Direct Numerical Simulation analysis of the Flame Surface Density transport equation in the context of Large Eddy Simulation

doi:10.1016/j.proci.2008.06.028

Proceedings of the Combustion Institute

Volume 32, Issue 1, 2009, Pages 1445-1453

https://doi.org/10.1016/j.proci.2008.06.028 Get rights and content

Abstract

Turbulent reaction rate closure modelling using the concept of Flame Surface Density (FSD) is now well-established in the context of Reynolds Averaged Navier Stokes (RANS) simulations. In the present study, three-dimensional DNS data is explicitly filtered in order to evaluate different terms of the FSD transport equation in the context of Large Eddy Simulation (LES). Existing sub-models for these unclosed terms are assessed with respect to the filtered DNS data, resulting in a recommended modelled form for each term. Previously assessed modelling of the propagation and curvature terms is combined with new model formulations for the turbulent transport and strain rate terms to produce a complete modelled FSD transport equation for use in LES. Displacement speed is shown to play a pivotal role in the modelling of several terms and its curvature dependence is explicitly accounted for in order to ensure that the FSD model will be valid in both the corrugated flamelets and the thin reaction zones regimes.

Introduction

Large Eddy Simulation (LES) has become an important tool for the analysis of turbulent combustion especially in problems involving significant large-scale unsteadiness, e.g. in thermo-acoustic instability in gas turbine combustors. Modelling of turbulent premixed flame propagation using the concept of Flame Surface Density (FSD) is well established in the context of Reynolds Averaged Navier Stokes (RANS) simulation. By contrast, extension of the FSD concept for LES combustion modelling is relatively recent [1], [2], [3], [4], [5], [6]. Other LES combustion models [7], [8], [9], [10], [11], [12], including those based on artificially thickened flames and the sub-grid scale wrinkling factor, are closely related to models for the sub-grid FSD.

Modelling the FSD transport equation is expected to have advantages over simpler algebraic FSD models in cases where the level of sub-grid wrinkling is high and the flame propagation is highly unsteady [13]. Moreover, straining and curvature effects can be represented directly in the FSD transport equation by using a suitable model for the displacement speed of the flame. At present, most FSD modelling does not account for the strain rate and curvature dependence of displacement speed and is valid only for the corrugated flamelets regime [14]. In the thin reaction zones regime the curvature contribution to displacement speed becomes a leading order effect [14], and hence cannot be ignored.

In LES, the level-set approach has proved successful in addressing flame propagation behaviour in the thin reaction zones regime [15], while Sankaran and Menon [16] have recently proposed a development of the Linear Eddy Model (LEM) for LES in the same context. By contrast, FSD based models have yet to be extended properly to the thin reaction zones regime. Some FSD and wrinkling factor models [2], [3], [8], [9], [10] do include straining and curvature effects through the use of an efficiency function, although the applicability of this approach within the thin reaction zones regime is not yet clear. Recent work has provided a detailed examination of the curvature and propagation terms of the FSD transport equation [17], and has indicated the importance of the displacement speed in closing these terms.

In this paper, FSD transport equation modelling is extended to the thin reaction zones regime based on a priori DNS analysis. Three-dimensional DNS with single-step Arrhenius chemistry has been carried out for freely propagating statistically planar turbulent premixed flames. The DNS data is explicitly filtered for LES using a Gaussian filter. The modelling assumptions for the FSD transport equation are assessed by comparing the LES-modelled terms with filtered DNS data.

The rest of the paper is organised as follows. The mathematical background is presented in Section 2, followed by a brief description of the numerical implementation in Section 3. The results are presented and discussed in Section 4, and the main conclusions are summarised in the final section of the paper.

Section snippets

Mathematical background

Combustion DNS in 3D with detailed chemistry has become feasible only recently and remains immensely computationally expensive. For the present investigation, three-dimensional DNS with a single step irreversible Arrhenius reaction mechanism is used. A reaction progress variable c is defined in terms of product mass fraction Y_P as c = (Y_P − Y_P0)/(Y_P∝ − Y_P0), which increases monotonically from zero in fresh gas (subscript 0) to unity in fully burned products (subscript ∞). The LES filtered transport

Numerical implementation

A series of DNS runs has been carried out for freely propagating statistically planar turbulent premixed flames in a cubical domain. The boundaries normal to the mean direction of flame propagation were specified as acoustically non-reflecting using the NSCBC formulation [19], while the transverse boundaries were specified as periodic. The combustion DNS code SENGA [17] was used, in which spatial differentiation is carried out using 10th-order central differences for interior points, decreasing

Results and discussion

All quantities required to model the terms of the generalised FSD transport equation have been evaluated from the DNS data using the Gaussian filter [1]: $G (\vec{r}) = (6 / π Δ^{2})^{3 / 2} \exp (- 6 \vec{r} \cdot \vec{r} / Δ^{2})$ , where Δ is the filter width. In this section, the modelling issues related to the curvature and propagation terms of the FSD transport equation are addressed first, followed by a discussion on the modelling of sub-grid FSD transport, and a discussion on the modelling of the strain rate term. The surface

Conclusions

Modelling of the unclosed terms of the FSD transport equation in the context of LES has been addressed in the light of a priori DNS analysis. Models for the unclosed terms are proposed and their performance is compared to the actual quantities obtained from DNS data. These modelled expressions are taken together to produce a complete modelled FSD transport equation. Modelling of surface averaged displacement speed ${\bar{(S_{d})}}_{s}$ has been shown to be of major importance for the closure of the sub-grid

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Direct Numerical Simulation analysis of the Flame Surface Density transport equation in the context of Large Eddy Simulation

Abstract

Introduction

Section snippets

Mathematical background

Numerical implementation

Results and discussion

Conclusions

Proc. Combust. Inst.

Combust. Flame

Proc. Combust. Inst.

Combust. Flame

Combust. Flame

Proc. Combust. Inst.

Proc. Combust. Inst.

Proc. Combust. Inst.

Int. J. Eng. Sci.

J. Comp. Phys.

Combust. Flame

Combust. Flame

Combust. Flame

Proc. Combust. Inst.

Joint Meeting of the British, German and French Sections of the Combustion Institute