Consistent large-eddy simulation of a temporal mixing layer laden with evaporating drops. Part 1. Direct numerical simulation, formulation and a priori analysis

Large-eddy simulation (LES) models are presented and evaluated on a database obtained from direct numerical simulation (DNS) of a three-dimensional temporal mixing layer with evaporating drops. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, momentum and energy source terms. The DNS database consists of transitional states attained by layers with different initial Reynolds numbers and initial liquid-phase mass loadings. Budgets of the LES equations at the transitional states show that, for the mass loadings considered, the filtered source terms (FSTs) are smaller than the resolved inviscid terms and some subgrid scale (SGS) terms, but larger than the resolved viscous stress, heat flux and mass flux terms. The irreversible entropy production (i.e. the dissipation) expression for a two-phase flow with phase change is derived, showing that the dissipation contains contributions due to viscous stresses, heat and species-mass fluxes, and source terms. For both the DNS and filtered flow fields at transition, the two leading contributions are found to be the dissipation due to the energy source term and that due to the chemical potential of the mass source. Therefore, the modelling effort is focused on both the SGS fluxes and the FSTs in the LES equations. The FST models considered are applicable to LES in which the grid is coarser than the DNS grid and, consistently, ‘computational’ drops represent the DNS physical drops. Because the unfiltered flow field is required for the computation of the source terms, but would not be available in LES, it was approximated using the filtered flow field or the filtered flow field augmented by corrections based on the SGS variances. All of the FST models were found to overestimate DNS-field FSTs, with the relative error of modelling the unfiltered flow field compared to the error of using computational drops showing a complex dependence on filter width and number of computational drops. For modelling the SGS fluxes and (where possible) SGS variances, constant-coefficient Smagorinsky, gradient and scale-similarity models were assessed on the DNS database, and calibrated coefficients were statistically equivalent when computed on single-phase or two-phase flows. The gradient and scale-similarity models showed excellent correlation with the SGS quantities. An a posteriori study is proposed to evaluate the impact of the studied models on the flow-field development, so as to definitively assess their suitability for LES with evaporating drops