## 1 Introduction  
Turbulence is a very frequent phenomenon in almost all engineering flows. For flows with a low Reynolds number, the viscous part (dissipative) strongly acts to stabilize such flows. However, an increase in Reynolds number disorganizes the flows, making them turbulent [1]. As an example of the complexity of turbulent phenomena, [PERSON] and [PERSON] [2] examined the substantial changes in turbulent flow structures and flow characteristics around a NACA 0012 airfoil using two-dimensional PIV measurements and concluded that the vortical interactions and consequent fluctuations in the near wake of airfoil were gradually intensified with an increasing Reynolds number for zero angle of attack (\(\lambda=0^{\circ}\)), whereas the reverse was the case for \(\lambda=5^{\circ}\). So, the crucial issue presented to the scientific community is to solve the flows when they are turbulent, and with such purpose, new models are frequently proposed in the literature [3]. The Navier-Stokes equations have existed since 1822, and it is necessary to handle different flow scales to solve the equations. In other words, it is mandatory to understand the vortical structure behavior for all frequencies and scale lengths. Nowadays, it is still impractical to solve the Navier-Stokes equations with that prism. The turbulence has been additionally studied for more than eight decades, and there is no easy/general solution to solve turbulent flows; as a consequence, lots of turbulence models have been developed by researchers in different study centers around the world [4]. A spectrum with all frequencies and scale lengths of the vortical structures into the flow field is available in large-eddy simulation (LES). The large scales can be directly solved in a typical direct numerical simulation (DNS), and the sub-grid scales must be modeled using a turbulence model. The use of LES theory blended with the second-order velocity structure function model is a tool that enables the development of Lagrangian vortex methods with roughness models. In this context, our research group reported in the literature that a two-dimensional surface roughness model is more sensitive to capture nonlinear hydrodynamic phenomena than a simple turbulence model [5; 6; 7; 8]. That research line is effective and produces interesting numerical results to be applicable in engineering conservative designs. Lagrangian vortex methods are a very interesting alternative to simulate unsteady and vortical flows because the technique provides a direct simulation of the real vortical structures in the most economical manner, being applicable to a wide range of industry problems [9; 10]. The vorticity created at solid boundaries is discretized and represented by computation points (vortex elements or vortex blobs), which are strategically positioned only where there is vorticity in the fluid domain. The cluster of vortex blobs must be submitted to the advection and diffusion mechanisms. Also, the Lagrangian description for the vorticity field does not require the use of mesh, and the boundary conditions far from the solid boundaries are automatically satisfied. The real history of the \"Vortex Methods\" began with the modelling of inviscid potential flows as presented by [PERSON] [11]. [PERSON] [12] and [PERSON] [13] contributed to the modern vortex method in terms of a time-stepping scheme to compute viscous effects and inviscid parts of the governing equations as successive sub-steps. [PERSON] [12] introduced the idea of the \"fractional step\" method or \"viscous splitting\" (a particular case of the general approach of \"operator splitting\") blended with the random walk diffusion scheme, the latter a Brownian-like motion of the vortex blobs. [PERSON] [13] introduced the core-spreading algorithm (CSA) in a purely Lagrangian manner to take into account viscous effects by changing the core size of the vortex blobs to exactly solve the diffusion equation.