# 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
[11]: the vortex blobs, with velocities that are considered in the Second-Order Velocity Structure Function, must be properly located in a circular crown bounded by \(\mathrm{r}_{\mathrm{int}}=0.1\sigma\) and \(\mathrm{r}_{\mathrm{ext}}=\mathrm{sm}\ \sigma\), which are, respectively, the inner and outer radii around the \(\mathrm{i}-\mathrm{th}\) vortex with core size \(\sigma\) and turbulent activity to be defined, as shown in Figure 3. The insertion of the inner radius is to eliminate the singularity in the velocity equation (Equation (8)). The turbulent viscosity is then defined locally and dynamically during the simulation. On one hand, the vortex cloud advection is solved by applying the First-Order Advance Scheme presented by [PERSON] [37], and, on the other hand, the molecular diffusion of the vorticity field is computed through a probabilistic technique known as the Random Advance Method [51]:  
\[\mathbf{x}(\mathrm{t}+\Delta\mathrm{t})=\mathbf{x}(\mathrm{t})+\mathbf{u}( \mathbf{x},\mathrm{t})\Delta\mathrm{t}+\mathbf{Z_{d}} \tag{15}\]  
where \(\mathbf{Z_{d}}\equiv(\mathrm{x_{d}},\mathrm{y_{d}})\) is the contribution of turbulent diffusion, and the components, \(\mathrm{x_{d}}\) and \(\mathrm{y_{d}}\), are defined in the following manner:  
\[\left\{ \begin{array}{l}\mathrm{x_{d}}=\sqrt{2\mathrm{k}\Delta\mathrm{t} \Big{(}\frac{1}{\mathrm{Re}}+\mathrm{v_{t}}\Big{)}\mathrm{ln}\Big{(}\frac{1}{ p}\Big{)}}[\cos(2\pi\mathrm{Q})]\\ \mathrm{y_{d}}=\sqrt{2\mathrm{k}\Delta\mathrm{t}\Big{(}\frac{1}{\mathrm{Re}}+ \mathrm{v_{t}}\Big{)}\mathrm{ln}\Big{(}\frac{1}{p}\Big{)}}[\sin(2\pi\mathrm{Q })]\end{array} \right. \tag{16}\]  
Here, P and Q are random numbers in the range between 0 and 1. In the form of Equation (16), the diffusion problem is approximately resolved as in several other random vortex methods that require a modification of the intensity of the vorticity blobs or base functions [53]. Accidentally, some vortex blobs can migrate to the interior of the polygon discretizing the cylinder or to the inner half-plane of the moving wall, because of the random diffusion simulation, violating the impermeability boundary condition. In this work it was opted to use the method of reflecting vortex blobs out of the cylinder or out of the semi-plane of the horizontal wall. Consequently, in the next time step, the new circulation inserted in the fluid domain to restore the no-slip condition is redistributed to the nascent vortex blobs, maintaining the criterion of total circulation conservation. Much of the literature suggests that vortex blobs entering the body are eliminated or reflected by mirroring the nearest panel [40]. Because of the rough geometry, the vortex blobs are reflected considering the closest reference panel and are randomly allocated on the surface of the roughness element of that base panel. It is noteworthy that the distribution of vortex blobs between the rough elements, although it requires a careful method of reflection, contributes to an accurate boundary layer condition [15]. Figure 3: Spatial average velocity model for computing the turbulent activity of a vorticity element. In this framework, large scales are solved through the interaction of the discrete vortex, and small scales can be defined by introducing eddy viscosity. The distributed aerodynamic loads (static pressure) and the integrated aerodynamic loads (drag force, \(F_{D}\), and lift force, \(F_{L}\)) are found by applying an integral formulation originating from a Poisson equation for the pressure. Thus, the contribution of all vortex blobs instantaneously constructing the viscous wake can be taken into account [58].