  The Arnold-Beltrami-Childress (ABC) flow and the Kolmogorov flow are three
dimensional periodic divergence free velocity fields that exhibit chaotic
streamlines. We are interested in front speed enhancement in G-equation of
turbulent combustion by large intensity ABC and Kolmogorov flows. First, we
give a quantitative construction of the ballistic orbits of ABC and Kolmogorov
flows, namely those with maximal large time asymptotic speeds in a coordinate
direction. Thanks to the optimal control theory of G-equation (a convex but
non-coercive Hamilton-Jacobi equation), the ballistic orbits serve as
admissible trajectories for front speed estimates. To study the tightness of
the estimates, we compute front speeds of G-equation based on a semi-Lagrangian
(SL) scheme with Strang splitting and weighted essentially non-oscillatory
(WENO) interpolation. The SL scheme is stable when the ratio of time step and
spatial grid size (in the propagation direction) is smaller than a positive
constant independent of the flow intensity. Numerical results show that the
front speed growth rate in terms of the flow intensity may approach the
analytical bounds from the ballistic orbits.
