  Predicting turbulent flame speed (the large time front speed) is a
fundamental problem in turbulent combustion theory. Several models have been
proposed to study the turbulent flame speed, such as the G-equations, the
F-equations (Majda-Souganidis model) and reaction-diffusion-advection (RDA)
equations. In the first part of this paper, we show that flow induced strain
reduces front speeds of G-equations in periodic compressible and shear flows.
The F-equations arise in asymptotic analysis of reaction-diffusion-advection
equations and are quadratically nonlinear analogues of the G-equations. In the
second part of the paper, we compare asymptotic growth rates of the turbulent
flame speeds from the G-equations, the F-equations and the RDA equations in the
large amplitude ($A$) regime of spatially periodic flows. The F and G equations
share the same asymptotic front speed growth rate; in particular, the same
sublinear growth law $A\over \log(A)$ holds in cellular flows. Moreover, in two
space dimensions, if one of these three models (G-equation, F-equation and the
RDA equation) predicts the bending effect (sublinear growth in the large flow),
so will the other two. The nonoccurrence of speed bending is characterized by
the existence of periodic orbits on the torus and the property of their
rotation vectors in the advective flow fields. The cat's eye flow is discussed
as a typical example of directional dependence of the front speed bending. The
large time front speeds of the viscous F-equation have the same growth rate as
those of the inviscid F and G-equations in two dimensional periodic
incompressible flows.
