# Effect of Surface Roughness on Aerodynamic Loads of Bluff Body in Vicinity of Smoothed Moving Wall
## 2 Mathematical Formulation
### Computational Arrangement  
The essence of the vortex method is the generation of vorticity blobs on the surface of the body and the subsequent advective and diffusive transport of these to the fluid domain (Figure 1). The discretized vorticity field is governed by the following expression [52]:  
\[\mathbf{\omega}(\mathbf{x},\mathbf{t})\approx\mathbf{\omega}^{\mathrm{h}}(\mathbf{x}, \mathbf{t})=\sum_{\mathrm{i=1}}^{\mathrm{Nv}}\mathbf{\Gamma}_{\mathrm{i}}(\mathbf{t })\varsigma_{\sigma\mathrm{i}}(\mathbf{x}-\mathbf{x}_{\mathrm{i}}(\mathbf{t})) \tag{6}\]  
where \(\mathbf{x}_{\mathrm{i}}\) is the position of i-th vortex blob; \(\mathbf{\Gamma}_{\mathrm{i}}\) is circulation strength; \(\varsigma_{\sigma\mathrm{i}}\) is the cutoff function; and \(\mathrm{Nv}\) is the number of discrete vortices. In taking \(\sigma\) to be the core size and k, the cut-off width, the cut-off function is inserted as a Gaussian distribution in the two-dimensional domain [53]:  
\[\varsigma_{\sigma\mathrm{i}}(\mathbf{x})=\frac{1}{\mathrm{k}\,\pi\,\sigma^{2}} \exp\left(\frac{-|\mathbf{x}|^{2}}{\mathrm{k}\,\sigma^{2}}\right) \tag{7}\]  
In the present paper, the cylinder surface and the ground plane (with the moving wall effect) are discretized and computationally represented by flat panels of length \(\Delta\mathrm{sp}\), over which sources with uniform density are distributed [54]. The number of panels of the cylinder's representative geometry, Nc, depends on the surface type, smooth or rough. The moving horizontal surface is divided into M modules, and each module is subdivided into Npm panels, totaling \(\mathrm{Np}=\mathrm{M.Npm}\) rectilinear elements were used to represent this solid boundary. The modules have a unit length represented by \(\Delta\mathrm{S}\). In this insertion, the relative position of the stagnation point of the cylinder, \(\mathrm{x}=0\) and \(\mathrm{y}=0\), coincides with the absolute position \(\mathrm{X}=3\mathrm{M}\) and \(\mathrm{Y}=(\mathrm{G}/\mathrm{D}+\mathrm{D}/2)\), according to the model set out in Figure 1. G/D is the vertical distance between the lowest point of the cylinder surface and the horizontal wall (gap-to-diameter ratio). Vortex blobs are conveniently used for the discretization of the vorticity field present in the fluid domain. The intensity of each nascent blob is obtained in order to guarantee the no-slip condition over the strategic points chosen on the solid boundary of the cylinder. These strategic points, called control points, are located at the midpoint of a surface segment. Singularities of type sources are generated at each time instant to guarantee the impenetrability condition (the Neumann boundary condition) on the control points of both boundaries, the cylinder and moving wall. Therefore, the boundary conditions for the problem are related to the solid borders--cylinder (S\({}_{1}\)) and moving horizontal wall (S\({}_{2}\))--and the boundary at great distances from  
Figure 1: Sketch of the computational fluid–structure model. the bodies (\(\mathbb{S}_{\infty}\)), namely, the no-slip condition on the cylinder surface is \(\mathbf{u}_{\tau}=\mathbf{u}\cdot\mathbf{e}_{\tau}=0\) in \(\mathbb{S}_{1}\); the impenetrability condition on the cylinder surface and moving wall, \(\mathbf{u}_{n}=\mathbf{u}\cdot\mathbf{e}_{n}=0\) in \(\mathbb{S}_{1}\) and \(\mathbb{S}_{2}\); and the undisturbed flow condition at large distances from the cylinder and on the moving wall, \(|\mathbf{u}|\to\mathrm{U}\) in \(\mathbb{S}_{\infty}\) and \(\mathbb{S}_{2}\). The undisturbed flow incident on the cylinder, \(\mathrm{U}\), is governed by the Reynolds number given by the relation \(\mathrm{Re}=\mathrm{UD}/v\). \(\Omega=\mathbb{S}_{1}\cup\mathbb{S}_{2}\cup\mathbb{S}_{\infty}\) is the semi-infinite fluid domain. The moving wall can be positioned distant or close to the cylinder. The first option is so as not to interfere with the vortex shedding in the cylinder.