The Mathematical Institute, University of Oxford, Eprints Archive

Exponential asymptotics and gravity waves

Chapman, S. J. and Vanden-Broeck, J. (2006) Exponential asymptotics and gravity waves. Journal of Fluid Mechanics, 567 . pp. 299-326. ISSN 0022-1120



The problem of irrotational inviscid incompressible free-surface flow is examined in the limit of small Froude number. Since this is a singular perturbation, singularities in the flow field (or its analytic continuation) such as stagnation points, or corners in submerged objects or on rough beds, lead to a divergent asymptotic expansion, with associated Stokes lines. Recent techniques in exponential asymptotics are employed to observe the switching on of exponentially small gravity waves across these Stokes lines.

As a concrete example, the flow over a step is considered. It is found that there are three possible parameter regimes, depending on whether the dimensionless step height is small, of the same order, or large compared to the square of the Froude number. Asymptotic results are derived in each case, and compared with numerical simulations of the full nonlinear problem. The agreement is remarkably good, even at relatively large Froude number. This is in contrast to the alternative analytical theory of small step height, which is accurate only for very small steps.

Item Type:Article
Subjects:D - G > Fluid mechanics
Research Groups:Oxford Centre for Industrial and Applied Mathematics
ID Code:609
Deposited By: Jon Chapman
Deposited On:24 May 2007
Last Modified:29 May 2015 18:25

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