# Research Topic

Multi-Phase Flow Simulation with an Eulerian approach and the Direct Quadrature Method of Moments (DQMOM)

# 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 detailed 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 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. However, available methods for the numerical simulation of strongly unsteady fuel sprays are not yet able to account for the influential occurring phenomena with a satisfactory accuracy and a practicable computational cost.

# Introduction

The fundamental mathematical description is the Williams spray equation:

\begin{equation}
\label{eq:williams}
\partial_{t}f + u_{i}\partial_{x_{i}}f + \partial_{v}(R_{v}f) + \partial_{u_{i}}(F_{i}f) = \Gamma,
\end{equation}

a kinetic Boltzmann type equation that describes the evolution of the droplet distribution function f.

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, resulting in a possibly lower computational cost and an optimal parallelization of the solver.

The aim of this project is the development of an Eulerian method for the numerical simulation of unsteady sprays using the Direct Quadrature Method of Moments (DQMOM) to describe droplet polydispersity.

# Method and Theory Moment methods compute the evolution of the moments of the droplet distribution function with an Eulerian approach. These methods allow the description of droplet polydispersity in an Eulerian framework.

The Direct Quadrature Method of Moments (DQMOM) is a moment method. The approach consists in approximating the droplet distribution function by weighted Dirac-delta functions:

\begin{equation}
\label{eq:f}
f(v,\mathbf{u}) \approx \sum_{n=1}^{N} w_{n}\delta(v-v_{n})\delta(\mathbf{u}-\mathbf{u}_{n})
\end{equation}

from a set of moments, where vn and un are volume and velocities abscissas, respectively, and wn their corresponding weights. Unlike other moment methods, which solve transport equations for the moments of f, DQMOM solves transport equations for the abscissas and their corresponding weights. These transport equations are derived from the Williams spray equation.

# Key Research Area

Multi-Physics; Multi-Phase Flow Simulation, Moment Methods, Direct Quadrature Method of Moments

# Contact

Werner Gumprich
M.Sc.

 Address: Dolivostraße 15 D-64293 Darmstadt Germany Phone: +49 6151 16 - 24401 or 24402 Fax: +49 6151 16 - 6555 Email: gumprich (at) gsc.tu...