  We perform direct numerical simulations (DNS) of an advected scalar field
which diffuses and reacts according to a nonlinear reaction law. The objective
is to study how the bulk burning rate of the reaction is affected by an imposed
flow. In particular, we are interested in comparing the numerical results with
recently predicted analytical upper and lower bounds. We focus on reaction
enhancement and quenching phenomena for two classes of imposed model flows with
different geometries: periodic shear flow and cellular flow. We are primarily
interested in the fast advection regime. We find that the bulk burning rate v
in a shear flow satisfies v ~ a*U+b where U is the typical flow velocity and a
is a constant depending on the relationship between the oscillation length
scale of the flow and laminar front thickness. For cellular flow, we obtain v ~
U^{1/4}. We also study flame extinction (quenching) for an ignition-type
reaction law and compactly supported initial data for the scalar field. We find
that in a shear flow the flame of the size W can be typically quenched by a
flow with amplitude U ~ alpha*W. The constant alpha depends on the geometry of
the flow and tends to infinity if the flow profile has a plateau larger than a
critical size. In a cellular flow, we find that the advection strength required
for quenching is U ~ W^4 if the cell size is smaller than a critical value.
