Homogeneous viscous flows about corners

Abstract

This thesis begins with a survey of the work completed by Moffatt (1964) where he deduced that for regions near sharp corners in a two dimensional homogeneous viscous fluid, an approximation for the flow is appropriate in an expansion of analytical functions. This approximation implies that for some flow schemes with corners, a singularity in the flow is present; especially when dealing with derivatives of the velocity field. It is the aim of this thesis to develop and incorporate numerical methods which use this analytical expansion so that the singularities can be resolved. We proceed to analyse and overview three methods that incorporate the Moffatt expansion in the lid-driven cavity. These methods have been appropriately named to be the subtraction method, the scaling method and the alternative finite difference method. Following this, we analyse the Moffatt eigenvalues for corners whose angles are greater than 180 degrees and then modify the above methods, so that they can be applied to solve the steady state Navier-Stokes equations in more general domains with corners. It is found for internal corners with non-homogeneous boundary conditions, the subtraction method works best. For external corners with homogeneous boundary conditions, the alternative finite difference equations work best. Following that, we investigate the use of conformal transformations especially for the solutions of the steady state Navier-Stokes equations. These conformal transformation methods are then compared to the methods mentioned earlier in terms of accuracy and resolution. We find that for certain geometries, conformal transformations can be used to significantly increase the resolution and accuracy of the flow near `external' corners. However, these transformations are not so well suited for `internal' corners as the resolution is comparatively poorly captured when solving the Navier-Stokes equations with a regular grid spacing. The central result of this thesis shows that in some geometries, changes to the accuracy of the flow around a corner with respect to the Navier-Stokes equations can significantly change and influence the contour curves in a sub-region of the geometry of the fluid.