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Wave drag on asymmetric bodies

  • G. P. Benham (a1), J. P. Boucher (a1), R. Labbé (a1), M. Benzaquen (a1) and C. Clanet (a1)...


An asymmetric body with a sharp leading edge and a rounded trailing edge produces a smaller wave disturbance moving forwards than backwards, and this is reflected in the wave drag coefficient. This experimental fact is not captured by Michell’s theory for wave drag (Michell Lond. Edinb. Dubl. Phil. Mag. J. Sci., vol. 45 (272), 1898, pp. 106–123). In this study, we use a tow-tank experiment to investigate the effects of asymmetry on wave drag, and show that these effects can be replicated by modifying Michell’s theory to include the growth of a symmetry-breaking boundary layer. We show that asymmetry can have either a positive or a negative effect on drag, depending on the depth of motion and the Froude number.


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Benham, G. P., Hewitt, I. J., Please, C. P. & Bird, P. A. D. 2018 Optimal control of diffuser shapes for non-uniform flow. J. Engng. Maths 113 (1), 6592.
Benzaquen, M., Chevy, F. & Raphaël, E. 2011 Wave resistance for capillary gravity waves: finite-size effects. Europhys. Lett. 96 (3), 34003.
Benzaquen, M. & Raphael, E. 2012 Capillary-gravity waves on depth-dependent currents: consequences for the wave resistance. Europhys. Lett. 97 (1), 14007.
Berberović, E., van Hinsberg, N. P., Jakirlić, S., Roisman, I. V. & Tropea, C. 2009 Drop impact onto a liquid layer of finite thickness: dynamics of the cavity evolution. Phys. Rev. E 79 (3), 036306.
Bezanson, J., Edelman, A., Karpinski, S. & Shah, V. B. 2017 Julia: a fresh approach to numerical computing. SIAM Rev. 59 (1), 6598.
Boucher, J. P.2018 Problèmes d’optimisation à la surface de l’eau. PhD thesis, Ecole polytechnique.
Boucher, J. P., Labbé, R., Clanet, C. & Benzaquen, M. 2018 Thin or bulky: optimal aspect ratios for ship hulls. Phys. Rev. Fluids 3, 074802.
Dambrine, J., Pierre, M. & Rousseaux, G. 2016 A theoretical and numerical determination of optimal ship forms based on Michell’s wave resistance. ESAIM: Control Optim. Calculus Variations 22 (1), 88111.
Darmon, A., Benzaquen, M. & Raphaël, E. 2014 Kelvin wake pattern at large froude numbers. J. Fluid Mech. 738, R3.
Dunning, I., Huchette, J. & Lubin, M. 2017 Jump: a modeling language for mathematical optimization. SIAM Rev. 59 (2), 295320.
Fourdrinoy, J., Caplier, C., Devaux, Y., Rousseaux, G., Gianni, A., Zacharias, I., Jouteur, I., Martin, P. M., Dambrine, J., Petcu, M. et al. 2019 The naval battle of actium and the myth of the ship-holder: the effect of bathymetry. In 5th MASHCON – International Conference on Ship Manoeuvring in Shallow and Confined Water, with non-exclusive focus on manoeuvring in waves, wind and current, pp. 104133. Flanders Hydraulics Research; Maritime Technology Division, Ghent University.
Gotman, A. S. 2002 Study of Michell’s integral and influence of viscosity and ship hull form on wave resistance. Ocean. Engng Intl 6 (2), 74115.
Havelock, T. H. 1919 Wave resistance: some cases of three-dimensional fluid motion. Proc. R. Soc. Lond. A 95 (670), 354365.
Havelock, T. H. 1932 The theory of wave resistance. Proc. R. Soc. Lond. A 138 (835), 339348.
Huan, J. & Modi, V. 1996 Design of minimum drag bodies in incompressible laminar flow. Inverse Problems Engng 3 (4), 233260.
Lazauskas, L. V.2009 Resistance, wave-making and wave-decay of thin ships, with emphasis on the effects of viscosity. PhD thesis, The University of Adelaide.
Maynord, S. T. 2005 Wave height from planing and semi-planing small boats. River Res. Appl. 21 (1), 117.
Menter, F. R. 1994 Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32 (8), 15981605.
Michell, J. H. 1898 XI. The wave-resistance of a ship. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 45 (272), 106123.
Newman, J. N. 2018 Marine Hydrodynamics. MIT Press.
Nocedal, J. & Wright, S. J. 2006 Numerical Optimization, 2nd edn. Springer.
Pethiyagoda, R., McCue, S. W. & Moroney, T. J. 2017 Spectrograms of ship wakes: identifying linear and nonlinear wave signals. J. Fluid Mech. 811, 189209.
Rabaud, M. & Moisy, F. 2014 Narrow ship wakes and wave drag for planing hulls. Ocean Engng 90, 3438.
Schlichting, H., Gersten, K., Krause, E., Oertel, H. & Mayes, K. 1960 Boundary-Layer Theory. Springer.
Stack, J. & Von Doenhoff, A. E. 1934 Tests of 16 Related Airfoils at High Speeds. NACA.
Theodorakakos, A. & Bergeles, G. 2004 Simulation of sharp gas–liquid interface using VOF method and adaptive grid local refinement around the interface. Intl J. Numer. Meth. Fluids 45 (4), 421439.
Tuck, E. O. 1989 The wave resistance formula of J. H. Michell (1898) and its significance to recent research in ship hydrodynamics. ANZIAM J. 30 (4), 365377.
Ubbink, O.1997 Numerical prediction of two fluid systems with sharp interfaces. PhD thesis, Imperial College London.
Videler, J. J. 2012 Fish Swimming. Springer.
Wächter, A. & Biegler, L. T. 2006 On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Progr. 106 (1), 2557.
Zakerdoost, H., Ghassemi, H. & Ghiasi, M. 2013 Ship hull form optimization by evolutionary algorithm in order to diminish the drag. J. Marine Sci. Appl. 12 (2), 170179.
Zhao, Y., Zong, Z. & Zou, L. 2015 Ship hull optimization based on wave resistance using wavelet method. J. Hydrodyn. 27 (2), 216222.
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Wave drag on asymmetric bodies

  • G. P. Benham (a1), J. P. Boucher (a1), R. Labbé (a1), M. Benzaquen (a1) and C. Clanet (a1)...


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