A Comparison of Evaluation Methodologies of the Fractal Dimension of Premixed Turbulent Flames in 2D and 3D Using Direct Numerical Simulation Data

A Direct Numerical Simulation (DNS) database of statistically planar flames ranging from the wrinkled flamelets to the thin reaction zones regime and DNS data for a Bunsen premixed flame representing the wrinkled flamelets regime have been utilised to evaluate the fractal dimensions of flame surfaces using the filtering dimension method, the box-counting algorithm and the correlation dimension approach. The fractal dimension evaluated based on the fully resolved three-dimensional data has been found to be reasonably approximated by adding unity to the equivalent fractal dimension evaluated based on two-dimensional projections irrespective of the methodology of extracting fractal dimension. This indicates that the flame surface can be approximated as a self-similar fractal surface for the range of Karlovitz and Damköhler numbers considered here. While all methods, provide results identical to each other for benchmark problems, it has been found that the fractal dimension evaluation based on box-counting method provides almost identical results as that obtained using the filtering dimension method for both three and two dimensions, while the fractal dimensions based on the correlation dimension tend to be slightly smaller. The findings of the current analysis have the potential to be used to reliably estimate the actual fractal dimension in 3D based on experimentally obtained 2D binarised reaction progress variable field. The inner cut-off scales estimated based on all three methodologies yield comparable results in terms of order of magnitude with the box-counting method predicting a smaller value of inner cut-off scale in comparison to other methods. The execution times for fractal dimension extraction based on filtering dimension and box-counting methodologies are found to be comparable but the correlation dimension method is found to be considerably faster than the two alternative approaches and provides results consistent with theoretical bounds in all cases.