Propagation of rate of fluid flow in a home

Question

Suppose I have a large home with two windows. Assume the home forms a perfect seal and does not leak fluid other than from open windows.

The window in the bottom of the home is open. The window on the top floor of the home is closed, with a fan pointed at it, blowing air at it. When I open the window in the top floor, how can I estimate how long it should take for air to flow into the bottom window?

My first thought is that the negative pressure propagates through the house at the speed of sound, at which point, upon reaching the bottom window, air begins to flow into the home, but I’m doubting this answer for some reason.

Answer 1

If your house is a perfect cylinder and the ventilator is not to strong, there may be a chance to observe a pressure wave propagating from one windows to the other and then a current of air establishing itself.

This scenario is however not realised in reality because the flow of air becomes turbulent as a result of the complicated geometry of houses. Then the main effect of the ventilator is to mix the air in the region where it is active. The overall flow is rapidly damped and very probably does not reach the other window. It might be possible to observe a turbulent front propagating from one room to the other. This may lead to a slow change of the temperature as the air inside the house gets mixed with the outside air.

You will however have to work very hard if you really want to answer your question quantitatively. You could set up a numerical simulation of Navier--Stokes equations with the geometry of your home (and the fan) accounted for, but I would advise against it. Turbulence is an extremely complex phenomena that is difficult to simulate and more or less precludes analytical solutions.

The easiest way to answer your questions would be do just do it: Set your ventilator on a timer, go to the other window with a stopwatch and measure how long it takes for the air to feel different.

Answer 2

Yes, in theory any change in pressure should propagate through your house at the speed of sound, because that's exactly what a sound wave is. However, to get a noticeable effect at the bottom window, I think your fan would have to be closely fitted ("ducted") to the top window, and other places where air could flow into the house would have to be carefully sealed off.