# Numerical Study of Aircraft Wake Vortex Evolution under the Influence of Vertical Winds
## 3 Results
### Analysis of Tail Vortex Structure  
The vorticity magnitude is an important parameter describing the strength of the wake vortex, where the vorticity magnitude \(\omega\) is expressed as follows [11]. The maximum vorticity magnitude in the vertical plane is the location of the vortex core, and the distance between the two vortex cores is the vortex core spacing. \[\omega=\sqrt{\omega_{\rm x}^{2}+\omega_{\rm y}^{2}+\omega_{\rm z}^{2}}, \tag{8}\]  
\[\omega_{\rm x}=\frac{\partial\rm w}{\partial\rm y}-\frac{\partial\rm v}{ \partial\rm z}, \tag{9}\]  
\[\omega_{\rm y}=\frac{\partial\rm u}{\partial\rm z}-\frac{\partial\rm w}{ \partial\rm x}, \tag{10}\]  
\[\omega_{\rm z}=\frac{\partial\rm v}{\partial\rm x}-\frac{\partial\rm u}{ \partial\rm y}, \tag{11}\]  
where \(\omega_{\rm x}\), \(\omega_{\rm y}\) and \(\omega_{\rm z}\), respectively, denote the components of vorticity along the X, Y, and Z axes. The rotational characteristic of the wake vortex is described using the tangential velocity; the expression for the tangential velocity \(\rm V_{\Earth}\) is given by:  
\[\rm V_{\Earth}=\sqrt{u^{2}+v^{2}}, \tag{12}\]  
where \(u,\ v\), respectively, denote the component of velocity on the X, Y axis. A vortex's entire lifecycle can be divided into four distinct stages: Formation Stage, Initial Diffusion Stage, Stable Diffusion and Decay Stage (also known as the mid-to-far-field stage), and Dissipation Stage [33]. Typically, the Formation Stage of a vortex is observed to extend along the wing span for approximately one wing span length. This stage is characterized by differences in lift, causing air to flow downward from around the wingtip to the lower surface, subsequently rolling upward, thereby giving rise to a rotating vortex. During this stage, the vortex exhibits high intensity and is primarily concentrated near the wingtip. Following formation, the vortex enters the Initial Diffusion Stage, spanning a range of approximately 2 to 10 times the wing span length. The vortex still preserves its rotating characteristics during this stage, albeit with diminishing strength and core spacing. Figure 9: Comparison between the results of references and numerical simulations. (**a**) Vertical displacement of the vortex core. (**b**) Horizontal displacement of the vortex core. The vortex undergoes a gradual process of stretching and thinning, elongating along the flight direction and reducing its cross-sectional area. The Stable Diffusion and Decay Stage, also referred to as the mid-to-far-field stage, sees the continued diffusion and weakening of the vortex. While rotation persists, the vortex becomes more diffuse, with a further decrease in intensity. This stage extends significantly from the aircraft and can be sustained considerably. The Dissipation Stage marks the eventual complete dissipation of the vortex, where its characteristics become indiscernible from the surrounding airflow. Figure 10, through a three-dimensional vorticity cloud map, illustrates the near-field vortex evolution of an A320 aircraft under static wind conditions as it extends along the wing by 260 m, equivalent to seven wing spans. At the one wing span position, the vortex core is already well-defined, and there is an apparent trend of separation between the two vortices. Within the two to seven wing span region, the attenuation in vorticity and vortex core spacing is evident. Figure 11 shows the vorticity contours at different distances behind the wing under different vertical wind speeds. The statistics show that the airflow will flow from the high-pressure region to the low-pressure area, forming a vortex structure extending backward from the tip of the wing. The vorticity is highest at the vortex core, gradually decreasing outward from the center. The vortex core radius also enlarges with increasing distance from the wing's trailing edge.