ELSA Model for Atomization: To Benefit of the Eulerian and Lagrangian Descriptions of the Liquid Phase

In the context of the Euler-Lagrange Spray Atomization (ELSA) model [1,2,3] for two-phase flows, a Eulerian approach is used to describe the dense part of the spray and it is completed elsewhere by a standard Lagrangian approach. A recurrent issue of the Lagrangian approach is the difficulty to reach a number of stochastic samples large enough, in each mesh cell, to achieve statistical convergence. It is necessary to propose methods that can benefit of the Lagrangian formulation to describe the statistical dispersion but that are converged for certain key quantities like the mass of liquid. Such ideas are already developed under the so-called Direct Quadrature Method Of Moments (DQMOM) [4]. This complex method is not directly used here, a more practical approach that used the usual formalism of Lagrangian methods for spray is proposed. The method will take into account the originally unused Eulerian equation in the diluted part of the spray to transport the key mean quantities of the spray. In each mesh cell that contains Lagrangian stochastic particles the liquid mass fraction can be obtained from the Eulerian equation. Equivalently, in the mesh cell considered, a Lagrangian liquid mass fraction is defined. It is clear that big statistical fluctuations of this variable can be expected if the number of particles in the cell is not high enough. The least noisy Eulerian variable will be used to correct the Lagrangian one. There are numerous possible ways to correct the Lagrangian variable, one method will be presented. In the same manner, the Eulerian equation for the density of liquid surface is used to correct the Sauter mean diameter that can be obtained from the Lagrangian description. Hence, the Lagrangian phase is linked to the Eulerian equations. But the Eulerian equations have to be linked also to the Lagrangian phase, this is achieved through the liquid turbulent diffusion flux closure and through the source terms of surface density transport equation, i.e. those modeling break-up and coalescence effects. By this way, the Lagrangian phase, whenever it is available, enables to take into account the joint probability density functions of fluctuating variables such as droplet diameter or droplet velocities.