A spectral closure for premixed turbulent combustion in the flamelet regime

Abstract

Premixed turbulent combustion in the flamelet regime is analysed on the basis of a field equation. This equation describes the instantaneous flame contour as an isoscalar surface of the scalar field G(x,t). The field equation contains the laminar burning velocity sL as velocity scale and its extension includes the effect of flame stretch involving the Markstein length [Lscr ] as a characteristic lengthscale of the order of the flame thickness. The scalar G(x,t) plays a similar role for premixed flamelet combustion as the mixture fraction Z(x,t) in the theory of non-premixed flamelet combustion.Equations for the mean $\overline{G}$ and variance $\overline{G^{\prime 2}}$ are derived. Additional closure problems arise for the mean source terms in these equations. In order to understand the nature of these terms an ensemble of premixed flamelets with arbitrary initial conditions in constant-density homogeneous isotropic turbulence is considered. An equation for the two-point correlation $\overline{G^{\prime}({\boldmath x},t)G^{\prime}({\boldmath x}+{\boldmath r},t)}$ is derived. When this equation is transformed into spectral space, closure approximations based on the assumption of locality and on dimensional analysis are introduced. This leads to a linear equation for the scalar spectrum function Γ(k,t), which can be solved analytically. The solution Γ(k,t) is analysed by assuming a small-wavenumber cutoff at k0 = lT−1, where lT is the integral lengthscale of turbulence. There exists a $k^{-\frac{5}{3}}$ spectrum between lT and LG, where LG is the Gibson scale. At this scale turbulent fluctuations of the scalar field G(x,t) are kinematically restored by the smoothing effect of laminar flame propagation. A quantity called kinematic restoration ω is introduced, which plays a role similar to the scalar dissipation χ for diffusive scalars.By calculating the appropriate moments of Γ(k,t), an algebraic relation between ω, $\omega,\overline{G^{\prime}({\boldmath x},t)^2}$, the integral lengthscale lT and the viscous dissipation ε is derived. Furthermore, the scalar dissipation χ[Lscr ], based on the Markstein diffusivity [Dscr ][Lscr ] = sL [Lscr ], and the scalar-strain co-variance Σ[Lscr ] are related to ω. Dimensional analysis, again, leads to a closure of the main source term in the equation for the mean scalar $\overline{G}$. For the case of plane normal and oblique turbulent flames the turbulent burning velocity sT and the flame shape is calculated. In the absence of flame stretch the linear relation sT ∼ u′ is recovered. The flame brush thickness is of the order of the integral lengthscale. In the case of a V-shaped flame its increase with downstream position is calculated.