Ship waves on uniform shear current at finite depth: wave resistance and critical velocity

Abstract

We present a comprehensive theory for linear gravity-driven ship waves in the presence of a shear current with uniform vorticity, including the effects of finite water depth. The wave resistance in the presence of shear current is calculated for the first time, containing in general a non-zero lateral component. While formally apparently a straightforward extension of existing deep water theory, the introduction of finite water depth is physically non-trivial, since the surface waves are now affected by a subtle interplay of the effects of the current and the sea bed. This becomes particularly pronounced when considering the phenomenon of critical velocity, the velocity at which transversely propagating waves become unable to keep up with the moving source. The phenomenon is well known for shallow water, and was recently shown to exist also in deep water in the presence of a shear current [Ellingsen, J.~Fluid Mech.\ {\bf 742} R2 (2014)]. We derive the exact criterion for criticality as a function of an intrinsic shear Froude number $S\sqrt{b/g}$ ($S$ is uniform vorticity, $b$ size of source), the water depth, and the angle between the shear current and the ship's motion. Formulae for both the normal and lateral wave resistance force are derived, and we analyse its dependence on the source velocity (or Froude number $Fr$) for different amounts of shear and different directions of motion. The effect of the shear current is to increase wave resistance for upstream ship motion and decrease it for downstream motion. Also the value of $Fr$ at which $R$ is maximal is lowered for upstream and increased for downstream directions of ship motion. For oblique angles between ship motion and current there is a lateral wave resistance component which can amount to $10$-$20\%$ of the normal wave resistance for side-on shear and $S\sqrt{b/g}$ of order unity. (Continues...)

arXiv comment27 pages, 14 figures. On the cover of JFM vol 791

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• Oblique waves on a vertically sheared current are rotational
• Transient wave resistance upon a real shear current
• Surface waves on arbitrary vertically-sheared currents
• Kelvin wake pattern at small Froude numbers
• Observation of surface wave patterns modified by sub-surface shear currents
• Three dimensional free-surface flow over arbitrary bottom topography
• Waves from an oscillating point source with a free surface in the presence of a shear current
• Multiple resonances of a moving, oscillating surface disturbance on a shear current
• Hysteretic wave drag in shallow water
• Sheared free-surface flow over three-dimensional obstructions of finite amplitude