Divergent low-Froude-number series expansion of nonlinear free-surface flow problems

Abstract

The low-Froude-number approximation in free-surface hydrodynamics is singular, and leads to formal series in powers of the Froude number with zero radius of convergence. The properties of these divergent series are discussed for several types of two-dimensional flows. It is shown that the divergence is of ‘n!' or exponential-integral character. A potential or actual lack of uniqueness is discovered and discussed. The series are summed by use of suitable nonlinear iterative transformations, giving good accuracy even for moderately large Froude number. Converged ' solutions ’ are obtained in this way, which possess jump discontinuities on the free surface. These jumps can be explained and, in principle, removed, by consideration of appropriate choices for the branch cut of the limiting exponential-integral solution. For example, we provide here a solution for a continuous wave-like flow, behind a semi-infinite moving body.