Comparison of 2D and 3D density-weighted displacement speed statistics and implications for laser based measurements of flame displacement speed using direct numerical simulation data

In a recent study, a light sheet imaging approach has been proposed (Hartung et al., J. Appl. Phys. B 96 (2009) 843–862) which permits measurement of a quantity Sd∗2D, which is the two-dimensional projection of the actual density-weighted displacement speed Sd∗ for turbulent premixed flames. Here the statistics of Sd∗ and Sd∗2D are compared using a direct numerical simulation database of statistically planar turbulent premixed flames. It is found that the probability density functions (pdfs) of Sd∗2D approximate the pdfs of Sd∗ satisfactorily for small values of root-mean-square turbulent velocity fluctuation u  ′, though the Sd∗2D pdfs are wider than the Sd∗ pdfs. Although the agreement between the pdfs and the standard-deviations of Sd∗2D and Sd∗ deteriorate with increasing u  ′, the mean values of Sd∗2D correspond closely with the mean values of Sd∗ for all cases considered here. The pdfs of two-dimensional curvature κm2D and the two-dimensional tangential-diffusion component of density-weighted displacement speed St∗2D are found to be narrower than their three-dimensional counterparts (i.e. κm   and St∗ respectively). It has been found that the pdfs, mean and standard-deviation of π/2×κm2D and π/2×St∗2D faithfully capture the pdfs, mean and standard-deviation of the corresponding three-dimensional counterparts, κm   and St∗ respectively. The combination of wider Sd∗2D pdfs in comparison to Sd∗ pdfs, and narrower St∗2D pdfs in comparison to St∗ pdfs, leads to wider Sr∗+Sn∗2D=Sd∗2D-St∗2D pdfs than the pdfs of combined reaction and normal-diffusion components of density-weighted displacement speed Sr∗+Sn∗. This is reflected in the higher value of standard-deviation of Sr∗+Sn∗2D, than that of its three-dimensional counterpart Sr∗+Sn∗. However, the mean values of Sr∗+Sn∗2D remain close to the mean values of Sr∗+Sn∗. The loss of perfect correlation between two and three-dimensional quantities leads to important qualitative differences between the Sr∗+Sn∗2D-κm2D and Sr∗+Sn∗-κm, and between the Sd∗2D-κm2D and Sd∗-κm correlations. For unity Lewis number flames, the Sd∗-κm correlation remains strongly negative, whereas a weak correlation is observed between Sd∗2D and κm2D. The study demonstrates the strengths and limitations of the predictive capabilities of the planar imaging techniques in the context of the measurement of density-weighted displacement speed, which are important for detailed model development or validation based on experimental data.