Stretch rate effects on displacement speed in turbulent premixed flame kernels in the thin reaction zones regime
Introduction
Premixed flames propagate in the direction of their local normal vector with a displacement speed Sd 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 YP aswhere subscripts 0 and ∞ correspond to reactants and products, respectively. Then the displacement speed Sd of the isosurface at c = c∗ is given bywhere 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 bywhere aT is the tangential strain rate, where κm is the local mean flame curvature and 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]Here, is the density-weighted displacement speed given by where ρ0 is the unburned gas density, SL 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 Sd into three components, namely the reaction component Sr, normal diffusion component Sn and tangential diffusion component St [3], [8]From Eqs. (3), (5) it is evident that the contribution to the stretch rate K due to tangential diffusion is given by which may lead to a non-linear response of the displacement speed to stretch rate.
Experimental measurements of Sd and stretch rate in a turbulent flame are often difficult to perform. This has led to measurements of Sd 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 Sd 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 = (Tad − T0)/T0 = 3, Prandtl number Pr = 0.7 and ratio of specific heats γ = Cp/Cv = 1.4. For the velocity scale and length scale ratios (u′/SL and l/δth) 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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