Application of the eddy damped quasi-normal Markovian spectral transport theory to premixed turbulent flame propagation

Abstract

The eddy damped quasi-normal Markovian (EDQNM) turbulence theory was applied to a modified Kuramoto–Sivashinsky field equation to develop a spectral model for investigating the single and two-point scalar statistics associated with a flame front (treated as a passive scalar interface) propagating through isotropic turbulence. As a result of the presence of a uniform mean gradient in the scalar field, all correlations involving the scalar were found to be functions of both the wave number, k, and μ, the cosine of the angle between the k vector and the mean gradient vector. An infinite Legendre expansion separated out the wave number and angle dependencies, where the first term in each series accounted for the isotropic contribution to the correlations and the higher order terms accounted for the anisotropy introduced as a result of the mean gradient. It was found that while strong anisotropy existed in the scalar field at short times, at steady state the scalar field became nearly isotropic. A parameter study was then conducted to ascertain the effect of independently varying u′/sL and Reλ (where u′ is the rms velocity, sL is the laminar burning velocity, and Reλ is the Reynolds number based on the Taylor microscale). The turbulent burning velocity increased with increases in either u′/sL or Reλ; however, the model predicted a finite turbulent burning velocity as u′/sL→∞, even though flame quenching was not accounted for. This finite asymptote for the burning velocity was traced to the constitutive relationship used for the flame thickness and the ratio of the Markstein length to the flame thickness. It was also shown that the dominant wrinkling of the flame surface and subsequent contribution to the turbulent burning velocity occurred at smaller and smaller length scales as the inertial range of the scalar spectrum increased. Single point models will therefore have great difficulty reproducing this significant result. Scalar spectra exhibited changes over all wave numbers as either u′/sL or Reλ was modified. Transfer spectra, which arose in the form of convolution integrals as a result of the advection and propagation processes, were also analyzed and separated into their pairwise spectral interactions to determine which nonlinear terms in each integral were dominant.