Ship waves in the presence of uniform vorticity

Abstract

Lord Kelvin's result that waves behind a ship lie within a half-angle 19 deg 28' is perhaps the most famous and striking result in the field of surface waves. We solve the linear ship wave problem in the presence of a shear current of constant vorticity S, and show that the Kelvin angles (one each side of wake) as well as other aspects of the wake depend closely on the "shear Froude number" Frs=VS/g (based on length g/S^2 and the ship's speed V), and on the angle between current and the ship's line of motion. In all directions except exactly along the shear flow there exists a critical value of Frs beyond which no transverse waves are produced, and where the full wake angle reaches 180 deg. Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed 90 deg. On the other hand, the angle of maximum wave amplitude scales as 1/Fr (Fr based on size of ship) when Fr >> 1, a scaling virtually unaffected by the shear flow.Comment: 10 pages, 6 figures, accepted for J. Fluid Mech. Rapid

Initial surface disturbance on a shear current: The Cauchy-Poisson problem with a twist

“…This theory does, however, rely on the use of a velocity potential, which in 3 dimensions limits the theory to situations of irrotational flow 14 . It was recently realised, however, that the Euler equations permit an exact solution for linear surface waves in the presence of uniform shear, allowing the study of ship waves in the presence of uniform vorticity 15 . For that case the presence of a shear current was found to be able to change a ship's train of waves quite radically.…”

“…15 shows that group and phase velocities can differ greatly in different directions of propagation when the uniform shear (in suitably non-dimensionalised form) is strong. Thus one must expect that the well known circular ring waves, such as seen when throwing a pebble into a quiescent pond, will no longer be circular and perhaps not even ring shaped, with a shear current present.…”

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 .…”