Campus Units

Mechanical Engineering, Chemical and Biological Engineering, Ames Laboratory

Document Type


Publication Version

Accepted Manuscript

Publication Date


Journal or Book Title

Proceedings of the Combustion Institute




Two solution algorithms are developed for the conditional moment closure (CMC) using quadrature-based moment methods (QBMM). Their primary purpose is to eliminate the necessity of the additional grid for the conditioning variable (e.g., mixture fraction). As in almost every probability-density-function (PDF)-based description, the main technical hurdle is the solution of the molecular-mixing term. Here the focus is on solving this term coupled with chemistry, as opposed to spatial transport. In the first algorithm (referred to as SA-CMC), this is done by semi-analytically solving an equation for the deviation variable of the conditional scalar mean in terms of Jacobi polynomials. The mixture-fraction space is represented by the Gauss–Lobatto quadrature rule and a β-PDF. In the second algorithm (referred to as QBMM-CMC), a closed set of differential equations is written for the joint moments of a scalar and the mixture fraction, thereby eliminating the need to assume a form for the mixture-fraction PDF. Both solution algorithms are tested for multi-step H2 combustion, and the expected results are achieved with at most six quadrature nodes. Since in the proposed algorithms the scalar mean is ensured to be constant during molecular mixing, it is further concluded that the proposed algorithms are accurate, computationally cost-effective, and straightforward alternatives for traditional CMC solution methods.


This is a manuscript of an article published as Ilgun, A. D., A. Passalacqua, and R. O. Fox. "Application of quadrature-based moment methods to the conditional moment closure." Proceedings of the Combustion Institute (2020). DOI: 10.1016/j.proci.2020.07.075. Posted with permission.

Copyright Owner

The Combustion Institute



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Available for download on Monday, September 19, 2022

Published Version