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Objective:

The overarching goal of the project is to understand the complex physical interactions in spray combustion. In this performance period, the research team at UT developed a novel probability density function approach for describing multiphase mixing and combustion. Combustion modeling, in single-phase flows, requires apriori knowledge of the flame structure. In gas-phase systems, the flame structure is determined by the boundary conditions and can be categorized as premixed, non-premixed, or partially premixed systems based on the fuel/oxidizer injection design. Spray systems, on the other hand, may contain different flame structures inside a single configuration. The flame structure or the combustion regime is determined based on the local evaporation rate, droplet inertia, gas-phase turbulence, and fuel-air mixing rates. For this reason, a spray combustion model should be able to describe multiple combustion regimes. The probability density function (PDF) approach, previously used in the single-phase context, obviates the need for presuming reaction regime by directly evolving the joint-PDF of the gas-phase scalars. However, the modeling problem now shifts to describing gas-phase mixing in the presence of fuel droplets. In terms of practical implementation, the transport equation describing the evolution the PDF is high dimensional and cannot be solved using conventional finite-difference/finite-volume methods. The research team funded through the PRF grant and supported by co-workers funded through NSF and DoD has developed a robust Lagrangian numerical method for the PDF approach. The findings from this development program are reported below.

Methodology:

This research work is conducted in the framework of large eddy simulations (LES). In LES, the large-scale features of the flow are resolved on a computational mesh while the unresolved small-scale physics have to be modeled explicitly. Since spray evolution including dispersion, mixing, and combustion are significantly influenced by interactions with the turbulent flow at the small scales, these processes have to be modeled. In this work, the spray population is described using a number density equation, and solved numerically using a Lagrangian approach. The gas-phase thermochemical evolution is handled using the PDF approach. The PDF approach solves a single transport equation for the joint PDF of the gas phase scalars. The PDF transport equation is N+4 dimensional, where N is the number of scalars used to describe the gas phase thermochemical state. This equation is solved using a Monte Carlo method. Here, the gas-phase fluid is represented by a large ensemble of notional particles, with each particle moving according to a set of stochastic differential equations. The transport in composition space occurs through mixing and chemical reaction. While the chemical source term is exactly described and requires no closure, the mixing term requires two-point information not available in the one-point PDF formulation used here and needs closure. Here, a simple interaction by exchange with the mean (IEM) model is used. It is noted that the development of the mixing model specific to spray combustion is the logical next step that the research team will focus beyond the end of this research grant. The LES approach is time dependent and requires consistent coupling of the spray and PDF solvers. All the solvers are evolved simultaneously and exchange mean field information. For instance, the spray droplet evaporation requires the gas phase composition at the droplet location. However, only the filtered gas phase composition is available from the PDF solver. Hence, a liner interpolation method is used to approximate the solution at the droplet location. In a turbulent flow, this approximation can lead to large errors, especially when large scalar changes are present due to the combustion process. This aspect is being studied currently, and an improved closure based on DNS-derived functional formulation is being developed. Due to the large computational cost, an MPI-based parallelization strategy was used. The code was run on 128-256 processors.

Results:

The LES/FDF approach for spray combustion was tested using a planar jet configuration. The spray droplets were injected through the central jet along with air. Two coflow streams on either side consisting of air but at velocities one-half of the central jet were present. The LES simulations yielded the following findings.

1. In the absence of chemical reactions, the PDF algorithm was found to reproduce the evolution of the gas-phase fuel composition accurately when compared with a baseline Eulerian method. In spite of very crude mixing and interpolation models used, the error in the gas phase scalar composition and its variance was within 3% over the entire domain. For the reacting case, the comparisons with direct numerical simulation (DNS) of the same system showed that the PDF method captures the mean reasonably accurately but the variance was underpredicted. This points to a problem with the mixing model. We are developing an improved closure based on a dynamic formulation to address this issue.

2. The numerical implementation of the PDF method revealed a very subtle but important feature of the momentum and continuity equations for spray problems. In the spray formulation, the continuity and momentum equations contain source terms that describe mass addition due to the evaporation and drag force due to droplet-fluid interactions, respectively. There are two approaches to solving flow equations based on a compressible or an incompressible formulation. While the compressible formulation directly solves the flow equations explicitly, the incompressible formulation typically uses a Poisson-equation based projection step to enforce continuity equation. This difference leads to different interpretations of the evaporation-generated mass addition. The incompressible formulation implies that mass addition happens at constant density, and the added mass is redistributed in the domain through a velocity correction step. On the other hand, the compressible formulation implies that mass addition leads to a density change, and there is no velocity change caused by evaporation. The PDF approach, due to its formulation, is incompatible with the incompressible formulation due to this implied assumption. Consequently, the LES-PDF approach is applicable only with a compressible formulation of the LES equations.