \right\} , \end{aligned}$$

(6)

where

$$\begin{aligned} \left. \begin{array}{lll} k_Y &{}=&{} (1+a_H)f_{\alpha }A_R/(Ld) \\ k_N &{}=&{} -(x_R'+a_Hx_H')f_{\alpha }A_R/(Ld) \end{array} \right\} . \end{aligned}$$

(7)

Derivatives with the superscript $*$ include the effects of the propeller and rudder. Here $\gamma _R$ denotes the flow-straightening coefficient, $\ell _R'$ denotes the non-dimensional effective longitudinal coordinate of rudder position, $u_R'$ represents the non-dimensional longitudinal inflow velocity components to rudder, $a_H$ indicates the rudder force increase factor, $f_{\alpha }$ denotes the rudder lift gradient coefficient, $x_H'$ is the non-dimensional longitudinal coordinate of the point at which the additional lateral force acts, and $x_R'$ is the non-dimensional longitudinal coordinate of the rudder position ($=-0.5$). $z_R'$ is the non-dimensional vertical coordinate of the position at which the rudder force acts.

After correction of propeller and rudder effects by the method mentioned above, we converted the corrected linear derivatives into derivatives about the center of gravity. Concrete values of the parameters for the correction of linear derivatives are shown in Table 7. The values were determined by referring to Yasukawa and Sano [10]. The $\gamma _R$ listed in the table is an averaged value for port and starboard motions. The $f_{\alpha }$ was obtained by applying Fujii’s formula [11] for a whole rudder, including the horn.

Table 7 Parameters for correction of linear derivatives