Skip to main content

Free surface flow past topography: A beyond-all-orders approach


The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech.567, 299–326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

Hide All
[1]Berry M. V. (1991) Asymptotics, superasymptotics, hyperasymptotics. In: Segur H., Tanveer S. & Levine H. (editors), Asymptotics Beyond All Orders, Plenum, Amsterdam, pp. 114.
[2]Binder B. J. (2010) Steady free-surface flow at the stern of a ship. Phys. Fluids, 22, 012104.
[3]Binder B. J., Dias F. & Vanden-Broeck J.-M. (2007) Influence of rapid changes in a channel bottom on free-surface flows. IMA J. Appl. Math. 73, 120.
[4]Binder B. J., Vanden-Broeck J.-M. & Dias F. (2005) Forced solitary waves and fronts past submerged obstacles. Chaos 15, 037106.
[5]Body G. L., King J. R. & Tew R. H. (2005) Exponential asymptotics of a fifth-order differential equation. Euro. J. Appl. Math. 16, 647681.
[6]Brower R. C., Kessler D. A., Koplik J. & Levine H. (1983) Geometrical approach to moving-interface dynamics. Phys. Rev. Lett. 51, 11111114.
[7]Chapman S. J. (1999) On the rôle of Stokes lines in the selection of Saffman-Taylor fingers with small surface tension. Euro. J. Appl. Math. 10, 513534.
[8]Chapman S. J. & King J. R. (2003) The selection of Saffman-Taylor fingers by kinetic undercooling. J. Eng. Math. 46, 132.
[9]Chapman S. J., King J. R. & Adams K. L. (1998) Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc. Roy. Soc. Lond. A 454, 27332755.
[10]Chapman S. J. & Mortimer D. B. (2005) Exponential asymptotics and Stokes lines in a partial differential equation. Proc. Roy. Soc. Lond. A 461, 23852421.
[11]Chapman S. J. & Vanden-Broeck J.-M. (2002) Exponential asymptotics and capillary waves. SIAM J. Appl. Math. 62, 18721898.
[12]Chapman S. J. & Vanden-Broeck J.-M. (2006) Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299326.
[13]Dagan G. (1975) Waves and wave resistance of thin bodies moving at low speed: The free-surface nonlinear effect. J. Fluid Mech. 69, 405416.
[14]Dingle R. B. (1973) Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York.
[15]Doctors L. J. & Dagan G. (1980) Comparison of nonlinear wave-resistance theories for a two-dimensional pressure distribution. J. Fluid Mech. 98, 647672.
[16]Forbes L. K. (1982) Non-linear, drag-free flow over a submerged semi-elliptical body. J. Eng. Math. 16, 171180.
[17]Forbes L. K. (1988) Critical free-surface flow over a semi-circular obstruction. J. Eng. Math. 22, 313.
[18]Forbes L. K. & Schwartz L. W. (1982) Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114, 299314.
[19]Gazdar A. S. (1973) Generation of waves of small amplitude by an obstacle placed on the bottom of a running stream. J. Phys. Soc. Japan 34, 530.
[20]Hocking G. C. & Forbes L. K. (1992) Subcritical free-surface flow caused by a line source in a fluid of finite depth. J. Eng. Math. 26, 455466.
[21]Howls C. J., Langman P. J. & Olde Daalhuis A. B. (2004) On the higher-order Stokes phenomenon. Proc. Roy. Soc. Lond. A 460, 22852303.
[22]King A. C. & Bloor M. I. G. (1987) Free-surface flow over a step. J. Fluid Mech. 182, 193208.
[23]King A. C. & Bloor M. I. G. (1990) Free-surface flow of a stream obstructed by an arbitrary bed topography. Q. J. Mech. Appl. Math. 43, 87106.
[24]Kruskal M. D. & Segur H. (1991) Asymptotics beyond all orders in a model of crystal growth. Stud. Appl. Math. 36, 129181.
[25]Lamb H. (1932) Hydrodynamics. Cambridge University Press.
[26]Maleewong M. & Grimshaw R. H. J. (2008) Nonlinear free surface flows past a semi-infinite flat plate in water of finite depth. Phys. Fluids 20, 062102.
[27]McCue S. W. & Forbes L. K. (1999) Bow and stern flows with constant vorticity. J. Fluid Mech. 399, 277300.
[28]McCue S. W. & Forbes L. K. (2002) Free-surface flows emerging from beneath a semi-infinite plate with constant vorticity. J. Fluid Mech. 461, 387407.
[29]McCue S. W. & Stump D. M. (2000) Linear stern waves in finite depth channels. Q. J. Mech. Appl. Math. 53, 629643.
[30]Mekias H. & Vanden-Broeck J.-M. (1991) Subcritical flow with a stagnation point due to a source beneath a free surface. Phys. Fluids A 3, 26522658.
[31]Ogilat O., McCue S. W., Turner I. W., Belward J. A. & Binder B. J. (2011) Minimising wave drag for free surface flow past a two-dimensional stern. Phys. Fluids 23, 072101.
[32]Ogilvie T. F. (1968) Wave Resistance. The Low Speed Limit. Technical Report, Michigan University, Ann Arbor, MI.
[33]Olde Daalhuis A. B., Chapman S. J., King J. R., Ockendon J. R. & Tew R. H. (1995) Stokes phenomenon and matched asymptotic expansions. SIAM J. Appl. Math. 55, 14691483.
[34]Scullen D. & Tuck E. O. (1995) Nonlinear free-surface flow computations for submerged cylinders. J. Ship Res. 39, 185193.
[35]Trinh P. H., Chapman S. J., & Vanden-Broeck J.-M. (2011) Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413439.
[36]Tuck E. O. & Scullen D. C. (1998) Tandem submerged cylinders each subject to zero drag. J. Fluid Mech. 364, 211220.
[37]Vanden-Broeck J.-M. (1980) Nonlinear stern waves. J. Fluid Mech. 96, 603611.
[38]Vanden-Broeck J.-M., Schwartz L. W. & Tuck E. O. (1978) Divergent low-Froude-number series expansion of nonlinear free-surface flow problems. Proc. R. Soc. Lond. A 361, 207224.
[39]Vanden-Broeck J.-M. & Tuck E. O. (1977) Computation of near-bow or stern flows using series expansion in the Froude number. In: 2nd International Conference on Numerical Ship Hydrodynamics, University of California, Berkeley, CA.
[40]Vanden-Broeck J.-M. & Tuck E. O. (1985) Waveless free-surface pressure distributions. J. Ship Res. 29, 151158.
[41]Wehausen J. V. & Laitone E. V. (1960) Surface waves. In: Handbuch der Physik, Springer, pp. 446778.
[42]Zhang Y. & Zhu S. (1996) A comparison of nonlinear waves generated behind a semicircular trench. Proc. Roy. Soc. Lond. A 452, 15631584.
[43]Zhang Y. & Zhu S. (1996) Open channel flow past a bottom obstruction. J. Eng. Math. 30, 487499.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 27 *
Loading metrics...

Abstract views

Total abstract views: 157 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th January 2018. This data will be updated every 24 hours.