The turbulent burning velocity for large-scale and small-scale turbulence

The level-set approach is applied to a regime of premixed turbulent combustion where the Kolmogorov scale is smaller than the flame thickness. This regime is called the thin reaction zones regime. It is characterized by the condition that small eddies can penetrate into the preheat zone, but not into the reaction zone. By considering the iso-scalar surface of the deficient-species mass fraction Y immediately ahead of the reaction zone a field equation for the scaler quantity G(x, t) is derived, which describes the location of the thin reaction zone. It resembles the levelset equation used in the corrugated flamelet regime, but the resulting propagation velocity st normal to the front is a fluctuating quantity and the curvature term is multiplied by the diffusivity of the deficient species rather than the Markstein diffusivity. It is shown that in the thin reaction zones regime diffusive effects are dominant and the contribution of st to the solution of the level-set equation is small. In order to model turbulent premixed combustion an equation is used that contains only the leading-order terms of both regimes, the previously analysed corrugated flamelets regime and the thin reaction zones regime. That equation accounts for non-constant density but not for gas expansion effects within the flame front which are important in the corrugated flamelets regime. By splitting G into a mean and a fluctuation, equations for the Favre mean (G) over tilde and the variance <(G "(2))over tilde> are derived. These quantities describe the mean flame position and the turbulent flame brush thickness, respectively. The equation for <(G "(2))over tilde> is closed by considering two-point statistics. Scaling arguments are then used to derive a model equation for the flame surface area ratio <(sigma)over tilde>. The balance between production, kinematic restoration and dissipation in this equation leads to a quadratic equation for the turbulent burning velocity. Its solution shows the 'bending' behaviour of the turbulent to laminar burning velocity ratio s(T)/s(L), plotted as a function of v'/s(L). It is shown that the bending results from the transition from the corrugated flamelets to the thin reaction zones regimes. This is equivalent to a transition from Damkohler's large-scale to his small-scale turbulence regime.