We present the modeling of the main facets of turbulence diffusion, i.e., diffusion of momentum, mass, density, and heat, within the smoothed particle hydrodynamics (SPH) method. The treatment is developed considering the large eddy simulation (LES) approach and is specifically founded on the δ-LES-SPH [A. Di Mascio et al., Phys. Fluids 29, 035102 (2017)], a model characterized by a turbulence closure for the continuity equation. The novelties introduced are the modeling of the advection–diffusion equation through turbulent mass diffusivity and the modeling of the internal energy equation through heat eddy diffusivity. Moreover, a calibration for the closure term of the continuity equation is also proposed, based on the physical assumption of equivalence between turbulent mass and density diffusion rates. Three test cases are investigated. The first test regards a two-dimensional (2D) problem with splashing and wave-breaking dynamics, which is used to investigate the proposed calibration for the turbulent density diffusion term. In the second test, a 2D jet in coflow condition without gravity is studied with particular emphasis on the advection–diffusion process. The last test regards the most general condition and reproduces three-dimensional (3D) jets in crossflow conditions, in which attention is given to both the mass and heat advection–diffusion processes. The proposed methodology, which allowed us to accurately reproduce the experimental tests considered, represents a promising approach for future investigation of problems characterized by complex dynamics with turbulence and mixing involved.
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