The simulation of fuel sprays using the moments of the drop number size distribution
Authors: Beck J. C. , Watkins A. P. 
Published: February 2004

The modelling of fuel sprays has for the past two decades been dominated by the mixed Eulerian-Lagrangian approach in which the gas-phase equations are expressed and solved in an Eulerian framework, whereas the spray is discretized into droplet parcels whose progress through the gas is characterized by Lagrangian equations. The Eulerian-Eulerian approach employed to date is computationally much more expensive as the spray is discretized into drop size ranges, resulting in equations having to be solved over size range space as well as physical space and time. An alternative Eulerian-Eulerian methodology has recently been receiving attention. In this method the polydispersed nature of the spray is captured through the use of probability density functions based on the moments of the drop number size distribution. Thus no discretization into drop sizes is required. The work described here involves the implementation of such a spray model. Transport equations are written for the two moments which represent the liquid mass and surface area, while two more moments representing total radius and droplet number are approximated via use of a truncated presumed distribution function which is allowed to vary in space and time. The velocities used in the two transport equations are obtained by defining moment-average velocities and solving transport equations for the relevant moment-average momentum. The model is completed by an equation for the energy of the liquid phase and standard gas-phase equations, including a k-ε turbulence model and a fuel vapour transport equation for evaporating sprays. All the equations are solved in an Eulerian framework using the finite volume approach and the phases are coupled through source terms. Effects such as drop drag, droplet break-up, droplet-droplet collisions and evaporation are also modelled through the use of source terms. All the source terms are derived in terms of the four moments of the droplet size distribution to find the net effect on the whole spray flow field and surrounding gas.

In previous journal publications, the model has been qualitatively assessed by examining the predicted structures of narrow-cone, wide-angle full-cone and hollow-cone sprays, and the dependence of the results on parametric changes. Quantitative verification using experimental data has largely been confined to macro features of the sprays, such as penetration rates.

In this paper all the calculations that have been made to date relating to fuel sprays are brought together. In total, the predictions of 24 cases are compared with experimental results. The data used for these comparisons include the spray penetrations, but also include more fine-scale data such as drop sizes and, for evaporating sprays, the liquid and vapour mass fluxes.

The model is applied to a wide variety of different fuel sprays, including high-pressure diesel sprays and hollow-cone sprays under both non-evaporating and evaporating conditions. The comparisons of the results with experimental data show that the model performs well. Good agreements in the spray penetration results for a wide variety of sprays indicate that the inter-phase drag model works well. The moment-average velocity approach gives good radial distributions of droplet size, showing the capability of the model to predict polydisperse behaviour. Droplet break-up, collisions and evaporation effects are also successfully modelled.