Ahh right, understood, that makes a bit more sense 
My “open system vs closed system” comments are still relevant here (since water can go around the vehicle), although I’m unsure how significant that would be in practice (it would need to be tested, and would likely vary depending on vehicle shape, rigidity, surface materials, and speed).
My “currents and waves” comment is relevant to this (e.g. speed relative to water may not be the same as speed relative to earth), as well as noting that a single pipe can only measure velocity in a single direction, which may not be the direction of travel (although that can perhaps be worked around by using multiple setups with pipe ends oriented in different directions).
If your lake has minimal currents, and is large enough that vehicle movements will have effectively negligible effects on the water body, that does definitely improve the likelihood that an inertial approach could work. If the vehicle is close enough to the bottom to see it, but far enough not to stir it up then a visual approach (e.g. optical flow) could also be worth considering (cameras are cheap, although the software side may be challenging).
DVLs can be purchased ‘off the shelf’, but at this stage don’t really meet your “low cost” requirement, at least relative to something like an IMU, a few flow meters, and/or a camera or two.