Leak from base of flue

Raising the flue may help a bit by increasing the draft. The direction of the smoke once it exits the chimney is totally dependent on the weather, winds and atmospheric pressure. You will notice that the smoke will go up when the atmospheric pressure is low, and will linger or drop in high pressure.

Raising the flue temp would require they open the damper and burn the wood more rigorously. This is often counter productive and sucks more heat out of the house than it puts in. Also in the case you site, a single wall metal flue could become extremely hot, especially close to the firebox. The smoke will exit the flue more vigorously, but not necessarily raise far enough before it cools to cure the problem.

Here in Maine where many folks heat with wood, the exhaust smoke is becoming an issue in more densely populated areas. There have been many stories on the news, and experts talking about new wood burning technology, but no one has found a way to reduce smoke pollution from older units, airtite stoves or folks that burn damp, or not fully dried wood.

I've had two masonry contractors out. They both agreed the flue is unused, both stated it is a dummy flue. I'm not convinced, but it doesn't matter.

Both noted the water getting inside. Booth discussed the cream of the crop solution, New crown. Neither oversold it.

One liked the idea of metal cap, one never heard of it, lol.

Both suggested to cementing cap the dummy flue and sealing cracks. As I know I'll never use this flue, it made enough sense, I went this approach.

Job done as it was the simplest most immediate solution. I suspect in the near future, maybe next season, I will do a stainless, or all out copper top, with a drip edge, as its astute top lady longer than concrete in wet snowy buffalo weather.