  We investigate the dispersion of a passive scalar such as the concentration
of a chemical species, or temperature, in homogeneous bubbly suspensions, by
determining an effective diffusivity tensor. Defining the longitudinal and
transverse components of this tensor with respect to the direction of averaged
bubble rise velocity in a zero mixture velocity frame of reference, we focus on
the convective contribution thereof, this being expected to be dominant in
commonly encountered bubbly flows. We first extend the theory of Koch et al.
(1989) (which is for dispersion in fixed beds of solid particles under Stokes
flow) to account for weak inertial effects in the case of ordered suspensions.
In the limits of low and of high P\'eclet number, including inertial effect of
the flow does not affect the scaling of the effective diffusivity with respect
to the P\'eclet number. These results are confirmed by direct numerical
simulations performed in different flow regimes, for spherical or very deformed
bubbles and from vanishingly small to moderate values of the Reynolds number.
Scalar transport in arrays of freely rising bubbles is considered by us
subsequently, using numerical simulations. In this case, the dispersion is
found to be convectively enhanced at low P\'eclet number, like in ordered
arrays. At high P\'eclet number, the Taylor dispersion scaling obtained for
ordered configurations is replaced by the one characterizing a purely
mechanical dispersion, like in random media, even if the level of disorder is
very low.
