  Statistical properties of $d$-dimensional incompressible flows with and
without cylindrical reduction are studied, leading to several explanations and
conjectures about turbulent flows and passive scalars, such as the
de-correlation between the flow and scalar, reduction of passive scalar
intermittency in the bottleneck regime, et al. The absolute-equilibrium
analyses assure the correctness of a recent numerical result. It is implied
that passive scalar(s) in two-dimensional (2D) space can be fundamentally
different to those in $d>2$, concerning the correlations with the flow, which
is not considered in the celebrated Kraichnan model. The possibility of genuine
inverse transfer to large scales of 2D passive scalar energy, together with the
advection energy, is indicated. The compressible situation is also briefly
remarked in the end, in particular the absence of density in a nontrivial
Casimir which, without boundary contribution, also vanishes for $d=4$.
