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 S d ∗ 2 D , which is the two-dimensional projection of the actual density-weighted displacement speed S d ∗ for turbulent premixed flames. Here the statistics of S d ∗ and S d ∗ 2 D are compared using a direct numerical simulation database of statistically planar turbulent premixed flames. It is found that the probability density functions (pdfs) of S d ∗ 2 D approximate the pdfs of S d ∗ satisfactorily for small values of root-mean-square turbulent velocity fluctuation u′, though the S d ∗ 2 D pdfs are wider than the S d ∗ pdfs. Although the agreement between the pdfs and the standard-deviations of S d ∗ 2 D and S d ∗ deteriorate with increasing u′, the mean values of S d ∗ 2 D correspond closely with the mean values of S d ∗ for all cases considered here. The pdfs of two-dimensional curvature κ m 2 D and the two-dimensional tangential-diffusion component of density-weighted displacement speed S t ∗ 2 D are found to be narrower than their three-dimensional counterparts (i.e. κm and S t ∗ respectively). It has been found that the pdfs, mean and standard-deviation of π / 2 × κ m 2 D and π / 2 × S t ∗ 2 D faithfully capture the pdfs, mean and standard-deviation of the corresponding three-dimensional counterparts, κm and S t ∗ respectively. The combination of wider S d ∗ 2 D pdfs in comparison to S d ∗ pdfs, and narrower S t ∗ 2 D pdfs in comparison to S t ∗ pdfs, leads to wider S r ∗ + S n ∗ 2 D = S d ∗ 2 D - S t ∗ 2 D pdfs than the pdfs of combined reaction and normal-diffusion components of density-weighted displacement speed S r ∗ + S n ∗ . This is reflected in the higher value of standard-deviation of S r ∗ + S n ∗ 2 D , than that of its three-dimensional counterpart S r ∗ + S n ∗ . However, the mean values of S r ∗ + S n ∗ 2 D remain close to the mean values of S r ∗ + S n ∗ . The loss of perfect correlation between two and three-dimensional quantities leads to important qualitative differences between the S r ∗ + S n ∗ 2 D - κ m 2 D and S r ∗ + S n ∗ - κ m , and between the S d ∗ 2 D - κ m 2 D and S d ∗ - κ m correlations. For unity Lewis number flames, the S d ∗ - κ m correlation remains strongly negative, whereas a weak correlation is observed between S d ∗ 2 D and κ m 2 D . 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.