Large Eddy Simulation of Turbulent Combustion: A Novel Multivariate Probability Density Function Approach

In this work a new method has been proposed and validated to approximate multivariate scalar probability density functions (PDFs) within the framework of large eddy simulation (LES) of turbulent combustion. One of the most powerful modeling concepts is the so-called flamelet generated manifolds (FGM) approach in which the one-dimensional laminar flamelet solutions are embedded in a statistical description of turbulent combustion. This made extraordinary progress possible, however, properly describing and predicting such processes with sufficient accuracy in a computationally efficient manner still remains a formidable challenge. To make a simulation feasible, the number of degrees of freedom characterizing turbulent reacting flows must be reduced by a statistical averaging or filtering technique. In LES, structures smaller than the grid spacing are removed by means of a low-pass filter. As both the reaction kinetics and the turbulence-chemistry interaction are highly non-linear, the filtering leads to the occurrence of unclosed terms that describe the effects of unresolved fluctuations and need to be modeled. One way of accounting for these effects is to describe the unresolved fluctuations in a statistical fashion by means of a PDF. A common approach is to assume the shape of the PDF and to parametrize it with the first and second statistical moments of the distribution. Most processes can only be characterized with more than one control variable, however, it is a daunting task to come up with a viable assumption regarding the shape of such a joint PDF. The common assumption is that the control variables appearing in the FGM tables are statistically independent, consequently the sought joint PDF can be expressed as the product of univariate PDFs. However, this approximation has been proven inaccurate under certain circumstances. Furthermore, in this case when coupling with the FGM method the second moments of the control variables have to be introduced as additional parameters and the table of laminar flamelet solutions has to be pre-integrated. It increases the table's dimensionality often causing storage and memory problems. In this work a new multivariate discrete joint scalar PDF approach has been proposed to overcome these challenges. The novelty of this method is that the covariances among the univariate samples drawn from the marginal distributions are set with Kirkpatrick's simulated annealing algorithm (SA), which ensures that all the first and second statistical moments match the specified values including the correlations of the fluctuating control variables. This is done in such an efficient manner that makes it possible to generate the samples on the fly during the simulations. Once the sample set in the parameter space with the desired statistical moments have been generated, the look-up table can be accessed by each sample individually. Then the mean values of the thermochemical properties of interest can be calculated by simple ensemble averaging. This eliminates the need of pre-integrating the look-up table and consequently the increase in its dimensionality. It is sufficient to store the variables as functions of only the first moments of the control variables since higher moments are accounted for through the distribution of the discrete samples. Furthermore, this method can be generalized and adjusted to many different conditions as it does not pose any constraints on either how the marginal PDFs can be chosen or the number of control variables. Decoupling the look-up table from the actual shape of the PDFs offers the necessary flexibility for evaluating different PDFs or multiple look-up tables. The approach has been validated on the Sandia Flame D and the Sydney Bluff-Body Burner configurations against detailed experimental data and it has been compared to a conventional tabulated chemistry approach (FGM) with very encouraging results and a modest increase in CPU time.