When air is made to move (e.g. by a fan) it has a lower pressure than when it is still. Essentially, this moving air should become a low pressure zone relative to the static air around it and the static air (high pressure) should flow into the path of the moving air (low pressure). But I think that doesn't happen although I am not sure why. What is happening?
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I remember from the 1970's that turbulent jets entrain five times their mass every diameter. This should/will be the same for fans, which are just really large diameter jets. Of course there will be detailed effects near the exit of the fan. This relationship should hold well away from the fan.
https://engineering.dartmouth.edu/~d30345d/books/EFM/chap9.pdf offers a discussion. I searched "jet entrainment Escudier" for info as I worked with M. P. Escudier in the past.