# Multi-Phase Flow Simulation with an Eulerian approach and the Direct Quadrature based Sectional Method of Moments (DQbSMoM)

# Introduction and Motivation

The air traffic is expected to double over the period between 2005 and 2020. This rapid growth and its impact on the environment results in sharper environmental regulations on civil aviation. Therefore, it is necessary to understand the complex processes occurring inside the combustion chamber of an aircraft turbine in order to optimize the combustion process, improve fuel efficiency and reduce emissions to meet e.g. future EU targets. In combustion processes involving a liquid fuel, combustion is predominantly influenced by the spray characteristics of the injected fuel. Hence, the numerical spray simulation is essential to the combustion optimization.

The fundamental mathematical description for spray simulations is the Williams spray equation, a kinetic Boltzmann type equation that describes the evolution of the droplet number density function (NDF). On a mesoscopic level, two general approaches, the Lagrangian and the Eulerian, are considered to solve the Williams spray equation. Although the stochastic Lagrangian methods are able to fully describe the Williams spray equation in a straightforward way, their computational cost strongly depends on the droplet loading and unsteadiness of the spray system. In Eulerian methods, the balance equations of both continuous and disperse phases share the same structure, allowing an optimal parallelization of the solver and a possibly lower computational cost.

The aim of this project is the development of an Eulerian method for the numerical simulation of unsteady sprays using the Direct Quadrature based Sectional Method of Moments (DQbSMoM) to describe droplet polydispersity, occuring phenomena, such as drag force, evaporation and combustion.

# Method and Theory

Moment methods compute the evolution of the moments of the droplet number density function (NDF) with an Eulerian approach. These methods allow the description of droplet polydispersity in an Eulerian framework. The Direct Quadrature based Sectional Method of Moments (DQbSMoM) is one of theses so called moment method. The approach consists in approximating the NDF by weighted Dirac-delta functions from a set of moments. Unlike other moment methods, which solve transport equations for the moments of the NDF, the DQbSMoM solves transport equations for the abscissas and their corresponding weights. These transport equations are derived from the Williams spray equation.

The ongoing attempts are aiming towards the coupling of the DQbSMoM with the Eddy break-up model (EBU) for combustion. The EBU is based on the assumption that the reaction rate of the evaporated fuel is controlled by turbulent mixing processes because of the dependence of reaction rate on the mixing of the turbulent structures. Although it is one among the most popular combustion models, a large number of equations need to be studied which include knowledge of both spray dynamics and chemistry of reactions during combustion.

# Proceeding

This project involves the following steps:

- Coupling the DQbSMoM with a combustion model
- Verification and Validation