Stretch rate effects on displacement speed in turbulent premixed flame kernels in the thin reaction zones regime

doi:10.1016/j.proci.2006.07.184

Proceedings of the Combustion Institute

Volume 31, Issue 1, January 2007, Pages 1385-1392

https://doi.org/10.1016/j.proci.2006.07.184 Get rights and content

Abstract

Direct numerical simulation (DNS) of three-dimensional turbulent premixed flame kernels is carried out in order to study correlations of displacement speed with stretch rate for a range of different flame kernel radii. A statistically planar back-to-back flame is also simulated as a special case of a flame kernel with infinite radius of curvature. In all the cases the joint pdf of displacement speed with stretch rate shows two distinct branches, whose relative strength is found to be dependent on the mean flame curvature. A positive (negative) correlation is prevalent when the flame is curved in a convex (concave) sense towards the reactants. The observed non-linearity of the displacement speed-stretch rate correlation, qualitatively consistent with previous two-dimensional DNS with complex chemistry, is shown to exist purely due to fluid-dynamical interactions even in the absence of detailed chemistry. It is demonstrated that the combined contribution of reaction and normal diffusion is important in the response of flame propagation to stretch rate, and the effect is found to increase with decreasing mean kernel radius. The implications of the stretch rate dependence of displacement speed are discussed in detail in the context of the flame surface density (FSD) approach to turbulent combustion modelling.

Introduction

Premixed flames propagate in the direction of their local normal vector with a displacement speed S_d relative to an initially coincident material surface [1]. Detailed information about the displacement speed is required for modelling of turbulent premixed flames using both the flame surface density (FSD) [2] and G-equation [3] formulations. For premixed flames a reaction progress variable c may be defined which increases monotonically from zero in fresh reactants to unity in fully burned products and which may be expressed in terms of the product mass fraction Y_P as $c = (Y_{P} - Y_{P 0}) / (Y_{P \propto} - Y_{P 0})$ where subscripts 0 and ∞ correspond to reactants and products, respectively. Then the displacement speed S_d of the isosurface at c = c^∗ is given by $S_{d} = {\frac{\dot{w} + \nabla \cdot (ρ D \nabla c)}{ρ | \nabla c |}|}_{c = c^{*}}$ where $\dot{w}$ is the reaction rate, ρ is the density and D is the mass diffusion coefficient. Flame stretch rate is defined as the fractional rate of change of flame surface area A, given by $K = (1 / A) d A / d t = a_{T} + S_{d} \nabla \cdot \vec{N}$ where a_T is the tangential strain rate, $\nabla \cdot \vec{N} = 2 κ_{m}$ where κ_m is the local mean flame curvature and $\vec{N} = - \nabla c / | \nabla c |$ is the local flame normal vector. In the present convention, flame curvature is positive when the flame is convex to the reactants. Theoretical studies [4], [5], [6] suggest that displacement speed is a linear function of stretch rate K in the limit of small stretch rates [7] $S_{d}^{*} / S_{L} = 1 - K ℓ / S_{L}$ Here, $S_{d}^{*}$ is the density-weighted displacement speed given by $S_{d}^{*} = ρ S_{d} / ρ_{0}$ where ρ₀ is the unburned gas density, S_L is the unstrained laminar flame speed and ℓ is the Markstein length for stretch rate.

The response of displacement speed to stronger variations in stretch rate is important in flamelet modelling [1], [2], [3]. Peters [3] has demonstrated that curvature effects on displacement speed become dominant in the thin reaction zones regime. It is possible to decompose the displacement speed S_d into three components, namely the reaction component S_r, normal diffusion component S_n and tangential diffusion component S_t [3], [8] $S_{r} = \frac{\dot{w}}{ρ | \nabla c |}; S_{n} = \frac{\vec{N} \cdot \nabla (ρ D \vec{N} \cdot \nabla c)}{ρ | \nabla c |}; S_{t} = - 2 D κ_{m}$ From Eqs. (3), (5) it is evident that the contribution to the stretch rate K due to tangential diffusion is given by $S_{t} \nabla \cdot \vec{N} = - 4 D κ_{m}^{2}$ which may lead to a non-linear response of the displacement speed to stretch rate.

Experimental measurements of S_d and stretch rate in a turbulent flame are often difficult to perform. This has led to measurements of S_d in planar strained flame configurations with small values of strain rate. As a consequence, the effects of curvature and unsteady strain rates are seldom taken into account [7]. Previous studies using direct numerical simulation (DNS), as well as experimental results, have indicated that the linear relationship of Eq. (4) might continue to hold at large stretch rates. However, a DNS study in two dimensions with complex chemistry [7] has shown significant non-linearity.

Here, the stretch rate effects on S_d are studied using three-dimensional DNS of turbulent premixed flames in the thin reaction zones regime. A broad range of stretch rates is explored by examining spherical flame kernels with different radii. A statistically planar flame is also considered, as a special case of a flame kernel with infinite radius. Stretch rate effects on displacement speed are found to be non-linear, which is qualitatively in agreement with the findings of Chen and Im [7]. However, the observed non-linear behaviour is explained in terms of the response of displacement speed to strain rate and curvature, and is shown to exist purely due to fluid-dynamical interactions between the flame and flow field even in absence of complex chemistry. The objectives of the paper are as follows:

1.
To understand the response of the displacement speed to local stretch rate.
2.
To understand the effects of the global flame curvature on this response.
3.
To assess the implications of the results for turbulent premixed flame modelling, by considering the curvature stretch contribution to FSD transport in a configuration where the flame has a finite global curvature.

Section snippets

Mathematical background and numerical implementation

The standard conservation equations of mass, momentum and energy are solved in three dimensions for compressible reacting flows along with a single equation for a reaction progress variable [12]. In all cases the global Lewis number is unity and transport properties such as viscosity (μ), thermal conductivity (λ) and density-weighted mass-diffusivity (ρD) are considered to be independent of temperature. A single step Arrhenius-type reaction mechanism is employed.

Simulations are carried out

Results and discussion

The numerical parameters of the DNS database are presented in Table 1. For all of the simulations, standard values are chosen for Zel’dovich number β = 6, heat release parameter τ = (T_ad − T₀)/T₀ = 3, Prandtl number Pr = 0.7 and ratio of specific heats γ = C_p/C_v = 1.4. For the velocity scale and length scale ratios (u′/S_L and l/δ_th) listed, the combustion corresponds to the thin reaction zones regime [3] where turbulent eddies may enter into the preheat zone of the flame. As a result, the preheat zone

Concluding remarks

The effects of stretch rate on flame propagation have been studied for premixed flame kernels using three-dimensional DNS with simple chemistry. The effects of mean curvature have been demonstrated by simulating flame kernels of different radius, and a statistically planar back-to-back flame has been included as a special case of a flame kernel with infinite radius. In all cases, the joint pdfs of stretch rate and displacement speed show two distinct branches. The branch displaying a negative

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Stretch rate effects on displacement speed in turbulent premixed flame kernels in the thin reaction zones regime

Abstract

Introduction

Section snippets

Mathematical background and numerical implementation

Results and discussion

Concluding remarks

Int. J. Eng. Sci.

Proc. Combust. Inst.

Proc. Combust. Inst.

Combust. Flame

Combust. Flame

Proc. Combust. Inst.

J. Comput. Phys.

Proc. Combust. Inst.

Combust. Flame

Proc. Combust. Inst.

Proc. Combust. Inst.

Combust. Flame