Hairpin vortex generation around a straight vortex tube in a laminar boundary-layer flow

TL;DR

This study uses direct numerical simulation to explore how a straight vortex tube in a laminar boundary layer can generate hairpin vortices, revealing the roles of circulation, stability modes, and nonlinear dynamics.

Contribution

It provides new insights into hairpin vortex formation mechanisms by analyzing linear and nonlinear disturbance evolution around vortex tubes in laminar flows.

Findings

Multiple hairpin vortices form when vortex circulation exceeds a threshold.

Two unstable modes, off-wall and near-wall, lead to corkscrew-like disturbances.

Nonlinear analysis clarifies the dynamics of hairpin vortex generation.

Abstract

Direct numerical simulation (DNS) was conducted to investigate the evolution of a straight vortex tube based on Hon & Walker's model, which was originally proposed for a symmetric hairpin vortex tube (Hon,T.L. & Walker, J.D.A, Computers & Fluids, 20(3), pp. 343-358, 1991), under the shear of a laminar boundary-layer flow and its response of the near-wall flows. Here, the streamwise vortex tube is a lifted streamwise vortex. The circulation and angle-to-wall of the vortex tube are varied. Circulation above a threshold produces multiple hairpin vortices that align in the streamwise direction over the vortex tube. Stability of the disturbances according to the situation of the vortex tube inside the boundary layer such as the fraction of the vortex tube inside the boundary layer in addition to the circulation and angle-to-wall is shown by a linear stability analysis. The contribution of linear and nonlinear terms in the dynamics of the disturbance evolution is clarified by a fully-nonlinear disturbance analysis that is conducted concurrently with the DNS. When the circulation is sufficiently large, two unstable modes, namely the `off-wall mode' and `near-wall mode', appear. The presence of these two modes generates a corkscrew-like disturbance around the vortex tube, and becomes a precursor for the generation of hairpin vortices. The dominant structure of disturbances appearing around the vortex tube along the elapse of time is also analyzed by the proper orthogonal decomposition. The present DNS of the model allowing continuous variation of the characteristics of the vortex tube clarifies this new aspect of the nonlinear boundary-layer stability regarding the generation of hairpin vortices.