Comparison of turbulent flame speeds from complete averaging and the G-equation

A popular contemporary approach in predicting enhanced flame speeds in premixed turbulent combustion involves averaging or closure theories for the G-equation involving both large-scale flows and small-scale turbulence. The G-equation is a Hamilton-Jacobi equation involving advection by an incompressible velocity field and nonlinear dependence on the laminar flame speed; this G-equation has been derived from the complete Navier-Stokes equations under the tacit assumptions that the velocity field varies on only the integral scale and that the ratio of the flame thickness to this integral scale is small. Thus there is a potential source of error in using the averaged G-equation with turbulent velocities varying on length scales smaller than the integral scale in predicting enhanced flame speeds. Here these issues are discussed in the simplest context involving velocity fields varying on two scales where a complete theory of nonlinear averaging for predicting enhanced flame speeds without any ad hoc approximations has been developed recently by the authors. The predictions for enhanced flame speeds of this complete averaging theory versus the averaging approach utilizing the G-equation are compared here in the simplest context involving a constant mean flow and a small-scale steady periodic flow where both theories can be solved exactly through analytical formulas. The results of this comparison are summarized briefly as follows: The predictions of enhanced flame speeds through the averaged G-equation always underestimate those computed by complete averaging. Nevertheless, when the transverse component of the mean flow relative to the shear is less than one in magnitude, the agreement between the two approaches is excellent. However, when the transverse component of the mean flow relative to the shear exceeds one in magnitude, the predictions of the enhanced flame speed by the averaged G-equation significantly underestimate those computed through complete nonlinear averaging, and in some cases, by more than an order of magnitude.