Unsteady free-surface waves due to a submerged body moving in a viscous fluid

Unsteady viscous free-surface waves generated by a three-dimensional submerged body moving in an incompressible fluid of infinite depth are investigated analytically. It is assumed that the body experiences a Heaviside step change in velocity at the initial instant. Two categories of the velocity change, (i) from zero to a constant and (ii) from a constant to zero, will be analyzed. The flow is assumed to be laminar and the submerged body is mathematically represented by an Oseenlet. The Green functions for the unbounded unsteady Oseen flows are derived. The solutions in closed integral form for the wave profiles are given. By employing Lighthill's two-stage scheme, the asymptotic representations of free-surface waves in the far wake for large Reynolds numbers are derived. It is shown that the effects of viscosity and submergence depth on the free-surface wave profiles are respectively expressed by the exponential decay factors. Furthermore, the unsteady wave system due to the suddenly starting body consists of two families of steady-state waves and two families of nonstationary waves, which are confined within a finite region. As time increases, the waves move away from the body and the finite region extends to an infinite V-shaped region. It is found that the nonstationary waves are the transient response to the suddenly started motion of the body. The waves due to a suddenly stopping body consist of a transient component only, which vanish as time approaches infinity.