Viscous-Inviscid Interaction Analysis in High-Reynolds Number Flows Using Complex Singularities

Interaction between the viscous boundary layer and inviscid outer flow occurs during unsteady separation on two distinct spatial scales depending upon the Reynolds number regime. Using the impulsively-started flow about a circular cylinder, it is illustrated how these regimes can be identified by tracking singularities in the complex plane based on numerical solutions of the unsteady Navier-Stokes equations. Such an analysis also allows for clear identification in Navier-Stokes solutions of the precursor to the van Dommelen singularity, which occurs in the classical non-interactive boundary-layer equations. The first interaction developing in the separation process is large-scale interaction that is visible for all the Reynolds numbers considered, and it signals the first relevant differences between the boundary-layer and Navier-Stokes solutions. For $Re\geq O(10^4)$, a small-scale interaction corresponding to the van Dommelen singularity follows the large-scale interaction.