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), 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 incline. 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.
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|Item Type:||Journal Article|
|Keywords:||exponential asymptotics, free surface flows, water waves, flow past topography, intermediate waves, wave cancellation, analytic continuation, low Froude number expansion, diverging series, Stokes phenomenon, asymptotics beyond all orders|
|Subjects:||Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Approximation Theory and Asymptotic Methods (010201)|
Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Theoretical and Applied Mechanics (010207)
|Divisions:||Past > QUT Faculties & Divisions > Faculty of Science and Technology|
Past > Schools > Mathematical Sciences
|Copyright Owner:||Copyright 2012 Cambridge University Press|
|Deposited On:||06 Jan 2012 08:45|
|Last Modified:||19 Jun 2012 18:14|
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