Scalar dissipation rate modelling for Large Eddy Simulation of turbulent premixed flames
Introduction
Scalar dissipation rate (SDR) plays a crucial role in the modelling of turbulent reacting flows and the mean reaction rate is directly proportional to the Favre-averaged SDR in turbulent premixed flames under high Damköhler number conditions [1] in the context of Reynolds Averaged Navier Stokes (RANS) simulations. Recent analyses [2], [3] showed that this dependence remains valid even for low Damköhler number premixed combustion and the justification was provided based on scaling arguments [3]. Although a number of studies [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], 17 focused on the SDR based modelling for turbulent premixed flames using RANS simulations, Large Eddy Simulation (LES) modelling of turbulent premixed flames using the SDR has rarely been addressed. The SDR, Nc, of reaction progress variable, c, is defined as: , where D is the progress variable diffusivity. The SDR is closely related to the generalised Flame Surface Density (FSD) [18] where the over-bar indicates a LES filtering operation. The equivalence between the FSD and SDR based modelling approaches has been discussed elsewhere in detail [5], 17, [19]. A wrinkling factor based on the FSD is often expressed as [20], [21], [22], [23], [24], [25], [26], [27]:where ηi and ηo are the inner and outer cut-off scales and DF is the fractal dimension based on the FSD. In LES, the filter width, Δ, can be taken to be the outer cut-of scale, ηo, and according to Peters [28], ηi, scales with the Gibson (Obukov–Corrsin) scale in the Corrugated Flamelets (CF) (Thin Reaction Zones (TRZ)) regime of combustion, which was subsequently supported by DNS data [27]. However, the inner cut-off scale, ηi, is also found to scale with the thermal flame thickness, δth, in turbulent premixed flames in a number of previous experimental and DNS studies [25], [27], [29].
In the context of RANS the mean reaction rate of reaction progress variable for high Damköhler number flames can be expressed as: [1] where {q} is the Reynolds averaged value of a general quantity q and with subscript ‘L’ referring to planar laminar flame conditions. It has recently been shown that the relation is approximately valid also for low Damköhler number combustion [2]. However, it is yet to be addressed if the filtered reaction rate, , can be closed in terms of the SDR for LES as:where is the Favre filtered value of a general quantity q. A wrinkling factor based on SDR can be defined in the following manner by drawing analogy with Eq. (1):
The density-weighting is used in Eq. (3) because the mean value of the density-weighted SDR is directly proportional to the mean reaction rate in the context of RANS 1, [2]. It remains to be seen if the wrinkling factors ΞD2 or ΞD1 can be expressed using the power-law, as:where ηiD1 and ηiD2 are the inner cut-off scales for ΞD1 and ΞD2, respectively. The outer cut-off scale, ηo, for LES can be taken to be Δ. Several previous analyses (e.g. Refs. [30], [31], [32], [33] and references therein) indicated that the SDR shows a multi-fractal nature in non-reacting flows and thus a single exponent α1 or α2 for the power-laws given by Eq. (4) may be inadequate. Although power-law models such as Eq. (1) have been used extensively for the FSD based closure in the context of LES [20], [21], [22], [23], [24], [25], [26], [27], its applicability for SDR (Eq. (4)) in LES have not yet been investigated. Moreover, the applicability of Eq. (2) for LES is yet to be assessed. Thus the main objectives of the present analysis are:
- 1.
To examine whether the relation between and , as given by Eq. (2), is valid for LES.
- 2.
To assess whether a power-law, as given by Eq. (4), can be used to model the SDR in the context of LES.
- 3.
To propose an algebraic model for the SDR in the context of LES, using the above two objectives.
These objectives are met by a priori analysis of a simplified chemistry based DNS database [34] of a statistically stationary turbulent V-flame. The DNS data is filtered using a Gaussian filter for the purpose of this analysis.
The rest of the paper is organised as follows. A brief description of mathematical background and numerical implementation of the analysis is provided in the next section. Following this, results are presented and discussed. Finally, the main findings are summarised and conclusions are drawn.
Section snippets
Mathematical background and numerical implementation
Three-dimensional combustion DNS with detailed chemistry [35] remains extremely expensive especially for non-canonical configuration, such as V-flame configuration, considered in this analysis. Thus single-step Arrhenius type chemistry is considered for the present analysis in which the species field is uniquely represented by c, which is defined here in terms of a suitable reactant mass fraction, YR, as c = (YR0 − YR)(YR0 − YR∝ where subscripts 0 and ∝ denote the values in unburned reactants and
Results and discussion
The volume rendered fields of instantaneous reaction progress variable, c, normalised reaction rate, , and normalised SDR, , are shown in Fig. 1a–c, respectively. The Favre-averaged reaction progress variable field (i.e. {ρc}/{ρ}), obtained in the context of Reynolds Averaging is also shown by the solid lines in Fig. 1a–c. It can be seen from Fig. 1a that c iso-surfaces corresponding to the preheat zone (i.e. c ⩽ 0.5) are more distorted than the iso-surfaces representing
Conclusions
In the present analysis the statistical behaviours of the SDR (i.e. and ) of the reaction progress variable, c, in the context of LES have been analysed in detail using a three-dimensional simplified chemistry based DNS data of a turbulent V-flame. The filter size dependences of and have been analysed and detailed physical explanations are provided for the observed filter size dependences. The statistical behaviours of the generalised FSD, Σgen, for different filter widths have
Acknowledgement
The financial assistance of EPSRC is gratefully acknowledged.
References (41)
- et al.
Combust. Flame
(2011) Combust. Flame
(1990)- et al.
Combust. Flame
(1994) - et al.
Combust. Flame
(2003) - et al.
Combust. Flame
(2005) - et al.
Proc. Combust. Inst.
(1998) - et al.
Prog. Combust. Sci.
(2002) - et al.
Combust. Flame
(1989) - et al.
Proc. Combust. Inst.
(1998) - et al.
Combust. Flame
(2002)
Combust. Flame
Combust. Flame
Flow Turb. Combust.
Proc. Combust. Inst.
Phys. Fluids
Phys. Fluids
Phys. Fluids
Combust. Theory Model.
Flow Turb. Combust.
Cited by (89)
Spatial characteristics and modelling of mixture fraction variance and scalar dissipation rate in steady turbulent round jets
2022, International Journal of Heat and Fluid FlowCitation Excerpt :Ma et al. (2014) determined that SDR closure predictions had comparable or better accuracy than two promising Flame Surface Density models (Ma et al., 2013, 2014). Butz et al. (2015) used the algebraic model of Dunstan et al. (2013) in LES simulations of swirl-stabilised premixed flames, obtaining good agreement with experimental data. The remainder of the paper is organised in the following way.
On the blow-off correlation for swirl-stabilised flames with a precessing vortex core
2022, Combustion and FlameEffect of flame–flame interaction on scalar PDF in turbulent premixed flames
2022, Combustion and Flame