# 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
 \tag{9}\]  
where \(\mathrm{K_{ik}}\) is the velocity induced over the \(\mathrm{j}\)-th pivotal point by the \(\mathrm{k}-\mathrm{th}\) vortex \((1<\mathrm{j}<\mathrm{Nc})\); \(\Gamma\) is the circulation vector of nascent blob; and \(\mathrm{\mathbf{RHSV}}\) is the vortex's right side column vector with a \(\mathrm{j}-\mathrm{th}\) element given by  
\[\mathrm{RHSV(j)}=\left\{-\mathrm{Ucos}(\mathrm{th_{j}})-(\mathrm{V})\mathrm{ sin}(\mathrm{th_{j}})-\mathrm{u_{j,i}}\cos(\mathrm{th_{j}})-\mathrm{v_{j,i}} \mathrm{sin}(\mathrm{th_{j}})\right\} \tag{10}\]  
where \(\mathrm{U}\) and \(\mathrm{V}\) are the components of the undisturbed flow velocity vector; \(\mathrm{u_{j,i}}\) and \(\mathrm{v_{j,i}}\) are the components of the total velocity induced by the viscous wake (discretized and represented by \(\mathrm{i}=1,\mathrm{NV}\) vortex blobs) at the \(\mathrm{j}-\mathrm{th}\) pivotal point; and \(\mathrm{th_{j}}\) is the orientation angle of the panel \(\mathrm{j}-\mathrm{th}\). A line should be added to Equation (10) for global circulation conservation, such as  
\[\sum_{\mathrm{j}=1}^{\mathrm{Nc}}\Gamma_{\mathrm{j}}=0 \tag{11}\]  
In heat transfer, the term advection is used to represent the macroscopic motion of a fluid mass. As an analogy, each vortex blob is now advected with its own velocity, which is induced on the fluid, characterizing, thus, a purely Lagrangian description. Furthermore, for the solution of the advective part of the vorticity transport equation (Equation (4)), it is necessary to calculate the velocity field. The contributions to the total velocity of a vortex blob, \(\mathbf{u}(\mathbf{x},\mathrm{t})\), are due to the undisturbed incident flow, the solid boundaries (source panels representing both the cylinder and ground plane surfaces), and the viscous wake. To compute the vortex-vortex interaction, the velocity at any point of interest in the two-dimensional fluid domain is obtained by summing over all vortex blobs and inserting the discretized vorticity in the expression representing the Biot-Savart law. \[\mathbf{u}(\mathbf{x},\mathrm{t})=\frac{-1}{2\pi}\int\frac{(\mathbf{x}- \mathbf{x}^{\prime})\times\omega(\mathbf{x}^{\prime},\mathrm{t})\hat{\mathbf{ k}}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{2}}\mathrm{d}\mathbf{x}^{\prime} \tag{12}\]  
The eddy viscosity evaluation is performed by applying the truncated Second-Order Velocity Structure Function [56], according to the following expressions:  
\[\mathrm{v_{t}}(\mathbf{x},\ \Delta,\mathrm{t})=0.105\mathrm{C_{k}^{-3/2}} \Delta\sqrt{\overline{\mathrm{F}_{2}}(\mathbf{x},\Delta,\mathrm{t})} \tag{13}\]  
\[\overline{\mathrm{F}_{2}}(\mathbf{x},\ \mathrm{r},\mathrm{t})=\overline{\left\| \overline{\mathbf{u}}(\mathbf{x},\mathrm{t})-\overline{\mathbf{u}}(\mathbf{x }+\mathbf{r},\ \mathrm{t})\right\|^{2}_{\left\|\mathbf{r}\right\|}} \tag{14}\]  
where \(\mathrm{v_{t}}(\mathbf{x},\ \Delta,\mathrm{t})\) is the eddy viscosity in the physical space; \(\mathrm{C_{k}}=1.4\) is the Kolmogorov constant; \(\overline{\mathrm{F}_{2}}\) is the Second-Order Velocity Structure Function; \(\overline{\mathbf{u}}(\mathbf{x},\mathrm{t})\) is the velocity at the point where turbulent activity must be defined; and \(\overline{\mathbf{u}}(\mathbf{x}+\mathbf{r},\mathrm{t})\) is the velocity in the vicinity of that point, with distance \(\mathbf{r}\) between each other. In Eulerian approaches, the simplest way to estimate the spatial average is with respect to the six points to the position point \(\mathbf{x}\)[57] for three-dimensional simulations and the four points in two-dimensional simulations. For the present Lagrangian approach, there is a convenient technique proposed by [PERSON] et al.