We perform experimental measurements of the instantaneous velocity profile of the flowing layer during granular avalanches. In the pile depth, the velocity profile follows a pure exponential decrease in contrast with steady flows that are known to exhibit a well developed upper linear part. The velocity profile in the pile width is a plug flow with two exponential boundary layers at the walls. Even though no steady state is observed during the avalanche, these velocity profiles are self-similar and build up almost instantaneously, with time independent characteristic lengths.