  We present an experiment set to address a standard specification aiming at
avoiding local turbulence inside the Coud\'e train of telescopes. Namely, every
optical surface should be kept within a 1.5$^\circ$ range around ambient
temperature. Such a specification represents an important concern and
constraint when developing optical systems for astronomy. Our aim was to test
its criticality in the context of the development of the VLTI/NAOMI and
VLTI/GRAVITY+ adaptive optics. This experiment has been conducted using the
hardware of the future Corrective Optics (CO) of GRAVITY+. Optical measurements
were performed in order to observe the evolution of turbulence in front of a
flat mirror for which the surface temperature was controlled in a range of
$22^\circ$ above ambient temperature. A time-dependent analysis of the
turbulence was led along with a spatial analysis. This experiment shows no
influence of temperature on local turbulence. It should be noted however that
this result is only applicable to the very specific geometry described in this
paper, which is representative of an adaptive optics (AO) system located inside
the Coud\'e train (facing-down mirror heated on its backface).

Instrumentation and Methods for Astrophysics