Scalar dissipation rate modelling for Large Eddy Simulation of turbulent premixed flames

Abstract

The statistical behaviours of scalar dissipation rate (SDR) in the context Large Eddy Simulations (LES) of turbulent premixed combustion have been analysed using a simplified chemistry based Direct Numerical Simulations (DNS) data of a turbulent V-flame. The filter size dependence of the SDR is analysed in detail and it has been demonstrated that the filtered reaction rate can be satisfactorily closed using the Favre filtered SDR provided the filter width, Δ, remains greater than the thermal flame thickness, δ_th. Due to the close relation between the SDR and generalised Flame Surface Density (FSD), the dependence of the FSD on filter size has also been addressed. It has been found that a fractal dimension based power-law model satisfactorily captures the global and local behaviours of the generalised FSD in the context of LES. The fractal dimension and the inner cut-off scale for flame surface based on the volume-averaged value of the FSD are found to be in good agreement with previous analytical, experimental and DNS studies. The ratio of the volume-integrated filtered value of density-weighted SDR to its resolved component exhibits a power-law in terms of Δ with an inner cut-off scale scaling with δ_th. A power-law based model with a global exponent and inner cut-off scale is found to be insufficient to capture the local variations of SDR possibly due to its multi-fractal nature. An algebraic model for SDR, which was originally proposed for Reynolds Averaged Navier Stokes simulations, has been extended here for LES, which is found to satisfactorily capture both the global and local behaviours of SDR.

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, N_c, of reaction progress variable, c, is defined as: $N_c=D\nabla c·\nabla c$, where D is the progress variable diffusivity. The SDR is closely related to the generalised Flame Surface Density (FSD) [18] ${\Sigma }_{\mathit{gen}}=\mid \overline{\nabla c}\mid =\overline{{(N_c/D)}^{1/2}}$ 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 ${\Xi }_{\Delta }={\Sigma }_{\mathit{gen}}/\mid \nabla \overline{c}\mid$ based on the FSD is often expressed as [20], [21], [22], [23], [24], [25], [26], [27]:

$${\Xi }_{\Delta }={\Sigma }_{\mathit{gen}}/\mid \nabla \overline{c}\mid ={({\eta }_o/{\eta }_i)}^{D_F-2}$$

where η_i and η_o are the inner and outer cut-off scales and D_F is the fractal dimension based on the FSD. In LES, the filter width, Δ, can be taken to be the outer cut-of scale, η_o, and according to Peters [28], η_i, 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, η_i, is also found to scale with the thermal flame thickness, δ_th, 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 $\{\dot{w}\}$ for high Damköhler number flames can be expressed as: $\{\dot{w}\}=2\{\rho N_{(}c)\}/(2c_m-1)$ [1] where {q} is the Reynolds averaged value of a general quantity q and $c_m={\int }_0^1{[\dot{w}c]}_{L}f(c)\mathit{dc}/{\int }_0^1{[\dot{w}]}_{L}f(c)\mathit{dc}$ with subscript ‘L’ referring to planar laminar flame conditions. It has recently been shown that the relation $\{\dot{w}\}=2\{\rho N_c\}/(2c_m-1)$ is approximately valid also for low Damköhler number combustion [2]. However, it is yet to be addressed if the filtered reaction rate, $\overline{\dot{w}}$, can be closed in terms of the SDR for LES as:

$$\overline{\dot{w}}=2\overline{\rho }{\tilde{N}}_c/(2c_m-1)$$

where $\tilde{q}=\overline{\rho q}/\overline{\rho }$ 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):

$${\Xi }_{D1}=\overline{\rho }{\tilde{N}}_c/\overline{\rho }\overline{D}\nabla \overline{c}·\nabla \overline{c}\text{and}{\Xi }_{D2}=\overline{\rho }{\tilde{N}}_c/\overline{\rho }\tilde{D}\nabla \tilde{c}·\nabla \tilde{c}$$

The density-weighting is used in Eq. (3) because the mean value of the density-weighted SDR $\{\rho N_c\}$ is directly proportional to the mean reaction rate $\{\dot{w}\}$ in the context of RANS 1, [2]. It remains to be seen if the wrinkling factors Ξ_D2 or Ξ_D1 can be expressed using the power-law, as:

$${\Xi }_{D1}={({\eta }_o/{\eta }_{\mathit{iD}1})}^{{\alpha }_1}\text{and}{\Xi }_{D2}={({\eta }_o/{\eta }_{\mathit{iD}2})}^{{\alpha }_2}$$

where η_iD1 and η_iD2 are the inner cut-off scales for Ξ_D1 and Ξ_D2, respectively. The outer cut-off scale, η_o, 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:

• To examine whether the relation between $\overline{\dot{w}}$ and ${\tilde{N}}_c$, as given by Eq. (2), is valid for LES.

• To assess whether a power-law, as given by Eq. (4), can be used to model the SDR in the context of LES.

• 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.