Tracking Birth of Vortices in Two-dimensional Flow

In order to avoid undesired effects from vortices in many industrial processes, it is important to know the set of operating parameters at which the flow does not have recirculation. The map of these conditions in the parameter space is called vortex-free operating window. Here, we propose an efficient way to construct such window automatically without expensively checking every possible flow states. The proposed technique is based on tracking a path in the parameter space at which the local kinematic condition at a stagnation point for vortex birth is satisfied. This multiparameter continuation is performed by solving an augmented Navier–Stokes system. In the augmented system, the birth condition and the governing equations were represented in Galerkin’s finite element context. We used the proposed method in two important coating flows with free surfaces: single-layer slot coating and forward roll coating.