## 3 Numerical Solution: The Temperature Particles Method (TPM)
### Turbulence Modeling  
The diffusive transport of the vorticity makes use of the Random-Walk Method (RWM) developed by [PERSON] [7] and adapted by [PERSON] et al. [19] to include turbulence modeling in two dimensions. The RWM calculates the vorticity diffusion process through a displacement of each vortex particle in the following form:  
\[\xi_{i}^{V}(t)=\sqrt{4\Delta t\bigg{(}\frac{1}{Re}+\
u_{T_{i}}\bigg{)}\ln \bigg{(}\frac{1}{P}\bigg{)}[\cos(2\pi Q)_{x}+\sin(2\pi Q)_{y}]}, \tag{15}\]  
where P and Q are random numbers between 0 and 1, and \(\
u_{T_{i}}\) is the turbulent viscosity coefficient evaluated over the vortex blob \(i\); in which Equation (8) assumes the following form for the discrete vortex \(i\):  
\[\
u_{T_{i}}(t)=0.105C_{k}^{-3/2}\sigma_{v_{i}\backslash}\sqrt{\overline{F}_{ 2_{i}}(t)}. \tag{16}\]The second-order velocity structure function is evaluated for the discrete vortex \(i\) as follows [19; 42]:  
\[\overline{F}_{2_{i}}=\frac{1}{N}\sum_{j=1}^{N}\|\overline{u}_{t_{i}}(x_{i})- \overline{u}_{t_{j}}(x_{i}+r_{j})\|^{2}\left(\frac{\sigma_{V_{i}}}{r_{j}}\right) ^{2/3}, \tag{17}\]  
where the index \(i\) in \(\overline{u}_{t}\) refers to the discrete vortex under analysis and the index j refers to the inducing vortex; N is the number of discrete vortices inside a circular crown defined around the discrete vortex \(i\); \(r_{j}\) defines the distance between the \(j\)-th discrete vortex and the \(i\)-th discrete vortex belonging that group of \(N\) discrete vortices. In other words, N establishes the group of discrete vortices placed inside an annulus, centered at the \(i\)-th discrete vortex under analysis; the annulus is defined by \(r_{int}=0.1\sigma_{V_{i}}\) and \(r_{ext}=4.0\sigma_{V_{i}}\), \(r_{int}\) and \(r_{ext}\) being the internal and external radius of the circular crown, respectively (see Figure 3a). For more details about the numerical implementation, please see [PERSON] et al. [42]. The term \((\sigma_{V_{i}}/r_{j})^{2/3}\) is necessary in Equation (17) because every vortex blob existing in a circular crown is not equidistant from the vortex blob under analysis. In Lagrangian fashion, the eddy viscosity coefficient related to a vortex blob is calculated with each increment of time. This paper contributes to the literature including turbulence modeling to evaluate the heat diffusion process via RWM in the following manner:  
\[\xi_{i}^{T}(t)=\sqrt{\frac{4\Delta t}{Re}\bigg{(}\frac{1}{Pr}+\frac{\pi_{T_{i}} }{Pr}\bigg{)}\ln\bigg{(}\frac{1}{P}\bigg{)}[\cos(2\pi Q)_{x}+\sin(2\pi Q)_{y}]}, \tag{18}\]  
where \(P\) and \(Q\) are new random numbers between 0 and 1; \(\pi_{T_{i}}\) is the turbulent thermal diffusivity coefficient evaluated over every temperature particle due to the vortex cloud; and the turbulent Prandtl number adopted as 0.95, Equation (7). Figure 3b schematizes the idea of the vorticity field effect over temperature particles, which contributes to increasing the CPU time too. In this context, the second-order velocity structure function is evaluated in terms of the velocities difference between a determined particle temperature i under analysis and the vortex blobs that are present inside its circular crown according to Equation (17), which must be adapted for the cluster of temperature  
Figure 3: Schematic of the spatial average velocity calculation model for the TPM (vortex blobs are in blue and temperature particles are in red). particles. This calculation is repeated throughout the computational points where the turbulent activity is more intense, that is, where each temperature particle is placed. A boxes structure is also created, aiming to divide the fluid domain in a manner such that the neighborhood of each particle (vortex blob or temperature particle) under analysis can be taken into account directly in Equation (17) (see Figure 4). This strategy considerably reduces the CPU time, since the identification of which particle will be inside each circular crown is more efficient, rather than looking for the set of vortex blobs in the entire fluid domain.