Vorticity sign law in three-dimensional wake of bluff body at low Reynolds number

Bluff bodies possess many engineering applications. The flow past a bluff body is a classical issue in fluid mechanics. In most of previous studies, the role of streamwise vorticity is mainly stressed in three-dimensional wake flow, such as in the physical origin of streamwise vortices in the mode A, and the complete suppression of alternatively shedding Karman vortices under the effect of geometric disturbances introduced in the bluff body. However, through the careful investigation of two examples above, the vertical vorticity actually plays a key role. Furthermore, there is a physical phenomenon, special relationship among dominant vorticity components with specific signs or vorticity sign law, occurring in the wake of a bluff body with geometric disturbances. In the present paper, through direct numerical simulations at low Reynolds numbers, such a phenomenon is summarized in two kinds of cylinders, i.e. basic straight cylinder and geometrically disturbed cylinder. Two typical cross-sections are examined, including the circular and square sections. Three sub-regimes (front surfaces, shear layers and wake) are mainly investigated. The numerical results show that two generalized vorticity sign laws exist in the wake of a bluff body. For example, the first sign law shows that the sign of streamwise vorticity is always the same as that of vertical vorticity in the upper shear layer, but opposite in the lower shear layer. The second sign law shows that the sign combination of three components of vorticity is always negative in the shear layers and wake. As for the physical mechanism of sign laws appearing in the present two kinds of cylinders, the main difference between the small perturbance which induces the natural three-dimensional instability and the geometric disturbance leads to the evolution of generated surface vorticity under the effect of inertial forces. These generalized sign laws have been already verified theoretically in the previous work recently. Moreover, sign laws also indicate that the different vortex-shedding patterns in the wake of different bluff bodies are inherently identical from the point of vorticity sign. Considering the physical fact that the wall is the only source of new vorticity in the present vortex dynamics, the theoretical results also indicate that the P-type vortex with specific groups of vorticity components is a basic vortex pattern generated on the walls, once the three-dimensional wake first appears in the wake at low Reynolds number