As does the smoke from incense or fine particles of ash.
The former I did as a test a few winters ago. (No discernible change in the movement of the smoke)
The latter I do every day in the winter, because I place a pot on the stove top into which I empty the ash pan, before taking the pot outside to cool. The pot is positioned right in front of the stove top fan, with the fan "blowing" across the top of the pot from left to right. So the finer ash particles in the air ought to move to the right, or at least curl or circulate round. Here's the daily official Pud eco fan test results:
With the fan turning, the fine particles of ash (hot or cold) that puff up into the air from tipping out the ash pan into the pot, float vertically upwards following the rising heat from the stove top and flue, with a small amount of the ash particles moving up and left towards the closed but presumably draughty window nearby.
With no stove top fan, the fine particles of ash that puff up into the air from tipping out the ash pan into the pot, float vertically upwards following the rising heat from the stove top and flue, with a small amount of the ash particles moving up and left towards the closed but presumably draughty window nearby.
In other words, the fan makes no visible difference at all to the air flow, using either smoke to test it or fine ash particles.
These are the results of the BlueStringPudding jury. 😊
So if you don't already have an eco fan, I would suggest you don't waste your money on buying one. If you do have one, enjoy its pretty spinning propellers and the sense of comfort that you feel warmer at the other end of your boat, whether it's real or imagined. 😊