  The phenomenological models of convection use characteristic length scales
they do not determine but that are chosen to fit solar or stellar observations.
We investigate if changes of these length scales are required between the Sun
and low mass stars on the red giant branch (RGB). The question is addressed
jointly in the frameworks of the mixing length theory and of the full spectrum
of turbulence model. For both models, the convective length scale is assumed to
be a fixed fraction of the local pressure scale height. We use constraints
coming from the observed effective temperatures and linear radii independently.
We rely on a sample of 38 nearby giants and subgiants for which surface
temperatures and luminosities are known accurately and the radii are determined
through interferometry to better than 10%. For the few cases where the stellar
masses were determined by asteroseismological measurements, we computed
dedicated models. First we calibrate the solar models. Then, with the same
physics, we compute RGB models for masses between 0.9 Mo and 2.5 Mo and
metallicities ranging from $\rm [Fe/H]=-0.34$ to solar. The evolution is
followed up to 1000 Lo. A special attention is given to the opacities and to
the non grey atmosphere models used as boundary conditions for which the model
of convection is the same as in the interior. We find that for both the mixing
length theory and the full spectrum of turbulence model the characteristic
solar length scale for convection has to be slightly reduced to fit the lower
edge of the observed RGB. The corresponding models also better match the
expected mass distribution on the RGB and are in better agreement to the
seismic constraints. These results are robust whether effective temperatures
determined spectroscopically or radii determined interferometrically are used.
