I have a nice sized wood burning stove in my gameroom, unfortunately if I get a good fire going it is too hot downstairs and I can't get enough to rise. So far I cut a hole in the floor directly above the stove and installed a grate, I also installed a ceiling fan in my livingroom which is directly above the gameroom where the stove is, and I added an induction fan in the grate above the stove.
In my estimation, cold air is falling down the split entryway and trapping the heated air from rising, I hoped that the grate would have relieved the problem and promoted circulation and flow. I seem to get just enough heat to rise that it keeps my thermostat satisfied and therefore freezes the back bedrooms. I cut various lengths of thread and hung it from my doorway downstairs and it appears that the upper 18" of air is exiting the gameroom and the other 45" is entering the gameroom.