A flamelet-based Eulerian transported PDF method for the modeling and simulation of supersonic combustion

This paper presents a high-efficiency and high-fidelity approach to model supersonic combustion using the extended flamelet-generated manifold (FGM) and the Eulerian transported probability density function (PDF), also known as the Eulerian stochastic fields (ESF) method. The efficiency benefits from the FGM, where the compressibility effects induced by shock waves are considered using two extra control variables, i.e., the pressure ($p$) and the absolute internal energy of the oxidizer ($E_{ox}$), in addition to the mixture fraction ($Z$) and progress variable ($Y_c$) in traditional flamelet tables. The joint PDF for these control variables is modeled using transported PDF based on the ESF method. The ESF method enhances accuracy in the prediction of turbulence-chemistry interactions, avoiding complexity induced by the presumed PDF in the flamelet table and ad-hoc presumed and independent joint PDF assumptions, e.g., the typical presumed $\beta$-PDF for $Z$ and $\delta$-PDF for $Y_c$. This FGM-ESF method is tested in large eddy simulations of two canonical hydrogen supersonic flames: a strut-stabilized hydrogen supersonic flame (DLR case) and a transverse hydrogen jet flame in a high-enthalpy incoming flow (Stanford case). For both cases, results show that including the compressibility effects in the FGM table is essential for properly describing the flame behaviors near shock waves. The sub-grid PDF of control variables significantly influences near-wall and shear-layer combustion, and the ESF method demonstrates superior performance in predicting the near-wall reaction zone and the shear-layer reaction zone compared to the perfectly micro-mixed sub-grid model ($\delta$-PDF) for the Stanford case. This study marks the first application of the FGM-ESF approach with a $Z$-$Y_c$-$E_{ox}$-$p$ FGM table to simulate supersonic flames, offering a novel perspective for future modeling efforts in this domain.

Novelty and Significance Statement

The novelty of this research lies in the development and application of an efficient computational approach that for the first time couples the extended flamelet-generated manifold (FGM) method with the Eulerian stochastic fields (ESF) to simulate supersonic flames. This combined FGM-ESF method incorporates pressure and oxidizer energy as additional control variables in the FGM tabulation to account for compressibility effects. The ESF is used to describe the sub-grid scale joint probability density function without relying on complex presumed functions. Application to a canonical strut-stabilized hydrogen supersonic flame and transverse hydrogen jet flames in a high-enthalpy incoming flow demonstrate the superior predictive capabilities of the FGM-ESF method in capturing flame dynamics. The comparison between four-dimensional and two-dimensional FGM tables provides new insights into the importance of compressibility effects for these configurations.