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

Abstract

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. † Author to whom correspondence should be addressed: Josette.Bellan@jpl.nasa.gov IntroductionThe accurate modelling of two-phase (TP) turbulent flows has been a long-standing problem that is so far unresolved. In these flows, the interaction of the phases through momentum, energy and sometimes mass transfer, determines...

Explicit filtering to obtain grid-spacing-independent and discretization-order-independent large-eddy simulation of two-phase volumetrically dilute flow with evaporation

“…As in Okong'o & Bellan (2004), we define the vector of gas-phase conservative variables φ = {ρ, ρu i , ρe t , ρY V } and denote the flow field as φ, where ρ is the density, u i is the velocity in the x i coordinate direction, e t is the total energy and Y V is the vapour (subscript V) mass fraction (the carrier gas, subscript C, mass fraction is Y C ; Y C + Y V = 1). The gas-phase conservation equations are:…”

“…In the above equations (3.2)-(3.5), it is assumed that f (φ) can be replaced by f φ for terms such as pressure, viscous stresses and heat conduction, and that the error due to this assumption is negligible (see Okong'o & Bellan 2004). Quantities τ ij , ζ j and η j in (3.2)-(3.5) are the SGS terms that represent the effect of the unresolved component on the resolved component of the flow field 6) whereh =ẽ +p/ρ.…”

Large-eddy simulation of particle-laden turbulent flows

“…An aspect of primary importance in dense spray flows is the influence of the droplet dynamics on the dissipation rate. Apart for the a priori analysis of Okong'o & Bellan (2004), it appears that there has been only one LES (Yuu, Ueno & Umekage 2001) in which the modelled SGS stress tensor has been modified to account for dispersed phase effects on the subgrid scale turbulence.…”

“…The best known expression is that given by Maxey & Riley (1983) for an isolated particle in a Stokes flow. This is the expression invariably adopted in deterministic DNS studies in which the entire gas-phase turbulent fluctuations are resolved and various interpolation schemes are used for the calculation of the gas velocity at the particle positions (Ferrante & Elghobashi 2003;Okong'o & Bellan 2004). In the context of LES, the gas-phase velocity is not completely known and if the force were to be evaluated solely in terms of the resolved velocity field, the effect of the unresolved fluctuations upon the particle motion would be omitted.…”

“…In the context of LES, the gas-phase velocity is not completely known and if the force were to be evaluated solely in terms of the resolved velocity field, the effect of the unresolved fluctuations upon the particle motion would be omitted. In an LES of a dilute spray, the magnitude of any error associated with the use of an unmodified single phase SGS model is likely to be much smaller than that due to an inadequate representation of the dispersion effects of the unresolved gas fluctuations on the liquid dispersed phase, as is shown by Okong'o & Bellan (2004). However, quoting Mashayek & Pandya (2003), 'the above conclusion is despite the fact that the majority of the LES studies tend to (conveniently) neglect the subgrid velocity fluctuations'.…”