  Flows are omnipresent and govern the dynamics of plasma. Solar tornadoes are
a class of apparently rotating prominences, that might be formed by thermal
instability. In spectroscopic studies on thermal instability background flow is
commonly neglected. We here determine the effect of background flow on thermal
instability in cylindrical magnetic field configurations as the influence of
various parameters on the MHD spectrum. We investigate discrete thermal modes.
In an analytical study, we extend upon the literature by including a generic
background flow in a cylindrical coordinate system. The non-adiabatic MHD
equations are linearised, Fourier-analysed, and are examined to understand how
a background flow changes the continua. An approximate expression for discrete
thermal modes is derived using a WKB analysis. The analytical results are then
verified for a benchmark equilibrium using the eigenvalue code Legolas. The
eigenfunctions of discrete thermal modes are visualised in 2D and 3D. The
thermal continuum is Doppler-shifted due to the background flow, just like the
slow and Alfv\'en continua. Discrete modes are altered because the governing
equations contain flow-related terms. An approximate expression to predict the
appearance of discrete thermal modes based on the equilibrium parameters is
derived. All analytical expressions match the numerical results. The
distribution of the density perturbations of the discrete thermal modes is not
a singular condensation. 3D visualisation of the total velocity field shows
that the helical field is heavily influenced by the radial velocity
perturbation. We derived analytic expressions for non-adiabatic MHD modes of a
cylindrical equilibrium with background flow and verified them using a coronal
equilibrium. However, the equations are valid for and can be applied in other
astrophysical environments.
