Steady gravity waves due to a submerged source

Abstract

In the low-Froude-number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Steady linearized flow past a submerged source is considered, and exponential asymptotic methods are applied to determine the behaviour of the free-surface gravity waves. The free surface is found to contain longitudinal and transverse waves that switch on rapidly across curves known as Stokes lines on the free surface. The longitudinal waves are present everywhere downstream of the singularity, while the transverse waves are restricted to two downstream wedges. As the depth of the source approaches the surface, the familiar Kelvin-wedge wave behaviour is recovered.

Three-dimensional capillary waves due to a submerged source with small surface tension

“…This provides us with the means to check the accuracy of our approximation. We can compare the amplitude of the asymptotic results with those of numerically-calculated free surface profiles, calculated using an adaption of the method described [36].…”

“…Unsteady free-surface flow, such as the unsteady gravity wave problem considered in [37], does contain three interacting contributions (exponentially small steady and transient ripples, and algebraic effects); hence, higher-order Stokes lines must play a role in the solution. It is therefore not sufficient to find the ordinary Stokes lines in this problem, as the Stokes structure would be incorrect.…”

“…Following the approach of [36], we first expand the fluid potential and free-surface position as a power series in ,φ…”

“…Aside from [36,37], each of the previous exponential asymptotic studies on fluid flow behaviour have been performed in two dimensions. We will therefore follow the methodology of [36,37], in which we linearise the problem around the source strength in order to fix the position of the boundary in the linearised regime. We will then apply exponential asymptotic techniques directly to the flow equations in order to determine the fluid potential, and the free surface position.…”

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

“…These Stokes lines emerge from singularities in the analytical continuation of the leading order term and typically intersect the part of the complex plane corresponding to the physical problem of interest (which is the real axis in many problems, but the unit circle in our problem). Note that this method has been successfully applied to a variety of problems in fluid mechanics, including two-and three-dimensional water waves [13,37,38,39,40,57,58]. Finally, the third key idea is that of selection.…”