## 2 Mathematical Formulation  
Figure 1 illustrates a typical situation of a Fluid-Structure Interaction (FSI) problem. It can be identified by the presence of an incident flow \(U_{\infty}\) with constant temperature \(T_{\infty}\), a solid boundary heated surface with constant temperature \(T_{w}>T_{\infty}\), defined as \(S_{w}\), counter-rotating vortical structures simulated by superposition of computational points (in blue), and a far-away boundary defined by \(S_{\infty}\). Thus, the fluid domain \(\Omega\) is delimited by boundary \(S=S_{w}\cup S_{\infty}\). Furthermore, the presence of a thermal gradient \(T_{w}-T_{\infty}\) will transfer heat from the surface \(S_{w}\) to the fluid very close to it; this phenomenon will result in heat generation and, in consequence, a modification in the dynamics of the counter-rotating vortical structures. The essence of the Temperature Particles Method (TPM) is to discretize the vortical structures using particles (computational points) known as Lamb discrete vortices, and the heat by using temperature particles [1; 9; 11]. The flow is assumed to be two-dimensional, incompressible and unsteady, and the fluid to be Newtonian with moderate variation in the fluid density. The latter assumption is associated with the Boussinesq approximation [11]. Taking the curl of the Navier-Stokes equations supported by the continuity equation, the vorticity transport equation for the filtered velocity field \(\overline{u}\) can be obtained [19; 37] as follows:  
\[\frac{D\overline{\omega}}{Dt}=\frac{\partial\overline{\omega}}{\partial t}+ \frac{\partial\overline{u}_{j}\overline{\omega}}{\partial x_{j}}=\left(\frac{ 1}{Re}+\
u_{T}\right)\frac{\partial^{2}\overline{\omega}}{\partial x_{j}x_{j} }+Ri\frac{\partial\overline{\theta}}{\partial x_{j}}, \tag{1}\]  
where \(D/Dt\) is related to the Lagrangian derivative, \(\
abla\times\overline{u}=\overline{\omega}\), \(\
u_{T}\) is the turbulent viscosity coefficient. Furthermore, the Reynolds number, Richardson number and dimensionless temperature are defined, respectively, in the following manner:  
\[Re=\frac{U_{\infty}D}{\
u}=\frac{\Gamma}{\
u}, \tag{2}\]  
Figure 1: Example of problem geometry and important definitions. where \(D\) is the cylinder diameter and \(\Gamma\) is the circulation strength [11; 35]. \[Ri=\frac{Gr}{Re^{2}},\text{ being }Gr=\frac{\beta g(T_{W}-T_{\infty}b^{3})}{\
u^{2}} \tag{3}\]  
the Grashof number [11],  
\[\overline{\theta}=\frac{\overline{T}-T_{\infty}}{T_{W}-T_{\infty}}. \tag{4}\]  
The energy equation is defined as  
\[\frac{D\overline{\theta}}{Dt}=\frac{\partial\overline{\theta}}{\partial t}+ \frac{\partial\overline{u}_{j}\overline{\theta}}{\partial x_{j}}=\frac{ \partial}{\partial x_{j}}\Bigg{[}\bigg{(}\frac{1}{RePr}+\frac{\
u_{T}}{RePr_{ T}}\bigg{)}\frac{\partial\overline{\theta}}{\partial x_{j}}\Bigg{]}, \tag{5}\]  
where \(D/Dt\) is also related to the Lagrangian derivative, \(\alpha_{T}\) is the turbulent thermal diffusivity coefficient, and the Prandtl number and the turbulent Prandtl number are defined, respectively, as follows:  
\[Pr=\frac{\
u}{\alpha}, \tag{6}\]  
\[Pr_{T}=\frac{\
u_{T}(t)}{\alpha_{T}(t)}, \tag{7}\]  
where \(\
u\) is the kinematics viscosity, \(\alpha\) is the thermal diffusivity coefficient and \(0.7<Pr_{T}<0.95\)[18; 38]. The turbulent viscosity coefficient is defined in the following manner [18; 19]:  
\[\
u_{T}(\mathbf{x},\triangle,t)=0.105\text{C}_{k}^{-3/2}\triangle\sqrt{\overline {F}_{2}(\mathbf{x},\triangle,\mathbf{t})}, \tag{8}\]  
where \(C_{k}=1.4\) is the Kolmogorov constant [39] and the second-order velocity structure function is established as folllows [18; 19]:  
\[\overline{F}_{2}(\mathbf{x},\triangle,t)=\|\overline{\mathbf{\overline{u}}(\mathbf{x},t) -\overline{u}(\mathbf{x}+\mathbf{r},t)}\|_{\|\mathbf{r}\|=\triangle}^{2}. \tag{9}\]  
In Equation (9), the vorticity field induces velocity over every Lamb discrete vortex to compute the turbulent viscosity coefficient.