A poor, but sometimes inevitable, way to do a hydrotest, is the so-called "air over water" method. You fill the vessel with water, then introduce compressed air at a high point from a cylinder. Cheaper than a pump if you don't already have a pump capable of the pressure you need, but the resulting test is much less sensitive and can be more dangerous. While the water reduces the volume of compressed gas to a negligible amount IF you design the test properly, in any hydrotest the goal should be to have ZERO air in the vessel under test itself- limiting the stored air and hence the source of dangerous quick-release stored energy to the tubing or hose supplying the compressed air from the cylinder.
In a test entirely water-filled and without air, the test becomes truly sensitive: even a tiny amount of leakage will result in a large drop in pressure of the (nearly) Incompressible water and (somewhat) inelastic vessel.
It is better to think of both the water and the vessel as springs with really high spring rates, such that very small changes in distance (volume change resulting from compressing the water or elastic stretching of the vessel) result in huge reaction forces (pressure)- or vice versa. In practice the energy stored in those springs is quite small in a water-filled system, and is rapidly released by even small changes in dimension of the vessel under test, and hence that energy is not of much risk of danger upon failure- until the vessel under test becomes very large indeed. That is NOT true of a vessel tested pneumatically, which is why pneumatic tests must be done so much more carefully and to a smaller fraction of the design pressure.
An accumulator with diaphragm or piston can be used to allow you to apply air pressure to the water without having the air in direct contact with the water. This is important because air dissolves in water, which can tend to make pressure bleed away without the need for a leak, making the test look like a failure when there aren't any leaks or flaws in the vessel. But in this case, the diaphragm or piston system need to be capable of safely withstanding the results of a vessel total failure, where the air pressure on one side of the diaphragm/piston is the full pressure required in the test but the water side drops suddenly to near zero. It's not really all that practical unless the test pressure and/or volume are quite small. A pump is safer, and if you do a good job of removing air (sometimes that may require a vacuum pump) prior to the test, even a tiny high pressure pump will easily provide enough water to accomplish the test.
