  Topological insulator nanowires with uniform cross section, combined with a
magnetic flux, can host both a perfectly transmitted mode and Majorana zero
modes. Here we consider nanowires with rippled surfaces---specifically, wires
with a circular cross section with a radius varying along its axis---and
calculate their transport properties. At zero doping, chiral symmetry places
the clean wires (no impurities) in the AIII symmetry class, which results in a
$\mathbb{Z}$ topological classification. A magnetic flux threading the wire
tunes between the topologically distinct insulating phases, with perfect
transmission obtained at the phase transition. We derive an analytical
expression for the exact flux value at the transition. Both doping and disorder
breaks the chiral symmetry and the perfect transmission. At finite doping, the
interplay of surface ripples and disorder with the magnetic flux modifies
quantum interference such that the amplitude of Aharonov-Bohm oscillations
reduces with increasing flux, in contrast to wires with uniform surfaces where
it is flux-independent.
