Access by Xinjiang University

Thin or bulky: Optimal aspect ratios for ship hulls

Jean-Philippe Boucher, Romain Labbé, Christophe Clanet, and Michael Benzaquen^*

LadHyX, UMR 7646 du CNRS, École polytechnique, 91128 Palaiseau Cedex, France

^*Corresponding author: michael.benzaquen@polytechnique.edu

Phys. Rev. Fluids 3, 074802 – Published 30 July, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.074802

Abstract

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.

Physics Subject Headings (PhySH)

Article Text

References (43)

K. J. Rawson, Basic Ship Theory (Butterworth-Heinemann, Oxford, 2001), Vol. 1.
F. Fossati, Aero-hydrodynamics and the Performance of Sailing Yachts: The Science behind Sailing Yachts and Their Design (A&C Black, London, 2009).
L. Larsson, H. C. Raven, and J. R. Paulling, Ship Resistance and Flow, Principles of Naval Architecture (Society of Naval Architects and Marine Engineers, Jersey City, NJ, 2010).
R. Eliasson, L. Larsson, and M. Orych, Principles of Yacht Design (A&C Black, London, 2014).
J. McArthur, High Performance Rowing (Crowood Press, Ramsbury, 1997).
V. Nolte, Rowing Faster (Human Kinetics 1, Stanningley, Leeds, 2005).
S. F. Hoerner, Fluid-Dynamic Drag: Practical Information on Aerodynamic Drag and Hydrodynamic Resistance (Hoerner Fluid Dynamics, New York, 1965).
J. H. Michell, XI. The wave-resistance of a ship, London, Edinburgh, Dublin Philos. Mag. J. Sci. 45, 106 (1898).
T. H. Havelock, Wave resistance: Some cases of three-dimensional fluid motion, Proc. Royal Soc. London, Ser. A 95, 354 (1919).
T. H. Havelock, The theory of wave resistance, Proc. Royal Soc. London, Ser. A 138, 339 (1932).
F. E. Fish, Influence of hydrodynamic-design and propulsive mode on mammalian swimming energetics, Aust. J. Zool. 42, 79 (1994).
W. T. Gough, S. C. Farina, and F. E. Fish, Aquatic burst locomotion by hydroplaning and paddling in common eiders (Somateria mollissima), J. Exp. Biol. 218, 1632 (2015).
S. W. McCue and L. K. Forbes, Bow and stern flows with constant vorticity, J. Fluid Mech. 399, 277 (1999).
S. Percival, D. Hendrix, and F. Noblesse, Hydrodynamic optimization of ship hull forms, Appl. Ocean Res. 23, 337 (2001).
C.-C. Hsiung, Optimal ship forms for minimum wave resistance, J. Ship Res. 25, 95 (1981).
B.-J. Zhang, M. Kun, and Z.-S. Ji, The optimization of the hull form with the minimum wave making resistance based on rankine source method, J. Hydrodynam., Ser. B 21, 277 (2009).
M. Benzaquen, A. Darmon, and E. Raphaël, Wake pattern and wave resistance for anisotropic moving disturbances, Phys. Fluids 26, 092106 (2014).
C.-C. Hsiung and D. Shenyan, Optimal ship forms for minimum total resistance, J. Ship Res. 28, 163 (1984).
J. Dambrine, M. Pierre, and G. Rousseaux, A theoretical and numerical determination of optimal ship forms based on Michell's wave resistance, ESAIM 22, 88 (2016).
E. F. Campana, D. Peri, Y. Tahara, and F. Stern, Shape optimization in ship hydrodynamics using computational fluid dynamics, Comput. Methods Appl. Mech. Eng. 196, 634 (2006).
E. O. Tuck, The wave resistance formula of JH Michell (1898) and its significance to recent research in ship hydrodynamics, ANZIAM J. 30, 365 (1989).
T. Inui, Wave-Making Resistance of Ships (Society of Naval Architects and Marine Engineers, 1962).
J. V. Wehausen, The wave resistance of ships, Adv. Appl. Mech. 13, 93 (1973).
In dimensionless coordinates, the hull boundary $y = f (x)$ with $x \in [- ℓ / 2, ℓ / 2]$ becomes $\tilde{y} = y / ℓ = (w / ℓ) \tilde{f} (\tilde{x})$ with $\tilde{x} \in [- 1 / 2, 1 / 2]$ .
E. Raphaël and P.-G. De Gennes, Capillary gravity waves caused by a moving disturbance: Wave resistance, Phys. Rev. E 53, 3448 (1996).
A. Darmon, M. Benzaquen, and E. Raphaël, Kelvin wake pattern at large Froude numbers, J. Fluid Mech. 738, R3 (2014).
A. S. Gotman, Study of Michell's integral and influence of viscosity and ship hull form on wave resistance, Ocean. Eng. Int. 6, 74 (2002).
The immersed volume $Ω_{i}$ is related to $Ω$ through $Ω_{i} = 2 a_{\tilde{f}} Ω$ [see Eq. (5)].
This empirical expansion is expected to hold for $α ≳ 2$ (see Ref. [7]).
F. C. Michelsen, Wave resistance solution of Michell's integral for polynomial ship forms, Ph.D. thesis, The University of Michigan, 1960.
With no constraint on the geometry of the hull, the shape minimizing the wetted surface is a spherical cap.
Note that in order to avoid oscillations due to the sharp edges of the hull, the integral over $x$ in Eq. (2) was actually computed over $R$ (see Appendix pp1). We checked that doing so had negligible effect on the results.
R. H. Byrd, J. C. Gilbert, and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Math. Program. 89, 149 (2000).
R. H. Byrd, M. E. Hribar, and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim. 9, 877 (1999).
International Canoe Federation, Canoe sprint competition rules, www.canoeicf.com/rules.
The maximal lengths for canoes and kayaks are the same for C1 and K1 (5.2 m), C2 and K2 (6.5 m), but not for C4 (9 m) and K4 (11 m) (see also Table 1).
Note that for real hull design one should also address dynamic stability [1], but the latter falls beyond the scope of our study.
P. Y. Belly, 250 Réponses aux Questions d'un Marin Curieux (Le gerfaut, 2004).
Vertical momentum balance writes $M g ≃ ρ Ω_{i} g + \frac{1}{2} ρ C_{L} ℓ w sin (2 θ) U^{2}$ , where $M$ is the mass of the boat, $Ω_{i}$ is the immersed volume, and $θ (Fr)$ is the Froude-dependent angle of the hull with respect to the horizontal direction of motion [4, 7]. This leads to an immersed volume which depends on the Froude number through $Ω ≃ (M / ρ) / [1 + 0.5 C_{L} β sin (2 θ) {Fr}^{2}]$ . For low Froude number, $θ (Fr) ≃ 0$ and the volume is that imposed by static equilibrium, while for larger $Fr$ number $θ > 0$ and the volume $Ω$ is decreased. Note that foil devices also contribute to decreasing the immersed volume, by increasing the lift.
E. O. Tuck, Wave resistance of thin ships and catamarans, Applied Mathematics Report T8701, 1987.
R. B. Chapman, Hydrodynamic drag of semisubmerged ships, J. Basic Eng. 94, 879 (1972).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Courier Corporation, 1964).
J. B. Hadler, Coefficients for international towing tank conference 1957 model-ship correlation line, Technical Report, David Taylor Model Basin, Washington, DC, 1958 (unpublished).

Physical Review Fluids

Export citation

Export citation

Thin or bulky: Optimal aspect ratios for ship hulls

Abstract

Physics Subject Headings (PhySH)

Article Text

References (43)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation