Since the movement of the air is induced by buoyancy, i. e. there's a constant force acting on the air, so I would expect the velocity to increase during ascent, much like an object falling down due to gravity accelerates.
But wouldn't a velocity gradient tear the column apart along the axis? Or would it just lead to a corresponding pressure gradient and a deformation of the column (thinner at the top)?
If there is acceleration, there's probably a terminal velocity, just like with objects falling down through the atmosphere. What would that be limited by?
There's two effects that I would like to neglect for the purpose of this question, if possible, the dissipation of the warmer air into surrounding air and the cooling of the air as it rises. I suspect with a high enough $\Delta T$ and a column that is wide enough and not too tall, we can ignore those (for a bit, anyway)?
And just for context, what brought me to this was the idea of having hot air rising through a chimney to act as a pump via a constriction in the chimney that creates a Venturi effect. I was wondering whether it would be better to put that constriction further up in the chimney, so I get a higher pressure difference because of a higher velocity.