  Turbulence in the flow of fluid through a pipe can be suppressed by buoyancy
forces. As the suppression of turbulence leads to severe heat transfer
deterioration, this is an important and undesirable phenomenon in both heating
and cooling applications. Vertical flow is often considered, as the axial
buoyancy force can help drive the flow. With heating measured by the buoyancy
parameter $C$, our DNS show that shear-driven turbulence may either be
completely laminarised or transitions to a relatively quiescent
convection-driven state. Buoyancy forces cause a flattening of the base flow
profile, which in isothermal pipe flow has recently been linked to complete
suppression of turbulence (K\"{u}hnen et al. Nat. Phys., 2018), and the
flattened laminar base profile has enhanced nonlinear stability (Marensi et al.
JFM, 2019). In agreement with these findings, the nonlinear lower-branch
travelling-wave solution analysed here, which is believed to mediate transition
to turbulence in isothermal pipe flow, is shown to be suppressed by buoyancy. A
linear instability of the laminar base flow is responsible for the appearance
of the relatively quiescent convection driven state for $C\gtrsim 4$ across the
range of Reynolds numbers considered. In the suppression of turbulence,
however, i.e. in the transition from turbulence, we find clearer association
with the analysis of He et al. (JFM, 2016) than with the above dynamical
systems approach, which describes better the transition to turbulence. The
laminarisation criterion He et al. propose, based on an apparent Reynolds
number of the flow as measured by its driving pressure gradient, is found to
capture the critical $C=C_{cr}(Re)$ above which the flow will be laminarised or
switch to the convection-driven type. Our analysis suggests that it is the
weakened rolls, rather than the streaks, which appear to be critical for
laminarisation.
