Large Eddy Simulations of Two-Phase Turbulent Flows"

A two-phase subgrid combustion model has been developed for large-eddy simulations (LES). This approach includes a more fundamental treatment of the effects of the final stages of droplet vaporization, molecular diffusion, chemical reactions and small scale turbulent mixing than other LES closure techniques. As a result, Reynolds, Schmidt and Damkohler number effects are explicitly included. This model has been implemented within an Eulerian-Lagrangian two-phase formulation. In this approach, the liquid droplets are tracked using the Lagrangian approach up to a prespecified cut-off size. The evaporation of the droplets larger than the cut-off size and the evaporation and mixing of droplets smaller than the cut-off size are modeled within the subgrid using an Eulerian twophase model. It is shown that droplets with order unity Stokes number disperse more than small droplets in agreement with earlier numerical and experimental studies. Conventional and the present approach agree very well when droplets do not fall below the cut-off. However, the present approach gives consistently better results when the cut-off is increased, thereby, demonstrating an important advantage of the new method. The limitations of the current methodology are also highlighted and possible solutions are discussed. INTRODUCTION Combustion efficiency, reduced emissions and stable combustion in the lean limit are some of the desirable features in the next generation gas turbine engines. To achieve these capabilities, current research is focusing on improving the liquid fuel atomization t. GRA, Student Member, AIAA $. Professor, Senior Member, AIAA * Copyright © 1997 by S. Pannala and S. Menon. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission process and to increase fuel-air mixing downstream of the fuel injector. However, the structure of complex three-dimensional, fuel-air mixing layers is very difficult to resolve using current experimental and numerical methods. Since, fuel atomization and fuelair mixing are both highly unsteady, conventional steady state methods cannot be used to elucidate the finer details. On the other hand, although the unsteady mixing process can be studied quite accurately using direct numerical simulation (DNS) (e.g., Poinsot, 1996), the application of DNS is limited to low to moderate Reynolds numbers (Re) due to resolution requirements and therefore, cannot be used for high Re flows of current interest. In LES, the scales larger than the grid are computed using a timeand space-accurate scheme, while the unresolved smaller scales are modeled. Closure of momentum and energy transport equations can be achieved using a subgrid eddy viscosity model since the small scales primarily provide a dissipative mechanism for the energy transferred from the large scales. However, for combustion to occur, the species must first undergo mixing and come into molecular contact. These processes occur at the small scales which are not resolved in the conventional LES approach. As a result, conventional subgrid eddy diffusivity models cannot be used to model these features. To address these issues, recently (Menon et al., 1993; Menon and Calhoon, 1996; Calhoon and Menon, 1996, 1997) a subgrid combustion model was developed and implemented within the LES formulation. This model separately and simultaneously treats the physical processes of molecular diffusion and small scale turbulent convective stirring. This is in contrast to probability density function closure which phenomenologically treats these two processes by a single model, thereby removing experimentally observed Schmidt number variations of the flow. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. The gas-phase methodology was recently extended to two-phase flows (Menon and Pannala, 1997) to accurately capture the process of phase change and turbulent mixing. In the present paper, this approach has been further refined and used to study vaporization and the subsequent chemical reactions. Both infinite and finite-rate kinetics have been investigated and discussed in this paper. FORMULATION Both Eulerian and Lagrangian formulations have been used to simulate two-phase flows in the past (e.g., Mostafa and Mongia, 1983). However, most state-ofthe-art codes employ the Lagrangian form to capture the droplet dynamics, while the gas phase is computed in the Eulerian form (e.g., Oefelein and Yang, 1996). In this formulation, the droplets are tracked explicitly using Lagrangian equations of motion, and heat and mass transfer are computed for each droplet. Due to resource constraints (computer time and memory), only a limited range of droplet sizes are computed. Droplets below an ad hoc cut-off size are assumed to vaporize instantaneously and to become fully mixed in the gas phase. This is a flawed assumption, since even in pure gas flows small-scale mixing process is very important for quantitative predictions (Menon and Calhoon, 1996). Here, the gas-phase subgrid combustion methodology has been extended to allow proper simulation of the final stages of droplet evaporation and turbulent mixing. The two-phase subgrid process is implemented within the framework of the Eulerian-Lagrangian LES approach. Thus, droplets larger than the cut-off size are tracked as in the usual Lagrangian approach. However, once the droplets are smaller than the cut-off, a twophase subgrid Eulerian model is employed to include the effects of the small droplets within the LES cells. Gas Phase LES Equations The incompressible Navier Stokes equations in the zero Mach number limit are employed for the present study. Zero-Mach number approach involves using a series expansion in terms of Mach number to remove the acoustic component from the equations and is a well established method (McMurtry et al., 1989; Chakravarthy and Menon, 1997). The LES mass, momentum, energy and species equations in the zero-Mach number limit are: + U: aV. at,, . •^ + Fs,i (2)