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. 742 R2 (2014)]. We derive the exact criterion for criticality as a function of an intrinsic shear Froude number S 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 b/g of order unity.The theory is fully laid out and far-field contributions are carefully separated off by means of Cauchy's integral theorem, exposing potential pitfalls associated with a slightly different method (Sokhotsky-Plemelj) used in several previous works.

Observation of surface wave patterns modified by sub-surface shear currents

“…Recent theory has predicted that ship and ring waves are affected in striking ways by sub-surface shear; for instance, ring waves such as from a pebble thrown into the water no longer form concentric rings (Johnson 1990;Ellingsen 2014b;Khusnutdinova 2016), and Kelvin's famous result that ship waves in deep water always form an angle of approximately 39 • no longer holds (Ellingsen 2014a;Li & Ellingsen 2016). The physical understanding of ship waves more generally has been the focus of much recent attention (Rabaud & Moisy 2013;Darmon et al 2014;Noblesse et al 2014;Pethiagoda et al 2014).…”

Surface waves on currents with arbitrary vertical shear

“…The effects of a background shear flow of constant vorticity on the behavior of ship wakes has been studied recently 10,11 , yet many shear profiles encountered in reality have depthvariable vorticity. A model taking into account arbitrary vorticity depth-dependence may be necessary in obtaining quantitatively accurate results for ship wakes and related parameters such as wave resistance in the presence of realistic shear flows.…”

“…The ship wave problem can be solved numerically for arbitrary shear profiles using the piecewise linear approximation in a direct generalization of the recent theory for the constant vorticity profile 10,11 . Assuming a stationary wave solution in a reference frame moving with the ship, a coordinate transformation ξ = x − Vt is introduced where V is the velocity of a moving prescribed pressure sourcep ext (ξ) representing the ship relative to the undisturbed free surface.…”

“…The generalized case consists of a horizontal plane with waves propagating at an arbitrary oblique angle to the direction of the current, referred to three dimensional (3D). It was recently shown how 3D solutions to the Euler equations in the presence of a linear shear current (constant vorticity) can be used to solve classical problems such as ship waves and ring waves [10][11][12] . Shear currents were found to have the potential to significantly alter the characteristics of wave propagation in inherently 3D problems, evidenced by the behavior of ship waves as well as solutions to CauchyPoisson initial-value problems 12 .…”

Sheared free-surface flow over three-dimensional obstructions of finite amplitude

“…We review the extensive literature below, but note already that no study to date includes the possibility of a velocity of nontrivial depth-dependence, which is indicated by recent studies to affect the stationary waves behind obstacles significantly (e.g. Li & Ellingsen 2016).…”