# Effect of Surface Roughness on Aerodynamic Loads of Bluff Body in Vicinity of Smoothed Moving Wall
## 2 Mathematical Formulation
### The Discrete Vortex Method with Roughness and Turbulence Closure Models  
In order to impose a refined boundary layer condition, the roughness model developed by [PERSON] and [PERSON] was employed [PERSON] and [PERSON] (2015), i.e., the Lagrangian Dynamic Roughness Model--MLDR\({}_{\mathrm{VL}}\). This technique consists of the construction of a cylinder with rough geometry represented by the distribution of isosceles triangular elements of height \(\varepsilon\) (Figure 2) concomitant with the introduction of a small-scale perturbation in the nascent blobs for an adequate induction of the change in the momentum in the flow adjacent to the cylinder wall. The no-slip condition at the wall and the induction of a velocity profile in the normal direction of a discretizing panel result in a mesh-free boundary layer model with no wall function. The roughness model is not applied on the ground plane with a moving wall effect. The i-th nascent blob is initially generated with a circulation, \(\Gamma_{\mathrm{i}}\), and allocated with a specific distance, \(\sigma_{\mathrm{o}}\), in the normal direction, \(\mathbf{e}_{n}\), from the control point of its reference panel. \(\sigma_{\mathrm{o}}\) is the core size, which is uniform for conventional vortex methods, and set to be \(\sigma_{\mathrm{o}}=\sqrt{2\mathrm{vt}}\)Zhu and Zhu (2016). In a second stage, in the same computational time interval, the core size is increased by a value, \(\mathrm{k}_{\mathrm{p}}\
u_{\mathrm{t}}\), governed by the order of magnitude of the local effects to trigger the convenient modification of the intensity (\(\Gamma_{\mathrm{i}}\pm\Delta\Gamma_{\mathrm{i}}\)) and consequently change the momentum associated with this i-th element. \(\mathrm{k}_{\mathrm{p}}\) is an adjustable proportionality constant for the particularities of the flow. This was set at \(\mathrm{k}_{\mathrm{p}}=1\) according to [PERSON] and [PERSON] (2015). In the \(\mathrm{MLDR}_{\mathrm{VL}}\), the velocity induction of a vortex blob with a rough effect can be expressed as  
\[\mathrm{U}_{\theta}=\frac{\Gamma}{2\pi\mathrm{r}}\bigg{[}1-\exp\bigg{(}-\frac{ \mathrm{r}^{2}}{\mathrm{k}\sigma_{\mathrm{r}}{}^{2}}\bigg{)}\bigg{]} \tag{8}\]  
where \(\sigma_{\mathrm{r}}=\sqrt{2(\mathrm{v}+\
u_{\mathrm{t}})}\mathrm{t}\) is the core size with a rough effect. Therefore, each nascent vortex blob has a different core size at birth and this remains constant in the simulation. Figure 2: Cylinders with different numbers of rough elements. A linear algebraic equation for the vorticity generation using vortex blobs is set up to guarantee the no-slip boundary condition and can be expressed as  
\[ \begin{bmatrix}\mathrm{K_{11}}&\dots\mathrm{K_{ij}}&\dots\mathrm{K_{1k}}&\dots \mathrm{K_{1 Nc}}\\ &\dots&&\\ \mathrm{K_{jl}}&\dots\mathrm{K_{jj}}&\dots\mathrm{K_{jk}}&\dots\mathrm{K_{jlNc} }\\ &\dots&&\\ \mathrm{K_{Ncl}}&\dots\mathrm{K_{Ncl}}&\dots\mathrm{K_{Nckc}}&\dots\mathrm{K_{Nckc }}\end{bmatrix}\begin{Bmatrix}\Gamma_{1}\\ \dots\\ \Gamma_{\mathrm{j}}\\ \dots\\ \Gamma_{\mathrm{Nc}}\end{Bmatrix} = \begin{Bmatrix}\mathrm{RHV_{1}}\\ \dots\\ \mathrm{RHV_{j}}\\ \dots\\ \mathrm{RHV_{Nc}}\end{Bmatrix}