Theoretical analysis of vorticity in a hairpin vortex in the viscous sublayer of a laminar boundary layer

Abstract

Based on the vortex-induced vortex theory, the classic paradigm of Theodorsen’s hairpin vortex is used to analyze the disturbed vorticity and its sign relationship. The hairpin vortex is assumed to exist in the viscous sublayer of a laminar boundary layer, i.e. , the immediate neighborhood of the solid walls. Accordingly, the theoretical model is established mainly by ignoring the inertial forces. The disturbed vorticity field and associated two vorticity sign laws are obtained. The first sign law reveals that the wall-normal vorticity is actually induced by the streamwise vorticity in the two legs, as a kind of interaction between the vortex and solid wall. Such induction is completely a result of the viscous forces, identified as the third viscous effect. Both vorticity sign laws illustrate that the direction of a vorticity vector in the present hairpin vortex is specific, as is the inclined direction of the two legs. Furthermore, the intrinsic relationship between the hairpin vortex and other three-dimensional vortices appearing in external and internal flows at low and laminar Reynolds numbers is thus established by both sign laws. Then, the generalized Π-type vortex is defined as a category of a vortex in which the three vorticity components are consistent with both sign laws through quadrant analysis. In addition, it is theoretically confirmed that the lifting process of the hairpin vortex is predominantly attributable to the mechanism of the vortex-induced vortex and is thus the typical linear process due to viscous forces. Other features, such as the varied inclination, third sign law and self-similarity, are presented in detail.