A conformal mapping approach to modelling two-dimensional stratified flow

Herein we describe a new approach to modelling inviscid two-dimensional stratified flows in a general domain. The approach makes use of a conformal map of the domain to a rectangle. In this transformed domain, the equations of motion are largely unaltered, and in particular Laplace's equation remains unchanged. This enables one to construct exact solutions to Laplace's equation and thereby enforce all boundary conditions.

An example is provided for two-dimensional flow under the Boussinesq approximation, though the approach is much more general (albeit restricted to two-dimensions). This example is motivated by flow under a weir in a tidal estuary. Here, we discuss how to use a dynamically-evolving conformal map to model changes in the water height on either side of the weir, though the example presented keeps these heights fixed due to limitations in the computational speed to generate the conformal map.

The numerical approach makes use of contour advection, wherein material buoyancy contours are advected conservatively by the local fluid velocity, while a dual contour-grid representation is used for the vorticity in order to account for vorticity generation from horizontal buoyancy gradients. This generation is accurately estimated by using the buoyancy contours directly, rather than a gridded version of the buoyancy field. The result is a highly-accurate, efficient numerical method with extremely low levels of numerical damping.