Ship's Wake on a Finite Water Depth in the Presence of a Shear Flow

The ship's wake in the presence of a shear flow of constant vorticity at a finite water depth is investigated by expanding the Whitham-Lighthill kinematic theory. It has been established that the structure of a wave ship wake radically depends on Froude number Fr (in terms of water depth) and especially on the critical Froude number Fr-cr, which depends on the magnitude and direction of the shear flow. At its subcritical values Fr < Fr-cr two types of waves are presented: long transverse and short divergent waves inside the wedge area with an angle depending on Fr For the supercritical range Fr > Fr-cr only divergent waves are presented inside the wake region. Critical Froude number Fr-cr is variable and mostly depends on collinear with the ship path component of the shear flow: it decreases with the unidirectional shear flow and increases on the counter shear current. The wedge angle of the ship wake expands with an increase in the unidirectional shear flow and narrows in the oncoming flow in the subcritical mode of the ship's motion Fr < Fr-cr. Wake angle decreased with Froude number for Fr > Fr-cr and only divergent waves with the crests almost collinear with ship path are finally presented in the narrow ship wake. For a critical value of the Froude number Fr = Fr-cr, the ship's wake has a total wedge angle of 180 degrees with waves directed parallel to the ship's motion. The presence of a shear flow crossing the path of the ship gives a strong asymmetry to the wake. An increase in the perpendicular shear flow leads to an increase in the difference between the angles of the wake arms.