## 3 Theory and Numerical Method
### Numerical Method  
[PERSON] [12] proposed the \"viscous splitting algorithm\" for the Navier-Stokes equations that consists of sub-time-steps to successively compute the effects of advection and diffusion. Equation (1) is the well-known version of the Navier-Stokes equations with no pressure term. It is obtained by starting with the definition of vorticity (\(\overline{\omega}=\
abla\times\overline{u}\)) and applying the curl operator on Navier-Stokes equations (still assisted by the continuity equation). The Lagrangian version of Equation (1) is written as:  
\[\rm\frac{D\overline{\omega}}{Dt}=\bigg{(}\frac{1}{Re}+v_{t}\bigg{)}\
abla^{2} \overline{\omega} \tag{5}\]  
The vorticity advection problem is then governed by [12]:  
\[\rm\frac{D\overline{\omega}}{Dt}=0 \tag{6}\]  
and the viscosity is incorporated into vortex simulations as follows:  
\[\rm\frac{\partial\overline{\omega}}{\partial t}=\bigg{(}\frac{1}{Re}+v_{t} \bigg{)}\
abla^{2}\overline{\omega} \tag{7}\]  
The vorticity created by the slip flow at the ground plane is then shed freely into the fluid. Figure 2a exemplifies the instantaneous creation and shedding of the second vortex blob along the ground plane. It is important to emphasize that M vortex blobs are shedding at every time-stepping of a typical numerical simulation. The centre of every nascent vortex blob (viscous core) is exactly located at the \"shedding point\". Figure 2: Example of instantaneous shedding of a vortex blob from the second flat panel: (**a**) with no roughness model; (**b**) momentum injection model; and (**c**) with roughness model. Figure 2b illustrates the key idea of the model to inject momentum during vorticity shedding from a flat panel. The practical effect is obtained in three steps, whose purpose is to change the viscous core, \(\sigma_{0}\), and the vortex strength, \(\Gamma_{\rm v}\), of every nascent vortex blob. Firstly, the vortex strength is obtained by solving a linear system of algebraic equations to ensure the non-slip condition on every \"pivotal point\". The global circulation conservation (_[PERSON]'s theorem_[33]) must also be satisfied. Secondly, a semicircle of radius \(\|\mathbf{b}\|=2\varepsilon-\sigma_{0}\), centered on a \"shedding point\", is required to capture turbulent activities nearest a ground plane, where \(\varepsilon\equiv\varepsilon/\Delta\mathrm{s}\) simulates the interference of a selected relative roughness height [7]. [PERSON] et al. [7] adapted Equation (4), aiming to capture the velocity differences among the center of the i-th \"shedding point\" and NR = 21 \"rough points\", as shown in Figure 2b (see also in Figure 1), where:  
\[\overline{\mathrm{F}}_{2_{\mathrm{i}}}(\mathbf{t})=\frac{1}{\mathrm{NR}}\sum _{\mathbf{w}=1}^{\mathrm{NR}}\|\overline{\mathbf{u}}_{\mathrm{i}}(\mathbf{x} _{\mathrm{i}},\mathbf{t})-\overline{\mathbf{u}}_{\mathrm{tw}}(\mathbf{x}_{ \mathrm{i}}+\mathbf{b},\mathbf{t})\|_{\mathrm{w}}^{2}(1+\varepsilon) \tag{8}\]  
and \((1+\varepsilon)\) represents the kinetic energy gain because of surface roughness effects. [PERSON] et al. [8] concluded that a set of NR = 21 \"rough points\" on each semicircle is sufficient to compute the average velocity differences necessary to evaluate the second-order velocity structure function of the filtered field, \(\overline{\mathbf{u}}\). Thus, in response to Equation (8), the viscous core must be corrected accordingly to yield [7]:  
\[\sigma_{0_{\mathrm{G}_{\mathrm{i}}}}(\mathbf{t})=2\sqrt{\frac{\mathrm{At}}{ \mathrm{Re}}\left(1+\frac{\upnu_{\mathrm{i}}(\mathbf{t})}{\upnu}\right)} \tag{9}\]  
Thirdly, the vortex strength is also corrected by again solving the linear system of algebraic equations to ensure the non-slip condition on every \"pivotal point\". Figure 2c illustrates the effective creation and shedding of the second vortex blob with the surface roughness model. The roughness model can be physically interpreted as a new momentum source term for the Navier-Stokes equations because of additional inertial effects nearest the rough ground plane.