  Motivated by the thermal transport problem in the Kitaev spin liquids, we
consider a nearest-neighbor tight-binding model on the honeycomb lattice in the
presence of random uncorrelated $\pi$-fluxes. We employ different numerical
methods to study its transport properties near half-filling. The
zero-temperature DC conductivity away from the Dirac point is found to be
quadratic in Fermi momentum and inversely proportional to the flux density.
Localization due to the random $\pi$-fluxes is observed and the localization
length is extracted. Our results imply that, for realistic system size, the
thermal conductivity of a pure Kitaev spin liquid diverges as
$\kappa_\text{K}\sim T^3 e^{\Delta_v/k_BT}$ when $k_B T\ll \Delta_v$, and
suggest the possible occurrence of strong Majorana localization
$\kappa_\text{K}/T\ll k_B^2/2\pi\hbar$ when $k_B T\sim \Delta_v$, where
$\Delta_v$ is the vison gap.
