  We carry out direct numerical simulation of compressible square duct flow in
the range of bulk Mach numbers M_b = 0.2-3, and up to friction Reynolds number
Re_{\tau} = 500. The effects of flow compressibility on the secondary motions
are found to be negligible, as the typical Mach number associated with the
cross-stream flow is always less than 0.1. As in the incompressible case, we
find that the wall law for the mean streamwise velocity applies with good
approximation with respect to the nearest wall, upon suitable compressibility
transformation. The same conclusion also applies to a passive scalar field,
whereas the mean temperature does not exhibit inertial layers because of
nonuniformity of the aerodynamic heating. We further find that the same
temperature/velocity relation that holds for planar channels is applicable with
good approximation for square ducts, and develop a similar relation between
temperature and passive scalars.
