Just wondering why your turning the blank without the tube already inserted to begin with
I can't speak for Joe, but I usually turn segmented blanks round prior to drilling. My segmentations typically require a centered hole for the turned pattern to come out right. I get the best drill bit registration drilling on the lathe in a collet chuck. Therefore, after gluing the segments together, I identify the middle of the design, mark the ends, and turn the blank round between centers. Then I drill.
Before rounding particularly fragile blanks, I wrap them in twine (some people use cotton bandage wrap) and saturate the string with CA glue. That helps keep everything together while I'm knocking the edges off.
After drilling, I flood the hole with thin CA glue, just to insure that the segments are well bonded at the tube surface (it helps avoid blow-outs later). Once cured, I chase the hole and then glue in the tube.
I hope that helps,
Eric