Two-dimensional vector field topology and scalar fields in viscous flows: Reconstruction methods

Abstract

This paper discusses the reconstruction of the two-dimensional (2D) vector field topology (VFT) from a 2D scalar potential field and vice versa. The physical foundation of the proposed reconstruction method is the convection-type equation coupling a 2D vector field (e.g., skin friction) with a 2D potential field (e.g., surface pressure, temperature, or scalar concentration) in viscous flows. To reconstruct the VFT, a variational method is applied to this inverse problem, and then, an approximate method is proposed based on the linear superposition of some elemental potential field structures with simple analytical forms (source, vortex, saddle, etc.). As examples, the proposed method is applied to swept shock-wave/boundary-layer interaction and near-wall turbulence. Furthermore, in a reversed process to reconstruct a 2D potential field from a 2D vector field, a similar variational method is applied, and an approximate method with a constant source term in the convection-type relation is proposed, which is particularly applicable to reconstruction of a surface pressure field from global skin friction measurements in aerodynamics experiments. The significance of this work is that the complex VFT can be reconstructed based on a scalar potential field by using a semi-analytical approach. The proposed method can be used in fluid mechanics and other disciplines such as computer graphics and data visualization.