# Analysis of Pressure Effect on Three-Dimensional Flame Surface Density Estimation
## 1 Introduction  
Flame surface density (FSD or expressed as \(\Sigma\)), defined as flame surface area per unit volume, is a vital parameter under the flamelet concept for turbulent premixed combustion [1]. It can evaluate the fuel consumption rate of the combustion system, which is of primary practical importance when designing the combustor. In terms of large-scale premixed turbulent combustion, the flame thickness is much smaller than the characteristic length of the combustion system. Therefore, the significant increase in turbulent flame speed \(S_{T}\) is mainly due to the enlargement of the flame surface area caused by the wrinkled or corrugated flame surface, as firstly described by [PERSON] [2]. Thus, FSD is regarded as an effective measure of \(S_{T}\), i.e., \(S_{T}/S_{L}=\int\Sigma d\eta\)[3], as well as of the fuel consumption rate of the combustion system, i.e., \(\overline{W}=\delta_{T}\Sigma_{\text{max}}\)[4], in which \(S_{L}\) is the laminar flame speed and \(\delta_{T}\) represents the flame brush thickness. In theory, FSD is a well-developed approach for turbulent premixed combustion modeling on the basis of Reynolds averaged Navier Stokes theory [5]. In experiments, FSD could be calculated by detecting and extracting the turbulent flame front with planar laser-induced florescence (PLIF) [4; 6; 7; 8] or Rayleigh scattering [9; 10]. However, these laser diagnostic approaches can only provide flame information within a two-dimensional measurement plane, while the flame is three-dimensional in nature. As a result, estimating the flame surface density using its two-dimensional (2D) value as flame length per area would lead to underestimation of the fuel consumption rate. This is because the flame wrinkles have a fluctuating component that is not measurable in the measurement plane. The direct three-dimensional flame structure measurement requires a very complicated experimental apparatus, with high resource costs [11; 12]. Therefore, it is of primary importance to analyzevarious assumptions in FSD calculations invoked in the interpretation of 2D planar measurements. For instance, the 2D PSD approximation from 2D laser measurement technique assumes that the flame front is only wrinkled on the measurement plane. To obtain more reasonable approximations of 3D FSD, different models were proposed under the following basic assumptions: (1) The turbulent flow is isotropic; (2) The turbulent flow and flame wrinkles is axisymmetric; (3) The characteristics of the wrinkled/corrugated flame front out of the measurement plane similar with that on the plane. Starting from the definition of flame surface density, it can be deduced that the measured 2D FSD and 3D FSD value on the measurement plane is related to the surface averaged cosine value of the orientation angle of the measurement plane [13; 14; 15]. Numerous models in the literature were proposed to evaluate this parameter, including algebraic formulation from the statistical analysis of a Bunsen burner [16], isotropic wrinkled flame surface and the analysis of the flame front normal vector [13; 14], as well as numerical investigations [17]. The models show the ability to capture 3D FSD while they are mostly performed and verified under atmosphere pressure. Furthermore, the real combustors are in generally operated at high pressure. This will significantly affect the flame front character, laminar flame speed and laminar flame thickness, which, in turn, influence the reliability of the assumptions described above. There are still issues regarding of the pressure's effect on the reliability of the assumptions and the models. Based on the previous work and motivation above, the overall objective of this paper was to investigate pressure's effect on the 3D FSD estimation models. The CH4/air premixed turbulent flame was stabilized on a Bunsen burner and the measurement was performed with the OH-PLIF technique at a pressure up to 1.0 MPa. An in-house auto adaptive threshold binarization code was used to extract the flame front. The paper is composed as follows.