A quadrature closure for the reaction-source term in conditional-moment closure

doi:10.1016/j.proci.2006.08.011

Proceedings of the Combustion Institute

Volume 31, Issue 1, January 2007, Pages 1675-1682

https://doi.org/10.1016/j.proci.2006.08.011 Get rights and content

Abstract

A Gaussian-quadrature closure is introduced for the conditional reaction-source term in conditional-moment closure, motivated by the desire to accurately model the dynamics of homogeneous extinction and re-ignition in turbulent non-premixed flames. A priori analysis of the quadrature model was performed using results of direct numerical simulations (DNS) of isotropic turbulence which included a mixture fraction and reaction-progress variable with a reversible chemical-source term. The DNS calculations for three Damköhler numbers, each exhibiting a different degree of extinction and all exhibiting re-ignition, were used to demonstrate the shortcomings of the Taylor-series closure. A quantitative validation was completed, and the quadrature closure showed significantly less error than the Taylor-series closures for all of the conditions considered.

Introduction

As the modeling of mixing and reactions in non-premixed turbulent combustion steadily becomes more advanced, the diversity of physical phenomena that can be described accurately is naturally broadened. The advancement from mixing-limited chemistry to equilibrium chemistry allowed the ability to model the chemical thermodynamics. Steady laminar flamelets offered the ability to model the overlap of mixing and reaction timescales, and consequently flame extinction could be modeled in a qualitative way. Unsteady flamelets, conditional-moment closure (CMC) and the Lagrangian probability-density-function (PDF) approach have offered, in different ways, the ability to model the overlap of multiple reaction timescales with the mixing timescale(s). Yet, the accurate prediction of homogeneous extinction and re-ignition are phenomena that are challenging for even the most advanced combustion models. Stated simply, extinction is the consequence of strong mixing overbalancing the chemical reactions. Physically, homogeneous re-ignition is the consequence of un-extinguished reaction zones interacting with extinguished zones. Capturing this interaction is required to not only describe re-ignition, but also to accurately describe the dynamics of extinction. As Bilger et al. [1] stated, “In flames with significant local extinction and re-ignition it has been found that [the] first-order closure is not sufficiently accurate.” It is the opinion of the authors, of the current paper, that higher-order CMC offers the greatest potential to accurately model the dynamics of extinction and re-ignition for a reasonable computational cost. However, the higher-order CMC methods do require refinement. The purpose of this paper is to propose an adjustment to one aspect of the higher-order CMC model. More specifically, we suggest that the use of the Taylor series as a closure for the conditional reaction-source term is unsuitable, and that the source-term integral can be more accurately approximated by an appropriate Gaussian-quadrature formula.

Section snippets

Theory

The homogeneous, one-point, constant-density joint PDF, f_Y,ξ, of mixture fraction, ξ, and reaction-progress variable, Y, evolves in time, t, according to [2], [3] $\frac{\partial f_{Y, ξ}}{\partial t} = - \frac{\partial}{\partial y} (S_{Y} (y, ζ) f_{Y, ξ}) - \frac{1}{2} \frac{\partial^{2} 〈 ϵ_{Y} | y, ζ 〉 f_{Y, ξ}}{\partial y^{2}} - \frac{1}{2} \frac{\partial^{2} 〈 ϵ_{ξ} | y, ζ 〉 f_{Y, ξ}}{\partial ζ^{2}} - \frac{\partial^{2} 〈 ϵ_{Y, ξ} | y, ζ 〉 f_{Y, ξ}}{\partial y \partial ζ},$ where y and ζ are the sample-space variables for the reaction-progress variable and the mixture fraction, respectively, S_Y is the chemical-reaction source term for the reaction-progress variable, and the rates of dissipation by diffusion with

Direct numerical simulation data

In order to analyze the validity of CMC models to describe such phenomena, we chose to a priori test the models using the direct numerical simulations (DNS) of Sripakagorn et al. [4]. The DNS included the transport of two scalars: a passive mixture fraction and a reaction-progress variable. The two scalars were injected into isotropic decaying turbulence. The mixture fraction was initialized such that it was nearly segregated with a mean near 〈ξ〉 = 0.5, and the reaction-progress variable was

Modeling

In the process of analyzing the “standard” higher-order CMC models, we were surprised by how large the error was in the pivotal conditional reaction-source term. In this paper, we discuss the error of the closure for this term as presented by Klimenko and Bilger [2]. Additionally, we present a new closure based on Gaussian-quadrature integration.

Taylor-series approximation

Figure 3 shows the second- and third-order Taylor-series approximations to the reaction-source term using the DNS at a time near extinction. One can see that for a complex chemical-source term, the lower-order Taylor-series approximations, although accurate in the region near the conditional mean, do not approximate the source term well across the realizable region. This weakness is accentuated by the fact that the corresponding conditional PDF, shown in Fig. 4, is largest in regions where the

Conclusions

Based on the a priori analysis using DNS data for a relatively complex kinetic expression, we can conclude that the quadrature approximation for the conditional chemical-source term offers a powerful approach for modeling the reaction-source term in higher-order CMC. In practice, use of the quadrature approximation will require the numerical simulation of transport equations for the conditional moments 〈Y^α∣ζ〉, where the maximum value of α depends on the order of the approximation. For the case

Acknowledgments

This research was partially supported by Grants (CTS-0336435, CTS-0403864) from the US National Science Foundation. The authors thank Drs. Sripakagorn, Kosály and Riley for providing us access to their DNS data.

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There are more references available in the full text version of this article.

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