The air being heated naturally flows upwards, so blowing downwards in such a design is a very bad idea. Ideally (and for it to be truly a relevant design), the whole thing is silent at the very least while idle, using very little if any fan and relying on convection to do most of the cooling.
So how many cubic feet per minute will convection cause in this case? 1? 10? Fans will move a hundred cfm or more without much noise.
Both of my recent hardware failures have been caused by dust accumulating in heatsinks, so having something which contributes to that is also a "very bad idea". Especially since the air passes through a relatively confined space which you can't regularly clear out.
What constitutes much noise is subjective, I suppose, but any fan moving hundreds of cfm will be very loud. A 140mm fan moving 20 to 30 cfm is still audible, if you're in a quiet room.
A common approach is to have an intake filter that catches dust. Another is to elevate the case a few inches above the ground.
Unless you're supposed to have this on your desk to show it off. I suppose that wouldn't be unusual...