Is a flexible vent pipe permissible on a water heater.
Yes and No. They make B- flex vent pipes, double wall. You need to look for label/markings. Google B-flex vent pipes. Tried pasting but no good. Sorry
That looks like cheap dryer vent duct.
This is the good stuff.
http://www.woodlanddirect.com/core/media/media.nl?id=129316&c=483668&h=ef700c3ca440590839df
I would say no. If it were flex b-vent I would expect to see it looking different at connection to elbow.
a) would be slightly larger than elbow - and have collar to connection point.
b) would not expect to see tape at the connection point. You should be able to tell at bottom connection to the water heater… if it is a single layer flex tubing - then it is likely dryer vent.
Could be wrong - but my best guess is it is not correct.
I vote no on that one.
Flues like that can be great as secondary means of venting (chimney liners for example) but not stand alone. I like to see rigid venting with mechanical fasteners.
Chimney liners, installed correctly, reduce the volume in the chimney which allows for the flue to heat faster and allow for a better, more powerful draft.
A double walled b-vent should look similar to this.
http://www.hartandcooley.com/products/dfc/double-wall-flexible-gas-vent-connector