In liquid-driven combustion systems, the fuel is sprayed as a cloud of droplets into the combustion chamber. These droplets evaporate and promote the preparation of a suitable fuel-air mixture prior to combustion. Together with the injection process, the evaporation of droplets influences the evolution of droplet distribution that, in turn, strongly affects the flame structure, combustion efficiency and emissions. Therefore, it is of great interest to study how the droplet evaporation and combustion modeling, that includes heat, mass and momentum transfer processes in both gas and liquid phase, would influence the numerical prediction of spray properties in practical spray applications. On a mesoscopic level, numerical tools for the spray simulation are based on a kinetic description given in terms of a number density function (NDF). The evolution of this NDF is described by the Williams spray equation. In general, there are two approaches to solve the Williams equation: the stochastic Lagrangian and the Eulerian moment methods. In the present thesis the second one is used to solve this equation. The general approach is to calculate the evolution of the moments of the NDF with a set of Navier-Stokes-like transport equations.

The stochastic Lagrangian methods approximate the solution of the Williams equation by individually tracking numerical particles (parcels) within a Lagrangian approach. They are considered as the most robust, reliable and accurate methods for the spray simulation, allowing a straightforward description of the occurring physical phenomena. Thereby, as each parcel often represents several real droplets of same properties in the spray system, the computational cost are being held at a reasonable level. However, some major drawbacks are associated with the stochastic Lagrangian methods. First, even though a parcel approach is considered, the computational cost is high and strongly dependent on the unsteadiness of the flow and mass loading of the system. In order to avoid a high level of statistical noise and achieve an accurate and smooth continuous field of droplet averaged properties, a sufficient amount of sampled parcels is required within each relevant grid cell during each numerical time-step. This required amount of parcels increases significantly for unsteady simulations with fine time and spatial discretization. Especially in the case of Large Eddy Simulation (LES), the necessary amount of sampled parcels is considerably higher in comparison to RANS models. Indeed, LES requires a strong coupling between continuous and disperse phases, significantly increasing the implementation complexity of the Lagrangian method. Nevertheless, a RANS-approach is followed in this thesis. Finally, since most parcels are located in a small region of the computational domain, an optimal parallelization of the solver is hardly demanding. Due to these drawbacks, a great interest for alternative methods has emerged. Recently, the Eulerian moment methods have gained special attention. Their general strategy is to calculate the evolution of the moments of the NDF with a set of Navier-Stokes-like transport equations. These transport equations are directly derived from the Williams equation. Their structure and, consequently, their computational cost, do not depend on the droplet mass loading or the unsteadiness of the system. Since the equations of both disperse and continuous phase share the same structure, the coupling between both phases is straightforward and solver parallelization can be easily optimized. Therefore, the Eulerian moment methods are promising alternatives to the Lagrangian approach.

Numerous Eulerian approaches and their variations have been proposed, each with its advantages and drawbacks. Among prominent methods are the Quadrature Method of Moments (QMOM), the Direct Quadrature Method of Moments (DQMOM) and the Sectional Methods (SM). The DQMOM is a robust approach for describing the transport in phase space. However, a major challenge involves the accurate description of convective transport and drag effects. The general DQMOM approach is to approximate the NDF by a few quadrature points and assume that all droplets of a certain size locally share the same velocity (mono-kinetic assumption). Hence, regarding convective transport and drag, these quadrature points are handled as ordinary monodisperse phases of a multi-fluid method. Indeed, the range of droplet sizes and velocities is substantial in most spray systems. Thus, a significant error associated with convective transport and drag is observed if only a few droplet size classes are considered. This issue has been recently tackled with a so-called Coupled Size-Velocity Moments (CSVM) model, which addresses droplet size-velocity correlations in drag and convectional transport processes within a moment method framework. Within a DQMOM framework, the most immediate strategy to reduce the error related to convective transport and drag is to increase the amount of quadrature points. However, the amount of quadrature points cannot be arbitrarily increased without compromising the well conditioning of the DQMOM solver. Another challenge associated to DQMOM is the accurate prediction of the evaporative flux, since the NDF is not known. Recently, the DQbSMOM, which combines the Direct Quadrature Method of Moments (DQMOM) with a SM, was introduced. The strategy of the DQbSMOM is to split the droplet size domain into so-called sections and to approximate the NDF in each section using DQMOM. Including the droplet evaporation process, the evaporative flux between two adjacent sections is described by applying the Eulerian Multi-Size Moment (EMSM) method using the Ambramzon-Sirignano model, as well as using using the Abramzon-Sirignano model - an equlibrium evaporation model - and the Langmuir-Knudsen model, that numbers among the non-equilibrium evaporation models.

The aim of this thesis is to extend the DQbSMOM to the Flamelet Generated Manifold (FGM) model in order to describe reactive spray systems. For simulating reactive spray systems, a modified FGM model is used to account for multiphase combustion phenomena. Due to the combustion-induced strong decrease of the droplets' diameter and volume fraction, two additional, surface-based approaches have been implemented. The first approach concerns a dynamic subiteration within the evaporation model in case of exceeding a CFL-like criterion, based upon the ratio of evaporation rate and timestep, while the other approach deals with an adapted formulation of the fixed sections in each computational cell. The complete model is successfully evaluated in terms of droplet mean diameters, droplet number concentration and droplet flux, as well as droplet velocity components and temperature of the continuous phase using the so-called Sydney spray burner, with respect to a comparison between a configuration for evaporation and evaporation with subsequent combustion, in order to emphasize the capabilities of the DQbSMOM. The Sydney spray burner consists of a piloted jet in coflow configuration, whereas experimental results are available for non-reactive and reactive acetone, as well as reactive ethanol configurations. The presented numerical results show three configurations with variing boundary conditions. The numerical results have been achieved using \textit{Precise/UNS}, a Finite-Volume-Method-based (FVM) in-house code of \textit{Rolls-Royce Germany}.