Modelling of the Curvature Term in the Flame Surface Density Transport Equation: A Direct Numerical Simulations Based Analysis

Abstract

A simple chemistry based three-dimensional Direct Numerical Simulations (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different values of Karlovitz number Ka, turbulent Reynolds number Re t , heat release parameter τ and global Lewis number Le has been used for the modelling of the curvature term of the generalised Flame Surface Density (FSD) transport equation in the context of Reynolds Averaged Navier Stokes (RANS) simulations. The curvature term has been split into the contributions arising due to the reaction and normal diffusion components of displacement speed (i.e. T 1 ) and the term arising due to the tangential diffusion component of displacement speed (i.e. T 2 ). Subsequently, the subterms (i.e. T 1 and T 2 ) of the curvature contribution to the FSD transport have been split into the closed (i.e. T 1r and T 2r ) and unclosed (i.e. T 1ur and T 2ur ) components. It has been found that T 2 remains deterministically negative throughout the flame brush. However, the qualitative behaviour of T 1 changes significantly depending upon the values of Ka, Re t and Le. Detailed physical explanations have been provided for the observed behaviours of the components of the curvature term. Moreover, it has been observed that the closed contributions of T 1 and T 2 (i.e. T 1r and T 2r ) remains negligible in comparison to the unclosed contributions (i.e. T 1ur and T 2ur ). Suitable model expressions have been identified for T 1ur and T 2ur in the context of RANS simulations, which are shown to perform satisfactorily in all cases considered in the current analysis, accounting for the variations in Ka, Re t , τ and Le.Keywords: Direct Numerical Simulation, Flame Surface Density, Curvature, Lewis Number, Turbulent Reynolds Number, Reynolds Averaged Navier Stokes Modelling [14] demonstrated that algebraic models may not be adequate for simulating incylinder processes in Spark Ignition (SI) piston engines. Therefore, it may be necessary to consider a modelled FSD transport equation for simulating turbulent premixed flames under some conditions. Moreover, despite ever increasing computational power and capacity, Large Eddy Simulations (LES) are still not routinely used in industrial calculations due to large computational costs associated with it, and, consequently, RANS is still the most commonly used simulation method for industrial engineering problems. The accurate modelling of the chemical reaction rate is a necessity for highfidelity Computational Fluid Dynamics (CFD) simulations, which play important roles in the development of new generation energy-efficient and low-pollution SI engines and Lean Premixed Prevaporaised (LPP) gas turbines. Thus, it is important to have highfidelity models for FSD transport in the context of RANS simulations. The FSD transport equation has a number of unclosed terms, and previous analyses [1][2][3][4][5][6][7][8][9][10][11][12] indicated that the terms due to fluid-dynamic straining and flame curvature act as leading order c...

Direct Numerical Simulations of high Karlovitz number premixed flames for the analysis and modeling of the displacement speed.

“…The influence of curvature on the displacement speed can be considered through a Markstein length, which should depend on the Lewis number, allowing to take into account differential diffusion. From these observations and the study by Han and Huh [34], which were confirmed by several studies such as [35][36][37], Katragadda et al [38] proposed a model for the stretch due to curvature for flames in the TRZ regime. This model presents some encouraging results allowing to reproduce this stretch for different turbulent intensities.…”

“…Models for ρS d and S d κ , see Eq. ( 49) and (38), are obtained by integration of the proposed expression of S d , Eq. ( 28), over curvature, using a presumed Gaussian PDF for curvature.…”

“…Although qualitative agreement between the DNS data and Eq. ( 38), as well as with Katraggada's model [38] is obtained, an accurate description of S d κ s,c * is still not reached.…”

“…To close Eqs. ( 35), ( 37) and (38), models are needed for κ s,c * and κ 2 s,c * : • The curvature is modeled using the model proposed by Rymer [48] (see also [22]) adapted to Eqs. ( 33) and (34).…”

“…The proposed model for the stretch due to curvature, defined by Eq. (38), is compared to the DNS in Fig. 24.…”