As reported in a previous work by Lin et al. (Acta Mech Sin, 2018. 10.1007/s10409-018-0758-z In the present paper, the theory of a vortex-induced vortex is introduced to explore mechanisms in a wavy vortex and applicable sign laws for uniform and incompressible flow past a fixed bluff body. Based on the analysis of the nearest-wall flow, two vortex-induced models for streamwise and vertical vortex pairs, respectively, are proposed under two boundary cases, denoting the induced vorticity introduced or distributed on and near the walls. As a result, the first sign law, for only streamwise and vertical components of vorticity, and the second sign law, for three components of vorticity, are obtained under their own particular conditions. The first sign law reveals the intrinsic physical relationship between streamwise and vertical vorticities, independent of the distribution of spanwise vortices in the whole flow field. It is also confirmed that the spanwise vortices, as well as the shear layers and wake width, distributed wavily across the span, are attributed to the introduced streamwise or vertical vortices. The two sign laws for vorticity are independent of the disturbed spanwise wavelength and the Reynolds number. Through the analysis of flow past the conic shroud, the two sign laws are successfully used to summarize typical spacial distributions of vorticity in three flow regions: on and near the front cylinder surfaces, the separated shear layers and the near wake.