Compressed how? Axially, radially, filled with compressed air, etc. etc. Your description of the problem isn't sufficient
to even begin to understand the question let alone answer it.
As for simple equations, where are simple equations, and
then there are usefully correct equations; the two aren't
always overlapping when you're dealing with something
like drag due to motion in water. There's a reason that
engineers often build scale models of things and test them
"for real" in water tanks rather than relying on the accuracy
of estimates from fluid flow / drag calculations; it's
often easier to build and test the thing rather than model
it sufficiently accurately even with complex equations and
computer models.
If it's just a question of axial acceleration of a tube through
water given a certain 'rocket' thrust from compressed air
inside the tube released from an axial nozzle in the rear,
then you could probably find a good simple formula for
the drag of a cylinder of a given size in water, and the
thrust will just be the simple rocket equation due to
the momentum of the release of the gas given a certain
escaping gas flow rate due to the pressure differential of
the gas vs. the water and the size/performance of the
nozzle the gas escapes through. That's about as simple
as that gets, pretty simple to estimate with simple
math and a notepad, but still not trivial.
Unless you're talking about the mechanical relaxation /
oscillation / 'springing' / ringing (like a bell) /
expansion (like a bellows) of a structurally compressed tube
under water in which case, well, good luck, that's likely
a lot more complicated due to (in)elasticity, friction,
drag, heat, possibly turbulent fluid flow, etc. etc.
The simplest case would be like letting go of a coiled
tube spring underwater in which case you're back to
estimating the cylinder's drag and the the evolution of
the 'spiringing away' based on the water's drag and the
F=mx tension in the spring and how long the spring will
be able to push against its dock...
