On the Necessary Grid Resolution for Verified Calculation of Premixed Laminar Flames

Abstract

We consider the grid resolution necessary to resolve combustion in a mixture of calorically imperfect ideal gases described by detailed kinetics and multicomponent transport. Using the steady premixed laminar flame as a paradigm, the required spatial discretization to capture all detailed physics in the reaction zone is found via 1) determination of the finest grid used in a standard software tool which employs adaptive mesh refinement, 2) examination of peak values of intermediate species mass fractions in the flame zone as a function of grid size, 3) a formal grid resolution study, and 4) a robust new eigenvalue analysis developed to estimate the finest length scale. Application to laminar premixed flames in hydrogen-air flames reveals that the finest length scale is on the order of 10 −4 cm for combustion at atmospheric pressure. Resolution at this scale is shown to be necessary to capture detailed species mass fraction profiles; other features such as steady flame speeds and equilibrium thermochemical properties do not have such a stringent length scale requirement.

Analysis of the spatio-temporal scales of laminar premixed flames near equilibrium

“…Further details are given by Al-Khateeb et al [5]. In this spatially local analysis, we are not concerned with the length scale of any possible oscillations.…”

“…To compute all the length scales accurately, the robust method developed by Al-Khateeb et al [5] is used. In general, the complete set of ODEs, Equations (7), is reduced to a highly refined non-traditional system of 2N + 2 Ordinary Differential-Algebraic Equations (ODAEs).…”

“…Reaction-diffusion coupling is especially important in the thin zone of rapid temperature rise and species mass fraction adjustment embedded within any flame. In a recent study not employing operator splitting [5], it has been shown via so-called spatial eigenvalue analysis that the finest relevant continuum length scale for an atmosphericpressure laminar premixed flame is typically on the order of a few microns. Others are also finding it necessary to use such fine scales for direct numerical simulation of combustion phenomena.…”

Verified and validated calculation of unsteady dynamics of viscous hydrogen–air detonations

“…For a steady, travelling, inviscid, Chapman-Jouguet (CJ) hydrogen-air detonation propagating into ambient atmospheric conditions, the reaction length scales range from 10 −5 to 10 0 cm at equilibrium, and near the detonation front this breadth of scales can be even wider. These length scales are a manifestation of the averaged continuum representation of the collisions that occur between molecules, which have an underlying principal length scale of the mean free path (Al-Khateeb, Powers & Paolucci 2010). Resolving this range of scales is a difficult task, but to attain a mathematically verified prediction, this breadth of scales must be captured.…”

The Dynamics of Unsteady Detonation with Diffusion

“…These fine reaction scales are a manifestation of an averaged representation of the molecular collision model in which the fundamental length scale is the mean free path. 11 In order to have a mathematically verified prediction, this wide range of scales must be resolved, which poses a daunting task.…”