Coupling the Conditional Moment Closure Model to a Fully Compressible Large Eddy Simulation Algorithm

The Conditional Moment Closure (CMC) model provides a means of closing the subgrid terms for the reaction rates through the assumption that departures of the mean filtered reaction rate (conditional on a mixture fraction or progress variable) are small. Turbulentchemistry interaction is incorporated through a conditional scalar dissipation. To date, all Large Eddy Simulation implementations of the CMC model are with incompressible solvers and for non-premixed flames. This paper presents a methodology of coupling the CMC model with a fully compressible solver, and resultant validation against DNS data. This methodology can be employed for both premixed and non-premixed flames. Several challenges associated with premixed modeling using this approach are outlined. Introduction Computational power now permits the application of time-accurate unsteady methods to problems of industrial interest [1-3]. A key challenge when using Large Eddy Simulation (LES) is to incorporate the effects of turbulence-chemistry interactions through a detailed subgrid model which permits the use of complex reaction mechanisms. The majority of combustion LES involve the coupling of the combustion subgrid model with a variabledensity incompressible methodology via the continuity equation. However, this is not appropriate for flows which are influenced by compressible (acoustic) features. Examples of these types of flows include deflagrations, detonations, knock or ringing in homogeneous charge compression ignition engines at high load, knocking spark-ignition engines, combustion noise and acoustically driven instabilities in gas turbines. The Conditional Moment Closure (CMC) model [4-6] has been thoroughly validated in RANS simulations of turbulent non-premixed flames and has recently been successfully extended to LES (denoted LES-CMC) of turbulent diffusion flames and bluff body flames [79]. This modelling approach has several advantages; most notably that it provides a simple means to integrate finite rate complex chemistry effects into an LES. Closure is achieved through assuming that the variations around conditional means are small. Several additional terms must be closed, including expressions for the conditional velocities, conditional scalar dissipation and the Filtered Probability Density Function (FDF). The closure of these terms is expected to be dependent on the combustion regime encountered. This paper presents a new high-order accurate, fully compressible (shock capturing) implementation of the CMC modelling approach for lean premixed turbulent combustion. The CMC model is coupled through the energy and species equations using a fifth-order accurate in space modified Godunov method which has recently been derived specifically for problems including both compressible features and turbulent mixing [10-11]. In addition, the same proposed algorithmic structure can be used for non-premixed combustion using LES-CMC where the chosen conditioning variable would then be the mixture fraction and not a progress variable based on a reacting scalar. The proposed algorithm is validated against the low Damköhler number slot flame DNS of Sankaran et al. [12]. Governing Equations and Numerical Methods The governing equations chosen are the LES filtered equations for a compressible, reacting mixture [13-14]. This includes a single equation tracking the progress variable, the continuity, momentum and total energy equation. Using a direct numerical simulation database modelling a turbulent lean premixed methane slot flame due to Sankaran et al. [12], it has been shown in ‘a-priori’ tests [15] that the first order CMC closure assumption based on mass fraction of O2 gives good results for the reaction rate of the major species. Hence the progress variable in this work is based on the mass fraction of O2 and varies from 0 in the unburnt premixture to 1 in the burnt gases. Following Vremen et al (1995) the subfilter contributions to the momentum and energy equations from unresolved fluctuations of the viscous stress tensor, and subfilter contributions from unresolved heat flux are neglected. In this paper, the novel reconstruction method proposed by Thornber et al. [10,11] is employed to model terms the unclosed Reynolds stress. This numerical method is not kinetic energy conserving. Rather, the reconstruction method is designed to give a leading order dissipation of turbulent kinetic energy proportional to the velocity increment at the cell interface cubed (Δu) as expected from Kolmogorov’s analysis [16]. The improved interpolation approach helps overcome the typical poor high wavenumber performance of standard compressible Godunov methods [19, 20]. It acts as an implicit subgrid model [17,18] whilst naturally stabilising the numerical solution and retaining monotonicity. As with all LES models this relies on sufficient separation of the large scales from the scales where numerical dissipation acts strongly. In several previous test cases, this numerical method has demonstrated a good ability to represent the dissipation of turbulent kinetic energy [11, 21,22], most notably in flows requiring excellent resolution of turbulent scalar mixing parameters [23]. In an analysis of premixed and non-premixed methane air flames, Smooke and Giovangigli [24] demonstrated that the terms including enthalpy diffusion could be neglected by comparison to the other terms in the energy equation. Standard closures are employed for the remaining terms, utilising the Smagorinsky model [25] to provide an eddy viscosity type closure, with species and thermal diffusion given by a turbulent Schmidt number and the Prandtl analogy respectively. The reaction rates are given by the filtered CMC equations,