Thin or bulky: Optimal aspect ratios for ship hulls

Empirical data reveal a broad variety of hull shapes among the different ship categories. We present a minimal theoretical approach to address the problem of ship hull optimization. We show that optimal hull aspect ratios result—at given load and propulsive power—from a subtle balance among wave drag, pressure drag, and skin friction. Slender hulls are more favorable in terms of wave drag and pressure drag, while bulky hulls have a smaller wetted surface for a given immersed volume, thus reducing skin friction. We compare our theoretical results to real data and discuss discrepancies in the light of hull designer constraints, such as stability and maneuvrability.

Figure 1

Pictures of (a) a 44-ft sailing boat: length-to-width aspect ratio 3 and typical Froude number 0.6, (b) the Queen Mary 2 liner: length-to-width aspect ratio 8.4 and typical Froude number 0.26, and (c) a coxless quadruple scull rowing boat: length-to-width aspect ratio 31 and typical Froude number 0.54. See Table 1 for details and characteristics of other boats.

Figure 2

Length-to-width aspect ratio $\ell /w$ as a function of Froude number $U/\sqrt{g\ell }$ for different kinds of bodies moving at the water surface (see Table 1 for details). Solid symbols represent displacement hulls, whereas open symbols indicate planing hulls. The aspect ratio for multihulls is computed for each hull independently. The black line corresponds to the optimal aspect ratio; see Sec. 3. Solid lines indicate global optima, while dashed lines signify local optima.

Figure 3

Schematics of the simplified hull geometry considered in this study. The hull of length $\ell$, width $w$, and draft $d$ has a constant horizontal cross section, which is defined by $y=f\left(x\right){\mathbb{1}}_{z\in [-d,0]}$. Note that only the part of the hull immersed in the water is represented.

Figure 4

Contour plots of (a) the wave drag coefficient $C_{\mathrm{w}}$ and (b) the profile drag coefficient $C_{\mathrm{p}}$ as a function of the aspect ratios $\alpha$ and $\beta$. For the wave drag coefficient, we set $\mathrm{Fr}=0.5$. In both plots, black regions correspond to $C_{\mathrm{p}}/\mathrm{w}\ge {10}^{-2}$ and arrows indicate the direction of the gradient.

Figure 5

(a) Optimal aspect ratio ${\alpha }^{★}$, (b) optimal aspect ratio ${\beta }^{★}$, (c) optimal Froude number ${\mathrm{Fr}}^{★}$, and (d) corresponding value of the total drag coefficient $C^{★}=C({\alpha }^{★},{\beta }^{★},{\mathrm{Fr}}^{★})$, as a function of the dimensionless power $\mathrm{\Pi }$. The curves in orange and green represent the two optimal branches. Solid and dashed lines indicate global and local optima, respectively.

Figure 6

Wave-drag coefficient $C_{\mathrm{w}}$ as function of the Froude number $\mathrm{Fr}$ for a Gaussian hull and a parabolic hull for $\alpha =6.7$ and $\beta =2.3$. These results are compared to the theoretical curve from Tuck [40] and experimental data points from Chapman (black crosses) [41].

Figure 7

Cross section of the model hull (see Fig. 3) in (a) vertical position and (b) slightly inclined position.