Least Squares Minimization Closure Models for LES of Turbulent Combustion

Abstract

This work summarizes the development and testing of a new family of chemical closure models for large-eddy simulation (LES) of turbulent combustion using finite-rate chemistry. The goal of this research is to provide a simple, yet effective model that provides a correction to the ‘laminar chemistry’ prediction formed by evaluating chemical production terms using filtered-mean data. The general model takes the form $\overline {\dot {{\omega }}_{s} (q)} =f(\overline {{q}},{\Delta } ,{\ldots } )\dot {{\omega }}_{s} (\overline {{q}})$, where the enhancement factor, f, accounts for the effects of the subgrid fluctuations on apparent reactivity as expressed at a given mesh level. A form for the enhancement factor is derived by least-squares minimization (LSM) of a ‘reactivity functional’ connecting information at different mesh levels. A modified a priori analysis, in which simultaneous large-eddy simulations are performed on fine and coarse mesh levels, is used to identify candidate modeled forms for the enhancement factor. In the modified a priori analysis, coarse-mesh realizations are constrained by the filtered fine-mesh velocity, allowing eddy structures to be highly correlated. Several LSM variants are described and tested through comparisons with experimental data. The test cases include three experiments conducted at the University of Virginia’s supersonic combustion facility involving non-premixed hydrogen and partially-premixed ethylene combustion as well as a premixed propane-air flame in the Volvo Validation Rig. The results demonstrate the capability of the models to provide a consistent (if modest in some cases) improvement in predictive capability, relative to ‘laminar chemistry’, in a cost-effective manner.

Appendix

In this Appendix, analyses similar to those shown in Fig. 3 are repeated for the ‘positive’ chemical source terms (those acting to produce a particular species). Referring back to Eqs. 4 and 5, the ‘positive’ source terms can be calculated as

$$ \dot{{\omega }}_{s}^{+} =M_{w,s} \sum\limits_{i = 1}^{NR} {\left( {{v}^{\prime\prime}_{s,i} k_{f,i} \prod\limits_{j = 1}^{NS} {(\hat{{\rho }}_{j} )^{{v}^{\prime}_{j} }+} {v}^{\prime}_{s,i} k_{b,i} \prod\limits_{j = 1}^{NS} {(\hat{{\rho }}_{j} )^{{v}^{\prime\prime}_{j} }} } \right)} $$

(A-1)

Defining an L₂ norm of the ‘positive’ source-term vector as $\left \| {\dot {{\omega }}^{+}} \right \|\equiv (\sum \limits _{s} {\dot {{\omega }}_{s}^{+} \dot {{\omega }}_{s}^{+} } )^{1/2}$ and utilizing the same data fields as in Fig. 3 (q_{evolved, coarse}, q_{filtered, fine}, q_{filtered, coarse}), one obtains scatter plots of the form shown in Fig. 33.

Fig. 33

Scatter plots of ‘positive’ source-term norms (see text for descriptions of components of the figure

The first thing to note is that the degree of correlation is generally higher than evidenced in Fig. 3 for the net production rates. The left-most component of Fig. 33 compares directly with the left-most component of Fig. 3 and shows a similar degree of correlation between filtered-fine-mesh source terms and source terms evaluated using evolved coarse-mesh data. A clear trend toward diminishing reactivity due to the effects of unresolved information is apparent—this is the situation that the LSM models try to improve upon. The center component represents the results from a classical a priori analysis applied to the ‘positive’ source terms and is comparable to the same component in Fig. 3—both figures use fine-mesh data exclusively. The difference in the degree of correlation is striking, illustrating the point that classical a priori analysis as applied to the chemical production rates fails not because of nonlinearities in the reaction rates themselves but in the disruption of the balance between chemical production and depletion components. The right-most component of the figure utilizes only evolved coarse-mesh data and thus compares directly with the right-most component of Fig. 3. A greatly increased degree of correlation is evidenced, relative to Fig. 3. This again illustrates the balance-disruption effect that is present when the net production rates are considered. We noticed these trends early in our studies and developed some LSM-like models that used the positive components of the source terms. These models were highly effective in correlating the responses of the positive components across mesh levels in a priori testing but failed to provide a significant benefit when applied to the net production terms, both in a priori and in a posteriori settings [29, 30].