Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-20T00:13:50.971Z Has data issue: false hasContentIssue false
  • English
  • Français

On steady non-breaking downstream waves and the wave resistance – Stokes’ method

Published online by Cambridge University Press:  20 February 2017

Dmitri V. Maklakov Open the ORCID record for Dmitri V. Maklakov [Opens in a new window]  and
Alexander G. Petrov
Dmitri V. Maklakov*
Affiliation:
Kazan (Volga region) Federal University, N.I. Lobachevsky Institute of Mathematics and Mechanics, Kremlyovskaya 35, Kazan, 420008, Russia
Alexander G. Petrov
Affiliation:
Institute for Problems in Mechanics of the Russian Academy of Sciences, prosp. Vernadskogo 101, block 1, Moscow, 119526, Russia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work, we have obtained explicit analytical formulae expressing the wave resistance of a two-dimensional body in terms of geometric parameters of nonlinear downstream waves. The formulae have been constructed in the form of high-order asymptotic expansions in powers of the wave amplitude with coefficients depending on the mean depth. To obtain these expansions, the second Stokes method has been used. The analysis represents the next step of the research carried out in Maklakov & Petrov (J. Fluid Mech., vol. 776, 2015, pp. 290–315), where the properties of the waves have been computed by a numerical method of integral equations. In the present work, we have derived a quadratic system of equations with respect to the coefficients of the second Stokes method and developed an effective computer algorithm for solving the system. Comparison with previous numerical results obtained by the method of integral equations has been made.

JFM classification

Waves/Free-surface Flows: Channel flow Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 815 , 25 March 2017 , pp. 388 - 414
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balk, A. M. 1996 A Lagrangian for water waves. Phys. Fluids 8 (2), 416–420.CrossRefGoogle Scholar
Benjamin, T. B. 1995 Verification of the Benjamin–Lighthill conjecture about steady water waves. J. Fluid Mech. 295, 337–356.CrossRefGoogle Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448–460.Google Scholar
Bergman, S. 1950 The Kernel Function and Conformal Mapping, Mathematical Surveys, vol. 5. American Mathematical Society.CrossRefGoogle Scholar
Binder, B. J., Vanden-Broek, J.-M. & Dias, F. 2009 On satisfying the radiation condition in free-surface flows. J. Fluid Mech. 624, 179–189.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Proc. R. Soc. Lond. A 286, 183–230.Google Scholar
Dallaston, M. C. & McCue, S. W. 2010 Accurate series solutions for gravity-driven stokes waves. Phys. Fluids 22, 0821104.CrossRefGoogle Scholar
De, S. C. 1955 Contributions to the theory of Stokes waves. Math. Proc. Camb. Phil. Soc. 51, 713–736.CrossRefGoogle Scholar
Dias, F. & Vanden-Broek, J.-M. 2002 Generalised critical free-surface flows. J. Engng Maths 42, 291–301.Google Scholar
Dias, F. & Vanden-Broek, J.-M. 2004 Trapped waves between submerged obstacles. J. Fluid Mech. 509, 93–102.Google Scholar
Duncan, J. H. 1983 A note on the evaluation of the wave resistance of two-dimensional bodies from measurements of the downstream wave profile. J. Ship Res. 27 (2), 90–92.Google Scholar
Dyachenko, A. I., Zakharov, V. E. & Kuznetsov, E. A. 1996 Nonlinear dynamics of the free surface of an ideal fluid. Plasma Phys. Rep. 22 (10), 829–840.Google Scholar
Fenton, J. D. 1985 A fifth-order stokes theory for steady wave. ASCE J. Waterway Port Coastal Ocean Engng 111 (2), 216–234.Google Scholar
Forbes, L. K. 1988 Critical free-surface flow over a semi-circular obstruction. J. Engng Maths 22, 3–13.CrossRefGoogle Scholar
Keady, G. & Norbury, J. 1975 Waves and conjugate streams. J. Fluid Mech. 70, 663–671.Google Scholar
Kelvin, Lord 1887 On ship waves. Proc. Inst. Mech. Engrs 38, 409–434.Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157–174.Google Scholar
Longuet-Higgins, M. S. 1978 Some new relations between Stokes’s coefficients in the theory of gravity waves. J. Inst. Maths Applics. 22, 261–273.CrossRefGoogle Scholar
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397.Google Scholar
Maklakov, D. V. 2002 Almost highest gravity waves on water of finite depth. Eur. J. Appl. Maths 13, 67–93.CrossRefGoogle Scholar
Maklakov, D. V. & Petrov, A. G. 2015a On steady non-breaking downstream waves and the wave resistance. J. Fluid Mech. 776, 290–315.Google Scholar
Maklakov, D. V. & Petrov, A. G. 2015b Stokes coefficients and wave resistance. Dokl. Phys. 60 (7), 314–318.CrossRefGoogle Scholar
Petrov, A. G. 2009 Analytical Hydrodynamics. FizMatLit; (in Russian).Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Shen, S. S. P. & Shen, M. C. 1990 On the limit of subcritical free-surface flow over an obstacle. Acta Mechanica 82, 225–230.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Stokes, G. G. 1880 Supplement to a paper on the theory of oscillatory waves. In Mathematical and Physical Papers, vol. I, pp. 314–326. Cambridge University Press.Google Scholar
Wilton, J. R. 1914 On deep water waves. Phil. Mag. 27 (158), 385–394.Google Scholar
Wolfram, S. 2003 The Mathematica Book, 5th edn. Wolfram Media.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.Google Scholar
Supplementary material: File

Maklakov and Petrov supplementary material

Maklakov and Petrov supplementary material 1

Download Maklakov and Petrov supplementary material(File)
File 5.3 MB