# Airplane Vortices Evolution Near Ground
## 1 Introduction
On the other hand, some methods have been developed in order to reduce the strength of the vortex wake, for example, in [3] it is stated that the vortex rebound near the runway is produced by the apparition of secondary vortices induced by the boundary layer near the ground. Therefore, the objective of the study was to find a boundary condition that could reduce this phenomenon, concluding that a possibility could be to suc the vortices' boundary layer near the ground. This would avoid the apparition of the secondary vortexes, so the principal ones would have a similar behavior to the inviscid case, whose displacement is hyperbolic, meaning that they move away faster from the runway domain. Another methodology is proposed by [PERSON] et al. [4]. Their idea is to reduce the strength of the vortex by dividing the main ones into many small vortices, reducing their strength and therefore encouraging the vanishing of the vortex wake. A long set of measurements to evaluate the wake vortex behavior in ground proximity were performed by [PERSON] and [PERSON] [10]. They observed that the importance of turbulence and crosswind for wake vortex decay was weak, but light crosswind was found to be sufficient to cause pronounced asymmetric rebound. They concluded that wake rebound predictions could be significantly improved when considering the strength and time of generation of secondary vortices dependent on the crosswind's value. In a further paper [11], the impact of the meteorological parameters on vortices' evolution was experimentally investigated. A ranking of impact parameters affecting such evolution was presented, and it was observed that flow stratification was very important at cruise altitudes. The suitability of combining several independent wake vortex models to improve deterministic and probabilistic wake vortex forecast was analyzed in [PERSON] et al. [12]. In [PERSON] et al. [13], they determined for different types of aircraft and wind turbulence strengths, the limits of applicability of the radial velocity method for estimating wake vortex parameters. Recently there has been an increase in the use of CFD methods, which can give a very precise understanding of the phenomenon. However, there are two main problems associated with this methodology as stated in [14]. First, the simulations should be done in 3D, which involves large computational resources and time, and the results vary depending on the chosen turbulence model. Secondly, the simulations depend as well on the environmental conditions, meaning that to cover the whole range of possible cases it would be necessary to perform many 3D simulations. Although the second problem is understandable due to the characteristics of each case, the first problem is the main one, especially when the objective is to obtain fast results. In other CFD simulations [PERSON] et al. (2015), researchers tried to couple the wake's roll-up process with its own vortices to initialize the wake. But again, the main problem is the high computational time and computer resources required to undertake such simulations. One of the objectives of the present article is to develop a program based on potential flow theory that can produce accurate results (even though with a reasonable error) and needs less computational time when compared to existing CFD methods. The paper is outlined as follows: the mathematical process will start from the known complex potential equation of a vortex and its image. Then, by simplifying it and separating the real and the imaginary parts, the velocity potential equation can be obtained. Afterwards, from this equation, the temporal variation and the velocity field can be found. Finally, with the Bernoulli equation for potential flow, the pressure distribution across the entire domain will be obtained. The validation of the model and the results section are presented next, and the paper ends with the conclusions.