On small-time expansion of nonlinear free surface problems

Abstract

The small-time expansion of the 2-D problem of a heaving semi-submerged circular cylinder, starting from rest, in analyzed. The first three terms of the small-time expansion of the outer solution (away from the cylinder) are developed. A logarithmic singularity is found for the second derivative in time of the surface elevation at the intersection point. This singularity is of second order in the heaving velocity of the cylinder, and the inner expansion developed for the linear wavemaker problem therefore does not cover this case. The problem of the horizontal motion of the semi-submerged cylinder is analyzed as well. For the case with the initial condition for the velocity potential chosen as zero on the free surface, the logarithmic singularity found for the wavemaker is recovered. Selecting the initial condition as a vanishing normal velocity on the free surface leads to a solution where the singularity appears in the fifth derivative in time of the free-surface elevation. The solution of this problem shows the characteristic behaviour of formation of asymmetry, which finally would lead to formation of lee waves. The inner solution of the problem of the heaving cylinder is discussed, showing that the oscillatory behaviour reported for the wavemaker problem appears in the present solution as well.