  By using Direct Numerical Simulations (DNS) of bubbly flows with passive
scalars, we show a transition in the scalar spectra from a $k^{-5/3}$ to a
$k^{-3}$ scaling with the wavenumber $k$, in contrast with those of
single-phase isotropic turbulence. For cases with a mean scalar gradient in the
horizontal direction, the scalar spectrum decays faster than $k^{-3}$ at high
wavenumbers. While the $k^{-3}$ scaling is well established in the bubbly flow
velocity spectrum, the scalar spectrum behaviour is not fully understood. We
find that the transition length scale of the scalar spectra is comparable to or
below the bubble diameter and decreases with the molecular diffusivity of the
scalar in the liquid. We use DNS to compute the scalar spectra budget and show
that the scalar fluctuations are produced by the mean scalar gradient at length
scales above the bubble diameter, contrary to the velocity fluctuations. At
length scales below the bubble diameter, the net scalar transfer scales as
$k^{-1}$ inducing the $k^{-3}$ scaling of the scalar spectra. This finding is
consistent with the hypothesis proposed by Dung et al. (2023) about the
physical mechanism behind the $k^{-3}$ scaling. We also show dependencies of
the bubble suspension's convective scalar diffusivity on the gas volume
fraction and molecular diffusivity that differs based on the direction of the
mean scalar gradient. For a mean scalar gradient in the vertical direction, we
find and qualitatively explain a significant effect of the molecular
diffusivity in the gas on the convective scalar diffusivity.
