The 12th International Conference on Hydrodynamics
18 – 23 september 2016, Egmond aan Zee, The Netherlands
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OSCILLATING SOURCES IN A SHEAR FLOW WITH A FREE SURFACE


Go-down ichd2016 Tracking Number 49

Presentation:
Session: Linear and non-linear waves I
Room: Room 2
Session start: 14:00 Tue 20 Sep 2016

Simen Å. Ellingsen   simen.a.ellingsen@ntnu.no
Affifliation: Norwegian University of Science and Technology

Peder A. Tyvand   peder.tyvand@nmbu.no
Affifliation: Norwegian University of Life Sciences


Topics: - Linear and non-linear waves and current

Abstract:

In the domain of potential theory, oscillating line and point sources form the building blocks of the Green function theory which describes how immersed bodies interact with surface waves (e.g., [1]). The solutions are well established and in routine use when irrotational flow may be assumed. In the presence of a shear flow, however, the behaviour of free surface flow in the presence of point singularities remains relatively unexplored, although other 3D surface flow phenomena have recently been solved [2,3]. We report on progress on the free surface flow in the presence of submerged oscillating line sources (2D) or point sources (3D) when a simple shear flow is present varying linearly with depth. In both cases the Euler equations are solved to linear order in perturbation quantities using standard Fourier transform techniques. The sources are held at rest with respect to the surface for simplicity. Both in 2D and 3D an new type of solution appears compared to irrotational case, which we identify as a critical layer, whose surface manifestation (“wave”) drifts downstream from the source at the velocity of the flow at the source depth. A previous 2D solutions based on a potential theory [4] fails to capture this effect. We provide a simple physical argument why a critical layer is a necessary consequence of Kelvin’s circulation theorem. For the flow around a point source in 3D, no potential theory can exist. Again, critical layer-type solutions are found to drift downstream of the source. We provide, for the first time, illustrations of the velocity field downstream of the source to illustrate the shape of the critical layer flow structures. Our results all indicate that the formulation of a Green function theory able to describe bodies in a shear flow will be much complicated by the presence of critical layer solutions. REFERENCES [1] J. N. Newman, Marine Hydrodynamics (MIT Press, 1977) [2] S. Å. Ellingsen, “Initial surface disturbance on a shear current: The Cuahcy-Poisson problem with a twist” Phys. Fluids Vol. 26, 082104 (2014). [3] S. Å. Ellingsen, “Ship waves in the presence of uniform vorticity” J. Fluid Mech. Vol 742, R2 (2014) [4] P. A. Tyvand and M. E. Lepperød, “Oscillatory line source for water waves in shear flow” Wave Motion Vol 51, 505-516 (2014)