## 5 Conclusions  
The vortex-temperature interactions are the main characteristic of the Temperature Particles Method (TPM), a natural extension of the Discrete Vortex Method (DVM) in the Lagrangian manner. In the TPM, two clouds particles are used in order to represent the vorticity and heat, which are generated on the solid surface and develop into the boundary layer and the viscous wake. Every vortex blob and every temperature particle are followed during the numerical simulation in a typical Lagrangian scheme. Important features of the (Lagrangian) TPM are as follows: (i) it is a numerical technique suitable for the solution of advective/diffusion type equations like the Navier-Stokes equations and the energy equation; (ii) it is a mesh-free technique, where the vorticity field and the heat are respectively represented by a cloud of vortex blobs and by another cloud of temperature particles, both moving with the fluid velocity; (iii) the vortex blobs generated at the body surface are treated as material particles carrying vorticity; (iv) the temperature particles generated at the body surface are treated as material particles carrying heat; (v) it is a suitable technique for direct simulation and large-eddy simulation; and (vi) the turbulence modeling employs the computation of the second-order velocity structure function of the velocity filtered field; this computation is advantageous in the Lagrangian manner, since it utilizes the concept of velocity fluctuations (differences of velocity) instead of the rate of deformation (derivatives). Another advantage of working with the vorticity transport equation and the energy equation in Lagrangian fashion is that the pressure term is eliminated, which requires special treatment in most numerical methods. In this paper, the computation of the second-order velocity structure function model has been successively extended for the energy equation, aiming to include turbulence modeling. This calculation was initially adapted for construction of the two-dimensional vorticity field via DVM [19]. The key idea of the present methodology is to compute the turbulent viscosity coefficient over each temperature particle and relate it to the turbulent thermal diffusivity coefficient via definition of the turbulent Prandtl number, as defined in Equation (7). Two examples of flow configuration have been utilized to show the effectiveness of the new TPM. Firstly, the inclusion of buoyancy forces linked with turbulence modeling has somehow interfered with the aerodynamics of the circular cylinder. The present numerical results for Reynolds number of 100,000 and [PERSON] number of 0.168 indicated a drag coefficient of 1.21, Strouhal number of 0.199 and separation angle of 81\({}^{\circ}\). For the isothermal condition, the experimental results from [PERSON] [43], for a Reynolds number of 100,000, predict a drag coefficient of 1.20\({}^{\pm 10\%}\) and a Strouhal number of 0.19\({}^{\pm 10\%}\). The experimental result from [PERSON] and [PERSON] [44] predicts the separation angle for the hydrodynamic boundary layer to be about 78\({}^{\circ}\). This behavior tends to reflect the expected physics for the problem. In other words, the inclusion of buoyancy forces, as a result of the heating of the body, indicates the creation of a contraposition with the viscous forces around it, and, consequently, there is a delay in the boundary layer separation angle. As suggested by [PERSON] et al. [42], the effects of surface roughness are more sensitive for capturing the mentioned delay in two dimensions. Secondly, the combined effects of turbulence modeling, buoyancy forces and crosswind for the problem of wingtip vortices produced by a wing near a heated ground plane modified the expected trajectory for the pair of counter-rotating primary vortical structures. These vortical structures became closer during the numerical simulation in comparison with the isothermal experimental results from [PERSON] and [PERSON] [32].