## 1 Introduction
In the Lagrangian manner: (a) a cloud of free particles (computational points) is used to discretize some property of interest (e.g., vorticity or heat), which can be generated from the surface of a body and develops into the boundary layer and viscous wake; (b) the absence of a mesh eliminates problems of numerical stability and need of mesh refinement related to Eulerian schemes applied for higher Reynolds number flows; (c) it is not necessary to explicitly treat advective derivatives; (d) all the calculations are restricted to the rotational flow regions and no explicit choice of the outer boundaries is needed a priori; (e) no boundary conditions are required far from the solid boundaries; and (f) the use of two-dimensional LES-based turbulence modeling is suitable to stabilize the numerical solution and provides a basis for the development of a surface roughness model. The advantage of combining the LES model and TPM is that the concept of velocity fluctuations (differences in velocity) is used instead of the rate of deformation (derivatives). In other words, the local turbulence effect is computed during simulation of the diffusion of vorticity and heat. The calculations of the local eddy viscosity coefficient and turbulent thermal diffusivity coefficient are performed on every vortex blob and on every temperature particle, respectively. These computations include the effect of the small scales through the concept of differences of velocity among vortex blobs and differences of velocity between temperature particles and vortex blobs. The support to the turbulence modeling success is that every particle needs to move with the local fluid velocity to simulate the advection processes, and the velocity induced at every particle is preliminarily computed in the same time increment of the numerical simulation. This paper presents a numerical technique based on Lagrangian TPM for Fluid-Structure Interaction analysis considering incompressible flows with mixed convection heat transfer and turbulence modeling using a Large-Eddy Simulation (LES) type. In LES, a scale separation operator subdivides the scales into filtered field scales and unresolved scales. The main characteristic of LES for simulating turbulent flows is that large eddies of the flow depend on the geometry, while smaller ones are more universal [18]. [PERSON] et al. [19] motivated by this characteristic, presented an adaptation of the computation of the second-order velocity structure function model for the DVM in two dimensions. In this strategy, for each vortex blob, during each time increment, the turbulent viscosity coefficient must be computed, which depends on the second-order velocity structure function model. In other words, that approach explicitly solves the large eddies in a calculation using the viscous splitting algorithm [7] and implicitly accounts for the small eddies by using the sub-grid-scale model (SGS model) through the computation of the turbulent viscosity coefficient over every vortex blob. As a natural extension of the work of our research group, this study examines the performance of an in-house code based on a TPM applied for flows over a heated circular cylinder and trajectories of airplane wingtip vortices in the vicinity of a heated ground plane. The main contribution of this paper is the implementation of the SGS model for the energy equation; however, a surface roughness model is not yet employed. Two examples of solid boundary representation are presented; i.e., the circular cylinder surface is discretized and represented by flat panels with sources distribution of constant density and the ground plane is represented by the method of images [20]. This strategy guarantees the impermeability condition on the pivotal point of each flat panel and on all points of the runway ground surface; additionally, the mass conservation is ensured. The generation of Lamb discrete vortices [21] near each pivotal point of a solid boundary is necessary to ensure the no-slip condition and the global circulation conservation.