I wonder if it could be related to Bernoulli’s Principle as well?

Possibly this and also because fan without tube causes a circular, or more precisely donough-shaped vortex: In the front of the spinning "disk", the pressure becomes higher , in the back lower, so air flows around the disk from high to low pressure. The Bernoulli principle would lower pressure in the "jet" of air going out of the window, and suck in other surrounding air.