Assessment of crossed-plane tomography for flamelet surface normal measurements

A recently introduced method for measuring three-dimensional flamelet surface normal vectors, N , in turbulent, premixed flames is described and evaluated. Crossed-plane tomography allows the determination of N from two simultaneously acquired laser-tomographic images. The method is evaluated through comparisons with data from single-plane measurements made on a vortex-street perturbed v-flame, and through an error analysis. The standard deviation of the angle between N s measured using crossed-plane tomography and a vertical image-plane aligned perpendicular to the flame stabilizing rod is 3.7°, consistent with the instantaneous, two-dimensional structure of the perturbed flame. PDFs of the orientation of the projection of N onto this vertical plane are compared to PDFs of the orientation of N s measured using single-plane tomography, and the two are found to be, for the most part, within statistical uncertainty of one another. The results of an analysis of random error due to electronic noise and marker shot noise are consistent with experimental results, and thus uncertainty can be estimated through analytic expressions. The statistical weighting of PDFs generated by crossed-plane tomography measurements is discussed, as is the evaluation of the flamelet surface density in terms of the mean inverse cosine of the angle between N and an arbitrarily oriented line η. It is shown that this PDF is weighted by the probability that the flamelet crosses the line of intersection between the two illumination planes, a crossing-weighted PDF, and that the mean inverse cosine term is also a crossing-weighted average, where η is the line crossed. The flamelet surface density is written as the product of the mean inverse cosine term and the average number of flamelet crossings of η per unit length, the flamelet crossing density. It is argued that the preferred orientation of η is perpendicular to 〈 c 〉 constant surfaces, and a PDF fit for the N s from previous work (Bingham, Gouldin, and Knaus, 1998) is used to evaluate the mean inverse cosine term. It is found that the maximum value for this orientation of η is slightly more than two, indicating that large burning rate flames must have large corresponding values of the flamelet crossing density.