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In this paper, the physical origin of vortex stretching and twisting is theoretically investigated. The effects of inertial and viscous forces are mainly considered and discussed. Two key conditions, i.e., solid walls and three-dimensional (3D) disturbances, are adopted in three typical cases. Among them, the first two cases are straight and curved vortex lines at the initial time without any kind of disturbance. The third case is a straight vortex line at the initial time with introduced 3D natural disturbances. Through experimental observations, numerical simulations, and theoretical analysis in these cases, the first two cases illustrate that the straight or curved vortex lines are still straight or curved at the next time, respectively, regardless of whether solid walls are introduced. However, the third case clearly shows that once natural disturbances are introduced, the straight vortex lines near and at solid walls at the initial time are stretched and twisted mainly by viscous forces, instead of inertial forces, typically demonstrated by the 3D wake transition of a straight circular cylinder and the transition of the laminar boundary layer at a flat plate. Accordingly, based on definitions of generation and enhancement in vortex stretching and twisting, it is confirmed that the viscous forces with two key conditions are the generation mechanism, while the inertial forces alone are the enhancement mechanism.

In this paper, the indirect physical mechanism of viscous forces, i.e., vortex stretching and twisting, is theoretically explored. The local flow in the immediate neighborhood of a flat plate is analyzed. Only the viscous forces are considered. By introducing oppositely signed streamwise vorticity in such a region, the induced velocity and vorticity fields are obtained. Because the induced streamwise velocity varies along with both the wall-normal and spanwise directions, induced vertical and spanwise vorticities are generated. The induction sequence is first clarified. Then, two indirect effects of viscous forces are identified. The induced spanwise vorticity leads to the original two-dimensional spanwise vorticity increasing and decreasing alternately across the span, as the vortex stretching effect of viscous forces. The local spanwise vortex is distorted at positions where the streamwise vorticity is introduced, and the appearance of induced vertical vorticity at the same positions enhances this effect, as the vortex twisting effect of viscous forces.

Vortex-induced vortex theory, commonly used for a flow past a disturbed bluff body, is applied in this paper to analyze an incompressible flow through a circular-section pipe with the occurrence of a secondary flow. The disturbed flow field is solved based on the Stokes equations by introducing a vortex or vortex pair uniformly distributed along the axial direction and periodically varying along the azimuthal direction as a result of the secondary flow with the assumption of inertial force being neglected and the viscous force being dominant in the vicinity of the pipe walls. Two kinds of boundary cases are considered to simulate the introduced vorticity distributed on and near the walls, and two sign laws for vorticity are also derived and verified for the present internal flow. For original pipe flow with a specific velocity distribution, such as a paraboloid of revolution at lower Reynolds numbers, these two sign laws are all positive upstream but negative downstream and they physically reveal the intrinsic relationships of the vorticity sign among different vorticity components, which are generated inherently with specific signs in secondary flows. Furthermore, two basic viscous force mechanisms are identified: a direct effect for vorticity generated on the walls due to shear flows and an indirect effect for vorticity induced by a vortex with former vorticity near the walls. Examples to demonstrate these two sign laws under geometric disturbances at a laminar Reynolds number of 200 and physical meaning are also presented briefly.