Relation Between 3 and 2D Wrinkling Factors in Turbulent Premixed Flames

The magnitude of the wrinkled flame surface area in turbulent premixed flames divided by its projection in the direction of flame propagation, known as the wrinkling factor, is a fundamental quantity for the purpose of analysis and modelling premixed combustion, for example, in flame surface density based modelling approaches. According to Damköhler’s hypothesis it is closely related to the turbulent burning velocity, an equally important measure of the overall burning rate of a wrinkled flame. Three-dimensional evaluation of the area of highly wrinkled flames remains difficult and experiments are often based on planar measurements. As a result of this, model development and calibration require an extension of 2D measurements to 3D data. Different relations between 2D and 3D wrinkling factors are known in literature and will be discussed in the present work using a variety of direct numerical simulation (DNS) databases combined with theoretical arguments. It is shown, based on an earlier analysis, that the isotropic distribution of the surface area weighted probability density function of the angle between the normal vectors on the measurement plane and the flame surface, provides a very simple relationship, stating that the ratio between 3D and 2D flame surface area is given by \(4/\pi \), which is found to be in excellent agreement with DNS data of statistically planar turbulent premixed flames.