  Context. Gas in protoplanetary disks mostly cools via thermal accommodation
with dust particles. Thermal relaxation is thus highly sensitive to the local
dust size distributions and the spatial distribution of the grains. So far, the
interplay between thermal relaxation and gas turbulence has not been
dynamically modeled in hydrodynamic simulations of protoplanetary disks with
dust.
  Aims. We aim to study the effects of the vertical shear instability (VSI) on
the thermal relaxation times, and vice versa. We are particularly interested in
the influence of the initial dust grain size on the VSI and whether the
emerging turbulence is sustained over long timescales.
  Results. We find that the emergence of the VSI is strongly dependent on the
initial dust grain size. Coagulation also counteracts the emergence of
hydrodynamic turbulence in our simulations, as shown by others before. Starting
a simulation with larger grains (100 $\mu$m) generally leads to a less
turbulent outcome. While the inner disk regions (within $\sim$ 70 au) develop
turbulence in all three simulations, we find that the simulations with larger
particles do not develop VSI in the outer disk.
  Conclusions. Our simulations with dynamically calculated thermal
accommodation times based on the drifting and settling dust distribution show
that the VSI, once developed in a disk, can be sustained over long timescales,
even if grain growth is occurring. The VSI corrugates the dust layer and even
diffuses the smaller grains into the upper atmosphere, where they can cool the
gas. Whether the instability can emerge for a specific stratification depends
on the initial dust grain sizes and the initial dust scale height. If the
grains are initially $\gtrsim$ 100 $\mu$m and if the level of turbulence is
initially assumed to be low, we find no VSI turbulence in the outer disk
regions.
